This chapter discusses the optical properties of phonons in materials. It covers: 1) Optical and acoustic phonons - some interact directly with light, others cause light scattering. 2) Optical excitation of phonons - how phonons contribute to optical properties through the dielectric function. 3) Phonon polaritons - mixed phonon-photon excitations in crystals near resonance frequencies. 4) Light scattering - concepts of Brillouin, Raman, and Rayleigh scattering involving phonons. 5) Coherent Raman spectroscopy - an experimental technique that enhances weak Raman scattering signals.

1. Chapter 4
Optical properties of phonons
23-4-2019
Presented by
P. Kedhareswara Sairam

2. Chapter 4 : optical properties of phonons
1) This chapter concentrates on the optical properties of the ionic
part of the material response
2) mass of the ions is typically 103 times larger than that of the
electrons
3) the ionic movement is usually restricted to small oscillations
around the equilibrium position in the lattice.
4) The oscillation modes of these lattice vibrations are called
phonon modes.

3. Outline
4-1. Optical and Acoustical Phonons
- Some of phonon interact directly with light → absorption and reflection by
the medium
- Some phonons do not absorb light directly but are the origin of light
scattering
4-2. Optical Excitation of Phonons
- Optical properties associated with phonons, deriving the phonon dielectric
function
4-3. Phonon Polaritons
- The mixed phonon-photon excitations in a crystal
4-4. Light Scattering
- Concepts of Brillouin, Raman, and Rayleigh scattering
4-5. Coherent Raman Spectroscopy
- Experimental application

4. 4-1. Optical and Acoustical Phonons_ equation of motion
 Linear chain with two atoms per unit cell
We assume that the there is harmonic oscillation in time with
constant amplitude and angular frequency
For simplicity, we only need to consider nearest neighbor
interactions
• Atomic masses : M1, M2
• Interatomic distance : a
• Spring constant : f
• The displacement of the
jth atom from its
equilibrium position : Uj

5. Equation of motion
Atomic oscillation motion → equation of motion → Uj is expressed by Ω a
• Atomic masses : M1, M2
• Interatomic distance : a
• Spring constant : f
• The displacement of the jth atom from its equilibrium
position : Uj
 Equation of motion
 The solution for these equation

6. u-GaN
Equation of motion_ mathematical calculation !
 Equation of motion  The solution for these equation
These equations can be satisfied only if

7. 4-1. Optical and Acoustical Phonons
Dispersion relation for the lattice vibrations (phonon dispersion relation)
Ω is a periodic function of K
It is enough to consider the range of the first Brillouin zone
The first Brillouin zone is the region between -𝜋/2a ≤ K ≤ 𝜋/2a
Let’s see at the K = 0 and K =
𝝅
𝟐𝒂
!

8. Dispersion relation for the lattice vibrations (phonon dispersion relation)
Ω is a periodic function of K
It is enough to consider the range of the first Brillouin zone
The first Brillouin zone is the region between -𝜋/2a ≤ K ≤ 𝜋/2a
Let’s see at the K = 0 and K =
𝝅
𝟐𝒂
!

9. Ω +
optical phonon : high frequency phonons, almost dispersionless!
Ω −
acoustic phonon : low frequency phonons, dispersion!
For small K, sin(Ka) → Ka

10. Acoustical branch, K → 0, Ω −
→ 0, A1/A2 → 1
Two types of atoms move in the same direction with the
same amplitude.
Optical branch, K → 0, Ω +
→ 𝐹𝑖𝑔, A1/A2 → -M2/M1
Two types of atoms move in the opposite direction
and amplitude inversely proportional to the masses.

11. One dimensional diatomic chain so far is one optical branch and one acoustic
branch.
Three dimensional solid, the same procedures to calculating phonon dispersion
curve.
A three-dimensional vibrational amplitude may be written as
Wave vector K involve direction propagation of wave.
unit vector e is polarization of the waves.
K and e is parallel – longitudinal wave – atomic oscillation along the chain
direction
K and e is perpendicular – transverse wave – perpendicular to the chain
Generally phonon wave in a solid, a mixture of transverse and longitudinal type.

12. • 3D, mono atomic solid → Three equation of motion
• Describe three acoustic branch and optical branch
each
• Two are transverse and one is longitudinal
• The dispersion curves may have
different profiles in different
directions.
• TA branches degenerate along
[100] direction but not degenerate
[111]

13. 4-2. Optical Excitation of Phonons _ Disperasion relation near K=0
Dispersion relation of light :
Dispersion relation of acoustic
phonon: near K=0
Acoustic branch and photon
dispersion :
do not cross → no regions of the
energy and momentum conservation
laws matched
Optical branch and photon dispersion :
cross
→ photon can be converted to optical
phonon

14. An ionic crystal such as NaCl exhibits strong optical reflection and
absorption in the infrared region associated with optical phonons.
Let’s see how this happen
① The E-field forced vibration of atoms (optical mode)
② The equation of motion for the two ions ( - and + charged ion)
③ Try solution form
damping terms

15. Substituting Equations…
two algebraic equations for the two unknown quantities A1 and A2
The solution is
 Opposite sign of A1 and A2
→ direction of displacement of negative and positive ions.
 𝜔(light frequency) = Ω (optical-phonon frequency)
→ very large vibration amplitudes
→ strong absorption
→ resonance effect

16. Let’s see the optical properties of crystal using dielectric function
ionic and electronic contribution
N : ionic pairs per unit volume
: dipole moment for a
pair of ions
The dielectric function
is
reduced mass of the two ions

17. ① For low frequencies, when 𝜔 << Ω +
(0)
② For high frequencies, when 𝜔 ≫ Ω +
(0)
static dielectric constant.
Dielectric function

18. ① light with transverse polarization couples with TO
phonons
→
② light frequency = LO phonon :
Note that at 𝜔2 = Ω +
2
the dielectric function → resonance
Then dielectric function is
To get the longitudinal optical-phonon frequency w = Ω 𝐿 we recall that it is obtained by the
condition :

19. The behavior of the reflectivity of
an incident wave is shown

20. • infrared transmission in a thin film of NaCl.
• transmission drop occurs at w = Ω 𝑇,
• The phenomena of strong infrared absorption and reflection by the
lattice

21. 4-3. Phonon Polaritons Optical response of crystal → dielectric
function
For simple, damping 𝛾 = 0, and LST relation
Dispersion relation for the light inside the crystal in the
vicinity of the optical-phonon resonance

22. Phonon-polarition dispersion curves
• The two independent oscillators are the pure photon and pure phonon which
couple in the crystal to give two new modes ( polariton modes )
• Away from the photon-phonon crossing point, the two polariton modes
behave like pure modes.
• photon-phonon dispersion in the
presence of interaction between the
photon and phonon
• when a photon impinges on a crystal
with frequency in the vicinity of the
TO phonon frequency, it excites a
polariton.

23. 4-4. Light Scattering
 scattering process
incoming photon
frequency: 𝜔i ,wave vector: 𝑞i
outgoing photon
frequency: 𝜔 s , wave vector : 𝑞s
The energy difference h(𝜔i - 𝜔s ) and momentum, 𝑞s - 𝑞i is exchanged with the
medium.
Photons may be scattered inelastically by both optic and acoustic
phonons.
 When the 𝜔s < 𝜔i
the exchanged energy creates a phonon
and the process is referred phonon emission or Stokes process
 when 𝜔s > 𝜔i ,
a phonon is absorbed and the process is called phonon absorption or anti-
Stokes process
𝜔i , 𝑞i
𝜔s , 𝑞s
Κ, Ω
𝜔i , 𝑞i
𝜔s , 𝑞s
Κ, Ω

24.  Raman scattering : optical phonon is
involved in the scattering
→ Optical phonon energy shift
 Brillouin scattering : photon-acoustic-
phonon interaction
 Rayleigh scattering : a process
whereby the scattered photons
fluctuate in frequency within a small
width of ~ 10-9 to 10-4 cm-1,
→ No shift

25. Selection rules have to be satisfied for the
scattering
Conservation of momentum and energy provides one set of such selection
rules.
phonon absorption (anti-Stokes process)

26.  frequency of acoustic phonon are small
 high resolution techniques are needed to
detect
conservation of momentum
∵
Brillouin scattering

27. IR or infrared active modes : phonons that are involved in the infrared absorption
Raman active modes : phonons that participate in the scattering process
This figure clarifies how acoustic phonon, which cannot be involved in the
infrared absorption, may still participate in the scattering process

28. A double monochromator or a Fabry-Perot
interferometer needed for Raman scattering
Brillouin scattering as a tunable spectral
filter
the Stokes and anti~Stokes lines~~

29. Angle resolved scattering experiments
By varying the scattering angle, the wave vector of the phonon is changed
By measuring the scattered photon frequency as a function of the scattering angle 𝜃
for small angle
Note that only the TO phonon shows the strong dispersion characteristic of a polariton,
in contrast to the LO phonon, which is more or less dispersionless

30. 4-5. Coherent Raman Spectroscopy
Because the probability of photon scattering is very small
the intensity of the scattered light (proportional to the phonon population) is
extremely weak
Bose-Einstein distribution function
~ 0.5 at room temperature for a phonon energy of
ℏΩ = 30 meV
• Two intense laser beams are adjusted such that their energy difference, ℏ(𝜔1 −
𝜔2) equals the phonon energy ℏΩ
• The intensity of the scattered anti-Stokes signal at ℏ(2𝜔1 − 𝜔2) is typically nine
orders of magnitude stronger than a spontaneous anti-Stokes signal.

31. • Two intense laser beams are adjusted such that their energy difference, ℏ(𝜔1 −
𝜔2) equals the phonon energy ℏΩ
• The intensity of the scattered anti-Stokes signal at ℏ(2𝜔1 − 𝜔2) is typically nine
orders of magnitude stronger than a spontaneous anti-Stokes signal.

32. Summary
4-1. Optical and Acoustical Phonons
- Some of phonon interact directly with light
- Some phonons do not absorb light directly
4-2. Optical Excitation of Phonons
- Optical properties associated with phonons, deriving the phonon dielectric function

33. 4-3. Phonon Polaritons
- The mixed phonon-photon excitations in a crystal
4-4. Light Scattering
- Concepts of Brillouin, Raman, and Rayleigh scattering
4-5. Coherent Raman Spectroscopy
- Experimental application

34. Thanks for listening