Download Phys 774: Raman Scattering

Introduction: Brillouin and Raman spectroscopy
Inelastic light scattering mediated by the electronic polarizability of the medium
• a material or a molecule scatters irradiant light from a source
• Most of the scattered light is at the same wavelength as the laser source
(elastic, or Raileigh scattering)
• but a small amount of light is scattered at different wavelengths
(inelastic,
or Raman scattering)
Phys 774:
Raman Scattering
α
β
ћωi
Fall 2007
β
α
ћΩ
ћωs
Stokes
ћωi
0
0
Raileigh
ћΩ
I
ћωs
AntiStokes
Stokes
Raman
Scattering
ωi- Ω(q)
Elastic
(Raileigh)
Scattering
ω
ωi Anti-Stokes
Raman
Scattering
ωi+ Ω(q)
Analysis of scattered light energy, polarization, relative intensity
provides information on lattice vibrations or other excitations 2
1
Raman scattering:
how does it look like?
Raman scattering in crystalline solids
Not every crystal lattice vibration can be probed by Raman
scattering. There are certain Selection rules:
1. Energy conservation:
=ωi = =ωs ± =Ω
=Ω Excitation energy
2. Momentum conservation in crystals:
ki = k s ± q ⇒ 0 ≤ q ≤ 2 k ⇒ 0 ≤ q ≤
λi ~ 5000 Å, a0 ~ 4-5 Å ⇒ λphonon >> a0
4πn
λi
ks
ki
q≈0
q ≈ 2k
ki
ks
ks
q
ki
Phonons
⇒ only small wavevector (cloze to BZ center) phonons are seen in
the 1st order (single phonon) Raman spectra of bulk crystals
3. Selection rules determined by crystal symmetry and by
symmetry of excitations
3
4
Excitations
History of Raman scattering
1. Molecular vibrations
Before Raman … : Rayleigh and Mie
=ωi = =ωs
2. Phonons: acoustic and optical
3. Magnons
4. Collective excitations: Electrons+phonons = Plasmons
5. Resonant excitations: Spin-flip excitations
Experimental setups
1. Excitation source: Lasers or Synchrotron radiation
2. Sample: optical cryostats, magnets, pressure cells, etc
Scattered intensity
3. Polarizers for selection rules analysis
4. Spectrometers
Cross section
5. Detectors
5
6
History of Raman scattering
History of Raman scattering
Mandelstam-Brillouin scattering, 1928
Hg lamp
=ωi = =ωs ± =Ω
Photo plate
=ωi = =ωs
Molecular
vibrations:
7
8
What is Raman shift ?
Mandelstam-Brillouin vs. Raman scattering
Examples of Phonons in Crystals
=ωi = =ωs ± =Ω
Raman
scattering
Raman Shift = =ωs − =ωi = ± =Ω
Photo plate
Phonon Energy
Hg lamp
Molecular
vibrations:
Mandelstam-Brillouin
scattering
9
Phonons
• Quantum mechanics: energy levels of the harmonic oscillator are quantized
• Similarly the energy levels of lattice vibrations are quantized.
• The quantum of vibration is called a phonon
(in analogy with the photon - the quantum of the electromagnetic wave)
G
G G
G
q = ±∆k = ± | ki − ks |
Phonon wavevector
10
1D Model of lattice vibrations
one-dimensional lattice: linear chain of atoms
harmonic approximation:
force acting on the nth atom is
Allowed energy levels of the harmonic oscillator:
where n is the quantum number
equation of motion (nearest neighbors interaction only):
A normal vibration mode of frequency ω is given by
u = Ae
i ( q⋅r −ωt )
mode is occupied by n phonons of energy ħω; momentum p = ħq
Number of phonons is given by :
(T – temperature)
n=
1
e
=ω kT
−1
The total vibrational energy of the crystal is the sum of the energies of the
individual phonons:
(p denotes particular
11
phonon branch)
M
∂ 2u
= Fn = C (u n +1 − un ) + C (un −1 − un ) = −C (2u n − un +1 − un −1 )
∂t 2
M is the atomic mass, C – force constant
Now look for a solution of the form
u ( x, t ) = Aei ( qxn −ωt )
where xn is the equilibrium position of the n-th atom → xn = na
obtain
12
→
⇒ the dispersion relation is
Note: we change q → q + 2π/a the atomic displacements and
frequency ω do not change ⇒ these solutions are physically identical
⇒ can consider only
i.e. q within the first Brillouin zone
The maximum frequency is 2 C
M
13
14
15
16
At the boundaries of the Brillouin
zone q = ±π/a → u n = A(−1) e
n
− iωt
standing wave
Phase and group velocity
phase velocity is defined as v p =
group velocity
vg =
ω
q
dω
dq
vg = a
C
qa
cos
M
2
vg = 0 at the boundaries of the Brillouin zone (q = ±π/a) ⇒
no energy transfer – standing wave
Long wavelength limit: λ >> a ; q = 2π/λ << 2π/a ⇒ qa << 1
one-dimensional linear chain, atoms of two types: M1 and M2
small q - close to the center of Brillouin zone
ω=
4C
qa
C
sin
≈
qa
M
M
2
v p = vg = a
C
M
Diatomic lattice
- linear dispersion
Optical Phonons
- sound velocity for the one dimensional
lattice
can interact with light
For diamond
Optical phonon
frequency is ≈ 1300 cm-1
λ≈ 7700 nm
17
(far-IR)
18
Note: the first Brillouin zone is now
from -π/2a to +π/2a
The lower curve - acoustic branch, the upper curve - optical branch.
at q = 0 for acoustic branch ω0 = 0; A1 = A2
⇒ the two atoms in the cell have the same amplitude and the phase
dispersion is linear for small q
for optical branch
at q = 0
⎛ 1
1 ⎞
⎟⎟
ω0 = 2C ⎜⎜
+
⎝ M1 M 2 ⎠
M1A1 +M2A2 = 0
⇒ the center of mass of the atoms remains fixed. The two atoms move out of
19
phase. Frequency is in infrared – that's why called optical
20
Quasi-Classical Theory of Raman Scattering
=ωi = =ωs ± =Ω
G G G
ki = k s ± q
GG
G G
Q = Q0 ⋅ ei ( k ⋅r −ω0t )
Elastic scatt.
Lattice Deformation
Deformation gradient
External E-field
2-phonon process
21
Selection rules for Raman Scattering by phonons
Quantum Theory of Raman Scattering
e
e'
22
GG
G G
Q = Q0 ⋅ ei ( k ⋅r −ω0t )
A1g B2 g
=ωi = =ωs ± =Ω
G G G
ki = k s ± q
A2u Fu
G
G
Iˆ − ⋅ (Q) = −Q
G
G
Iˆ − ⋅ (Q) = Q
"odd "
"even "
Symmetry of Raman-active modes, “g” – gerade (de.) = even
Symmetry of IR-active modes, “u” – ungerade (de.) = odd
Center of inversion
between atoms
Center of inversion
at the atom
No center of
inversion
23
24
Atomic displacement and Raman / IR-activity of optical
phonons in crystals with the center of inversion
x → −x
y → −y
z → −z
GG
G G
Q = Q0 ⋅ ei ( k ⋅r −ω0t )
GG
G G
E = E0 ⋅ ei ( k ⋅r −ωt )
G
G
Iˆ − ⋅ (Q) = −Q
G
G
Iˆ − ⋅ ( E ) = − E
G
G
Iˆ − ⋅ (Q) = Q
G
G
Iˆ − ⋅ ( E ) = − E
No IR-active
phonons in Ge, Si,
Diamond
Atomic displacement and Raman / IR-activity of optical
phonons in crystals with the center of inversion
Phonons in Ge, Si, Diamond
are Raman active
GG
G G
Q = Q0 ⋅ ei ( k ⋅r −ω0t )
GG
G G
E = E0 ⋅ ei ( k ⋅r −ωt )
G
G
Iˆ − ⋅ (Q) = Q
G
G
Iˆ − ⋅ ( E ) = − E
e
e'
No Raman-active phonons in
NaCl, MgO, SrTiO3
25
Polarization Selection rules for Raman Scattering
26
Example of selection rules for Raman scattering by optical
phonons in III-V semiconductor crystals
(no center of inversion)
by phonons
G G G G
ki ( Ei , Es )ks
Raman tensor
G
ki
G
ks
Forward scattering
G
ks
G
ki
Back- scattering
Polarization configuration of
experiment
G
ki
G
ks
90 deg scattering
27
28
Experimental setups for Raman scattering
Experimental setups for Raman scattering
SPEX – “Triplemate”
29
Stress analysis in microstructures using Raman scattering
31
30
Composition analysis in
microstructures
Micro-Raman scattering
32
Raman scattering in Low-dimensional periodic structures
(folded acoustic phonons in GaAs/AlAs MQW’s)
Raman scattering in Low-dimensional periodic structures
(Confined optical phonons in GaAs/AlAs MQW’s)
33
Resonant Raman scattering
34
Raman scattering by coupled Plasmons: plasmon+LOph
ω+
ω−
35
36
Raman scattering and IR reflectivity by Plasmons: e+ph
Example of E-field induced Raman scattering in
crystalline solids
2 × TO1 + LO1
3S = 15 modes
2 × TO2 + LO2
3 acoustic modes
12 optical modes; 3 × 4
37
2 × TO3 + LO3
2 × TO4 + LO4
38
Raman thermometer
HW: Which crystal has this phonon dispersion diagram ?
Classical formula
is not complete!
You need to take
Planck‘s statistics
for phonons into
account to
calculate the
scattering
intensity
HW:
What is St/aSt
ratio ???
Stokes ~ n + 1
39
40
anti − Stokes
~n