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Phys 774: Raman Scattering

Fall 2007

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Introduction: Brillouin and Raman spectroscopy

•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)

Stokes Raman

Scatteringωi- Ω(q)

Anti-Stokes Raman Scatteringωi+ Ω(q)

ωi

Elastic (Raileigh) Scattering

ω

I

0

αβ

ћΩ

ћωiћωsStokes

0

αβ

ћΩ

ћωi

Raileigh

ћωsAnti-Stokes

Analysis of scattered light energy, polarization, relative intensity provides information on lattice vibrations or other excitations

Inelastic Inelastic light scatteringscattering mediated by the electronic polarizabilitypolarizability of the medium

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Not every crystal lattice vibration can be probed by Raman scattering. There are certain Selection rules:

1. Energy conservation:

2. Momentum conservation in crystals:

λi ~ 5000 Å, a0 ~ 4-5 Å ⇒ λphonon >> a0

⇒ 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

Raman scattering in crystalline solids

i sω ω= ± Ω

kq 20 ≤≤qkk ±= si ⇒i

nqλπ40 ≤≤⇒ q

ki

ks q ≈ 0ks

ki

ks ki

q ≈ 2k

Excitation energyΩ

Phonons

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Raman scattering: how does it look like?

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1. Molecular vibrations

2. Phonons: acoustic and optical

3. Magnons

4. Collective excitations: Electrons+phonons = Plasmons

5. Resonant excitations: Spin-flip excitations

Excitations

Experimental setups 1. Excitation source: Lasers or Synchrotron radiation

2. Sample: optical cryostats, magnets, pressure cells, etc

3. Polarizers for selection rules analysis4. Spectrometers

5. Detectors

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History of Raman scattering

Before Raman … : Rayleigh and Mie

Scattered intensity

Cross section

i sω ω=

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History of Raman scattering

i sω ω= i sω ω= ± Ω

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History of Raman scattering

Mandelstam-Brillouin scattering, 1928

Phot

o pl

ate

Molecular vibrations:

Hg lamp

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What is Raman shift ?

Phot

o pl

ate

Molecular vibrations:

Hg lamp

i s

s iRaman Shiftω ω

ω ω= ± Ω

= − = ± Ω

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Mandelstam-Brillouin vs. Raman scatteringExamples of Phonons in Crystals

| |i sq k k k= ±∆ = ± −

Phonon wavevector

Pho

non

Ene

rgy

Mandelstam-Brillouinscattering

Ramanscattering

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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)

Allowed energy levels of the harmonic oscillator:

where n is the quantum number

A normal vibration mode of frequency ω is given by

mode is occupied by n phonons of energy ħω; momentum p = ħqNumber of phonons is given by :

(T – temperature)

The total vibrational energy of the crystal is the sum of the energies of the individual phonons:(p denotes particular

phonon branch)

)( tie ω−⋅= rqAu

11−

= kTen ω

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1D Model of lattice vibrationsone-dimensional lattice: linear chain of atoms

harmonic approximation: force acting on the nth atom is

)2()()( 11112

2

−+−+ −−−=−+−==∂∂

nnnnnnnn uuuCuuCuuCFtuM

equation of motion (nearest neighbors interaction only):

M is the atomic mass, C – force constant

Now look for a solution of the form )(),( tqxi nAetxu ω−=

where xn is the equilibrium position of the n-th atom → xn = na

obtain

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→

⇒ 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 CM 14

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At the boundaries of the Brillouin

zone q = ±π/a →standing wave

tinn eAu ω−−= )1(

Phase and group velocity

phase velocity is defined as

group velocity

qvp

ω=

dqdvgω

=2

cos qaMCavg =

vg = 0 at the boundaries of the Brillouin zone (q = ±π/a) ⇒no energy transfer – standing wave 16

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Long wavelength limit: λ >> a ; q = 2π/λ << 2π/a ⇒ qa << 1

qaMCqa

MC

≈=2

sin4ω - linear dispersion

small q - close to the center of Brillouin zone

MCavv gp == - sound velocity for the one dimensional

lattice

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Diatomic lattice

one-dimensional linear chain, atoms of two types: M1 and M2

Optical Phonons can interact with light

For diamond Optical phonon

frequency is ≈ 1300 cm-1

λ≈ 7700 nm(far-IR)

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Note: the first Brillouin zone is nowfrom -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 phasedispersion is linear for small q

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

210

112MM

Cω M1A1 +M2A2 = 0

⇒ the center of mass of the atoms remains fixed. The two atoms move out of phase. Frequency is in infrared Frequency is in infrared –– that's why called opticalthat's why called optical

for optical branchat q = 0

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21 22

i s

i sk k q

ω ω= ± Ω

= ±

Quasi-Classical Theory of Raman Scattering

0( )0

i k r tQ Q e ω⋅ −= ⋅

External E-field

2-phonon process

Lattice Deformation

Deformation gradient

Elastic scatt.

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'

e

ei s

i sk k q

ω ω= ± Ω

= ±

Quantum Theory of Raman Scattering

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Selection rules for Raman Scattering by phonons

1 2g gA B Symmetry of Raman-active modes, “g” – gerade (de.) = even

2u uA F Symmetry of IR-active modes, “u” – ungerade (de.) = odd

Center of inversion at the atom

Center of inversion between atoms

No center of inversion

ˆ ( ) " "ˆ ( ) " "

I Q Q odd

I Q Q even

−

−

⋅ = −

⋅ =0( )

0i k r tQ Q e ω⋅ −= ⋅

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Atomic displacement and Raman / IR-activity of optical phonons in crystals with the center of inversion

x xy yz z

→−→ −→ −

0( )0

( )0

ˆ ˆ( ) ( )ˆ ˆ( ) ( )

i k r t

i k r t

Q Q e

E E e

I Q Q I Q Q

I E E I E E

ω

ω

⋅ −

⋅ −

− −

− −

= ⋅

= ⋅

⋅ = − ⋅ =

⋅ = − ⋅ = −

No IR-active phonons in Ge, Si, Diamond

No Raman-active phonons in NaCl, MgO, SrTiO3 26

Atomic displacement and Raman / IR-activity of optical phonons in crystals with the center of inversion

0( )0

( )0

ˆ ( )ˆ ( )

i k r t

i k r t

Q Q e

E E e

I Q Q

I E E

ω

ω

⋅ −

⋅ −

−

−

= ⋅

= ⋅

⋅ =

⋅ = −

Phonons in Ge, Si, Diamond are Raman active

'

e

e

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Polarization Selection rules for Raman Scattering by phonons

( , )i i s sk E E kPolarization configuration of experimentRaman tensor

Forward scattering Back- scattering 90 deg scattering

skikik

iksksk

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Example of selection rules for Raman scattering by optical phonons in III-V semiconductor crystals

(no center of inversion)

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Experimental setups for Raman scattering

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Experimental setups for Raman scattering

SPEX – “Triplemate”

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Stress analysis in microstructures using Raman scattering

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Micro-Raman scatteringComposition analysis in microstructures

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Raman scattering in Low-dimensional periodic structures (folded acoustic phonons in GaAs/AlAs MQW’s)

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Raman scattering in Low-dimensional periodic structures (Confined optical phonons in GaAs/AlAs MQW’s)

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Resonant Raman scattering

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Raman scattering by coupled Plasmons: plasmon+LOph

ω +

ω −

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Raman scattering and IR reflectivity by Plasmons: e+ph

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Example of E-field induced Raman scattering in crystalline solids

3 15 modes3 acoustic modes12 optical modes; 3 4

S =

×

1 1

2 2

3 3

4 4

2222

TO LOTO LOTO LOTO LO

× +× +× +× +

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HW: Which crystal has this phonon dispersion diagram ?

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Raman thermometer 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 ???

~ 1~

Stokes nanti Stokes n

+−