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: ћΩ I Elastic ћΩ β (Raileigh) Raman Scattering α α Scattering β ћωs Anti- ћω ћωs i ћωi Stokes ω Fall 2007 Stokes Stokes ωi Anti-Stokes 0 0 Raman Raman Scattering Scattering Raileigh ωi- Ω(q) ωi+ Ω(q) Analysis of scattered light energy, polarization, relative intensity 1 provides information on lattice vibrations or other excitations 2

Raman scattering: Raman scattering in crystalline solids how does it look like? Not every crystal lattice vibration can be probed by Raman scattering. There are certain Selection rules: 1. Energy conservation:

===ωis=±Ωω =Ω Excitation energy

2. Momentum conservation in crystals: k k 4πn s s q k = k ± q ⇒ 0 ≤ q ≤ 2k ⇒ 0 ≤ q ≤ q ≈ 0 i s ki λ ki i q ≈ 2k

ks ki λi ~ 5000 Å, a0 ~ 4-5 Å ⇒ λphonon >> a0 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 ==ωis= ω 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 ==ωis= ω ===ωis=±Ωω Photo plate Molecular vibrations: 7 8 What is Raman shift ? Mandelstam-Brillouin vs. Raman scattering Examples of Phonons in Crystals ===ωis=±Ωω Raman scattering Raman Shift =−=±Ω==si = ωω Hg lamp G GGG qkkk=±∆ =±||is − Phonon Energy Photo plate Molecular vibrations: Mandelstam-Brillouin Phonon wavevector 9 10 scattering

Phonons 1D Model of lattice vibrations • Quantum mechanics: energy levels of the harmonic oscillator are quantized one-dimensional lattice: linear chain of atoms • Similarly the energy levels of lattice vibrations are quantized. • The quantum of vibration is called a phonon harmonic approximation: th (in analogy with the photon - the quantum of the electromagnetic wave) force acting on the n 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 = Aei(q⋅r−ωt) ∂ 2u M 2 = Fn = C(un+1 − un ) + C(un−1 − un ) = −C(2un − un+1 − un−1) mode is occupied by n phonons of energy ħω; momentum p = ħq ∂t M is the atomic mass, C – force constant Number of phonons is given by : 1 (T – temperature) n = i(qxn −ωt) e=ω kT −1 Now look for a solution of the form u(x,t) = Ae

The total vibrational energy of the crystal is the sum of the energies of the where xn is the equilibrium position of the n-th atom → xn = na individual phonons: (p denotes particular obtain phonon branch) 11 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

At the boundaries of the Brillouin n −iωt zone q = ±π/a → un = A(−1) e standing wave Phase and group velocity ω phase velocity is defined as v = p q dω C qa group velocity vg = v = a cos dq g M 2

vg = 0 at the boundaries of the Brillouin zone (q = ±π/a) ⇒ no energy transfer – standing wave 15 16 Long wavelength limit: λ >> a ; q = 2π/λ << 2π/a ⇒ qa << 1 Diatomic lattice small q - close to the center of Brillouin zone one-dimensional linear chain, atoms of two types: M1 and M2

4C qa C ω = sin ≈ qa - linear dispersion M 2 M

C vp = vg = a - sound velocity for the one dimensional Optical Phonons M lattice can interact with light

For diamond Optical phonon frequency is ≈ 1300 cm-1 λ≈ 7700 nm 17 18 (far-IR)

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 ⎛ 1 1 ⎞ ω = 2C⎜ + ⎟ M A +M A = 0 at q = 0 0 ⎜ ⎟ 1 1 2 2 ⎝ M1 M 2 ⎠ ⇒ the center of mass of the atoms remains fixed. The two atoms move out of phase. Frequency is in infrared – that's why called optical 19 20 Quasi-Classical Theory of Raman Scattering

===ωis=±Ωω GGG kkqis=±

GG G G ikr()⋅−ω0 t QQe=⋅0 Elastic scatt.

Lattice Deformation

Deformation gradient

External E-field

2-phonon process 21 22

Selection rules for Raman Scattering by phonons Quantum Theory of Raman Scattering GG − GG G G Iˆ ⋅=−( Q ) Q " odd " ikr()⋅−ω0 t QQe=⋅0 GG IQQˆ− ⋅=( ) " even " e A1gg B 2 Symmetry of Raman-active modes, “g” – gerade (de.) = even ===ωis=±Ωω GGG Symmetry of IR-active modes, “u” – ungerade (de.) = odd e' A2u Fu kkqis=± 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 Atomic displacement and Raman / IR-activity of optical phonons in crystals with the center of inversion phonons in crystals with the center of inversion GG G G GG G G ikr()⋅−ω0 t Phonons in Ge, Si, Diamond ikr()⋅−ω0 t QQe=⋅0 QQe=⋅0 xx→− GG G G GG G G ikr()⋅− tω are Raman active ikr()⋅− tω EEe=⋅0 No IR-active EEe=⋅0 GG GG GG yy→− IQˆˆ−−⋅=−( ) Q IQQ ⋅= ( ) phonons in Ge, Si, IQQˆ− ⋅=() G GGG GG z →−z IEˆˆ−−⋅=−( ) E IE ⋅=− ( ) E Diamond IEˆ− ⋅=−() E

e e'

No Raman-active phonons in

NaCl, MgO, SrTiO3 25 26

Polarization Selection rules for Raman Scattering Example of selection rules for Raman scattering by optical by phonons phonons in III-V semiconductor crystals (no center of inversion)

GGGG kEEkiiss(, ) Polarization configuration of Raman tensor experiment G G ks k G s k G G G i k k i s ki

Forward scattering Back- scattering 90 deg scattering 27 28 Experimental setups for Raman scattering Experimental setups for Raman scattering

SPEX – “Triplemate”

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Stress analysis in microstructures using Raman scattering Composition analysis in Micro-Raman scattering microstructures

31 32 Raman scattering in Low-dimensional periodic structures Raman scattering in Low-dimensional periodic structures (folded acoustic phonons in GaAs/AlAs MQW’s) (Confined optical phonons in GaAs/AlAs MQW’s)

33 34

Resonant Raman scattering 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×+TO LO 3S = 15 modes 11 2×+TO LO 3 acoustic modes 22 2×+TO33 LO 12 optical modes; 3× 4 2×+TO44 LO

37 38

Raman thermometer Classical formula HW: Which crystal has this phonon dispersion diagram ? 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~1 n + 39 anti− Stokes40 ~ n