Introduction to Crystal Symmetry and Raman Spectroscopytddft.org/bmg/files/seminarios/127440.pdf ·...

Introduction to Crystal Symmetry and RamanSpectroscopy

Jose A. Flores Livas

Thursday 13th December, 2012

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Outline

Introduction to Raman spectroscopy

Symmetry of crystals and normal modes

Experimental set-up

Calculation of Raman intensities

Example: Raman intensities for Wurtzite-ZnO

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Outline

Experimental set-up

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Outline

Experimental set-up

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Outline

Experimental set-up

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Outline

Experimental set-up

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First, what is ... ?

SymmetrySymmetry is when one shape becomes exactly like another if you flip,slide or turn it. The simplest type of symmetry is “reflection” (mirrorsymmetry). An “orthogonal” transformation varying orientation.

SpectroscopyThe Science concerned with the investigation and measurement ofspectra produced when matter interacts with or emits electromagneticradiation.

Raman spectroscopySpectroscopic technique used to observe vibrational, rotational, andoptical quasiparticles in matter.

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Applications

Raman Spectroscopy is used for

X Identification of phases (minerals, and composition of materials).X Identification of crystalline polymorphs (Olivine, andalusite, etc).X Measurement of stress in solids, at nanosclae (nanotubes).X High-pressure and High-temperature in-situ studies possibleX Water content of silicate glasses and minerals, liquid phase, etc.X Suitable for biological samples in native state.

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Applications

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Applications

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Applications

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Applications

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Applications

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Applications

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Applications

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In-situ planetary Raman spectroscopy

Instruments are small enough to fit in a human hand, it is now feasibleto apply Raman spectroscopy as a field tool for geology and planetaryexploration.1

On Mars Surveyor 2003, 2005 and curiosity 2009 missionsSpectra were obtained from rocks and soils Martians. Detailedmineralogy information identified trace of minerals and water.2

1A. Wang, et al. Journal of Geophysical research, 108, 5005, (2003).2See http://mars.jpl.nasa.gov/msl/

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In-situ planetary Raman spectroscopy

Instruments are small enough to fit in a human hand, it is now feasibleto apply Raman spectroscopy as a field tool for geology and planetaryexploration.1

On Mars Surveyor 2003, 2005 and curiosity 2009 missionsSpectra were obtained from rocks and soils Martians. Detailedmineralogy information identified trace of minerals and water.2

1A. Wang, et al. Journal of Geophysical research, 108, 5005, (2003).2See http://mars.jpl.nasa.gov/msl/

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Table of contents

Experimental set-up

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Spectroscopy of matter with photons

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Raman spectroscopy in dates:

1871 Lord Baron-Rayleigh (elastic scattering3) I∝ 1/λ4.1923 Inelastic light scattering is predicted by Adolf Smekal.4

1928 Landsberg seen frequency shifts in scattering from quartz.1928 C.V. Raman and K.S. Krishnan see feeble fluorescence in liquids.5

1930 Nobel prize to C.V. Raman for his work on the scattering of light.1931 Q.M. theory, Georges Placzek: “Zur theorie des Ramaneffekts”.6

1960 Patent of “laser” by Bell laboratory.1975 Shorygin, resonance Raman and use of micro-Raman techniques.2012 More than 25 types of linear and non-linear Raman spectroscopy.

3Lord Rayleigh, Philosophical Magazine, 47, p. 375, 18994A. Smekal, Naturwissenschaften, II, (1923).5Raman and Krishnan, Nature, 121 (1928).6G. Placzek, Z. Phys. 58 585, (1931).

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First measure of an inelastic scattered light (1928)

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Origin of the Raman scattering

A exciting wave (electromagnetic radiation) induce a polarization ofmatter. The induced dipole moment (or its density) is proportional tothe electric field of the wave:

−→P ω(t) = α(ω)

−→E 0e

−iωt,

where α is the (complex) tensor of polarizability (susceptibility). Theelectronic contribution α depends on the positions Q of the atomic nuclei.

−→P ω(t, Q) = [α(ω,Q)I0 +

∂α(ω,Q)∂Q

I0Q+12∂2α(ω,Q)∂Q2

I0Q2 + ...]

−→E 0e

−iωt

For example the polarizability of the H2 molecule, changes for differentinter-atomic distances (see in following slides).

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−→P ω(t) = α(ω)

−→E 0e

−iωt,

−→P ω(t, Q) = [α(ω,Q)I0 +

∂α(ω,Q)∂Q

I0Q+12∂2α(ω,Q)∂Q2

I0Q2 + ...]

−→E 0e

−iωt

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−→P ω(t) = α(ω)

−→E 0e

−iωt,

−→P ω(t, Q) = [α(ω,Q)I0 +

∂α(ω,Q)∂Q

I0Q+12∂2α(ω,Q)∂Q2

I0Q2 + ...]

−→E 0e

−iωt

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For a periodic motion of the nuclei with the frequency Ω,

Q(t) = Qme−iΩt +Qpe

−iΩt,

The electric susceptibility splits into a static and a dynamic Q-dependentpart. The leading terms of the induced moment are,

−→P ω(t, Q) ≈ α(ω,Q)I0

−→E 0e

−iωt+12∂α(ω,Q)∂Q

I0−→E 0[Qpe−i(ω−Ω)t+Qme−i(ω+Ω)t]

containing the unmodified frequency ω, and the sidebands:ω − Ω (stokes) and ω + Ω (anti-Stokes).

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−iΩt,

−→P ω(t, Q) ≈ α(ω,Q)I0

−→E 0e

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−iΩt,

−→P ω(t, Q) ≈ α(ω,Q)I0

−→E 0e

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Selection rules of vibrational modes

Rule of thumb: symmetric=Raman active, asymmetric=IR active 11 of 30

Polarisability tensor matrix

Since−→P and

−→E are both vectors, in general for molecules and crystals

the correct expression is:PxPyPz

αxx αxy αxzαyx αyy αyzαzx αzy αzz

ExEyEz

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Energy diagrams of scattered light

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Table of contents

Experimental set-up

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Crystal symmetry

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Phonon symmetry in crystals

The phonon symmetry description in crystals is given by the analysis ofthe eigenvector e(κ|ki), obtained from the dynamical matrix:

Dαβ(κκ′|k)

Is the displacement pattern of the atoms vibrating in the mode (ki).Knowledge of the forms of these eigenvectors and of their transformationproperties under the symmetry operations is often useful for the solutionof certain types of lattice dynamical problems.7 Among:

• Selection rules for processes such as two-phonon lattice.• Absorption and the second-order Raman effect (or higher order).• Phonon-assisted electronic transitions.

7Vosko, Rev. Mod. Phys., 40, pp. 137, Jan (1968).15 of 30

Dαβ(κκ′|k)

Character table: IR, Rmn and H-Raman modes

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Assignment of normal modes

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Mulliken symbols

A,B Non-degenerate (single) mode: one set of atom displacements.Subscripts g and u: symmetric or a-symmetric to inversion −1.

E Doubly degenerate mode. Two sets of atom displacements.Superscripts ’ and “: symmetric or anti-symmetric to a m.

T Triply degenerate mode. Three sets of atom displacements.Subscripts 1 and 2: symmetric or anti-symmetric to m or Cn.

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Mulliken symbols

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Mulliken symbols

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Mulliken symbols

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Mulliken symbols

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On summary for Raman scattering

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Table of contents

Experimental set-up

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Experimental set-up

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Experimental geometry: Porto notation

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Example for cubic crystals

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Table of contents

Experimental set-up

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Using Kramers-Kroing relation

Born and Huang derived8 an expression which reduces the computationof the integral Raman intensities to the evaluation of the imaginary partof the linear Raman susceptibility.9

Ii,γβ =2π~(ωL − ωi)4

c4ωi[n(ωi)− 1](αi,γβ)2, (1)

polarisation along γ, and field along β, for the i-mode, ωL is the laserfrequency of excitation source, and the Bose occupation number

n(ω)− 1 =[1− e(

−~ωikBT )

where ωi the frequency of mode i and kB the Boltzmann constant. TheRaman susceptibility tensor (αi,γβ) is then, defined as

8Born and Huang, ”Dynamical Theory of Crystal Lattice“, Oxford U. press (1969).9S. A. Prosandeev, et al., Phys Rev. B 71, 214307, (2005).

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αi,γβ =√

∑nγi

Riαβ,nγeinγM−1/2n , Riγβ,nν =

∂εγβ(ωL)∂uinν

Mn mass, einγ eigenvector and Ω unit cell volume. Two cases: a) Forsingle crystal Eq. 1 is applicable. b) For poly-crystal, an average (usingthe ellipsoid) of intensity.10 Reduced intensity for polarised ‖ anddepolarised ⊥ light (backscattering) is,

Ipolyi‖ ∼ (ωL − ωi)4 [1 + n (ωi) /30ωi][10G(0)

i + 4G(2)i

Idepoli⊥ ∼ (ωL − ωi)4 [1 + n (ωi) /30ωi][5G(1)

i + 3G(2)i

Isumi,reduced = Ipoli‖ + Idepoli⊥ and Iratioi = Ipolyi‖ /Idepoli

10Poilblanc et Crasnier, ”Spectroscopies Infrarouge et Raman“ (2006).24 of 30

αi,γβ =√

∑nγi

i + 4G(2)i

i + 3G(2)i

αi,γβ =√

∑nγi

i + 4G(2)i

i + 3G(2)i

αi,γβ =√

∑nγi

i + 4G(2)i

i + 3G(2)i

On summary for the ab inito calculation:

Third-order derivatives of the energy can be calculated

• DFPT for χ + Frozen phonon (finite differences).• DFPT using (2n+ 1) theorem. The th derivative of energy depends

only on derivatives up to order n of the charge density.• DFPT + second-order response to electric filed.• Finite electric fields + frozen phonon.