Introduction to Crystal Symmetry and Raman Spectroscopy

Introduction to Crystal Symmetry and Raman Spectroscopy José A. Flores Livas Thursday 13 th December, 2012 1 of 30

Outline Introduction to Raman spectroscopy Symmetry of crystals and normal modes Experimental set-up Calculation of Raman intensities Example: Raman intensities for Wurtzite-ZnO 2 of 30

First, what is...? Symmetry Symmetry is when one shape becomes exactly like another if you flip, slide or turn it. The simplest type of symmetry is reflection (mirror symmetry). An orthogonal transformation varying orientation. Spectroscopy The Science concerned with the investigation and measurement of spectra produced when matter interacts with or emits electromagnetic radiation. Raman spectroscopy Spectroscopic technique used to observe vibrational, rotational, and optical quasiparticles in matter. 3 of 30

Applications Raman Spectroscopy is used for Identification of phases (minerals, and composition of materials). Identification of crystalline polymorphs (Olivine, andalusite, etc). Measurement of stress in solids, at nanosclae (nanotubes). High-pressure and High-temperature in-situ studies possible Water content of silicate glasses and minerals, liquid phase, etc. Suitable for biological samples in native state. 4 of 30

In-situ planetary Raman spectroscopy Instruments are small enough to fit in a human hand, it is now feasible to apply Raman spectroscopy as a field tool for geology and planetary exploration. 1 On Mars Surveyor 2003, 2005 and curiosity 2009 missions Spectra were obtained from rocks and soils Martians. Detailed mineralogy information identified trace of minerals and water. 2 1 A. Wang, et al. Journal of Geophysical research, 108, 5005, (2003). 2 See http://mars.jpl.nasa.gov/msl/ 5 of 30

Table of contents Introduction to Raman spectroscopy Symmetry of crystals and normal modes Experimental set-up Calculation of Raman intensities Example: Raman intensities for Wurtzite-ZnO 5 of 30

Spectroscopy of matter with photons 6 of 30

Raman spectroscopy in dates: 1871 Lord Baron-Rayleigh (elastic scattering 3 ) I 1/λ 4. 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. 3 Lord Rayleigh, Philosophical Magazine, 47, p. 375, 1899 4 A. Smekal, Naturwissenschaften, II, (1923). 5 Raman and Krishnan, Nature, 121 (1928). 6 G. Placzek, Z. Phys. 58 585, (1931). 7 of 30

First measure of an inelastic scattered light (1928) 8 of 30

Origin of the Raman scattering A exciting wave (electromagnetic radiation) induce a polarization of matter. The induced dipole moment (or its density) is proportional to the electric field of the wave: P ω (t) = α(ω) E 0 e iωt, where α is the (complex) tensor of polarizability (susceptibility). The electronic contribution α depends on the positions Q of the atomic nuclei. P ω (t, Q) = [α(ω, Q)I 0 + α(ω, Q) Q I 0Q + 1 2 2 α(ω, Q) Q 2 I 0 Q 2 +...] E 0 e iωt For example the polarizability of the H 2 molecule, changes for different inter-atomic distances (see in following slides). 9 of 30

Origin of the Raman scattering For a periodic motion of the nuclei with the frequency Ω, Q(t) = Q m e iωt + Q p e iωt, The electric susceptibility splits into a static and a dynamic Q-dependent part. The leading terms of the induced moment are, P ω (t, Q) α(ω, Q)I 0 E 0 e iωt + 1 α(ω, Q) 2 Q I 0 E 0 [Q p e i(ω Ω)t +Q m e i(ω+ω)t ] containing the unmodified frequency ω, and the sidebands: ω Ω (stokes) and ω + Ω (anti-stokes). 10 of 30

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: P x α xx α xy α xz E x P y = α yx α yy α yz E y P z α zx α zy α zz E z 12 of 30

Energy diagrams of scattered light 13 of 30

Table of contents Introduction to Raman spectroscopy Symmetry of crystals and normal modes Experimental set-up Calculation of Raman intensities Example: Raman intensities for Wurtzite-ZnO 13 of 30

Crystal symmetry 14 of 30

Phonon symmetry in crystals The phonon symmetry description in crystals is given by the analysis of the eigenvector {e(κ ki)}, obtained from the dynamical matrix: {D αβ (κκ k)} Is the displacement pattern of the atoms vibrating in the mode (ki). Knowledge of the forms of these eigenvectors and of their transformation properties under the symmetry operations is often useful for the solution of 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. 7 Vosko, Rev. Mod. Phys., 40, pp. 137, Jan (1968). 15 of 30

Character table: IR, Rmn and H-Raman modes 16 of 30

Assignment of normal modes 17 of 30

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 C n. 18 of 30

On summary for Raman scattering 19 of 30

Table of contents Introduction to Raman spectroscopy Symmetry of crystals and normal modes Experimental set-up Calculation of Raman intensities Example: Raman intensities for Wurtzite-ZnO 19 of 30

Experimental set-up 20 of 30

Experimental geometry: Porto notation 21 of 30

Example for cubic crystals 22 of 30

Table of contents Introduction to Raman spectroscopy Symmetry of crystals and normal modes Experimental set-up Calculation of Raman intensities Example: Raman intensities for Wurtzite-ZnO 22 of 30

Using Kramers-Kroing relation Born and Huang derived 8 an expression which reduces the computation of the integral Raman intensities to the evaluation of the imaginary part of the linear Raman susceptibility. 9 I i,γβ = 2π (ω L ω i ) 4 c 4 ω i [n(ω i ) 1](α i,γβ ) 2, (1) polarisation along γ, and field along β, for the i-mode, ω L is the laser frequency of excitation source, and the Bose occupation number n(ω) 1 = [ ] 1 1 e ( ω i k B T ) where ω i the frequency of mode i and k B the Boltzmann constant. The Raman susceptibility tensor (α i,γβ ) is then, defined as 8 Born and Huang, Dynamical Theory of Crystal Lattice, Oxford U. press (1969). 9 S. A. Prosandeev, et al., Phys Rev. B 71, 214307, (2005). 23 of 30

Calculation of Raman intensities α i,γβ = Ω 4π nγi R iαβ,nγ e inγ M 1/2 n, R iγβ,nν = ɛ γβ(ω L ) u inν. M n mass, e inγ eigenvector and Ω unit cell volume. Two cases: a) For single crystal Eq. 1 is applicable. b) For poly-crystal, an average (using the ellipsoid) of intensity. 10 Reduced intensity for polarised and depolarised light (backscattering) is, [ ] I poly i (ω L ω i ) 4 [1 + n (ω i ) /30ω i ] 10G (0) i + 4G (2) i, [ ] I depol i (ω L ω i ) 4 [1 + n (ω i ) /30ω i ] 5G (1) i + 3G (2) i, I sum i,reduced = I pol i + Idepol i and I ratio i = I poly i /I depol i 10 Poilblanc et Crasnier, Spectroscopies Infrarouge et Raman (2006). 24 of 30

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. References. 11 12 13 14 15 16 17 11 S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58, 1861 (1987). 12 X. Gonze, J.-P. Vigneron, Phys. Rev. B 39, 13120 (1989). 13 X. Gonze, Phys. Rev. A 52, 1096 (1995). 14 S. de Gironcoli, Phys. Rev. B 51, 6773 (1995). 15 X. Gonze, Phys. Rev. B. 55, 10337 (1997). 16 S. Baroni, S. de Gironcoli, A. Dal Corso, Rev. Mod. Phys. 73, 515 (2001). 17 M. Veithen, X. Gonze, Ph. Ghosez, Phys. Rev. B 71, 125107, (2005). 25 of 30

Table of contents Introduction to Raman spectroscopy Symmetry of crystals and normal modes Experimental set-up Calculation of Raman intensities Example: Raman intensities for Wurtzite-ZnO 25 of 30

The case of Wurtzite-ZnO 26 of 30

Polycrystalline Wurtzite-ZnO 27 of 30

Single crystal of ZnO aligned on X(YX)Z 28 of 30

Single crystal of ZnO aligned on X(YY)Z 29 of 30

General comments IR: induced dipole moment due to the change in the atomic positions. Raman: induced dipole moment due to deformation of the e- shell. Simultaneous IR and Raman, only in non-centrosymmetric structures. Further reading Transformation of polarizability tensors. Disorder effects on Raman peaks. Temperature dependence of the Raman scattering. Phonon lifetimes and linewidths (Γ i ). Fröhlich interactions (shape of response) anharmonic effects. Fluctuations of the spin density: Magnon. 18 18 You can download this presentation here: http://www.fysik-aztek.net/ 30 of 30