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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