Raman Effect The Scattering of electromagnetic radiation by matter with a change of frequency

Outline Introduction Classical Description Quantum Description Resonant Raman Scattering Conservation of energy and momentum Symmetry of Raman Tensor Selection Rules Experimental Setup and results

Introduction When light enters a medium it is part reflected part refracted part scattered and part absorbed. Scattering is due to inhomogeneities inside the medium. When these inhomogeneities are not static (density fluctuations) the scattered light can have a change of frequency. This is called Raman scattering.

Phonons dispersion relation of Si Introduction There are many non static inhomogeneities, due to Temperature, that can be described as elementary excitations of the medium: Phonons, Plasmons, Spin-Waves, Electronic states etc. Phonons dispersion relation of Si

The Dielectric function Of a collection of simple harmonic oscillators with density N, charge Q, Mass M and natural frequencies ωi is: The SHO can be electronic or from lattice vibrations. The response of the SHO to an electric field with frequency ω depends on the difference (ω-ωi). if Lattice contribution is negligible, and the electronic contribution doesn't depend on ω

Classical Macroscopic theory Definitions and results from electrodynamics In a dielectric medium the electric force is different from the one in vacuum due to polarizability: Radiation of an oscillating dipole P

Classical Macroscopic theory Polarizability Atomic thermal vibrations (or any other density fluctuations) denoted (r,t) can be expanded as plane waves: The electric susceptibilty is fluctuating due to these thermal vibrations and can be expanded from zero temperature value (treating separately each normal mode) as

Classical Macroscopic theory Polarizability Since the polarizability also fluctuate anti Stokes Stokes Pind is an oscillating dipole and therefore it radiates. This radiation is Raman Scattering.

Classical Macroscopic theory Polarizability There are two frequencies of oscillation, which give two different scattering lines: Stokes Anti-Stokes Power radiated by Pind Raman Tensor

Oscillating dipole doesn’t radiate. Quantum transitions do. Quantum description Oscillating dipole doesn’t radiate. Quantum transitions do. transition probabilities are calculated with fermi golden rule with

Quantum Description Who interacts with what phonon-photon interaction is weak, since semi classical approach – ignore Hrad HeR is treated in the electric dipole approximation. adiabatic approximation. Electrons are in the ground state before and after the scattering The state of the crystal is separated to a product of electrons state and phonon states.

Quantum description schematic representation Incoming Photon interacts with an electron. the Photon is annihilated and the electron is excited to an intermediate virtual state |b>. The excited electron interact with a phonon, and returns to the electronic ground state creating a scattered Photon.

Quantum Description Feynman Diagrams There are six processes that contributes to the one phonon stokes Raman scattering. Three of the m are shown.

Quantum description transition probability

Resonance term the resonance term in the transition probability leads to an enhancement of the scattering intensity when the incident light is close to an electronic energy level. This allows to explore the energy spectrum of the mater in the light energy range. Only one term contribute the most, because it is the multiply of two resonances: that of the incoming beam and that of the outgoing beam.

Energy and momentum conservation one phonon process Conservation of energy and crystal momentum requires (for one-phonon process) Sizes of k,q,i and  0 Wavevector of a visible light photon ~ 105cm-1 Wavevector of phonons range typically 0-107cm-1 Photons can exchange momentum only with zone center phonons (q~0) and Q=0

What is the Raman Tensor In the classical viewpoint, the induced dipole moment is proportional to the Raman tensor, and to the fluctuation amplitude. Quantum mechanics replaces the amplitude with occupancy. The scattering intensity of a certain process (certain Phonon branch) is proportional to the Raman tensor squared of that process. To find the intensity of a certain frequency shift we need to find the Raman tensors for all phonons which give that shift. l is the incident photon polarization m is the scattered photon polarization and k is the phonon polarization

Raman Tensor example The third rank tensor for the diamond structure crystal (for even-parity Phonons belonging to 25’ representation) is: For scattering from yz plane (100). From wavevector conservation q is along the x axis. If ki , ks are also along the x axis, then the Raman tensor will be Ryz and scattering intensity will be proportional to dLO2 and the scattering is only from LO phonons. If the photons goes in (110) direction q will also be in (110) the Raman tensor will be a combination of Ryz and Rzx and TO Phonons also participate in the scattering.

Selection Rules Phonons wavefunction symmetry for q=0 can be characterized by the irreducible representation of the crystal symmetry group. A Phonon can participate in a scattering process only if its symmetry X the symmetry of the third rank tensor contains the A1 fully symmetric represnentation. Therefore, Certain polarizations and geometries gives no Raman scattering, because of symmetry requirements. For example: odd-parity Phonons in a crystal with center of inversion symmetry (diamond) are forbidden.

What can we learn from Raman Scattering The investigation of the Raman spectrum of a crystal should include the angular and polarization dependence of the scattering intensity, and also the width of peak and the efficiency. From this information can be extracted: The frequency of an optic phonon The symmetry of the phonon Electron-Phonon interaction Two Phonon scattering process give information about Phonon density of states From the incident light frequency dependence of the intensity we can find electronic energy levels

Experimental setup

Ratio of the anti Stokes to Stokes. Temperature dependence of Raman scattering in silicon T. R. Hart, R.T Aggarwal, Benjamin Lax, Phys Rev B. 1 638 (1970) Stokes and anti Stokes lines In different T. Ratio of the anti Stokes to Stokes.

Scattering intensity as a function of photon energy in GaP

Exciton mediated RRS in CdTe

Scattering from electronic states of a Doped GaP Energy levels of Zn acceptor in GaP Raman Spectrum of GaP dopped with Zn

Spin Flip Raman Scattering in CdS Florescence spectrum that shows the bound exciton lines, which are close to the Ar+ 4880Å laser line. Raman Spectrum due to magnetically split ground state of the exciton

Spin Flip Raman Scattering in CdS

Spin Flip Raman Scattering in CdS Measurement of the electron g factor, with the separation of the Stokes and anti Stokes lines vs. the magnetic field

Results Si and C from Modi’s Lab