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Raman Spectroscopy Raman spectroscopy 1 1. DESCRIPTION OF THE RAMAN EXPERIMENT 1.1 General Description Raman spectroscopy is a technique for observing vibrational, rotational, electronic and other low-frequency modes in a system. It relies on inelastic scattering (i.e Raman scattering) of monochromatic light, usually from a laser in the visible, near infrared, or near ultraviolet range. The laser light indices transitions between different energy states in crystals or molecules, resulting in an energy loss (Stokes process) or an energy gain (anti-Stokes process) in the scattered light [1]. These energy shifts of the photons correspond to specific excitations in a material. They are detected by a special optical spectrometer. Light can interact with matter in a number of different ways. Refraction, reflection and absorption, that you are familiar with, are the most common ones. Refracted light changes direction, which is used in lens and prism optics. Absorbed photons disappear, giving off their energy to the absorbing material. During reflection light bounces off the surfaces or interfaces of materials. When thinking of light in terms of photons, reflection from surfaces is a type of scattering. There are other types. You are probably familiar with Compton scattering when high energy photons scatter from free electrons exchanging energy and momentum with them. This lab focuses on Raman scattering, where visible light interacts with crystals while exchanging energy with them. The simplest (but not quite correct as described below) way to understand Raman scattering is to imagine that a photon gives off some but not all of its energy to the material. In the process the photons do not disappear, but loose or gain some energy, i.e. they change their wavelength. By measuring the spectrum of the Raman-scattered light one can learn a great deal about the excitation spectrum of the material. Raman scattering is used widely in chemistry to identify materials based on their Raman spectra. In physics Raman scattering is used for investigating a wide variety of materials. It provides information about how atoms vibrate and about electronic excitations. In this lab we will perform experiments on two quantum materials, Silicon, which is widely used in electronics and YBaCuO, which is closely related to high temperature superconductors. 2 1.2 Basic Theory Raman scattering can be understood on an intuitive level starting from basic quantum mechanics. Due to Heisenberg energy-time uncertainty principle, a photon can be absorbed without conservation of energy as long as it is emitted a short time after. This process is called virtual absorption and the shortly-lived (virtual) excited state is called the intermedi- ate state. Most photons that are absorbed like this are reemitted a very short time later with the same wavelength and no excitation in the material is created or destroyed in the end. This process is called Rayleigh scattering. It actually interferes with Raman measurements and Raman spectrometer we are going to use is designed to filter it out. During Raman scattering the intermediate virtual state is involved in the same way, but wavelength of the emitted photon is different from the incident one. In this case the energy has to be conserved between the initial and the final state, because both the incident and scattered photon have a long lifetime. In order to conserve energy an excitation is created inside the material. For example an electron can jump into a different orbital or an atomic lattice vibration is induced or annihilated. The energy of this excitation must equal to the difference between the incident and scattered photon. If there is no excitation that would satisfy this rule, then the scattering process is impossible. Therefore, Raman scattering can occur only if the difference between the incident and scattered photon energies is equal to the energy of an excitation inside the material. Let's see how this works starting from the quantum mechanical Hamiltonian describing the interaction of light with electrons. If you don't understand the following equations, you will still be able to do the lab. First, consider the coupling between photons and electrons. The hamiltionian is shown as follows [2][3]: ^ 2 X [^pi + eA(~ri; t)=c] H^ = = H^ + H^ + H^ (1) 2m 0 1 2 i ^ 2 2 Z 2 X [eA(~ri; t)=c] e H^ = = d~rρ(~r) A^(~r ; t) (2) 1 2m 2mc2 i i X e e Z H^ = (^p · A^(~r ; t) + A^(~r ; t) · p^ ) = d~r^j(~r) · A^(~r; t); (3) 2 2mc i i i i c i ^ wherep ^i is the momentum operator, A(~ri; t) is the electromagnetic vector potential, P ^ charge density operator ρ(~r) = i δ(~r − ~ri), electron particle current operator j(~r) = 3 P p^iδ(~r−~ri)+δ(~r−~ri)^pi i 2m , and sum is over all electrons and we omit the term without the vector potential. As a result of one scattering event, the scattering medium makes a transition from state jii to state jfi and the laser light from jNLki to jNLk − 1;NSk0 = 1i. Here L and S are the polarization of light and k is the wavevector. Let X denote the Raman active excitation. If X is coupled to ~ρ(~r) and ~j(~r) and satisfies the symmetry requirements (see section 1.5), it will be Raman active. From perturbation theory, in the absence of a significant number of photons in the scattered channel, we get the following second order scattering cross section: d2σ ! !3 n V 2 L S S X −βi 2 = 4 e jhfj χLS jiij δ(!L − !S − !) (4) d!dΩ nLc i;f in which 2 ~ ~ ~ ~ ~ ~ ~ ~ V e X hfj j(kS) · ~eS jmi hmj j(−kL) · ~eL jii hfj j(−kL) · ~eL jmi hmj j(−kS) · ~eS jii hfj χ jii = ( + ) LS ! ! ! − ! − ! − iη ! − ! + ! − iη L S m m i L m i L e2 − me!L!S~eL · ~eS hfj ρ(−~q) jii (5) ^ ^ where H1 contributes to the first order expansion and H2 to the second order term, ~eL and ~eS are polarization vectors of the incident and scattered photons respectively, !L and !S are their frequencies, jii ; jfi and jmi are the initial, final and (virtual) intermediate electronic states of second order perturbation theory, and !i;!f and !m are their energies. η is a positive infinitesimal, which is replaced by the damping parameter for state jmi, where appropriate nL and nS are the refractive index at !L and !S, me is the mass of the electron. 1.3 Notation for describing polarization geometries Notation for describing the result of a Raman experiment is a(bc)d, where a(d) is direction of momentum of incident (scattered) photons, and b(c) is the direction of polarization of incident (scattered) photons. For example, z(xy)¯z means that momentum of the incident photon is in the z direction, of scattered photon in the z direction; incident photon is polarized along x, scattered along y. In our experiment we use a quasi backscattering geometry, so the polarization configuration is z(ab)¯z. We just use ab below for simplicity. 4 1.4 Resonant Raman Scattering From equation (5), when either of the denominators goes to zero, we will get the resonant Raman Scattering. At the resonance condition, Raman signals are greatly enhanced. 1.5 Selection Rules Crystals are highly symmetric. There are two types of symmetries in crystals: Discrete translational symmetry and discrete point group symmetry. The translational symmetry means that certain translations take the crystal into itself. In addition certain rotations reflections, and inversion performed while keeping one point fixed also take the crystal into itself. For example, in a simple cubic crystal 90 degree rotations around any one atomic position take the crystal into itself, etc. Turns out that these symmetry operations have deep consequences for the physical properties of materials, including for what is observed in the Raman spectra. The formal theory is rather involved and is outside the scope of this lab. For our purposes it is important to realize that the wavelength of both incident and outgoing light is much longer than the interatomic distances, so one can think of every unit cell sitting in the oscillating effectively spatially uniform electric field of the incident light plus the electric field of the Raman-scattered light. Now, for example, consider an atomic vibration in a crystal with inversion symmetry induced by the Raman process. Inversion takes the equilibrium structure into itself, however, if you think of the phonon as a deviation from the equilibrium structure, inversion can take the phonon into minus itself (odd phonon) or into itself (even phonon) depending on the specifics of how the atoms vibrate. Photons in the incident and scattered light together are even under inversion, because the relative angle between their polarizations does not change. The scattering process starts out with the incident electric field only, and ends with the phonon and scattered electric field. But the symmetry of the entire system (electric field plus crystal) has to remain the same before and after scattering. Let's consider, for example, inversion. Originally, crystal is even under inversion and the incident electric field is odd, so the total system before scattering is odd under inversion. After the scattering, if the phonon is even under inversion, the system is still odd (because the scattered photon is odd), so this scattering process is allowed. If 5 the phonon is odd under inversion, then the symmetry after scattering is even, and such a process is forbidden. Each symmetry operation of the crystal imposes strict constraints on which incident/scattered photon polarization allows scattering from which phonons.
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