Tag Archives: Spectroscopy

Symmetry, selection rules and reduction to a bare-bones model

When I was a graduate student, a group of us spent long hours learning group theory from scratch in effort to understand and interpret our experiments. One of our main goals back then was to understand Raman and infrared selection rules of phonons. We pored over the textbook and notes by the late Mildred Dresselhaus (the pdf can be found for free here). It is now difficult for me to remember what it was like looking at data without the vantage point of symmetry, such was the influence of the readings on my scientific outlook. Although sometimes hidden behind opaque mathematical formalism, when boiled down to their essence, the ideas are profound in how they simplify certain problems.

Simply stated, symmetry principles allow us to strip unnecessary complicating factors away from certain problems as long as the pertinent symmetries are retained. In this post, I will discuss Raman and IR selection rules in a very simple model that illustrates the essence of this way of thinking. Although this model is cartoonish, it contains the relevant symmetries that cut right to the physics of the problem.

To illustrate the concepts, I will be using the following harmonic oscillator-based model (what else would I use?!). Let’s consider the following setup, depicted below:

Figure 1: Two identical masses, two walls, three springs

It’s relatively intuitive to see that this system possesses two normal modes (see image below). One of these normal modes is inversion symmetric (i.e. maintains the symmetry about the dashed vertical line through the entirety of its oscillatory motion), while the other normal mode is manifestly inversion asymmetric (i.e. does not maintain symmetry about the dashed vertical line through the entirety of its oscillatory motion). In particular, this latter mode is anti-symmetric. These considerations lead us to label the symmetric mode “even” and the anti-symmetric mode “odd”. (I should mention that in group theoretical textbooks, even modes are often labelled gerade (German for even), while odd modes are labelled ungerade (German for odd), from which the u and g subscripts arise). Normal modes for the system of two oscillators are depicted below:

Figure 2: (Left) An even (gerade) normal mode. (Right) An odd (ungerade) normal mode.

These are the “natural” modes of the system, but our ability to observe them requires us to excite these modes in some way. How do we do this? This is where the analogy to IR and Raman spectroscopy comes in. I’ll describe the analogy in more detail below, but for now consider the possibility that we can move the walls in the oscillator system. Consider moving the walls in the following way. We can oscillate the walls back and forth, moving them closer and farther apart as depicted in the image below. Clearly, this oscillatory wall motion is also symmetric.

Figure 3: A symmetric perturbation that will excite the symmetric mode. (Left) Depiction of the wall motion at time t and (Right) at time t+T/2, where T is the period of oscillation.

This obviously isn’t the only way that we can move the walls. We could just as easily move them like this, which is anti-symmetric:

Figure 4: An anti-symmetric perturbation that will excite the anti-symmetric mode. (Left) Depiction of the wall motion at time t and (Right) at time t+T/2, where T is the period of oscillation.

While there are many other ways we could move the walls, it turns out that the above images essentially capture how Raman (gerade) and infrared (ungerade) modes are excited in a solid. Infrared modes are excited using an odd perturbation (proportional to the electric field \vec{E}), while Raman modes are excited with an even perturbation (proportional to two instances of the electric field \vec{E}\vec{E})^{**}. (Under an inversion operation, the electric field switches sign, thus the infrared perturbation is odd while the Raman perturbation is even). And that’s basically it — you can see from the images that an even (odd) perturbation will result in the excitation of the even (odd) normal mode!

While this model is unlikely to be taught in classrooms any time soon in reference to Raman and IR selection rules, it does capture the physical picture in a (what I would consider) meaningful way through the use of symmetry. You can even imagine changing the relative masses of the two blocks, and you would then start to see that the formerly IR and Raman modes start to “mix”. The normal modes would no longer be purely even and odd modes, and the perturbations would then excite linear combinations of these new modes (e.g. the even perturbation would excite both modes). The analogous spectroscopic statement would be that in a system that lacks inversion symmetry, normal modes are not exclusively Raman or IR active.

While pictures like this won’t give you a precise solution to most questions you’re trying to answer, they will often help you identify obviously wrong lines of reasoning. It’s been said that physics is not much more than the study of symmetry. While that’s not exactly true, it’s hard to overstate its importance.

^{**} Why is the Raman excitation even? There are many ways to explain this, but on a cartoon level, the first photon (the electric field vector) induces a dipole moment, and the second photon (the other electric field) interacts with the induced dipole. Because this is a two-photon process (i.e. photon-in, photon-out), the excitation is even under inversion. (I should mention that the strength of the induced dipole moment is related to the polarizability of system, which is often why folks talk about the polarizability in relation to Raman spectroscopy).

Why is the infrared excitation odd? Contrary to the Raman excitation, the infrared excitation requires the absorption of the incoming photon. Thus, infrared spectroscopy is a single photon process and requires only a single electric field vector to couple to a dipole moment. The excitation is thus odd under inversion.

Electron-Hole Droplets

While some condensed matter physicists have moved on from studying semiconductors and consider them “boring”, there are consistently surprises from the semiconductor community that suggest the opposite. Most notably, the integral and fractional quantum Hall effect were not only unexpected, but (especially the FQHE) have changed the way we think about matter. The development of semiconductor quantum wells and superlattices have played a large role furthering the physics of semiconductors and have been central to the efforts in observing Bloch oscillations, the quantum spin Hall effect and exciton condensation in quantum hall bilayers among many other discoveries.

However, there was one development that apparently did not need much of a technological advancement in semiconductor processing — it was simply just overlooked. This was the discovery of electron-hole droplets in the late 60s and early 70s in crystalline germanium and silicon. A lot of work on this topic was done in the Soviet Union on both the theoretical and experiment fronts, but because of this, finding the relevant papers online are quite difficult! An excellent review on the topic was written by L. Keldysh, who also did a lot of theoretical work on electron-hole droplets and was probably the first to recognize them for what they were.

Before continuing, let me just emphasize, that when I say electron-hole droplet, I literally mean something quite akin to water droplets in a fog, for instance. In a semiconductor, the exciton gas condenses into a mist-like substance with electron-hole droplets surrounded by a gas of free excitons. This is possible in a semiconductor because the time it takes for the electron-hole recombination is orders of magnitude longer than the time it takes to undergo the transition to the electron-hole droplet phase. Therefore, the droplet can be treated as if it is in thermal equilibrium, although it is clearly a non-equilibrium state of matter. Recombination takes longer in an indirect gap semiconductor, which is why silicon and germanium were used for these experiments.

A bit of history: The field got started in 1968 when Asnin, Rogachev and Ryvkin in the Soviet Union observed a jump in the photoconductivity in germanium at low temperature when excited above a certain threshold radiation (i.e. when the density of excitons exceeded \sim 10^{16}  \textrm{cm}^{-3}). The interpretation of this observation as an electron-hole droplet was put on firm footing when a broad luminescence peak was observed by Pokrovski and Svistunova below the exciton line (~714 meV) at ~709 meV. The intensity in this peak increased dramatically upon lowering the temperature, with a substantial increase within just a tenth of a degree, an observation suggestive of a phase transition. I reproduce the luminescence spectrum from this paper by T.K. Lo showing the free exciton and the electron-hole droplet peaks, because as mentioned, the Soviet papers are difficult to find online.

EHD-Lo.JPG

From my description so far, the most pressing questions remaining are: (1) why is there an increase in the photoconductivity due to the presence of droplets? and (2) is there better evidence for the droplet than just the luminescence peak? Because free excitons are also known to form biexcitons (i.e. excitonic molecules), the peak may easily interpreted as evidence of biexcitons instead of an electron-hole droplet, and this was a point of much contention in the early days of studying the electron-hole droplet (see the Aside below).

Let me answer the second question first, since the answer is a little simpler. The most conclusive evidence (besides the excellent agreement between theory and experiment) was literally pictures of the droplet! Because the electrons and holes within the droplet recombine, they emit the characteristic radiation shown in the luminescence spectrum above centered at ~709 meV. This is in the infrared region and J.P. Wolfe and collaborators were actually able to take pictures of the droplets in germanium (~ 4 microns in diameter) with an infrared-sensitive camera. Below is a picture of the droplet cloud — notice how the droplet cloud is actually anisotropic, which is due to the crystal symmetry and the fact that phonons can propel the electron-hole liquid!

Pic_EHD.JPG

The first question is a little tougher to answer, but it can be accomplished with a qualitative description. When the excitons condense into the liquid, the density of “excitons” is much higher in this region. In fact, the inter-exciton distance is smaller than the distance between the electron and hole in the exciton gas. Therefore, it is not appropriate to refer to a specific electron as bound to a hole at all in the droplet. The electrons and holes are free to move independently. Naively, one can rationalize this because at such high densities, the exchange interaction becomes strong so that electrons and holes can easily switch partners with other electrons and holes respectively. Hence, the electron-hole liquid is actually a multi-component degenerate plasma, similar to a Fermi liquid, and it even has a Fermi energy which is on the order of 6 meV. Hence, the electron-hole droplet is metallic!

So why do the excitons form droplets at all? This is a question of kinetics and has to do with a delicate balance between evaporation, surface tension, electron-hole recombination and the probability of an exciton in the surrounding gas being absorbed by the droplet. Keldysh’s article, linked above, and the references therein are excellent for the details on this point.

In light of the recent discovery that bismuth (also a compensated electron-hole liquid!) was recently found to be superconducting at ~530 microKelvin, one may ask whether it is possible that electron-hole droplets can also become superconducting at similar or lower temperatures. From my brief searches online it doesn’t seem like this question has been seriously asked in the theoretical literature, and it would be an interesting route towards non-equilibrium superconductivity.

Just a couple years ago, a group also reported the existence of small droplet quanta in GaAs, demonstrating that research on this topic is still alive. To my knowledge, electron-hole drops have thus far not been observed in single-layer transition metal dichalcogenide semiconductors, which may present an interesting route to studying dimensional effects on the electron-hole droplet. However, this may be challenging since most of these materials are direct-gap semiconductors.

Aside: Sadly, it seems like evidence for the electron-hole droplet was actually discovered at Bell Labs by J.R. Haynes in 1966 in this paper before the 1968 Soviet paper, unbeknownst to the author. Haynes attributed his observation to the excitonic molecule (or biexciton), which he, it turns out, didn’t have the statistics to observe. Later experiments confirmed that it indeed was the electron-hole droplet that he had observed. Strangely, Haynes’ paper is still cited in the present time relatively frequently in the context of biexcitons, since he provided quite a nice analysis of his results! Also, it so happened that Haynes died after his paper was submitted and never found out that he had actually discovered the electron-hole droplet.

Ruminations on Raman

The Raman effect concerns the inelastic scattering of light from molecules, liquids, or solids. Brian has written a post about it previously, and it is worth reading. Its use today is so commonplace, that one almost forgets that it was discovered back in the 1920s. As the story goes (whether it is apocryphal or not I do not know), C.V. Raman became entranced by the question of why the ocean appeared blue while on a ship back from London to India in 1921. He apparently was not convinced by Rayleigh’s explanation that it was just the reflection of the sky.

When Raman got back to Calcutta, he began studying light scattering from liquids almost exclusively. Raman experiments are nowadays pretty much always undertaken with the use of a laser. Obviously, Raman did not initially do this (the laser was invented in 1960). Well, you must be thinking, he must have therefore conducted his experiments with a mercury lamp (or something similar). In fact, this is not correct either. Initially, Raman had actually used sunlight!

If you have ever conducted a Raman experiment, you’ll know how difficult it can be to obtain a spectrum, even with a laser. Only about one in a million of the incident photons (and sometimes much fewer) actually gets scattered with a change in wavelength! So for Raman to have originally conducted his experiments with sunlight is really a remarkable achievement. It required patience, exactitude and a great deal of technical ingenuity to focus the sunlight.

Ultimately, Raman wrote his results up and submitted them to Nature magazine in 1928. Although these results were based on sunlight, he had just obtained his first mercury lamp to start his more quantitative studies by then. The article made big news because it was a major result confirming the new “quantum theory”, but Raman immediately recognized the capability of this effect in the study of matter as well. After many years of studying the effect, he came to realize that the reason that water is blue is basically the same as why the sky is blue — Rayleigh scattering goes as 1/\lambda^4.

Readers of this blog will actually notice that I have written about Raman scattering in several different contexts on this site, for instance, in measuring the Damon-Eschbach mode and the Higgs amplitude mode in superconductorsilluminating the nature of LO-TO splitting polar insulators and measuring unusual collective modes in Kondo insulators demonstrating its power as probe of condensed matter even in the present time.

On this blog, one of the major themes I’ve tried to highlight is the technical ingenuity of experimentalists to observe certain phenomena. I find it quite amazing that the Raman effect had its rather meager origins in the use of sunlight!

A View From an X-ray Beam

X-rays have become a rather commonplace tool for people within the art world for several reasons. The example I gave in a previous post was for its use in exposing artistic sketches beneath the final image. These revelations gave a window into Picasso’s artistic process.

X-ray spectroscopy has also become an important method by which to verify the authenticity of a painting. It can determine the materials used, elements within the paint, and the type of paper utilized. This can help pinpoint the painting geographically as well as revealing its age.

To the left below is one of Georges Seurat’s pointilist masterpieces entitled Young Woman Powdering Herself. Apparently, this woman was Seurat’s mistress. X-rays revealed that Seurat had originally painted himself watching her from the window, but later covered this up. It turns out that this would have been Seurat’s only self-portrait.

seurat

There are several more of these images here accompanied by other interesting anecdotes. Be sure to click on the images to reveal the sketches beneath!

A First-Rate Experiment: The Damon-Eshbach Mode

One of the things I have tried to do on this blog is highlight excellent experiments in condensed matter physics. You can click the following links for posts I’ve written on illuminating experiments concerning the symmetry of the order parameter in cuprate superconductors, Floquet states on the surface of topological insulators, quantized vortices in superfluid 4He, sonoluminescence in collapsing bubbles and LO-TO splitting in doped semiconductors, just to highlight a few. Some of these experiments required some outstanding technical ingenuity, and I feel it important to document them.

In a similar vein, there was an elegant experiment published in PRL back in 1977 by P. Grunberg and F. Metawe that shows a rather peculiar spectral signature observed with Brillouin scattering in thin film EuO. The data is presented below. For those who don’t know, Brillouin scattering is basically identical to Raman scattering, but the energy scale observed is much lower, typically a fraction of a cm^{-1} ~ 5 cm^{-1} (1 cm^{-1} \approx 30GHz). Brillouin scattering is often used to observe acoustic phonons.

Damon-Eshbach

From the image above there is immediately something striking in the data: the peak labeled M2 only shows up on either the anti-Stokes side (the incident light absorbs a thermally excited mode) or the Stokes side (the incident light excites a mode) depending on the orientation of the magnetic field. In his Nobel lecture, Grunberg revealed that they discovered this effect by accident after they had hooked up some wires in the opposite orientation!

Anyway, in usual light scattering experiments, depending on the temperature, modes are observed on both sides (Stokes and anti-Stokes) with an intensity difference determined by Bose-Einstein statistics. In this case, two ingredients, the slab geometry of the thin film and the broken time-reversal symmetry give rise to the propagation of a surface spin wave that travels in only one direction, known as the Damon-Eshbach mode. The DE mode propagates on the surface of the sample in a direction perpendicular to the magnetization, B, of the thin film, obeying a right-hand rule.

When one thinks about this, it is truly bizarre, as the dispersion relation would for the DE mode on the surface would look something like the image below for the different magnetic field directions:

Damon-Eshbach

One-way propagation of Damon Eshbach Mode

The dispersion branch only exists for one propagation direction! Because of this fact (and the conservation of momentum and energy laws), the mode is either observed solely on the Stokes or anti-Stokes side. This can be understood in the following way. Suppose the experimental geometry is such that the momentum transferred to the sample, q, is positive. One would then be able to excite the DE mode with the incident photon, giving rise to a peak on the Stokes side. However, the incident photon in the experiment cannot absorb the DE mode of momentum -q, because it doesn’t exist! Similar reasoning applies for the magnetization in the other direction, where one would observe a peak in only the anti-Stokes channel.

There is one more regard in which this experiment relied on a serendipitous occurrence. The thin film was thick enough that the light, which penetrates about 100 Angstroms, did not reach the back side of the film. If the film had been thin enough, a peak would have shown up in both the Stokes and anti-Stokes channels, as the photon would have been able to interact with both surfaces.

So with a little fortune and a lot of ingenuity, this experiment set Peter Grunberg on the path to his Nobel prize winning work on magnetic multilayers. As far as simple spectroscopy experiments go, one is unlikely to find results that are as remarkable and dramatic.