# Tag Archives: Phonons

## LST Relation – The Physical Picture

In 1941, Lydanne, Sachs and Teller wrote a paper entitled “On the Polar Vibrations of Alkali Halides”, where they derived a result now known as the Lydanne-Sachs-Teller (LST) relation. It has wide applicability for polar insulators. I reproduce the relation below:

$\frac{\omega_{LO}^2}{\omega_{TO}^2} = \frac{\epsilon(o)}{\epsilon(\infty)}$

In the equation above, $\omega_{LO}$ and $\omega_{TO}$ refer to the frequencies of the longitudinal and transverse optical phonons respectively. $\epsilon(0)$ and $\epsilon(\infty)$ refer to the static and high frequency (above the phonon frequencies, but below any electronic energy scale) dielectric constants. All these quantities are understood to be the values in the long-wavelength limit (i.e. $q \approx 0$).

The beautiful thing about the LST result is that it is independent of any microscopic description, which is quite unusual in solid-state physics. Therefore, the result can be derived from classical electrodynamics, without resorting to any quantum mechanics. It is an interesting question as to whether or not quantum mechanics plays a role in the long-wavelength optical response in general.

Regardless, it turns out that all quantities in the LST relation are experimentally accessible! I find this relation quite remarkable and deep. Not only that, the agreement with experiment in many polar semiconductors is excellent. Take a look at the table below to get an idea of how well this relation holds for a few materials (reproduced from Mark Fox’s textbook Optical Properties of Solids):

I have found textbook derivations don’t give a good intuition of why this relation holds, so here is my attempt to rectify this situation. First, let me state an important assumption that goes into the LST relation:

The phonons are assumed to be in the harmonic limit (i.e. no phonon anharmonicity) and as a result, the dielectric constant has the following form:

$\epsilon(\omega) = \epsilon(\infty) + \frac{C}{\omega_{TO}^2-\omega^2}$

where $C$ is a constant. This form of the dielectric constant can be arrived at using either classical electrodynamics or quantum mechanics (see e.g. Ashcroft and Mermin, Kittel or Ziman).

Now, with this result under our belts, it turns out that it is quite simple to understand why the LST relation holds. In a simple polar semiconductor, we have two atoms per unit cell that are oppositely charged like so:

Therefore, for the longitudinal optical phonon we have an extra polarization effect due to the long-range nature of the Coulomb interaction. This extra polarization results in an extra restoring force (in addition to the springy restoring force between the ions), yielding a higher longitudinal phonon frequency compared to the transverse optical phonon. I have discussed this a little more extensively in a previous post. This extra restoring force (which is only present for the longitudinal oscillation) is pictured below:

The longitudinal optical phonon is at a higher energy because of the extra Coulombic polarization effect

More precisely, we can write the following when including this extra restoring force:

$\omega_{LO}^2 = \omega_{TO}^2 + \frac{C}{\epsilon(\infty)}$

There is an $\epsilon(\infty)$ in the formula above because this polarization will necessarily be screened by higher energy (electronic) processes. Dividing both sides by $\omega_{TO}^2$, we can write the above equation suggestively as:

$\frac{\omega_{LO}^2}{\omega_{TO}^2} = \frac{\epsilon(\infty)+C/\omega_{TO}^2}{\epsilon(\infty)}$

Looking at the equation for the dielectric constant from earlier, this is precisely the LST relation! In effect, the same extra restoring due to the long-range Coulomb interaction leads to the extra screening in the static limit, yielding, in my mind, a delightful little result.

Using the LST relation, we can deduce a property of ferroelectric materials. Namely, we know that at the transition temperature between the normal state and a ferroelectric ground state, the static dielectric constant, $\epsilon(0)$, diverges. Therefore, we can surmise from the LST relation that a zone center transverse optical phonon must go to zero energy (soften) at the transition temperature (see here for PbTiO3). This is a totally non-trivial consequence of the LST relation, demonstrating again its far-reaching utility.

Did I mention that I think this result is pretty excellent?

I’d like to acknowledge Zhanybek Alpichshev for enlightening some aspects regarding this topic.

## Interactions, Collective Excitations and a Few Examples

Most researchers in our field (and many outside our field that study, e.g. ant colonies, traffic, fish schools, etc.) are acutely aware of the relationship between the microscopic interactions between constituent particles and the incipient collective modes. These can be as mundane as phonons in a solid that arise because of interactions between atoms in the lattice or magnons in an anti-ferromagnet that arise due to spin-spin interactions.

From a theoretical point of view, collective modes can be derived by examining the interparticle interactions. An example is the random phase approximation for an electron gas, which yields the plasmon dispersion (here are some of my own notes on this for those who are interested). In experiment, one usually takes the opposite view where inter-particle interations can be inferred from the collective modes. For instance, the force constants in a solid can often be deduced by studying the phonon spectrum, and the exchange interaction can be backed out by examining the magnon dispersions.

In more exotic states of matter, these collective excitations can get a little bizarre. In a two-band superconductor, for instance, it was shown by Leggett that the two superfluids can oscillate out-of-phase resulting in a novel collective mode, first observed in MgB2 (pdf!) by Blumberg and co-workers. Furthermore, in 2H-NbSe2, there have been claims of an observed Higgs-like excitation which is made visible to Raman spectroscopy through its interaction with the charge density wave amplitude mode (see here and here for instance).

As I mentioned in the post about neutron scattering in the cuprates, a spin resonance mode is often observed below the superconducting transition temperature in unconventional superconductors. This mode has been observed in the cuprate, iron-based and heavy fermion superconducting families (see e.g. here for CeCoIn5), and is not (at least to me!) well-understood. In another rather stunning example, no less than four sub-gap collective modes, which are likely of electronic origin, show up below ~40K in SmB6 (see image below), which is in a class of materials known as Kondo insulators.

Lastly, in a material class that we are actually thought to understand quite well, Peierls-type quasi-1D charge density wave materials, there is a collective mode that shows up in the far-infrared region that (to my knowledge) has so far eluded theoretical understanding. In this paper on blue bronze, they assume that the mode, which shows up at ~8 cm$^{-1}$ in the energy loss function, is a pinned phase mode, but this assignment is likely incorrect in light of the fact that later microwave measurements demonstrated that the phase mode actually exists at a much lower energy scale (see Fig. 9). This example serves to show that even in material classes we think we understand quite well, there are often lurking unanswered questions.

In materials that we don’t understand very well such as the Kondo insulators and the unconventional superconductors mentioned above, it is therefore imperative to map out the collective modes, as they can yield critical insights into the interactions between constituent particles or couplings between different order parameters. To truly understand what is going on these materials, every peak needs to be identified (especially the ones that show up below Tc!), quantified and understood satisfactorily.

As Lestor Freamon says in The Wire:

All the pieces matter.

## Another Lesson from the Coupled Oscillator: Fano Lineshape

Every once in a while, I read a paper that exhibits a rather odd spectral signature. This odd spectral signature is asymmetric and looks something like this (taken from Wikipedia):

Fano Resonance

This is called a Fano lineshape and occurs when an excitation interferes with a background process. “Background process” is pretty vague, but I hope that the notion becomes clearer below. Here are a couple papers that have observed Fano lineshapes in different contexts, including a review paper:

The reason I chose the first two papers was because a couple peaks in their respective spectra exhibit a Fano lineshape as a function temperature. In the Raman paper, after the charge density wave gap in the single-particle spectrum opens up, the lineshapes of the phonons are no longer Fano-like, but become Lorentzian. This indicates that the Fano lineshape likely resulted from strong electron-phonon coupling.

Anyway, there is a pedagogical paper on the Fano resonance that is worth a read here (pdf!), and is where the example below comes from. This paper in fact illustrates the simplest and most enlightening case of the Fano resonance I’ve come across. The basic idea is that one has a pair of coupled oscillators, and someone is driving only one of the oscillators (ball 1 below). This is pictured below for clarity:

Only the ball 1 is driven

Now, if one looks at the amplitude of the first oscillator, one gets the following plot:

The first oscillator was chosen to have a natural resonance frequency of 1, while the second oscillator was chosen to have a resonance frequency of 1.5. You can see that this is not exactly where the peaks show up, however. This is because of the level repulsion due to the coupling between the two oscillators.

Now the \$64k question: why does the asymmetric lineshape appear? Well, there are a few ingredients. The first ingredient is that the first oscillator is being driven by an extrenal source and the second oscillator is not. Secondly, there is a strong coupling between the first oscillator and second one. As one decreases the coupling between the two, the amplitude of the second (asymmetric) peak starts to shrink. Thirdly, the natural frequencies of the oscillators need to be relatively close to one another otherwise again the second peak decreases in amplitude.

What is remarkable is that this simple coupled oscillator model is able to exhibit the Fano lineshape while serving as a very instructive toy model. Because of this simplicity, the Fano lineshape is extremely general and is used to explain data in a wide variety of contexts across all areas of physics. A spectroscopist should always keep a look out for an asymmetric lineshape, as it is usually the result of a non-trivial coupling or interaction with another degree of freedom.

## Lessons from the Coupled Oscillator

In studying solid state physics, one of the first problems encountered is that of phonons. In the usual textbooks (such as Ashcroft and Mermin or Kittel), the physics is buried underneath formalism. Here is my attempt to explain the physics, while just quoting the main mathematical results. For the simple mass-spring oscillator system pictured below, we get the following equation of motion and oscillation frequency:

Simple harmonic oscillator

$\ddot{x} = -\omega^2x$

and      $\omega^2 = \frac{k}{m}$

If we couple two harmonic oscillators, such as in the situation below, we get two normal modes that obey the equations of motion identical to the single-oscillator case.

Coupled harmonic oscillator

The equations of motion for the normal modes are:

$\ddot{\eta_1} = -\omega^2_1\eta_1$      and

$\ddot{\eta_2} = -\omega^2_2\eta_2$,

where

$\omega_1^2 = \frac{k+2\kappa}{m}$

and   $\omega_2^2 = \frac{k}{m}$.

I should also mention that $\eta_1 = x_1 - x_2$$\eta_2 = x_1 + x_2$. The normal modes are pictured below, consisting of a symmetric and antisymmetric oscillation:

Symmetric normal mode

Antisymmetric normal mode

The surprising thing about the equations for the normal modes is that they look exactly like the equations for two decoupled and independent harmonic oscillators. Any motion of the oscillators can therefore be written as a linear combination of the normal modes. When looking back at such results, it seems trivial — but I’m sure to whoever first solved this problem, the result was probably unexpected and profound.

Now, let us briefly discuss the quantum case. If we have a single harmonic oscillator, we get that the Hamiltonian is:

$H = \hbar\omega (a^\dagger a +1/2)$

If we have many harmonic oscillators coupled together as pictured below, one would probably guess in light of the classical case that one could obtain the normal modes similarly.

Harmonic Chain

One would probably then naively guess that the Hamiltonian could be decoupled into many seemingly independent oscillators:

$H = \sum_k\hbar\omega_k (a^\dagger_k a _k+1/2)$

This intuition is exactly correct and this is indeed the Hamiltonian describing phonons, the normal modes of a lattice. The startling conclusion in the quantum mechanical case, though, is that the equations lend themselves to a quasiparticle description — but I wish to speak about quasiparticles another day. Many ideas in quantum mechanics, such as Anderson localization, are general wave phenomena and can be seen in classical systems as well. Studying and visualizing classical waves can therefore still yield interesting insights into quantum mechanics.

## Why are the quantum mechanical effects of sound observed in most solids but not most liquids?

Well, if liquids remained liquids down to low temperatures, then the quantum mechanical effects of sound would also occur in them as well. There is actually one example where these effects are important and this is in liquid helium.

Therefore the appropriate questions to ask then are: (i) when are quantum mechanical effects significant in the description of sound? and (ii) when does quantum mechanics have any observable consequences in matter at all?

The answer to this question is probably obvious to most people that read this blog. However, I would still think it needs to be reiterated every once in a while. When does the wave nature of “particles” become relevant? Usually, when the wavelength, $\lambda$, is on the order of some characteristic length, $d$:

$\lambda \gtrsim d$

What is this characteristic length in a liquid or solid? One can approximate this by the interparticle spacing, which one can take to be the inverse of the cube root of the density, $n^{-1/3}$. Therefore, quantum mechanical effects can be said to become important when:

$d \sim n^{-1/3}$

Now, lastly, we need an expression for the wavelength of the particles. One can use the deBroglie expression that relates the wavelength to the momentum:

$\lambda \sim \frac{h}{p}$,

where $h$ is Planck’s constant and $p$ is the momentum. And one can approximate the momentum of a particle at temperature, $T$, by:

$p \sim \sqrt{mk_BT}$    (massive)    OR      $p \sim k_BT/v_s$     (massless),

where $k_B$ is Boltzmann’s constan, $m$ is the mass of the particle in question, and $v_s$ is the speed of sound. Therefore we get that quantum mechanics becomes significant when:

$n^{2/3}h^{2}/m \gtrsim k_BT$   (massive)    OR     $n^{1/3}h v_s \gtrsim k_BT$     (massless).

Of course this expression is just a rough estimate, but it does tell us that most liquids end up freezing before quantum mechanical effects become relevant. Therefore sound, or phonons, express their quantum mechanical properties at low temperatures — usually below the freezing point of most materials. By the way, the most celebrated example of the quantum mechanical effects of sound in a solid is in the $C_v \sim T^3$ Debye model. Notice that the left hand side in formula above for massless particles is, within factors of order unity, the Boltzmann constant times the Debye temperature. Sound can exhibit quantum mechanical properties in liquids and gases, but these cases are rare: helium at low temperature is an example of a liquid, and Bose condensed sodium is an example of a gas.

## Kohn Anomalies and Fermi Surfaces

Kohn anomalies are dips in phonon dispersions that arise because of the presence of a Fermi surface. The presence of the Fermi surface renormalizes the bare phonon frequencies and causes an anomly in the phonon dispersion, as seen below for lead (taken from this paper):

Why this happens can be understood using a simplified physical picture. One can imagine that the ions form some sort of ionic plasma in the long-wavelength limit and we can use the classical harmonic oscillator equation of motion:

$m\frac{d^2\textbf{x}}{dt^2} = \frac{-NZ^2e^2\textbf{x}}{\epsilon_0}$

One can take into account the screening effect of the electrons by including an electronic dielectric function:

$m\frac{d^2\textbf{x}}{dt^2} = \frac{-NZ^2e^2\textbf{x}}{\epsilon(\textbf{q},\omega)\epsilon_0}$

The phonon frequencies will therefore be renormalized like so:

$\omega^2 = \frac{\Omega_{bare}^2}{\epsilon(\textbf{q},\omega)}$

and the derivative in the phonon frequency will have the form:

$\frac{d\omega}{d\textbf{q}} \propto -\frac{d\epsilon(\textbf{q},\omega)}{d\textbf{q}}$.

Therefore, any singularities that arise in the derivative of the dielectric function will also show up in the phonon spectra. It is known (using the Lindhard function) that there exists such a weak logarithmic singularity that shows up in 3D metals at $\textbf{q} = 2k_F$. This can be understood by noting that the ability of the electrons to screen the ions changes suddenly due to the change in the number of electron-hole pairs that can be generated below and above $\textbf{q}=2k_F$.

The dip in the phonon dispersion can be thought of as the phonon analogue of the “kinks” that are often seen in the electron dispersion relations using ARPES (e.g. see here). In the case here, the phonon dispersion is affected by the presence of the electrons, whereas in the “kink” case, the electronic dispersion is affected by the presence of the phonons (though kinks can arise for other reasons as well).

What is remarkable about all this is that before the advent of high-resolution ARPES, it was difficult to map out the Fermi surfaces of many metals (and still is for samples that don’t cleave well!). The usual method was to use quantum oscillations measurements. However, the group in this paper from the 60s actually tried to map out the Fermi surface of lead using just the Kohn anomalies! They also did it for aluminum. In both cases, they observed pretty good agreement with quantum oscillation measurements — quite a feat!

## Screening, Plasmons and LO-TO Splitting: One Last Time

I hope you are not by now fed up with my posts on this topic, but there is a great paper by Mooradian and Wright, which I’ve actually linked before in a different context, that is worth tackling. In this paper, they discuss the concept of plasmon-longitudinal optical (LO) phonon coupling.

To my mind, there is a significant aspect of the data which they do not explicitly address in their paper. Of course, I’m referring to degeneracy, LO-TO splitting, and screening. An image of their data is shown below (click to enlarge):

A quick run-down of the experiment: they are using Raman scattering on several different samples of GaAs with different doping levels. The carrier density can be read off in the image above.

It can be seen that for lower doped samples of GaAs, that there is strong LO-TO splitting. This is because of the long-ranged nature of the Coulomb interaction, as detailed here. As the carrier concentration is increased beyond the plasmon-phonon mixing region, the LO-TO splitting starts to disappear. This observation is noteworthy because there exists a “critical carrier density”, beyond which the LO and TO phonons are degenerate.

One can think of this in the following way: the plasmon energy is a measure of how quickly the free carriers can respond to an electric field. Therefore, for the highly doped GaAs samples, where the plasmon is at a significantly higher energy than the phonons, the free carriers can quickly screen the Coulomb field set up by the polar lattice. The electric field that is set up by the phonons can hence be approximated by a screened electric field (of the Thomas-Fermi kind) in this limit, and the Coulomb interaction is hence no longer long-ranged.

While the points I have made above will be quite obvious to many of you, I still find the data and its implications from a historical perspective quite profound.

Aside: I was heartened by Sarang’s post on the concept of emergence and upward heritability. One tends to think harder about one’s stance when there is an opposing view. He made some extremely important points regarding this topic, though I have to admit that I still lean towards Wilczek-ian concepts at present.