# Tag Archives: Metals

## Plasma Frequency, Screening Response Time and the Independent Electron Approximation

The plasma frequency in the study of solids arises in many different contexts. One of the most illuminating ways to look at the plasma frequency is as a measure of the screening response time in solids. I’ve discussed this previously in reference to the screening of longitudinal phonons in semiconductors, but I think it is worth repeating and expanding upon.

What I mean by “screening response time” is that in any solid, when one applies a perturbing electric field, the electrons take a certain amount of time to screen this field. This time can usually be estimated by using the relation:

$t_p = \frac{2\pi}{\omega_p}$

Now, suppose I introduce a time-varying electric field perturbation into the solid that has angular frequency $\omega$. The question then arises, will the electrons in the solid be able to respond fast enough to be able to screen this field? Well, for frequencies $\omega < \omega_p,$ the corresponding perturbation variation time is $t = 2\pi/\omega > t_p$. This means the the perturbation variation time is longer than the time it takes for the electrons in the solid to screen the perturbation. So the electrons have no problem screening this field. However, if $\omega > \omega_p$ and $t < t_p$, the electronic plasma in the solid will not have enough time to screen out the time-varying electric field.

This screening time interpretation of the plasma frequency is what leads to what is called the plasma edge in the reflectivity spectra in solids. Seen below is the reflectivity spectrum for aluminum (taken from Mark Fox’s book Optical Properties of Solids):

One can see that below the plasma edge at ~15eV, the reflectivity is almost perfect, resulting in the shiny and reflective quality of aluminum metal in the visible range. However, above $\hbar\omega$=15eV, the reflectivity suddenly drops and light is able to pass through the solid virtually unimpeded as the electrons can no longer respond to the quickly varying electric field.

Now that one can see the effect of the screening time on an external electric field such as light, the question naturally arises as to how the electrons screen the electric field generated from other electrons in the solid. It turns out that much of what I have discussed above also works for the electrons in the solid self-consistently. Therefore, it turns out that the electrons near the Fermi energy also have their electric fields, by and large, screened out in a similar manner. The distance over which the electric field falls by $1/e$ is usually called the Thomas-Fermi screening length, which for most metals is about half a Bohr radius. That the Thomas-Fermi approximation works well is because one effectively assumes that $\omega_p \rightarrow \infty$, which is not a bad approximation for the low-energy effects in solids considering that the plasma frequency is often 10s of eV.

Ultimately, the fact that the low-energy electrons near the Fermi energy are well-screened by other electrons self-consistently permits one to use the independent electron approximation — the foundation upon which band theory is built. Therefore, in many instances that the independent electron approximation is used to describe physical phenomena in solids, it should be kept in mind the hidden role the plasmon actually plays in allowing these ideas to work.

Naively, from my discussion above, it would seem like the independent electron approximation would then break down in a band insulator. However, this is not necessarily so. There are two things to note in this regard: (i) there exists an “interband plasmon” at high energies that plays essentially the same role that a free-carrier plasmon does in a metal for energies $E_g << E < \hbar\omega_p$ and (ii) whether the kinetic or Coulomb energy dominates will determine the low energy phenomenology. An image below is taken from this paper on lithium fluoride, which is a band insulator with a band gap of about 5eV and exhibits a plasmon at ~22eV:

The interband plasmon ultimately contributes to the background dielectric function, $\epsilon$, which reduces the Coulomb energy between the electrons in the form:

$V_{eff} = \frac{e^2}{\epsilon_0 \epsilon r}$

For example, this is the Coulomb interaction felt between an electron and hole when an exciton is formed (with opposite sign), as can be seen for LiF in the above image.

Now, the kinetic energy can be approximated by the band width, $W$, which effectively gives the amount of “wavefunction overlap” between the neighboring orbitals. Now, if $W >> V_{eff}$, then the independent electron approximation remains a good approximation. In this limit, one can get a band insulator, that is adequately described using the independent electron approximation. In the opposite limit, however, often one gets what is called a Mott insulator. Because d- and f-electrons tend to be closely bound to the atomic site, there is usually less wavefunction overlap between the electrons, leading to a small band width. This is why Mott insulators tend to occur in materials that have d- and f-electrons near the Fermi energy

Most studies on strongly correlated electron systems tend to concentrate on low-energy phenomenology.  While this is no doubt important, in light of this post, I think it may be worth looking up from time to time as well.

## How do we define states of matter?

Historically, many people seemed to lean towards defining a phase of matter by its (broken) symmetries. For instance, a ferromagnet has broken rotational symmetry and time-reversal symmetry, a solid has broken translational and rotational symmetry, etc. In light of the discoveries of the Quantum Hall Effect and topological insulators, it seems like this symmetry classification does not encompass all states of matter.

The symmetry classification is largely a theoretical construct, however. I would think that one defines a state of matter by particular experimental properties that it exhibits. For example, one could define a superconductor by requiring it to exhibit the following:

1. Zero Resistivity
2. Meissner Effect
3. Zero Peltier Coefficient

Put another way, to verify that one has discovered a superconductor, these three criteria must be satisfied.

Let us take another example: a simple metal. The criterion that must be satisfied for this case is the existence of a Fermi surface. This can be measured by quantum oscillation measurements, angle-resolved photoemission, or a few other probes.

Yet another example: a 2D topological insulator. What one must observe is:

1. The Fermi energy intersects an odd number of topologically protected edge states in half the edge Brillouin zone (which was shown by transport in this classic paper)
2. The existence of a spin-polarization associated with the edge states

While these three examples were chosen because they were simple, I have remaining doubts. Are these observations necessary and sufficient to define these states of matter? Are there cases where one can better define a state of matter theoretically?

For instance, a theorist may define a 2D topological insulator by the existence of a non-trivial topological number, which seems like a perfectly valid criterion to me. This topological number cannot be experimentally observed in a very direct way (to my knowledge) and has to be inferred from the edge states, band structure, etc.

The reason I started thinking about this is because I did not find the definition of a charge density wave in this widely-cited paper by Johannes and Mazin appropriate. It states:

[A charge density wave is a] Peierls-like instabilit[y] that occur[s] due to a divergency in the real part of the electronic susceptibility, so that the electronic subsystem would be unstable per se, even if the ions were clamped at their high symmetry positions.

This definition bothers me in particular because it defines a charge density wave by its cause (i.e. Peierls-like instability due to a divergence in the real part of the electronic susceptibility).  The main qualm I have is that one should not define a state of matter by its origin or cause. This is like trying define a superconductor by the mechanism that causes its existence (i.e. phonon-mediated electron-electron interaction for superconductors, which would exclude unconventional superconductors from its definition). This is obviously problematic. Therefore, shouldn’t one define a charge density wave by its experimentally measured properties?

So I come back to the original question: how does one define a state of matter?