# 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.