# Why isn't the frequency dependence of permittivity monotonic?

1. Sep 29, 2014

### FillBill

I naively thought that most materials were transparent to radiation of frequencies above their plasma frequency, and opaque to radiation below it. The most intuitive (and analyzed lightly in Griffiths' E&M book) reason I've heard is that opaqueness is caused by electrons in the material responding to the incident radiation by getting oscillated by it, in which they produce their own radiation that cancels the incident radiation. However, when the incident radiation is above the plasma frequency, they can't respond quickly enough and it can start to penetrate the material.

Mathematically, $ϵ(ω)>0$ for $ω<ω_{plasma}$ and $ϵ(ω)<0$ for $ω>ω_{plasma}$, and the wave vector $k∝n∝\sqrt ϵ$, so when $ω$ is above $ω_{plasma}$, $k$ is imaginary and $e^{ik⃗ ⋅r}$ decays quickly in the material.

But I was recently told that as the frequency keeps rising, the material again at some point becomes opaque to it. What is this called, and what's the physical reason for it?

Last edited: Sep 29, 2014
2. Sep 29, 2014

### DrDu

This description is at best qualitatively correct for one class of materials: metals. Most non-conducting materials show some absorption in the IR part of the spectrum due to vibrational resonances and then stronger absorption lines in the UV part of the spectrum due to electronic transitions. Far from these resonances, the materials are transparent. The position of the absorption lines depends strongly on the material. E.g. for a gas of hydrogen atoms, you know all these series of absorption lines like Bragg, Balmer etc. At very high frequencies, you will find transitions from the bound states into the continuum. But the imaginary part due to these transitions is very small.

3. Sep 29, 2014

### FillBill

Alright, but let me take a simple example like a semiconductor with a direct band gap, like GaAs. It starts absorbing light when the incident photo energy is ~1.4eV, and it can absorb higher energies than that as well. Why does it stop absorbing higher energy photons at some point? Is it because, to absorb higher energies, the transition has to happen from a spot in the valence band to a spot in the conduction band with the same k value, but the conduction band doesn't increase indefinitely?

4. Sep 29, 2014

### DrDu

Yes, exactly. But note that in general there will be more than one band and for photons of high enough energy, you can get photoionization.

5. Oct 1, 2014

### FillBill

Thank you for the response. As a follow up question: The band gap energy is a well defined concept, but is there a name for the upper limit on energy that the semiconductor can absorb because of the mechanism I just said?

6. Oct 2, 2014

### DrDu

I am not sure. The point is the following: The lower band gap is a reasonably well defined concept as it is the lower limit for one particle and consequently also many particle excitations (this is already simplified as there may be excitations inside the gap, like bound electron hole pairs). However there may be electronic excitations which correspond to multi-particle excitations and may be excited by absorption of light, which fall into a higher band gap of single particle excitations.
The point is that electron electron interaction is very strong, so that it renders a single particle picture invalid in most parts of the spectrum. The only exception being the excitations of lowest energy.
You may want to read about the Landau Fermi liquid theory to better understand what I am talking about.