In a famous 1962 paper entitled Plasmons, Gauge Invariance and Mass (pdf!), P.W. Anderson described the relationship between the gap in the plasmon spectrum and the idea of spontaneous symmetry breaking. It is an interesting historical note that Higgs cites Anderson’s paper in his landmark paper concerning the Higgs mechanism.

While there are many different formulations, including Anderson’s, of why the plasmon is gapped at zero momentum in a 3D solid, they all rely on one crucial element: the long-range nature of the Coulomb interaction (i.e. the electrons are charged particles). Of these formulations, I prefer one “cartoon-y” explanation which captures the essential physics well.

Before continuing, let me stress that it is quite unusual for a fluid medium (such as the electrons in a metal) to possess no zero frequency excitations at long wavelengths. For instance, the dispersion relation for surface gravity waves on water (pdf!) is:

.

Now, in 3D and in the long-wavelength limit, the plasmon sets up opposite charges on the surfaces of the solid as pictured below:

The long-wavelength plasmon therefore sets up the same electric field as in a capacitor. The electric field for a capacitor is . This expression is surprisingly *independent* of the distance separating the surfaces of the solid. Therefore, it takes a finite amount of energy to set up this electric field, even in the limit of infinite distance. This finite energy results in the gapping of the plasmon.

This cartoon can be extended further to 2D and 1D solids. In the 2D case, the electric field for the 1D “lines of charge” bounding the solid falls off like . Therefore, in the infinite distance limit, it takes no energy to create this electric field and the plasmon is not gapped at . Similarly, for the 1D case, the electric field from the points bounding the solid falls of as , and the plasmon is again gapless.

This reasoning can be applied further to the phenomenon known as LO-TO splitting in a polar solid. Here, the longitudinal optical phonon (LO) and the transverse optical phonon (TO) branches are non-degenerate down to the very lowest (but non-zero!) momenta. Group theory predicts these modes to be degenerate at for the zincblende crystal structure of typical semiconducting compounds. Below is the phonon dispersion for GaAs demonstrating this phenomenon:

Again, the splitting occurs due to the long-ranged nature of the Coulomb interaction. In this case, however, it is the polar ionic degree of freedom that sets up the electric field as opposed to the electronic degrees of freedom. Using the same reasoning as above, one would predict that the LO-TO splitting would disappear in the 2D limit, and a quick check in the literature suggests this to be the case as reported in this paper about mono-layer Boron Nitride.

I very much appreciate toy models such as this that give one enough physical intuition to be able to predict the outcome of an experiment. It has its (very obvious!) limitations, but is valuable nonetheless.

It seems like in this picture you could still get arbitrarily low-energy plasmons by letting the charge density on your “capacitor” go to zero. Is this not allowed for some reason?

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In this picture, the charge density is only present if there exist mobile electrons in the solid. Perhaps I did not make that clear in the post. Yes, the plasmon energy would go to zero in the limit of zero charge density, but this is akin to removing the medium (i.e. you can’t have water waves without water!).

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That’s clear, it’s just that naively you can have whatever charge you like on an ordinary capacitor and it’s not obvious why that’s fixed by the material in this case.

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Ah, I understand your question better now. This charge density in this picture is fixed by the mobile electron density, which will be determined by the material.

Keeping all other things fixed, that is why increasing the electron density in a material will increase the plasmon energy (i.e. it increases the electric field set up in the capacitor).

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I found that Sec. 3 of this paper (http://rmf.smf.mx/pdf/rmf/39/4/39_4_640.pdf) also offers a neat toy model for plasmon dispersion (long wavelength limit) in 1D, 2D, and 3D.

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You still need to consider the ability of the electric field to permeate the material. This is given by the complex dielectric functions for that specific material. For example, in the infrared regime, the dielectric response of Silicon is such that infrared frequency photons can barely propagate through the material. This means that the coulomb field cant permeate the material and seperate the charges or move dipoles. As a result polariton is created. and you have degenerate TO/LO bands. This seems to be the case in elemental materials (even graphene has this property)

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Thanks for your comment. You actually can’t get LO-TO splitting in a monatomic solid. There needs to be a dipole set up between the ions of different species. Your point about the internal dielectric environment will change the amount of splitting, but will not get rid of it, as long as there are at least two different atomic species in an insulating medium.

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