## Excitonium

In certain crystals you can knock an electron out of its favorite place and leave a hole: a place with a missing electron. Sometimes these holes can move around like particles. And naturally these holes attract electrons, since they are places an electron would want to be.

Since an electron and a hole attract each other, they can orbit each other. An orbiting electron-hole pair is a bit like a hydrogen atom, where an electron orbits a proton. All of this is quantum-mechanical, of course, so you should be imagining smeared-out wavefunctions, not little dots moving around. But imagine dots if it’s easier.

An orbiting electron-hole pair is called an exciton, because while it acts like a particle in its own right, it’s really just a special kind of ‘excited’ electron—an electron with extra energy, not in its lowest energy state where it wants to be.

An exciton usually doesn’t last long: the orbiting electron and hole spiral towards each other, the electron finds the hole it’s been seeking, and it settles down.

But excitons can last long enough to do interesting things. In 1978 the Russian physicist Abrikosov wrote a short and very creative paper in which he raised the possibility that excitons could form a crystal in their own right! He called this new state of matter excitonium.

In fact his reasoning was very simple.

Just as electrons have a mass, so do holes. That sounds odd, since a hole is just a vacant spot where an electron would like to be. But such a hole can move around. It has more energy when it moves faster, and it takes force to accelerate it—so it acts just like it has a mass! The precise mass of a hole depends on the nature of the substance we’re dealing with.

Now imagine a substance with very heavy holes.

When a hole is much heavier than an electron, it will stand almost still when an electron orbits it. So, they form an exciton that’s very similar to a hydrogen atom, where we have an electron orbiting a much heavier proton.

Hydrogen comes in different forms: gas, liquid, solid… and at extreme pressures, like in the core of Jupiter, hydrogen becomes metallic. So, we should expect that excitons can come in all these different forms too!

We should be able to create an exciton gas… an exciton liquid… an exciton solid…. and under the right circumstances, a metallic crystal of excitons. Abrikosov called this metallic excitonium.

People have been trying to create this stuff for a long time. Some claim to have succeeded. But a new paper claims to have found something else: a Bose–Einstein condensate of excitons:

• Anshul Kogar, Melinda S. Rak, Sean Vig, Ali A. Husain, Felix Flicker, Young Il Joe, Luc Venema, Greg J. MacDougall, Tai C. Chiang, Eduardo Fradkin, Jasper van Wezel and Peter Abbamonte, Signatures of exciton condensation in a transition metal dichalcogenide, Science 358 (2017), 1314–1317.

A lone electron acts like a fermion, so I guess a hole does do, and if so that means an exciton acts approximately like a boson. When it’s cold, a gas of bosons will ‘condense’, with a significant fraction of them settling into the lowest energy states available. I guess excitons have been seen to do this!

There’s a fairly good simplified explanation at the University of Illinois website:

• Siv Schwink, Physicists excited by discovery of new form of matter, excitonium, 7 December 2017.

However, the picture on this page, which I used above, shows domain walls moving through crystallized excitonium. I think that’s different than a Bose-Einstein condensate!

I urge you to look at Abrikosov’s paper. It’s short and beautiful:

• Alexei Alexeyevich Abrikosov, A possible mechanism of high temperature superconductivity, Journal of the Less Common Metals
62 (1978), 451–455.

(Cool journal title. Is there a journal of the more common metals?)

In this paper, Abrikoskov points out that previous authors had the idea of metallic excitonium. Maybe his new idea was that this might be a superconductor—and that this might explain high-temperature superconductivity. The reason for his guess is that metallic hydrogen, too, is widely suspected to be a superconductor.

Later, Abrikosov won the Nobel prize for some other ideas about superconductors. I think I should read more of his papers. He seems like one of those physicists with great intuitions.

Puzzle 1. If a crystal of excitons conducts electricity, what is actually going on? That is, which electrons are moving around, and how?

This is a fun puzzle because an exciton crystal is a kind of abstract crystal created by the motion of electrons in another, ordinary, crystal. And that leads me to another puzzle, that I don’t know the answer to:

Puzzle 2. Is it possible to create a hole in excitonium? If so, it possible to create an exciton in excitonium? If so, is it possible to create meta-excitonium: an crystal of excitons in excitonium?

### 12 Responses to Excitonium

1. allenknutson says:

What sort of material is it, that a hole could be more massive than an electron?

Also, I vaguely understand why an electron doesn’t merge with a proton. But wouldn’t electrons love to fall into holes? How long-lived are these states of excitonium suppose to be?

• John Baez says:

Allen wrote:

What sort of material is it, that a hole could be more massive than an electron?

I don’t know. I’m quite ignorant of the masses of holes in various kinds of materials, and how large they can be.

But wouldn’t electrons love to fall into holes?

Yes.

How long-lived are these states of excitonium supposed to be?

I don’t know this either. I imagine it could depend quite a bit on the material… but anyway, googling ‘exciton lifetime’ I got the paper Exciton radiative lifetime in transition metal dichalcogenide monolayers, which mainly mentions lifetimes roughly on the order of a picosecond (10-12 seconds), though it also seems to say that at temperatures above 50 kelvin some excitons come into some sort of approximate thermal equilibrium and last a thousand times longer:

Our detailed analysis suggests the following scenario: at low temperature (T ≤ 50 K), the exciton oscillator strength is so large that the entire light can be emitted before the time required for the establishment of a thermalized exciton distribution. For higher lattice temperatures, the photoluminescence dynamics is characterized by two regimes with very different characteristic times. First the photoluminenscence intensity drops drastically with a decay time in the range of the picosecond driven by the escape of excitons from the radiative window due to exciton-phonon interactions. Following this first non-thermal regime, a thermalized exciton population is established gradually yielding longer photoluminescence decay times in the nanosecond range.

One thing I learned from this paper is that an exciton can stick to another electron, or perhaps another hole, and formed a charged quasiparticle called a trion. This happens enough that the paper also studies the lifetimes of trions.

This is again like hydrogen. A hydrogen atom can stick to another electron and form a hydrogen anion—but just barely, because this combination of one proton and two electrons has just one bound state: put in any more energy and it falls apart. (This was proved mathematically in 1977.)

Can a hydrogen atom stick to another proton? In other words, can we remove one electron from H2 and get an ion H2+? I don’t know—I’m not seeing information about this online. Maybe this collection of particles has no bound states.

There’s a fun but probably quite hard math problem here, concerning Schrödinger’s equation:

Puzzle. Suppose we have two particles of charge 1 and one particle of charge -1 interacting via a Coulomb potential as described by Schrödinger’s equation. How many bound states does this system have, as a function of the masses of the 3 particles?

• In semiconductor material, p-type doping (majority hole carriers) shows a lower mobility than n-type doping (majority electron carriers). This is because holes have a higher effective mass than electrons. It’s easy to understand if you consider that hole motion is actually a group of electrons moving in the opposite direction. Holes are really an abstraction of the Fermi level occupancy of states.

What I find interesting is the transition from Fermi-Dirac statistics for the electrons to Bose-Einstein stats for excitonium.

• John Baez says:

Do you have any estimate of typical effective masses of holes? Thanks to Allen’s question, I’m wondering how big (or small) they get.

• For the GaAs energy band structure, the effective mass of an electron is 0.067 of the free electron mass and a hole is 0.34. So the hole is heavier and shows a slower mobility, as that goes as 1/m.

This is an interesting discussion thread for EEs, asking the question “mass of electron and holes-why mass of hole is more?”

• John Baez says:

Thanks! While working out at the gym last night I realized I should also ask for the effective electron mass, since that’s what really matters here.

(For nonexperts: an electron moving in a material like the commonly used semiconductor GaAs acts like it has a different mass than its mass in vacuum, since it’s interacting with all the other charges in the material. This is called the electron’s effective mass.)

So, the hole’s mass is about 5 times the effective electron mass in gallium arsenide. Do we get excitons in this material? If so, they might resemble muonic hydrogen, where a negatively charged muon 207 times the mass of an electron orbits a proton. The proton’s mass is 1836 times the electron’s mass, so the mass ratio in muonic hydrogen is just 9.

However, I don’t know if the potential attracting a hole to an electron is of the usual Coulomb form,

$V \propto 1/r$

It could be quite different, since the hole is not a pointlike entity (is it?).

People do chemistry with muonic hydrogen, and its lifetime is 2 microseconds, vastly longer than the exciton lifetimes I’ve seen so far—though there might be materials where excitons last longer, and these would be very nice to develop.

2. An Excitronium hole? Interesting.
I suspect that this is totally possible, somehow. Though what would that mean? Are we talking that an electron-hole-pair, a single excitron, is gone missing? In that case, the resulting structure would remain neutral, right? (Though the missing pair might weirdly act like a dipole, just like a hole might weirdly act like an electron?)

If we’re, instead, talking about a mismatch between electrons and holes, then the resulting structure would, in fact, have charge, and thus electromagnetic attraction, and thus your meta-excitronium seems like it should be possible.

Another puzzle: Is it possible to keep multiple holes close together somehow? – Perhaps by using appropriate excitons as a glue similarly to how neutrons might work?
And if that’s the case, can we get a whole set of analogous exciton-based chemistry? Even exciton-based life forms?

if so, that sounds like an… exciting venture for some hard science fiction novel.

3. Bob says:

There’s an article in October 2017 Physics Today titled The new era of polariton condensates which discusses the excitation of excitons to generate a resonance in the polarizability of a medium which then non-linearly amplifies photo-photon interactions producing the thermalization of photons leading to the full thermal equilibration necessary for superfluidity of a polariton BE condensate. It discusses producing an exciton with small Bohr radius for sufficient binding energy/effective mass thus generating longer exciton persistence times at room temperature.

“Electrical Engineering professors have, yes, invented lots of smoke and mirror ways of “explaining” holes in terms of classical physics, but these ways are all, really, seductive frauds.”
– Paul J. Nahin in a preface to his book,The Science of Radio.

How are massive holes not like anti-muons?
muonium

dropleton

\$10 has been donated to Wikipedia in honor of the Azimuth blog.

4. Puzzle 2. Koch snowflakes made of Menger sponge!

5. @whut says:

Regarding the qualifier “In certain crystals”, I just wanted to point out that knowledge of excitons is widespread in the semiconductor industry. Millions of research dollars have been spent over the years in characterizing high-speed semiconductor and laser materials through techniques such as photoluminescence (PL) analysis to evaluate the lifetime of excitons. I recall going to conferences years ago where every presentation had the requisite PL analysis.

Here’s a review article:
Use of excitons in materials characterization of semiconductor system

So if some semiconductor researcher is reading this post, it might give them some ideas for a new research path.

6. Terry Bollinger says:

Puzzle 2 about meta levels of excitonium brought an instant smile to my face. Stranger things than that happen in quantum mechanics, so who knows?

7. […] ¿Para qué sirve el excitonio? Por ahora no tiene ningún aplicación práctica concebible; su uso en investigación básica es lo más relevante, ya que permite estudiar en laboratorio condensados de Bose-Einstein a grandes temperaturas (en lugar de en ultrafrío). El artículo es Anshul Kogar, Melinda S. Rak, …, Peter Abbamonte, “Signatures of exciton condensation in a transition metal dichalcogenide,” Science 358: 1314-1317 (08 Dec 2017), doi: 10.1126/science.aam6432, arXiv:1611.04217 [cond-mat.str-el]; la predicción teórica es de Bert I. Halperin, Thomas M. Rice, “Possible Anomalies at a Semimetal-Semiconductor Transistion,” Rev. Mod. Phys. 40: 755 (1968), doi: 10.1103/RevModPhys.40.755; más información divulgativa en John Baez, “Excitonium,” Azimuth, 10 Dec 2017. […]

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