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,

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.

]]>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?”

]]>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.

]]>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.

]]>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 H_{2} and get an ion H_{2}^{+}? 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?

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.

]]>*“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

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

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