Positrons versus holes as positive charge carriers

Question

From Wikipedia: [The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.]

and:

[Dirac's idea is completely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice — that cancels out the electric charge of the sea.]

It always confused me to think of holes as positive charge carriers in semi-conductors as not being real: real electrons move from one lattice-position to another lattice-position, which effectively looks like a positive hole in the lattice that is moving in the other direction, but in reality a real electron moves, the hole is kind of an "illusion".

On the other hand the positrons are always introduced as real hard-core particles.

The quotes from the Wikipedia article make me unsure: how should I look upon these phenomena?

Edit: holes in a Dirac sea give rise to real pos. entities in one case and to unreal pos. entities in another - how can we distinguish, is it a matter of formalism?

Answer 1

I want to come back to this answer in a bit and expand a bit more, but it seems to me that the notion that you are reaching for is that of quasiparticles, and I want to argue that these might be best understood in the context of effective field theories.

Roughly: the electrons and holes in solid state physics are not actually necessarily true single particles, but actually collective excitations of the material (you can think of an electron quasiparticle as being an electron plus some minor deformation of the underlying lattice plus effects due to interactions with other electrons; the picture for holes is that of an empty electron quasiparticle state).

However, the same is true for electrons and positrons in vacuum (though the energy scales are much different)! The mass of the electron has contributions from a cloud of virtual electron-positron pairs (this is an effect called mass renormalization).

This is a bit confusing -- it turns out that e.g. talking about an electron as a single particle that doesn't interact with itself is a classical notion that breaks down when quantum field theory comes into the picture.

Indeed, quantum field theories (i.e. the best physical pictures that we currently have) are actually agnostic in a certain strong sense as to whether an electron is a single particle, or a particle surrounded by a cloud, or whether a hole is a true particle or a quasiparticle, etc. What do I mean by this? Well, the mathematical models that currently describe the systems best do not talk about particles at all, they talk about excitations in quantum fields, and for interacting quantum fields, it turns out that the coupling constants that go into the Lagrangians for these field theories will get (in general divergent!) corrections from self-interactions. Luckily, we now understand that this is not something to worry about -- there's a technique called "renormalization" which fixes the mathematical divergences, and there's a philosophy of "effective field theories" which explains how to think about these cancellations. Roughly speaking, depending on what energy scale you want to work in, you might interpret the electron as being a single particle, or a fuzzy cloud, but with regards to calculating measurable quantities, it doesn't really matter. For a better take on this, I like the first few sections of Howard Georgi's review "Effective field theory".

Sorry if this doesn't make much sense at the moment. I hope I've given you a few key words to look up. Perhaps someone who understands all this a bit better can come by and explain it in easier terms, or I'll come back and edit it in a few days.

Answer 2

Holes in semi-conductors are considered to be quasiparticles. A quasiparticle is a group of particles that end up acting and having many of the properties of a single particle for a period of time. They are normally considered to have a lifetime tau and decay exponentially.

Are positrons and electrons that we obverse quasiparticles too? QPs with a very long decay time? That goes beyond our current level of physics. I have not heard any evidence that they are but the conjecture has been considered.

Answer 3

In either case the particles remain virtual until promoted to real by conversion of energy via pair production. The Feynman diagrams seem quite similar whether they are describing mev processes with thermal phonons as gauge bosons (holes in solid states) or Mev processes with photons (positrons).

Of course with holes in the solid state, promotion to real free electrons (n type semiconductor) or free holes (p type) comes from the thermal environment, so the excess charge carriers are assumed real from the outset.

Thats what I recall from long ago, anyway.

Answer 4

The key to understanding when the hole picture is appropriate vs. when the particle picture is appropriate is the bending of the E vs. k curve for the particle. When higher momentum particle states have more energy, then the particle picture gives physical behavior--- the excitations accelerate in the direction of the force. If the higher momentum states have less energy, then the hole picture is typically the physical one, because the holes have positive mass.

This doesn't mean that you can't use a hole description in the positive mass regime, or a particle picture in the negative mass regime, but it means that you have to consider that the negative mass particles accelerate the wrong way in response to forces. This is physically weird, so people work in the picture that gives a normal force/acceleration dependence.

The negative energy solutions of the Dirac equation are on the negative mass shell hyperboloid, and this goes down with k. So the hole picture is the correct positive mass excitation. In the modern treatment of Dirac fields, the creation and annihilation operators are defined to be such by their frequency, so this condition automatically comes out of a relativistic formalism.

See also this answer: What are "electron holes" in semiconductors?

Answer 5

A very simple picture for understanding the propagation of a positive hole would be those puzzles consisting of 15 square tiles, held together tongue-and-groove fashion in a square frame. There's always one empty space big enough to slide a tile into. This allows one to solve the puzzle by rearranging the tiles.

In this picture, the "hole" is the empty space. If the empty space is in position 16 at bottom right, it's easy to see that the "hole" can be shifted to position 4 at the top-right by three successive downward moves of the tiles occupying positions 12, then 8, then 4. Although the "virtual" space may not have the same physical reality as the "real" tile, anyone attempting to solve the puzzle quickly adapts to thinking in terms of "shifting the space" by a series of retrograde moves. Although the space has no surface that we can touch, it has a location, which can be manipulated indirectly.

For many purposes, a real electron and a virtual positive hole can be thought of as similar, oppositely charged particles. This equivalency breaks down when one imagines building a cathode ray tube for positive holes. An electron gun can easily fire an electron across the vacuum. But how could a gun propagate a positive hole? There's no real mass to accelerate across the wide gap between gun and phosphor screen. For the positive hole to breach the gap, there would have to be (as within a crystal) a whole chain of filled electron states, stretching unbroken from gun to phosphor, and each capable of sequentially yielding to the hole's forward progress by individual retrograde motions. Not gonna happen!

Even so, it's fun to think about this: A "virtual" positive hole has no rest mass. Yet it can exert a force and have a measurable velocity. And so, it follows that we can calculate a "virtual" mass!

These simple pictures of course do not convey quantum reality. When a positive hole is activated within an oxide crystal, for instance, its absence isn't necessarily localized as in the tile puzzle as "the absence of a negative charge where one would normally expect one to be." The effect of that absent charge might be smeared across the nearest 100 or 1,000 atoms, affecting conductivity of the whole neighborhood along the valence band. So that instead of one atom missing a single electron, we might think of a hundred atoms missing .01 electrons, or a thousand atoms missing .001 electrons.

Which depending on your point of view may be amusingly counterintuitive, spookily surreal, or just the way things are. ;-)

PS: Of course, the positron is like the proverbial 800 lb. gorilla by comparison to the hole. It has a very real mass equal to the electron. And one could imagine, with proper containment, accelerating a beam of positrons across a CRT, where by virtue of being an anti-particle, they would commence to obliterate themselves and the screen.

Answer 6

Positrons are not holes in a Dirac sea of filled electron orbitals. This concept is untenable. Holes in semiconductors and certain metals however do meet this description. This application of Dirac's idea is very useful.