Electrons, holes and positrons

[Ranku]

Do positively charged holes have any other quantum numbers assigned to them? What are the similarities and differences between a hole and a positron?

 

[SpectraCat]

Do positively charged holes have any other quantum numbers assigned to them? What are the similarities and differences between a hole and a positron?

Electron "holes" are a semantic convenience used to represent electronic vacancies in explanations of physical phenomena. You can think of them like the placeholder zeros in a number like, 18057604. In principle there could be values in the $$10^{1}$$ and $$10^{5}$$ places, but in this particular instance, there aren't. Depending on the specific context in which they are used, it sometimes makes sense to associate quantum numbers with holes. For example, a chlorine atom is one 3p electron short of having a complete n=3 shell. Therefore it is semantically consistent to say that it has an "n=3, l=1" hole.

Having said all of that, holes do not really have any independent reality. Positrons, on the other hand, are real particles ... they are the antiparticles of electrons, and have precisely the same set of quantum numbers, but with a positive charge. They have independent reality, that is, they exist and can be isolated experimentally.

 

[Ranku]

Electron "holes" are a semantic convenience used to represent electronic vacancies in explanations of physical phenomena. You can think of them like the placeholder zeros in a number like, 18057604. In principle there could be values in the $$10^{1}$$ and $$10^{5}$$ places, but in this particular instance, there aren't. Depending on the specific context in which they are used, it sometimes makes sense to associate quantum numbers with holes. For example, a chlorine atom is one 3p electron short of having a complete n=3 shell. Therefore it is semantically consistent to say that it has an "n=3, l=1" hole.

Having said all of that, holes do not really have any independent reality. Positrons, on the other hand, are real particles ... they are the antiparticles of electrons, and have precisely the same set of quantum numbers, but with a positive charge. They have independent reality, that is, they exist and can be isolated experimentally.

So holes are independently introduced conceptual constructs, that are not part of the particle standard model?

 

[f95toli]

So holes are independently introduced conceptual constructs, that are not part of the particle standard model?

Yes, holes are not "real" as such . They are mathematical constructs that are useful because they simplify our calculations.

 

[Galap]

Yes, holes are not "real" as such . They are mathematical constructs that are useful because they simplify our calculations.

You can think of them pretty much like holes. Imagine a piece of paper with a hole in it. Is the hole a real thing? No, it's just the fact that the paper envelops a region where there is no paper. However, in certain situations, it's more meaningful to talk about the hole than the paper, for example when you are trying to fit it in a 3 ring binder. The same is true about electron holes: they're not real, but sometimes it's better to use that description.

 

[Frame Dragger]

Holes are very much like Virtual Particles in that their effect seems obvious, but they are really just a trick of the math. That said, they are a trick which does a good job describing the situation for people who don't think in dimensions higher than 3

 

[Galap]

Holes are very much like Virtual Particles in that their effect seems obvious, but they are really just a trick of the math. That said, they are a trick which does a good job describing the situation for people who don't think in dimensions higher than 3

I'd like to hear more in depth about the more-than-3d explanation you imply :)

 

[Frame Dragger]

I'd like to hear more in depth about the more-than-3d explanation you imply :)

? I mean to say that human beings can't imagine structures in more than the 3 spatial dimensions we experience daily. Given that, we get gravity "wells" and and electron "holes". What else can we say unless people simply spoke in equations?

 

[ArjenDijksman]

I see no inconsistency in thinking of holes as ordinary locations left vacant by electrons in a lattice. An electron is attracted to a hole, because there's more place for the electron around holes then elsewhere in the lattice, it can reoccupy the hole ("recombine"), it can "circle" around it (forming an exciton)... It is above all a physical reality and incidentally a mathematical trick...

 

[ZapperZ]

I see no inconsistency in thinking of holes as ordinary locations left vacant by electrons in a lattice. An electron is attracted to a hole, because there's more place for the electron around holes then elsewhere in the lattice, it can reoccupy the hole ("recombine"), it can "circle" around it (forming an exciton)... It is above all a physical reality and incidentally a mathematical trick...

I think this is an excellent view of it.

Claiming that holes aren't real is analogous to saying a "bubble" in water isn't real. It is as much a "real" object as the glob of water that left it and went up above the surface. The fact that we can "renormalize" the environment in the filled band and to consider such holes as having not only a positive charge, but also to have spin, etc., means that below the Fermi level, they are as real as the "vacuum excitation" that we call 'electron'.

In condensed matter physics, there is never any demotion of the concept of "holes" with respect to "electrons". YBCO and LACO and BCCO are all hole-doped superconductors, while NCCO is an electron-doped superconductor, for example. In Andreev scattering, it does makes a difference if a hole is reflected at the interface. There are plenty more examples where this came from.

Zz.

 

[Galap]

? I mean to say that human beings can't imagine structures in more than the 3 spatial dimensions we experience daily. Given that, we get gravity "wells" and and electron "holes". What else can we say unless people simply spoke in equations?

I was just unaware that there was a higher dimension interpretation of specifically electron holes.

 

[SpectraCat]

I think this is an excellent view of it.

Claiming that holes aren't real is analogous to saying a "bubble" in water isn't real. It is as much a "real" object as the glob of water that left it and went up above the surface. The fact that we can "renormalize" the environment in the filled band and to consider such holes as having not only a positive charge, but also to have spin, etc., means that below the Fermi level, they are as real as the "vacuum excitation" that we call 'electron'.

In condensed matter physics, there is never any demotion of the concept of "holes" with respect to "electrons". YBCO and LACO and BCCO are all hole-doped superconductors, while NCCO is an electron-doped superconductor, for example. In Andreev scattering, it does makes a difference if a hole is reflected at the interface. There are plenty more examples where this came from.

Zz.

I think this is kind of a "potato/potahtoe" thing, and the differences in description are mostly semantic. Holes are semantically and mathematically convenient for sure, but they are an "extra" concept. What I mean by this is that the properties of what we call a hole or vacancy could be obtained simply from the properties of the electrons that are actually there. This is true in the atomic case (where it is simple to prove), and while I am less familiar with solid-state physics, I suspect it is true there as well, although it is probably *way* less tractable to deal with.

Note that I am not disputing the reality of holes ... of course they are real, as much as the hole in a doughnut is real ... but it requires the doughnut for its existence. If you excite an electron out of a filled band, then you create a higher energy configuration, which has a vacancy in the filled band that "wants" to have an electron in it. Therefore it seems to have a positive charge, and will act as ArjenDijksman described. If the band is spin polarized, then only an electron with a particular spin can fill the hole. The properties of spin and charge are also only relevant with respect to some lower energy reference state. That is what I meant earlier when I said they have no independent reality.

 

[ZapperZ]

I think this is kind of a "potato/potahtoe" thing, and the differences in description are mostly semantic. Holes are semantically and mathematically convenient for sure, but they are an "extra" concept. What I mean by this is that the properties of what we call a hole or vacancy could be obtained simply from the properties of the electrons that are actually there. This is true in the atomic case (where it is simple to prove), and while I am less familiar with solid-state physics, I suspect it is true there as well, although it is probably *way* less tractable to deal with.

Note that I am not disputing the reality of holes ... of course they are real, as much as the hole in a doughnut is real ... but it requires the doughnut for its existence. If you excite an electron out of a filled band, then you create a higher energy configuration, which has a vacancy in the filled band that "wants" to have an electron in it. Therefore it seems to have a positive charge, and will act as ArjenDijksman described. If the band is spin polarized, then only an electron with a particular spin can fill the hole. The properties of spin and charge are also only relevant with respect to some lower energy reference state. That is what I meant earlier when I said they have no independent reality.

But within context to condensed matter physics, an "electron" also relies on the vacuum state, i.e. a background of "holes" for its existence! That's why I called an electron a "vacuum excitation"! In fact, both holes and electrons are "quasiparticles" in a many-body interaction. They are not different.

Zz.

 

[Frame Dragger]

Hmmm, I've learned to look at Holes in a very new way. Thanks ZapperZ and ArjenDijksman!

 

[Physics Monkey]

Although this is certainly speculative, it's amusing to note that even our beloved "real" positrons could themselves be holes in a material we call the vacuum. Imagine a hypothetical person living inside the low energy world of a semiconductor without access to the high energy world of the lattice. Such a person might be inclined to view holes in the same way we view positrons.

And in fact, the analogy isn't just for fun. The low energy electrons and positrons we experience are different from the bare particles that we suppose exist at much higher energies (greater than a TeV say) where the electroweak symmetry is unbroken.

 

[SpectraCat]

But within context to condensed matter physics, an "electron" also relies on the vacuum state, i.e. a background of "holes" for its existence! That's why I called an electron a "vacuum excitation"! In fact, both holes and electrons are "quasiparticles" in a many-body interaction. They are not different.

Zz.

Ok .. I guess I see what you are saying, but there still seems to be a significant difference to me. Electrons and holes both "exist" in a solid state electrical conductor, however, if we use that conductor as the emission element of a beam source, it is straightforward to make a beam of electrons, but to my knowledge, no one has ever made a beam of holes.

I am a rank novice when it comes to high-energy physics, so I will have to take the word of those more knowledgeable than myself about "vacuum holes". I am also willing to accept that we can only make an electron beam because we live in and experience the "low energy" case, as suggested by Physics Monkey.

 

[Frame Dragger]

Ok .. I guess I see what you are saying, but there still seems to be a significant difference to me. Electrons and holes both "exist" in a solid state electrical conductor, however, if we use that conductor as the emission element of a beam source, it is straightforward to make a beam of electrons, but to my knowledge, no one has ever made a beam of holes.

I am a rank novice when it comes to high-energy physics, so I will have to take the word of those more knowledgeable than myself about "vacuum holes". I am also willing to accept that we can only make an electron beam because we live in and experience the "low energy" case, as suggested by Physics Monkey.

Well, given the description of holes a la ZapperZ not being able to find holes outside the lattice isn't any odder than not finding free quarks. Doping the material for holes seems to be the equivalent within the medium.

 

[Bob S]

Dirac ("Principles of Quantum Mechanics" fourth edition) specifically discusses electrons falling into the negative energy states (holes) which he identifies as positrons, with "the electron and positron disappearing simultaneously and emitting radiation". (This is the positron annihilation process). He also states that "the converse process would consist of the creation of an electron and a positron from electromagnetic radiation." (This is the pair production process.)

The first statement (but not the second) also appears in Dirac's first edition (1930), before the positron was discovered.

Bob S

 

[Ranku]

The fact that we can "renormalize" the environment in the filled band and to consider such holes as having not only a positive charge, but also to have spin, etc., means that below the Fermi level, they are as real as the "vacuum excitation" that we call 'electron'.

So holes can be assigned quantum numbers like spin?

 

[DrDu]

Recently there was a discussion in the Atomic and Solid State Physics forum "What actually are electron holes?", which may be interesting in this context.

 

[ArjenDijksman]

Ok .. I guess I see what you are saying, but there still seems to be a significant difference to me. Electrons and holes both "exist" in a solid state electrical conductor, however, if we use that conductor as the emission element of a beam source, it is straightforward to make a beam of electrons, but to my knowledge, no one has ever made a beam of holes.

Well said. That's clear indeed apart from the fact that the term "holes" applies generally to semiconductors. In conductors (read: good conductors), it doesn't make much sense to speak of holes, because the electron is almost free. As an analogy, if soccer players are electrons, the soccer field is a metal where the player has plenty of space to circulate (occasionally colliding with other electrons or objects that compose the lattice). But if the player is in the crowded changing room, he will be attracted to the vacant places (=the holes). The changing room can then be seen as a semiconductor.

Arjen

 

[DrDu]

Even in a metal, an electron behaves only like a free electron for energies sufficiently close to the Fermi surface. If you excite an electron several eV above the Fermi surface, it will start to emit all kind of electron hole pairs, phonons etc in sharp contrast to a free particle with the same kinetic energy.
That makes the difference between particles and quasi-particles.
What tends to make the hole concept very usefull both in particle and in solid state physics is the fact that an electron near the upper end of the valence (negative energy) band, has a negative effective mass. A hole then has a positive mass, which is what we are more used to from free particles.

 

[ZapperZ]

Ok .. I guess I see what you are saying, but there still seems to be a significant difference to me. Electrons and holes both "exist" in a solid state electrical conductor, however, if we use that conductor as the emission element of a beam source, it is straightforward to make a beam of electrons, but to my knowledge, no one has ever made a beam of holes.

Ignoring a bit the "free particles" scenario that involves positron, the idea of "electrons and holes" are valid within the condensed matter scenario. So asking about such a thing for free particles would be extrapolating it to where it isn't applicable. Having said that, I would refer to you Physics Monkey's post in this thread.

Furthermore, if we actually can set up a particle accelerator in the Fermi sea, I bet you we can make a beam of holes. When you are in vacuum, you are already in the "holes' Fermi sea", so you can only make electron beam!

Zz.

 

[ZapperZ]

Well said. That's clear indeed apart from the fact that the term "holes" applies generally to semiconductors. In conductors (read: good conductors), it doesn't make much sense to speak of holes, because the electron is almost free. As an analogy, if soccer players are electrons, the soccer field is a metal where the player has plenty of space to circulate (occasionally colliding with other electrons or objects that compose the lattice). But if the player is in the crowded changing room, he will be attracted to the vacant places (=the holes). The changing room can then be seen as a semiconductor.

Arjen

That isn't quite true. The cuprate superconductors, for example, in the normal state, are "bad metals". They get to become that by doping the insulating parent compound with holes, not electrons (that's why I earlier called some of them as hole-doped superconductors). So the charge carrier that is moving in this "conductor" are holes, not electrons. We do not always have to have a semiconductor to get the presence of holes.

Zz.

 

[ArjenDijksman]

That isn't quite true. The cuprate superconductors, for example, in the normal state, are "bad metals". They get to become that by doping the insulating parent compound with holes, not electrons (that's why I earlier called some of them as hole-doped superconductors). So the charge carrier that is moving in this "conductor" are holes, not electrons. We do not always have to have a semiconductor to get the presence of holes.

I'm not well acquainted with superconductivity, so I didn't know that. Isn't that hole doping just analogous to doping with atoms with unfilled electron shells, like for p-type semiconductors?

 

[ZapperZ]

I'm not well acquainted with superconductivity, so I didn't know that. Isn't that hole doping just analogous to doping with atoms with unfilled electron shells, like for p-type semiconductors?

It certainly is, but it is still not a "semiconductor". It is a metal (i.e. the resistivity vs. temp. curve has a positive slope), but yet, it becomes more conducting with more hole doping.

Zz.

 

[Frame Dragger]

It certainly is, but it is still not a "semiconductor". It is a metal (i.e. the resistivity vs. temp. curve has a positive slope), but yet, it becomes more conducting with more hole doping.

Zz.

Is there a sense of the maximum doping that can be applied to any given metal? If you go beyond it, do you slowly reduce efficacy or does something else occur? CAN you overdope the metal?

 

[SpectraCat]

What tends to make the hole concept very usefull both in particle and in solid state physics is the fact that an electron near the upper end of the valence (negative energy) band, has a negative effective mass. A hole then has a positive mass, which is what we are more used to from free particles.

Really? I find that hard to understand, but I guess it must be all in the definition of "effective mass". I will check it out in my Ashcroft and Mermin when I have time.

 

[Galap]

Really? I find that hard to understand, but I guess it must be all in the definition of "effective mass". I will check it out in my Ashcroft and Mermin when I have time.

I think it's the same thing as the fact that while photons have zero rest mass, they have an effective mass due to the fact that they have kinetic energy/momentum. Now if you had an electron with negative energy greater than its rest mass equivalent, you would have negative effective mass.

I'm not sure whether that's right or not, so someone who knows more about this can correct.

 

[DrDu]

The effective mass is given by the second derivative of the energy with respect to k, the crystal momentum (in crystals, it may be a tensorial quantity). It is clear that this curvature is negative both for at the top of the valence band and for the negative energy solutions of the Dirac equation. This is all very consistent. Given that the relativistic rest mass is the energy of a particle at rest, it has to be negative for the negative energy solutions. Hence the hole (positron) has positive rest mass.