What is a 'hole' in the context of solid state?

[Nusc]

Can someone please explain to me the concept of the hole as it pertains to solid state physics?

I don't understand how it works.

 

[mgb_phys]

It's just the absence of an electron - because current flow was defined as positive to negative before electrons were discovered - it's convenient to have a positive electron.

 

[Phrak]

It's just the absence of an electron - because current flow was defined as positive to negative before electrons were discovered - it's convenient to have a positive electron.

How would you explain the Hall effect in P type material?

 

[LydiaAC]

It's just the absence of an electron - because current flow was defined as positive to negative before electrons were discovered - it's convenient to have a positive electron.

I think that defining a hole as "the absence of an electron" is one of the most misleading sentences that I have heard.

A hole is the absence of an electron... but not of any electron. The absent electron must be one of the most energetic electrons in a almost-filled band, like a valence band in a semiconductor.

When electrons are inside crystals, they must obey Schroedinger equation for a periodic potential. The effective mass is defined to relate an external force with the group velocity of the wave packages. Electrons in an almost empty band, have an effective mass more or less normal, but different from the mass of electron in vacuum. Electrons in a almost filled band, have a NEGATIVE EFFECTIVE MASS. This feature, allows us to work them as positive particles with positive effective mass, that is, as holes.

To change a negative particle moving to left for a positive particle moving to right is a very common trick. It is used when defining conventional current. However, nobody talk about holes in this situation. If you want to keep the illusion that conventional current is made from positive particles, you would be obliged to accept a negative mass.

That is what is special about holes. Since electrons in the top of the valence band has a negative effective mass, when we change the sign of charge, direction and mass, we end with a perfect positive particle, with positive charge, moving in the opposite direction as the original electron and a positive effective mass.

How would you explain the Hall effect in P type material?

An electron with negative effective mass will accelerate in a direction opposite to that of the force applied to it. From rest, this will require to go in the opposite direction in which it is pushed. It is easier to imagine that the current is made of holes which respond to the force in a conventional way.

You are right. The Hall effect is the better proof that the hole cannot be simply the absence of an electron.

Lydia Alvarez

 

[Phrak]

An electron with negative effective mass will accelerate in a direction opposite to that of the force applied to it. From rest, this will require to go in the opposite direction in which it is pushed. It is easier to imagine that the current is made of holes which respond to the force in a conventional way.

You are right. The Hall effect is the better proof that the hole cannot be simply the absence of an electron.

Lydia Alvarez

Thanks for answering, LydiaAC.

I don't get the negative mass idea. Is it a convenient fiction of some sort? It's hard to imagine a change in the effective mass of the electron, which is something like 500 eV positive to begin with.

 

[LydiaAC]

Dear Phrak:
You are right, effective mass is an abstract concept and it is only called "mass" for analogy. It is not actually a mass.
Newton's second law, relates force and derivative of velocity (acceleration) through a parameter called mass. As you already know, it has been found that this is the same parameter that controls the gravitational force.
For particles inside a crystal, there is an analogous relationship between "external force" and the derivative of "group velocity". On my very basic knowledge of Quantum Mechanics, I remember that you can identify "group velocity" with the velocity of a particle. However, I have the feeling that this is a collective effect and you cannot identify a specific "group velocity" with a specific "electron velocity".
On the other hand, "external force" IS NOT "force". Inside the crystal you have many additional forces, exerted from ions and other electrons. This "force" or "total force" is often unknown, contrary to the "external force" that you know because you set it, applying a potential difference by means of a power supply.
The parameter relating "external force" and "group velocity" is known as "effective mass". It is called "mass" for analogy with real mass but it is called "effective" to point out that it is not the real mass.
It must be some process by means of which, when an electron is carried by an external force to one of the highest states in a band, the electron becomes vulnerable to the effect of an "internal force" opposite to the external force. This net force is actually pushing it in the same direction in which is accelerated, but as we are only aware of the external force, we can see it accelerating in the opposite direction in which is pushed.
In conclusion, the price to forget "internal force" is to accept a negative mass. If you want to keep the mass positive, you must deal with the very complicated internal force.
Lydia Alvarez

 

[Phrak]

Thanks, Lydia. I guess its a question a few solid state physicists would have something to say about. It's one of those open quesitons I've unresolved over the years.

 

[zhanghe]

hi , Lydia Alvarez
i've read your post here carefully, and i think you OMIT one thing so that you make a mistake about this popular conclution in almost all textbook --" a hole is the absence of an electron " ,though easily misleading but it is right, just at large in some degree.

According to your post, "a negtively charged and negtively massed electron equals a hole having positive charge and positive mass", be careful , this is NOT true.

The thing you omit is that the hole is a collective phenomenon of electrons in different motions in the valence band (VB), not a single electron in VB itself.

If the valence is full, there is no current when applied a electric field. There must be a empty postition ,that is, behind which an electron must leave the valence band, if you want to get a current.

And of course we cannot say the empty position is just the hole concept. In effect,
the behaivor of a hole in the crystal applied E field is accomplished by all the electron, left in the VB, having the two negtive item ( charge and mass:) ).

Check the E-k diagram in the view of velocity (hk reprent the velocity),carefully,
you will find that ,at every monent there is an electron having a velocity, opposite the E field. at this time "he" is the hole, at the next monent another electron having a another velocity will act as the hole. so in general, all electron that remain in VB, are acting as a hole. in other words, the hole is the absence of the electron that leaving VB.

maybe i failed to say it clearly. hope your reply,Lydia Alvarez. and don't go crazy after reading my post, Phrak.:)
it is not my original intention,you know.:)

 

[zhanghe]

some complements:
in applicatin, a hole is just treated as a + mass point with uint + charge, and says the hole is a collective behaviour of many electrons. it is enough for learning semiconductor device physics (including hall effect) etc.

 

[LydiaAC]

Hello Zhange:
I think you are right in all your explanation, a "hole" is a collective phenomenon caused by electrons in valence band.
However, I believe that an "electron" is also a collective phenomenon caused by electrons in conduction band.
I will check carefully your explanation to see if this symmetry really applies. I have never thought things in that way and it will be interesting to check this vantage point.
About the "misleading explanation", I have found tutorials in internet which say that "metals have the same number or electrons and holes because they have a half filled band". You will agree that you cannot count electrons AND holes in the same band. This is only an example of a misconception caused by the simple definition of "absence of an electron".
As to the "crazy" thing. In this forum it is difficult to know who is asking to know and who is asking rhetoric questions. I cannot do otherwise but explain things as I understand them. If someone understand more than I, it is OK. I registered in this forum to learn and I was surprised to see that I could, sometimes, give answers.
I think that you observation is very keen, and your footnote shows that you are a sensitive person, so I do not have a reason to go crazy.
If you want to defy all the explanations I have put in this forum, go ahead. I will try to defend what I can, but if I see that I cannot, I will change my mind, and will be benefited
in the process. I am a staunch believer in the positive power of debate.
Lydia Alvarez

 

[Defennder]

Consider this in 1D:

We all know that J=nev where n is the concentration of electrons, v is the drift velocity.

Suppose the valence band is completely full of N electrons, then it goes without saying there can be no current flow:


J_{x} = \sum_i^N -ev_{i} = 0

But on the other hand suppose there is one missing electron, one vacancy in the valence band where an electron should be:

J_{x}=\sum_i^{N-1}-ev_{i} = \sum_i^N -ev_{i} -(-ev_{j}) = ev_{j}

Notice that the final expression on the right can be thought of as current due to the drift of a single positive charge, since there is no minus sign. That is why we are justified in thinking that we can treat absence of negative charges as positively charged holes.

 

[Defennder]

About the "misleading explanation", I have found tutorials in internet which say that "metals have the same number or electrons and holes because they have a half filled band". You will agree that you cannot count electrons AND holes in the same band. This is only an example of a misconception caused by the simple definition of "absence of an electron".

I really don't see any problem with this statement. Isn't this like saying you can either see the half-full glass of water either half-full or half-empty?

 

[Cthugha]

I really don't see any problem with this statement. Isn't this like saying you can either see the half-full glass of water either half-full or half-empty?

Well, that depends on what one is talking about.
If you model your system using electrons and holes at the same time, you have to keep in mind, that only one of these can be used to calculate the current - holes or electrons. Beginners are often confused by unclear textbooks at this point and try "adding up" electronic and hole current.

On the other hand you can of course consider holes and electrons at the same time, if you do it correctly. For example in low dimensional nanostructures excitons (in layman terms a bound complex of an electron and a hole) are very important.

 

[zhanghe]

hi , LydiaAC:
:) you mistake my sentence, see it again here,
" maybe i failed to say it clearly. hope your reply,Lydia Alvarez. and don't go crazy after reading my post, Phrak.:)
it is not my original intention,you know.:)"
you can see that in fact
the sentence with "crazy" is going to be given to Phrak, not you.
because i talked more and complexly and i am not a native English speaker, so i am afraid Phrak will can't understand
what i talked about as a new learner in this area.
but you are quite excellent, truly, i think.:)
and of course you give me a good lesson that i should not comment anyone when i don't understand him/her clearly
thank you for your supporting my thought about the hole, again. thank you:)

 

[LydiaAC]

Hello Zanghe:
My mistake. I am sorry for this. I thought that Phrak was your signature and this was in conflict with my knowledge that your nickname was Zhanghe. Now I can see it, but it was a sort of "cognitive dissonance" or something. I am sorry again.
I am also a non-native English speaker. Actually, forums (fora?) are very good to me because I can write much better than I can speak. I am not so good at Physics. I studied Electrical Engineering in Mexico and somebody decided we did not need Dynamics or Thermodynamics, so I never studied them. In graduate school I catched up with a little of Optics and began to study hard Solid State and Semiconductor Physics.
I know a little about holes, electrons, diodes and bipolar transistors but I can make big mistakes in other fields of physics. I am not happy about this because I am old, not young as surely you are. I am very happy to discuss about these fascinating concepts and I am really open to discover new things or even that I am completely wrong.
Lydia Alvarez

 

[LydiaAC]

On the other hand you can of course consider holes and electrons at the same time, if you do it correctly. For example in low dimensional nanostructures excitons (in layman terms a bound complex of an electron and a hole) are very important.

Hello Cthugha:
Do you think you can give a "for dummies" explanation of excitons? I do not know anything about them but their definition.
It would be nice to know a little more.
Lydia Alvarez

 

[Cthugha]

Hello Cthugha:
Do you think you can give a "for dummies" explanation of excitons? I do not know anything about them but their definition.
It would be nice to know a little more.
Lydia Alvarez

Well, I can try.

For example in absorption experiments insulators and especially semiconductors show an absorption line below the energy, which is supposed to arise due to the creation of an free electron and a hole. This line arises due to the excitation of an electron to the conduction band, which is still subject to Coulomb-correlations with the hole, which is created at the same time. The result is a hydrogen-like complex of an electron and a hole.

Excitons can be described in two limits: Wannier excitons (large radius, rather small binding energy) and Frenkel excitons (small radius, rather large binding energy). As one can imagine, excitons are rather important in semiconductors at low temperatures, especially if the exciton binding energy is higher than the thermal energy.

From a more theoretical point of view, excitons are used to define the eigenstates of the em-field in material. In a classical treatment a polarization field is introduced in order to explain the effects of interaction of the em-field with material. In an qm-approach, it can be shown that excitons behave like a quantized polarization field in the limit of low exciton density. So one can model the eigenstates of the em-field in material as being quite polariton-like, which means, that there is a propagating mode of a "mixed" photon and exciton.

A classic reference is:
J.J. Hopfield, "Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals", Phys. Rev. 112, 1555 - 1567 (1958)

One can easily imagine, that excitons are even more important in low-dimensional nanostructures as the strong spatial confinement of the electrons and holes leads to stronger Coulomb interactions and therefore higher binding energies.