What is a hole and how does it relate to electricity?

[I_am_learning]

I never actually understood what exactly is a Hole? Could you explain it clearly.

 

[mgb_phys]

It's simply the absence of an electron.
Because of historical accident, before the discovery of the electron, current was assumed to flow from positive to negative.
It's a bit error prone to have electrons moving one way and current moving the opposite way - so we imagine the movement of the absence of electrons, which go in the same direction as current

 

[I_am_learning]

But that don't seem to explain the whole story. If movement of holes were actually movement of negatively charged electron in the opposite direction then the Hall effect shouldn't have shown positive hall voltage. But as far as I know, the holes behave totally like positive charges, in every respect. This is what bothering me for years!

 

[Modey3]

I never actually understood what exactly is a Hole? Could you explain it clearly.

Consider a string of hydrogen atoms. If you take away one electron on a particular hydrogen atom you would get a string of something like: ... H, H, H+, H, H ...

If the current moves in the right-direction the positive-charge (not the hydrogen cation!) moves in the left-direction. As you can see the hole is the positive charge induced by the un-screened nucleus.

modey3

 

[vin300]

But that don't seem to explain the whole story. If movement of holes were actually movement of negatively charged electron in the opposite direction then the Hall effect shouldn't have shown positive hall voltage.

If B is along +ve z and either velocity of holes is in +ve x or velocity of electrons is in -ve x, Hall voltage is +ve.
If B is along +ve z and either velocity of electrons is in +ve x or holes in -ve x, Hall voltage is -ve

 

[I_am_learning]

If B is along +ve z and either velocity of holes is in +ve x or velocity of electrons is in -ve x, Hall voltage is +ve.
If B is along +ve z and either velocity of electrons is in +ve x or holes in -ve x, Hall voltage is -ve

I don't think that's correct. Look at the following illustration I sketched

If the movement of holes were actually movement of electrons, then should always find -ve hall voltage (Right Figure) . Shouldn't we?

 

[Phrak]

If the movement of holes were actually movement of electrons, then should always find -ve hall voltage (Right Figure) . Shouldn't we?

I understand your problem but not the solution. Apparently the others are unfamiliar with the the Hall effect in P-type materials in violation of classical elecrtromagnetism. Perhaps someone can enlighten us both.

 

[DrDu]

Maybe you should take into account that electrons at the upper end of the band have a negative effective mass. Hence a hole (the absence of one of these electrons) has a positive mass.

 

[josephajain]

I agree with the critic.

 

[I_am_learning]

Maybe you should take into account that electrons at the upper end of the band have a negative effective mass. Hence a hole (the absence of one of these electrons) has a positive mass.

O.K. I guess you mean that the holes are not just the absence of electron but absence of electron with negative effective mass (Someone somewhere has already told me this). But if that is the case then; when the electric Field is applied in upward direction (the right side figure of my sketch some posts above), then electrons having negative mass should have moved in the same direction (upward). O.K yes, That could produce and hence explain the +ve hall voltage, but the absurd thing is that, that would result in current in downward direction (opposite to direction of applied field).
But we always see the current in the direction of applied field, don't we?

 

[Studiot]

In post#6 you have a contradiction.

This is because you are trying to show the current in three different ways
That is not a good idea.

You have shown a conventional current. This current is in one direction only, regardess of the movement of its carriers and is one of the determining vectors of the Hall voltage.

This conventional current arrow may be made of all holes, all electrons or a mixture.
Regardless the arrow is still the same.

So the two important vectors in your diagram are the magnetice field and the current (with its associated electric field of course).

This leads to build up of negative charge on the left and positive on the right which in turn makes the polarity Hall voltage, developed across the section negative on the left and positive on the right.

Are you also aware that this voltage develops its own electric field so that the final electric field is no longer parallel to the current direction but the vector sum of the two fields?

I think this is a fine example of trying to work in terms of charge carriers rather than a (ficticious) entity called conventional current brings difficulties. All the other equations of physics are adjusted to be in sync with this.

 

[I_am_learning]

In post#6 you have a contradiction.
The derived force vector is correct in the right hand diagram but should be reversed in the left one as positive and negative charges are deflected in opposite directions.

Yes, I have considered that positive and negative charges are deflected in opposite directions; but since their direction of motion is also opposite; they are finally deflected in the same same direction, as shown in the figure (post 6).
Either you missed it or I am in serious trouble with the Left-Hand Rule.

 

[Studiot]

But your current vector is the same in both diagrams.

The LH rule expects this vector (as you have it) irrespective of the polarity of the carriers.

This is a sign convention thing. You also have to start introducing -ve signs into the vector cross product if you must use charge carrier direction. Don't forget that the current vector also defines a direction in space, ahich you have chosen by using conventional current (wisely in my opinion).

 

[Modey3]

Folks,

This subject is being over-analyzed. IMO the Hall Effect experiment doesn't adequately describe what a hole is. A hole is not a particle. Yes, I said it! A hole is an absence of a particle. In semiconductors the absence of an electron within (not at the top) the DOS is manifested locally on an atom. Thus, it may seem like a hole is a particle, but in reality it is not. The positive charge of the hole comes from the nucleus. Holes are created by adding localized p-type defect states near the VBM, which pulls electrons out of the VB and this creates holes in the VB. The p-type defect gets neutralized while an atom in the parent lattice gets ionized . This is why holes are localized around an atom. In metals, a hole has a different meaning because as the temperature is raised you are technically creating holes in the VB, but the holes are not localized to a particular atom, which is why it's impossible to describe current in terms of hole-flow in metals.

I hope this helps

modey3

 

[Studiot]

Yes you have shown the electron and hole flow in opposite directions.

Your diagram in post 6 is both correct and compatible with vin300 statement in post 5.

The Lorenz force pushes both carriers to the same side. This is fairly easy to show mathematically.

As a result the Hall voltage across the section will be either positive or negative on the left depending upon whether the flow is holes or electrons.

This situation when there are both carriers present (semiconductors) is much more complicated. It is not true to say that just because there are more holes or electrons the left side will end up more positive or more negative. This is because the Lorenz force equation has to be modified to take into account the diffeent mobilities of holes v electrons. This can lead to the minority carrier determining the Hall voltage polarity.

You can have the equations if you want them, but it is a lot of algebra.

 

[I_am_learning]

Your diagram in post 6 is both correct

Its a great relief to hear that.
Now, I am at point at asking what I intended to ask.

In some material Hall Voltage is found to be positive. This is explained by saying that the current is constituted by holes (Left figure, Post 6).

But the contradiction arises when we study about the nature of holes. If we are to say that holes are actually absence of electrons (and there is no such thing as positively charged hole) and that the motion of holes is accomplished by transfer of electrons in the opposite direction, then we are forced to model the current flow in the material by Rightside Figure in Post 6. And now to the arousal of great contradiction the Hall Voltage is Found to be -ve.

Please for gods sake, let be get through this. I am dying.

 

[I_am_learning]

Your diagram in post 6 is both correct

Its a great relief to hear that.
Now, I am at point at asking what I intended to ask.

In some material Hall Voltage is found to be positive. This is explained by saying that the current is constituted by holes (Left figure, Post 6).

But the contradiction arises when we study about the nature of holes. If we are to say that holes are actually absence of electrons (and there is no such thing as positively charged hole) and that the motion of holes is accomplished by transfer of electrons in the opposite direction, then we are forced to model the current flow in the material by Rightside Figure in Post 6. And now to the arousal of great contradiction the Hall Voltage is Found to be -ve.

Please for gods sake, let be get through this. I am dying.

 

[Studiot]

First let us look at the model you are employing in post#6.

The lattice is electrically neutral. So if we take a negative electron from point A and shove it over to point B, point B becomes –1 negative and this leaves a corresponding +1 positive charge at point A, supplied by the positive charges in the area that were formerly balancing the -1 charge on the electron we have moved.

However, once the hole and electron have separated (ie the pair has been 'created') there is nothing in the model to continue to link a particular hole to a particular electron. If the electron moved on again to point C it would not leave another hole at B, just a neutral point.
Equally the hole could move by ‘displacing’ a different electron, or if you like a different electron could occupy point A which would again be neutral, leaving a +1 hole somewhere else in the lattice.

Now we know that all charges, holes and electrons, moving as part of the current, are subject to the leftwards Lorenz force.
But all other electrons in the lattice are not subject to this force as they are not moving as part of the current. And there are plenty available.
So a leftwards movement of the hole at point A can be accompanied by a rightwards movement of electrons without violating any laws.
This, of course, is movement at right angles to the current flow so no work is required from the electric field causing the current ie there is no interference with it.

Note also my earlier comment about the different ‘mobilities’ of holes and electrons.

Now to address the title question of your thread.

A simple explanation of holes is positional. This is because electrons are regarded as point particles. As such they have a location in space and it makes sense to talk about electrons moving one way and 'holes' moving the other.

However when an electron moves it does not leave behind ‘nothing = absence of electron’
It is rather like pigeon holes in a letter rack and letters or pages in a book and writing. Just because there are no letters in a particular pigeonhole or no writing on a particular page does not mean that the pigeon hole or page does not exist. They are still there, available for letters or writing.
So if you imagine the solid block as the rack or book and the pigeonholes or pages as the wave states and the letters or writing as the electron perhaps you can see the analogy.
Remember the analogy is not perfect, just an aid to visualisation.

Further electrons and holes are not point particles. They have no location in space. In solids the wavefunctions extend across the entire block of material.
Some are filled some are not, but all are present.
This is how the electron can actually ‘move’ – It jumps from state to state. This is a quantum thing and the quantum Hall effect is much more complicated than the classical smooth one.

 

[rootX]

I don't think that's correct. Look at the following illustration I sketched
...

If the movement of holes were actually movement of electrons, then should always find -ve hall voltage (Right Figure) . Shouldn't we?

1. Could the confusion be that you mixing n-type materials with p-type? I believe current is solely due to the particles in the conduction band, so it might depend whether you have n or p majority.I did not read through the whole thread ... I would have to review the Hall's effect before coming with a valid answer

 

[I_am_learning]

I read and re-read your reply for some hints, but please forgive me, I couldn't still get it.

The lattice is electrically neutral. So if we take a negative electron from point A and shove it over to point B, point B becomes –1 negative and this leaves a corresponding +1 positive charge at point A, supplied by the positive charges in the area that were formerly balancing the -1 charge on the electron we have moved.

This is explaining how conduction begins, Right?

However, once the hole and electron have separated (ie the pair has been 'created') there is nothing in the model to continue to link a particular hole to a particular electron. If the electron moved on again to point C it would not leave another hole at B, just a neutral point.
Equally the hole could move by ‘displacing’ a different electron, or if you like a different electron could occupy point A which would again be neutral, leaving a +1 hole somewhere else in the lattice.

This is explaining how conduction continues.

Now we know that all charges, holes and electrons, moving as part of the current, are subject to the leftwards Lorenz force.
So a leftwards movement of the hole at point A can be accompanied by a rightwards movement of electrons without violating any laws.

Same story repeats, I think. How can electrons be made to move rightwards?
I would like to put it this way
Be it n-type or p-type material,
1. the only thing that can move is electrons, right? (Am considering Classical Model)
2. If the current is in upward direction, No matter how you try to explain the phenomenum, The net electron flow must be in downward direction, Right?
3. If the elctron flows in downward direction, then Lorentz Force pushes it left, and Hence
-ve Hall Volatge, Right?

4.I must be wrong at least Somewhere, Right? But where (1,2,3,4)?

 

[Pythagorean]

A hole is NOT merely an absence of an electron. By that definition, no charge would be a hole.

A hole IS the absence of an electron in a formerly charge-balanced material, giving the material a net positive charge.

 

[Studiot]

If the current is in upward direction, No matter how you try to explain the phenomenum, The net electron flow must be in downward direction, Right?
3. If the elctron flows in downward direction, then Lorentz Force pushes it left, and Hence
-ve Hall Volatge, Right?

Your problem is in number 2/3


The crux of it is here

Now we know that all charges, holes and electrons, moving as part of the current, are subject to the leftwards Lorenz force.
But all other electrons in the lattice are not subject to this force as they are not moving as part of the current. And there are plenty available.
So a leftwards movement of the hole at point A can be accompanied by a rightwards movement of electrons without violating any laws.

etc

The electrons that need to move rightwards are not the electrons that take part in the upward flowing current.

Consequently they are not the ones being forced leftwards by the Lorenz force.
(You are aware that the force you have labelled 'Thrust' is called the Lorenz Force?)

Don't forget there are no electrons moving directly leftwards. These are the ones whose trajectory has two components, viz the upward flow of current and the leftwards push, resulting in a path curving towards the left.

 

[I_am_learning]

(You are aware that the force you have labelled 'Thrust' is called the Lorenz Force?)

Yes, I am aware of that.

Don't forget there are no electrons moving directly leftwards. These are the ones whose trajectory has two components, viz the upward flow of current and the leftwards push, resulting in a path curving towards the left.

I am aware of that too.

The electrons that need to move rightwards are not the electrons that take part in the upward flowing current.
Consequently they are not the ones being forced leftwards by the Lorenz force.

Then which electrons are they? And which force causes them to move rightwards?
(I think I have reached to the root point, now. Wow!)

 

[DrDu]

Ok, here's my view:

ma=qE+q vxB
is the equation of motion. Due to friction, the acceleration will lead to an equilibrium velocity v that will show in the same direction as the acceleration a.
Case a: m>0, q<0 (electrons at the lower end of the band)
v is antiparallel E, j =q v is parallel E. If E is in x-direction and B in z-direction, then the electrons will be deflected in direction of -y.
Case b: m<0, q<0 (electrons at the upper end of the band with negative effective mass)
v is parallel E, j is antiparallel E. If E is in x-direction and B in z-direction, then the
electrons will be deflected in direction of -y, as in case a. However, as the direction of the current is reversed in comparison with case a, the hall coefficient will be of alternate sign.
case c) m>0, q>0 (holes):
v is parallel E, j is parallel E and holes are deflected in direction of -y. The Hall coefficient equals the one in case b.

In the band picture, the situation in a nearly full band is complicated by the fact that most levels are filled and "Pauli block" the mutual motion. In a completely full band, the electrons cannot react at all to the applied field and acceleration a=0. In a nearly full band, the mean acceleration of the electrons will be very small, but will on the mean follow case b. Alternatively, the situation can be described in terms of some holes of case c.

 

[Phrak]

Thanks for that DrDu. I was hoping you'd come up with more. You are the one who has done the work, not I, so I'm a little embarassed to mention this. In case b) where electron mass is less than zero, j is antiparallel with E. The positive current is moving counter to the applied field! So we get perpetual motion due to negative resistance (which would be awfully nice to have...)

I think we might consider the equilibrium condition where an electric charge gradiant has already accumulated in the ±Y-direction so that there is an additional E_y field due to this charge.

The drift velocity, v_x = (½d)ma_x = k^-1ma_x, where d is a some nominal positive distance between collisions, so that k is positive and

sign(v_x) = sign(a_x). (I'm just repeating what you’ve established in this.)

kv_x = ma_x = qE_x + q(v_yB_z)

v_y=0 by conservation of charge in the equilibrium condition, so that

1) kv_x = qE_x. Now the electrons are not giving us free energy. Bummer.

kv_y = 0 = ma_y = qE_y – v_xB_z so that

2) qE_y = v_xB_z

Equations 1 and 2 are as far as I've gotten. Maybe the negative electron mass model works. I have no idea. ---or did I make a fundamental error somewhere?

 

[I_am_learning]

Ok, here's my view:

ma=qE+q vxB
is the equation of motion. Due to friction, the acceleration will lead to an equilibrium velocity v that will show in the same direction as the acceleration a.
Case a: m>0, q<0 (electrons at the lower end of the band)
v is antiparallel E, j =q v is parallel E. If E is in x-direction and B in z-direction, then the electrons will be deflected in direction of -y.
Case b: m<0, q<0 (electrons at the upper end of the band with negative effective mass)
v is parallel E, j is antiparallel E. If E is in x-direction and B in z-direction, then the
electrons will be deflected in direction of -y, as in case a. However, as the direction of the current is reversed in comparison with case a, the hall coefficient will be of alternate sign.
case c) m>0, q>0 (holes):
v is parallel E, j is parallel E and holes are deflected in direction of -y. The Hall coefficient equals the one in case b.

Are you sure case b, exist? That E field and j can sometimes be anti-parallel?
And for case c, I need to repeat the same story, If we in fact do believe that holes are actually absence of electrons (in formerly charge neutral place) and that holes are abstract things and that they are really no particles and that really no force acts on them; can't we always talk in terms of electrons (at least for the sake of this discussion)? Then Case C should be modeled with case b, I think?

 

[DrDu]

Yes, I do believe that case b exists and that E and j can be anti-parallel.
Of course holes (case c) are abstract things, but they precisely model situation b.

 

[ZapperZ]

There's a similar discussion on this already in another thread:

https://www.physicsforums.com/showthread.php?t=374651

For anyone claiming that "holes" aren't as "real" as "electrons", keep in mind that these are done in the context of many-body interaction. If you think the electrons you get in such a system are "real" while holes are not, then you need to convince me why the Landau's Fermi Liquid theory is wrong, why these are actually "quasiparticles" and not bare electrons (look at the mass renormalization of these things), etc.. etc.

Zz.

 

[Studiot]

You have to show consistency in a model.

The critic wanted to discuss the hall effect with a point charge model of current.
This is considered difficult to impossible to use to explain all the observed phenomena.

However you cannot both say that a hole is a convenient fiction or non real entity and that the Lorenz force acts on it.

So if we discuss the diagrams in post#6 we are tacitly giving substance to holes.

Allowing this argument the point charge theory goes as follows

1) Point charges do not interfere with each other.

2) All point charges positive or negative, are swept in the same direction by the Lorenz force.

3) At equilibrium the net buildup of charge to one side is manifest as +ve or -ve voltage at right angles to the direction of flow, depending upon whether there are more +ve or negative charges diverted.

With regard to the existence of holes or otherwise, the molecular orbital exists whether it is occupied or not.

 

[DrDu]

I finally found the time to have a look to a book (how old fashioned!), Ashcroft Mermin, solid state physics.
They discuss the situation in detail, backing up my rather hand-waving argumentation.
The most difficult part is the justification of the equation of motion. Ashcroft and Mermin refer only to the literature on that point for the general formula (especially for the Lorentz term) although it is rather easy to prove for the reaction to a static electric field. That the current must be opposite for some electron levels compared to what one would expect for a free electron follows from the fact, that the total current from a full band must vanish.

 

[Studiot]

My copy of Ashcroft and Thingy expounds a quantum explanation which quantitatively accounts for various values of the Hall voltage.

But they do not provide a motivation for an answer to the OP's question.

Here is a mathematical instance of the same question.

When multiplying two ordinary numbers
+ times + makes +
- times + or + times - makes -

But why does - times - make +?

The best answer I have seen to this (due to Gelfand)

3x5 = 15
Getting 5$ three times is gaining 15$

3x(-5) = -15
paying 5$ three times is losing 15$

(-3)x5 = -15
not getting 5$ three times is losing 15$

(-3)x(-5) = 15
not paying 5$ three times is gaining 15$

The question about the hole is similar to the last one.

 

[Phrak]

Yes, I do believe that case b exists and that E and j can be anti-parallel.

I don't see how this is justifiable. With E and j antiparallel the P type Hall effect device is a source of emf; energy is being put into the system--but from where? This is a perpeptual motion machine of the first kind.

And more, it is not equivalent to case c) which behaves like a resistor.

 

[DrDu]

The point is that without taking scattering which leads to resistance into account, an electron of initial crystal momentum k_0 would move trough the whole Brillouin zone (1 dimensional case). During that motion, its velocity would also change sign so that finally it would end where it started, so that no energy is deposed in the system on the average.

 

[Naty1]

Here's an interesting description:

The electron–hole pair is the fundamental unit of generation and recombination, corresponding to an electron transitioning between the valence band and the conduction band. ...

the valence band is so nearly full, its electrons are not mobile, and cannot flow as electrical current...However, if an electron in the valence band acquires enough energy to reach the conduction band, it can flow freely among the nearly empty conduction band energy states. Furthermore it will also leave behind an electron hole that can flow as current exactly like a physical charged particle. Carrier generation describes processes by which electrons gain energy and move from the valence band to the conduction band, producing two mobile carriers; while recombination describes processes by which a conduction band electron loses energy and re-occupies the energy state of an electron hole in the valence band.

http://en.wikipedia.org/wiki/Electron-hole_pair

 

[Studiot]

That's exactly what I said before.

If you allow quantum explanations, the pigeon hole is already there, whether occupied or not.

 

[ZapperZ]

Here's an interesting description:
http://en.wikipedia.org/wiki/Electron-hole_pair

While this is certainly true, it is missing something extremely fundamental that is a common practice in condensed matter physics. I'll illustrate this first using a simple metal case.

In metals, we think of there being these conduction electrons. However, are they really "true" electrons? We know that the free electron approximation is really just that, and approximation. A more detailed look at this requires us to include the many-body electron-electron interactions that occurs between all of the conduction electrons. This is where the Landau's Fermi Liquid theory comes in. This theory shows that, in the weak-coupling limit, we can make use of the mean-field approach and "renormalize" this problem from one many-body problem, into a many one-body problem. The latter is simpler, because we already know how to deal with that when we did the free-electron case (which is a many one-body problem). However, the trade-off here is that, what we have is not really a "bare" electron as in the free-electron case, but rather a "quasiparticle", which is considered as a single-particle excitation out of this many-body interaction. We lump all the many-body interactions by renormalizing the quasiparticle mass, i.e. the mass of the 'electron' you measure in a conductor can differ from the electron bare mass. It is tied to the dispersion.

So even for these electrons that people think are "real", they are really not your bare electrons, but rather electrons as quasiparticles. These are what you measure in solids.

Now, let's go back to these pesky holes. In the valence band, rather than complicate things and try to describe the dynamics of the system by describing ALL of the electrons in that band, we can simply say "Ah, let's renormalize everything and consider the holes as charge carrier". After all, we did that for the electrons already in the conduction band. So via the same method, these holes emerge as also being quasiparticles - i.e. they are endowed with mass, spin, charge, etc. They are no better nor worse than the "electron quasiparticles" that one deals with in the conduction band.

It is why, in condensed matter physics, we never argue that "electrons" are real while "holes" are not. We know better. These electrons are not "bare" electrons. They are as "real" as the holes.

Zz.

 

[I_am_learning]

I followed this thread and also the thread that zapperz linked to. But, Pherabs my unability, I couldn't find exactly, what I wanted. However, I got a general sense like this

In a hall effect scenario post #6, although Electrons move as shown in figure post #6, right, if the material is p-type, then by some complex mysterious (at least for me) forces, it is deflected to the opposite direction, i.e. to the right, due to the Lorentz force, apparently to the contradiction of my classical picture of Lorentz force. However this is no contradiction again due to some complex mysterious phenomenums. :).

Any one interested to non-mystrify the mysterious things, may proceed. :)

 

[bjacoby]

I followed this thread and also the thread that zapperz linked to. But, Pherabs my unability, I couldn't find exactly, what I wanted. However, I got a general sense like this

In a hall effect scenario post #6, although Electrons move as shown in figure post #6, right, if the material is p-type, then by some complex mysterious (at least for me) forces, it is deflected to the opposite direction, i.e. to the right, due to the Lorentz force, apparently to the contradiction of my classical picture of Lorentz force. However this is no contradiction again due to some complex mysterious phenomenums. :).

Any one interested to non-mystrify the mysterious things, may proceed. :)

OK, It's true the nature of holes does involve complexity. They are not "real" particles. You can't build a "hole" gun shooting positive charged holes into space. Indeed the complex math giving rise to holes relies on the nature of periodic structures which explains why semiconductor devices are made from single crystal materials.

But with that said and understood, let's demystify holes. Consider a neutral semiconductor material that has been doped with a "P" type atom. This involves the valence etc of things used, but the bottom line is that sprinkled throughout the material are spots where electrons are "missing". We've already noted this. Let us apply an electric field to that material to move the electrons. They move in a given direction. Did you ever notice how bubbles rise in a liquid? Ever notice how a balloon in a car goes the OPPOSITE way that you are apparently thrown when the car goes around a curve? Same idea. The holes move in the OPPOSITE direction from the electrons. Think of it as an array of electrons with a few holes (missing electrons) in it. As you apply a field forcing the electrons one way they fill the holes and given that the empty space has a net positive charge it "sucks" the electron into the space leaving the next space empty. In P type materials the energy required to do this is actually much lower than expected due to the structure of the matter. Hence the vacancies are actually traveling opposite to the electrons. And the vacancies have a net positive charge. And even though the nuclei don't actually move, on a larger scale there clearly appears to be ON AVERAGE a seeming net transport of positive charge. And that flow is Opposite to electron flow. And it turns out that that such a flow acts as if it were a real positive particle with a real positive mass!

Now let's go to the Hall effect. One can easily determine that if the material has electrons carrying a current one polarity develops from a magnetic field, but if the carriers are positive then the opposite polarity develops. The confusion arises by thinking that there are equal numbers of electrons and holes available for conduction. Such is not the case. In the semiconductors involved the periodicity of the crystal structures give rise to energy structures known as "bands". Because of these bands electrons and hence holes can be "locked" in and not be free to move. In insulators nothing is allowed to move. In semiconductors depending on the valence etc. of impurities introduced in the crystals one can create a situation where the density of apparent "holes" is much great than that of free electrons. That is known as P type material. Because there are no "free" electrons able to conduct current, current is apparently transported from one end to the other by positive holes. We do know that the valence electrons actually ARE shifting from atom to atom to create the "hole" motion but these valence electrons are not free. They do not conduct and hence do not produce a Hall voltage. One can by changing materials also create a material where electrons are able to conduct. That is called N type material. That would have the opposite polarity.

Kittel [Intro to solid state physics] Observes: "The circumstance that vacant states near the top of an otherwise filled band behave in every way as particles of positive charge +|e| is one of the most interesting features of the band theory of solids."

If you want to understand all this in detail, get busy! It's going to take you some time!

 

[Studiot]

@bjacoby

I am not disputing your observations, but your model does not explain the OP's original question which refers to the fact that holes and electrons subject to the Lorenz force in the Hall effect are deflected in the same direction.

This is why we can deduce the polarity of the majority carrier from the polarity of the Hall voltage. It depends whether holes or electrons accumulate on one particular side of the material.

 

[DrDu]

I think the interesting point is that the electrons near the top of the band which we are removing when forming holes don't behave as we would expect from our experience with free electrons. What happens can be seen even for the case of nearly free electrons in an extended zone scheme: Far away from the band gaps, the electronic wavefunction is exp(ipx). However, when approaching the band gap (but so far away that perturbation theory is still a good approximation), a small amount A(p) exp(-ipx) of the reflected wave will appear in perturbation theory. If we now calculate the change of theexpectation value of the momentum operator with the parameter p, something interesting happens: The expectation value is something like <P> =p-A(p)^2 p. Importantly, this will be the expectation value of momentum for a wavepacket made up of states in a region Delta p with Delta p <<(p-p_Gap).
Now A increases very rapidly near the band gap so that the derivative of the second term with respect to p will dominate the first contribution (This can be imagined to happen already when the absolute value of A is still small). The effective momentum drops although the parameter p increases. The effective energy is still p^2/2m to a good approximation. For the wavepacket considered, the group velocity will be v=dE/d<P>=dE/dp * dp/d<P> . The first term is proportional to p (or approximately <P>) while the second is a negative constant, hence v and <P> will be anti-parallel.
It is clear that the electromagnetic field will couple to p, at least as a lowest approximation, and not to v. So the electric field really accelerates the wavepacket in the opposite direction as to what we would expect for a free electron.

 

[Studiot]

Dr Du I understand where you are coming from, it's just that the OP was trying to resolve the question in terms of point charges with mass and (force) vectors.

This is the 'Classical Model'

I have read that it is impossible to resolve this issue using this model, but I thought I would have a go anyway.

I agree that if you delocalise the charge to 'bands' or zones or molecular orbitals, which is the logical next step, these extend to the full limits of the whole crystal.
Current is then modeled, not as moving points of charge, but perturbations moving through the bands. Electric and magnetic fields are seen as modifying the disposition of these bands, by generating these perturbations.

Using this model you do not have to account for apparently perverse statements about some force moving charges the 'wrong' way.

Incidentally you did refer to the charges being 'locked in place' in one post, and later suggested that they wer affected by the Lorenz force. You cannot have it both ways, if they are not moving, they are not affected.

I think the present vogue for teaching current as a string of little bead like electrons flowing from negative to positive is just as damaging to future deeper understanding as the former convention of current as some kind of 'juice' flowing the other way.

 

[Studiot]

Just for the record here is the classical calculation for holes and electrons.
Establish a right handed coordinate system with holes flowing in the +y direction ans electrons flowing in the -y. Establish a magnetic field in the -z direction.

Then since there is no E field established the Lorenz Force is given by

\begin{array}{l}<br /> F\quad = \quad q(v \otimes B) \\ <br /> {F_x}\quad = \quad q({v_y}{B_z} - {v_z}{B_y}) \\ <br /> {F_y}\quad = \quad q({v_z}{B_x} - {v_x}{B_z}) \\ <br /> {F_z}\quad = \quad q({v_x}{B_y} - {v_y}{B_x}) \\ <br /> \end{array}

For holes

\begin{array}{l}<br /> q = + e;\quad {B_z} = - B;\quad {B_y} = {B_x} = 0 \\ <br /> {v_y} = + v;\quad {v_x} = {v_z} = 0 \\ <br /> {F_x} = + e\{ ( + v)( - B) - 0\} = \quad - evB \\ <br /> {F_y} = + e\{ 0 - 0\} = \quad 0 \\ <br /> {F_y} = + e\{ 0 - 0\} = \quad 0 \\ <br /> \end{array}

For electrons

\begin{array}{l}<br /> q = - e;\quad {B_z} = - B;\quad {B_y} = {B_x} = 0 \\ <br /> {v_y} = - v;\quad {v_x} = {v_z} = 0 \\ <br /> {F_x} = - e\{ ( - v)( - B) - 0\} = \quad - evB \\ <br /> {F_y} = - e\{ 0 - 0\} = \quad 0 \\ <br /> {F_y} = - e\{ 0 - 0\} = \quad 0 \\ <br /> \end{array}

It can readily be seen that this model gives the same force acting on holes or electrons in the same direction. It can be also seen that this is because we have changed the sign of two quantities which are multiplied together viz velocity and charge.

 

[bjacoby]

@bjacoby

I am not disputing your observations, but your model does not explain the OP's original question which refers to the fact that holes and electrons subject to the Lorenz force in the Hall effect are deflected in the same direction.

This is why we can deduce the polarity of the majority carrier from the polarity of the Hall voltage. It depends whether holes or electrons accumulate on one particular side of the material.

I have no "model" here! The point is that in the Hall effect the force IS in the same direction given negative or positive carriers. Since the sign of the carriers is opposite the polarity of the Hall effect is reversed according to the charge of the current carriers.

You'd like a quick simple "model" to quickly explain this. There is none. As you yourself note there is no "classical" explanation. Trying to "explain" this as a gas or plasma theory simply won't work. There are no "gaseous" transistors. The explanation of why holes act like real positive charges comes out of the band theory of solids. That is why I included a quote from the classic Kittel book. I"m sorry but a quick hand-waving to "explain" the band theory of solids simply won't do.

 

[Studiot]

I"m sorry but a quick hand-waving to "explain" the band theory of solids simply won't do.

In fact you were so quick you missed the question entirely.

Perhaps if you took the time to read it properly you might be able to discuss it properly.

 

[bjacoby]

In fact you were so quick you missed the question entirely.

Perhaps if you took the time to read it properly you might be able to discuss it properly.

OK, What was the question of the OP? Let me get it exactly right: "...what exactly is a Hole? Could you explain it clearly?"

So you are saying that nobody me included explained what a "hole" was? That's simply not true. We all pointed out that a "hole" represents a missing valence electron in a material. And we noted that since the missing spot leaves a net positive charge as a negative electron is missing and since that missing spot can jump from atom to atom, the missing spot when viewed from a macroscopic viewpoint seems to be a positive charge that is moving around the material. So please explain how we "missed the question entirely"?

The problem is that asking for a clear explanation of something as complex as a "hole" requires details that can't easily be answered in a forum like this. Details such as "how can the absence of an electron, when obviously electric fields are ONLY acting on electrons to move them, act as if it were a real positive particle with real honest to goodness mass. We know all the positive charges are fixed. Or more relevant to this discussion the question was raised: How can a "hole" act as if it is a real positive particle in the Hall effect?

That answer is NOT so simple nor obvious. Apparently you don't accept that. So hey, I've got a minute. Could you give me a clear "proper" explanation of all cosmology and the nature of the universe in all it's aspects?

 

[Studiot]

If movement of holes were actually movement of negatively charged electron in the opposite direction

The crux of the OP's question lies here in his post#3 and again in the diagram in post#6, as confirmed by several subsequent posters who could not supply an explanation either.

My summary of the question is

If the Lorenz force deflects both negative electrons and positive holes in the same direction, and movement of positive holes is really movement of electrons in the opposite direction,

What moves the electrons that cause the hole movement?

And I added an allied thought.

Which electrons move in the direction opposite to the Lorenz Force?

 

[I_am_learning]

OK, What was the question of the OP? Let me get it exactly right: "...what exactly is a Hole? Could you explain it clearly?"

So you are saying that nobody me included explained what a "hole" was? That's simply not
true.

Your arguments are certainly legal. I am sorry, if that was a mistake, for not including what I intended to actually ask, in the very first post, what some of you have been calling the OP.
It is indeed a vague question what I asked in post #1.
But What I supposed was that everybody will follow the later posts, especially post #3 and Post #6 and then recognize what actually I wanted.
I certainly don't expect full account of the question of post #1 but what I wan't is the answer to post #1 that is sufficient to explain post #3 / #6.

 

[I_am_learning]

The crux of the OP's question lies here in his post#3 and again in the diagram in post#6, as confirmed by several subsequent posters who could not supply an explanation either.

My summary of the question is

If the Lorenz force deflects both negative electrons and positive holes in the same direction, and movement of positive holes is really movement of electrons in the opposite direction,

What moves the electrons that cause the hole movement?

And I added an allied thought.

Which electrons move in the direction opposite to the Lorenz Force?

Its pleasure to have at least you to have understood my problem. Since I am long tired asking and searching the answer for this question, Please do me a favor. If you get the answer (either in this place or elsewhere) please do inform me at therajendraadhikrai@gmail.com.

 

[DrDu]

Dear thecritic,
I actually thought to have provided an answer to your question, but I will try to answert the summary of your questions you posed above again:
"What moves the electrons that cause the hole movement?"
On a microscopic level all electrons in the lattice respond in the same way to the electromagnetic forces, including the Lorenz force, however, the forces due to the electric field of the lattice will be magnitudes of order stronger than the external fields. In the absence of electric fields, the electrons will be in so-called Bloch states which are superpositions of states corresponding to electrons moving in opposite directions. The question is now how these Bloch states will change under the influence of an external field. It can be shown that an attempt to increase the momentum of the electrons may result in an increased rate of reflection from the lattice which may look as a decrease rather than an increase of the effective velocity of the Bloch states on scales larger than the lattice spacing. This will be the case for electrons sufficiently near to the upper edge of a band. It is these electrons whose absence is most naturally described in terms of holes with positive charge and mass.
The situation is similar to the motion of photons in a medium described by an effective index of refraction. This description will also only be valid on a scale much larger than the spacing of the atoms in the material, while on a microscopic scale, the motion between the scatterings on the lattice will still be described by the Maxwell equations in vacuum. Incidentally, the analogy is much closer than this. To the electronic states with negative mass correspond materials with a negative index of refraction, which have been a hot topic in research ultimately. In these materials, basically the derivative of the wave number dependent dielectric constant with respect to k becomes negative \partial \epsilon(k,\omega)/\partial k&lt;0 while in the electronic states giving rise to a hole, the derivative of energy with respect to k is negative.

 

[Per Oni]

however, the forces due to the electric field of the lattice will be magnitudes of order stronger than the external fields.

This remark might directly explain the problem in a classical way. (It’s therefore not meant to invalidate anything you subsequently said!)

My idea is this: the Lorenz force causes slightly more electrons to flow along the left of the conductor (ref post # 6). This would mean that more “holes” are formed at the left side, where now the probability of an electron knocking other electrons is higher. This way the fields of the static +ve holes at the left will dominate the Hall field generated by the electron current.