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

[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

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

For holes

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

For electrons

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

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 $$\partial \epsilon(k,\omega)/\partial k<0$$ 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.

 

[I_am_learning]

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.

Can I understand what you said like this:
Considering figure post #6, right; Yes the current moves upwards, so due to Lorentz force/field more electrons flows in the left hand side than on the right. However for the more electrons to flow there must be more available holes (since in p-type materials the only way electrons move is my jumping from one hole to the other). So, in addition to more electrons on the left side there is also more holes.
But due to some * reasons, the holes dominate and hence we get the positive polarity!

If this summary is incorrect, you can entirely skip, what I have written below!

However one question has struck to me. Does the existence of hole makes the region spontaneously +ve? I mean to ask, suppose there are two blocks of silicon wafer. One is heavily doped with holes. One isn't doped. Is there a Voltage Deference between them? I guess no, because both are electrically neutral, although the p-type wafer has tendency to accept electrons in expense of its neutrality!
So, I think I got the answer, a Voltage develops after sometime if we connect them with a wire but not initially, is this correct?
If that's correct, then in above scenario, how can holes dominate and give positive polarity? Having holes in a region doesn't spontaneously makes that region +ve!

 

[Per Oni]

thecritic.
You are giving me a real hard time here.

The way I see it is that the slightly increased +ve lattice points at the left are fixed points and can’t go nowhere. Therefore it’s +ve field will stay locally. Conducting electrons on the other hand are infinitely more moveable and will spread their fields through the whole material, left as well as right.

This answer will perhaps still not satisfy you but you must remember that my classical explanation must at some point fail. It’s in a way the same as wondering “why doesn’t an electron slam into the nucleus” In the end the quantum people have the answers.

 

[I_am_learning]

Well, if the correct explanation (which won't evoke any contradictions) can't be provided based on the classical picture, I have no way but to go for Quantum Mechanics.
By the way, I am not too new at Quantum Mechanics (Know as much as the solution of Wave functions for hydrogen atom; but no further).
So, anyone interested at providing Quantum explanation is welcome! After all what I need is an answer that could explain what I have been question for so long!

 

[Studiot]

Here is a quantum explanation that sets out to avoid using holes (although they authors do mention them).

They also provide a basic explanation of Dr Du's comments about negative mass.

http://www.journal.lapen.org.mx/jan09/LAJPE%20225%20Lianxi%20Ma%20Preprint%20f.pdf

 

[I_am_learning]

http://www.journal.lapen.org.mx/jan09/LAJPE%20225%20Lianxi%20Ma%20Preprint%20f.pdf

Thanks for that, I hope that's what I had searching for.
Though I currently have no time to go through the details, one line struct me-

We discuss the details about two situations and point out that both quantum and classical mechanics give same current direction under external electric field. However, under the influence of external magnetic field, because the mass of electrons is negative at valence band, electrons move to the opposite direction of its Lorentz force, which behave like a positive charge and give positive RH.

So, the explanation in bulk is I think about showing that the electrons behave as having +ve Effective Mass for the electric Field but -ve Effective Mass for the Magnetic Field. Thats amazing! I will look at it this afternoon!
Thanks again.

 

[DrDu]

This is exactly my argumentation of post #24, only that I was arguing in terms of the electrons to be removed while in the article they consider the electrons that remain.
The effective mass is negative in both point of views for the reaction to both the electric and the magnetic field, but, if you argue in terms of the electrons that remain, dk/dt and the average k in the stationary state have different directions, here the relevant quote from the article:
"When electric field E is introduced, Eq. (5)
shows that k decreases (dk/dt is negative). But electrons
going out of -k border come back from +k border [3], and
so there are more electrons with +k, which means that
there are more electrons with negative velocity (opposite
to the electric field E)."

 

[I_am_learning]

This is exactly my argumentation of post #24,

O.K, I apologize for not getting you back then.

The effective mass is negative in both point of views for the reaction to both the electric and the magnetic field, but, if you argue in terms of the electrons that remain, dk/dt and the average k in the stationary state have different directions, here the relevant quote from the article:
"When electric field E is introduced, Eq. (5)
shows that k decreases (dk/dt is negative). But electrons
going out of -k border come back from +k border [3], and
so there are more electrons with +k, which means that
there are more electrons with negative velocity (opposite
to the electric field E)."

I followed the whole text but I am no further in position than in my previous post.
Is it that the effective Mass of electrons for electric and magnetic is different because they act along different directions??
Otherwise I have even read that there is not a absolute distinction between electric and magnetic field, what appears as electric field may appear as magnetic field for some other frame of reference!

 

[DrDu]

I again thought about the problem and now I am convinced, that most of the mind boggling effects are due to the utilization of the reduced zone scheme. In an extended zone scheme, we are talking about electrons (or holes) living in a region where the velocity decreases with increasing momentum, but with force and momentum being always parallel. This decrease of velocity is due to increased scattering of the electrons from the lattice with increasing momentum. As a classical analog, think of pulling a parachute behind you. As long as you pull weakly, the parachute is closed and its velocity does increase the stronger you pull. But then there is a region where the parachute starts to open and it gets slower with increasing force and momentum due to its increasing air resistance. A magnetic field would only change the direction of the momentum, but not its absolute value, so the velocity would increase normally with momentum in case of the parachute for a force tangential to the velocity.
This would only describe a differential hall effect. It becomes a true Hall effect if we go over to the reduced zone scheme.

I would like to predict a new kind of Hall effect ( which probably has been observed and described one hundred years ago): Take a light bulb. The current is known to be higher at low voltages when the filament is cold than at higher voltages when the filament is hot. So there is a region where the current gets smaller with increasing temperature. However, this should have no effect on the reaction of the charge carriers to an applied magnetic field. So there is some kind of a negative Hall effect, at least a differential Hall effect, in that system. Obviously, the true Hall effect is defined under isothermal conditions.

 

[Per Oni]

I followed the whole text but I am no further in position than in my previous post.
Is it that the effective Mass of electrons for electric and magnetic is different because they act along different directions??
Otherwise I have even read that there is not a absolute distinction between electric and magnetic field, what appears as electric field may appear as magnetic field for some other frame of reference!

I read the article in post #54 as well and now in addition to thecritc’s valid point I want to add another equally valid objection.

Take the quote from top left of the last page:

Rather, they accelerate toward the direction opposite to the Lorentz force!

I explain my objection:
With the strip of material on right side of the picture in post #6 in mind, imagine underneath this strip of +ve doped material run 2 fixed parallel conducting rails in the x-direction. (Current is in the y and magnetic field in the z direction as shown). Say the distance between these rails is L. The set up is such that the rails and strip are in good electrical contact but the strip can slide, say over a small film of mercury.

Now I can take any material I like to replace the strip but the force F on the strip is always BIL and the direction of this force is always to the left (for the picture shown). So if I make F big enough the strip starts to slide to the left and not right. Therefore the assertion made in this article that the Lorentz force is now in the opposite direction is false!

 

[Per Oni]

Therefore the assertion made in this article that the Lorentz force is now in the opposite direction is false!

Sorry folks, I have clouded this issue. After reading the article again a bit more careful this time I realize I should not have made that claim. Their Lorenz is in the correct direction although the electrons end up on the opposite site they would normally end up. It ‘s very confusing but I still support thecritic in his claims.

 

[Studiot]

I would just like to point out the the Left Hand Rule is predicated upon the use of the direction of conventional current. It is the motor rule and will give the correct direction for force or motion only when the direction of the second finger is aligned with the direction of conventional current.

The Right Hand Rule is the generator rule and gives correct (voltage) polarity, regardless of charge carriers because it defines a voltage, not a current.

When considering the vector cross products the right hand rule should be used in any case.