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What is a hole? And how should we describe it to study it properly?

Many textbooks refer to it as an empty state that carries a positive charge, but how can an empty state carry a positive charge? And other textbooks refer to it an physical particle with positive charge and positive effective mass, but how do they just consider it like this?

And why do we just calculate the current of the moving electrons? Why is there the concept of a hole?

I'm so confused about it. What is really a hole and how should we describe it?

Peter Mortensen
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amin
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    why we just calculate the current of the moving electrons - this is not correct. If we use hole description, we also calculate the hole current. – Roger V. Mar 13 '23 at 08:18
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    As much as the solid state text by Kittel is difficult to learn from, his discussion of holes in semiconductors, how to think about them, and why they can be treated as having positive charge is excellent. – march Mar 13 '23 at 15:45
  • Do you know what the effective mass is, and how it is related to the curvature of the dispersion relation E vs k? – Peltio Mar 13 '23 at 22:47
  • @Peltio ,,yes I do – amin Mar 14 '23 at 14:08

3 Answers3

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Properly holes are introduced as quasiparticles, i.e., poles in the Green's function. In this sense they are no different from electrons in a semiconductors/insulators, which are not real electrons, but also quasiparticles - with dispersion relation determined by the crystal band structure and the complex interactions with other electrons and the lattice. Thus, electrons are the excitations above the Fermi level, while the holes are below.

Simple hand-waving description of a hole is as a vacancy in the valence band filled with electrons - which for practical purposes behaves as a particle. A close analogy is a bubble of gas in a sparkling drink - it is really an (nearly) empty space moving in the liquid, but we do speak of it as a particle (bubble) rather than about liquid moving into an empty space.

Related:
Why do Drude/Sommerfeld models even work?
Vacuum state in particle hole symmetric Hamiltonian
Do holes have wavefunctions?
Electrons and holes vs. Electrons and positrons

Roger V.
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A thorough understanding of holes, why they are useful and why classical analogies fail requires a rudimentary understanding of energy band structures in solids. The classical "bubble" picture alone, while it might be useful for introducing the idea of a hole, fails to explain the positive Hall or Seebeck coefficients of p-type semiconductors (and some metals).

Semiconductors (and some metals) have a valence band that is mostly filled. Electrons in this band often have the peculiar property that when a force acts upon them, they accelerate in the opposite direction: they have a negative effective mass. Since the forces due to electric and magnetic fields are proportional to charge, valence band electrons thus respond to forces as if they were positively charged.

Suppose the current density due to the valence band electrons is $\vec J_\text{occupied}$. Let's define $\vec J_\text{vacant}$ as the current density that the vacant valence band electron states would yield if they were occupied. When the valence band is completely filled, for each electron moving in one direction, there is another moving in the opposite direction, so the net current is zero:

$$\vec J_\text{occupied}+\vec J_\text{vacant}=0$$ $$\vec J_\text{occupied} = -\vec J_\text{vacant}.$$ The valence band current is the negative of the current that would result from electrons if they occupied the vacant states. We can thus regard this current as being due to positively charged particles occupying the states free of electrons. Since there are much fewer vacant states in the valence band, they are so much easier to keep track of in calculations. Finally, the vacant states respond to forces (i.e. accelerate) precisely as occupied states do: like positive charges, as mentioned above in the second paragraph.

In summary, vacant states in the valence band move as if they were positively charged particles, and contribute to current as if they were positively charged particles, and consequently it is convenient to attribute the dynamics and the current of the valence band to fictitious positively charged particles.

Puk
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  • but the electrons in the valence band behave as positive charged , then $J_\text{occupied}$ will be with the direction of electric field , the the current density of vacant will be opposite to the field ? can you clarify why did you introduce the negative effective mass theory and how it will help the problem ? – amin Mar 13 '23 at 09:21
  • If the vacant states were occupied by electrons, then yes, their current density would oppose the E-field, precisely because of the negative $m_\text{eff}$ of the electrons. But the actual current is $\vec J_\text{occupied}$, which is in the E-field direction. The negative $m_\text{eff}$ is key to understanding phenomena like the Hall and thermoelectric effects. In the classical "bubble" picture, each bubble (the lack of an electron) accelerates in the same direction as the force on a negative charge. Holes however move like positive charges, because electrons have $m_\text{eff} < 0$. – Puk Mar 13 '23 at 09:35
  • but if the electron which have a negative effective mass move with the direction of the field it behaves as a positive charged particle then how it will produce J Opposite to the field ? – amin Mar 13 '23 at 09:45
  • i will sum up what I actually know and could you try to clear my confustion?

    an electron on the top of the valence band has a negative effective mass so if an electric field is applied the electron will accelerate in the direction of the electric field behaving like a positive charged particle then the current density due to this moving electron will be $J=|e|.v$ where $J$ and $v$ is in the direction of the electric filed

    ,,according to this information how the concept of the hole is produced ?

     – amin Mar 13 '23 at 09:57
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    The description of the collective motion of the valence band electrons is difficult, so we look at the vacant states at the top of the band. An electron in such a state moves like a positive charge, but it is a negative charge so its current (and hence $\vec J_\text{vacant}$) opposes the E-field. I think maybe this is where your confusion is. Since the actual current is $J_\text{occupied}=-\vec J_\text{vacant}$, we think of the current as being due to the electron vacancies, which move like positive charges and (unlike electrons) also contribute to current like positive charges. – Puk Mar 13 '23 at 10:17
  • my confusion is as you said ,I was thinking if an electron moves like a positive charge it's current will be just like the positive charge in the direction of the field – amin Mar 13 '23 at 10:22
  • But did the $J$ occupied when there is just thermal energy equal to zero or has some value? No electric field is supplied to the material – amin Apr 01 '23 at 14:22
  • @amin With no E-field applied, there is nothing driving electrons to move one way or another, so $\vec J_\text{occupied}=0$ at steady-state. – Puk May 05 '23 at 20:12
  • can you elaborate how the vacant states respond to the forces because I can't grasp it ? I know that electrons in valence band respond forces according to it's effective mass,but how the vacant states respond to it ? – amin May 13 '23 at 08:18
  • @amin Like the answer says, vacant states react to forces exactly as occupied states do: like electrons with negative $m^*$. I think the easiest way to visualize this is to think of the vacant state like a "bubble": as liquid in a tube gets pushed in one direction under the influence of some force, the bubble gets dragged along, moving just like the rest of the liquid. This happens because in a short time step, the liquid just ahead of the bubble moves forward to make space for it, and the fluid behind the bubble takes up its former place. For more details, see e.g. Ashcroft & Mermin p. 226. – Puk May 13 '23 at 09:20
  • I visualize the band like this: a plenty of electrons are in the valence band when electric field applied all these electrons moves in the direction of the field ( because of negative effective mass) and the empty states moves opposite to the electrons ,so it seems that holes move opposite to the field ,but that isn't the case as you mentioned it respond to forces as electron does. Am I visualizing it incorrectly due to mixing picture of electrons and holes at the same time in the same band ? And if I want to talk about holes I must ignore the electrons? – amin May 13 '23 at 21:18
  • @amin Your description is fine if you just amend it to realize the empty states (holes) move just like the valence band electrons, not in the opposite direction (think of them like as a bubble being dragged along in the direction of the current). This means the holes also move in the direction of the field, just like a positive charge with positive mass would, and this is why we think of holes as positive charges. – Puk May 13 '23 at 21:27
  • But the main reason we think of the hole as positive charge is the reason you wrote in your answer, right? Which is the $J$occupied =$-J$ vacant, so we think of it as due to positive charges,is that right? Because that what I read in Ashcroft solid state physics – amin May 14 '23 at 22:54
  • @amin There are two reasons: (1) it moves like a positive charge (what I said in my previous comment), and (2) it contributes to current like a positive charge (what you said). I mentioned both of these in my answer. – Puk May 15 '23 at 00:01
  • ,one last question,many people say that the hole is positive charged due to that the silicon atom loses an electron so it became positively charged and this positive charge belongs to the hole. But I don't think that a correct reason because the positive charge is due to the unbalanced charges between protons and electrons and when the atom loses the electron the positive charge due to the unbalanced proton , and the real reason that a hole is positively charged is what you wrote in your answer (which is exactly what I read in Ashcroft solid state physics) ,,Am I right about it ?? – amin May 20 '23 at 11:43
  • @amin I wouldn't say this is incorrect. It's just not complete because it doesn't consider the subtle transport properties of valence band vacancies that makes them behave like ordinary positive charges. However, it is yet another reason that the description of holes as positively charged particles is appropriate. A solid with a full valence band and an empty conduction band is neutral. If one atom is ionized and if the valence band has $N$ states, the valence band charge can be thought of as a $+qN$ charge due to nuclei and $-qN$ due to valence electrons. – Puk May 20 '23 at 17:05
  • @amin If one atom is ionized (one vacancy in the valence band), you can either say the charge in the valence band is $q[N-(N-1)]$, or attribute the net charge to a positively charged hole in an otherwise neutral band and say the charge is $q$. It doesn't matter how you count the charges. Each valence band vacancy (i.e. hole) contributes a positive charge, so there is no harm in thinking of them as positively charged, especially since holes do move and contribute to current like positive charges. – Puk May 20 '23 at 17:05
  • But the reason itself in thinking about them as positive charged particles is due to the transport properties of the valence band ,I mean we can consider it as due to positive charged particles filling the unoccupied states, that the reason I read in Ashcroft (the book you recommend) ,my question is some people say that is due to the unbalanced charges they attribute it that when atom is ionized it's charge goes to hole not the unbalanced proton ? – amin May 20 '23 at 18:30
  • I mean when atom is ionized it leaves an empty state (hole) we think of hole as positive charged particle because if we consider the current of this band we find it due to positive charges and equal to the number of unoccupied states so we think of it as being particles filling the unoccupied states and having positive charge, not because that just the atom is ionized so we attribute that positive charge to the empty state not to the unbalanced proton – amin May 20 '23 at 18:35
  • @amin I'm trying to tell you the transport properties aren't necessarily the only reason a valence band vacancy is thought of as a positive charge. Valence band vacancies are the ionized atoms carrying a net positive charge because of the charge imbalance you speak of, so this is a valid description. Even in the absence of a net current in the valence band, holes are still treated as positive charges when calculating the space charge density. If you are still confused, consider asking a separate question about this because comments are not suitable for carrying on such extended discussions. – Puk May 20 '23 at 18:50
  • My confusion is really about what you said ,that I consider hole due to the transport properties only ,not in the general case I will ask another question,hope you answer it – amin May 20 '23 at 19:33
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Did you ever solve one of these? The puzzle consists of fifteen tiles and a tile-sized hole. Each move involves moving an adjacent tile into the hole. Another way of looking at it is each move consists of moving the hole to an adjacent position.

Now, if we assign each position on the board a unit positive charge (attached, say, to the stationary backing), and each tile a unit negative charge, then the entire board has a charge of positive one.

This charge appears to be localized at the hole. So, when you slide a tile into the hole, you effectively move the location of the positive charge an equal distance in the opposite direction.

Note that although the apparent location of the positive charge has changed, none of the fixed positive charges have moved at all.

Anyone observing the board only when your fingers operating on the tiles will see that negative charges are actually moving. Anyone observing the board only between moves will see a positive charge moving.

A. I. Breveleri
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    Those puzzles are a visual analogy I hadn't thought of before, and I like it. It has all the inherent flaws of any other visual analogy (such as the ones mentioned in the first paragraph of Puk's answer), but I don't think a "real life" introductory visual analogy can really overcome those weaknesses anyways. – Arthur Mar 14 '23 at 14:56
  • This answer is basically the same as my answer to an older question. – DrSheldon Mar 16 '23 at 17:29