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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?

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    $\begingroup$ 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. $\endgroup$
    – Roger V.
    Mar 13, 2023 at 8:18
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    $\begingroup$ 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. $\endgroup$
    – march
    Mar 13, 2023 at 15:45
  • $\begingroup$ Do you know what the effective mass is, and how it is related to the curvature of the dispersion relation E vs k? $\endgroup$
    – Peltio
    Mar 13, 2023 at 22:47
  • $\begingroup$ @Peltio ,,yes I do $\endgroup$
    – amin
    Mar 14, 2023 at 14:08

3 Answers 3

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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

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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.

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  • $\begingroup$ 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 ? $\endgroup$
    – amin
    Mar 13, 2023 at 9:21
  • $\begingroup$ 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 $m_\text{eff} < 0$. $\endgroup$
    – Puk
    Mar 13, 2023 at 9:35
  • $\begingroup$ 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 ? $\endgroup$
    – amin
    Mar 13, 2023 at 9:45
  • $\begingroup$ 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 ? $\endgroup$
    – amin
    Mar 13, 2023 at 9:57
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    $\begingroup$ 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. $\endgroup$
    – Puk
    Mar 13, 2023 at 10:17
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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.

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    $\begingroup$ 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. $\endgroup$
    – Arthur
    Mar 14, 2023 at 14:56
  • $\begingroup$ This answer is basically the same as my answer to an older question. $\endgroup$
    – DrSheldon
    Mar 16, 2023 at 17:29

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