What is the concept of hole in semiconductor physics?

Question

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?

Answer 1

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

Answer 2

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, except at the very "top" (highest energy states) where some vacant electron states may exist. Electrons near the top of the valence band 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. Moreover, each vacant state (or "bubble") gets dragged along with nearby electrons, and moves precisely as if it were occupied by an electron, i.e. like a positive charge. The common simplified explanation (like the analogy in A. I. Breveleri's answer) gets this wrong when it claims the electrons move in one direction and the "bubbles" move in the opposite direction.

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 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:

$$\vec J_\text{occupied}+\vec J_\text{vacant}=0$$ $$\vec J_\text{occupied} = -\vec J_\text{vacant}.$$ This says that 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.

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.

Answer 3

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.