# Electron hole

An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics and chemistry. The concept describes the lack of an electron. It is different from the positron, which is the antimatter duplicate of the electron.

The electron hole was introduced into calculations for the following two situations:

## Solid state physics

File:ElectronHole.JPG
When an electron leaves a helium atom, it leaves an electron hole in its place. This causes the helium atom to become positively charged.

In solid state physics, an electron hole (usually referred to simply as a hole) is the absence of an electron from the otherwise full valence band. A full (or nearly full) valence band is present in semiconductors and insulators. The concept of a hole is essentially a simple way to analyze the electronic transitions within the valence band.

Hole conduction can be explained by the following analogy. Imagine a row of people seated in an auditorium, where there are no spare chairs. Someone in the middle of the row wants to leave, so he jumps over the back of the seat into an empty row, and walks out. The empty row is analogous to the conduction band, and the person walking out is analogous to a free electron.

Now imagine someone else comes along and wants to sit down. The empty row has a poor view; so he does not want to sit there. Instead, a person in the crowded row moves into the empty seat the first person left behind. The empty seat moves one spot closer to the edge and the person waiting to sit down. The next person follows, and the next, et cetera. One could say that the empty seat moves towards the edge of the row. Once the empty seat reaches the edge, the new person can sit down.

In the process everyone in the row has moved along. If those people were negatively charged (like electrons), this movement would constitute conduction. If the seats themselves were positively charged, then only the vacant seat would be positive. This is a very simple model of how hole conduction works.

In reality, due to the crystal structure properties, the hole is actually not localized to a single position as described in the previous example. Rather, the hole is delocalized and spans an area in the crystal lattice covering many hundreds of unit cells. This is equivalent to the idea that we cannot tell which broken bond corresponds to the 'missing' electron, and is supported by uncertainty theorems from quantum mechanics.

Instead of analyzing the movement of an empty state in the valence band as the movement of billions of separate electrons, physicists propose a single imaginary particle called a "hole". In an applied electric field, all the electrons move one way, so the hole moves the other way. If a hole associates itself with a neutral atom, that atom loses an electron and becomes positive. The physicists therefore say that the hole must have positive charge—in fact, they assign a charge of +e, precisely the opposite of the electron charge.

Using Coulomb's law, we can calculate the force on the "hole" due to an electric field. Physicists then propose an effective mass which will relate the (imaginary) force on the (imaginary) hole to the acceleration of that hole. In some semiconductors, such as silicon, effective mass is dependent on direction (anisotropic), however a value averaged over all directions can be used for some macroscopic calculations.

In most of the semiconductors, the effective mass of a hole is larger than that of an electron. This results in less mobility for holes under the influence of an electric field and this may slow down the speed of the electronic device made of that semiconductor. This is one major reason for adopting electrons as the primary charge carriers, whenever possible in semiconductor devices instead of holes.

## Holes in quantum chemistry

An alternate meaning for the term electron hole is used in computational chemistry. In coupled cluster methods, the ground (or lowest energy) state of a molecule is interpreted as the "vacuum state"—conceptually, in this state there are no electrons. In this scheme, the absence of an electron from a normally-filled state is called a "hole" and is treated as a particle, and the presence of an electron in a normally-empty state is simply called an "electron". This terminology is almost identical to that used in solid-state physics.