A completely filled band carries no electric current:
We distinguish three different band filling types: filled, empty and partially filled.
Let us start with the most extreme cases - filled and empty bands.
We treat these two cases together, because a completely filled band is very similar to a completely empty band.
For example, both filled and empty bands lead to zero current:
$$
\begin{align}
j = 2e \frac{1}{2\pi} \int_{-\pi/a}^{\pi/a} v(k) dk = 2e \frac{1}{2\pi \hbar} \int_{-\pi/a}^{\pi/a} \frac{dE}{dk} \times dk = \\
2e \frac{1}{2\pi \hbar} [E(-\pi/a) - E(\pi/a)] = 0
2e \frac{1}{2\pi \hbar} [E(-\pi/a) - E(\pi/a)] = 0.
\end{align}
$$
An empty band has no electrons and thus no current.
On the other hand, a filled band has an equal number of electrons going forwards and backwards which thus cancel and lead to zero current.
Similar results apply to many other physical quantities such as heat capacity and magnetisation.
Therefore, filled and empty bands do not affect most physical properties and can be disregarded.
As a result, rather than to consider thousands of bands that a material contains, we neglect most of them and just focus on partially filled bands around Fermi level.
## From electrons to holes
A completely filled band $\approx$ completely empty band. The idea of introducing **holes** is to transform an *almost* completely filled band $\Rightarrow$ almost empty one. Instead of describing a lot of electrons that are present, we focus on those that are absent.
Definition: a **hole** is a state of a completely filled band with one particle missing.
In order to understand partial filling, we begin with a simple analogy.
Let's say we have 100 chairs: 99 are occupied and 1 is empty.
To keep track which chair is occupied/empty, we could write down the occupation of each and every chair.
However, that would require a lot of unnecessary writing!
Instead, we only need to write down which chair is empty, because we know the rest wil be occupied.
The same philosophy is applied to band filling.
Instead of describing a lot of electrons that are present in an almost filled band, we focus on those that are absent.
The absence of an electron is called a **hole**: a state of a completely filled band with one particle missing.
In this schematic we can either say that 8×2 electron states are occupied (the system has 8×2 electrons counting spin), or 10×2 hole states are occupied. A useful analogy to remember: glass half-full or glass half-empty.
In this schematic we can either say that 8×2 electron states are occupied (the system has 8×2 electrons counting spin), or 10×2 hole states are occupied.
A useful analogy to remember: glass half-full or glass half-empty.
Electron and hole pictures correspond to two different, but equivalent ways of describing the occupation of a band.
In practice, we choose to deal with electrons whenever a band is almost empty and holes when a band is almost full.
## Properties of holes
The probability for an electron state to be occupied in equilibrium is $f(E)$:
Let us compare the properties of electrons and holes.
The probability for an electron state to be occupied in equilibrium is given by $f(E)$:
$$f(E) = \frac{1}{e^{(E-E_F)/kT} + 1}$$
$$f(E) = \frac{1}{e^{(E-E_F)/kT} + 1}.$$
The probability for a hole state to be occupied:
On the other hand, since a hole is a missing electron, the probability for a hole state to be occupied is
therefore for holes both energy $E_h= -E$ and $E_{F,h} = -E_F$.
The **momentum** $p_h$ of a hole should give the correct total momentum of a partially filled band if one sums momenta of all holes. Therefore $p_h = -\hbar k$, where $k$ is the wave vector of the electron.
The **momentum** $p_h$ of a hole should give the correct total momentum of a partially filled band if one sums momenta of all holes.
Therefore $p_h = -\hbar k$, where $k$ is the wave vector of the electron.
Similarly, the total **charge** should be the same regardless of whether we count electrons or holes, so holes have a positive charge $+e$ (electrons have $-e$).
On the other hand, the velocity of a hole is **the same**:
Finally, we derive the hole effective mass from the equations of motion:
$$m_h \frac{d v}{d t} = +e (E + v\times B)$$
$$m_h \frac{d v}{d t} = +e (E + v\times B).$$
Comparing with
$$m_e \frac{d v}{d t} = -e (E + v\times B)$$
$$m_e \frac{d v}{d t} = -e (E + v\times B),$$
we get $m_h = -m_e$.
## Semiconductors: materials with two bands.
In semiconductors the Fermi level is between two bands. The unoccupied band is the **conduction band**, the occupied one is the **valence band**. In the conduction band the **charge carriers** (particles carrying electric current) are electrons, in the valence band they are holes.
In semiconductors the Fermi level is between two bands.
The unoccupied band is the **conduction band**, the occupied one is the **valence band**.
In the conduction band the **charge carriers** (particles carrying electric current) are electrons, in the valence band they are holes.
We can control the position of the Fermi level (or create additional excitations) making semiconductors conduct when needed.
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@@ -133,8 +156,13 @@ Therefore we can approximate the dispersion relation of both bands as parabolic.
