Boltzmann distribution: $$ \frac{n_i}{n_j} = \frac{g_i}{g_j} e^{(E_j-E_i) / (kT)} $$

where $g$ is the number of distinct quantum mechanical states at a specific energy level.

The conductivity of a substance depends on the gap and characteristics of the conduction and valence bands. Below you see the inner, valence and conduction energy bands of some hypothetical insulators, metals and semiconductors. The vertical axis represents energy where $E_G$ is the band gap. The horizontal axis represents position in the material.

In an insulator $E_G$ is large and, hence, electrons are forbidden to jump the band gap in order to go to the conduction band from the valence band. The band gap is large compared to the thermal energy $kT=0.026$ eV at 300 K. In metals, the two bands are literally attached to each other, so electrons in the valence band can easily move to the conduction band by absorbing even a small amount of energy.

According to Pauli’s exclusion principle, an electron can occupy a new state only if the state is permitted and unoccupied. In the figure the filled states are shown in green and the permitted but unoccupied states in blue.

Three different semiconductors $a$, $b$ and $c$ are shown where $a$ and $b$ are intrinsic semiconductors and $c$ is an extrinsic semiconductor. The material $a$ looks like an insulator except that it has a small band gap. This is the case for silicon at zero temperature. All states in the valence band are occupied and there are unoccupied permitted states in the conduction band where the electrons can go by absorbing some energy when global electric fields or local kinetic energies are introduced. In $b$ we see the same semiconductor at a higher temperature. Some electrons have jumped to the conduction band leaving behind holes in the valence band. The direction of the the flow of electrons is opposite to that of the holes which are like mobile positive charges.

The semiconductor $c$ has electrons in the conduction band without any corresponding holes in the valence band. These are extrinsic semiconductors widely used to make electronic devices. Another class of extrinsics exhibits holes in the valence band without any corresponding electrons in the conduction band.

$$ P(T,E) = \frac{1}{1+e^{(E-E_F) / (kT)}} $$

$$ n_e (T,E) = P(T,E) S(E) $$

where $S$ is the number density of available states at an energy $E$.