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.

At room temperature, all elements of the column IVA are non-conductors, but they become capable of ionization (promotion of electrons to the conduction band) as temperature increases.

How does an electron get enough energy to emancipate itself? One way is to collide with each other and escape from the valence band together. But the more important way is to absorb or emit a phonon, another name for the particle associated with the quantized lattice vibration energies. A lattice can vibrate by oscillating bond length and angle. An electron can absorb these vibrational kinetic energy and jump to the conduction band.

At a particular temperature $T$, the number of electrons in at energy $E$ is

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

where $P(T,E)$ is the probability of an electron having an energy $E$ at temperature $T$, and $S(E)$ is the number density of the available states at energy $E$. In case of a semiconductor, the valence band states are completely filled, so the electrons are in a degenerate condiciton. In such a condiction, the aforementioned probability is given by the famous Fermi-Dirac distribution:

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

where $E_F$ is the Fermi energy, the energy at which the probability of finding an electron in a permitted state is 1/2. Fermi-Dirac reduces to Maxwell-Boltzmann at high temperatures ($T\gg 0$). At zero temperature, however,

$$ P(E) = \begin{cases} 1, & \text{if } E \lt E_F \\ 0, & \text{if } E \gt E_F \end{cases}$$

which is shown in Panel $b$ of the diagram below.

The Panel $a$ shows the bands of Si at absolute zero where $E_F$ (dashed line) is exactly in the middle of the band gap. The solid black line of Panel $b$ makes it clear that at $T=0$ K, the probability of finding electrons at $E\lt E_F$ is always 1, meaning all the permitted states are filled. At higher temperatures, the probability distribution follows the green or blue line.

Panel $c$ shows $S(E)$ for the electrons in the valence and conduction bands of Si. This function is almost quadratic near the edges of the bands, but vanishes in the gap. Finally, Panel $d$ shows the number density of electrons $n_e$ and also the number density of holes

$$ n_h = [1-P(E)] S(E) $$

at a non-zero temperature and we see that $n_e$ peaks near the edge of the valence band, but $n_h$ peaks inside the valence band just as $n_e$ peaks inside the conduction band. The total number of charge carriers can be calculated as

$$ N_n = \int_{E_F}^\infty n_e dE $$

$$ N_p = \int_{-infty}^{E_F} n_h dE. $$

In intrinsic semiconductors the positive and negative charge carriers are equal in number in equilibrium; $N_p=N_n$. The number depends on temperature as

$$ N_p = N_n = AT^3 e^{-E_G/kT}. $$

Periodic table of chemical elements near column IVA:

These elements are used as semiconductors either separately or in combination with each other. For example, the elements in column IVA create diamond lattice structures where each atom has 4 covalent bonds and 8 electrons in the outer shell. Each covalent bond is created by 2 electrons.

Elements on either side of the IVA column can create binary compounds with the IVA elements. The binary compounds also have stable covalent bonds and create a zinc blend structure. Examples include GaAs, CdTe. There are also ternary and quarternary compounds some of which are listed in the table below.

The table below shows some common semiconductors where $\lambda_c=(hc/E_G)$ is the cutoff wavelength at room temperature. The resistivity or conductivity depend critically on temperature and band gap. The resistivity of semiconductors is in the range $10^{-3}$ to $10^9$ $\Omega$ m which is between that of conductors ($\sim 10^{-6}$ $\Omega$ m) and insulators ($\gt 10^{14}$ $\Omega$ m).

As seen above, the diamond allotrope of carbon is an insulator because its band gap is huge (5.48 eV). The allotropes graphite and carbon nanostructures are conductors.

So far we have talked about the emancipation of electrons via lattice vibration and collision, but an a photon with a wavelength larger than the curoff wavelength can also emancipate an electron. The cutoff wavelength

$$ \lambda_c = \frac{hc}{E_G} = \frac{1.24 \ \mu\text{m}}{E_G \text{ eV}}. $$

For Si, the value of 1.1 $\mu$m. In astronomy we use semiconductors as photoabsorbers. A simple example is shown below.

The photon stream promotes electrons to the conduction band leaving behind an equal number of holes in the valence band. This is a basic detector or sensor or receiver that converts energy into matter, photons into electrons. The greater the stream the higher the conductivity of the detector. If the voltage across the semiconductor is constant, the electrical current $i$ through the resistor $R_L$ would depend on the number of photons absorbed per second.

So the voltage measured at $V_o$ will be directly related to the intensity of light.