Silicon Crystal Lattice

In a silicon crystal lattice, each silicon atom forms 4 covalent bonds with 4 other silicon atoms. At low temperatures, there are no free electrons, making the silicon an insulator. At higher temperatures, the increase in energy causes covalent bonds to break, resulting in negatively charged free electrons and positively charged holes. Holes and electrons are capable of movement, which makes them capable of conducting current. The properties of a silicon crystal can be modified through the process of doping.

Recombination

Recombination is when electrons “fill” holes back in. The recombination rate is proportional to the number of carriers, which depends on thermal generation, which strongly depends on temperature.

Thermal Equilibrium

Thermal equilibrium is the state where the recombination rate is equal to the generation rate. In thermal equilibrium, $n=p=n_i$.
$n$ - electron concentration
$p$ - hole concentration.
$n_i$ - intrinsic carrier concentration

Intrinsic Carrier Concentration

The intrinsic carrier concentration is the number of electrons or holes per ${\text{cm}}^3$ of the intrinsic material.
$n_i=B{T}^{3/2}{e}^{-E_g/2kT}\cong 1.5\times 10^{10}c{m}^{-3}$
$pn=n_i^2$
$B$ - material parameter ($7.3\times 10^{15}c{m}^{-3}{K}^{-3/2}$ for Si)
$T$ - temperature in kelvin
$E_g$ - bandgap energy ($1.12eV$ for Si), or minimum energy to break covalent bond and generate electron hole pair
$k$ - Boltzmann’s constant ($8.62\times 10^{-5}eV\cdot K^{-1}$)