$$n=N_c\cdot e^{\frac{-(E_c-E_F)}{kT}}$$
$$p=N_v\cdot e^{\frac{-(E_F-E_v)}{kT}}$$
- $E_c$ is the position of the conduction band minimum
- $E_v$ is the position of the valence band maxmimum
- $k$ is Boltzmann's constant
- $N_x$ are the effective density of states

$$np=n_i^2$$
- $n_i$ = Intrinsic carrier concentration

$$n_i=\sqrt{N_cN_v}e^{\frac{-E_g}{2kt}}$$
- $E_g$ = Band Gap = $E_c-E_v$

## Substitutional Doping

- Donated electrons are delocalised
- Ions are immobile

$$N_c \equiv 2 \left[ \frac{2\pi m_nkT}{h^2}\right]^{3/2}$$
$$N_v \equiv 2 \left[ \frac{2\pi m_pkT}{h^2}\right]^{3/2}$$