Semiconductor Equations 

The semiconductor equations that are relating these variables are shown below:

Carrier density:

n = n_i exp (E_FN - E_i / KT)        (1)

p = ni exp (E_i - E_FP / KT)        (2)

In which E_FN is the electron quasi Fermi level and E_FP is the hole quasi Fermi level. These above 2 equations lead to 

Np = n­_i² exp (E_FN - E_FP/ KT)   (3)

In equilibrium E_FN = E_FP = Constant

Current:

There are two mechanism of current; electron current density and hole current density. There are various mechanisms of current flow:

1. Drift
2. Diffusion
3. Thermionic emission
4. Tunnelling

The final two mechanisms are significant frequently only at the interface of two different materials like a metal-semiconductor junction or a semiconductor-semiconductor junction where the two semiconductors are of dissimilar materials. Tunneling is as well significant in the case of PN junctions in which both sides are heavily doped.

The dominant conduction mechanisms include drift and diffusion in the bulk of semiconductor. The current densities because of these two mechanisms can be written as

J_N = qnμ_Nε + qD_N dn/dx   (4)

J_P = qnμ_Pε + qD_P dP/dx   (5)

In which μ_N and μ_P are electron and hole mobilities correspondingly and D_N, D_P are their diffusion constants.

Potential:

The potential and electric field in a semiconductor can be described in the following ways:

1. Ψ = - E_C /q + constant ; ε =  (1/q) (dE_c / dX)
2. Ψ = - E_V /q + constant ; ε =  (1/q) (dE_V / dX)
3. Ψ = - E_i /q + constant ; ε =  (1/q) (dE_i / dX)
4. Ψ = - E_O /q + constant ; ε =  (1/q) (dE_O / dX)

All these definitions are equal and one or the other may be selected on the basis of convenience. The potential is connected to the carrier densities through the Poisson equation: -

∂² Ψ / ∂X² = - q/ε (p-n+ N⁺_D - N^-_A)      (6)

In which the last two terms present the ionized donor and acceptor density.