Semiconductor in Equilibrium

Equilibrium in semiconductors implies the following:

(i) Steady state: ∂Z / ∂t = 0

In which Z is any physical quantity like charge, voltage electric field etc.

(ii) No net thermal Currents and electrical current:

Because current can be carried via both electrons and holes, equilibrium involves zero values for both of the net electron current and net hole current. The drift and diffusion components of electron and hole currents require not be zero.

(iii) Constant Fermi energy:  dE_F / dX = 0

The only one equations that are relevant (another being zero!) for analysis in equilibrium are:

Poisson Equation:

∂²Ψ₀ / ∂x² = - q/ε (p₀ - n₀ + N_D⁺ - N_A^-)

n₀ = n_i exp (E_F - E_i / KT)

P₀ = n_i exp (E_i - E_F / KT)

n₀ p₀ = n²_i                

In equilibrium, there is only single independent variable out of the three variables:  

n_0, p_0, Ψ₀

If one of them is well known, all the rest can be measured from the equations listed above. We shall consider this independent variable to be potential. The analysis trouble in equilibrium is hence determination of potential or equivalently, energy band diagram of the semiconductor device. This is the cause why we begin discussions of all semiconductor devices with a sketch of the energy band diagram of it in equilibrium.