Figure 8.6: (a) The space-charge density ρ(x), (b) the electric field ξ(x), and (c) the electrostatic potential ψ(x) across the
depletion region of a n-p junction under equilibrium.
where N D and N A is the concentration of donor and acceptor atoms, respectively. Outside
the space-charge region the space-charge density is zero. The electric field which is
calculated from the Poisson’s equation, in one dimension can be written as
where ψ is the electrostatic potential, ξ is the electric field, ρ is the space-charge density,
∈ r is the semiconductor dielectric constant and ∈ 0 is the vacuum permittivity. The vacuum
permittivity is ∈ 0 = 8.854 × 10
−14
F/cm and for crystalline silicon ∈ r = 11.7. The electric
field profile can be found by integrating the space-charge density across the space-charge
region,
Substituting the space-charge density with Eqs. (8.3) and using the boundary
conditions
we obtain as solution for the electric field