The metallurgical junction between p-type and n-type semiconductor regions is the foundational building block of modern solid-state electronics. Driven by carrier concentration gradients across the interface, mutual diffusion forms an ionized space-charge depletion region whose built-in electric field establishes asymmetric, rectifying current-voltage characteristics governed by Shockley’s ideal diode equation.
1. The Abrupt p-n Junction in Thermal Equilibrium
Consider an abrupt junction at (x = 0) separating a p-type region ((x 0), donor density (N_D)). The steep majority carrier gradients cause electrons to diffuse into the p-side and holes into the n-side. This leaves behind uncompensated, immobile ionized donors ((N_D^+)) on the n-side and ionized acceptors ((N_A^-)) on the p-side, forming a space-charge depletion region of width (W = x_p + x_n).
The space-charge density (
ho(x)) is expressed under the depletion approximation as:
[
ho(x) = begin{cases}
-e N_A, & -x_p le x < 0
+e N_D, & 0 < x le x_n
0, & x x_n
nd{cases}
]
Global electrostatic neutrality mandates that total positive space charge matches negative space charge:
[
e N_A x_p = e N_D x_n implies rac{x_n}{x_p} = rac{N_A}{N_D}
]
2. Poisson’s Equation and Built-in Potential
The electrostatic potential (phi(x)) and electric field (E(x) = -rac{dphi}{dx}) are governed by the 1D Poisson equation:
[
rac{d^2phi}{dx^2} = -rac{
ho(x)}{psilon_s}
]
Integrating across the depletion boundary with boundary condition (E(-x_p) = E(x_n) = 0):
[
E(x) = begin{cases}
-rac{e N_A}{psilon_s} (x + x_p), & -x_p le x le 0
rac{e N_D}{psilon_s} (x – x_n), & 0 le x le x_n
nd{cases}
]
The peak electric field occurs precisely at the metallurgical interface (x = 0):
[
E_{max} = -rac{e N_A x_p}{psilon_s} = -rac{e N_D x_n}{psilon_s}
]
Integrating (E(x)) from (-x_p) to (x_n) yields the total potential drop, designated the built-in potential (V_{ ext{bi}}):
[
V_{ ext{bi}} = phi(x_n) – phi(-x_p) = rac{e}{2psilon_s} left( N_A x_p^2 + N_D x_n^2
ight)
]
In thermal equilibrium, the Fermi level must be flat across the entire structure ((dE_F/dx = 0)). Relating (V_{ ext{bi}}) to the equilibrium carrier densities (p_p approx N_A) and (n_n approx N_D):
[
V_{ ext{bi}} = rac{k_B T}{e} lnleft( rac{N_A N_D}{n_i^2}
ight)
]
Under an applied external bias voltage (V_a), the total electrostatic barrier becomes (V_{ ext{bi}} – V_a). Solving for the total depletion layer width (W) yields:
[
W(V_a) = x_p + x_n = sqrt{ rac{2psilon_s}{e} left( rac{1}{N_A} + rac{1}{N_D}
ight) (V_{ ext{bi}} – V_a) }
]
3. Derivation of the Shockley Ideal Diode Equation
When a forward bias (V_a > 0) is applied, the potential barrier is reduced to (V_{ ext{bi}} – V_a). Majority carriers can now surmount the lowered barrier, leading to minority carrier injection at the depletion edges. In accordance with Boltzmann statistics, the injected minority carrier densities at the edges are:
[
n_p(-x_p) = n_{p0} e^{e V_a / k_B T}, quad p_n(x_n) = p_{n0} e^{e V_a / k_B T}
]
where (n_{p0} = n_i^2 / N_A) and (p_{n0} = n_i^2 / N_D) are the equilibrium minority concentrations. The excess minority carrier densities are:
[
Delta n_p(-x_p) = n_{p0} left( e^{e V_a / k_B T} – 1
ight), quad Delta p_n(x_n) = p_{n0} left( e^{e V_a / k_B T} – 1
ight)
]
In the neutral quasi-neutral bulk regions, the electric field is negligible, and transport is purely diffusive. Solving the steady-state diffusion equation (rac{d^2 Delta p_n}{dx^2} = rac{Delta p_n}{L_p^2}) with boundary conditions (Delta p_n(infty) = 0) yields exponential decay:
[
Delta p_n(x) = Delta p_n(x_n) e^{-(x – x_n)/L_p}
]
The hole diffusion current injected into the n-region is evaluated at the depletion edge (x = x_n):
[
J_p(x_n) = -e D_p left. rac{d Delta p_n}{dx}
ight|_{x_n} = rac{e D_p p_{n0}}{L_p} left( e^{e V_a / k_B T} – 1
ight)
]
Symmetrically, the electron diffusion current injected into the p-region is:
[
J_n(-x_p) = rac{e D_n n_{p0}}{L_n} left( e^{e V_a / k_B T} – 1
ight)
]
Neglecting recombination inside the narrow depletion region, the total current density (J = J_p(x_n) + J_n(-x_p)) yields the famous Shockley Ideal Diode Equation:
[
J = J_0 left( e^{e V_a / k_B T} – 1
ight)
]
where the reverse saturation current density (J_0) is given explicitly by:
[
J_0 = e n_i^2 left( rac{D_n}{L_n N_A} + rac{D_p}{L_p N_D}
ight)
]
This explains the asymmetric exponential conduction under forward bias ((V_a > 0)) and small, temperature-dependent saturation current (-J_0) under reverse bias ((V_a < 0)).
