A uniformly doped PN junction is examined, where one region of the semiconductor is doped with acceptor impurities (P region), and the adjacent region is doped with donor impurities (N region).
$$ N_A, \quad N_D $$
A space charge (or depletion) region forms on both sides of the junction, which is depleted of free charge carriers (electrons or holes). Positive charge is present in the N region (due to donor ions), and negative charge in the P region (due to acceptor ions).
$$ W = \sqrt{\frac{2 \epsilon_s (V_bi - V)}{q} \left(\frac{1}{N_A} + \frac{1}{N_D}\right)} $$
An electric field is created in the depletion region, directed from the N region to the P region due to the net space charge.
$$ E_{max} = \frac{q N_D W}{\epsilon_s} $$
A potential difference exists across the space charge region (known as the built-in potential barrier), which maintains thermal equilibrium and prevents majority carriers from crossing the junction without external influence.
$$ V_{bi} = \frac{kT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right) $$
When a reverse-bias voltage is applied (N region positive with respect to the P region), the potential barrier, depletion region width, and electric field all increase.
This reverse-bias voltage alters the amount of charge in the depletion region, which changes the junction capacitance.
$$ C = \frac{\epsilon_s}{W} $$
Avalanche breakdown occurs when a large reverse-bias voltage is applied, resulting in a significant reverse current at the breakdown voltage, which depends on the doping concentration.
$$ V_{BR} \propto \frac{1}{N_D^{2/3}} $$
A linearly graded junction, representing a nonuniformly doped PN junction, has different functional relationships for the electric field, potential barrier, and junction capacitance compared to a uniformly doped junction.
$$ E(x) = \frac{q N_A x}{\epsilon_s} $$
Specific doping profiles can be used to control capacitance characteristics. A hyperabrupt junction, where doping decreases away from the junction, is advantageous in applications like varactor diodes used in resonant circuits.
$$ C \propto V^{-m} $$