Solution (source code)

= Solution

In a <p-n junction>, <diffusion> and <electron-hole recombination> near the contact leave a <depletion region> of ionized dopants. Its built-in <electric field> opposes further majority-carrier <diffusion>. At <thermodynamic equilibrium> drift and <diffusion> currents cancel. Forward bias lowers the electrostatic barrier and injects <electrons> and <electron holes> across the junction; reverse bias raises the barrier and suppresses that majority-carrier injection. For an ideal diode below breakdown,
$$
\boxed{I=I_s\left(e^{eV/(k_BT)}-1\right).}
$$
Thus a modest forward voltage gives a rapidly increasing current, whereas reverse current remains approximately $-I_s$. This asymmetric current-voltage relation makes the junction a <rectifier>. Real leakage, <electron-hole recombination> and breakdown limit the ideal expression.