PN Junction, Zener, LED and Photodiode

The PN junction is the building block of diodes, transistors and optoelectronic devices.

Equilibrium

• At the junction, diffusion leaves a depletion region of fixed ions and a built-in field.
• Built-in potential: V_bi = V_T ln(N_A N_D/n_i²).
• Depletion width W = √(2ε(V_bi − V)/q × (1/N_A + 1/N_D)); it extends mostly into the lightly doped side: x_n N_D = x_p N_A.
• Reverse bias widens W; forward bias narrows it. W ∝ √(V_bi + V_R).

Current

I = I_s(e^(V/(ηV_T)) − 1)

• I_s depends on n_i², so it rises very fast with temperature (doubles about every 10 °C for Si).
• A 60 mV increase in forward voltage multiplies the current by about 10 (at η = 1).
• At fixed current, V_D falls by about 2 mV/°C.

Capacitance

• Junction (depletion) capacitance C_j = εA/W, dominant in reverse bias: C_j ∝ (V_bi + V_R)^(−1/2) for an abrupt junction. Used in varactor diodes for tuning.
• Diffusion capacitance C_d = τ I/V_T, dominant in forward bias.

Breakdown

• Zener breakdown: heavily doped junctions, thin depletion region, tunnelling; below about 5 V; negative temperature coefficient.
• Avalanche breakdown: lightly doped, impact ionisation; above about 6 V; positive temperature coefficient.
• Around 5–6 V the two effects cancel, giving the most temperature-stable Zeners.

Optoelectronic devices

• LED: forward-biased direct-gap junction; photon wavelength λ (µm) = 1.24/E_g (eV).
• Photodiode: reverse-biased; light creates carriers that add a current proportional to light intensity. PIN and avalanche photodiodes are faster and more sensitive.
• Solar cell: unbiased junction in the fourth quadrant; open-circuit voltage, short-circuit current and fill factor describe it.

Worked example

A Si junction has N_A = 10¹⁷ and N_D = 10¹⁶ cm⁻³, n_i = 10¹⁰ cm⁻³, V_T = 0.0259 V.

• V_bi = 0.0259 ln(10³³/10²⁰) = 0.0259 × 29.93 = 0.775 V.
• The depletion region extends 10 times further into the n side than the p side.

Practice