Junction Theory
Part of Semiconductors
The physics of p-n junctions — depletion zones, built-in potentials, and current flow mechanisms.
Why This Matters
The p-n junction is the fundamental building block of all semiconductor electronics. Transistors are two junctions; solar cells are one junction; LEDs are one junction engineered for light emission. Every junction-based device behavior flows from the same underlying physics: what happens when p-type and n-type semiconductor come into contact.
Understanding junction theory lets you predict device characteristics from material parameters, diagnose failure modes, and design circuits that stay within device limits. Without this foundation, semiconductor devices are empirical black boxes — you follow rules without knowing why they work or what happens when operating conditions push outside the documented range.
For a rebuilding civilization, junction theory connects the atomic-level physics of doping to the measured I-V curves of fabricated devices. When your diodes show unexpected leakage, when your transistors have lower-than-expected gain, junction theory points toward the physical cause: contamination raising recombination rate, excessive base width, doping levels that produce too-wide depletion zones.
Equilibrium Junction: Depletion Zone and Built-In Field
At equilibrium (no applied voltage), the p-n junction reaches a state where no net current flows. Reaching this state from the initial contact involves:
Diffusion: Electrons from the n-side (high electron concentration) diffuse toward the p-side (low electron concentration). Holes from the p-side diffuse toward the n-side. This is driven by concentration gradients.
Charge exposure: As electrons leave the n-side, they expose positively charged donor ions (atoms that donated their electrons to become carriers). As holes leave the p-side, they expose negatively charged acceptor ions. A dipole layer of fixed charges builds up at the junction.
Electric field: The exposed charges create an electric field pointing from n to p (from positive charges in n-region to negative charges in p-region). This field opposes diffusion — carriers trying to diffuse across the junction are repelled by the field.
Equilibrium: Diffusion tendency balanced by field repulsion. Net current is zero. The region depleted of mobile carriers is the depletion zone.
Built-in potential (V_bi): The potential difference across the depletion zone:
V_bi = (kT/q) × ln(N_A × N_D / n_i²)
For silicon with N_A = 10^17 cm^-3 (p-side doping) and N_D = 10^16 cm^-3 (n-side doping), at room temperature: V_bi ≈ 0.75V.
For germanium with same doping: V_bi ≈ 0.36V (because n_i is much larger for Ge, reducing the log term).
Depletion zone width: W = √(2ε × V_bi × (N_A + N_D) / (q × N_A × N_D))
where ε is the dielectric permittivity of the semiconductor. For lightly doped germanium (N_A = N_D = 10^14 cm^-3): W ≈ 10 µm. For heavily doped silicon (10^17 cm^-3 both sides): W ≈ 10 nm.
The depletion zone extends further into the more lightly doped side — the asymmetry is important for designing specific breakdown voltages and capacitances.
Forward Bias: Minority Carrier Injection
Applying forward voltage reduces the barrier. When forward voltage V is applied (positive to p-side):
Effective barrier = V_bi - V
At V = V_bi, the barrier is gone. But in practice, significant current flows well before this — the exponential I-V relationship means current doubles every ~18 mV (for n=1), so measurable current appears at V ≈ 0.6-0.7V for silicon even though V_bi ≈ 0.75V.
The physics of current flow in forward bias is minority carrier injection:
- Reduced barrier allows electrons to cross from n to p (they are minority carriers in p). Similarly, holes cross from p to n.
- These injected minority carriers exist far above equilibrium concentration near the junction. They diffuse away from the junction into the quasi-neutral regions.
- As they diffuse, they recombine with majority carriers. The distribution decays exponentially with distance: n_p(x) = n_p0 × e^(V/V_T) × e^(-x/L_n)
- This recombination draws majority carriers from the contacts to maintain neutrality — this is the external current.
The current density: J = J_s × (e^(V/V_T) - 1) where J_s = q × n_i² × (D_n/(L_n × N_A) + D_p/(L_p × N_D))
This is the Shockley diode equation. Notice J_s increases with n_i² — exponentially with temperature. This is why reverse leakage doubles every ~10°C.
Reverse Bias: Depletion Zone Widening
Reverse bias increases the barrier and widens the depletion zone. No majority carriers cross. The only current is from thermally generated minority carriers swept across by the strong field — the reverse saturation current I_s.
I_s is small because thermal carrier generation is slow (controlled by the generation rate in the depletion zone and diffusion length in quasi-neutral regions). But it is not zero. For silicon at room temperature, I_s is typically 1-100 nA for small junctions. For germanium, 0.1-10 µA.
Depletion capacitance: The widening depletion zone acts like a parallel-plate capacitor. As reverse voltage increases, the zone widens and capacitance decreases:
C_j = C_j0 / √(1 + V_R/V_bi)
This reverse-bias dependent capacitance is exploited in varactor diodes (voltage-controlled capacitors used in tunable oscillators and filters). For a rebuilding civilization, varactors enable electronically tuned radios — the frequency changes by adjusting a reverse-biased junction voltage rather than mechanically adjusting a capacitor.
Breakdown Mechanisms
At high reverse voltage, one of two breakdown mechanisms generates large current:
Avalanche breakdown: Carriers crossing the wide depletion zone accelerate under the strong field. At sufficient energy, they ionize lattice atoms upon collision, generating new electron-hole pairs. These accelerate and ionize more atoms. The multiplication factor increases rapidly until a runaway avalanche occurs. Avalanche breakdown voltage typically 15-200V depending on doping — lighter doping gives wider depletion zone and higher breakdown voltage.
Doping-breakdown relationship: V_BD ≈ C × N^(-3/4) where N is doping concentration and C is a material constant. Reducing doping by 10× increases breakdown voltage roughly 5.6×. This guides design: if you need a 50V diode, calculate the required doping level.
Zener (tunneling) breakdown: In heavily doped junctions, the depletion zone is very thin (< 10 nm). Quantum mechanical tunneling allows electrons to cross the thin barrier directly without ionizing collisions. This occurs at lower voltages (below ~5V) than avalanche. Temperature coefficient is opposite: Zener voltage decreases with temperature (negative TC), while avalanche voltage increases (positive TC). A junction with both mechanisms active near 5-6V has near-zero temperature coefficient — the basis of precision Zener voltage references.
Junction Recombination and Real Diode Non-Ideality
Real diodes deviate from the ideal Shockley equation in two ways:
Recombination-generation current: In the depletion zone, thermal generation and recombination via trap states (from crystal defects and metallic impurities) creates an additional current component. This component varies as e^(V/2V_T) rather than e^(V/V_T). At low forward voltages, this component dominates; at high voltages, the diffusion current component dominates. The ideality factor n in I = I_s × e^(V/nV_T) quantifies the blend: n=1 is pure diffusion; n=2 is pure recombination. Real diodes have n between 1 and 2.
High injection effects: At high forward current, minority carrier concentration approaches majority carrier concentration. The assumptions of the Shockley derivation break down. Junction resistance increases beyond the theoretical value. This limits the maximum practical forward current density before efficiency drops.
For fabrication quality control: measure n from the slope of a log(I) vs V plot. A slope of q/kT (1/0.026 = 38.5 V^-1) corresponds to n=1. Slope of q/2kT (19.25 V^-1) corresponds to n=2. If your fabricated diodes show n > 1.5 at low currents, your material has high defect density — improve purification and crystal quality.