Unit 1: PN Junction

A PN junction is formed when a single crystal of semiconductor (usually silicon, with a bandgap of 1.1 eV) is doped so that one region is P-type and the adjacent region is N-type. It is the elementary building block of almost every analog device, and its rectifying behaviour underlies diodes, rectifiers and voltage regulators.

• P-type material: silicon doped with trivalent acceptors (boron): majority carriers are holes, minority carriers are electrons.
• N-type material: silicon doped with pentavalent donors (phosphorus): majority carriers are electrons, minority carriers are holes.
• Barrier potential (V₀): built-in voltage across the junction at equilibrium; ≈ 0.7 V for silicon, ≈ 0.3 V for germanium.
• Two conduction processes: drift (carrier motion under an electric field) and diffusion (carrier motion down a concentration gradient).
• Rectification convention: conventional current flows easily from P (anode) to N (cathode) under forward bias, and is blocked under reverse bias.

II. Formation of the Junction — Construction, Depletion and Capacitance

A. Construction and Working

A PN junction is a metallurgical boundary across which majority carriers diffuse until an internal field halts them.

• Construction: P and N regions are grown in one continuous crystal (by diffusion, alloying or epitaxy); a metal contact is bonded to each side to form the anode and cathode terminals.
• Working at equilibrium: holes diffuse into the N-side and electrons into the P-side, recombining near the boundary and leaving fixed ionised impurities.
• Forward bias: P connected to +, N to −. The applied field opposes V₀, narrows the barrier, and majority carriers cross freely once V > V₀.
• Reverse bias: P connected to −, N to +. The applied field aids V₀, widening the barrier; only a tiny minority-carrier current flows.

B. Depletion region

The depletion region is the carrier-free zone of fixed ionised charge either side of the junction.

• Formation: diffusing carriers recombine near the boundary, exposing immobile positive donor ions (N-side) and negative acceptor ions (P-side).
• Internal field: the exposed ions set up an electric field pointing N→P that opposes further diffusion, establishing equilibrium.
• Width dependence: narrows under forward bias, widens under reverse bias; approximately

W = sqrt( (2·ε·V₀) / q · (Nₐ + N_d)/(Nₐ·N_d) )

where ε = permittivity, q = electronic charge (1.6×10⁻¹⁹ C), Nₐ, N_d = acceptor and donor concentrations.

• Asymmetry: the region penetrates deeper into the more lightly doped side, since charge neutrality requires Nₐ·xₚ = N_d·x_n.

C. Junction capacitance

Because the depletion region stores charge across an insulating gap, the junction behaves as a voltage-dependent capacitor.

1. Transition (depletion) capacitance C_T: dominant in reverse bias.
   • Model: parallel-plate capacitor with the depletion width W as separation: C_T = ε·A / W.
   • Behaviour: as reverse voltage rises, W increases and C_T falls — exploited in varactor (varicap) diodes for tuning.
2. Diffusion (storage) capacitance C_D: dominant in forward bias.
   • Cause: injected minority carriers stored in the neutral regions: C_D = τ·I / (η·V_T), where τ = carrier lifetime, I = forward current, V_T = thermal voltage.
   • Significance: C_D is large and limits the diode's switching speed.

III. Electrical Behaviour — Diode Equation and Characteristics

A. Diode equation

The Shockley equation describes the current through an ideal junction as a function of applied voltage.

I = I₀ ( e^(V / (η·V_T)) − 1 )

• I: total diode current (A).
• I₀: reverse saturation current (A), set by minority-carrier concentrations.
• V: applied voltage (+ forward, − reverse).
• η: ideality factor ≈ 1 for germanium, ≈ 2 for silicon.
• V_T: thermal voltage = kT/q ≈ 26 mV at 300 K, where k = Boltzmann's constant, T = absolute temperature.
• Interpretation: for V ≫ V_T the "−1" is negligible and current rises exponentially; for large reverse V the exponential vanishes, leaving I ≈ −I₀.

B. Effect of temperature on reverse saturation current

I₀ arises from thermally generated minority carriers, so it is strongly temperature-sensitive.

• Empirical rule: I₀ approximately doubles for every 10 °C rise in temperature.
• Expression: I₀(T₂) = I₀(T₁) · 2^((T₂−T₁)/10).
• Barrier voltage drift: forward voltage falls by about 2 mV/°C for silicon as temperature rises.
• Consequence: reverse leakage grows and can cause thermal runaway if self-heating feeds back; a major reliability concern in power circuits.

C. V-I characteristics

The V-I curve plots diode current against applied voltage across all three operating regions.

• Forward region: negligible current until V reaches the cut-in (knee) voltage (≈ 0.7 V Si, ≈ 0.3 V Ge), then a steep exponential rise.
• Reverse region: a small, near-constant leakage current (≈ I₀) flows, essentially independent of reverse voltage.
• Breakdown region: beyond the reverse breakdown voltage the current increases sharply; destructive in ordinary diodes unless current-limited.
• Static resistance: R = V/I at an operating point; dynamic (AC) resistance: r = ΔV/ΔI ≈ η·V_T / I.

IV. Junction Diode Rectifiers — AC-to-DC Conversion

A. Junction diode rectifiers

Rectifiers exploit the diode's one-way conduction to convert alternating current into unidirectional (pulsating DC) output.

• Principle: the diode conducts only on the polarity that forward-biases it, clipping the opposite half-cycle.
• Key figures of merit: DC output voltage V_dc, ripple factor γ, rectification efficiency η, and Peak Inverse Voltage (PIV) — the maximum reverse voltage a diode must withstand.
• Ripple factor: γ = V_ac(rms) / V_dc; lower γ means smoother DC.

B. Half wave rectifiers

A single diode passes one half of each AC cycle to the load.

• Operation: conducts on the positive half-cycle, blocks the negative; output is a train of half sine pulses.
• Output values: V_dc = V_m/π, I_dc = I_m/π, RMS I_rms = I_m/2, where V_m and I_m are peak values.
• Ripple factor: γ = 1.21 — a very rippled output.
• PIV: equals V_m.
• Ripple frequency: equal to the supply frequency (50 Hz), since only one pulse per cycle appears.

C. Full wave rectifiers

Both half-cycles are directed through the load in the same direction, doubling the pulses.

1. Centre-tapped configuration: two diodes with a centre-tapped transformer; each diode conducts on alternate half-cycles.
   • PIV: 2V_m across each diode.
2. Bridge configuration: four diodes, no centre tap; pairs conduct on alternate half-cycles.
   • PIV: V_m per diode — its main advantage over the centre-tapped type.
   • Output values (both types): V_dc = 2V_m/π, I_dc = 2I_m/π, RMS I_rms = I_m/√2.
   • Ripple factor: γ = 0.482 — far smoother than half-wave.
   • Ripple frequency: twice the supply frequency (100 Hz), easing filter design.

D. Efficiency

Rectification efficiency measures how much AC input power is delivered as DC output power.

η = P_dc / P_ac = (I_dc² · R_L) / (I_rms² · (R_L + r_f))

• r_f: forward resistance of the diode; R_L: load resistance.
• Half-wave maximum: η = 40.6 % (ignoring r_f), because half the input is discarded.
• Full-wave maximum: η = 81.2 %, roughly double, since both half-cycles are used.
• Worked example: a full-wave rectifier with V_m = 20 V into R_L gives V_dc = 2×20/π ≈ 12.7 V, versus 20/π ≈ 6.37 V for half-wave — the same peak delivers nearly twice the DC.

V. Zener Diode — Controlled Reverse Breakdown

A. Definition and principle

A Zener diode is a heavily doped junction designed to operate stably in the reverse breakdown region.

• Heavy doping: narrows the depletion region, sharpening and lowering the breakdown voltage V_Z (available from ≈ 2.4 V upward).
• Breakdown mechanisms:
  1. Zener breakdown: dominant below ≈ 5 V; the intense field directly tears electrons from covalent bonds; V_Z has a negative temperature coefficient.
  2. Avalanche breakdown: dominant above ≈ 6 V; carriers accelerate and ionise atoms by collision; V_Z has a positive temperature coefficient.
• Symbol and biasing: operated in reverse; once V_Z is reached, voltage stays nearly constant while current varies widely.

B. Characteristics and application as regulator

In breakdown the Zener holds an almost fixed voltage, making it a simple reference and regulator.

• Sharp knee: a small increase in reverse voltage past V_Z causes a large current change; dynamic resistance r_Z is very low (a few ohms).
• Regulator action: a series resistor R_s absorbs the excess: R_s = (V_in − V_Z)/(I_Z + I_L), holding the load at V_Z despite input or load variation.
• Current limits: operation must stay between I_Z(min) — to remain in breakdown — and I_Z(max) = P_Z/V_Z, the power-limited maximum.
• Applications: voltage reference, clipping/clamping of waveforms, and overvoltage protection.