Part II: Semiconductor Physics | Chapter 5

PN Junctions & Diodes

The fundamental building block of semiconductor devices: a single p-n interface that rectifies, regulates, and switches.

The PN Junction in Equilibrium

When p-type and n-type semiconductors are brought into contact, electrons diffuse from n to p and holes diffuse from p to n. This leaves behind ionised donor/acceptor atoms forming a depletion region — a region swept free of mobile carriers.

The electric field in the depletion region creates a built-in potential \( V_{bi} \)that opposes further diffusion, establishing equilibrium:

\[ V_{bi} = \frac{k_B T}{q}\ln\!\left(\frac{N_A N_D}{n_i^2}\right) \]

For silicon with \( N_A = N_D = 10^{16} \;\text{cm}^{-3} \) at 300 K, \( V_{bi} \approx 0.72 \;\text{V} \).

Shockley Diode Equation

Under an applied voltage \( V \), the current through an ideal pn junction is given by the Shockley diode equation:

\[ I = I_s\!\left(\exp\!\frac{V}{nV_T} - 1\right) \]

\( I_s \): saturation current (10⁻¹²–10⁻⁹ A), \( n \): ideality factor (1–2), \( V_T = k_BT/q \approx 25.85 \text{ mV at 300 K} \)

Forward Bias (\( V > 0 \))

Applied voltage opposes the built-in field, narrowing the depletion region. The exponential term dominates: current rises steeply. Silicon diodes have a threshold of ~0.6–0.7 V.

Reverse Bias (\( V < 0 \))

Widens the depletion region. Current saturates at \( -I_s \) — extremely small. The diode is essentially an open circuit until breakdown.

Breakdown Mechanisms

Zener Breakdown

Quantum tunnelling of electrons across a thin, highly-doped junction at \( |V| < 5 \) V. Sharp, well-defined breakdown voltage; exploited in Zener regulators.

Avalanche Breakdown

Impact ionisation — carriers accelerated by the field collide with the lattice, creating electron-hole pairs that trigger a cascade. Dominant at \( |V| > 7 \) V.

Diode Circuit Applications

Half-Wave Rectifier

A single diode passes only the positive half-cycles of AC. The output is pulsating DC with average value \( V_{avg} = V_m/\pi \).

Full-Wave Bridge Rectifier

Four diodes rectify both half-cycles. Average output \( V_{avg} = 2V_m/\pi \); ripple frequency is doubled, making filtering easier.

Clipper Circuit

Limits the output voltage to a specified level, clipping the signal above (or below) a threshold voltage set by a biased diode.

Clamper Circuit

Shifts the DC level of a signal without changing its shape. A capacitor and diode add or subtract a DC offset equal to the peak voltage.

A large filter capacitor \( C \) placed across the load reduces AC ripple. For half-wave rectification the ripple voltage is approximately\( V_r \approx V_m / (f R_L C) \); for full-wave the factor of \( f \) becomes \( 2f \), halving the ripple.

Python: Diode I-V & Rectifier Simulation

Plot the full Shockley I-V characteristic, then simulate half-wave and full-wave bridge rectifiers with a capacitor filter, computing ripple voltage for each topology.

Shockley Diode I-V and Rectifier Circuits

Python

pn_diode_rectifier.py129 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ---- Shockley diode equation: I = Is*(exp(V/n*Vt) - 1) ----
q = 1.602e-19    # C
k_B = 1.381e-23  # J/K
T = 300.0        # K
Vt = k_B * T / q  # thermal voltage ~25.85 mV
Is = 1e-12       # A (saturation current)
n = 1.0          # ideality factor

print('=== Shockley Diode Equation ===')
print(f'T = {T:.0f} K, Vt = {Vt*1000:.3f} mV')
print(f'Is = {Is:.1e} A, n = {n:.1f}')
print()

V_fwd = np.linspace(0, 0.8, 1000)
V_rev = np.linspace(-2.0, 0, 500)
V_all = np.concatenate([V_rev, V_fwd])
I_all = Is * (np.exp(V_all / (n * Vt)) - 1)

# Key values
for V_val in [0.3, 0.5, 0.6, 0.7, 0.75]:
    I_val = Is * (np.exp(V_val / (n * Vt)) - 1)
    print(f'V = {V_val:.2f} V: I = {I_val*1000:.4f} mA')

# ---- Half-wave rectifier with capacitor filter ----
f_line = 60.0    # Hz
omega = 2 * np.pi * f_line
Vpeak = 10.0     # V
R_load = 1000.0  # Ohm
C_filter = 100e-6  # 100 uF

t = np.linspace(0, 5/f_line, 5000)
V_in = Vpeak * np.sin(omega * t)

# Ideal diode rectification
V_half = np.maximum(V_in, 0)

# Approximate RC filter effect on half-wave
tau = R_load * C_filter
V_cap = np.zeros_like(t)
dt = t[1] - t[0]
V_cap[0] = 0.0
for i in range(1, len(t)):
    if V_half[i] >= V_cap[i-1]:
        V_cap[i] = V_half[i]   # capacitor charges quickly
    else:
        V_cap[i] = V_cap[i-1] * np.exp(-dt / tau)   # capacitor discharges through R

# ---- Full-wave rectifier (bridge) ----
V_full = np.abs(V_in)
V_cap_full = np.zeros_like(t)
V_cap_full[0] = 0.0
for i in range(1, len(t)):
    if V_full[i] >= V_cap_full[i-1]:
        V_cap_full[i] = V_full[i]
    else:
        V_cap_full[i] = V_cap_full[i-1] * np.exp(-dt / tau)

ripple_half = V_cap.max() - V_cap[len(t)//2:].min()
ripple_full = V_cap_full.max() - V_cap_full[len(t)//2:].min()
print()
print('=== Rectifier Performance ===')
print(f'Half-wave ripple voltage: {ripple_half:.3f} V')
print(f'Full-wave ripple voltage: {ripple_full:.3f} V')
print(f'Ripple reduction (full/half): {ripple_full/ripple_half:.3f}x')

# ---- Plots ----
fig, axes = plt.subplots(1, 3, figsize=(16, 5))
fig.patch.set_facecolor('#0a0a0a')
fig.suptitle('PN Junction Diode Characteristics', color='#d1d5db', fontsize=13, y=0.98)
for ax in axes:
    ax.set_facecolor('#111827')
    for spine in ax.spines.values():
        spine.set_color('#374151')

# Plot 1: Diode I-V (Shockley)
ax = axes[0]
I_mA = I_all * 1000
V_plot = V_all
# Only show reasonable range
mask = (I_mA > -0.01) & (I_mA < 50)
ax.plot(V_plot[mask], I_mA[mask], color='#34d399', lw=2.5)
ax.axhline(0, color='#6b7280', lw=0.8)
ax.axvline(0, color='#6b7280', lw=0.8)
ax.annotate('Forward bias', xy=(0.6, 10), color='#6ee7b7', fontsize=9)
ax.annotate('Reverse bias
(~-Is)', xy=(-1.5, 0.005), color='#f87171', fontsize=9)
ax.set_xlabel('Voltage (V)', color='#9ca3af')
ax.set_ylabel('Current (mA)', color='#9ca3af')
ax.set_title('Shockley I-V Characteristic', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.set_xlim(-2.1, 0.85)
ax.set_ylim(-0.5, 50)
ax.grid(True, color='#1f2937', alpha=0.8)

# Plot 2: Half-wave rectifier
t_ms = t * 1000
ax = axes[1]
ax.plot(t_ms, V_in,   color='#6b7280', lw=1.2, alpha=0.5, label='V_in (AC)')
ax.plot(t_ms, V_half, color='#34d399', lw=1.5, alpha=0.7, label='Half-wave rect.')
ax.plot(t_ms, V_cap,  color='#fbbf24', lw=2.2, label=f'Filtered (C={C_filter*1e6:.0f}uF)')
ax.set_xlabel('Time (ms)', color='#9ca3af')
ax.set_ylabel('Voltage (V)', color='#9ca3af')
ax.set_title(f'Half-Wave Rectifier (ripple={ripple_half:.2f}V)', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8)

# Plot 3: Full-wave rectifier
ax = axes[2]
ax.plot(t_ms, V_in,      color='#6b7280', lw=1.2, alpha=0.5, label='V_in (AC)')
ax.plot(t_ms, V_full,    color='#34d399', lw=1.5, alpha=0.7, label='Full-wave rect.')
ax.plot(t_ms, V_cap_full, color='#fbbf24', lw=2.2, label=f'Filtered (C={C_filter*1e6:.0f}uF)')
ax.set_xlabel('Time (ms)', color='#9ca3af')
ax.set_ylabel('Voltage (V)', color='#9ca3af')
ax.set_title(f'Full-Wave Bridge Rectifier (ripple={ripple_full:.2f}V)', color='#d1d5db', pad=8)
ax.tick_params(colors='#9ca3af')
ax.legend(fontsize=8, facecolor='#1f2937', edgecolor='#374151', labelcolor='#d1d5db')
ax.grid(True, color='#1f2937', alpha=0.8)

plt.tight_layout(pad=2.0)
plt.savefig('output.png', dpi=140, bbox_inches='tight', facecolor='#0a0a0a')
print()
print('Plot saved to output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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