Pn Junction Physics Analysis

Assume complete depletion in junction region with widths:

• lp: depletion width in p-region
• ln: depletion width in n-region

3. Apply Charge Neutrality

Na * lp = Nd * ln

4. Calculate Electric Field

Maximum field at junction (x=0):

F_max = e * Nd * ln / (ε * ε0) = e * Na * lp / (ε * ε0)

Field distribution is triangular.

5. Calculate Diffusion Potential

ψ_D = (kT/e) * ln(Na * Nd / ni²)

Potential distribution is parabolic.

6. Calculate Depletion Widths

Solve for lp and ln using neutrality condition and diffusion potential.

Linearly Graded Junctions

For junctions with substantial doping gradient:

Field Distribution

F(x) = (q * a / ε) * (x * W/2 - x²/2)
F_max = (q * a * W²) / (8 * ε)

Barrier Width

W = [(12 * ε * (Vd - V)) / (q * a)]^(1/3)

Diffusion Potential

Vd = (kT/q) * ln((a * W) / (2 * n_i))

Junction Capacitance and Parameter Extraction

1. Capacitance per Unit Area

C = dQ/dV = (ε * ε0 * e * Na * Nd) / [(Na + Nd) * (ψn,D - V)]

2. Linear Plotting for Extraction

1/C² = [2 * (Na + Nd) / (ε * ε0 * e * Na * Nd)] * (ψn,D - V)

3. Extract Diffusion Voltage

• Plot 1/C² versus bias voltage V
• Intercept on voltage axis (where 1/C² = 0) gives ψn,D

4. Extract Doping Density (Asymmetric Junctions)

For Na >> Nd:

Slope ≈ 2 / (ε * ε0 * e * Nd)
Nd = 2 / (ε * ε0 * e * slope)

Model Applicability Guidelines

Simplified Model Use Cases

Use simplified box-like space charge model for:

• First estimates of junction fields, barrier heights, capacitance
• Understanding importance of junction recombination
• Qualitative I-V characteristics

Simplified Model Limitations

DO NOT use for:

• Quantitative agreement with actual devices
• Space charge distribution analysis from capacitance measurements
• Critical evaluation of reverse saturation currents
• Detailed diode curve shape analysis

Wide Bandgap (Si) Considerations

Frozen-in Equilibrium

• Si band gap: 1.16 eV
• When quasi-Fermi level in reverse bias > 1 eV from band edge, consider frozen-in equilibrium
• Minimum minority carrier density ~10² cm⁻³
• Quasi-Fermi approach may overestimate density if ignored

Highly Doped Thin Layers

• Thin layer (e.g., 500 Å) near contact with high doping (10¹⁸ cm⁻³) requires surface recombination consideration
• Minority carrier density ~200 cm⁻³ (near frozen-in equilibrium)

Lack of Saturation

• Total current may lack saturation due to expanding GR region with bias
• Current increases until junction fills entire bulk
• Field-enhanced diffusion shifts saturation onset to higher bias

Complex Junction Factors

Consider these factors for non-ideal/complex devices:

Primary Effects

• Doping gradients
• Inhomogeneous recombination center distribution
• Series resistance contribution
• High injection effects

Position-Dependent Parameters

Caused by inhomogeneous heavy doping, graded composition, surface treatments:

• Ec(x), Ev(x): band edge distributions
• Eg(x): band gap distribution
• gc(x), gv(x): density-of-states distributions
• μn(x), μp(x): carrier mobilities
• ε(x), κ(x), nk(x): optical constants

Output

• Electric field profile (triangular or rounded)
• Potential distribution (parabolic)
• Depletion widths (lp, ln or W)
• Diffusion voltage (ψD) from C-V intercept
• Doping density from C-V slope
• Assessment of model applicability