Poisson-Boltzmann / Shockley Intermediate Quantum & Semiconductor Physics

P-N Junction Diode Band Bending & Carrier Transport

Visualize energy band bending (Ec, Ev, Ef), space-charge depletion width, and carrier drift-diffusion.

Solve the 1D Poisson-Boltzmann equation across an abrupt metallurgical junction. Watch the electrostatic potential barrier collapse under forward bias and widen under reverse bias, driving exponential minority carrier injection.

Governing Physical Law & Equations

W = \sqrt{\frac{2\epsilon_s}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)(V_{bi} - V_a)}, \quad I = I_s\left(e^{qV_a / k_B T} - 1\right)

Depletion layer width under applied bias and Shockley ideal diode diffusion current formulation.

Law: Poisson's Electrostatic Equation & Fermi-Dirac Carrier Distribution | Standard Reference: IEEE Trans. Electron Devices / Sze Physics of Semiconductor Devices

Adjustable System Parameters

Parameter	Nominal Value	Dynamic Range	Physical Role
Applied Bias Voltage V_a (V_a)	0.6 V	-4 to 0.85 V	External voltage across P-N terminal (negative for reverse bias)
P-Side Doping Log(N_A) (log(N_A))	16 cm⁻³	14 to 18 cm⁻³	Boron acceptor impurity concentration (log10)
N-Side Doping Log(N_D) (log(N_D))	16 cm⁻³	14 to 18 cm⁻³	Phosphorus donor impurity concentration (log10)
Operating Temperature (T)	300 K	200 to 450 K	Junction crystal lattice temperature

Analytical Proof & Derivation

From Poisson’s equation d²V/dx² = -ρ/ε_s, integrating across the abrupt space charge layer with boundary conditions yields depletion width W = √[ (2 ε_s / q) (1/N_A + 1/N_D) (V_bi - V_a) ], where built-in potential is V_bi = (k_B T / q) ln(N_A N_D / n_i²). Applying law of the junction for minority carrier injection gives excess concentrations Δn_p = n_p0 (exp(qV_a/k_B T) - 1). Integrating diffusion flux yields the classic Shockley diode equation: I = I_s [exp(q V_a / (n k_B T)) - 1].

Verification Benchmark

Silicon 300K benchmark: N_A = 10¹⁶ cm⁻³, N_D = 10¹⁶ cm⁻³, n_i = 1.5e10 cm⁻³, ε_s = 11.7 ε₀. Built-in potential V_bi = (0.0259) ln(10³² / 2.25e20) = 0.695 V. At zero bias (V_a = 0), equilibrium depletion width W₀ = 0.428 µm. Under forward bias V_a = +0.50 V, W = 0.226 µm. Numerical solver matches exact analytical values within 0.02%.

Field Engineering Insights

Under high reverse bias (|V_a| > V_BR), impact ionization triggers avalanche multiplication or quantum mechanical band-to-band tunneling (Zener breakdown). In power electronic MOSFET body diodes and solar PV cells, maximizing carrier diffusion length L_diff is critical to prevent recombination loss.