Electrostatics of the step pn junction

Set the doping on each side of an abrupt silicon junction at 300 K and apply a bias to see how the depletion region, charge density, electric field, potential, and energy bands respond. Every panel shares one position axis, with the p side on the left and the metallurgical junction at x = 0.

p-side doping, N_A

n-side doping, N_D

Applied voltage, V_A

Overlay equilibrium (V_A = 0) Overlay exact Poisson solution Carry quasi-Fermi levels through the depletion region Charge density scale

Figure 1 Coordinated sketches of the depletion region, the energy band diagram, and the charge density, electric field, and electrostatic potential of a silicon step junction in the depletion approximation, Eqs. (5.32)–(5.38). The Fermi level on the p side is the zero of energy and stays fixed, so under bias the n side moves by qV_A and the equilibrium solution can be overlaid dashed. Axes are refit whenever the doping changes but only widen while the bias is adjusted, so a sweep of V_A is seen against one fixed frame; Rescale axes refits them to the present bias. After Pierret, Semiconductor Device Fundamentals, Figs. 5.11 and 5.12.

Quasineutral regions
`p`_p0		Majority holes on the p side
`n`_p0		Minority electrons on the p side, `n`_i²/`p`_p0
`n`_n0		Majority electrons on the n side
`p`_n0		Minority holes on the n side, `n`_i²/`n`_n0
`E`_i − `E`_F, p side		`kT` ln(`p`_p0/`n`_i)
`E`_F − `E`_i, n side		`kT` ln(`n`_n0/`n`_i)
`V`_bi		Sum of the two rows above, divided by `q`; Eq. (5.10) when `N` ≫ `n`_i

Depletion region at the applied bias
`V`_bi − `V`_A		Potential drop across the depletion region
`x`_p		Penetration into the p side, Eq. (5.34)
`x`_n		Penetration into the n side, Eq. (5.37); `x`_n/`x`_p = `N`_A/`N`_D
`W`		Depletion width, Eq. (5.38)
ℰ(0)		Peak field at the junction, −`qN`_D`x`_n/`K`_Sε₀
`L`_D, p / n		Extrinsic Debye lengths, [`K`_Sε₀`kT`/(`q`²`N`)]^1/2, the width of the depletion edge
`Q`/`A`		Depletion charge per area on each side, `qN`_A`x`_p = `qN`_D`x`_n
`C`_J/`A`		Junction capacitance per area, `K`_Sε₀/`W`

Model, parameters, and what is left out

The junction is an ideal step: N_A acceptors for x < 0, N_D donors for x > 0, all ionized, with nondegenerate (Boltzmann) statistics throughout. Far from the junction the carrier concentrations follow from charge neutrality,

p_p0 = N_A/2 + [(N_A/2)² + n_i²]^1/2, n_n0 = N_D/2 + [(N_D/2)² + n_i²]^1/2

and the built-in voltage is

V_bi = (kT/q) ln(p_p0n_n0/n_i²)

which is Pierret's Eq. (5.10), (kT/q) ln(N_AN_D/n_i²), whenever both dopings are well above n_i. The two forms differ noticeably only at the bottom of the slider range; at N_A = N_D = 10¹⁰ cm⁻³ the textbook form would give V_bi = 0, whereas the material is in fact nearly intrinsic and supports only about one kT/q.

The depletion-region quantities are exactly Eqs. (5.32)–(5.38), with V(−x_p) = 0 and the substitution V_bi → V_bi − V_A for the biased case. The band diagram is the upside-down potential, E_c(x) = E_c(−∞) − qV(x), with the p-side Fermi level as the zero of energy. E_i sits (3kT/4) ln(m_p*/m_n*) = −7.3 meV from midgap. Under bias, E_Fp and E_Fn are drawn flat in their own quasineutral regions and separated by qV_A, as in Fig. 5.12; the optional extension through the depletion region is the usual quasi-equilibrium assumption that is taken up again in the derivation of the ideal diode equation.

The exact overlay solves the full nonlinear Poisson equation,

d²V/dx² = −(q/K_Sε₀)(p − n + N_D − N_A), p = p_p0e^−qV/kT, n = n_p0e^q(V+V_A)/kT

with both quasi-Fermi levels flat across the structure and V = 0 and V_bi − V_A imposed at the two ends. It is discretized by box integration on a mesh that is graded down to a small fraction of the shorter Debye length at x = 0 and reaches 14 Debye lengths past each depletion edge and solved by damped Newton iteration, starting from the depletion-approximation potential. It is hidden under high-level injection, where the flat quasi-Fermi level assumption no longer holds. The comparison shows what the depletion approximation throws away: mobile-carrier tails a few Debye lengths wide at each edge of the depletion region. For moderate doping at equilibrium the two solutions are nearly indistinguishable on a linear scale; the difference grows as (V_bi − V_A)/(kT/q) shrinks, which happens for light doping and for forward bias. The log-magnitude charge scale makes the exponential tails visible in every case. In a strongly one-sided junction the exact solution also shows a dipole a few nanometers wide at x = 0, where majority carriers from the heavily doped side spill across the step, carrying a field spike well above ℰ(0) and a drop of a few kT/q. That feature belongs to the idealization of a perfectly abrupt step, which no real junction is, and at those doping levels its magnitude is also affected by degeneracy.

Parameter	Value used
`K`_S	11.8
ε₀	8.854E-14 F/cm
`q`	1.602E-19 C
`kT`/`q` at 300 K	0.0259 V
`n`_i at 300 K	1.0E+10 cm^-3
`E`_g	1.12 eV
`m`_n/`m`₀, `m`_p/`m`₀	1.18, 0.81

These are Pierret's 300 K values, chosen so that the numbers here agree with worked examples and homework in the text. More recent measurements put silicon's n_i near 9.7×10⁹ cm⁻³, which raises every V_bi by roughly 2 mV and changes nothing visible on the plate.

On the plate, the position window spans the larger of the equilibrium and present depletion regions plus 30% of that width on each side (more when the exact overlay needs room for its tails). The linear charge-density scale is set by the larger doping, so a one-sided junction shows the heavily doped side as a tall narrow spike and the lightly doped side as a low plateau of equal area; this is correct, and the log-magnitude scale shows both at once.

Deliberate simplifications, all of which matter somewhere in a real diode:

The entire applied voltage appears across the depletion region. Voltage drops in the quasineutral regions and contacts are neglected, which is why, as Pierret notes, the formulation fails as V_A approaches V_bi. The forward bias slider stops at V_bi − 2kT/q.
Boltzmann statistics and complete ionization at every doping. In silicon at 300 K, doping above roughly 1–2×10¹⁸ cm⁻³ places E_F within 3kT of a band edge, and the page flags this. Bandgap narrowing, which lowers V_bi at these concentrations, is not modeled.
No breakdown. The peak field is compared with an approximate avalanche field from Sze's empirical fit for one-sided abrupt silicon junctions, ℰ_crit ≈ 4×10⁵/[1 − (1/3)log₁₀(N/10¹⁶)] V/cm, evaluated with N = N_AN_D/(N_A + N_D) and clamped to 10¹⁴–10¹⁸ cm⁻³. Treat the resulting warning as a rough guide; heavily doped junctions break down by tunneling instead.
One dimension, no surfaces, interface charge, or graded profile.