Semiconductor Equations: Poisson, Continuity & Drift-Diffusion

The complete mathematical description behind device simulators. Learn, step by step, Poisson's equation (charge → field → potential), the electron and hole continuity equations (conservation with generation and recombination), the drift-diffusion current equations, how they couple, boundary conditions (ohmic contacts, Schottky contacts, depletion edges, surfaces), the minority-carrier diffusion equation and its standard solutions, and how TCAD tools solve them numerically — with Python solving the steady-state diffusion equation by finite differences (matching the analytic sinh solution) and integrating Poisson's equation in a depletion layer.

Every device in this course obeys the same small set of equations. Poisson's equation relates charge to potential; continuity equations track carriers; drift-diffusion equations give currents. Solving them with the right boundary conditions yields diode, transistor and solar-cell behaviour.

You rarely solve them fully by hand — simplifying assumptions (depletion approximation, low-level injection, quasi-neutrality) give the analytic device equations, while TCAD tools solve them numerically.

An analogy

The semiconductor equations are like the accounting rules of a bank: Poisson tracks where the money (charge) is and the pressure it creates (potential); continuity says balance changes = deposits − withdrawals (generation − recombination − outflow); drift-diffusion describes how money moves between branches.

Poisson's equation

Gauss's law in one dimension: the slope of the field equals the local charge divided by permittivity. In quasi-neutral regions ρ ≈ 0 and the field is small; in depletion regions ρ = ±qN, so the field changes linearly and the potential quadratically — the basis of junction and MOS analysis.

Continuity

Carriers are conserved: their concentration at a point changes only if more flow in than out, or if they are generated/recombined locally. Combined with the diffusion current (for minority carriers in a field-free region) it gives the minority-carrier diffusion equation, whose exponential solutions with length L describe injected carrier profiles in diodes and BJTs.

Standard approximations

These turn PDEs into device equations:

Boundary conditions and TCAD

Ohmic contacts: equilibrium carrier densities and applied potential. Schottky contacts: barrier height and thermionic emission velocity. Surfaces: recombination velocity and fixed charge. TCAD (Sentaurus, Silvaco, open-source DEVSIM) adds process simulation, quantum corrections, mobility models and self-heating to predict device characteristics before fabrication.

Formulas

Worked example: Field from charge

Given: N_A 10¹⁷, depletion 0.1 µm, Si

Find: Peak field.

Answer: ≈ 1.55 × 10⁵ V/cm

Worked example: Profile decay

Given: L = 20 µm, long diode

Find: Δp at 60 µm.

Answer: 5% of Δp(0)

Worked example: Continuity check

Given: J_p falls by 1 A/cm² over 10 µm, steady state

Find: Net recombination rate.

Answer: ≈ 6.2 × 10²¹ cm⁻³s⁻¹

How the equations connect

  Doping N_D, N_A  +  carriers n, p  ──► charge density ρ
                        ▲                            │ Poisson
                        │                            ▼
   continuity (G − R, ∇·J)                 potential ψ → field E
                        │                            │ drift-diffusion
                        └──────── currents J_n, J_p ◄┘
  Solve self-consistently (Gummel or Newton iteration) → terminal I–V, C–V

The loop is self-consistent: carriers set charge, charge sets the field, the field (and gradients) set currents, currents change carriers. Device simulators discretise these equations on a mesh and iterate until they agree.

Numerical diffusion equation and Poisson in a depletion layer

import math
# Solve the steady-state minority-carrier diffusion equation d2p/dx2 = p/L^2 numerically (finite differences)
L = 20e-4                    # diffusion length (cm)
W = 100e-4                   # device length
N = 200
dx = W / N
p0 = 1e14                    # injected excess holes at x = 0; p = 0 at the far contact
# tridiagonal system: p[i-1] - (2 + dx^2/L^2) p[i] + p[i+1] = 0
a = [1.0] * (N - 1); b = [-(2 + dx * dx / (L * L))] * (N - 1); c = [1.0] * (N - 1); d = [0.0] * (N - 1)
d[0] = -p0
for i in range(1, N - 1):                       # Thomas algorithm
    m = a[i] / b[i - 1]
    b[i] -= m * c[i - 1]; d[i] -= m * d[i - 1]
p = [0.0] * (N - 1)
p[-1] = d[-1] / b[-1]
for i in range(N - 3, -1, -1):
    p[i] = (d[i] - c[i] * p[i + 1]) / b[i]
for x_um in (10, 20, 40):
    i = int(x_um * 1e-4 / dx) - 1
    exact = p0 * math.sinh((W - x_um * 1e-4) / L) / math.sinh(W / L)
    print(f"x = {x_um:2d} um: numeric {p[i]:.3e}, analytic {exact:.3e}")
# Poisson in a uniformly charged depletion layer: E(x) = qN(W - x)/eps, max at the junction
q, eps, Na, Wd = 1.602e-19, 11.7 * 8.854e-14, 1e16, 0.3e-4
print(f"Poisson: Emax = qNW/eps = {q*Na*Wd/eps:.3e} V/cm, potential drop qNW^2/(2eps) = {q*Na*Wd**2/(2*eps):.3f} V")
# x = 10 um: numeric 6.065e+13, analytic 6.065e+13
# x = 20 um: numeric 3.678e+13, analytic 3.678e+13
# x = 40 um: numeric 1.350e+13, analytic 1.350e+13
# Poisson: Emax = qNW/eps = 4.639e+04 V/cm, potential drop qNW^2/(2eps) = 0.696 V

A 200-point finite-difference solution of d²p/dx² = p/L² reproduces the analytic sinh profile to four significant figures — the same method TCAD tools use, only in 2-D/3-D with coupled Poisson. Integrating Poisson across a uniformly charged 0.3 µm depletion layer (10¹⁶ cm⁻³) gives a peak field of 46 kV/cm and a 0.7 V potential drop — the built-in voltage scale of a PN junction.

Common mistakes

Applications

Key takeaways

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