Continuity Equation

When a diode is switched on, minority carriers are injected into both sides. The continuity equation tracks what happens to them over time and space. Every transient analysis in a circuit simulator solves this equation numerically.

Core Concept

The continuity equation is a conservation law for carriers. It says: the rate of change of carrier concentration at any point equals generation minus recombination plus the net flow of carriers into that point by drift and diffusion. It applies to both electrons and holes, and it links the current equation with the carrier dynamics.

For minority holes in an N-type region, the continuity equation is: ∂p/∂t = Dp * ∂²p/∂x² - (p - p0)/τp + G, where p0 is the equilibrium hole concentration, τp is the minority carrier lifetime, and G is the optical generation rate. Under steady state with no illumination (G=0, ∂p/∂t = 0), this simplifies to an ordinary differential equation with an exponential solution.

The steady-state solution gives pn(x) = pn0 + A * exp(-x/Lp), where Lp = sqrt(Dp * τp) is the diffusion length. Minority holes injected at a forward-biased p-n junction decay exponentially with distance. A longer τp (less recombination) means a longer Lp and more minority carriers reaching deeper into the N-region, increasing diode current. This is why carrier lifetime is a critical parameter in power rectifiers like the 1N4007.

Key Equations

Continuity equation for minority holes in N-type:

∂p/∂t = Dp * ∂²p/∂x² - (p - p0)/τp + G

Continuity equation for minority electrons in P-type:

∂n/∂t = Dn * ∂²n/∂x² - (n - n0)/τn + G

Steady-state solution (G=0, ∂p/∂t=0):

pn(x) = pn0 + [pn(0) - pn0] * exp(-x / Lp)

Diffusion length:

Lp = sqrt(Dp * τp) and Ln = sqrt(Dn * τn)

Minority carrier concentration at junction under forward bias V:

pn(0) = pn0 * exp(V / VT)

Given:
  N-type silicon with equilibrium hole concentration pn0 = 10^4 cm^-3
  Forward bias V = 0.5 V
  Hole diffusion coefficient Dp = 10 cm^2/s
  Minority hole lifetime τp = 1 μs = 10^-6 s
  VT = 0.026 V at 300K

Why this formula:
  Steady-state continuity equation gives exponential decay.
  Boundary condition at x=0 uses the Law of the Junction.

Formula:
  pn(0) = pn0 * exp(V / VT)
  Lp = sqrt(Dp * τp)
  pn(x) = pn0 + [pn(0) - pn0] * exp(-x / Lp)

Substitution:
  pn(0) = 10^4 * exp(0.5 / 0.026)
  Lp = sqrt(10 × 10^-6)

Calculation:
  0.5 / 0.026 = 19.23
  exp(19.23) = 2.24 × 10^8
  pn(0) = 10^4 × 2.24 × 10^8 = 2.24 × 10^12 cm^-3
  Lp = sqrt(10^-5) = 3.162 × 10^-3 cm = 31.6 μm
  pn(x) ≈ 2.24 × 10^12 * exp(-x / 31.6μm)  [since pn(0) >> pn0]

Final Answer:
  Hole concentration at junction pn(0) = 2.24 × 10^12 cm^-3
  Diffusion length Lp = 31.6 μm
  Profile decays to 1/e at x = 31.6 μm from the junction

Exam Tip: GATE consistently asks about the steady-state solution of the continuity equation. The key boundary conditions are: at x=0 (junction), use pn(0) = pn0 * exp(V/VT), the Law of the Junction. At x = infinity (or ohmic contact), p returns to pn0. The solution is always exponential with decay constant Lp. Students forget to subtract pn0 from the full expression and write pn(x) = pn(0)*exp(-x/Lp) instead of pn0 + [pn(0)-pn0]*exp(-x/Lp). This error gives wrong diode current calculations.

Key Properties

• The continuity equation combines diffusion, drift, generation, and recombination into one equation. It is the most general carrier transport equation.
• Diffusion length Lp = sqrt(Dp * τp). For silicon with τp = 1 μs and Dp = 10 cm^2/s, Lp = 31.6 μm.
• At x = Lp from the junction, excess minority carrier concentration falls to 1/e (about 37%) of its value at x = 0.
• Longer minority carrier lifetime τp means longer Lp and larger diode forward current at the same voltage.
• Under optical illumination, the G term adds a uniform generation throughout the semiconductor. This is the basis of solar cell operation.
• In a BJT like the BC547, the base width W must be much less than the minority carrier diffusion length Ln for high current gain.
• Excess minority carrier concentration Δp = pn(x) - pn0 satisfies the simplified equation: d²(Δp)/dx² = Δp / Lp^2 in steady state.

Quick Revision

• Continuity equation: ∂p/∂t = Dp*∂²p/∂x² - (p-p0)/τp + G.
• Steady-state, no light: d²(Δp)/dx² = Δp / Lp^2.
• Solution: Δp(x) = Δp(0) * exp(-x/Lp) where Lp = sqrt(Dp*τp).
• Boundary condition at junction: pn(0) = pn0 * exp(V/VT). This is the Law of the Junction.
• Diffusion length Lp = 31.6 μm for τp = 1 μs, Dp = 10 cm^2/s.
• BJT gain improves when base width W << Ln (minority electron diffusion length in base).
• Illumination adds the G term, enabling solar cell and photodiode analysis.
• Exam trap: Writing pn(x) = pn(0)*exp(-x/Lp) without adding the equilibrium term pn0. The correct expression is pn(x) = pn0 + [pn(0) - pn0]*exp(-x/Lp). The error changes the calculated diode saturation current by orders of magnitude.