Diffusion Current

When you charge a capacitor through a resistor, the current that flows inside the silicon is largely diffusion current, not drift current. Understanding diffusion current explains why BJT gain depends on base width.

Core Concept

Diffusion is the random thermal motion of carriers from a region of high concentration to a region of low concentration. No electric field is needed. The driving force is the concentration gradient, dn/dx or dp/dx. This is the same principle as perfume spreading in a room.

In a p-n junction diode like the 1N4007, when forward bias is applied, minority carriers are injected across the junction. Electrons injected into the P-side and holes injected into the N-side diffuse away from the junction. This diffusion current is what makes up most of the forward current in a diode. The diffusion current density for electrons is Jn = q * Dn * dn/dx, where Dn is the electron diffusion coefficient.

In a BJT like the BC547, minority carrier electrons diffuse across the thin P-type base from emitter to collector. A narrower base means a steeper gradient and higher diffusion current, which means higher gain (hFE). This is why base width directly controls transistor current gain in every BJT you will ever use.

Key Equations

Electron diffusion current density:

Jn(diff) = q * Dn * (dn/dx) in A/cm^2, where q = 1.6 × 10^-19 C, Dn = electron diffusion coefficient in cm^2/s

Hole diffusion current density:

Jp(diff) = -q * Dp * (dp/dx) where the negative sign accounts for positive charge moving down the gradient

Total current density (drift + diffusion):

Jn = q * n * μn * E + q * Dn * (dn/dx)

Diffusion coefficient for electrons in silicon at 300K:

Dn ≈ 25 cm^2/s, Dp ≈ 10 cm^2/s

Given:
  N-type silicon bar
  Electron concentration at x = 0: n(0) = 10^17 cm^-3
  Electron concentration at x = 2 μm: n(2μm) = 10^15 cm^-3
  Electron diffusion coefficient Dn = 25 cm^2/s
  q = 1.6 × 10^-19 C

Why this formula:
  A concentration difference over a distance creates a gradient.
  Diffusion current uses Jn = q * Dn * (dn/dx).

Formula:
  dn/dx = (n(2μm) - n(0)) / (2 × 10^-4 cm)
  Jn = q * Dn * (dn/dx)

Substitution:
  dn/dx = (10^15 - 10^17) / (2 × 10^-4)
  dn/dx = (-9.9 × 10^16) / (2 × 10^-4)
  dn/dx = -4.95 × 10^20 cm^-4

Calculation:
  Jn = 1.6 × 10^-19 × 25 × (-4.95 × 10^20)
  Jn = 1.6 × 10^-19 × 1.2375 × 10^22
  Jn = -1.98 A/cm^2

Final Answer:
  Diffusion current density Jn = -1.98 A/cm^2
  Magnitude = 1.98 A/cm^2
  Negative sign means electron diffusion is in the +x direction
  (conventional current flows in -x direction)

Exam Tip: The sign convention for diffusion current confuses most students in GATE. Electron diffusion is in the direction of falling electron concentration, but conventional current is opposite to electron flow. So Jn = +q * Dn * dn/dx (positive q because the result gives conventional current direction). For holes, Jp = -q * Dp * dp/dx. Memorise that the signs are opposite for electrons and holes in the diffusion term, unlike drift where both carry the same sign convention with their respective charge.

Key Properties

• Diffusion current requires no electric field. It is driven purely by concentration gradient dn/dx or dp/dx.
• Electron diffusion coefficient in silicon: Dn ≈ 25 cm^2/s. Hole diffusion coefficient: Dp ≈ 10 cm^2/s at 300K.
• In a BJT base, minority carriers (electrons in NPN) diffuse from emitter-base junction to collector-base junction. Shorter base = steeper gradient = larger Ic.
• Diffusion current and drift current coexist in a semiconductor. At thermal equilibrium, they exactly cancel each other, giving zero net current.
• Diffusion current increases with temperature because Dn and Dp increase with temperature (D is proportional to μkT/q and μ decreases, but kT increases faster at lower doping).
• The minority carrier diffusion current in a diode is the dominant component of the total forward current at low to moderate forward bias voltages.

Quick Revision

• Diffusion current: carriers move from high to low concentration region.
• Jn(diff) = q * Dn * dn/dx. Jp(diff) = -q * Dp * dp/dx.
• Dn ≈ 25 cm^2/s, Dp ≈ 10 cm^2/s for silicon at 300K.
• No electric field is required for diffusion current to flow.
• At equilibrium, diffusion current = drift current in magnitude. Net current = 0.
• BJT current gain (hFE) increases as base width decreases because diffusion gradient steepens.
• Diffusion length Ln = sqrt(Dn * τn) tells how far minority carriers diffuse before recombining.
• Exam trap: Students apply the same sign formula for both electron and hole diffusion currents. The signs are different. Jn = +q*Dn*dn/dx but Jp = -q*Dp*dp/dx. Getting this wrong reverses the direction of current.