Diffusion of carriers and conductivity

Diffusion of charge carriers is the movement of electrons or holes from a region of high concentration to low concentration due to their random thermal motion, even when no electric field is present. It occurs whenever the carrier concentration is non-uniform, e.g. near a pn junction or when light injects extra carriers at one end of a bar.

Why it happens

Carriers move randomly with thermal velocity. Across any plane, more carriers cross from the crowded side than from the sparse side, so there is a net flow down the gradient. By Fick's law the flux is

$$F = -D\frac{dn}{dx}$$

where $D$ (cm² s⁻¹) is the diffusion coefficient.

Diffusion current

$$J_{e,diff} = eD_e\frac{dn}{dx},\qquad J_{h,diff} = -eD_h\frac{dp}{dx}$$

 n high  o o o o o o  -->  o   o    n low
         electrons diffuse -->
         electron current  <--

Contribution to conduction

• Diffusion adds a second current component to the drift current, so the total is

$$J_e = en\mu_e\mathcal{E} + eD_e\frac{dn}{dx}$$

• It lets current flow where there is little or no field, e.g. minority carriers crossing the base of a transistor or the neutral regions of a forward-biased diode. The forward current of a diode is essentially a diffusion current.
• $D$ and $\mu$ are linked by the Einstein relation $D/\mu = kT/e$, so carriers with high mobility also diffuse quickly.
• In equilibrium (e.g. a pn junction with no bias) diffusion is balanced by an equal and opposite drift current, so the net current is zero.

Thus diffusion does not change the bulk conductivity $\sigma = e(n\mu_e + p\mu_h)$ of a uniform sample, but in devices with concentration gradients it is often the main way current is carried.