Drift and diffusion current density

In a semiconductor, current is carried by electrons and holes, and each carrier can move by two independent mechanisms: drift (due to an electric field) and diffusion (due to a concentration gradient). Metals have only drift current because their electron density is uniform; in semiconductors carrier concentrations can vary strongly with position (doping profiles, junctions, light injection), so both terms matter.

1. Drift current

When a field $\mathcal{E}$ is applied, electrons drift opposite to it and holes along it with velocities $v_d = \mu\mathcal{E}$:

$$J_{e,drift} = en\mu_e\mathcal{E},\qquad J_{h,drift} = ep\mu_h\mathcal{E}$$

2. Diffusion current

Carriers in random thermal motion spread from high to low concentration (Fick's law: flux $= -D\,dn/dx$):

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

(The signs differ because electrons are negative.)

 high n  ======>  low n     diffusion (no field)
 e- <---  E --->           drift (field)

3. Total current density

$$\begin{aligned} J_e &= en\mu_e\mathcal{E} + eD_e\frac{dn}{dx}\\ J_h &= ep\mu_h\mathcal{E} - eD_h\frac{dp}{dx}\\ J &= J_e + J_h \end{aligned}$$

The two mechanisms are linked by the Einstein relation $D/\mu = kT/e$, because both come from the same random thermal motion and scattering.

Justification with examples

So the current density in semiconductor devices is set by both drift and diffusion of electrons and holes.