Diffusion Current

The hole concentration profile across a Si bar can be described with $p (x)$ , and the electron concentration profile with $n (x)$ . From this distribution, the movement of charges produces a diffusion current.

$D_{p}$ - Hole diffusion constant ( $D_{p} = 12 \frac{c m ^{2}}{s}$ for intrinsic Si)
$D_{n}$ - Electron diffusion constant ( $D_{n} = 35 \frac{c m ^{2}}{s}$ for intrinsic Si)

Einstein Relation
The diffusion constants and charge mobility are related by the thermal voltage.
$\frac{D _{n}}{μ _{n}} = \frac{D _{p}}{μ _{p}} = V_{T}$
$V_{T} = \frac{k T}{q}$ (~25.9mV at room temperature of $T = 300 K$ )
$k = 8.617 \times 1 0^{- 5} e V K^{- 1}$

Diffusion current is proportional to the concentration gradient at a point.
Hole diffusion current density
$J_{D, p} = - q D_{p} \frac{d p ( x )}{d x}$

Electron diffusion current density
$J_{D, n} = q D_{n} \frac{d n ( x )}{d x}$

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Diffusion Current