Diffusion

For non degenerated semiconductors we can take the relation

$n_{e} = n_{i} exp (- \frac{E_{i} - μ^{*}}{k T}),$

(5.39)

or after transformation

$μ^{*} = E_{i} + k T ln (\frac{n_{e}}{n_{i}}) .$

(5.40)

$E_{i}$ is the energy in the middle of the band.
So we find

$\vec{\nabla} μ^{*} = k T \frac{\vec{\nabla} n_{e}}{n_{e}} .$

(5.41)

Neglecting electrical fields the second term in Eq. (5.34) is written as

${\vec{j}}_{d i f f} = - k T μ_{e} \vec{\nabla} n_{e} = - q D \vec{\nabla} n_{e} .$

(5.42)

Comparing the left and the right hand side of Eq. (5.42) we get the Einstein relation

$D = \frac{k T}{q} μ_{e} .$

(5.43)