Diffusion

For non degenerated semiconductors we can take the relation

\begin{equation*} n_e = n_i \exp\left(- \frac{E_i-\mu^*}{kT} \right) \qquad , \end{equation*}

(5.39)

or after transformation

\begin{equation*} \mu^* = E_i + k T \ln \left(\frac{n_e}{n_i} \right) \qquad . \end{equation*}

(5.40)

\(E_i\) is the energy in the middle of the band.
So we find

\begin{equation*} \vec{\nabla} \mu^* = k T \frac{\vec{\nabla} n_e}{n_e} \qquad . \end{equation*}

(5.41)

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

\begin{equation*} \vec{j}_{diff} = - k T \mu_e \vec{\nabla}n_e = - q D \vec{\nabla}n_e \label{j_diff} \qquad . \end{equation*}

(5.42)

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

\begin{equation*} D = \frac{k T}{q} \mu_e \qquad . \end{equation*}

(5.43)