Fermi level of a semiconductor

Question

When calculating the career concentrations in the conduction band of a intrinsic semiconductor we consider the integral $\int_{E_c}^\infty g_C(E)f_{FD}(E,T)dE$ where $g_c$ is the density of states in conduction band and $f_{FD}$ is the Fermi-Dirac distribution. $$f_{FD}=\frac1{1+e^{(E-E_F)/k_BT}}$$ where $E_F$ is the intrinsic fermi level. I have a problem that, does the intrinsic fermi level $E_F$ depend on temperature ?

Answer 1

The Fermi energy $E_F$ is defined as the chemical potential $\mu$ at $T=0\,$K, so it doesn't depend on the temperature.

Answer 2

Yes, it does depend on temperature. Fermi level is defined as, $$ \tag 1 \mu (\left\langle N\right\rangle ,T)=\left({\frac {\partial F}{\partial \left\langle N\right\rangle }}\right)_{T}, $$

Substituting Helmholtz free energy, $F=U-TS$ into it , we get : $$\begin{align} \tag 2 \mu (\left\langle N\right\rangle ,T)&= {\frac {\partial (U-TS)}{\partial \left\langle N\right\rangle }}\\&= \frac{\partial U}{\partial \langle N \rangle}-\frac{\partial (TS)}{\partial \langle N \rangle}\\&= \frac{\partial U}{\partial \langle N \rangle}- T\frac{\partial S}{\partial \langle N \rangle}-S\frac{\partial T}{\partial \langle N \rangle} \end{align}$$