Calculate the charge carrier densities $n$ and $p$ for $E_{\text{c}}-\mu_{\text{F}} >> k_{\text{B}}T$ and $\mu_{\text{F}}-E_{\text{v}} >> k_{\text{B}}T$.\\\\
{\bf Hint:}
-Consider the influence of these two conditions for the energy of the states, which are situated in the conduction and valence band.
-The parabolic approximation of the density of states of electrons in the conduction and holes in the valence band with the effective masses $m_n$ and $m_p$ is given by
+Consider the influence of these two conditions for the energy of the states, which are situated in the conduction and valence band, and the consequences for the respective occupation described by the Fermi distribution.
+The parabolic approximation of the density of states of electrons in the conduction band and holes in the valence band (effective masses $m_n$ and $m_p$) is given by
$D_c(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_n}{\hbar^2})^{3/2}(\epsilon-E_c)^{1/2}$ and
$D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$.
-If you do not calculate the non-simplified Fermi-integral the substitutions $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$ and $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$ can be used. Furthermore use the equality $\int_0^{\infty} x^{1/2} e^{-x} dx = 1/2 \sqrt{\pi}$.
+Simplify the Fermi function before calculating the integral and use the substitutions $x=(\epsilon - E_{\text{c}})/k_{\text{B}}T$ and $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$.
+Furthermore use the equality $\int_0^{\infty} x^{1/2} e^{-x} dx = \frac{\sqrt{\pi}}{2}$.
\end{document}
projects / lectures / latex.git / blobdiff