$\Rightarrow$
$n=\frac{1}{2\pi^2}(\frac{2m_nk_{\text{B}}T}{\hbar^2})^{3/2}
\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})
- \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
+ \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
\underbrace{2(\frac{m_nk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{c}}}
\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})=
N_{\text{c}}\exp(-\frac{E_{\text{c}}-\mu_{\text{F}}}{k_{\text{B}}T})$
$\mu_{\text{F}}-\epsilon >> k_{\text{B}}T$
$\Rightarrow$
$1-f(\epsilon,T)=
- 1-\frac{1}{\exp(\frac{\mu_{\text{F}}-\epsilon}{k_{\text{B}}T})+1}\approx
+ 1-\frac{1}{\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})+1}\approx
\exp(\frac{\epsilon-\mu_{\text{F}}}{k_{\text{B}}T})$\\
Parabolic approximation:
$D_v(\epsilon)=\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}(E_v-\epsilon)^{1/2}$
\frac{1}{2\pi^2}(\frac{2m_p}{\hbar^2})^{3/2}
\exp(-\frac{\mu_{\text{F}}}{k_{\text{B}}T})
\int_{-\infty}^{E_{\text{v}}}(E_v-\epsilon)^{1/2}
- \exp(-\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
+ \exp(\frac{\epsilon}{k_{\text{B}}T})d\epsilon$\\
Use: $x=(E_{\text{v}}-\epsilon)/k_{\text{B}}T$
$\Rightarrow\epsilon=E_{\text{v}}-xk_{\text{B}}T$ and
- $d\epsilon=-k_{\text{B}}Tdx$\\
+ $d\epsilon={\color{red}-}k_{\text{B}}Tdx$\\
$\Rightarrow$
$p=\frac{1}{2\pi^2}(\frac{2m_pk_{\text{B}}T}{\hbar^2})^{3/2}
\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})
- \underbrace{\int_0^{\infty}x^{1/2}e^{-x}dx}_{=1/2\sqrt{\pi}}=
+ \underbrace{\int_{{\color{red}0}}^{{\color{red}\infty}}x^{1/2}e^{-x}dx}_{=\frac{\sqrt{\pi}}{2}}=
\underbrace{2(\frac{m_pk_{\text{B}}T}{2\pi\hbar^2})^{3/2}}_{=N_{\text{v}}}
\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})=
N_{\text{v}}\exp(\frac{E_{\text{v}}-\mu_{\text{F}}}{k_{\text{B}}T})$