In your derivation you may use the following expressions for the hole density of states g, perunit volume and the electron Fermi-Dirac distribution f(E): g_{h}=\frac{4 \pi\left(2 m_{\mathrm{h}}^{*}\right)^{\frac{3}{2}}}{h^{3}} \sqrt{-E} f(E)=\frac{1}{1+\exp \left(\frac{E-E_{F}}{k_{B} T}\right)}

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