Numerical recipes

The concentration of free carriers in the conduction band can be calculated by (6.1)$$\begin{gathered} n = 4\pi {\left( {\frac{{2{m^*}{k_B}T}}{{{h^2}}}} \right)^{3/2}}\int_0^\infty {\frac{{{x^{1/2}}dx}}{{\exp (x - \eta ) + 1}}} \hfill \\ = \frac{{2{N_c}}}{{\sqrt \pi }}{F_{1/2}}(\eta ) \hfill \\ \end{gathered} $$ where N _c is the effective density of states in the conduction band, m* is the density-of-state effective mass, x = E/k _B T is the carrier energy in unit of k _B T, η =E _f /k _B T is the Fermi level in unit of k _B T F_1/2 (η) is the Fermi-Dirac integral of order of 1/2.