Analytical evaluation of the charge carrier density of organic materials with a Gaussian density of states revisited

An analytical solution for the calculation of the charge carrier density of organic materials with a Gaussian distribution for the density of states is presented and builds upon the ideas presented by Mehmeto?lu (J Comput Electron 13:960–964, 2014) and Paasch et al. (J Appl Phys 107:104501-1–104501-4, 2010). The integral of interest is called the Gauss–Fermi integral and can be viewed as a particular type of integral in a family of the more general Fermi–Dirac-type integrals. The form of the Gauss–Fermi integral will be defined as

$$\begin{aligned} G\left( \alpha ,\beta ,\xi \right) =\mathop {\displaystyle \int }\limits _{-\infty }^{\infty }\frac{ e^{-\alpha \left( x-\beta \right) ^{2}}}{1+e^{x-\xi }}\hbox {d}x\text {,} \end{aligned}$$

where \(G\left( \alpha ,\beta ,\xi \right) \) is a dimensionless function. This article illustrates a technique developed by Selvaggi et al. 3] to derive a mathematical formula for a complete range of parameters \(\alpha \), \(\beta \), and \(\xi \) valid \(\forall \) \(\alpha \) \( \varepsilon \) \( {\mathbb {R}} \ge 0\), \(\forall \) \(\beta \) \(\varepsilon \) \( {\mathbb {R}} \), and \(\forall \) \(\xi \) \(\varepsilon \) \( {\mathbb {R}} \).