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

n 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 Mehmetoğlu (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 G( α ,β ,ξ) =∫ _-∞^∞ e^-α( x-β) ^2/1+e^x-ξdx, where G( α ,β ,ξ) 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 α , β , and ξ valid ∀ α ε ℝ≥ 0 , ∀ β ε ℝ , and ∀ ξ ε ℝ .