Charge carrier density of organics with Gaussian density of states: Analytical approximation for the Gauss–Fermi integral

Active layers in organic devices prepared in solution based preparation routes are usually disordered. Their highest occupied molecular orbitals and lowest unoccupied molecular orbitals, or the valence and conduction band states, show an energetic distribution which can be approximated by a Gaussian density of states (DOS). The resulting dependency of the (electron and hole) mobility on temperature, carrier density, and field can be easily implemented into advanced device simulation programs. However, in addition the charge carrier density is needed as the integral over the DOS multiplied with the Fermi–Dirac distribution. We denote this normalized quantity as the Gauss–Fermi integral. Since it cannot be evaluated analytically, similarly as in the case of the Fermi–Dirac integral F1/2, an analytical approximation is needed for efficient device simulation. In the present article, such an approximation is proposed with different expressions in the nondegenerate and degenerate regions with a continuous and differentiable transition between both regions. The approximation is also applicable to traps with a Gaussian DOS.