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

Analytical evaluation of the charge carrier density of organic materials with a Gaussian density... 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 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 \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} G α , β , ξ = ∫ - ∞ ∞ e - α x - β 2 1 + e x - ξ d x , where $$G\left( \alpha ,\beta ,\xi \right)$$ 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 $$\alpha$$ α , $$\beta$$ β , and $$\xi$$ ξ valid $$\forall$$ ∀ $$\alpha$$ α $$\varepsilon$$ ε $${\mathbb {R}} \ge 0$$ R ≥ 0 , $$\forall$$ ∀ $$\beta$$ β $$\varepsilon$$ ε $${\mathbb {R}}$$ R , and $$\forall$$ ∀ $$\xi$$ ξ $$\varepsilon$$ ε $${\mathbb {R}}$$ R . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational Electronics Springer Journals

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

, Volume 17 (1) – Nov 23, 2017
7 pages

/lp/springer_journal/analytical-evaluation-of-the-charge-carrier-density-of-organic-0GZKFcdS9s
Publisher
Springer US
Subject
Engineering; Mathematical and Computational Engineering; Electrical Engineering; Theoretical, Mathematical and Computational Physics; Optical and Electronic Materials; Mechanical Engineering
ISSN
1569-8025
eISSN
1572-8137
D.O.I.
10.1007/s10825-017-1113-5
Publisher site
See Article on Publisher Site

### Abstract

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 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 \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} G α , β , ξ = ∫ - ∞ ∞ e - α x - β 2 1 + e x - ξ d x , where $$G\left( \alpha ,\beta ,\xi \right)$$ 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 $$\alpha$$ α , $$\beta$$ β , and $$\xi$$ ξ valid $$\forall$$ ∀ $$\alpha$$ α $$\varepsilon$$ ε $${\mathbb {R}} \ge 0$$ R ≥ 0 , $$\forall$$ ∀ $$\beta$$ β $$\varepsilon$$ ε $${\mathbb {R}}$$ R , and $$\forall$$ ∀ $$\xi$$ ξ $$\varepsilon$$ ε $${\mathbb {R}}$$ R .

### Journal

Journal of Computational ElectronicsSpringer Journals

Published: Nov 23, 2017

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