Generalized effective-medium model for the carrier mobility in amorphous organic semiconductors

Vadim Rodin, Franz Symalla, Velimir Meded, Pascal Friederich, Denis Danilov, Angela Poschlad, Gabriele Nelles, Florian von Wrochem, and Wolfgang Wenzel

Phys. Rev. B 91, 155203 – Published 10 April 2015

Abstract

Electronic transport through disordered organic materials is relevant in many applications, including organic light-emitting diodes and organic photovoltaics. The charge-carrier mobility is one of the most important material characteristics that must be optimized to make organic devices competitive. Here we introduce a general effective-medium model for the analytic calculation of zero-field mobilities on the basis of material-specific parameters that are obtained from extensive ab initio simulations. By means of kinetic Monte Carlo simulations, we generalize the model to also include the strong disorder limit. As a proof of concept the model is applied to two different disordered organic materials exhibiting medium and strong disorder, respectively. Surprisingly, even at strong disorder the hole mobilities computed with the effective-medium model in its original form are found to agree best with the experimental data. Seeking a possible explanation for this result, we investigate the strong dependence of the mobility on the connectivity of the model topology and show that the distribution of hopping matrix elements in the material is indeed much broader than assumed in simple lattice models. As the input parameters of the model can be computed on the basis of relatively small samples, this model may be used for materials’ screening without adjustable parameters.

Received 9 December 2014
Revised 5 March 2015

DOI:https://doi.org/10.1103/PhysRevB.91.155203

Authors & Affiliations

Vadim Rodin^1,*, Franz Symalla², Velimir Meded², Pascal Friederich², Denis Danilov², Angela Poschlad³, Gabriele Nelles¹, Florian von Wrochem¹, and Wolfgang Wenzel²

¹Materials Science Laboratory, SONY Deutschland GmbH, Hedelfinger Strasse 61, D-70327 Stuttgart, Germany
²Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), D-76021 Karlsruhe, Germany
³Steinbuch Centre for Computing, Karlsruhe Institute of Technology (KIT), D-76021 Karlsruhe, Germany

^*Vadim.Rodin@eu.sony.com

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Vol. 91, Iss. 15 — 15 April 2015

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Images

Figure 1
Left: zero-field mobilities of the GEMM for varying disorder strength βσ from 1 to 5 with $C = 0.25$ , 0.36, and 0.44, respectively; the dashed lines represent the KMC extrapolation to zero-field mobility. Right: KMC results for a simple cubic lattice with varying disorder strength βσ between 1 and 5 and a coordination of 6 as a function of the square root of the applied electric field. For 0.44 and 0.25 the discrepancy rapidly increases with increased disorder with 0.44 underestimating and 0.25 overestimating the mobility. $C = 0.36$ provides the best fit to the KMC results.
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Figure 2
The dependence of zero-field hole mobilities Eq. (12) on the reduced disorder parameter ${(β σ)}^{2}$ at $T = 290 K$ for the effective medium ( $C = 0.25$ , solid lines), the KMC ( $C = 0.36$ , dashed lines), and Bässler's ( $C = 0.44$ , dotted lines) models for $Al q_{3}$ (red/top lines) and α-NPD (blue/bottom lines). The circles denote experimental data for α-NPD [46, 47], and the squares denote experimental data for $Al q_{3}$ [46, 47, 48, 49, 50]. On the $x$ axis the experimental data points are positioned at the corresponding calculated energy disorder. The horizontal error bars of the experimental data points illustrate the stochastic uncertainties of energy disorder resulting from the limited sample size of our microscopic input data. The insets display the chemical formulas of the corresponding materials.
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Figure 3
Temperature dependence of zero-field hole mobilities in $Al q_{3}$ (red/three bottom lines) and α-NPD (blue/three top lines) according to the GEMM (solid lines: $C = 0.25$ ; dashed lines: $C = 0.36$ ; and dotted lines: $C = 0.44$ ). The experimental data for $Al q_{3}$ is presented as red crosses [48] and red squares [46, 47, 48, 49, 50], while for α-NPD it is shown as blue spheres [46, 47] and triangles [51]. For α-NPD, mobilities from fit-based GDM (green, dashed-dot-dot line) and correlated disorder model (CDM) (green, dashed-dashed-dot line) models [52] are additionally displayed.
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Figure 4
Ratio of zero-field mobilities obtained from GEMM and KMC for an SC lattice with disorder strength of $β σ = 5$ (typical material with such disorder: α-NPD) as function of the number of links (for each hopping site) for three values of $C$ [see Eq. (12): 0.25 (black), 0.36 (red), and 0.44 (blue)]. As indicated by the dashed line, with an increased number of links the GEMM increasingly agrees with the KMC results. The opposite trend is observed for $C = 0.36$ and 0.44 where the discrepancy increases.
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Figure 5
Scatter plot of the logarithm of the absolute values of the transfer integrals as a function of intermolecular distance for α-NPD. Although the linear behavior (exponential decay) is clearly observable, the spread indicates that an efficient connectivity exists until up to 20-Å (center-of-mass) distance between α-NPD molecules.
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