Understanding morphology-mobility dependence in PEDOT:Tos

Nicolas Rolland, Juan Felipe Franco-Gonzalez, Riccardo Volpi, Mathieu Linares, and Igor V. Zozoulenko

Phys. Rev. Materials 2, 045605 – Published 30 April 2018

Abstract

The potential of conjugated polymers to compete with inorganic materials in the field of semiconductor is conditional on fine-tuning of the charge carriers mobility. The latter is closely related to the material morphology, and various studies have shown that the bottleneck for charge transport is the connectivity between well-ordered crystallites, with a high degree of $π − π$ stacking, dispersed into a disordered matrix. However, at this time there is a lack of theoretical descriptions accounting for this link between morphology and mobility, hindering the development of systematic material designs. Here we propose a computational model to predict charge carriers mobility in conducting polymer PEDOT depending on the physicochemical properties of the system. We start by calculating the morphology using molecular dynamics simulations. Based on the calculated morphology we perform quantum mechanical calculation of the transfer integrals between states in polymer chains and calculate corresponding hopping rates using the Miller-Abrahams formalism. We then construct a transport resistive network, calculate the mobility using a mean-field approach, and analyze the calculated mobility in terms of transfer integrals distributions and percolation thresholds. Our results provide theoretical support for the recent study [Noriega et al., Nat. Mater. 12, 1038 (2013)] explaining why the mobility in polymers rapidly increases as the chain length is increased and then saturates for sufficiently long chains. Our study also provides the answer to the long-standing question whether the enhancement of the crystallinity is the key to designing high-mobility polymers. We demonstrate, that it is the effective $π − π$ stacking, not the long-range order that is essential for the material design for the enhanced electrical performance. This generic model can compare the mobility of a polymer thin film with different solvent contents, solvent additives, dopant species or polymer characteristics, providing a general framework to design new high mobility conjugated polymer materials.

Received 23 January 2018
Revised 15 March 2018

DOI:https://doi.org/10.1103/PhysRevMaterials.2.045605

Physics Subject Headings (PhySH)

Crystallization Electric field effects Percolation Polymer conformation & topology Polymer conformation changes

Polymers & Soft Matter

Authors & Affiliations

Nicolas Rolland¹, Juan Felipe Franco-Gonzalez¹, Riccardo Volpi^2,3, Mathieu Linares^4,5, and Igor V. Zozoulenko^1,*

¹Laboratory of Organic Electronics, Department of Science and Technology (ITN), Campus Norrköping, Linköping University, SE-60174 Norrköping, Sweden
²Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-58183 Linköping, Sweden
³Machine Learning and Optimization Group, Romanian Institute of Science and Technology (RIST), Cluj-Napoca, Romania
⁴School of Biotechnology, Division of Theoretical Chemistry & Biology, KTH Royal Institute of Technology, 114 21 Stockholm, Sweden
⁵Swedish e-Science Research Centre, KTH Royal Institute of Technology, 104 50 Stockholm, Sweden

^*igor.zozoulenko@liu.se

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Issue

Vol. 2, Iss. 4 — April 2018

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Images

Figure 1
Workflow to transform molecular dynamics simulations into resistive networks. (a) Morphology obtained from MD simulation $(N = 6$ and $W = 43)$ and corresponding resistive network. Only thiophene rings (white) are shown for clarity. The red lines are the edges connecting the network nodes (PEDOT geometrical centers). (b) Zoom on a small part of the structure. (c) $N / 3$ nodes are superimposed on each PEDOT chain. For each edge between two nodes a hopping rate is calculated according to a modified Miller-Abrahams expression. (d) Transfer integrals needed to calculate the hopping rates are calculated by the projection method at the ZINDO level. (e) Molecular orbital (MO) energies are drawn from a truncated gaussian distribution that represents the tail of the valence band DOS. The energy difference between MOs takes into account the electric field intensity. Molecular images were prepared using Visual Molecular Dynamics (VMD) [49].
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Figure 2
Workflow to calculate mobility from the resistive network. [(a) and (b)] The stationary state of the carriers under an electric field $F$ is described by the master equation for the occupation probabilities $p_{i}$ at every transport units $i$ . Solution of this equation is obtained iteratively; once all $p_{i}$ are calculated, the mobility is accessed via a equation given in (c) [57].
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Figure 3
Dependence of the mobility on the chain length $N$ of the PEDOT oligomers. The inset represents the same data but focusing on the dependence of the mobility on the water content $W$ for different chain length $N$ . Points represent values averaged over 13 frames (i.e., different MD simulations) and three electric field directions (along the $x, y$ , and $z$ axis) and 100 realizations of the Gaussian DOS, and the error bars are the associated standard deviations.
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Figure 4
Correlation between morphology and transfer integrals distribution. (a) Representative molecular dynamics snapshot $(N = 6, W = 0$ , only the thiophene rings are represented for clarity) where the three types of possible hops are indicated together with a set of geometrical features. (b) Representative transfer integrals distribution corresponding to the system $N = 6$ and $W = 0$ (statistics over 13 frames, the distribution is normalized to the maximum value). (c) Correlation between polymers arrangement and transfer integrals values.
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Figure 5
(a) Calculation of the percolation curve and network clustering. In this example, $ρ = 9 / (8 + 9) = 0.53$ . (b) Representative percolation curve (chain length $N = 3$ , water content $W = 0)$ . The curve exhibits a steplike behavior defining a so-called percolation threshold that corresponds to an onset for the charge carriers transport through the system. (c) Illustration of the network clustering in a real morphology (chain length $N = 3$ , water content $W = 0)$ . Left: One cluster with color scale indicating the transfer integral values between the nodes. Right: The cluster is divided in two smaller clusters when a threshold is set for the transfer integral.
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Figure 6
[(a)–(c)] Filled areas correspond to the transfer integrals distributions in various systems (statistics over 13 frames, normalized to the maximum value). Solid lines are the corresponding percolation curves. (d) Evolution of the peak A as $N$ changes for the water content $W = 0$ . (e) Schematic view of the link between microstructure and transfer integral features.
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