Carrier hopping in disordered semiconducting polymers: How accurate is the Miller–Abrahams model?

Abstract

We performed direct calculations of carrier hopping rates in strongly disordered conjugated polymers based on the atomic structure of the system, the corresponding electronic states and their coupling to all phonon modes. We found that the dependence of hopping rates on distance and the dependence of the mobility on temperature are significantly different than the ones stemming from the simple Miller-Abrahams model, regardless of the choice of the parameters in the model. A model that satisfactorily describes the hopping rates in the system and avoids the explicit calculation of electron-phonon coupling constants was then proposed and verified. Our results indicate that, in addition to electronic density of states, the phonon density of states and the spatial overlap of the wavefunctions are the quantities necessary to properly describe carrier hopping in disordered conjugated polymers.The carrier mobility of conjugated polymer materials is the most important physical property 1-6 for their application in organic electronic devices. Realistic polymer materials contain both crystalline (ordered) and amorphous (disordered) regions.7 Charge carrier transport is often limited by the presence of amorphous regions. In these regions the electronic states are localized due to presence of disorder and carrier transport takes place by phononassisted carrier hopping 8 between localized states. Such transport is traditionally modeled by assuming a certain density of electronic states in energy and space and a certain form of hopping probabilities between them.6,9-12 Different models are distinguished by the electronic density of states (DOS) assumed in the model, which is usually the tail of the Gaussian 9,10,12 or the exponential 11 distribution. The transition rates are typically assumed to decay exponentially with the distance between localized states, 9-12 in the Miller-Abrahams (MA) form.13 Free parameters that appear in the models are fitted to the experimental mobility measurements.From the applications of such models, it is widely understood that the mobility of the material strongly depends on the electronic DOS which is therefore believed to be the most important material property when charge transport is concerned.14 On the other hand, much less effort has been put into understanding how different forms of the transition rates affect the transport and in particular whether the MA expression is suitable at all. To address these questions, in this letter we perform direct ab-initio calculations of the transition rates between electronic states starting from atomic structure of the system, followed by explicit calculation of electronic state wavefunctions and their coupling to phonons. We find strong deviations of the transition rate dependence on distance from the MA form and analyze the consequences of this on electronic transport.Electronic structure calculations were performed using the plane wave pseudopotential approach and the charge patching method for organic systems introduced in Ref. 15 which gives the ac...

A very general rate expression for charge hopping in semiconducting polymers

“…It was shown for example that the distance dependence normally included in the Miller-Abrahams rate is too approximate. 25 This was later found also in ref. 23 where the importance of variable localization of states was highlighted.…”

“…A similar conclusion was achieved with different arguments in ref. 25 . Moreover, if quantum inducing modes are not too important, one can neglect them altogether using the equation valid in the classical limit of inducing modes (eq.…”

“…The most used rate expression to study charge transport in polymers is the Miller-Abrahams rate 12 k for the hopping between two electronic states (say 1 and 2): 8,[24][25][26] have used an alternative expression where the hopping between states is promoted via electron-phonon coupling by a phonon matching the energy difference between them. The results depend strongly on the phonon spectrum, unlike the other two models mentioned above.…”

How Many Parameters Actually Affect the Mobility of Conjugated Polymers?

“…Following ref. [14] we incorporate the effect of the distance between initial and final states by parametrizing the coupling as …”

Hole Transport in Diketopyrrolopyrrole (DPP) Small Molecules: A Joint Theoretical and Experimental Study

“…Notably, recent work has suggested that deviations from these hopping rates can be observed from first-principles simulations. 26 To capture the dynamic disorder, model Hamiltonians such as Holstein and Peierls types have been widely used but with mixed success, 27−29 partly because these model Hamiltonians are based on perturbation theories and thus are restricted to specific ranges of microscopic parameters. More importantly, the aforementioned models are not materialspecific and contain empirical parameters that need to be predetermined; thus they lack predictability that is crucial for computational materials design.…”