A map of high-mobility molecular semiconductors
Résumé
The charge mobility of molecular semiconductors is limited by the large fluctuation of intermolecular transfer integrals, often referred to as oï¿¿-diagonal dynamic disorder, which causes transient localization of the carriers' eigenstates. Using a recently developed theoretical framework, we show here that the electronic structure of the molecular crystals determines its sensitivity to intermolecular fluctuations. We build a map of the transient localization lengths of high-mobility molecular semiconductors to identify what patterns of nearest-neighbour transfer integrals in the two-dimensional (2D) high-mobility plane protect the semiconductor from the eï¿¿ect of dynamic disorder and yield larger mobility. Such a map helps rationalizing the transport properties of the whole family of molecular semiconductors and is also used to demonstrate why common textbook approaches fail in describing this important class of materials. These results can be used to rapidly screen many compounds and design new ones with optimal transport characteristics. I n the comparison of the charge transport properties of families of materials it is very desirable to identify a few key parameters that can be used to rationalize the observed dierences in charge mobility. For example, eective masses and mean free paths can be used for wide-band semiconductors, hopping rates between nearest-neighbour sites would characterize molecular solids where molecular orbital overlaps are very weak, and relatively simple phe-nomenological theories are available to describe transport in highly disordered materials. Such a simple reduction is not yet established for the class of high-mobility molecular semiconductors-that is, those displaying mobilities exceeding ⇠1 cm 2 V 1 s 1 and the most interesting from the technological point of view. As noted many times in the past 1-4 , band transport models are unsatisfactory for these materials because of a too short mean free path and, similarly, hopping theories yield unphysically high hopping rates 5 alongside incorrect temperature dependences, even when they reproduce the absolute mobility. A number of authors have contributed over the years to develop a transport model that seems suitable for this class of materials and that has now reached a high level of predictive power 4,6. A starting point is the observation that the transfer integrals between nearest neighbouring molecules undergo large fluctuations 7 : due to the softness of the intermolecular interaction and the sensitivity of the transfer integrals to small nuclear displacements 8-10 , the amplitude of these fluctuations is comparable to the average value of the transfer integrals. Developing on this idea, the carrier dynamics has been studied in models that capture the essential physics, both numerically and analytically 7,11-18 , or performing non-adiabatic molecular dynamics simulations in models with atomistic detail 19,20. All these studies revealed a common microscopic origin for the unconventional charge transport of organic semiconductors: the dynamic disorder broadens the density of states (DOS) and causes a localization of the instantaneous eigenstates. This phenomenon, whose origin is genuinely quantum mechanical, is especially strong at the band edges where charge carriers reside 12 , and is intuitively associated with a suppressed mobility. As disorder fluctuates in time, however, one cannot speak of localization in the traditional sense 21. Based on extensive numerical evidence 6,7,11-14,16 it was proposed 13,14 that the eect of dynamic disorder is to cause a transient localization over a length L ⌧ within a fluctuation time given by the inverse of the typical intermolecular oscillation frequency, ⌧ ⇠ 1/! 0. The fact that this initial localization time must be overcome before charge diusion can actually take place is responsible for the long-known breakdown of semiclassical transport, causing the mobility to fall below the Mott-Ioe-Regel limit (apparent mean free paths shorter than the intermolecular distance) 3. The eects of transient localization were given a mathematical basis using a relaxation time argument 13,14 , resulting in the following analytical formula for the charge mobility: µ = e k B T L 2 ⌧ 2⌧ (1) with e the electron charge, k B the Boltzmann constant and T the temperature. The theory embodied in equation (1), which has been shown to agree quantitatively with the most accurate numerical studies presently available 4,15 , goes beyond the semiclassical band transport approaches (as it contains quantum localization corrections) and it is alternative to traditional hopping theories (which assume that the wavefunction coherence is lost at each hop, an assumption that does not hold in high-mobility materials). This allows us to reliably and eciently compute the mobility of actual semiconducting materials, treating all possible molecular structures of interest on the same footing and with an aordable numerical eort. As the transient localization length L ⌧ is what ultimately determines how a given material performs, regardless of the detail of the model it would be particularly useful to be able to assess this quantity-and, at once, make quantitative predictions for the charge mobility-without going each time through a complex quantum dynamics simulation. To this aim we proceed to study systematically an ensemble of models, encompassing in practice all the dierent physical situations that can be encountered in organic semiconductors. We start by observing that, with the exception of fullerene