Organic semiconductors have attracted comprehensive interest among researchers all over the world, in particular as active components in organic electronic devices and molecular electronics. Compared with inorganic semiconductors, organic semiconductor materials possess advantages such as low cost, versatility of chemical synthesis, ease of processing, low weight, and flexibility. However, the performances of most current organic electronic devices are still limited by the relatively low carrier mobility of the organic material. Investigating the structure/property relationships for the ultimate design of new organic semiconductor materials with higher mobility has been a subject of great interest for many years.1-2 Nevertheless, the successful relations of microscopic molecular and structural characteristics to their macroscopic mobility properties, especially for the intrinsic anisotropic carrier mobility, are yet to be fully clarified. Only recently, with the development of single-crystal organic field-effect transistors (SCOFETs), the direct measurement of carrier mobility as an explicit function of intermolecular proximity and orientation within an organic crystal structure is allowed,3-5 which facilitate the direct comparison between theory and experiment, and provide the opportunity for theoretician to establish the quantitative relationship even the explicit analytical expression of the intrinsic anisotropic mobility of organic semiconductor crystals. For the final “functionality by design” of organic materials, a successful analytical expression of anisotropic mobility is very helpful and important since most organic crystals show the pronounced anisotropy which has to be taken into account for device design in the commercial application.
From the 1950s, significant progress has been made toward improved understanding of intrinsic charge-transport phenomena in organic materials, and several models, such as band model, tight-binding model and hopping model, have been proposed for the analysis and simulation of low-density intrinsic transport behavior in the organic crystals observed in OFET experiments.6 In most cases, the hopping model is one of the most appropriate method to describe carrier transport in organic semiconductor materials, especially at room temperature, due to the fact that organic molecules are usually aggregated by weak van der Waals forces and thus the intermolecular electronic couplings (electron transfer integral) are much weaker than the electron-vibration couplings (reorganization energy) for the majority of conjugated organic oligomers. In the hopping model, the intrinsic charge-transport rates for electron and hole transport mainly rely on two contributions. The first is the geometric relaxation of the molecule (inner reorganization energy) and its surroundings (outer reorganization energy) on movement of the charge carriers. The second is the magnitude of the intermolecular electronic coupling, which is intimately related to crystal packing. The former is mostly the energy change of a single molecule on charge addition/removal (inner reorganization energy), because contributions from the electronic and nuclear polarization/relaxation of the surrounding medium are significantly smaller.7 The latter can be approximated as nearest-neighbor contributions, as the electronic couplings fall off rapidly with intermolecular separation. In addition, electronic couplings between adjacent molecules in crystals are also highly sensitive to the molecular packing motif, such as the relative positions of the interacting molecules and intermolecular orientations.8
Another significant aspect of organic crystal transport properties is the anisotropy of charge transport on organic surfaces, which mainly originates from the sensitivity of electronic couplings to the mode of packing.9 In the last few years, the mobility anisotropy has received more and more attention as the continuous development of organic field-effect transistors (OFETs). In 2004, Sundar et al. investigated the dependence of the field-effect mobility on the orientation of the transistor channel relative to the crystallographic axes, and observed for the first time a strong anisotropy of the intrinsic hole mobility within the a-b plane of single crystals of rubrene in field-effect experiment.5 Later on, scientists found that the anisotropic field effect is common in various organic crystals, such as linear acenes and their derivatives/analogues.4, 9 These experimental results measured through single-crystal devices not only give us an opportunity to completely understand the charge transport mechanisms, but also provide references for the further theoretical study on relationships between the microscopic molecular packing and macroscopic charge transport of the materials since the crystal packing and molecular orientation are clearly fixed. On the theoretical side, there are several theoretical studies about anisotropic hole/electron mobilities and these investigations gave a detailed analysis and qualitative simulation of the anisotropy of charge transport behavior.10-12 However it should be noted that even though the Holstein–Peierls model can well describe the temperature-dependence and anisotropy of charge-carrier mobilities in organic molecular crystals, it does not give quantitatively predictive values, and in some cases the calculation results from the Holstein–Peierls model is about one to two orders of magnitude larger than that of the single-crystal experimental measurements.7 Moreover, the master equation method coupled with the Marcus–Hush electron transfer theory provides with an efficient method to numerically solve the anisotropic charge-carrier mobility from the molecular packing structure,11 nevertheless, master-equation method is always complex and does not present the inherent relationship between molecular packing architecture parameters and the mobility anisotropy in organic materials, which is very important for the design of new molecular materials and the improvement of the device performance. It will be very helpful for “design” if we can use a simpler and more intuitive model to offer clear physical insight. For this purpose, by ignoring diffusion pathways interaction in the one-dimensional diffusion equations, we developed a first-principles-based simulation model predicting anisotropic hole/electron mobility of organic crystals with only crystal structures needed.13 The model leads to the first analytical expression for predicting angular resolution anisotropic mobility in the organic crystal, and the mobility orientation function µΦ explicitly shows how the hole/electron mobility correlates with the molecular packing and the underlying atomistic electronic properties, which is very useful in aiding synthetic design of organic semiconductors. In spite of the approximations and the simple one-dimensional diffusion model, the analytical expression of our angular resolution anisotropic mobility function made surprising good predictions for the anisotropic mobility distributions in many organic molecular semiconductors such as linear acene, acene derivates, perylene bisimide derivatives, and oligothiophenes as well as their derivatives/analogues. Recently, it has been more and more widely used in the description of charge transfer behaviour and provides a guideline for “tailoring” new organic compounds for organic electronics.13-28
In this procotol, we mainly describe efforts in our team over the past few years to propose a theoretical model to establish the quantitative relationship between angular resolution anisotropic mobilities and molecular packing architecture parameters (r, θ, and γ) as well as underlying electronic properties of organic materials, and to simulate the anisotropic hole/electron mobilities of typical p-/n-type organic semiconductor materials. In Section II, we briefly describe our simulation model based on quantum chemical approaches to compute the molecular parameters that govern the intermolecular charge transfer process, followed by our proposed mobility orientation function μΦ(V, λ, r, θ, γ; Φ) that describes the mobility in a specific conducting direction on a specific surface in the organic crystal. Section III and Section IV focus on the applications of our simulation model to the p-type and n-type organic semiconductor materials, respectively. The conduction mechanism and the resistances at the TTF-TCNQ organic hetero-interface, based on our calculations, are discussed in Section V. A summary and outlook are present in the last section.