Effect of microscopic Coulomb interactions on concentration dependence mobility of charge carrriers in organic materials

The filling of deep states is considered usually to be the reason of increase of the mobility of disordered organic materials with increasing concentration. However, at moderate concentrations the effect of microscopic Coulomb interactions (MCI) could be significant, because these interactions can reduce the activation energy of a hop. It is shown, that MCI results in significant additional increase of mobility along with concentration, in addition to the effect of the filling of deep states. The results are in qualitative agreement with existing theoretical and experimental studies.


Introduction
The mobility of charge carriers is one of the most important characteristics of transport. In recent years, considerable attention is paid to experimental [1] and theoretical [2][3][4][5][6] studying of transport in disordered organics in case, where the effect of charge carrier concentration on the mobility is significant. It was shown, that the mobility is increasing significantly with increasing concentration. This dependence is essential for organic field-effect transistors, where the relative concentrations could reach the value of 0,01 [1], but in case of ohmic contacts it should be also taken into account for organic light-emitting diodes, although the concentration values there are much lower.
The filling of deep states is considered to be the main reason of increase of the mobility with increasing concentration. In that case these states stop acting as "traps". This mechanism is studied in a couple of works, see for example [2][3][4][5][6]. However, at moderate concentrations the other effect could be more significant -strong Coulomb repulsion of randomly nearing charges, which accelerates the release of charges from deep states. This effect is considered in present work analytically.

Model for mobility calculation considering microscopic Coulomb interactions
Transport in organic materials occurs due to hopping of charge carriers between localized states, randomly distributed in energy [2,7,8]. Consider a carrier (a test charge), jumping on a typical distance r  (in proximity to average distance between hopping centers). Energy of Coulomb interaction between the test charge and neighboring charges of the same sign, U  , varies randomly after a hop, since the direction of the hop is subjected to random scattering. The effect of the nearest neighbour, located at a distance r, is only significant at moderate carrier concentrations.
where  -the angle between field and jump directions, 0 F is a projection of electric field on the direction of a jump. The term ( Apparently, in this approximation U  is nothing but interaction energy between the test charge and small-size dipole p e r , situated on a distance r from it, see Figure. 1. where   wr is the distribution function of the distance to nearest neighbour, which is selected on the base of the following reasons. Long-range asympotics corresponds to Poisson's distribution (random distribution of the nearest neighbors). At small values of r the Coulomb repulsion should be taken into account. Indeed, owing to energetic disorder, one of the carriers occupies deeper state, while the other moves adiabatically promptly. Therefore at small r the dependence   wr could be considered as quasi- where A -normalization factor. Moving from potential energy to module of electric field strength in eq.
(2), one gets the following distribution function with normalization factor СF: where   0 ,, ex G U F F is a distribution function of energy ex U , which, in general, includes ex F and Here, the transport level EC dependence on field strength F0 yields well-known phenomenological, so called Pool-Frenkel field dependence of mobility, and is in accord with correlated disorder theory by Novikov et al. [12]: where 0 C E is an effective transport level in a low field limit [13], 5 8.85 10 C  (m/V) 1/2 and Г=2.

Results and Discussion
Results of calculations of mobility dependence on concentration in a weak external electric field in comparison with experimental data (from well-known work [1]) and with results of Extended Gaussian Disorder Model (EGDM) [2] for two different values of disorder parameter are shown in Figures. 2 and  3. Note, that the experimental results (see empty symbols) cover only the low and the high nm, 2 10 r  -are in proximity to those used in the work [1] for analysis of experimental data. The qualitative agreement with the experimental data is achieved by the use the same values in eq. (8), as in the ref. [12]. Figure. 2 shows, that the model considering MCI gives result in qualitative agreement with experimental data (except of very high concentrations region), whereas other models do not provide sufficient mobility increase, at moderate value of disorder parameter, /4 kT   .  The conclusion about quantitative agreement in moderate concentrations region (from 10 -5 to 10 -3 ) could not be made due to the lack of experimental results in that region. The discrepancy between MCI model and the experimental data at high concentrations ( 0.01) c  is not surprising, since the approximation (1) in this region is not applicable, and one need take into account not only the effect from the nearest neighbor, but also from other neighboring charges. From the Figure. 3

 
, used in [2]), good fit is obtained at moderate and high concentrations without considering MCI, while MCI provide too strong increase of mobility in this case. However, models not considering MCI provide too small mobility in limiting case of low concentrations, than the experiment.

Conclusion
Theoretical model of mobility dependence on carrier concentration in disordered organic materials at weak electric field, considering not only the filling of deep "tale" of DOS, but also Coulomb interactions between randomly nearing charges, was built. Microscopic Coulomb interactions (MCI) result in considerable increase of mobility at moderate concentrations, if the field dependence of mobility is rather strong. Previously, the effect of MCI ("short-range Coulomb interactions") on J-V characteristics (not directly on mobility) was studied by the means of Monte-Carlo modelling, and considerable reduction of current, at low voltages and small layer thickness, was demonstrated [14]. One has to note, that it is hard to distinguish effects of MCI on transport and injection in the case of thin (22 nm) layer and low (about 0.1 V) voltage [15]. Thus, further investigations are necessary. The analytic treatment of MCI in the present work do not claim to quantitative accuracy, being only the upper estimation, since only the interaction with the nearest charge of the same sign is considered. One has to stress, that various sets of parameters could be derived from analyses of experimental data, considering or not considering MCI. One of the aims of this work is to get the attention to a problem of independent (apart from J-V characteristics measurements) and reliable definition of disorder parameter, / kT  , in order to validate one or another model of field dependence of the mobility, and to make conclusions about the effect of MCI on concentration dependence, using experimental concentration dependence of mobility.