ChemicalPhysics 146 (1990) 381-388 North-Holland An effective medium treatment of the random bias problem Dora Izzo ‘, David H. Dunlap *, Kalyan Kundu 3 and Philip Phillips Department of Chemistry, Room 6223, A4assachusetts institute of Technology, Cambridge. MA 02139, USA Received 2 January 1990 We consider here a general model for the incoherent transport of an electron or a one-dimensional disordered lattice. No special symmetry is assumed to hold between the hopping rates. This model is applicable then to charge transport in highly anisotropic disordered systems in the presence of a static electric field. We show how an effective medium theory for the difFusion constant and the velocity of an initially local&d particle can be formulated directly from the equations of motion. The theory is shown to predict accurately the velocity for all values of the applied static field including the observed vanishing of the velocity at low values of the static field. The results for the diffusion constant are shown to be in agreement with known results for intermediate to high values of the applied field. We also discuss how the effective medium can easily be constructed in higher dimensions. zyxwvutsrqp 1. Introduetlon We focus here on a ubiquitous model for incoher- ent hopping conduction in a disordered environ- ment. Hopping transport between localised states is the basic mechanism underlying electronic conduc- tion in numerous organic conductors, incoherent ex- citon and spin migration and transport of vibrational energy. As a result of the extreme physical impor- tance of this problem, numerous physical models have been constructed to describe such processes. Two of the more widely used models are the random trap- ping (RT) and the random hopping (RH) models [ 11. The RT model describes the motion of a charge carrier on a site-disordered lattice. The site disorder arises from symmetric wells of varying depth which are placed at random on sites throughout the lattice. The depth of the well on each site is determined by a single probability distribution. On the other hand, the RH model is a bond-disordered model in which sym- metric barriers of varying heights separate adjacent lattice sites. Here again, the height of each barrier is ’ Also at Department of Physics, MIT. * Present address: Department of Physics, University of New Mexico, Albuquerque, NM 87 13 1, USA. a Present address: Institute of Physics, Sachivalaya Marg, Bhu- baneswar 751005 Orissa, India. uncorrelated and determined by a single probability distribution. The crucial difference between these models is that the diffusion constant in the RT model vanishes if a finite fraction of the hopping rates are identically zero. This result is true regardless of the spatial dimension. Stated differently, the RT model does not display a dynamical percolation transition in any spatial dimension. In fact, the RH model is the standard model for bond percolation whereas the RT model is applied solely to systems in which trapping effects dominate as in the case of anomalous diffu- sion [ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ 1 ] and transient photoconductivity experi- ments [2,3]. In this paper we consider a model for incoherent hopping transport which is a generalisation of the RT and RH models. No special symmetry is assumed to hold between the nearest neighbour hopping rates. The model we consider then is applicable to trans- port in highly anisotropic materials. We refer to the model we treat here as the random bias model. The source of the anisotropy can arise either from exter- nally applied electric or magnetic fields or the intrin- sic disorder in the material. In the latter case, the hopping rates will vary along the different axes of the crystal. Recent experiments by Devreux and Lecav- elier [ 41 have observed anomalous diffusion in the conducting polymer polypyrrole perchlorate as a function of temperature. They attribute their obser- 0301-0104/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)