Reformulated space-charge-limited current model and its application to disordered organic systems.

We have reformulated a traditional model used to describe the current-voltage dependence of low mobility materials sandwiched between planar electrodes by using the quasi-electrochemical potential as the fundamental variable instead of the local electric field or the local charge carrier density. This allows the material density-of-states to enter explicitly in the equations and dispenses with the need to assume a particular type of contact. The diffusion current is included and as a consequence the current-voltage dependence obtained covers, with increasing bias, the diffusion limited current, the space-charge limited current, and the injection limited current regimes. The generalized Einstein relation and the field and density dependent mobility are naturally incorporated into the formalism; these two points being of particular relevance for disordered organic semiconductors. The reformulated model can be applied to any material where the carrier density and the mobility may be written as a function of the quasi-electrochemical potential. We applied it to the textbook example of a nondegenerate, constant mobility material and showed how a single dimensionless parameter determines the form of the I(V) curve. We obtained integral expressions for the carrier density and for the mobility as a function of the quasi-electrochemical potential for a Gaussianly disordered organic material and found the general form of the I(V) curve for such materials over the full range of bias, showing how the energetic disorder alone can give rise, in the space-charge limited current regime, to an I∝V(n) dependence with an exponent n larger than 2.