Space-charge-limited currents in organics with trap distributions: Analytical approximations versus numerical simulation

Analytical approximations for space-charge-limited currents (SCLCs) in systems with exponential or Gaussian trap distributions were widely used in analyzing organic diodes. The current follows a power law with a transition into the trap-free SCLC at high voltages and an Ohmic low voltage limit. The power coefficient $γ$ is connected with either the decay constant or the variance of the distributions. Within these formulations, it is not possible to check the relevance of the numerous approximations needed to derive them. This concerns especially the relations of the contact work functions and of the layer thickness with the trap concentration, the position of the center of the trap distribution and its maximum value. Application of the analytical approximations to results of full numerical simulations allows one to set limits for the parameter ranges in which the approximations can be applied. In the case of the exponential distribution the analytical approximation is rather good for high trap concentrations and thicker layers. However, the simulations reveal a number of additional peculiarities. Such, the high voltage limit is usually not the trap-free SCLC but Ohmic and determined only by the anode barrier, the low voltage limit leads to a diodelike dependence with a large ideality factor and scaling with layer thickness and position of the trap distribution is extremely limited. In the case of the Gaussian trap distribution the simulations show indeed that the formula together with the connection between the power coefficient and the variance of the distribution fails completely. Thus, in principle, earlier analyzes of experimental data should be revised by using numerical simulations.