Self-consistent analytical solution of a problem of charge-carrier injection at a conductor/insulator interface

摘要

We present a closed description of the charge carrier injection process froma conductor into an insulator. Common injection models are based on singleelectron descriptions, being problematic especially once the amount ofcharge-carriers injected is large. Accordingly, we developed a model, whichincorporates space charge effects in the description of the injection process.The challenge of this task is the problem of self-consistency. The amount ofcharge-carriers injected per unit time strongly depends on the energy barrieremerging at the contact, while at the same time the electrostatic potentialgenerated by the injected charge- carriers modifies the height of thisinjection barrier itself. In our model, self-consistency is obtained byassuming continuity of the electric displacement and the electrochemicalpotential all over the conductor/insulator system. The conductor and theinsulator are properly taken into account by means of their respective densityof state distributions. The electric field distributions are obtained in aclosed analytical form and the resulting current-voltage characteristics showthat the theory embraces injection-limited as well as bulk-limitedcharge-carrier transport. Analytical approximations of these limits are given,revealing physical mechanisms responsible for the particular current-voltagebehavior. In addition, the model exhibits the crossover between the twolimiting cases and determines the validity of respective approximations. Theconsequences resulting from our exactly solvable model are discussed on thebasis of a simplified indium tin oxide/organic semiconductor system.