5.1 Introduction

Over the past fifteen years, there has been a surge of interest in the development and application of organic semiconductors, such as organic light-emitting diodes (OLED) and organic field effect transistors [25,105]. The processes of charge injection and transport play an extremely important role for OLED. Metal electrodes inject electrons and holes into opposite sides of the emissive organic layer(s), and this injection process, in most cases, governs the overall efficiency of the device. However, on the theoretical side there is still a lack of satisfactory description of the physical process underlying the charge injection in organic light-emitting diodes. One difficulty in extending our knowledge from crystalline to amorphous organic semiconductors arises because charge transport occurs no longer by free propagation in extended states, but rather by hopping in a manifold of localized states. This is reflected in the fact that there is little theoretical work that gives the electrical current at the interface in terms of experimentally obtainable parameters. Another difficulty arises from the fact that the nature of the interface in terms of composition and structure is not always understood. The sample preparation conditions, for example, have been shown to have a dramatic influence on charge injection.

The barrier height that controls hole or electron injection plays an important role in determining a measured current to be injection limited or transport limited, such as trapped charge limited transport [106,107] or space-charge-limited (SCL) transport with a field and temperature-dependent mobility [108]. The SCL transport needs the injection barrier to be Ohmic, i.e. it must be able to supply more carriers per unit time than the sample can transport [109], which requires the injection barrier to be small enough. The bulk-limited model predicts a dependence of the current density $J$ on the thickness $d$ following $J\propto 1/d^x$ $\left(x\leq 1\right)$ at a constant field, where $x=1$ in the absence of deep trap (Child's law). In the presence of an exponential distribution of traps, $x=5$ [110].

The present work is concerned with injection-limited conduction at high electric field. The text book models to describe injection into a semiconductor are the Fowler-Nordheim (FN) model for tunneling injection and the Richardson-Schottky (RS) model for thermionic emission [111]. The FN model ignores image charge effect and invokes tunneling of electrons from a metal through a triangular barrier into unbound continuum states. It predicts a current independent of temperature.

$$J\left(F\right)=BF^2\exp\left[-\frac{4\left(2m_{eff}\right)^{1/2}\Delta^{3/2}}{3\hbar qF}\right].$$

Here $\Delta$ is the barrier height in the absence of both the external field and the image effect, $F$ is the external field and $m_{eff}$ is the effective mass of the carriers in the semiconductor. The RS model is based on the assumption that an electron from the metal can be injected once it has acquired a thermal energy sufficient to cross the potential maximum that results from the superposition of the external and the image charge potential. The $J\left(F\right)$ characteristic is predicted as

$$J\left(F\right)=CT^2\exp\left[-\frac{\left(\Delta-\left(\frac{q^3F}{4\pi\epsilon_r\epsilon_0}\right)^{1/2}\right)}{k_BT}\right],$$

where $\epsilon_r$ is the relative dielectric constant . These two models, however, are insufficient to handle disordered organic materials, where the density of states is a Gaussian distribution, with localized carriers and discrete hopping within a distribution of energy states [9]. Arkhipov presented an analytical model based on hopping theory [112] and Wolf performed detailed Monte Carlo simulations of charge injection from a metal to an organic semiconductor layer [113]. In this chapter we will present two injection models, one is based on drift-diffusion theory and the other on a master equation.