6.2 Theory

The SCLC problem in dielectrics can be described by the following equations [107]

$$\frac{dF}{dx}=\frac{q}{\epsilon_r\epsilon_0}\left(n_f+n_t\right),$$ (6.1)

$$j=qn_f\mu F,$$ (6.2)

where $F$ is the electric field intensity, $\epsilon_r$ is the relative dielectric permittivity, $n_f$ and $n_t$ are the concentrations of the mobile and trapped carriers, respectively, $j$ is the current density, and $\mu$ is the drift mobility of carriers.

In this model, a Gaussian DOS function is assumed. Analysis of the optical adsorbtion spectrum and mobility for PPV indicates that the DOS can be fitted well to a Gaussian distribution with $\sigma\approx 0.1$eV. In other disordered molecular materials $\sigma$ typically lies between $0.07$ and $0.13$eV [9].

Figure 6.1: Gaussian density of states with zero mean energy. The vertical axis corresponds to energy, the horizontal axis reflects the site density. The center of the Gaussian DOS is at zero energy.

A schematic representation of the Gaussian DOS is shown in Fig 6.1. In the tail of the distribution few sites are available for hopping and their nearest neighbors are many $kT$ away in energy, so that they serve as trap centers. Site-selective fluorescence of PPV has also shown that the sites in the tail of the distribution act as traps [125]. While the sites towards the center of DOS more neighbors are accessible and the energy between them is very close. So they provide the mobile carriers. Here we define a conduction edge [126] at about $2\sigma$ below the Gaussian center. We do not rigorously justify such edge position and it is done only for illustration purpose, though it is similar to the method applied to absorption spectrum or STM measurements [127]. So the concentrations of mobile and trapped carriers can be calculated as

$$n_f=\int_{-2\sigma}^{\infty}g\left(E\right)f\left(E\right)dE,$$ (6.3)

$$n_t=\int_{-\infty}^{-2\sigma}g\left(E\right)f\left(E\right)dE,$$ (6.4)

with $g\left(E\right)$ being the DOS function and $f\left(E\right)=\left(1+\exp\left[E-E_F\right]\right)^{-1}$ the Fremi-Dirac distribution. Substituting (6.3) and (6.4) into (6.1) and (6.2), we obtain

$$\frac{dF}{dx}=\frac{q}{\epsilon_0\epsilon_r}\int_{\infty}^{\infty}g\left(E\right)f\left(E\right)dE.$$ (6.5)

$$F=\frac{j}{q\mu}\left(\int_{-2\sigma}^{\infty}g\left(E\right)f\left(E\right)dE\right)^{-1}.$$ (6.6)

Then differentiate (6.6) with respect to $x$ to obtain the equation

$$\frac{dF}{dx}=\frac{j}{e\mu}\frac{\int_{-2\sigma}^{\infty}g\left(... ...\left(1+\exp\left(\epsilon-\epsilon_F\right)\right)^{-1}d\epsilon}\right)^{2}},$$ (6.7)

where $\epsilon=E/{k_BT}$ and $\epsilon_F=E_F/{k_BT}$. Substituting (6.7) into (6.5), we obtain the differential equation for quasi Fermi-energy as

$$\frac{d\epsilon_F}{dx}=-\frac{q^2\mu N_t^2\exp\left(\epsilon_F\ri... ...)}{\left[1+\exp\left(\epsilon-\epsilon_F\right)\right]^2}d\epsilon\right]^{-1}.$$

where $\sigma_0=\sigma/{kT}$. Combing (6.6) and (6.8), we obtain the position-dependent electric field. the $j/V$ characteristics can be calculated by integrating the field over the coordinate.