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

(6.1) 

(6.2) 
where is the electric field intensity,
is the relative dielectric permittivity, and are the concentrations of the mobile and trapped
carriers, respectively, is the current density, and 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
eV. In other disordered
molecular materials typically lies between and 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 away in energy, so that they serve as trap
centers. Siteselective 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 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

(6.3) 

(6.4) 
with
being the DOS function and
the FremiDirac
distribution. Substituting (6.3) and
(6.4) into (6.1)
and (6.2), we obtain

(6.5) 

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

(6.7) 
where
and
. Substituting (6.7) into (6.5), we obtain the
differential equation for quasi Fermienergy as
where
. Combing (6.6) and (6.8), we obtain the
positiondependent
electric field. the characteristics can be
calculated by integrating the field over the coordinate.
Next: 6.3 Results and Discussion
Up: 6. Space Charge Limited
Previous: 6.1 Introduction
Ling Li: Charge Transport in Organic Semiconductor Materials and Devices