# Space Charge Accumulation and Decay in Dielectric Materials with Dual Discrete Traps
## 4 Results and Discussion
### Space Charge Accumulation Characteristics  
Free charge and trapped charge densities are dependent on time and position. The BCT model outputs of free charge density and net charge density distributions in the material at various times show that the accumulated negative charges are much larger than the accumulated positive charges in the bulk of LDPE. Since the contact potential barrier between the cathode and the dielectric material is lower than that between the anode and the material, the emission rate of electrons is much higher than the emission rate of holes according to the Schottky thermionic emission equation. For example,the injection current densities of electrons and holes are about \(7.36\times 10^{-3}\) Am\({}^{-2}\) and \(3.21\times 10^{-6}\) Am\({}^{-2}\), respectively, at the voltage of 6.0 kV at the beginning of voltage application. Accordingly, the injected negative charges are much larger than the injected positive charges.  
When the space charges are only caused by charge injections from contact electrodes, homogenous space charges will be accumulated in the dielectric material. The injected electrons drift in the bulk of the material towards the anode and are captured by trapping level 1 and trapping level 2 gradually. If the trapping rates are larger than the de-trapping rates, most of the injected electrons are trapped in the trapping centers. The trapped electrons can distort the electric fields in the vicinity of the cathode, reducing the Schottky thermionic emission current. This will lead to the saturation of accumulated space charges in the dielectric material.  
Figure 5a indicates the average net charge density as a function of time in the dielectric material subjected to various voltages. The average net charge density equals to the integration of net charge density over position \(x\) divided by the length of the dielectric material. The average net charge densities increase very fast at the beginning of the application of voltage. Then, the increasing rates of the average net charge densities decline gradually with time. In addition, the time needed for the space charge accumulation to reach steady-state decreases with increasing the applied voltage. Figure 5b shows the external current densities as a function of elapsed time after the application of voltages. Since no free charges are assumed inside the dielectric material before the application of voltage, the current density is very small at the beginning after applying voltage to the sample due to low density of free charges. With increasing the elapsed time of voltage application, more and more charges are injected into the dielectric material, so the current density increases with time until the front of injected charges reaches the opposite electrode. The peaks correspond to the transit time of free charges, which is 4.8 s at 6.0 kV, 4 s at 7.5 kV, and 3.4 s at 9.0 kV. After the injected charges reach the opposite electrode charge trapping process will dominate the charge transport in the dielectric material and the current density decrease monotonically with time. Stepwise reductions can be observed in the time-dependent current density, which are caused by charge trapping-de-trapping dynamics between the extended states and two deep traps.