# Space Charge Accumulation and Decay in Dielectric Materials with Dual Discrete Traps
## 2 Dynamic Responses of Charge Trapping and De-trapping  
### Charge Trapping/De-trapping Dynamics  
Free charges in extended states or quasi-free charges in localized states produced by injection, irradiation, or ionization can be captured by traps in dielectric materials with a trapping probability. The charge trapping properties were investigated theoretically and experimentally by considering the relationship between the Langevin recombination model and the Shockley-Read-Hall (SRH) recombination model. It was found that the probability of trapping of charge carriers \(P_{T}\) in deep trap centers is proportional to the mobility carriers \(\mu_{0}\) and the trap density of the material \(N_{T}\), and is inversely proportional to the permittivity [23; 24]. Namely, \(P_{T}\) can be given by,  
\[P_{T}\ =\ \mu_{0}eN_{T}/\varepsilon_{0}\varepsilon_{r} \tag{1}\]  
where \(\mu_{0}\) is the shallow-trap-controlled carrier mobility in m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\), \(e\) is the elementary charge in C, \(\varepsilon_{0}\) is the permittivity of vacuum in Fm\({}^{-1}\), and \(\varepsilon_{r}\) is the relative permittivity of dielectric materials. For LDPE, \(\varepsilon_{r}\) equals to 2.3. The charge trapping probability given by Equation (1) is proportional to the carrier mobility, which is consistent with the comments of [PERSON] et al. [15]. It was pointed out that the charge trapping probability would be related to the thermal velocity of carriers in the case of extended state conduction, nevertheless, this would depend on the drift velocity in the case of charge hopping in localized states. When the shallow-trap-controlled carrier mobility is set as \(10^{-13}\)-\(10^{-14}\) m\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\), the charge trapping probability estimated by Equation (1) will be \(4.91\times 10^{-1}\)-\(4.91\times 10^{-2}\) s\({}^{-1}\), which are in good agreement with the previous findings [15; 25; 26; 27]. The trapped charges can be released to the transport states after having been trapped in trap centers for a period. The thermally-assisted de-trapping probability is given as [26],  
\[P_{D}(E_{T})\ =\ \upsilon_{ATE}\exp(-E_{T}/k_{B}T) \tag{2}\]  
where \(\upsilon_{ATE}=k_{B}T/h_{p}\), is the attempt-to-escape frequency in s\({}^{-1}\)[22; 28]. Here, \(h_{p}\) is the Planck constant in Js. The charge trapping and de-trapping dynamics in insulators with multiple trap levels can be described by the first-order charge trapping and de-trapping dynamic equation. It is assumed that charge trapping events into various trap levels are incompatible and charge de-trapping events from various traps are mutually independent, as shown in Figure 1. In the present work, we just consider the charge trapping/de-trapping dynamics at two discrete trap levels, namely, trap level 1 and trap level 2. The energies and densities of deep trap level 1 and deep trap level 2 are \(E_{T1}\) and \(E_{T2}\), \(Q_{T1}\) and \(Q_{T2}\), respectively. \[\frac{\mathrm{d}q_{free}}{\mathrm{d}t}=-q_{free}\sum_{i\ =\ 1}^{n}P_{T}\bigg{(}1-\frac{q_{trapi}}{Q_{Ti}}\bigg{)}+\sum_{i\ =\ 1}^{n}P_{Di}q_{trapi} \tag{3}\]  
\[\frac{\mathrm{d}q_{trapi}}{\mathrm{d}t}\ =\ \ q_{free}P_{Ti}\bigg{(}1-\frac{q_{trapi }}{Q_{Ti}}\bigg{)}-P_{Di}q_{trapi},i\ =\ 1,2,\ldots,n \tag{4}\]  
where \(q_{free}\) and \(q_{trap}\) are the densities of free and trapped charges in Cm\({}^{-3}\), while \(Q_{T}\) equals to \(eN_{T}\) and \(n\) is the number of deep trap levels, where \(n=2\) in the present work; furthermore, \(t\) is the time. We will analyze the charge trapping and de-trapping dynamic responses in the following two cases. Namely, in the first case, free charges are assumed to be injected continuously to compensate for the depletion of free charges in the bulk caused by trapping. In other words, the free charges are injected to keep the total number of free charges in the bulk at a constant value. In contrast, a certain number of free charges are injected instantaneously only for a short period in the second case.