# Space Charge Accumulation and Decay in Dielectric Materials with Dual Discrete Traps
## 2 Dynamic Responses of Charge Trapping and De-trapping
### Steady State Characteristics in the Case of Instantaneous Charge Injection  
When the charge trapping/de-trapping dynamics reach steady-state \(t_{st}\), namely \(\mathrm{d}q_{trap1}(t)/\mathrm{d}t=0\), the exact relationship between trapped charge densities and free charge density can be obtained from the charge trapping/de-trapping differential equations.  
\[q_{trapi}(t_{st}) = \frac{P_{Ti}q_{free}(t_{st})q_{e}N_{Ti}}{P_{Ti}q_{free}(t_{st})+P_ {Di}q_{e}N_{Ti}} \tag{21}\]  
Using the above equation, we can calculate trapped charge densities in each trap center as a function of free charge density at steady state, as shown in Figure 3a. For small free charge densities, the relations between the trapped charge densities and the free charge densities are linear, \(q_{trapi}(t_{st})\propto q_{free}(t_{st})\). For high free charge densities, the traps will be fully filled. Moreover, the deeper traps will be fully filled first, then the shallower traps.  
The quantitative relationship among \(q_{free}(t_{st})\), \(q_{trap1}(t_{st})\), and \(q_{trap2}(t_{st})\) can be derived from Equations (13)-(15) for small free charge densities. The relationship is,  
Figure 3: Trapped charge densities (**a**) and total charge density and effective carrier mobility (**b**) as a function of free charge density at steady state in dielectric materials containing two discrete trap levels.  
Figure 2: Time-dependent free charge density and trapped charge densities in the dielectric material containing two discrete trap levels in the case of instantaneous charge injection. The total charge densities are 2 cm\({}^{-3}\) (**a**) and 10 cm\({}^{-3}\) (**b**). Solid lines are numerical results, and dots are analytical results.  
\[q_{free}(t_{st})/q_{trap1}(t_{st})/q_{trap2}(t_{st})\ =\ P_{D1}P_{D2}/P_{T1}P_{D2}/P_{T2}P_{D1} \tag{22}\]  
Substituting the charge trapping probability and charge de-trapping probability equations into Equation (22), we can obtain the relationships between \(q_{free}(t_{st})\) and \(q_{trap1}(t_{st})\), and between \(q_{trap1}(t_{st})\) and \(q_{trap2}(t_{st})\).  
\[\frac{q_{free}(t_{st})}{q_{trap1}(t_{st})}\ =\ \frac{\varepsilon_{0} \varepsilon_{F}\
u_{ATE}}{q_{e}N_{Ti}\mu_{0}}\exp\biggl{(}-\frac{E_{Ti}}{k_{B} T}\biggr{)} \tag{23}\]  
\[\frac{q_{trap1}(t_{st})}{q_{trap2}(t_{st})}\ =\ \frac{N_{T1}}{N_{T2}}\exp\biggl{(}- \frac{E_{T2}-E_{T1}}{k_{B}T}\biggr{)} \tag{24}\]  
Equations (23) and (24) show that the trapped charge densities increase exponentially with the increasing trap energy at a given \(q_{free}(t_{st})\). Accordingly, at steady state deeper traps will be fully filled first, as demonstrated in Figure 3b. However, ratios of \(q_{trap1}(t_{st})\) to \(q_{trap2}(t_{st})\) at transient states are different from the results of Equation (24). As indicated in Figure 2, \(q_{trap1}(t_{st})\) could be much larger than \(q_{trap2}(t_{st})\) at transient states. The exact relationship between \(q_{trap1}(t_{st})\) and \(q_{trap2}(t_{st})\) is demonstrated in Appendix B.  
Figure 3b demonstrates the dependences of total charge density, \(q_{total}(t_{st})\), and effective carrier mobility, \(\mu_{eff}(t_{st})\), on free charge density. The total charge density increases with increasing free charge density. The two convex inflection points in \(q_{total}(t_{st})\)-\(q_{free}(t_{st})\) curve imply the fully filling of trap level 1 and trap level 2. The effective carrier mobility is a constant for small free charge densities. Then, the effective carrier mobility rises as the free charge density increases.