## 4 Influence of Ageing on Charge Transport Properties and Discussion
### Influence of Ageing on Carrier Mobility Properties  
Figure 8 shows the band structure and thermal-assisted hopping conduction model of insulating materials [21]. When there is no externally applied electric field, the carriers will thermally move in six directions of the three-dimensional coordinate system under thermal activation. When there is no electric field force, the probabilities of carriers jumping in all directions are equal, and the carriers do not undergo directional migration, as shown in Figure 8b. The probability of the carrier jumping in all directions is related to the barrier height and temperature, that is, \(p_{\text{AB}}=p_{\text{BA}}=v_{\text{ATE}}\text{exp}(-E_{\text{T}}/k_{\text{B}}T)\)[21]. Here, \(p_{\text{AB}}\) is the probability of the electron jumping from the position A to the position B, \(p_{\text{BA}}\) is the probability of the electron jumping from the position B to the position A, \(E_{\text{T}}\) is the trap energy level, and \(T\) is the temperature of the material. After an electric field is applied, the potential barrier tilts and the carriers jump directionally. As shown in Figure 8c, the potential barrier for the electrons to jump from the position A to the position B will decrease \(\Delta E\) (\(\Delta E=e\lambda_{\text{hop}}F/2\)). The potential barrier for electrons to jump from position B to position A will increase by \(\Delta E\). Here, \(\lambda_{\text{hop}}\) is the average distance between shallow traps in m, and \(F\) is the applied electric field strength in Vm\({}^{-1}\). The potential barriers for electrons to jump from the position A to the position B and from the position B to the position A will become \(E_{\text{T}}-e\lambda_{\text{hop}}F/2\) and \(E_{\text{T}}+e\lambda_{\text{hop}}F/2\), respectively. Substituting the potential barriers under the electric field into the charge hopping probability equation, we can get the probabilities \(p_{\text{AB}}=v_{\text{ATE}}\text{exp}[-(E_{\text{T}}-e\lambda_{\text{hop}}F/2 )/k_{\text{B}}T]\) and \(p_{\text{BA}}=v_{\text{ATE}}\text{exp}[-(E_{\text{T}}+e\lambda_{\text{hop}}F/2 )/k_{\text{B}}T]\) for electrons from position A to position B and from position B to position A, respectively. It is shown that the potential barrier for electrons to jump from position A to position B is reduced due to the effect of the electric field. So, the probability of electrons jumping to the anode will increase, while their jumping towards the cathode will decrease. This will cause a net negative charge to flow from the cathode to the anode, or a net positive charge to flow from the anode to the cathode, creating a current in the external circuit. In the presence of an externally applied electric field, the probability of a carrier jumping from the position A to the position B per unit time will increase by \(\text{d}n_{\text{AB}}/\text{d}t\), but the probability of the carrier jumping from the position B to the position A will decrease by \(\text{d}n_{\text{BA}}/\text{d}t\). This will cause a net charge to migrate from the position A to the position B, which is \(\text{d}n_{\text{A}}/\text{d}t=\text{d}n_{\text{AB}}/\text{d}t-\text{d}n_{ \text{BA}}/\text{d}t=(n_{0}/3)\)\(v_{\text{ATE}}\text{exp}(-E_{\text{T}}/k_{\text{B}}T)\text{sinh}(e \lambda_{\text{hop}}F/2k_{\text{B}}T)\). The migration speed of charge carriers toward anode \(v\) is \(\lambda_{\text{hop}}/n_{0}\cdot\text{d}n_{\text{A}}/\text{d}t\). Then, we can obtain the charge carrier mobility from the migration speed divided by the electric field [21]:  
\[\mu=\frac{\lambda_{\text{hop}}v_{\text{ATE}}}{3F}\exp\left(-\alpha\lambda_{ \text{hop}}\right)\exp\left(-\frac{E_{\text{T}}}{k_{\text{B}}T}\right)\text{ sinh}\left(\frac{e\lambda_{\text{hop}}F}{2k_{\text{B}}T}\right), \tag{4}\]  
where \(\alpha\) is a constant. The relation between the average carrier hop distance and the trap density \(N_{\text{T}}\) is \(\lambda_{\text{hop}}=(1/N_{\text{T}})^{1/3}\). Thus, the relation between trap density and carrier mobility can be obtained.