# A Comparison of Electrical Breakdown Models for Polyethylene Nanocomposites
## 2 Electrical Breakdown Models
### Charge Injection and Charge Transport in Nanocomposites  
Figure 1 presents a schematic diagram of the charge transport inside a polymer nanocomposite [25]. The one-dimensional coordinate \(x\) is set in the thickness direction of the nanocomposite, and the thickness of the material is \(L\). The left side of the nanocomposite is the cathode and the right side is the anode. Under the action of an applied voltage, the cathode injects electrons into the nanocomposite, and the anode injects holes. Potential barriers exist between the cathode and anode and the nanocomposite interface, which are \(u_{in(e)}\) and \(u_{in(b)}\), respectively. The interfacial barrier hinders the transfer of charges from the electrode into the nanocomposite. Under an externally applied strong electric field, a potential barrier lowering \(u_{Sch}\) appears in the interface barrier, which promotes the transfer of charges to the bulk of the nanocomposite. When the applied electric field is \(E\), the Schottky barrier reduction \(u_{Sch}\) is proportional to the square root of the electric field, \(u_{Sch}=(eE/4\pi\varepsilon_{0}\varepsilon_{r})^{1/2}\). Here, \(e\) is the electron charge, \(\varepsilon_{0}\) is the permittivity of the vacuum in F/m, and \(\varepsilon_{r}\) is the dielectric constant of the nanocomposite. We adopt the Schottky thermal emission model to describe the charge injections, \(j_{in(e)}\) and \(j_{in(h)}\), per unit time and unit area of the cathode and anode, into the nanodielectrics [28]. \[j_{in(e)}(0,t) =AT^{2}\exp\Bigl{(}-u_{in(e)}/k_{B}T\Bigr{)}\exp(u_{Sch}(0,t)/k_{B}T) \tag{1}\] \[j_{in(h)}(L,t) =AT^{2}\exp\Bigl{(}-u_{in(h)}/k_{B}T\Bigr{)}\exp(u_{Sch}(L,t)/k_{B }T) \tag{2}\]  
where \(A\) is the Richardson constant, \(T\) is the temperature of materials, and \(t\) is the elapsed time of applied voltage. Under an electric field \(E\), there will be a certain concentration of mobile electrons and mobile holes in extended states after the charges in the electrodes are injected into the nanocomposite. Their concentrations are \(u_{M(e)}\) and \(u_{M(h)}\), respectively. These mobile electrons and mobile holes migrate in the extended states driven by the applied electric field. The mobilities of electrons and holes in the extended states are \(u_{0(e)}\) and \(u_{0(h)}\), respectively. Due to the existence of polar groups in polymer nanodielectrics, a certain concentration of deep traps is formed. It is assumed that the energy levels of deep traps of electrons and holes are \(u_{T(e)}\) and \(u_{T(h)}\), respectively, and their concentrations are \(N_{T(e)}\) and \(N_{T(h)}\), respectively. Due to the strong trapping ability of deep traps, mobile electrons and mobile  
Figure 1: Schematic of the bipolar charge carrier injection and charge carrier transport in polymer nanocomposites under a ramp voltage. holes may be captured by them during the migration process. The trapping probabilities of electrons and holes in extended states in the electron deep traps and hole deep traps are \(P_{T(e)}\) and \(P_{T(h)}\), respectively. After a period of time, the deep trapped electron and deep trapped hole densities are \(n_{T(e)}\) and \(n_{T(h)}\), respectively. In addition, the charges in the deep traps gradually gain energy from the heat bath due to thermal excitation. When the trapped charges gain sufficient energy, trapped electrons and trapped holes can transition to extended states with probabilities of \(P_{D(e)}\) and \(P_{D(h)}\), respectively. The detrapping probabilities \(P_{D(e)}\) and \(P_{D(h)}\) decrease exponentially with the increase in the deep trap levels \(u_{T(e)}\) and \(u_{T(h)}\). When positive and negative charges meet inside the sample, the recombination between these charges occurs. Mobile electrons may recombine with mobile and trapped holes, and mobile holes may recombine with mobile and trapped electrons. The trapped electrons mainly recombine with the mobile holes, and the trapped holes mainly recombine with the mobile electrons.