# Theory of Electrical Breakdown in a Nanocomposite Capacitor
## 4 Comparison with Experiments and Discussion  
The above theoretical findings can be compared with the experimental results available in the literature. One should note, however, that the quantities \(\gamma\), \(W\) and \(j_{B}\), which determine the breakdown field strength, are unknown and do not allow to calculate the _absolute_ value of \(E_{B}^{\prime}\). Instead, one can consider the _relative_ value \(E_{B}^{\prime}/E_{B}\) with \(E_{B}\) being the breakdown field strength in an NP-free capacitor and compare it with the predicted trend which follows from Equation (18). First, we consider the experiments on the breakdown field in nanocomposites represented by BaTiO\({}_{3}\) NPs with an average size of 100 nm dispersed in a polyvinylidene fluoride (PVDF) matrix [6]. We restrict ourselves by the range of relatively small volume fractions of BaTiO\({}_{3}\) NPs (\(f\leq 0.2\)) where our approach can be applied. One can see that the observed decrease with \(f\) in the breakdown strength \(E_{B}^{\prime}\) measured at different temperatures can be well fitted by straight lines (see Figure 2) which is in agreement with Equation (18). The quantity \(\beta\) found from the slopes of these dependencies nonmonotonically varies between 0.34 and 0.46 when the temperature changes from 20 to 120 \({}^{\circ}\)C which can be attributed to the error bars of the measurements. These results can be compared with the value \(\beta\approx 0.86\) calculated under the assumption that \(\epsilon_{i}=200\), which is typical for thin BaTiO\({}_{3}\) films [30], and \(\epsilon_{h}=10\)[31]. A significant difference between the calculated and fitted values of \(\beta\) can be ascribed to the fact that the dielectric properties of BaTiO\({}_{3}\) NPs are distinct from those of BaTiO\({}_{3}\) films [32; 33]. A remarkable temperature dependence of the breakdown field indicates that the thermionic current, Equation (13), is the dominant mechanism of the electron injection. As it follows from Equations (12) and (13), the temperature dependence of the quantity \(E_{B}^{\prime}/(1-3f\beta)\) should be the same for different \(f\). This can indeed be obtained from the  
Figure 2: The dependence of the breakdown field strength on the volume fraction of BaTiO\({}_{3}\) NPs embedded in PVDF matrix for different temperatures indicated in the inset. The straight lines are the best linear fits to the experimental data shown by symbols and taken from Figure 5a in Ref. [6]. The experimental error bars are not available. experimental plots of \(E_{B}^{\prime}\) versus \(T\) which, being recalculated to \(E_{B}^{\prime}/(1-3f\beta)\) with the mean value \(\beta=0.4\), coincide with each other within the possible error bars (see Figure 3). Taking \(\beta=0.4\) and the dielectric permittivity of PVDF \(e_{h}=10\)[31], one finds for the effective dielectric permittivity of BaTiO\({}_{3}\) NPs \(\epsilon_{i}\approx 30\). In another set of experiments, BaTiO\({}_{3}\) NPs of 30-50 nm in diameter were surface modified using a pentafluorobenzyl phosphonic acid (PFBPA) and incorporated into a poly(vinylidene fluoride-cohexafluoropropylene) (PVDF-HFP)) matrix [7]. Again, for relatively small volume fractions of NPs (\(f\leq 0.2\)), the breakdown field strength linearly decreases with \(f\) (see Figure 4). The value of \(\beta\) deduced from the slope of this dependence is about 0.57, and taking \(\epsilon_{h}=12.6\)[7], one obtains \(\epsilon_{i}\approx 70\). For comparison, the calculated value of \(\beta\) obtained with \(\epsilon_{i}=200\) is about 0.83. Finally, a decrease in the breakdown field strength was also observed for BaTiO\({}_{3}\) NPs of 7 nm in diameter dispersed in polystyrene (PS) [8]. A linear fit of this dependence (Figure 5) gives \(\beta\approx 0.54\) which, together with the value \(\epsilon_{h}=2.4\) for the PS [8], provides \(\epsilon_{i}\approx 11\). The calculated value with \(\epsilon_{i}=200\) is \(\beta\approx 0.96\).