# Theory of Electrical Breakdown in a Nanocomposite Capacitor
## 2 Methods
### Local Field in a Nanocomposite  
When applying the above equations to a nanocomposite capacitor, one should regard the field strength \(E\) as the microscopic (local) field which is determined not only by the applied voltage but also by the field of the polarized NPs. For extended media and relatively small volume fractions of NPs, this field can be found in the Maxwell Garnett approximation which models the inclusions by polarizable spheres [14; 15]. However, this approach is not applicable to the operating wavelengths \(\lambda\) which are comparable with or larger than the capacitor dimensions and especially for static fields which correspond to the limit \(\lambda\rightarrow\infty\)[16]. A proper approach to this problem involves an account of the image potential which originates from the polarization of the electrodes and can be obtained by an infinite summation of all the induced NP dipole images in the two electrodes [20]. However, a more convenient way of calculation is using the Fourier transform of the field derived for a dipole oscillating with frequency \(\omega\) between two reflecting surfaces and taking the limit \(\omega\to 0\)[21]. In the adopted approximation, the induced dipole moment of an NP located at the point \(\mathbf{r}=(x,y,z)\) is given by [22]  
\[\mathbf{p}(\mathbf{r})=\alpha\mathbf{E}(\mathbf{r})=\epsilon_{h}R^{3}\frac{ \epsilon_{i}-\epsilon_{h}}{\epsilon_{i}+2\epsilon_{h}}\mathbf{E}(\mathbf{r}), \tag{4}\]  
where \(\alpha\) is the sphere polarizability and \(\mathbf{E}(\mathbf{r})\) is the microscopic (local) field [23] at the position of the dipole. A random distribution of NPs allows one to formally consider their polarization, \(\mathbf{P}(\mathbf{r})=N\mathbf{p}(\mathbf{r})\), as a continuous function of the radius vector \(\mathbf{r}\). Then, the local field can be written in the form [23; 24]  
\[\mathbf{E}(\mathbf{r})=\mathbf{E}_{0}+N\int_{V^{\prime}}\bar{\mathbf{F}}( \mathbf{r},\mathbf{r}^{\prime};\omega)\mathbf{p}(\mathbf{r}^{\prime})d\mathbf{ r}^{\prime}, \tag{5}\]  
where \(\mathbf{E}_{0}\) is the external electric field applied to the capacitor and directed along the \(z\) axis and the integral term represents a collective action of the induced dipoles. Here, \(N\) is the volume number density of NPs, the quantity \(\bar{\mathbf{F}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\) is the so-called field susceptibility tensor that relates the electric field at the point \(\mathbf{r}\) generated by a classical dipole, oscillating at frequency \(\omega\), with the dipole moment itself, located at \(\mathbf{r}^{\prime}\)[25], and the symbol \(V^{\prime}\) denotes the gap volume after removal of a small volume around the NP under consideration that excludes its self-action. As far as the dipole moment in the integrand depends on the local field, Equation (5) is an integral equation which provides a self-consistent solution for the electric field in the capacitor. The tensor \(\bar{\mathbf{F}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\) can be expressed in terms of the dyadic Green's function for Maxwell's equations [26] and allows a decomposition into the direct contribution, which describes the field of a dipole in free space [27], and the reflected contribution, which provides the dipole field reflected from the parallel plates [21]. For a dipole near a flat surface, it is convenient to write it in the form of the 2D spatial Fourier integral as follows [28]  
\[\bar{\mathbf{F}}(\mathbf{r},\mathbf{r}^{\prime};\omega)=\frac{1}{(2\pi)^{2}} \times\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\bar{\mathbf{f}}(z,z^{ \prime};k_{x},k_{y};\omega)e^{ik_{x}(x-x^{\prime})}e^{ik_{y}(y-y^{\prime})}dk_ {x}dk_{y}, \tag{6}\]  
where \(\bar{\mathbf{f}}(z,z^{\prime};k_{x},k_{y};\omega)\) is the Fourier transform of the tensor \(\bar{\mathbf{F}}(\mathbf{r},\mathbf{r}^{\prime};\omega)\), \(x^{\prime},y^{\prime}\) and \(z^{\prime}\) are the Cartesian coordinates of the point \(\mathbf{r}^{\prime}\), \(k_{x}\) and \(k_{y}\) are the spatial frequencies along the \(x\) and \(y\) axes, respectively.