*2.3. 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 *λ* which are comparable with or larger than the capacitor dimensions and especially for static fields which correspond to the limit *λ* → ∞ [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 *ω* between two reflecting surfaces and taking the limit *ω* → 0 [21].

In the adopted approximation, the induced dipole moment of an NP located at the point **r** = (*x*, *y*, *z*) is given by [22]

$$\mathbf{p}(\mathbf{r}) = \boldsymbol{\kappa}\mathbf{E}(\mathbf{r}) = \boldsymbol{\epsilon}\_{\hbar}\boldsymbol{\mathcal{R}}^{3}\frac{\boldsymbol{\epsilon}\_{i} - \boldsymbol{\epsilon}\_{\hbar}}{\boldsymbol{\epsilon}\_{i} + 2\boldsymbol{\epsilon}\_{\hbar}}\mathbf{E}(\mathbf{r}),\tag{4}$$

where *α* is the sphere polarizability and **E**(**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, **P**(**r**) = *N***p**(**r**), as a continuous function of the radius vector **r**. Then, the local field can be written in the form [23,24]

$$\mathbf{E}(\mathbf{r}) = \mathbf{E}\_0 + \mathrm{N} \int\_{V'} \mathbf{F}(\mathbf{r}, \mathbf{r'}; \omega) \mathbf{p}(\mathbf{r'}) d\mathbf{r'},\tag{5}$$

where **E**<sup>0</sup> 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 **F**¯(**r**,**r** ; *ω*) is the so-called field susceptibility tensor that relates the electric field at the point **r** generated by a classical dipole, oscillating at frequency *ω*, with the dipole moment itself, located at **r** [25], and the symbol *V* 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 **F**¯(**r**,**r** ; *ω*) 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]

$$\mathbf{F}(\mathbf{r}, \mathbf{r}'; \omega) = \frac{1}{(2\pi)^2} \times \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \mathbf{\tilde{f}}(z, z'; k\_x, k\_y; \omega) e^{i\mathbf{k}\_z(x - x')} e^{i\mathbf{k}\_y(y - y')} dk\_x dk\_y \tag{6}$$

where ¯ **f**(*z*, *z* ; *kx*, *ky*; *ω*) is the Fourier transform of the tensor **F**¯(**r**,**r** ; *ω*), *x* , *y* and *z* are the Cartesian coordinates of the point **r** , *kx* and *ky* are the spatial frequencies along the *x* and *y* axes, respectively.

For the purposes of the present derivation, one needs the limiting value of the tensor component *fzz*(*z*, *z* ; *kx*, *ky*; *ω*) when *ω* → 0, which we denote as *fzz*(*z*, *z* ; *kx*, *ky*; 0). This quantity determines the *z*-component of the local field which dictates the potential barrier for injected electrons. In this limit, the reflected field is reduced to the field originating from the images of the NP-induced dipole in the two electrodes.

Let us introduce the Fourier transform of the local field in the capacitor,

$$E\_z(\mathbf{r}) = \frac{1}{(2\pi)^2} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} e\_z(z; k\_x, k\_y) e^{ik\_x x} e^{ik\_y y} dk\_x dk\_y,\tag{7}$$

and substitute it into Equation (5). The obtained equation involves the quantity *fzz*(*z*, *z* ; *kx*, *ky*; 0) whose variation with *z* and *z* is determined by the factors exp(±*κz*) and exp(±*κz* ) with *κ* = (*k*<sup>2</sup> *<sup>x</sup>* + *k*<sup>2</sup> *<sup>y</sup>*)1/2. Assuming the inequality *<sup>κ</sup><sup>d</sup>* 1, which will be justified in what follows, and the model of a perfect conductor for the electrodes, one obtains (see Ref. [29] for the detail)

$$f\_{zz}(0,0;k\_x,k\_y;0) \approx \frac{4\pi}{\epsilon\_h d}.\tag{8}$$

We then obtain

$$e\_z(k\_x, k\_y) = \frac{e\_0(k\_{x\prime} k\_y)}{1 - (4\pi/\epsilon\_h)\text{Na}^{\prime}}\tag{9}$$

where

$$\begin{split} e\_0(k\_\lambda, k\_y) &= \int\_{-L\_2/2}^{L\_2/2} \int\_{-L\_1/2}^{L\_1/2} E\_0 e^{-ik\_x x} e^{-ik\_y y} dx dy \\ &= E\_0 L\_1 L\_2 \text{sinc}\left(\frac{k\_x L\_1}{2}\right) \text{sinc}\left(\frac{k\_y L\_2}{2}\right) \end{split} \tag{10}$$

is the Fourier transform of the applied electric field. Here, sinc*x* ≡ sin *x*/*x*, *L*<sup>1</sup> and *L*<sup>2</sup> are the lateral dimensions of the electrodes along the *x* and *y* axes, respectively, and we have assumed that the field is zero outside the capacitor. Taking into account that the function sinc*x* is essentially nonzero within the range |*x*| ≤ 3, one concludes that the essential range of integration in Equation (7) is limited to the values |*kx*| ≤ 6/*L*<sup>1</sup> and |*ky*| ≤ 6/*L*2. This means that the condition *κd* 1, which we have used above, is justified provided *d L*1, *L*2.

Finally, the local field, Equation (7), takes the form

$$E\_z = \frac{E\_0}{1 - 3f\beta} \tag{11}$$

with *<sup>f</sup>* = (4*π*/3)*NR*<sup>3</sup> being the volume fraction of NPs and *<sup>β</sup>* = ( *<sup>i</sup>* −  *<sup>h</sup>*)/( *<sup>i</sup>* + <sup>2</sup> *<sup>h</sup>*). Let us note that the applicability of this approach which models NPs by point dipoles is limited to the range *f* ≤ 0.2 [16].

## **3. Results**

*3.1. Breakdown Parameters in a Nanocomposite Capacitor*

Equations (1)–(3), which describe the electrical breakdown, can be written in terms of the applied voltage, *V*, and the applied field strength, *E*<sup>0</sup> = *V*/*d*, as

$$j\_{\varepsilon}(d) = j\_{\varepsilon}(0) \exp\left(\frac{\gamma V}{1 - 3f\beta}\right),\tag{12}$$

$$j\_{\varepsilon}(0) = AT^2 \exp\left(-\frac{W}{kT}\right) \exp\left(\frac{B'E\_0^{1/2}}{kT}\right) \tag{13}$$

and

$$j\_\epsilon(0) = C' E\_0^2 \exp\left(-\frac{D'}{E\_0}\right),\tag{14}$$

respectively. Here, the new parameters

$$B' = \frac{B}{(1 - 3f\beta)^{1/2}},\tag{15}$$

$$\mathcal{C}' = \frac{\mathcal{C}}{(1 - 3f\beta)^2} \tag{16}$$

and

$$D' = D(1 - 3f\beta)\tag{17}$$

determine the field and temperature dependencies of the injection current in a nanocomposite capacitor.

The breakdown voltage for a nanocomposite capacitor, *V <sup>B</sup>*, is related with the one for an NPs free capacitor, *VB*, as follows

$$V\_B' = (1 - 3f\beta)V\_B.\tag{18}$$

For NPs with  *<sup>i</sup>* >  *<sup>h</sup>*, the quantity *β* is positive and therefore Equation (18) describes a lowering of the breakdown voltage with an addition of NPs.

#### *3.2. Maximum Energy Density*

An important characteristic of a capacitor is the electromagnetic energy density, *U*, which it can accumulate. This quantity is given by

$$
\mathcal{U} = \frac{1}{2} \epsilon\_0 \epsilon E^2 \tag{19}
$$

with <sup>0</sup> being the permittivity of the vacuum and  being an effective permittivity of the dielectric in the capacitor. The maximum value of the energy density which can be attained in a nanocomposite capacitor is determined by the breakdown field strength *E <sup>B</sup>* = *V <sup>B</sup>*/*d*, i.e.,

$$
\mathcal{U}'\_{\text{max}} = \frac{1}{2} \epsilon\_0 \varepsilon E'^2\_B. \tag{20}
$$

For a nanocomposite capacitor in the static limit [16]

$$
\epsilon = \epsilon\_{\mathbb{H}} \frac{1 + \mathfrak{H}\beta}{1 - \mathfrak{H}\beta} \tag{21}
$$

that, together with Equation (18), gives

$$\mathcal{U}\_{\max}^{\prime} = \left[1 - (\Im f \beta)^2\right] \mathcal{U}\_{\max} \tag{22}$$

where *Umax* is the maximum energy density in an NPs free capacitor. Equation (22) predicts a slower decrease in the maximum energy density with *f* than the decrease in the breakdown voltage, Equation (18).

#### **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 *γ*, *W* and *jB*, which determine the breakdown field strength, are unknown and do not allow to calculate the *absolute* value of *E <sup>B</sup>*. Instead, one can consider the *relative* value *E <sup>B</sup>*/*EB* with *EB* 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 BaTiO3 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 BaTiO3 NPs (*f* ≤ 0.2) where our approach can be applied. One can see that the observed decrease with *f* in the breakdown strength *E <sup>B</sup>* measured at different temperatures can be well fitted by straight lines (see Figure 2) which is in agreement with Equation (18). The quantity *β* found from the slopes of these dependencies nonmonotonically varies between 0.34 and 0.46 when the temperature changes from 20 to 120 ◦C which can be attributed to the error bars of the measurements.

These results can be compared with the value *β* ≈ 0.86 calculated under the assumption that  *<sup>i</sup>* = 200, which is typical for thin BaTiO3 films [30], and  *<sup>h</sup>* = 10 [31]. A significant difference between the calculated and fitted values of *β* can be ascribed to the fact that the dielectric properties of BaTiO3 NPs are distinct from those of BaTiO3 films [32,33].

**Figure 2.** The dependence of the breakdown field strength on the volume fraction of BaTiO3 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.

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 <sup>B</sup>*/(1 − 3 *f β*) should be the same for different *f* . This can indeed be obtained from the

experimental plots of *E <sup>B</sup>* versus *T* which, being recalculated to *E <sup>B</sup>*/(1 − 3 *f β*) with the mean value *β* = 0.4, coincide with each other within the possible error bars (see Figure 3). Taking *β* = 0.4 and the dielectric permittivity of PVDF  *<sup>h</sup>* = 10 [31], one finds for the effective dielectric permittivity of BaTiO3 NPs  *<sup>i</sup>* ≈ 30.

**Figure 3.** The dependence of the quantity *E <sup>B</sup>*/(1 − 3 *f β*) on the temperature for different volume fractions of BaTiO3 NPs embedded in PVDF matrix indicated in the inset. The experimental data are taken from Figure 5b in Ref. [6]. The experimental error bars are not available.

In another set of experiments, BaTiO3 NPs of 30–50 nm in diameter were surface modified using a pentafluorobenzyl phosphonic acid (PFBPA) and incorporated into a poly(vinylidene fluoride-cohexafluoropropylene) (P(VDF-HFP)) matrix [7]. Again, for relatively small volume fractions of NPs (*f* ≤ 0.2), the breakdown field strength linearly decreases with *f* (see Figure 4). The value of *β* deduced from the slope of this dependence is about 0.57, and taking  *<sup>h</sup>* = 12.6 [7], one obtains  *<sup>i</sup>* ≈ 70. For comparison, the calculated value of *β* obtained with  *<sup>i</sup>* = 200 is about 0.83.

Finally, a decrease in the breakdown field strength was also observed for BaTiO3 NPs of 7 nm in diameter dispersed in polystyrene (PS) [8]. A linear fit of this dependence (Figure 5) gives *β* ≈ 0.54 which, together with the value  *<sup>h</sup>* = 2.4 for the PS [8], provides  *<sup>i</sup>* ≈ 11. The calculated value with  *<sup>i</sup>* = 200 is *β* ≈ 0.96.

As one can see from the above consideration, the values of the effective dielectric permittivity of BaTiO3 NPs derived from different measurements differ from each other significantly which points at a dominant role of the interface polarization at the boundary between an NP and a polymer matrix. The obtained values of  *<sup>i</sup>* are in the range between 10 and 70 which is much less than the values typical for thin BaTiO3 films [30]. This discrepancy can be attributed to the existence of a nonferroelectric surface layer in the NPs which has a low dielectric permittivity, its relative contribution being that it is increasing with a decrease in the NP size [32,33].

**Figure 4.** The dependence of the breakdown field strength on the volume fraction of BaTiO3 NPs embedded in P(VDF-HFP) matrix. The straight line is the best linear fit to the experimental data shown by circles and taken from Figure 8b in Ref. [7]. The experimental error bars are not available.

**Figure 5.** The dependence of the breakdown field strength on the volume fraction of BaTiO3 NPs embedded in PS matrix. The straight line is the best linear fit to the experimental data shown by circles and taken from Figure 5 in Ref. [8].
