*2.2. Electrical Breakdown in Conventional Capacitors*

We assume that the main dependencies which govern the electrical breakdown in capacitors remain valid when a small volume fraction of NPs is embedded into the host dielectric. Namely, the avalanching effect in the dielectric in a strong electric field leads to an exponential increase in the electron current density with the dielectric film thickness as follows [17]

$$j\_\varepsilon(d) = j\_\varepsilon(0) \exp(\gamma Ed),\tag{1}$$

where *je*(0) is the injected current density at the electrode with a negative bias (at *z* = 0), the constant *γ* can be determined in terms of the electron energy sufficient for impact ionization and the recombination rate, *E* is the electric field strength inside the dielectric and *d* is the dielectric film thickness. The breakdown occurs when the electron current density reaches the critical value *jB* which is sufficient for the dielectric destruction. For a given thickness, this happens at a certain value of the field strength, *EB*, which determines the breakdown voltage *VB* = *EBd*.

The quantity *je*(0) is dictated, in its turn, by the work function of the electrode and the potential barrier for electrons formed by both the electric field potential and the image potential for an electron originating from the polarization of the electrode. Two processes contribute to this current: the quantum-mechanical tunneling through the barrier and the thermionic current above the barrier.

For relatively weak electric fields, the latter one prevails over the first one and is given by the Richardson–Schottky equation [18]

$$j\_{\varepsilon}(0) = AT^2 \exp\left(-\frac{W}{kT}\right) \exp\left(\frac{BE^{1/2}}{kT}\right),\tag{2}$$

where *A* and *B* are constants, *A* being known as the Richardson constant, *W* is the work function of the electrode, *E* is the electric field strength inside the dielectric film, *k* is the Boltzmann constant and *T* is the temperature.

In strong electric fields, the tunneling mechanism is the dominant one and the injection current is given by the Fowler–Nordheim formula [19]

$$j\_\varepsilon(0) = \mathbb{C}E^2 \exp\left(-\frac{D}{E}\right),\tag{3}$$

where the constants *C* and *D* are determined by the work function of the electrode and do not depend on the temperature.
