**1.3. Mechanisms for electrical conduction**

The pre rupture region is also named as linear region and has an ohmic behavior when the material is under operation normal tension. The varistor acts as a resistor in this case with a small amount of current (known as leakage current) passing through the material due to the action of the potential barrier formed at grain boundary and preventing the electronic conduction between the grains. The conductivity in this region is of thermionic emission type, i.e., the electrical conduction is strongly dependent of temperature, thus being possible to

The rupture region showed nonlinear behavior, i.e., non-ohmic behavior between the applied voltage and the current that the material is submitted. The conductivity of the material increases with a small variation in the applied voltage, indicating the varistor efficiency that starts to act as a conductor from a specific breakdown electric field (*E*R). Recombination of electron-hole pair at grain boundary interfaces, thermionic emission, and electron tunneling

In the post rupture region, the ohmic behavior between the current and the applied voltage is observed once again and is characterized by high current density. The electric conduction in

The *V*R value provides the varistor voltage application, and it is a function of a grain size of sintered material. If the composition is fixed, the microstructure becomes strongly dependent

The varistor efficiency determined by the breakdown region can be evaluated by the *α* nonlinear coefficient of the curve in Figure 1, which is used in Eq. 6, derived from Eq. 3, which allows the

21 2 1

The electric field and the current density are obtained from the measurements of the electric current (I) generated when the sample is submitted to a potential difference (V), according to

*d* is the thickness of the sample and *A* is the electrode area deposited on the film surface. For *α* calculation, the interval of 1 and 10 mA/cm2 of current density was used, i.e., *J*<sup>1</sup> = 1 and *J*2 =

2 1

a

1

=- - ( )/( ) *logJ logJ logE logE* (3)

*<sup>d</sup>* <sup>=</sup> (4)

*<sup>I</sup> <sup>J</sup> <sup>A</sup>* <sup>=</sup> (5)

( ) *logE logE* - = - (6)

calculation of the value of *α* by the field data electric (*E*) and current density (*J*) [18,19]:

*E V*

retrieve information about the resistivity of the material [8,16,17].

are suggested as electric conduction mechanisms of this region [8,16].

this region is controlled by the impedance of the grains [2,8].

a

on the processing conditions [12,15].

Eqs. 4 and 5 [18,19]:

28 Advanced Ceramic Processing

10 [18,19]:

The potential barrier is the determining factor on the electrical properties of varistors. Several models have been proposed to better understand the potential barrier formed in the grain boundary region [20,21].

Gupta et al. [22] proposed the first potential barrier model for ZnO-based varistor of the Schottky-type with an intergranular layer acting as insulator between the grains. In this model, negative charge densities (formed by Zn vacancies) were trapped between the grain boundary being balanced by two depletion layers that are positively charged. Leite et al. [23] proposed the accumulation of oxygen species adsorbed as new origin of negative defects [22,23].

Based on the ZnO potential barriers model, Bueno et al. [14] suggested a modification for formation of potential barrier in SnO2 varistors systems, whereas the sides of the barrier are in contact since there is no precipitated phase in the grain boundary, as observed Figure 2. In this model, the oxygen adsorbed species in the grain boundary region generate the negative charges defects, counterbalanced by the positive defects in the depletion layer. Pianaro et al. [1] proposed a potential barrier model, which has a large presence of negative charges on the SnO2 surface generated by tin vacancies (*Vsn* " ), adsorbed oxygen atoms and substitutional cobalt ions (*CoSn* '' ), and positive defects in the depletion layer formed by interstitial tin (*Sn* ••••, *Sn* ••), oxygen vacancies (*VO* ••, *VO* •), and niobium taking place of tin on the crystal lattice (*N bSn* • ) [1,14].

**Figure 2.** Potential barrier model to ZnO varistors proposed by Gutpa et al. [22] and Leite et al. [23] and to SnO2 varis‐ tors base proposed by Bueno et al. [14] and Pianaro et al. [1].

#### *1.3.1. Schottky type*

In this model, the electrons are emitted and pass through the potential barriers particularly due to the action of temperature distorting the energy band diagram, near the interface. This distortion modifies the potential barrier favoring the thermal emission. The equation that describes this behavior is [24,25]

$$J\_S = \left. A^\* . T\_2 . exp \left[ -\frac{\phi\_b - \beta E\_{\psi\_b}}{kT} \right] \right| \tag{7}$$

where *A*\* is the Richardson constant, *φ*b is the potential barrier height, *E* represents the electric field, *T* is the ambient temperature in Kelvin, and *β* is a constant related to the width of the potential barrier in accordance with the following equation [25]:

$$
\beta = (no)^{-1/2} \tag{8}
$$

where *n* is the grain number per unit length and *ω* is the width of the barrier.

#### *1.3.2. Poole–Frenkel type*

The emission of the Poole–Frenkel type assumes the formation of coulombian centers in the grain–intergranular layer interface region. The relationship that describes this type of emission is on Equation 9, where the external electric field variations are more relevant than for issue of Schottky type [26]:

$$J\_P = c.E.exp\left[-\frac{\phi\_b - 2\beta E\_{\forall}}{kT}\right] \tag{9}$$

where *c* is a constant of the material, *T* is the room temperature, *E* is the electric field, *k* is the Boltzmann constant, and *φ*b is the height of the potential barrier. The thermionic emission cannot explain the high nonlinear coefficients observed in varistors. In the post rupture zone with the presence of high electric fields, the possibility that distortion of the energy levels and, therefore, the possibility that electrons pass through the potential barrier by tunneling must be considered [27].
