## 3 Results and Discussion
### Dielectric Properties  
It is well known that ferroelectrics exhibit spontaneous polarization and reversal without an external electric field. If DC bias is applied, spontaneous reversal and polarization direction are governed by the electric field direction, which leads to a decrease in the capacitance value [9]. Perovskite ceramic materials such as BaTiO\({}_{3}\) have a high dielectric constant (D\({}_{\rm k}\)) and therefore capacitance value but as D\({}_{\rm k}\) increases, they show increased sensitivity to DC bias. It is known that up to 1 \(\upmu\)m, D\({}_{\rm k}\) sharply increases with the grain size [25]. BT02-MOD-Y showed by the order of magnitude higher dielectric constant value (2132), compared to KZM100-MOD-Y (182). However, consequently, it has the highest capacitance value decrease with DC bias. Figures 9 and 10 show the capacitance drop with DC bias for KZM100-MOD-Y and BT02-MOD-Y respectively. Capacitance of KZM100-MOD-Y remained stable, with negligible decrease up to 73 V, where a slow drop can be observed to 67% at 96 V.  
Figure 8: FESEM images of sintered KZM-100-MOD-Y (**left**) and BT02-MOD-Y (**right**).  
Figure 7: Transmission electron microscopy (TEM) images of core-shell structure in BaTiO\({}_{3}\)-MOD-Y.  
On the other side, BT02-MOD-Y showed a sharp drop around 4 V. This indicates that the amount of dopant in smaller grains induces higher capacitance stability of BaTiO\({}_{3}\) after the modification.  
Capacitance change dependence on DC bias for KZM100-MOD-Y could be modeled within the class of reliability functions.  
\[R(x)=1-F(x)=1-\int\limits_{-\infty}^{x}f(u)du,\]  
where \(F(x)\) is cumulative distribution of random variable. We chose Normal (Gaussian) distribution with scale 100 and got Scaled reliability function, \(\Delta\)C. Here, we used relative capacitance change in function of voltage, which is the result of experimental research.  
We propose a scaled reliability function of normal distribution:  
\[\Delta C(x)=100\cdot\left(1-\int\limits_{-\infty}^{x}\frac{1}{\sqrt{2\pi\delta ^{2}}}e-\frac{(u-\mu)^{2}}{2\sigma^{2}}du\right),\]  
with \(\mu=90,\sigma=10\). Variable \(x\) corresponds to DC bias (V). Figure 11 presents a Scaled reliability function of normal distribution for KZM100-MOD-Y.  
The selected model with a scaled reliability function was the first attempt and was found to be appropriate in comparison with experimental results presented in Figures 9 and 10 diagrams. Up to this research stage, the limiting case of Gauss distribution fits experimental results but it could be more  
Figure 10: Capacitance change with DC bias for BT02-MOD-Y.  
Figure 9: Capacitance change with DC bias for KZM100-MOD-Y.  
refined in the future. Possible other results for which we can improve the optimal selection of the function for a further adequate model could be expected. The physical meaning behind this starting model will be analyzed within the new experimental research.  
Capacitance change with DC bias for BT02-MOD-Y could be modeled with the above scaled reliability function when \(\sigma\to 0\) and it is practically inverted Dirac Delta function (Figure 12).  
Increased grain size causes an increase in dielectric constant and loss tangent, from 0.023 to 0.069 for BT02-MOD-Y, which is in accordance with the literature [15; 26]. The loss tangent value decrease is the consequence of intergranular free charges conduction.