# Reasons for the High Electrical Conductivity of Bismuth Ferrite and Ways to Minimize It
## 3 Results
### About the Causes of the High Conductivity of BiFeO\({}_{3}\)
second component is much higher than that of the first one. A number of works [33; 34] develop the ideas presented in [31]. Meanwhile, the physical nature of the gigantic increase in the effective dielectric permittivity is understandable: it is connected, as mentioned above, with the charge accumulation at the interfaces of the components of a heterogeneous dielectric. At the initial moment, after applying an electric voltage to the sample, the electric induction in the layers is the same, which means the distribution of the field strength is inversely proportional to the dielectric permittivity of the layers. However, after completion of the processes, the densities of the conduction currents are equalized, and the field strength in the layers becomes inversely proportional to the specific conductivities of the layers, which leads to an increase in the charge accumulated at the boundaries precisely in the \"reciprocal composite\". The same physical mechanisms lead to an increase in effective electrical conductivity. In the work of [PERSON] and co-workers [35] it is quite rightly pointed out that in the works [30; 36] and even in the most detailed work [31] devoted to the giant growth of \(\varepsilon\), the possibility of achieving giant electrical conductivities is not even mentioned. It should be noted that in the work [37] an analysis of this possibility is performed, but only for some types of heterogeneous dielectrics. A complete analysis of this problem in the literature is not available, which has served as an incentive to focus on this issue in greater detail. To predict the necessary conditions for giant dielectric amplification, it is sufficient to use the simplest model of an electrically inhomogeneous dielectric considered by [PERSON] [28]. Using the methods of impedance spectroscopy, for the electrical conductivity of a dielectric sample consisting of two parallel layers with significantly different capacitances and resistances, in \(ac\) fields with a circular frequency \(\omega\)[35], we obtained the following formula:  
\[G=[R_{1}+R_{2}+\omega^{2}\Big{(}\tau_{1}^{2}R_{2}+\tau_{2}^{2}R_{1}\Big{)}]/ [(R_{1}+R_{2})^{2}+\omega^{2}(\tau_{2}R_{1}+\tau_{1}R_{2})]^{2}, \tag{1}\]  
and for capacity  
\[C=[\tau_{1}R_{1}+\tau_{2}R_{2}+\omega^{2}(\tau_{1}\tau_{2}(\tau_{2}R_{1}+\tau _{1}R_{2}))]/[(R_{1}+R_{2})^{2}+\omega^{2}(\tau_{2}R_{1}+\tau_{1}R_{2})]^{2}, \tag{2}\]  
where \(\tau_{1}=R_{1}C_{1}\); \(\tau_{2}=R_{2}C_{2}\). Here \(R_{1}\) and \(C_{1}\)--resistance and capacitance of the first layer, \(R_{2}\) and \(C_{2}\)--resistance and capacitance of the second layer. From (1) and (2) for static values of \(G\) and \(C\) at \(\omega\to 0\) it can be written:  
\[G_{s}=1/(R_{1}+R_{2}),\;C_{s}=[\tau_{1}R_{1}+\tau_{2}R_{2}]/[\Big{(}R_{1}+R_{ 2})^{2}+R_{1}+\tau_{1}R_{2}\Big{)}]^{2}, \tag{3}\]  
and for high-frequency values of \(G\) and \(C\) at \(\omega\rightarrow\infty\) it can be written  
\[G_{\infty}=\Big{(}\tau_{1}^{2}R_{2}+\tau_{2}^{2}R_{1}\Big{)}/(\tau_{2}R_{1}+ \tau_{1}R_{2})^{2},\;C_{\infty}=\tau_{1}\tau_{2}/(\tau_{2}R_{1}+\tau_{1}R_{2} )^{2}, \tag{4}\]  
Thus, the two-layer Maxwell-Wagner capacitor is a dielectric with monotonically decreasing real parts of the capacitance with an increase in frequency and the complex permittivity. In this case, the imaginary part of the dielectric permittivity has a relaxation maximum [31; 35]. One of the mechanisms of the appearance of enormous electrical conductivity in BiFeO\({}_{3}\) is the peculiar structure of BiFeO\({}_{3}\) conditioned by the features of the state diagram of the Bi\({}_{2}\)O\({}_{3}\)-Fe\({}_{2}\)O\({}_{3}\) binary system (Figure 3) with three bismuth ferrites of different quantitative elemental composition [31; 38].