## 3 Results and Discussion
\(\text{Si}_{3}\text{N}_{4}\) ceramic materials mainly exhibit ionic conductivity in terms of carrier migration. The \(\text{Si}_{3}\text{N}_{4}\) ceramic materials prepared in this study have a density close to the theoretical density, and the impurity ions mainly consist of a small amount of glass-phase residue of sintering aids in the grains. The general expression for conductivity is as follows:  
\[\sigma=N\times\frac{q^{2}\delta^{2}}{6 kT}\times\
u_{0}\exp(-W/kT) \tag{1}\]  
where \(N\) represents the logarithm of ion pairs per unit volume; \(q\) represents the charge of the ion; \(\delta\) represents the distance between adjacent semi-stable positions; \(k\) represents the Boltzmann constant; \(T\) represents the thermodynamic temperature; \(\
u_{0}\) represents the vibration frequency of impurity ions at semi-stable positions; \(W\) represents the activation energy of conductivity. The conductivity depends on the carrier concentration and mobility. The carriers in \(\text{Si}_{3}\text{N}_{4}\) ceramics mainly come from impurities among grains, which are residual glass-phase materials remaining at the grain boundaries after sintering. The migration of carriers is also closely related to the process of ion diffusion, according to [PERSON]'s first law. \[J=-D\frac{\partial c}{\partial x} \tag{2}\]  
where \(J\) represents the diffusion flux; \(c\) represents the solute concentration; \(x\) represents the coordinate in the diffusion direction; the diffusion coefficient \(D\) is a fundamental parameter that describes the diffusion process, with unit m\({}^{2}\)/s. The relationship between \(D\) and ionic conductivity is  
\[\frac{\sigma}{D}=\frac{nq^{2}}{fkT} \tag{3}\]  
where \(n\) represents the number of ions per unit volume; \(q\) represents the charge of the ions; \(f\) represents the correlation coefficient; \(T\) represents the thermodynamic temperature. Therefore, the migration behavior of carriers in \(\text{Si}_{3}\text{N}_{4}\) ceramic materials can also be explained by ion diffusion behavior. In these materials, the diffusion path of ions has an important influence on diffusion behavior. Diffusion inside a normal crystal is collectively referred to as lattice diffusion, volume diffusion, or bulk diffusion. In contrast, diffusion along dislocations, grain boundaries, or surfaces is called dislocation core diffusion, grain boundary diffusion, or surface diffusion, respectively. Diffusion along line defects and surface defects is generally faster than bulk diffusion, so this phenomenon is referred to as \"short-range diffusion\" in the literature, referring to diffusion that occurs along paths that are much faster than bulk diffusion, including dislocations, grain boundaries, and solid-phase boundaries [22]. In summary, on the one hand, under the action of an electric field, the diffusion rate of carriers in \(\text{Si}_{3}\text{N}_{4}\) ceramics is slow, causing carriers to tend to migrate along the grain boundaries. On the other hand, the \"short-range diffusion\" path is determined by the direction of the grain boundary, which does not completely coincide with the direction of the electric potential gradient, resulting in a decrease in the electric field force and an increase in the migration path during the actual migration process of carriers, thereby resulting in higher resistivity of the ceramic body. At room temperature, impurity ionization inside the ceramic provides carriers for the conduction process, and the carriers mainly migrate along the grain boundaries. With the increase in temperature, more impurity ions are ionized from the glass-phase impurities. At the same time, the lattice vibration of the ceramic intensifies, but its basic framework structure remains unchanged; so, the migration path of carriers is less affected, and most carriers still migrate along the grain boundaries. The increase in temperature increases the carrier concentration inside the material and enhances the migration speed of carriers along grain boundaries, with little influence on the migration path of carriers.