# Absence of Weak Localization Effects in Strontium Ferromolybdate
## 3 Results and Discussion  
Strontium ferromolybate (Sr\({}_{2}\)FeMo\({}_{6-\delta}\), SFMO) is a half-metallic, ferrimagnetic compound with a saturation magnetization of 4 \(\mu_{\text{B}}\)/f.u [36]. However, SFMO does not exhibit a general metallic conductivity mechanism. In the absence of a magnetic field, the temperature dependence of the conductivity of SFMO ceramics [36] is well-described by the fluctuation-induced tunneling (FIT) model 46, e.g., by the presence of conducting grains separated by nanosized energy barriers where large thermal voltage fluctuations occur when the capacitance of an intergrain junction is in the order of 0.1 F. Here, tunneling occurs between large metallic grains via the intergrain junctions with a width (\(w\)) and area (\(A\)). The FIT model is specified by three parameters [46]: (i) the temperature (\(T_{1}\)) characterizing the electrostatic energy of a parabolic potential barrier,  
\[kT_{1}=\frac{A\cdot w\cdot e_{\phi}E_{0}^{2}}{2}, \tag{21}\]  
where \(k\) is the Boltzmann constant and the characteristic field (\(E_{0}\)) is determined by:  
\[E_{0}=\frac{4V_{0}}{e\cdot w}; \tag{22}\]  
(ii) the temperature (\(T_{0}\)) representing \(T_{1}\) divided by the tunneling constant,  
\[T_{0}=T_{1}\cdot\left(\frac{\pi\chi w}{2}\right)^{-1}, \tag{23}\]  
with the reciprocal localization length of the wave function  
\[\chi=\sqrt{\frac{2m^{*}V_{0}}{\hbar^{2}}}, \tag{24}\]  
where \(m^{*}\)\(\varepsilon\) is the effective electron mass; and (iii) the residual resistivity (\(\rho_{0}\)). The resulting resistivity of this model is then given by [46]:  
\[\rho(T)=\rho_{0}\exp\left(\frac{T_{1}}{T_{0}+T}\right), \tag{25}\]  
In our case, the model parameters amount to \(T_{0}\) = 141.1 K, \(T_{1}\) = 25.6 K and \(\sigma_{0}\) = 1/\(\rho_{0}\) = 37.16 S/cm (cf. Figure 1). The FIT model was recently applied to intergrain tunneling in polycrystalline Sr\({}_{2}\)CrMoO\({}_{6}\) and Sr\({}_{2}\)FeMoO\({}_{6}\) ceramics [47], half-metallicdouble perovskite Sr\({}_{2}\)BB\({}^{\prime}\)O\({}_{6}\) (BB\({}^{\prime}\)-FeMo, FeRe, CrMo, CrW, CrRe) ceramics [48] and in Ba\({}_{2}\)FeMoO\({}_{6}\) thin films [49]. One feature attributed to weak localization in SFMO ceramics is the decrease in resistivity in the temperature range between 405 K and 590 K [50]. This resistivity behavior is considered more in detail below. According to [50], the resistivity behavior of vacuum-annealed SFMO ceramics above room temperature is separated into three regions: (i) from 300 K up to the Curie temperature of about 405 K, the electrical resistivity increases with temperature and shows metallic behavior; (ii) above 405 K up to approximately 590 K, the resistivity decreases with temperature; and (iii) finally, from 590 K up to 900 K, the resistivity increases as the material becomes metallic again [50]. Another report on the electrical resistivity of SFMO indicates metallic behavior up to 420 K, a decrease in resistivity in the temperature range of 420-820 K and reversion to metallic behavior between 820 and 1120 K [51]. A similar resistivity behavior with a resistivity maximum at about 450 K--far above the Curie temperature of \(\sim\)330 K [52]--was obtained for Ba\({}_{2}\)FeMoO\({}_{6-\delta}\), while Ca\({}_{2}\)FeMoO\({}_{6-\delta}\) shows solely metallic behavior in the whole temperature range of 320-1120 K [51]. A more detailed consideration of the reported resistivity behavior of Sr\({}_{2}\)FeMoO\({}_{6-\delta}\) and Ba\({}_{2}\)FeMoO\({}_{6-\delta}\) above \(T_{\rm C}\)[50; 51] reveals a convincing fit to the adiabatic small polaron hopping model [53] (Figure 2). In the small polaron model, electrical conduction of perovskites at higher temperatures, i.e., above a certain transition temperature, occurs as a result of small polarons moving through the lattice by thermally activated jumps between neighboring sites.