# Absence of Weak Localization Effects in Strontium Ferromolybdate
## 3 Results and Discussion
The transition temperature from small polaron motion in a conduction band to small polaron hopping was estimated to be in the order of \(\theta_{\rm D}/2\) (with \(\theta_{\rm D}\) the Debye temperature), which amounts to 338 K for SFMO [54]. The adiabatic small polaron hopping model yields a resistivity of [53]  
\[\rho=\rho_{0}T\exp\left(\frac{E_{a}}{kT}\right), \tag{26}\]  
where \(E_{a}\) is the thermal activation energy. The obtained \(E_{a}\) values are 0.045-0.08 eV for Sr\({}_{2}\)FeMoO\({}_{6-\delta}\) and about 0.13 eV for Ba\({}_{2}\)FeMoO\({}_{6-\delta}\) and are in the order of the values of other perovskites and double perovskites, i.e., La\({}_{1-x}\)Sr\({}_{x}\)Co\({}_{1-y}\)Fe\({}_{y}\)O\({}_{3}\)[55], Sr\({}_{1.6}\)Sm\({}_{0.4}\)MgMoO\({}_{6-\delta}\),  
Figure 1: Fitting of the conductivity data of SFMO ceramics at zero magnetic flux [36] to the fluctuation-induced tunneling model [46]. Fitting parameters are \(T_{0}=141.1\) K, \(T_{1}=25.6\) K and \(\sigma_{0}=37.16\) S/cm. Weak localization effects were taken into account in order to explain the presence of minima in the \(\rho\)-\(T\) curves of perovskite oxides exhibiting metallic conductivity [25; 27; 33; 58; 59]. It has been suggested [29; 30] that the resistivity minimum and, consequently, the resistivity upturn at lower temperatures arise from the competition of two contributions--one usual increase combined with a decrease with the increase in the temperature of the other. In Equation (15), the corresponding terms are \((B^{\prime}\ln T)^{-1}\) and \(C_{n}T^{n}\). The resistivity-versus-temperature plots of SFMO ceramics in a magnetic field [36] are very similar to those of La\({}_{0.7}\)Ca\({}_{0.3}\)MnO\({}_{3}\) and La\({}_{0.7}\)Sr\({}_{0.3}\)MnO\({}_{3}\) thin films [25; 27]. A resistivity minimum also appears in polycrystalline SFMO ceramics at low temperatures [36], which was explained by weak localization [59]. In this case, a quantum correction term (\(A_{W}T^{p/2}\)) [25; 59] was added to the residual resistivity (\(\sigma_{0}\)), and an electron interaction term (\(A_{p}T^{n}\)) was added to the resistivity. A similar approach was applied to perovskite ceramics (LaNi\({}_{x}\)Co\({}_{1-x}\)O\({}_{3}\) and Na\({}_{x}\)Ta\({}_{y}\)W\({}_{1-y}\)O\({}_{3}\)[60], La\({}_{0.5}\)Pb\({}_{0.5}\)MnO\({}_{3}\) and La\({}_{0.5}\)Pb\({}_{0.5}\)MnO\({}_{3}\) ceramics containing 10 at% Ag in a dispersed form [23]). To evaluate the origin of the low-temperature resistivity minimum in SFMO, we fitted the experimental data [36] to Equation (15), assuming \(A^{\prime}\) = 0 (Figure 3, Table 1). The increase in the \(\sigma_{0}(B)\) values corresponds to a negative magnetoresistance obtained in ferrimagnetic SFMO ceramics, which arises due to the suppression of spin disorder by the magnetic field [35; 36; 50]. The fitted \(\sigma_{0}\) values describe a power-law magnetic flux dependence of the magnetoresistance (\(-MR\propto B^{\rm m}\)) with a power of \(m\) = 0.284. This value lies in between the values of \(m\) = 0.5 for metallic behavior and electron-electron interaction in the temperature range between the temperature of minimum resistivity and the Curie temperature and \(m\) = 0.1 for semiconducting conductivity behavior, both at high magnetic fluxes [61]. The fitted \(B^{\prime}\) values are three orders of magnitude lower than those calculated above. Consequently, the elastic scattering time should be reduced by six orders of magnitude. This is physically nonsensical. Furthermore, the \(C_{n}T^{n}\) term does not correspond to physically meaningful quantities. The values of the exponent (\(n\)) cannot be attributed to a specific electron-scattering mechanism. On the other hand, electron scattering should sufficiently change depending on the magnetic flux. When assuming \(n\) = 2 for electron-electron scattering and assuming \(C_{n}\) = 2.16 \(\times\) 10\({}^{-11}\)\(\Omega\)mk\({}^{-2}\) from [41], the fit becomes of much worse quality (Figure 4). Here, a satisfactory fit occurs only for \(B\) = 0.