# Absence of Weak Localization Effects in Strontium Ferromolybdate
## 1 Introduction  
Ordered double perovskite oxides of the general formula A\({}_{2}\)BB\({}^{\prime}\)O\({}_{6}\) have drawn scientific interest for several decades to due to their interesting magnetic properties and their high potential for use in novel electronic devices. Recently, we published a review on resistivity and tunnel magnetoresistance in double perovskite [1]. Compounds with high spin polarization dramatically enhance spintronic device performance and are required for a new generation of non-volatile memories [2; 3]. The room-temperature magnetization and large dielectric response of SFMO enable intriguing applications as microwave absorbers [4; 5]. Superior catalytic properties, as well as stability in both oxidizing and reducing atmospheres, enable the use of SFMO not only as an anode but also as a cathode and in symmetrical solid oxide fuel cells. SOFC provides both electron and ionic conductivity, extending the triple-phase boundary compared to purely electronic or purely ionic conductors. These properties can be tailored to a particular application by the substitution of different metal cations at different sites on SFMO lattices. Moreover, iron-rich SFMOmaterials are excellent catalysts in hydrocarbon oxidation and can prevent carbon deposition due to their ability to exchange lattice oxygen with the gaseous phase. They are also sulfur-tolerant. This opens the way to direct hydrocarbon-fueled SOFCs, eliminating the need for external fuel reforming and sulfur removal components. As such, SOFCs can be greatly simplified and operate with much higher overall efficiency, contributing to a solution to the lack of energy in our modern world [6]. The huge magnetic entropy change at the ferromagnetic transition makes SFMO a candidate for magnetocaloric cooling [7, 8]. In a disordered electronic system, the electron motion is diffusive due to scattering at non-uniformities. Weak localization is a physical effect that occurs in disordered electronic systems at very low temperatures. This effect manifests itself as a correction (\(\Delta\sigma\)) to the conductivity (or, correspondingly, the resistivity) of a metal or semiconductor arising in cases in which the mean free path (\(I\)) is in the order of the wavelength (\(\lambda_{F}=2\pi/k_{F}\)) of the carrier wavefunctions, i.e., \(k_{F}\)\(\sim\)1, with \(k_{F}\) denoting the Fermi wave vector. The weak localization correction comes from quantum interference of backscattered electrons. Labeling all the trajectories according to the time (\(t\)) it takes for a classical particle to go around a loop, the classical probability (\(dP\)) that the diffusing particle returns into the phase volume (\(dV\)) at a given time (\(t\)) is given by [9] (p. 153)  
\[dP=\frac{dV}{\left(4\pi Dt\right)^{d/2}} \tag{1}\]  
where \(P\) is the probability that a quantum mechanical particle will return to the starting point (the wave functions of both time-reversed paths constructively interfere with each other), \(D=v_{F}{}^{2}\tau/d\) is the diffusion constant, \(\tau\) is the elastic scattering time and \(d\) is the dimension. The relevant phase volume can be estimated as \(v_{F}\ dt\ (\delta\rho\delta\varphi)^{d-1}\)[10, 11], where \(v_{F}=(2E_{F}/m_{e})^{1/2}\) is the Fermi velocity; \(E_{F}\) is the Fermi energy; \(m_{e}\) is the electron mass; \(\delta\rho\) and \(\delta\varphi\) characterize the transverse distance between electron trajectories 1 and 2 at the intersection point, respectively; and \(\delta\varphi\) is the intersection angle between the two trajectories. For the interference between paths 1 and 2 to be effective, the uncertainty relation should hold:  
\[p_{F}\delta\rho\delta\varphi=\hbar k_{F}\delta\rho\delta\phi=h \tag{2}\]  
where \(p_{F}=m_{e}v_{\rm F}\) is the Fermi momentum, and \(h\) is the Planck constant. This leads to a relative change in conductivity of two- and three-dimensional systems, yielding [9] (p. 183):  
\[