# Absence of Weak Localization Effects in Strontium Ferromolybdate
## 1 Introduction
\begin{array}{c}\frac{\delta\sigma_{D}}{\sigma_{0}}=-\frac{2}{\pi k_{F}} \ln\frac{l_{\Phi}}{\tau_{e}},\\ \frac{\Delta\sigma_{D}}{\sigma_{0}}=-\frac{2}{2(k_{F}l_{e})^{2}}\end{array}  \tag{3}\]  
where \(\sigma_{0}\) is the residual low-temperature conductivity taken as the Drude conductivity, i.e., \(\sigma_{\rm Dr}=e^{2}n_{e}\tau_{e}/m_{e}\), as determined by the relaxation time (\(\tau\)) of the dominating charge-scattering mechanism, \(e\) is the electron charge, \(n_{e}\) is the electron density, \(\tau_{e}\) is the elastic scattering time and \(m_{e}\) is the electron mass. In Equation (3), the negative sign is due to the fact that the returning trajectory should arrive at the intersection point with the momentum almost opposite to the initial trajectory. This means that interference lowers the conductivity. Note that for small sizes (\(b\)), the volume (\(Dt\))\({}^{d/2}\) should be replaced by (\(Dt\))\({}^{d/2}\)\(b^{3-d}\), since the charge carrier has a chance to diffuse repeatedly from one wall to the other, and the probability of finding it at any point across films or wires limited in size is the same [12]. However, we are considering macroscopic sizes rather than sizes of the order of atomic distances. The lower cutoff (\(\tau_{e}\)) is justified by the fact that within a given time (\(\tau_{e}\)), no elastic scattering occurs, resulting in no closed trajectories. The upper limit is given by [13]:  
\[\tau_{\Phi}^{-1}\sim\left(\frac{T}{D^{d/2}N_{0}a^{3-d}}\right)^{2/(4-d)}, \tag{4}\]where \(D=(v_{F}{}^{2}\tau_{e}/d)\) is the diffusion constant, \(N_{0}\) is the one-spin density of states and \(d\) is the characteristic sample dimension. The phase coherence time (\(\tau_{\phi}\)) defines the phase coherence length (\(l_{\phi}=(D\tau_{\phi})^{1/2}\)). This yields conductivity corrections for two- and three-dimensional systems, which, in terms of quantum conductance (\(e^{2}/h\)), amounts to [9] (p. 282):  
\[ \begin{array}{c}\Delta\sigma_{2D}=-\frac{2^{2}}{\pi h}\ln\frac{l_{e}}{l_{e}} =-\frac{e^{2}}{\pi h}\ln\left(\frac{l_{e}}{l_{e}}\right)^{2},\\ \Delta\sigma_{3D}=-\frac{e^{2}}{\pi h}\left(\frac{1}{l_{e}}-\frac{1}{l_{\phi}} \right),\end{array}  \tag{5}\]  
where \(l_{e}\) is the mean free path of the electron. Temperature modifies the phase coherence length (\(l_{\phi}\)), as well as the weak localization correction. The temperature dependence of \(l_{\phi}\) can be described as [14; 15]  
\[\frac{1}{l_{\phi}^{2}(T)}=\frac{1}{l_{\phi}^{2}(0)}+A_{ee}T^{p_{1}}+A_{ep}T^{p_ {2}}, \tag{6}\]  
where \(l_{\phi}(0)\) is the zero-temperature phase coherence length, and \(A_{\text{ee}}T^{p1}\) and \(A_{\text{ep}}T^{p2}\) represent the contributions of two different dephasing mechanisms, e.g., electron-electron (ee) and electron-phonon (ep) interactions, respectively. Note that the temperature exponents \(p_{1}\) and \(p_{2}\) change fundamentally with the system dimensionality [16]. If electron-electron interaction is the dominant dephasing mechanism (also denoted as Nyquist dephasing mechanism), this gives rise to \(\tau_{\text{ee}}\propto T^{-p}\), with values of \(p\) equal to 0.66, and 1 for \(d=1\) and \(d=2\), respectively [9] (chapter 13.6.4), [13]. For \(d=3\), Equation (4) yields \(p=2\), while the result in [9] (p. 509) is \(p=1.5\). The electron-phonon interaction decoherence mechanism would provide a temperature dependence of the dephasing time (\(\tau_{\phi}\propto T^{-3}\)) [17]. Typically, one finds that \(\tau_{\phi}\approx\tau_{\text{ep}}\propto T^{-p}\) with the exponent of temperature \(p\approx 2\)-4 [16]. In metals and semiconductors at low temperatures, quasielastic (small energy transfer) electron-electron scattering is the dominant dephasing process.