Low-temperature electrical transport in bilayer manganite La${}_{1.2}$Sr${}_{1.8}$Mn${}_2$O${}_7$

The temperature $T$ and magnetic field $H$ dependence of anisotropic in-plane ${\rho }_{ab}$ and out-of-plane ${\rho }_c$ resistivities have been investigated in single crystals of the bilayer manganite La${}_{1.2}$Sr${}_{1.8}$Mn${}_2$O${}_7$. Below the Curie transition temperature $T_c=$ 125 K, ${\rho }_{ab}$ and ${\rho }_c$ display almost the same temperature dependence with an up-turn around 50 K. In the metallic regime (50 K $\le T\le$ 110 K), both ${\rho }_{ab}(T)$ and ${\rho }_c(T)$ follow a $T^{9\mathrm{/}2}$ dependence, consistent with the two-magnon scattering. We found that the value of the proportionality coefficient $B_{ab}^{fit}$ and the ratio of the exchange interaction $J_{ab}\mathrm{/}{J}_c$ obtained by fitting the data are in excellent agreement with the calculated $B_{ab}$ based on the two-magnon model and $J_{ab}\mathrm{/}{J}_c$ deduced from neutron scattering, respectively. This provides further support for this scattering mechanism. At even lower $T$, in the non-metallic regime ($T<$ 50 K), {\it both} the in-plane ${\sigma }_{ab}$ and out-of-plane ${\sigma }_c$ conductivities obey a $T^{1\mathrm{/}2}$ dependence, consistent with weak localization effects. Hence, this demonstrates the three-dimensional metallic nature of the bilayer manganite La${}_{1.2}$Sr${}_{1.8}$Mn${}_2$O${}_7$ at $T<T_c$.