Variable-range-hopping conductivity of half-doped bilayer manganite LaSr$_{2}$Mn$_{2}$O$_{7}$

We report measurements of in-plane $\rho_{ab}$ and out-of-plane $\rho_{c}$ resistivities on a single crystal of the half-doped bilayer manganite LaSr$_{2}$Mn$_{2}$O$_{7}$. In the temperature $T$ range 220 to 300 K, the resistive anisotropy $\rho_{c}/\rho_{ab}=A+B/T$ ($A$ and $B$ constants), which provides evidence for the variable-range-hopping conduction in the presence of a Coulomb gap. This hopping mechanism also accounts for the quadratic magnetic field $H$ and $\sin^{2}\phi$ dependences of the negative magnetoresistivity $\ln [\rho_{i}(T,H,\phi)/\rho_{i}(T,H=0)]$ ($i=ab,c$), where $\phi$ is the in-plane angle between the magnetic field and the current.