Hysteresis and memory in the magnetoresistance of underdoped La$_{2-x}$Sr$_{x}$CuO$_4$ thin films

We have studied the in-plane magnetoresistance (MR) in atomically smooth, MBE grown La$_{2-x}$Sr$_{x}$CuO$_4$ thin films with $x=0.03$, 0.05, 0.055, and 0.06. These dopings span the region from non-superconducting to superconducting samples. The MR was measured at temperatures $0.3$~K~$ \leq T \leq 20$~K and in magnetic fields $0 \leq $~B~$ \leq 9$~T, applied both parallel and perpendicular to the $c$-axis. At low $T$, the MR shows the emergence of a strong, positive contribution, which exhibits hysteresis and memory. At the same time, the zero-field resistances obtained after field cooling and zero-field cooling are found to be different. These glassy features, which seem to reflect the slow dynamics of holes, become more pronounced as $T$ is reduced. The evolution of this glassy, low-$T$ state with doping is discussed in detail.