Quantum Criticality in Electron-Doped La$_{2-x}$Ce$_{x}$CuO$_{4}$*

We studied the in-plane resistivity of over-doped La$_{2-x}$Ce$_ {x}$CuO$_{4}$ thin films as a function of temperature (down to 20 mK) and in magnetic fields of up to 17 T. In zero field, we found a $\rho \sim T$ region above $T_{c}$ for superconducting films and a $\rho \sim T^{2}$ region for non-superconducting films as $T\rightarrow 0$. The boundaries of $\rho \sim T $ and $\rho \sim T^{2}$ regions merge at a critical doping $x=0.175\pm 0.005$, which is also the boundary of the superconducting dome. In magnetic fields, the $\rho \sim T$ to $\rho \sim T^{2}$ transition as $T\rightarrow 0$ is shifted to lower doping with increasing field. We compare these results to related behavior in hole-doped cuprates [1] and to a recent phase diagram proposed by Sachdev [2].\\[4pt] [1] R.A. Cooper \textit{et al.}, Science \textbf{323}, 603 (2009).\\[0pt] [2] S. Sachdev, http://arxiv.org/abs/0907.0008V7 (2009).