Electronic competing phases and their magnetic field dependence in electron-doped nonsuperconducting and superconducting Pr$_{0.88}$LaCe$_{0.12}$CuO$_4$

We present inelastic neutron scattering studies of electron- doped Pr$_{0.88}$LaCe$_{0.12}$CuO$_4$ (PLCCO). Previous elastic neutron scattering results have shown when superconductivity is first induced by annealing in PLCCO, a quasi-two-dimensional spin density wave also appears and coexists with three- dimensional (3D) antiferromagnetic order. As a function of increasing $T_c$, the Neel temperature of the 3D order decreases and vanishes when optimal superconductivity is achieved, suggesting the potential presence of a quantum critical point at zero temperature. We have performed careful inelastic neutron scattering experiments on a $T_{c}= 21$ K and $T_{N}= 40$ K sample of PLCCO. We find that magnetic excitations in this material scale according to the proximity of a quantum critical regime. We compare the results with hole-doped La$_2$CuO$_4$.