Since high-T_c superconductivity in copper oxide (cuprate) and Fe-based material arises near the anti-ferromagnetic (AF) order phase boundary, relation between the magnetic quantum criticality and superconductivity has attracted extensive interest. On the other hand, there is an argument in many novel superconductors that the key role in inducing superconductivity is played by Lifshitz transition which leads to Fermi surface transformation without symmetry breaking.
We performed systematic doping and temperature dependent muon spin rotation (μSR) measurement and angle resolved photoemission spectroscopy (ARPES) on the electron-doped cuprate Pr_1-xLaCe_xCuO_4-δ in order to resolve the interplay among the magnetic quantum criticality, Lifshitz criticality, and superconductivity. Surprisingly, while the long-range AF order phase boundary is located around heavily under doped regime, the Lifshitz transition associated with nodal hole pocket takes place in the vicinity of optimal doping concentration. In this talk, we will present the magnetic and electronic phase diagram of electron doped cuprate Pr_1-xLaCe_xCuO_4-δ, and discuss the implication of our results on the generic phase diagram of high-T_c superconductors.