Negative Gradient Energy Facilitates Charged Domain Walls in HfO_{2}

Charged domain walls (CDWs) in ferroelectrics are often characterized as excited states, even after full bound charge compensation at domain walls (DWs). Here, we propose a mechanism where negative gradient energy (E_{grad}) counteracts the large positive electrostatic energy (E_{el}) induced by bound charge at DWs, stabilizing the CDWs over the bulk state, even with partial or no bound charge compensation. As proof of scheme, we demonstrate that negative E_{grad} of modes, which reverse their sign during the formation of CDWs in HfO_{2}, partially offsets E_{el}. Their corresponding remaining E_{el} can be neutralized by partial bound charge compensation. Exceeding this partial compensation, achieved through the substitution of Nb^{5+} at tail-to-tail and Y^{3+} at head-to-head DWs for Hf^{4+}, CDWs exhibit unprecedented stability over the bulk state. The 1.33 eV band gap of the most stable CDW is particularly suitable for photovoltaic applications due to their potential for efficient electron-hole separation. We further reveal that negative phonon band curvatures are useful descriptors to uncover the origin of negative E_{grad} in such CDWs, which can be estimated simply by phonon band fitting, making our scheme easy to implement for realizing CDWs beyond HfO_{2}.