Fig. 4: Oxygen vacancy ordering process and simulations.

a, b Cycling number dependences of the D-E hysteresis loop and the imprint field for the 5.42 μm-long HZO capacitor when using square fatigue pulses of ±4 V/500 ns at a repetition frequency of 1 MHz. The solid line in b represents the fit of an oscillating imprint field according to Eq. (1) that accompanies the V_O^·· ordering illustrated in the inset figures. c Schematic of the preferred V_O^·· accumulation near the top/bottom electrodes during fatigue that is interdependent on the E_i direction (thin arrows). The built-up internal field caused by the V_O^·· accumulation contradicts E_i and changes the dipole orientation (thick arrows). d Minimum energy paths for the O-anion movement within the (001) crystal plane in a 96-atom supercell for Cases i–iv. The dotted balls represent oxygen vacancies, the red balls represent the displaced oxygen anions, and the arrows indicate the movement directions. e Minimum energy paths for bipolar polarization switching with different numbers of three-coordinated oxygen vacancies within a 96-atom supercell. The arrows indicate the polarization (P) orientations in the initial and final states.