Fig. 2: Ultrahigh dielectric permittivities.

a Frequency dependences of ε′ and tanδ when E_osc = 50 kV/cm for the size-scaled capacitors after the occurrence of polarization fatigue. Larger capacitor sizes generally result in smaller values of ε′. The solid lines represent data fittings according to Eqs. (5) and (6). b Frequency dependences of the dielectric permittivity and loss tangent at various temperatures for the 4.39-μm-long fatigued capacitor. The solid lines represent data fittings according to Eqs. (5) and (6). c D-E hysteresis loops at 252 kHz for the 4.39-μm-long fatigued capacitor at different temperatures. d Cycling number dependences of the stored charge density within the 4.39-μm-long virgin/fatigued capacitors under application of unipolar square pulses with voltage/width characteristics of 1.2 V/50 ns with a repetition frequency of 10 MHz at room temperature. The charge density increases by 10 times after fatigue. e Comparison of the ultrahigh dielectric permittivity realized in this work with the corresponding values for the pure ZrO₂ and HfO₂^21,23, Al-doped HfO₂ (HAO)^55,56, Si-doped HfO₂ (HSO)¹⁴, Y-doped HfO₂ (HYO)¹⁶, Zr-doped HfO₂ (HZO)^57,58, and HfO₂-ZrO₂ superlattices^13,29,59 reported previously in the literature. f Synchrotron Laue micro-diffractions of the O \((111)\) reflections extracted from the virgin and fatigued areas nearby a 4.35 μm-long capacitor at a wavelength of 0.6209 Å (Supplementary Fig. S13a–c).