Fig. 1
From: Harnessing ferroelectric domains for negative capacitance
The ferroelectric capacitor (FE) with negative capacitance (NC). a The plot of the Landau functional describing the free energy of the FE capacitor, F as function of its charge, Q. b The charge–voltage characteristic, Q–V, corresponding to the Landau free energy shown in a. The blue lines in both a, b correspond to the monodomain state, while the segments corresponding to the two-domain state with static NC are shown in orange. The dashed green curves on b depict the polarization switching process manifesting the transient NC. c Ferrolelectric monodomain sample. Termination of polarization, P (black arrow), at the surface of the sample implies the emergence of the surface depolarization charges, which induce the depolarization field E shown in red, directed opposite to the polarization. d Formation of the periodic domain structure with the up-/down-oriented polarization and alternating surface depolarization charge, reduces the energy of a FE as compared to the energy of the uniformly polarized state. e The FE sample with short-circuited electrodes. Redistribution of the electric charge between electrodes screens the depolarization field, thus zeroing the depolarization energy. As a result, the monodomain structure with uniform polarization is recovered. f In the FE sample confined between two neutral disconnected electrodes, screening of the depolarization charges occurs if the uniform polarization transforms into two equal-size domains with the oppositely oriented polarizations. Free electrons redistribute inside the electrodes to preserve their electroneutrality. g Top view of the two-domain structure with disconnected electrodes. Both FE domains have equal surface areas S1 = S2. h Adding the charge, Q, reconfigures the depolarization charges that tend to maintain the field from the electrodes screened. This is achieved by displacing and bending the domain wall (DW) accompanied by the repartition of the areas of the respective domains. Since the DW shrinks upon approach to the edge of the nanodot, the surface tension pulls the DW out of the system to decrease its surface energy. This additional force drags DW slightly beyond its location that would have ensured the complete screening, leading to the peculiar “overscreening effect”. In such a configuration, the compensating depolarization field exceeds the field from the electrodes, hence the residual field appears to be directed oppositely to the polarization of the nanodot. This is what constitutes the NC effect. i Top view of the FE capacitor with the displaced DW. j Negative differential permittivity εf of the cylindrical FE PbTiO3 nanodot capacitors of different radii as functions of the charge density, Q/S. The jumps in εf from the negative to positive value mark the full polarization of the nanodot to the monodomain state
