Nano-world of domains
The famous Landau expression: "No one can cancel the Coulomb law" is often overlooked in attempts to understand the spontaneous polarization in Ferroelectric materials that, paradoxically, should be completely destabilized by the backward depolarizing electrostatic field produced by the charge 4πρ=divP of the polarization surface breakdown. This puzzle is partially resolved in samples of >500nm where the unfavorable depolarizing field is screened by the free semi-conducting charges. Alternatively, the smaller samples are segregated onto up- and down- polarized domains, as was initially proposed by Landau and Kittel for magnetic materials. This alternates the surface charge and vanishes the bulk depolarizing field.
The current tendency of miniaturization of ferroelectric-based computer memories challenges the study of non-uniform polarization distribution in nano-samples. However, the first-principle modeling by the ensemble of electrostatic dipoles is possible only for the very small systems < 50nm because of the non-local Coulomb interaction.
So, how the surface of the nanodevice of the realistic size of 50-500nm drives the polarization texture inside? To understand this we use the analytical approach of solution of nonlinear equations of condensation to ferroelectric state coupled with electrostatic (Maxwell) equations. Such an approach was proposed in the early 80s by Chensky and Tarasenko but never explored after... Periodic domains, vortices, skyrmions, and other exotic formations can be created inside of finite-size ferroelectric devices by long-range Coulomb forces of the surface. For sure they can be useful as the memory units of future memory devices.
And this is not all! Integration of ferroelectric elements into nano-electronic silicon environment again produces the polarization domains to compensate the interface-junction-created elastic stress. Surprisingly, such ferroelastic domains exist even inside the samples with no interface. This discovery, confirmed by our modeling, implies that the several-atom-thickness surface skin can trigger the intrinsic stress and produce the drastic domain-structuring of the device.