Abstract
The electrostatics arising in ferroelectric/dielectric two-dimensional heterostructures and superlattices is revisited within a Kittel model in order to define and complete a clear paradigmatic reference for domain formation. The screening of the depolarizing field in isolated ferroelectric or polar thin films via the formation of 180° domains is well understood, where the width of the domains w grows as the square-root of the film thickness d, following Kittel’s Law for thick enough films (w ≪ d). For thinner films, a minimum is reached for w before diverging to a monodomain. Although this behaviour is known to be qualitatively unaltered when the dielectric environment of the film is modified, we consider the quantitative changes in that behaviour induced on the ferroelectric film by different dielectric settings: as deposited on a dielectric substrate, sandwiched between dielectrics, and in a superlattice of alternating ferroelectric/dielectric films. The model assumes infinitely thin domain walls, and therefore is not expected to be reliable for film thickness in the nanometre scale. The polarization field P(r) does vary in space, deviating from ±PS, following the depolarizing field in linear response, but the model does not include a polarization-gradient term as would appear in a Ginzburg–Landau free energy. The model is, however, worth characterizing, both as paradigmatic reference, and as applicable to not-so-thin films. The correct renormalization of parameters is obtained for the thick-film square-root behaviour in the mentioned settings, and the sub-Kittel regime is fully characterized. New results are presented alongside well-known ones for a comprehensive description. Among the former, a natural separation between strong and weak ferroelectric coupling in superlattices is found, which depends exclusively on the dielectric anisotropy of the ferroelectric layer.
Highlights
The electrostatics arising in ferroelectric/dielectric twodimensional heterostructures and superlattices is revisited within a Kittel model in order to define and complete a clear paradigmatic reference for domain formation
For a free-standing thin film on a substrate, it was claimed that the electrostatic description is the same as for a thin film of half the thickness sandwiched between two paraelectric media [22]
By placing a ferroelectric thin film together with a paraelectric layer between two short-circuited capacitor plates, it was found that the domain structure could be controlled by tuning the properties of the paraelectric layer, and the stability of the ferroelectric film could be improved [23,24,25,26,27,28]
Summary
The fundamental model used in this work is based on the following free energy per unit volume of a ferroelectric thin film in a vacuum with a 180° stripe domain structure [1,5]. In the Kittel model, instead of solving for P in equation (2.2), the total polarization field P(r) is taken to deviate from the spontaneous polarization ±PS in linear response to the electric depolarizing field, according to the dielectric susceptibilities normal and parallel to the film, κc and κa, respectively This model makes significant approximations about the form of P(r), such as neglecting domain-wall width and surface/interface effects. In figure 3, we see that the domain width follows Kittel’s Law at large values of d, but, for decreasing d, w reaches a minimum at dm and diverges at d∞ We can understand this behaviour by studying the shape of the energy curves as a function of domain width and thickness, which is done in figure 4. We can obtain an analytic approximation to the domain width at all thicknesses by replacing equation (2.4) with a simpler expression which has the correct behaviour in the monodomain and Kittel limits. For that purpose a more general expression for the electrostatic energies, similar to equation (2.4) is needed
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