Electrostatics and domains in ferroelectric superlattices

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.


Comments to the Author(s)
This article studies a free energy model for the electrostatic problem of a finite-size ferroelectric layer under several boundary conditions (overlayer, sandwich, superlattice). The model describes the domain formation and domain widths of the ferroelectric layer as a function of the film thickness, and is a generalization of the Kittel-Mitsui-Furiuchi model. The authors describe in detail the model, compare it with previous models in the literature and assess it's validity in different limits. The article is thorough, very interesting, and well written. It provides a better understanding of similar (but less general) models previously reported in the literature, making also a connection between systems with different boundary conditions, thus serving as a paradigmatic model. Also, the authors obtain some new results for overlayer and superlattice geometries. I certainly recommend it for publication, although I suggest some minor changes.
1) In the third paragraph of the introduction the authors state that in Ref. [22] it was claimed that a free-standing thin film on a substrate has the same electrostatic description as a thin film of half the width sandwiched between two paraelectric media. I believe this is not correct, it was claimed that it is equivalent to a thin film of half the thickness. I assume this is a typo, otherwise I think it deserves further explanation.
2) In the spirit of setting a paradigm for electrostatic models with this work, could the authors further clarify (possibly with an equation) the connection between the more complex form of F_0(P) shown in Eq.(2) and its more simple form (mostly used throughout the text) in Eq. (3)?
3) When describing Fig. 5 the authors state that "By increasing kappa_c for a fixed value of d, the total energy minimum again becomes shallower and then disappears". Could the authors make a brief comment on the physical implications of this finding? 4) When describing Eq. (15) the authors state that "For each case in Eq. (15), Eq. (6) is recovered in the limit kappa_s → 1." In the case of the superlattice this is true only for alpha=0. I believe this should be stated explicitly. 5) kappa_s is introduced in the main text in Eq. (14) (Section III. A) but is defined later (first paragraph of Section III. B). The authors should define it as soon as it is introduced.
6) The acronym "IF" is never defined, it should be defined the first time it is used (Section III. B). Does it correspond to "isolated film"? 7) In the second paragraph of the Discussion and Conclusion section the authors state that the domain width increases with dielectric permittivity in the Kittel limit. I believe this should be rephrased to emphasize that this discussion is restricted to systems with non-periodic boundary conditions, since in the superlattice case the above statement is only true for a particular range of values of alpha (as the authors have found, and as they discuss in the next paragraph). 8) In the first paragraph of Appendix A the authors present the derivation of the electrostatic energy for the SL system, the method applying also to the OL and SW systems. I do not believe that the derivation for the three systems should be included, since the three of them are very similar. I do believe, however, that the statement regarding the difference in the boundary conditions for the three geometries is a bit too vague ("the only difference being that the boundary conditions change from periodic to infinite."). I suggest that this sentence is rephrased for the text to be more precise and to allow the readers to reproduce the results for the cases not shown explicitly. Alternatively, the authors could write down explicitly the boundary conditions in a mathematical form. 9) In Appendix C the authors give the expressions for the electrostatic energies of the SW, SL and OL geometries in their generalized Kittel model. When describing them (page 9, line 57, first column) there are two minor mistakes that may misguide the reader: the authors state that "The first and third lines are the electrostatic energies of the SW and SL cases respectively", but there are only two lines in Eq(30). Also, the SL case is the one for which the derivation has been presented step by step. I believe this minor mistake should be corrected for clarity (e.g. the sentence could read something like "The first line in Eq. (30) and Eq. (31) correspond to the electrostatic energies of the SW and OL cases respectively", or, alternatively, Eq.(31) could become the third line of Eq. (30)). 10) In Appendix C, when discussing the effect of the prefactor (1+alpha)^(-1), the authors state that both the electrostatic energy and the "energy cost of creating a domain structure" are scaled by it. The authors then claim that "the equilibrium domain width will be unaffected by this prefatcor, and we can neglect it". I believe the very last statement ("we can neglect it") is partially vague and somehow misleading, since in the next sentence the effect of the prefactor alpha on the (renormalized) permittivity is addressed, and later it is shown how it modifies other quantities like d_m and d_inf. I suggest either omitting it or rephrasing it with more precision. 11) In Fig. 4, could the authors show the saturation value for the energy (F_mono) for reference?
Finally, I attach a list of typos I found while reading the manuscript: -Mixed use of American and British spelling of behavior/behaviour throughout the text.
-Page 1, line 46 -48, first column: "however" is used twice in the last sentence of this paragraph for the same proposition.

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Accept as is

Comments to the Author(s)
The formation of ferroelectric domain structures in ferroelectric thin films and multilayers is a topic of active research. A key model that is frequently applied in these studies is the Kittel model, but it is most often applied without a great deal of thought about whether it should actually apply, where it might break down and how it's applicability and predictions might differ depending on the environment surrounding the ferroelectric material.
In the present paper the authors have carried out a very rigorous investigation of these issues. The presentation of their results is excellent, with a good pedagogical style that explains well the background to the model. The authors have also made a good effort to present their results in a way that will be useful to experimentalists. The results that pertain to how the dielectric constant of the ferroelectric material shift the deviation from the Kittel law are particularly interesting.
Overall, considering the care the authors have taken in developing their model and presenting their results I am happy to recommend publication without revisions.

Decision letter (RSOS-201270.R0)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.

Dear Mr Bennett
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-201270 "Electrostatics and domains in ferroelectric superlattices" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
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Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers). This article studies a free energy model for the electrostatic problem of a finite-size ferroelectric layer under several boundary conditions (overlayer, sandwich, superlattice). The model describes the domain formation and domain widths of the ferroelectric layer as a function of the film thickness, and is a generalization of the Kittel-Mitsui-Furiuchi model. The authors describe in detail the model, compare it with previous models in the literature and assess it's validity in different limits. The article is thorough, very interesting, and well written. It provides a better understanding of similar (but less general) models previously reported in the literature, making also a connection between systems with different boundary conditions, thus serving as a paradigmatic model. Also, the authors obtain some new results for overlayer and superlattice geometries. I certainly recommend it for publication, although I suggest some minor changes.
1) In the third paragraph of the introduction the authors state that in Ref. [22] it was claimed that a free-standing thin film on a substrate has the same electrostatic description as a thin film of half the width sandwiched between two paraelectric media. I believe this is not correct, it was claimed that it is equivalent to a thin film of half the thickness. I assume this is a typo, otherwise I think it deserves further explanation.
2) In the spirit of setting a paradigm for electrostatic models with this work, could the authors further clarify (possibly with an equation) the connection between the more complex form of F_0(P) shown in Eq.(2) and its more simple form (mostly used throughout the text) in Eq. (3)?
3) When describing Fig. 5 the authors state that "By increasing kappa_c for a fixed value of d, the total energy minimum again becomes shallower and then disappears". Could the authors make a brief comment on the physical implications of this finding? 4) When describing Eq. (15) the authors state that "For each case in Eq. (15), Eq. (6) is recovered in the limit kappa_s → 1." In the case of the superlattice this is true only for alpha=0. I believe this should be stated explicitly. 5) kappa_s is introduced in the main text in Eq. (14) (Section III. A) but is defined later (first paragraph of Section III. B). The authors should define it as soon as it is introduced.
6) The acronym "IF" is never defined, it should be defined the first time it is used (Section III. B). Does it correspond to "isolated film"? 7) In the second paragraph of the Discussion and Conclusion section the authors state that the domain width increases with dielectric permittivity in the Kittel limit. I believe this should be rephrased to emphasize that this discussion is restricted to systems with non-periodic boundary conditions, since in the superlattice case the above statement is only true for a particular range of values of alpha (as the authors have found, and as they discuss in the next paragraph). 8) In the first paragraph of Appendix A the authors present the derivation of the electrostatic energy for the SL system, the method applying also to the OL and SW systems. I do not believe that the derivation for the three systems should be included, since the three of them are very similar. I do believe, however, that the statement regarding the difference in the boundary conditions for the three geometries is a bit too vague ("the only difference being that the boundary conditions change from periodic to infinite."). I suggest that this sentence is rephrased for the text to be more precise and to allow the readers to reproduce the results for the cases not shown explicitly. Alternatively, the authors could write down explicitly the boundary conditions in a mathematical form. 9) In Appendix C the authors give the expressions for the electrostatic energies of the SW, SL and OL geometries in their generalized Kittel model. When describing them (page 9, line 57, first column) there are two minor mistakes that may misguide the reader: the authors state that "The first and third lines are the electrostatic energies of the SW and SL cases respectively", but there are only two lines in Eq(30). Also, the SL case is the one for which the derivation has been presented step by step. I believe this minor mistake should be corrected for clarity (e.g. the sentence could read something like "The first line in Eq. (30) and Eq. (31) correspond to the electrostatic energies of the SW and OL cases respectively", or, alternatively, Eq.(31) could become the third line of Eq. (30)).
10) In Appendix C, when discussing the effect of the prefactor (1+alpha)^(-1), the authors state that both the electrostatic energy and the "energy cost of creating a domain structure" are scaled by it. The authors then claim that "the equilibrium domain width will be unaffected by this prefatcor, and we can neglect it". I believe the very last statement ("we can neglect it") is partially vague and somehow misleading, since in the next sentence the effect of the prefactor alpha on the (renormalized) permittivity is addressed, and later it is shown how it modifies other quantities like d_m and d_inf. I suggest either omitting it or rephrasing it with more precision. Fig. 4, could the authors show the saturation value for the energy (F_mono) for reference?

11) In
Finally, I attach a list of typos I found while reading the manuscript: -Mixed use of American and British spelling of behavior/behaviour throughout the text. -Page 1, line 46 -48, first column: "however" is used twice in the last sentence of this paragraph for the same proposition. -Page 4, line 40, second column: "systems parameters" → "system's parameters". -Page 5, line 38, first column: "a more general expressions" → "a more general expression" -Page 5, line 22, second column: "refactor" → "prefactor" -Page 5, line 37, second column: "This expected" → "This is expected" -Page 6, line 48, first column: "the width at which the domain width diverges" → "the thickness at which the domain width diverges" -Page 6, line 41, first column: "of the substrate material, kappa_s for the OL[. The formation of ferroelectric domain structures in ferroelectric thin films and multilayers is a topic of active research. A key model that is frequently applied in these studies is the Kittel model, but it is most often applied without a great deal of thought about whether it should actually apply, where it might break down and how it's applicability and predictions might differ depending on the environment surrounding the ferroelectric material.
In the present paper the authors have carried out a very rigorous investigation of these issues. The presentation of their results is excellent, with a good pedagogical style that explains well the background to the model. The authors have also made a good effort to present their results in a way that will be useful to experimentalists. The results that pertain to how the dielectric constant of the ferroelectric material shift the deviation from the Kittel law are particularly interesting.
Overall, considering the care the authors have taken in developing their model and presenting their results I am happy to recommend publication without revisions.

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Decision letter (RSOS-201270.R1)
We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Mr Bennett, It is a pleasure to accept your manuscript entitled "Electrostatics and domains in ferroelectric superlattices" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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Comments by Reviewer 1
This article studies a free energy model for the electrostatic problem of a finite-size ferroelectric layer under several boundary conditions (overlayer, sandwich, superlattice). The model describes the domain formation and domain widths of the ferroelectric layer as a function of the film thickness, and is a generalization of the Kittel-Mitsui-Furiuchi model. The authors describe in detail the model, compare it with previous models in the literature and assess it's validity in different limits. The article is thorough, very interesting, and well written. It provides a better understanding of similar (but less general) models previously reported in the literature, making also a connection between systems with different boundary conditions, thus serving as a paradigmatic model. Also, the authors obtain some new results for overlayer and superlattice geometries. I certainly recommend it for publication, although I suggest some minor changes.
Thank you for taking the time to review our manuscript. We very much appreciate the care and attention to detail taken. We agree with all of the minor changes suggested and think they will help to make the manuscript clearer and overall more consistent. We will not respond to each one individually, but have implemented all of the suggested changes in the revised manuscript, highlighted in red.

Comments by Reviewer 2
The formation of ferroelectric domain structures in ferroelectric thin films and multilayers is a topic of active research. A key model that is frequently applied in these studies is the Kittel model, but it is most often applied without a great deal of thought about whether it should actually apply, where it might break down and how it's applicability and predictions might differ depending on the environment surrounding the ferroelectric material.
In the present paper the authors have carried out a very rigorous investigation of these issues. The presentation of their results is excellent, with a good pedagogical style that explains well the background to the model. The authors have also made a good effort to present their results in a way that will be useful to experimentalists. The results that pertain to how the dielectric constant of the ferroelectric material shift the deviation from the Kittel law are particularly interesting.
Overall, considering the care the authors have taken in developing their model and presenting their results I am happy to recommend publication without revisions.
Thank you for taking the time to review our manuscript.