Editing Ferroelectricity

{{Short description|Property of materials which both possess and are affected by electric fields}}
{{Technical|date=September 2010}}

In [[physics]] and [[materials science]], '''ferroelectricity''' is a characteristic of certain materials that have a [[Spontaneous process|spontaneous]] [[Polarization density|electric polarization]] that can be reversed by the application of an external [[electric field]].<ref name=Seitz>{{Cite book|chapter-url=https://books.google.com/books?id=7yFWuc_YL3UC&q=ferroelectricity&pg=PA5 |volume=4 |page= 5 |author=Werner Känzig |chapter=Ferroelectrics and Antiferroelectrics |editor1=Frederick Seitz |editor2=T. P. Das |editor3=David Turnbull |editor4=E. L. Hahn |isbn=978-0-12-607704-9 |year=1957 |publisher=Academic Press |title=Solid State Physics}}</ref><ref name=Lines>{{Cite book|author1=M. Lines |author2=A. Glass |title=Principles and applications of ferroelectrics and related materials |publisher=Clarendon Press, Oxford |year=1979|isbn=978-0-19-851286-8}}</ref> All ferroelectrics are also [[Piezoelectricity|piezoelectric]] and [[Pyroelectricity|pyroelectric]], with the additional property that their natural electrical polarization is reversible. The term is used in analogy to [[ferromagnetism]], in which a material exhibits a permanent [[magnetic moment]]. Ferromagnetism was already known when ferroelectricity was discovered in 1920 in [[Rochelle salt]] by American physicist [[Joseph Valasek]].<ref name=Valasek>See {{Cite journal|author= J. Valasek |title= Piezoelectric and allied phenomena in Rochelle salt |journal=Physical Review |volume=15 |issue= 6 |page=537 |year=1920|doi=10.1103/PhysRev.15.505|bibcode = 1920PhRv...15..505. |url= https://zenodo.org/record/1960072 }} and {{Cite journal|author= J. Valasek |journal=Physical Review |volume=17 |page=475 |year=1921|bibcode = 1921PhRv...17..475V |doi = 10.1103/PhysRev.17.475|title= Piezo-Electric and Allied Phenomena in Rochelle Salt|issue= 4 |url=https://zenodo.org/record/1528280 |hdl=11299/179514 |hdl-access=free }}</ref> Thus, the prefix ''ferro'', meaning iron, was used to describe the property despite the fact that most ferroelectric materials do not contain iron. Materials that are both ferroelectric ''and'' ferromagnetic are known as [[multiferroics]].

==Polarization==
[[Image:Dielectric polarisation.svg|right|frame|Linear dielectric polarization]]
[[File:Paraelectric polarisation DE.svg|frame|Paraelectric polarization]]
[[File:Ferroelectric polarisation DE.svg|frame|Ferroelectric polarization]]
When most materials are [[Polarization density|electrically polarized]], the polarization induced, ''P'', is almost exactly proportional to the applied external electric field ''E''; so the polarization is a linear function.  This is called linear dielectric polarization (see figure). Some materials, known as [[Paraelectricity|paraelectric]] materials,<ref>Chiang, Y. et al.: Physical Ceramics, ''[[John Wiley & Sons]]'' 1997, New York</ref> show a more enhanced nonlinear polarization (see figure). The electric [[permittivity]], corresponding to the slope of the polarization curve, is not constant as in linear dielectrics but is a function of the external electric field.

In addition to being nonlinear, ferroelectric materials demonstrate a spontaneous nonzero polarization (after [[entrainment (physics)|entrainment]], see figure) even when the applied field ''E'' is zero. The distinguishing feature of ferroelectrics is that the spontaneous polarization can be ''reversed'' by a suitably strong applied electric field in the opposite direction; the polarization is therefore dependent not only on the current electric field but also on its history, yielding a [[hysteresis]] loop.  They are called ferroelectrics by analogy to [[ferromagnetic]] materials, which have spontaneous [[magnetization]] and exhibit similar hysteresis loops.

Typically, materials demonstrate ferroelectricity only below a certain phase transition temperature, called the [[Curie temperature#Curie temperature in ferroelectric materials|Curie temperature]] (''T''<sub>C</sub>) and are paraelectric above this temperature: the spontaneous polarization vanishes, and the ferroelectric crystal transforms into the paraelectric state. Many ferroelectrics lose their pyroelectric properties above ''T''<sub>C</sub> completely, because their paraelectric phase has a centrosymmetric crystal structure.<ref>{{cite book|last1=Safari|first1=Ahmad|title=Piezoelectric and acoustic materials for transducer applications|date=2008|publisher=Springer Science & Business Media|isbn=978-0387765402|page=21|bibcode=2008pamt.book.....S}}</ref>

==Applications==
The nonlinear nature of ferroelectric materials can be used to make capacitors with adjustable capacitance. Typically, a [[ferroelectric capacitor]] simply consists of a pair of electrodes sandwiching a layer of ferroelectric material. The permittivity of ferroelectrics is not only adjustable but commonly also very high, especially when close to the phase transition temperature. Because of this, ferroelectric capacitors are small in physical size compared to dielectric (non-tunable) capacitors of similar capacitance.

The spontaneous polarization of ferroelectric materials implies a [[hysteresis]] effect which can be used as a memory function, and ferroelectric capacitors are indeed used to make [[ferroelectric RAM]]<ref name=Scott>{{Cite book|author=J.F. Scott |title= Ferroelectric Memories |publisher=Springer |year=2000|isbn=978-3-540-66387-4}}</ref> for computers and [[RFID]] cards. In these applications thin films of ferroelectric materials are typically used, as this allows the field required to switch the polarization to be achieved with a moderate voltage. However, when using thin films a great deal of attention needs to be paid to the interfaces, electrodes and sample quality for devices to work reliably.<ref name=Dawber>{{Cite journal|author1=M. Dawber |author2=K.M. Rabe|author2-link= Karin M. Rabe |author3=J.F. Scott |title= Physics of thin-film ferroelectric oxides |journal=Reviews of Modern Physics |volume=77 |page= 1083 |year=2005|arxiv = cond-mat/0503372 |bibcode = 2005RvMP...77.1083D |doi = 10.1103/RevModPhys.77.1083|issue=4 |s2cid=7517767}}</ref>

Ferroelectric materials are required by symmetry considerations to be also piezoelectric and pyroelectric. The combined properties of memory, [[piezoelectricity]], and [[pyroelectricity]] make ferroelectric capacitors very useful, e.g. for sensor applications. Ferroelectric capacitors are used in medical ultrasound machines (the capacitors generate and then listen for the ultrasound ping used to image the internal organs of a body), high quality infrared cameras (the infrared image is projected onto a two dimensional array of ferroelectric capacitors capable of detecting temperature differences as small as millionths of a degree Celsius), fire sensors, sonar, vibration sensors, and even fuel injectors on diesel engines.

Another idea of recent interest is the ''[[ferroelectric tunnel junction]]'' (''FTJ'') in which a contact is made up by nanometer-thick ferroelectric film placed between metal electrodes.<ref name=ftj>{{Cite journal
|doi=10.1103/PhysRevLett.94.246802
|author1=M.Ye. Zhuravlev |author2=R.F. Sabirianov |author3=S.S. Jaswal |author4=E.Y. Tsymbal |title=Giant Electroresistance in Ferroelectric Tunnel Junctions
|journal=Physical Review Letters
|volume=94
|pages=246802–4
|year=2005
|bibcode=2005PhRvL..94x6802Z|arxiv = cond-mat/0502109
|issue=24 |s2cid=15093350 }}</ref> The thickness of the ferroelectric layer is small enough to allow tunneling of electrons. The piezoelectric and interface effects as well as the depolarization field may lead to a giant electroresistance (GER) switching effect.

Yet another burgeoning application is [[multiferroics]], where researchers are looking for ways to couple magnetic and ferroelectric ordering within a material or heterostructure; there are several recent reviews on this topic.<ref name=ferroics>{{Cite journal|first1=R.|last1=Ramesh|first2=N.A|last2=Spaldin|author2-link=Nicola Spaldin|journal= Nature Materials |volume= 6 |year=2007|bibcode = 2007NatMa...6...21R |doi = 10.1038/nmat1805 |pmid=17199122|issue=1|pages=21–9|title=Multiferroics: Progress and prospects in thin films}}{{Cite journal|author1=W. Eerenstein |author2=N.D. Mathur |author3=J.F. Scott |journal= Nature |volume= 442 |year=2006|bibcode = 2006Natur.442..759E |doi = 10.1038/nature05023 |pmid=16915279 |issue=7104|pages=759–65 |title=Multiferroic and magnetoelectric materials|s2cid=4387694 }}, {{Cite journal|first1=N.A.|last1=Spaldin|author1-link=Nicola Spaldin|first2=M.|last2=Fiebig |journal= Science |volume= 309 |issue=5733|pages=391–2 | doi=10.1126/science.1113357  |year=2005|title=The renaissance of magnetoelectric multiferroics |pmid=16020720|s2cid=118513837}} {{Cite journal|author=M. Fiebig |journal= Journal of Physics D: Applied Physics|title=Revival of the magnetoelectric effect |volume=38 |issue= 8|page=R123 |year=2005|doi=10.1088/0022-3727/38/8/R01|bibcode = 2005JPhD...38R.123F |s2cid= 121588385}}</ref>

[[Catalysis|Catalytic]] properties of ferroelectrics have been studied since 1952 when Parravano observed anomalies in CO oxidation rates over ferroelectric sodium and potassium niobates near the [[Curie temperature]] of these materials.<ref>{{cite journal |last1=Parravano |first1=G. |title=Ferroelectric Transitions and Heterogenous Catalysis |journal=The Journal of Chemical Physics |date=February 1952 |volume=20 |issue=2 |pages=342–343 |doi=10.1063/1.1700412 |bibcode=1952JChPh..20..342P }}</ref> Surface-perpendicular component of the ferroelectric polarization can dope polarization-dependent charges on surfaces of ferroelectric materials, changing their chemistry.<ref>{{Cite journal|title=Ferroelectrics: A pathway to switchable surface chemistry and catalysis|pages=302–316|journal=Surface Science|volume=650|doi=10.1016/j.susc.2015.10.055|date=August 2016|bibcode=2016SurSc.650..302K|last1=Kakekhani|first1=Arvin|last2=Ismail-Beigi|first2=Sohrab|last3=Altman|first3=Eric I.|doi-access=free}}</ref><ref>{{Cite journal|last1=Kolpak|first1=Alexie M.|last2=Grinberg|first2=Ilya|last3=Rappe|first3=Andrew M.|date=2007-04-16|title=<nowiki>Polarization Effects on the Surface Chemistry of ${\mathrm{PbTiO}}_{3}$-Supported Pt Films</nowiki>|journal=Physical Review Letters|volume=98|issue=16|pages=166101|doi=10.1103/PhysRevLett.98.166101|pmid=17501432}}</ref><ref>{{cite journal |last1=Yun |first1=Yang |last2=Altman |first2=Eric I. |title=Using Ferroelectric Poling to Change Adsorption on Oxide Surfaces |journal=Journal of the American Chemical Society |date=December 2007 |volume=129 |issue=50 |pages=15684–15689 |doi=10.1021/ja0762644 |pmid=18034485 }}</ref> This opens the possibility of performing catalysis beyond the limits of the [[Sabatier principle]].<ref name=":0">{{cite journal |last1=Kakekhani |first1=Arvin |last2=Ismail-Beigi |first2=Sohrab |title=Ferroelectric-Based Catalysis: Switchable Surface Chemistry |journal=ACS Catalysis |date=29 June 2015 |volume=5 |issue=8 |pages=4537–4545 |doi=10.1021/acscatal.5b00507 |bibcode=2015APS..MARY26011K |doi-access=free }}</ref> Sabatier principle states that the surface-adsorbates interaction has to be an optimal amount: not too weak to be inert toward the reactants and not too strong to poison the surface and avoid desorption of the products: a compromise situation.<ref>{{cite journal |last1=Laursen |first1=Anders B. |last2=Man |first2=Isabela Costinela |last3=Trinhammer |first3=Ole L. |last4=Rossmeisl |first4=Jan |last5=Dahl |first5=Søren |title=The Sabatier Principle Illustrated by Catalytic H<sub>2</sub>O<sub>2</sub> Decomposition on Metal Surfaces |journal=Journal of Chemical Education |date=December 2011 |volume=88 |issue=12 |pages=1711–1715 |doi=10.1021/ed101010x |bibcode=2011JChEd..88.1711L }}</ref> This set of optimum interactions is usually referred to as "top of the volcano" in activity volcano plots.<ref>{{cite journal |last1=Seh |first1=Zhi Wei |last2=Kibsgaard |first2=Jakob |last3=Dickens |first3=Colin F. |last4=Chorkendorff |first4=Ib |last5=Nørskov |first5=Jens K. |last6=Jaramillo |first6=Thomas F. |title=Combining theory and experiment in electrocatalysis: Insights into materials design |journal=Science |date=13 January 2017 |volume=355 |issue=6321 |pages=eaad4998 |doi=10.1126/science.aad4998 |pmid=28082532 |s2cid=217918130 |url=https://backend.orbit.dtu.dk/ws/files/131069434/aad4998_Review_Article_Manuscript.pdf }}</ref> On the other hand, ferroelectric polarization-dependent chemistry can offer the possibility of switching the surface—adsorbates interaction from strong [[adsorption]] to strong [[desorption]], thus a compromise between desorption and adsorption is no longer needed.<ref name=":0" /> Ferroelectric polarization can also act as an [[Energy harvesting|energy harvester]].<ref>{{cite journal |last1=Zhang |first1=Yan |last2=Xie |first2=Mengying |last3=Adamaki |first3=Vana |last4=Khanbareh |first4=Hamideh |last5=Bowen |first5=Chris R. |title=Control of electro-chemical processes using energy harvesting materials and devices |journal=Chemical Society Reviews |date=2017 |volume=46 |issue=24 |pages=7757–7786 |doi=10.1039/c7cs00387k |pmid=29125613 |doi-access=free }}</ref> Polarization can help the separation of photo-generated [[electron-hole pairs]], leading to enhanced photocatalysis.<ref>{{cite book |doi=10.1002/9783527807505.ch9 |chapter=Ferroelectrics in Photocatalysis |title=Ferroelectric Materials for Energy Applications |year=2018 |last1=Fang |first1=Liang |last2=You |first2=Lu |last3=Liu |first3=Jun-Ming |pages=265–309 |isbn=9783527807505 |s2cid=104740681 }}</ref> Also, due to [[Pyroelectricity|pyroelectric]] and [[Piezoelectricity|piezoelectric]] effects under varying temperature (heating/cooling cycles)<ref>{{cite journal |last1=Benke |first1=Annegret |last2=Mehner |first2=Erik |last3=Rosenkranz |first3=Marco |last4=Dmitrieva |first4=Evgenia |last5=Leisegang |first5=Tilmann |last6=Stöcker |first6=Hartmut |last7=Pompe |first7=Wolfgang |last8=Meyer |first8=Dirk C. |title=Pyroelectrically Driven •OH Generation by Barium Titanate and Palladium Nanoparticles |journal=The Journal of Physical Chemistry C |date=30 July 2015 |volume=119 |issue=32 |pages=18278–18286 |doi=10.1021/acs.jpcc.5b04589 }}</ref><ref>{{cite journal |last1=Kakekhani |first1=Arvin |last2=Ismail-Beigi |first2=Sohrab |title=Ferroelectric oxide surface chemistry: water splitting via pyroelectricity |journal=Journal of Materials Chemistry A |date=2016 |volume=4 |issue=14 |pages=5235–5246 |doi=10.1039/C6TA00513F }}</ref> or varying strain (vibrations) conditions<ref>{{cite journal |last1=Starr |first1=Matthew B. |last2=Shi |first2=Jian |last3=Wang |first3=Xudong |title=Piezopotential-Driven Redox Reactions at the Surface of Piezoelectric Materials |journal=Angewandte Chemie International Edition |date=11 June 2012 |volume=51 |issue=24 |pages=5962–5966 |doi=10.1002/anie.201201424 |pmid=22556008 |doi-access=free }}</ref> extra charges can appear on the surface and drive various [[Chemical reaction|(electro)chemical reactions]] forward.

[[Photoferroelectric imaging]] is a technique to record optical information on pieces of ferroelectric material. The images are nonvolatile and selectively erasable.<ref name="land">{{cite encyclopedia |title=Photoferroelectric imaging |encyclopedia= McGraw-Hill Concise Encyclopedia of Science and Technology |edition=5 |date= |year=2004 |last=Land |first=Cecil |publisher=McGraw-Hill |location=New York |id= |url= |access-date= }}</ref>

==Materials==
The internal electric dipoles of a ferroelectric material are coupled to the material lattice so anything that changes the lattice will change the strength of the dipoles (in other words, a change in the spontaneous polarization).  The change in the spontaneous polarization results in a change in the surface charge.  This can cause current flow in the case of a ferroelectric capacitor even without the presence of an external voltage across the capacitor. Two stimuli that will change the lattice dimensions of a material are force and temperature. The generation of a surface charge in response to the application of an external stress to a material is called [[piezoelectricity]]. A change in the spontaneous polarization of a material in response to a change in temperature is called [[pyroelectricity]].

Generally, there are 230 [[space group]]s among which 32 [[Crystal system#Crystal classes|crystalline classes]] can be found in crystals. There are 21 non-centrosymmetric classes, within which 20 are [[piezoelectricity|piezoelectric]]. Among the piezoelectric classes, 10 have a spontaneous electric polarization which varies with temperature; thus they are [[pyroelectricity|pyroelectric]]. Ferroelectricity is a subset of pyroelectricity, which brings spontaneous electronic polarization to the material.<ref>{{Citation |last=Whatmore |first=R. W. |title=Piezoelectric and Pyroelectric Materials and Their Applications |date=1991 |url=https://doi.org/10.1007/978-1-4615-3818-9_19 |work=Electronic Materials: From Silicon to Organics |pages=283–290 |editor-last=Miller |editor-first=L. S. |place=Boston, MA |publisher=Springer US |language=en |doi=10.1007/978-1-4615-3818-9_19 |isbn=978-1-4615-3818-9 |access-date=2022-09-22 |editor2-last=Mullin |editor2-first=J. B.}}</ref>

{| class="wikitable centre" style="border:5px #E00;" width="90%"
! colspan="5" | 32 [[Crystallographic point group|Crystalline class]]es
|-
! colspan="4" |21 noncentrosymmetric
!11 [[Centrosymmetry|centrosymmetric]]
|-
! colspan="3" align="center" width="75%" | 20 classes [[Piezoelectricity|piezoelectric]] 
| rowspan="4" align="center" |
| rowspan="4" align="center" |non piezoelectric
|- 
!width="50%" colspan="2" align="center"| 10 classes [[pyroelectricity|pyroelectric]] 
| align="center" rowspan="2"| non pyroelectric
|-
!width="25%" align="center"| ferroelectric
| align="center" |non ferroelectric
|-
|e.g. : [[Lead zirconate titanate|PbZr/TiO<sub>3</sub>]], [[Barium titanate|BaTiO]]<sub>3</sub>, [[Lead titanate|PbTiO]]<sub>3</sub>, [[Aluminium nitride|AlN]]<ref>Wanlin Zhu, John Hayden, Fan He, Jung-In Yang, Pannawit Tipsawat, Mohammad D. Hossain, Jon-Paul Maria, and Susan Trolier-McKinstry, "Strongly temperature dependent ferroelectric switching in AlN, Al1-xScxN, and Al1-xBxN thin films", Appl. Phys. Lett. 119, 062901 (2021) https://doi.org/10.1063/5.0057869</ref>
|e.g. : [[Tourmaline]], [[Zinc oxide|ZnO]], 
|e.g. : [[Quartz]], [[Lanthanum gallium silicate|Langasite]]
|}

Ferroelectric phase transitions are often characterized as either displacive (such as BaTiO<sub>3</sub>) or order-disorder (such as NaNO<sub>2</sub>), though often phase transitions will demonstrate elements of both behaviors. In [[barium titanate]], a typical ferroelectric of the displacive type, the transition can be understood in terms of a polarization catastrophe, in which, if an ion is displaced from equilibrium slightly, the force from the local [[electric field]]s due to the ions in the crystal increases faster than the elastic-restoring [[force (physics)|force]]s. This leads to an asymmetrical shift in the equilibrium ion positions and hence to a permanent dipole moment. The ionic displacement in barium titanate concerns the relative position of the titanium ion within the oxygen octahedral cage.  In [[lead titanate]], another key ferroelectric material, although the structure is rather similar to barium titanate the driving force for ferroelectricity is more complex with interactions between the lead and oxygen ions also playing an important role. In an order-disorder ferroelectric, there is a dipole moment in each unit cell, but at high temperatures they are pointing in random directions. Upon lowering the temperature and going through the phase transition, the dipoles order, all pointing in the same direction within a domain.

An important ferroelectric material for applications is [[lead zirconate titanate]] (PZT), which is part of the solid solution formed between ferroelectric lead titanate and [[anti-ferroelectric]] lead zirconate. Different compositions are used for different applications; for memory applications, PZT closer in composition to lead titanate is preferred, whereas piezoelectric applications make use of the diverging piezoelectric coefficients associated with the morphotropic phase boundary that is found close to the 50/50 composition.

Ferroelectric [[crystals]] often show several [[transition temperature]]s and [[Hysteresis#Electrical hysteresis|domain structure hysteresis]], much as do [[ferromagnetism|ferromagnetic]] crystals. The nature of the [[phase transition]] in some ferroelectric crystals is still not well understood.

In 1974 R.B. Meyer used symmetry arguments to predict ferroelectric [[liquid crystals]],<ref name=Clark>{{cite journal |last1=Clark |first1=Noel A. |last2=Lagerwall |first2=Sven T. |title=Submicrosecond bistable electro‐optic switching in liquid crystals |journal=Applied Physics Letters |date=June 1980 |volume=36 |issue=11 |pages=899–901 |doi=10.1063/1.91359 |bibcode=1980ApPhL..36..899C }}</ref> and the prediction could immediately be verified by several observations of behavior connected to ferroelectricity in smectic liquid-crystal phases that are chiral and tilted. The technology allows the building of flat-screen monitors. Mass production between 1994 and 1999 was carried out by Canon. Ferroelectric liquid crystals are used in production of reflective [[LCoS]].

In 2010 [[David Field (astrophysicist)|David Field]] found that prosaic films of chemicals such as [[nitrous oxide]] or propane exhibited ferroelectric properties.<ref>{{Cite journal |last=Plekan |first=Oksana |date=2010 |title=Novel ferroelectric behaviour of N2O films: spontaneous potentials of up to 40 V. |url=https://pure.au.dk/portal/en/persons/richard-balog(31665d74-6b65-49f2-9bf1-ff81d3c50ef7)/publications/novel-ferroelectric-behaviour-of-n2o-films-spontaneous-potentials-of-up-to-40-v(137cba00-900d-11df-8c1a-000ea68e967b)/export.html |journal=Poster Session Presented at ECAMP 2010, Salamanca, Spain. |via=Aarhus University}}</ref> This new class of ferroelectric materials exhibit "[[spontelectrics|spontelectric]]" properties, and may have wide-ranging applications in device and nano-technology and also influence the electrical nature of dust in the interstellar medium.

Other ferroelectric materials used include [[triglycine sulfate]], [[polyvinylidene fluoride]] (PVDF) and [[lithium tantalate]].<ref name="Aggarwal">{{cite web |url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20110008068_2011008855.pdf |title=Pyroelectric Materials for Uncooled Infrared Detectors: Processing, Properties, and Applications |last=Aggarwal |first=M.D. |author2=A.K. Batra |author3=P. Guggilla |author4=M.E. Edwards |author5=B.G. Penn |author6=J.R. Currie Jr. |date=March 2010 |publisher=[[NASA]] 
|page=3 |access-date=26 July 2013}}</ref> A single atom thick ferroelectric monolayer can be created using pure [[bismuth]]. <ref>{{cite web |url=https://www.science.nus.edu.sg/blog/2023/04/06/discovery-of-ferroelectricity-in-an-elementary-substance/ |title=Discovery of ferroelectricity in an elementary substance|date=April 2023 |publisher=[[National University of Singapore]]  |access-date=10 April 2023}}</ref>

It should be possible to produce materials which combine both ferroelectric and metallic properties simultaneously, at room temperature.<ref>{{Cite web|url=https://www.rutgers.edu/news/rutgers-physicists-create-new-class-2d-artificial-materials|title = Rutgers Physicists Create New Class of 2D Artificial Materials}}</ref> According to research published in 2018 in ''Nature Communications'',<ref>{{cite journal |last1=Cao |first1=Yanwei |last2=Wang |first2=Zhen |last3=Park |first3=Se Young |last4=Yuan |first4=Yakun |last5=Liu |first5=Xiaoran |last6=Nikitin |first6=Sergey M. |last7=Akamatsu |first7=Hirofumi |last8=Kareev |first8=M. |last9=Middey |first9=S. |last10=Meyers |first10=D. |last11=Thompson |first11=P. |last12=Ryan |first12=P. J. |last13=Shafer |first13=Padraic |last14=N’Diaye |first14=A. |last15=Arenholz |first15=E. |last16=Gopalan |first16=Venkatraman |last17=Zhu |first17=Yimei |last18=Rabe |first18=Karin M.|author18-link= Karin M. Rabe |last19=Chakhalian |first19=J. |title=Artificial two-dimensional polar metal at room temperature |journal=Nature Communications |date=18 April 2018 |volume=9 |issue=1 |pages=1547 |doi=10.1038/s41467-018-03964-9 |pmid=29670098 |pmc=5906683 |arxiv=1804.05487 |bibcode=2018NatCo...9.1547C }}</ref> scientists were able to produce a two-dimensional sheet of material which was both ferroelectric (had a polar crystal structure) and which conducted electricity.

==Theory==
An introduction to Landau theory can be found here.<ref name="P. Chandra">{{Cite arXiv|author1=P. Chandra |author2=P.B. Littlewood |title=A Landau Primer for Ferroelectrics|eprint=cond-mat/0609347 |year=2006 }}</ref>
Based on [[Ginzburg–Landau theory]], the free energy of a ferroelectric material, in the absence of an electric field and applied stress may be written as a [[Taylor series|Taylor expansion]] in terms of the order parameter, {{mvar|P}}.  If a sixth order expansion is used (i.e. 8th order and higher terms truncated), the free energy is given by:

<math display=block>\begin{align}
  \Delta E =&\quad\, \tfrac{1}{2}\alpha_0 (T - T_0) (P_x^2 + P_y^2 + P_z^2) \\[4pt]
  &+ \tfrac{1}{4} \alpha_{11} (P_x^4 + P_y^4 + P_z^4) \\[4pt]
  &+ \tfrac{1}{2} \alpha_{12} (P_x^2 P_y^2 + P_y^2 P_z^2 + P_z^2P_x^2) \\[4pt]
  &+ \tfrac{1}{6} \alpha_{111} (P_x^6 + P_y^6 + P_z^6) \\[4pt]
  &+ \tfrac{1}{2} \alpha_{112} \bigl[ P_x^4(P_y^2 + P_z^2)
 + P_y^4(P_x^2 + P_z^2) + P_z^4(P_x^2 + P_y^2) \bigr] \\[4pt]
  &+ \tfrac{1}{2} \alpha_{123} P_x^2P_y^2P_z^2
\end{align}</math>

where {{math|''P<sub>x</sub>'', ''P<sub>y</sub>'', ''P<sub>z</sub>''}} are the components of the polarization vector in the {{math|''x'', ''y'', ''z''}} directions respectively, and the coefficients, {{math|''&alpha;{{sub|i}}'', ''&alpha;{{sub|ij}}'', ''&alpha;{{sub|ijk}}''}} must be consistent with the crystal symmetry.  To investigate domain formation and other phenomena in ferroelectrics, these equations are often used in the context of a [[Phase field models|phase field model]].  Typically, this involves adding a gradient term, an electrostatic term and an elastic term to the free energy.  The equations are then discretized onto a grid using the [[finite difference method]] or [[finite element method]] and solved subject to the constraints of [[Gauss's law]] and [[Linear elasticity]].

In all known ferroelectrics, {{math|''&alpha;''{{sub|0}} > 0}} and {{math|''&alpha;''{{sub|111}} > 0}}.  These coefficients may be obtained experimentally or from ab-initio simulations.  For ferroelectrics with a first order phase transition, {{math|''&alpha;''{{sub|11}} < 0}},  whereas {{math|''&alpha;''{{sub|11}} > 0}} for a second order phase transition.

The ''spontaneous polarization'', {{mvar|P<sub>s</sub>}} of a ferroelectric for a cubic to tetragonal phase transition may be obtained by considering the 1D expression of the free energy which is:

<math display=block>
\Delta E = \tfrac{1}{2}\alpha_0 (T-T_0)P_x^2 + \tfrac{1}{4}\alpha_{11}P_x^4 + \tfrac{1}{6}\alpha_{111}P_x^6
</math>

This free energy has the shape of a double well potential with two free energy minima at {{math|1=''P{{sub|x}}'' = ''P{{sub|s}}''}}, the spontaneous polarization.  We find the derivative of the free energy, and set it equal to zero in order to solve for {{mvar|P<sub>s</sub>}}:

<math display=block>\begin{align}
  \frac{\partial \Delta E}{\partial P_x} &= \alpha_0(T-T_0)P_x + \alpha_{11}P_x^3 + \alpha_{111}P_x^5 \\[4pt]
  \implies 0 = \frac{\partial \Delta E}{\partial P_x} &= P_s \bigl[ \alpha_0(T-T_0) + \alpha_{11}P_s^2 + \alpha_{111}P_s^4 \bigr]
\end{align}</math>

Since the {{math|1=''P<sub>s</sub>'' = 0}} solution of this equation rather corresponds to a free energy ''maxima'' in the ferroelectric phase, the desired solutions for {{mvar|P<sub>s</sub>}} correspond to setting the remaining factor to zero:
<math display=block>
\alpha_0(T-T_0) + \alpha_{11}P_s^2 + \alpha_{111}P_s^4 = 0
</math>

whose solution is:

<math display=block>P_s^2 = \frac{1}{2\alpha_{111}} \left[-\alpha_{11} \pm \sqrt{\alpha_{11}^2 + 4\alpha_0\alpha_{111} (T_0-T)} \;\right]</math>

and eliminating solutions which take the square root of a negative number (for either the first or second order phase transitions) gives:

<math display=block>P_s = \pm \sqrt{\frac{1}{2\alpha_{111}} \left[-\alpha_{11}+\sqrt{\alpha_{11}^2 + 4\alpha_0\alpha_{111} (T_0-T)} \;\right]}</math>

If <math>\alpha_{11}=0</math>, the solution for the spontaneous polarization reduces to:

<math display=block>P_s = \pm\sqrt[4]{\frac{\alpha_0 (T_0-T)}{\alpha_{111}}}</math>

The hysteresis loop ({{mvar|P<sub>x</sub>}} versus {{mvar|E<sub>x</sub>}}) may be obtained from the free energy expansion by including the term {{mvar|&minus;E<sub>x</sub>P<sub>x</sub>}} corresponding to the energy due to an external electric field {{mvar|E<sub>x</sub>}} interacting with the polarization {{mvar|P<sub>x</sub>}}, as follows:

<math display=block>
\Delta E = \tfrac{1}{2} \alpha_0(T-T_0)P_x^2 + \tfrac{1}{4} \alpha_{11}P_x^4 + \tfrac{1}{6} \alpha_{111}P_x^6 - E_x P_x
</math>

We find the stable polarization values of {{mvar|P<sub>x</sub>}} under the influence of the external field, now denoted as {{mvar|P<sub>e</sub>}}, again by setting the derivative of the energy with respect to {{mvar|P<sub>x</sub>}} to zero:
<math display=block>\begin{align}
  \frac{\partial \Delta E}{\partial P_x} &= \alpha_0(T-T_0)P_x + \alpha_{11}P_x^3 + \alpha_{111}P_x^5 - E_x = 0 \\[4pt]
  E_x &= \alpha_0(T-T_0)P_e + \alpha_{11}P_e^3 + \alpha_{111}P_e^5
\end{align}</math>

Plotting {{mvar|E<sub>x</sub>}} (on the X axis) as a function of {{mvar|P<sub>e</sub>}} (but on the Y axis) gives an S-shaped curve which is multi-valued in  {{mvar|P<sub>e</sub>}} for some values of {{mvar|E<sub>x</sub>}}.  The central part of the 'S' corresponds to a free energy [[Second derivative test|local maximum]] (since <math>\tfrac{\partial^2 \Delta E}{\partial P_x^2}<0</math> ).  Elimination of this region, and connection of the top and bottom portions of the 'S' curve by vertical lines at the discontinuities gives the hysteresis loop of internal polarization due to an external electric field.

==Sliding ferroelectricity==
Sliding ferroelectricity is widely found but only in two-dimensional (2D) van der Waals stacked layers. The vertical electric polarization is switched by in-plane interlayer sliding.<ref>{{cite journal |last1=Wu |first1=Menghao |last2=Li |first2=Ju |title=Sliding ferroelectricity in 2D van der Waals materials: Related physics and future opportunities |journal=Proceedings of the National Academy of Sciences |date=14 December 2021 |volume=118 |issue=50 |pages=e2115703118 |doi=10.1073/pnas.2115703118 |pmid=34862304 |pmc=8685923 |bibcode=2021PNAS..11815703W |s2cid=244872105 |doi-access=free }}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
*[[:Category:Ferroelectric materials]]
*{{annotated link|Ferroelectric capacitor}}
*{{annotated link|Ferroelectric RAM}}
*{{annotated link|Ferroelectric polymer}}
*{{annotated link|Piezoresponse force microscopy}}
*{{annotated link|Photoferroelectric imaging}}
{{Col-2-of-3}}
'''Physics'''
*{{annotated link|Paraelectricity}}
*{{annotated link|Piezoelectricity}}
*{{annotated link|Pyroelectricity}}
*{{annotated link|Antiferroelectricity}}
*{{annotated link|Ferroelasticity}}
*{{annotated link|Flexoelectricity}}
*{{annotated link|Condensed matter physics}}
*{{annotated link|Spintronics}}
*{{annotated link|Ceramic}}s
*{{annotated link|Multiferroics}}
{{Col-2-of-3}}
'''Lists'''
*{{annotated link|Electrical phenomena}}
*{{annotated link|Outline of physics}}
*{{annotated link|Index of electronics articles}}
{{col-end}}

==References==
{{Reflist|30em}}

==Further reading==
*{{Cite book|title=Domain Structure in Ferroelectrics and Related Materials |author=A. S. Sidorkin |year=2006 |publisher=Cambridge University Press |isbn=978-1-904602-14-9}}
*{{Cite book|title=Physics of Ferroelectrics: A modern perspective |author1=Karin M Rabe |author1-link= Karin M. Rabe |author2=Jean-Marc Triscone |author3=Charles H Ahn |year= 2007 |publisher =Springer |isbn=978-3-540-34591-6}}
*{{Cite book|title=Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics |author=Julio A. Gonzalo |isbn=978-981-256-875-5 |year=2006 |publisher=World Scientific}}

==External links==
*[http://www.doitpoms.ac.uk/tlplib/ferroelectrics/index.php Ferroelectric Materials] at [[University of Cambridge]]

{{Polarization states}}

{{Authority control}}

[[Category:Ferroelectric materials]]
[[Category:Electric and magnetic fields in matter]]
[[Category:Electrical phenomena]]
[[Category:Phases of matter]]