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DOMAIN WALLS Series on Semiconductor Science and Technology SeriesEditors R.J.Nicholas UniversityofOxford H.Kamimura UniversityofTokyo SeriesonSemiconductorScienceandTechnology 1.M.Jaros:Physicsandapplicationsofsemiconductormicrostructures 2.V.N.DobrovolskyandV.G.Litovchenko:Surfaceelectronictransportphenomenain semiconductors 3.M.J.Kelly:Low-dimensionalsemiconductors 4.P.K.Basu:Theoryofopticalprocessesinsemiconductors 5.N.Balkan:Hotelectronsinsemiconductors 6.B.Gil:GroupIIInitridesemiconductorcompounds:physicsandapplications 7.M.Sugawara:Plasmaetching 8.M.Balkanski,R.F.Wallis:Semiconductorphysicsandapplications 9.B.Gil:Low-dimensionalnitridesemiconductors 10.L.Challis:Electron-phononinteractionsinlow-dimensionalstructures 11.V.Ustinov,A.Zhukov,A.Egorov,N.Maleev:Quantumdotlasers 12.H.Spieler:Semiconductordetectorsystems 13.S.Maekawa:Conceptsinspinelectronics 14.S.D.Ganichev,W.Prettl:Intenseterahertzexcitationofsemiconductors 15.N.Miura:Physicsofsemiconductorsinhighmagneticfields 16.A.V.Kavokin,J.J.Baumberg,G.Malpuech,F.P.Laussy:Microcavities 17.S.Maekawa,S.O.Valenzuela,E.Saitoh,T.Kimura:Spincurrent 18.B.Gil:III-nitridesemiconductorsandtheirmoderndevices 19.A.Toropov,T.Shubina:PlasmonicEffectsinmetal-semiconductornanostructures 20.B.K.Ridley:Hybridphononsinnanostructures 21.A.V.Kavokin,J.J.Baumberg,G.Malpuech,F.P.Laussy:Microcavities,Secondedition 22.S.Maekawa,S.O.Valenzuela,E.Saitoh,T.Kimura:Spincurrent,SecondEdition 23.M.M.Glazov:Electronandnuclearspindynamicsinsemiconductornanostructures 24.D. Meier, J. Seidel, M. Gregg, R. Ramesh: Domain walls: from fundamental properties to nanotechnologyconcepts Domain Walls: From Fundamental Properties to Nanotechnology Concepts Dennis Meier, Jan Seidel, Marty Gregg, and Ramamoorthy Ramesh 1 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©OxfordUniversityPress2020 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2020 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2019957865 ISBN978–0–19–886249–9 DOI:10.1093/oso/9780198862499.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. Preface Technological evolution and revolution are both driven by the discovery of new func- tionalities,newmaterialsandthedesignofyetsmaller,faster,andmoreenergy-efficient components. Progress is being made at a breathtaking pace, stimulated by the rapidly growingdemandformorepowerfulandreadilyavailableinformationtechnology:high- speedinternetanddata-streaming,homeautomation,tabletsandsmartphonesarenow “necessities”foroureverydaylives.Consumerexpectationsforprogressivelymoredata storageandexchangeappeartobeinsatiable. Oxide electronics is a promising and relatively new field, that has the potential to trigger major advances in information technology.Oxide materials offer a multitude of applications including spintronics, thermoelectrics and power harvesting, which arise from the broad spectrum of tunable phenomena they exhibit, including magnetism, multiferroicity,andsuperconductivity. Oxide interfaces are particularly intriguing. Here, low local symmetry combined with an increased susceptibility to external fields leads to unusual physical properties distinctfromthoseofthehomogeneousbulk.Inthiscontext,butnotlimitedtooxides, ferroelectric domain walls have attracted recent attention as a completely new type of functional interface. In addition to their functional properties, such walls are spatially mobile and can be created, moved, and erased on demand. This unique degree of flexibility enables domain walls to take an active role in future devices and hold great potential as multifunctional 2D systems for nanoelectronics. With domain walls as reconfigurableelectronic2Dcomponents,anewgenerationofadaptivenanotechnology and flexible circuitry becomes possible, that can be altered and upgraded throughout thelifetimeofthedevice.Thus,whatstartedoutasfundamentalresearch,atthelimitof accessibility,isfinallymaturingintoapromisingconceptfornext-generationtechnology. This book provides a state-of-the-art overview about the significant progress that has been made in ferroelectric domain wall research over the last decade and evaluates emerging application possibilities in information technology. Bringing together world- leading scientists from complementary disciplines, the book gives a broad overview of how domain walls can be used as functional nano-objects with distinct physical properties; it also illustrates how domain walls have shifted from being muses for scientific curiosity into becoming key objects of interest to technology developers. Different chapters also highlight the close relationship between the progress and the developmentofcutting-edgeexperimentalandtheoreticalanalysistools. Going beyond the currently available literature, the book identifies major questions andchallengesthatwillinfluenceresearchondomainwalls,refiningthereader’spicture ofthestateoftheart. vi Preface Sharing our excitement about ferroelectric domain walls with you, we and our co- authors are hoping that you will enjoy reading this comprehensive work and become curioustofindouthowfarwecango,intheyearstocome,toestablishanewtechnology paradigm. DennisMeier, JanSeidel, J.MartyGregg,and RamamoorthyRamesh 1 Physical Properties inside Domain Walls Basic Principles and Scanning Probe Measurements G. Catalan and N. Domingo CatalanInstituteofNanoscienceandNanotechnology(ICN2),CSICandBIST,Campus UAB,Bellaterra08193,Barcelona,Catalonia. ICREA-InstitucioCatalanadeRecercaiEstudisAvançats,Barcelona,Catalonia. 1.1 Introduction Although domain wall properties are material specific,two features are common to all ofthem. First:bysymmetry,adomainwallcannotjuststopinthemiddleofacrystal.Anydomain wallendsataninterface(thesurfaceofthecrystal,grainboundary),inanotherdomain wall (forming a needle domain), or on itself (forming a bubble domain) (Figure 1.1). Thus, despite being nanoscopically thin, they can be macroscopically long, providing a continuous path between different interfaces of a crystal irrespective of how big the crystal is. This “topologically protected” percolation path is most useful for transport applications(LeeandSalje2005;Seideletal.2009). Second: domain walls are mobile; they shift their position as domains grow or shrink inresponsetoexternalfields.Thismobilitysetsdomainwallsapartfromothertypesof interfaces,andisausefulfeaturethatcanbeexploitedindeviceswherethewallismoved intoandoutofareadingunit,asinthe“racetrackmemory”conceptproposedbyStuart Parkin and co-workers (Parkin et al.2008),or the domain wall logic devices explored by the group of Russell Cowburn in Cambridge (Allwood et al. 2005). This mobility property means that domain walls need not be regarded as just a transport medium (aconnector)for,say,electricalcurrents,butalsoasa“container”ofinformationthatcan itselfbemovedintoandoutofthereadinghead,carryingwithitwhateverwall-specific physicalpropertyisofinterest,suchas,e.g.internalmagnetizationorpolarization. G.CatalanandN.Domingo,PhysicalPropertiesinsideDomain:BasicPrinciplesandScanningProbeMeasurements In:DomainWalls:From FundamentalPropertiestoNanotechnologyConcepts.Editedby:DennisMeier,JanSeidel,MartyGregg,andRamamoorthyRamesh,Oxford UniversityPress(2020).©G.CatalanandN.Domingo. DOI:10.1093/oso/9780198862499.003.0001 2 DomainWalls:FromFundamentalPropertiestoNanotechnologyConcepts (a) ((b) (c) (d) Figure1.1 Domainwallconfigurations.(a)Domainwallsendinginthemiddleofacrystalleadto unresolvedconfigurations.Domainswallsendingat(b)surfacesorinterfaces,and(c)inanotherdomain wallformingneedledomainsor(d)onitself,formingbubbledomains. The field of domain wall nanoelectronics (Catalan et al. 2012a) is predicated on the premise that the distinct physical properties of domain walls offer new conceptual possibilitiesfordevices.Thefirstpartofthischapterwilldealwiththebasicphysicsof domainwallproperties,andinparticularthecross-couplingthatallowsdomainwallsto displaypropertiesandorderparametersdifferentfromthoseoftheparentbulkmaterial. Thesecondpartwilldealwithscanningprobetechniquesformeasuringsomeofthese domainwallproperties,andspecificallyatomicforcemicroscopy(AFM).Togetherwith transmission electron microscopy, discussed in Chapter 10, AFM is one of the most importanttools we currentlyhave to probe and manipulatethe individualposition and physical properties of domain walls; although this book contains many chapters that discuss AFM probes, e.g. to inject domain walls and control their motion (Chapter 13), here we will focus on two recent developments that allow investigating hitherto overlooked properties of domain walls: their magnetotransport and their mechanical response. 1.2 Domain Wall Structure and Thickness 1.2.1 Domain Wall Thickness Domain walls are not the ground state of any material (with the possible exceptions of incommensurate materials and relaxors), so they cost energy. The minimization of this energy cost dictates their thickness and structure.First,there is a nearest neighbor interaction; in ferromagnets, this is called “exchange energy,” but the concept applies to any ferroic. The exchange energy penalizes differences between adjacent unit cells. Inotherwords:itpenalizesgradientsoftheorderparameter.Theassociatedenergycost ofthegradientoftheorderparameter,U ,foragenericferroicwithanorderparameter grad (cid:2)=(cid:2)(x)(whichcanbepolarization,orspontaneousstrain,ormagnetization)is (cid:2) (cid:3) k ∂(cid:2) 2 U = (1.1) grad 2 ∂x PhysicalPropertiesinsideDomainWalls 3 wherekplaystheroleofthe“exchangeconstant.”Thisenergycontributionisquadratic asitcannotdependonwhetherthegradientispositiveornegative:thewallenergymust beinvariantunderspaceinversion,asitobviouslydoesnotdependonwhetheryoucross thewallfromlefttorightorfromrighttoleft. Minimization of the gradient of the order parameter to minimize the energy favors broadeningofdomainwalls(broaderwallsmeanssmallergradients),butthisbroadening comesattheexpenseofhavingmorematerialmisalignedwithrespecttotheidealordered state,and this also costs energy.Since the Landau free energy,F,of the homogeneous ferroicstateis (cid:4) (cid:5) a b F = (cid:2)2+ (cid:2)4+O (cid:2)6 , (1.2) 2 2 where a and b are constants and (cid:2) is the order parameter, the equilibrium thickness of the domain wall δ can be found by variational minimization of the total energy (cid:5)G, includinggradientandhomogeneousterms,integratedacrossthedomainwallthickness: (cid:6)∞ (cid:2) (cid:3) a b k ∂(cid:2) 2 (cid:5)G= (cid:2)2+ (cid:2)4+ (1.3) 2 2 2 ∂x −∞ with the center of the wall at x = 0.Here,k is a constant and the boundary conditions are(cid:2)(∞)=−(cid:2)(−∞)=(cid:2) ,with(cid:2) beingthehomogeneousmonodomainstate.The 0 0 solutionforthisequationis(MitsuiandFuruichi1953;Zhirnov1959) (cid:4) (cid:5) x (cid:2)=(cid:2) tanh (1.4) 0 λ with (cid:7) a (cid:2) = − (1.5) 0 b andthecorrelationlengthλ (cid:7) (cid:7) 2k 2k λ=2(cid:2) −1 = . (1.6) 0 b −a From the polarization profile defined by Equation (1.4), the domain wall thickness δ caningoodapproximationbedefinedastwicethecorrelationlength,δ=2λ.Moreover, takingintoaccountthatthesecondderivativeofthefreeenergy,inthiscasewithrespect tothepolarizationP,yieldsthepermittivityχ: ∂2F 1 χ = =− , (1.7) ∂P2 2a 4 DomainWalls:FromFundamentalPropertiestoNanotechnologyConcepts thedomainwallthicknessbecomes (cid:7) (cid:8) 2k δ=2 =4 χk. (1.8) −a Anintuitivesimplificationthatbringsinquantitativelysimilarresultsconsistsinreplacing the hyperbolic tangent by a linear polarization profile across the wall (Catalan et al. 2012a): x (cid:2)(x)=(cid:2) (−δ/2<x<δ/2). (1.9) 0 δ/2 Inthisapproximation,thegradientoftheorderparameterisequaltothetotalchangeof theorderparameter,2(cid:2) ,dividedbythewallthickness,δ,sotheenergydensityU (per 0 unitvolume)associatedwiththegradientis (cid:9) (cid:4) (cid:9) (cid:5) U =1 k 2(cid:2)0 2. (1.10) gradient 2 δ The other term of the energy density is the standard electrostatic (or magnetostatic,or elastic)energy,whichisproportionaltothesquareoftheorderparameter: (cid:9) U =1 χ−1(cid:2)(x)2. (1.11) quadratic 2 Thetotalenergydensityperunitareaofthewall(σ)isobtainedbyintegratingthevolume energydensityacrossthedomainwallthickness: (cid:6) (cid:10) (cid:9) (cid:4) (cid:9) (cid:5) (cid:9) (cid:11) σ = δ/2 1 k 2(cid:2)0 2+1 χ−1(cid:2)(x)2 dx=2k(cid:2)02+1χ−1(cid:2) 2δ. (1.12) −δ/2 2 δ 2 δ 6 0 Minimizingthisenergywithrespecttothedomainwallthickness, ∂σ P 2 1 =0=−2k 0 + χ−1P 2, (1.13) ∂δ δ2 6 0 weobtainanexpressionthatissurprisinglyclosetoEquation(1.8)despitethesimplifi- cations: √ (cid:8) δ=2 3 kχ. (1.14) Allinall,theaforementionedequationstellusthat: