Domain Walls: From Fundamental Properties to Nanotechnology Concepts

363 Pages·2020·12.306 MB·English

by  Dennis Meier, Jan Seidel, Marty Gregg, Ramamoorthy Ramesh

DOMAIN WALLS
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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
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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: