Magnetoelastic energy in domains separated by 90° walls 1996 A. S. Arrott.pdf

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Citation:JournalofAppliedPhysics79,6529(1996);doi:
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PublishedbytheAmericanInstituteofPhysics
Magnetoelasticenergyindomainsseparatedby90°walls

SimonFraserUniversity,Burnaby,BritishColumbia,V5A1S6,Canada


,foundanalyticallyusing
thecalculusofvariations,accountfortheexperimentalobservationthatforeachcurrent,belowa
criticalcurrent,thereisanarrowrangeoffieldsforwhichtherearetwostablesizesforthecentral
core.©-8979͑96͒79908-7͔
Ironwhiskersmakeidealsubstratesforgrowingultra-
thinfilmsoftransitionmetalsbymolecularbeamepitaxyanalyticalandatthesametimetofindagreementwithex-

alsotoinfluencethemagneticresponseoftheultrathinfilms,problemsarenotgenerallyfamiliartoworkersinmicromag-
.,tomeasurethecouplingtoaferromagneticfilmsepa-netics,theexpositionbelowcontainssomepedagogicalele-
,,͑5͒–͑8͒,and͑10͒,and͑11͒,thatmightoth-
motivationforpassingacurrentalongtheaxisofthewhis-erwisebeskipped.
6
ker,withallsurfaces͓100͔,toswitchthedirectionoftheAccordingtoBrown,theHelmholtzfree-energydensity

-alongoneoranotherofthecubeedgesis
culationofthemagnetizationinthefourtransversedomains111
͑͒aϭkͫeͩ␣2Ϫͪϩeͩ␤2Ϫͪϩeͩ␥2Ϫͪͬ
referredtocollectivelyastheframeshouldbedeterminedmel1132333

centrallongitudinaldomain͑referredtoasthecore͒should166
ϩ͑͒
bedeterminedbythestrengthanddirectionofamagnetic͚͚cijeiej,1
2jϭ1iϭ1
-
sponseforvariousappliedfieldsandcurrentsshowedthatwherethedirectioncosinesofthemagnetizationare␣,␤,
-and␥andthesixcomponentsoftheelasticstrainaregiven
3
’
eventuallytracedtomagnetoelasticenergies,arisingbecauseisincorporatedintotheanisotropyconstant;seeBrown͓Eq.
͔͑͒͑͒
thecoreelongatesinthedirectionofitsmagnetizationwhile8-
,wherethederivatives
Thedcsusceptibilitydeducedfromtheacsusceptibilityofthemagnetizationwithrespecttoz,alongtheaxis,vanish.
measuredwithasingleturnaboutthecentralcrosssectionisThefullthree-dimensionalstrainproblemistreatedapproxi-


bythecompetitionbetweentheappliedfieldwhichexpands
thelongitudinalcoreandthecombinedeffectsofthedemag-
netizingfieldandthefieldfromthecurrentwhichshrinkthe

demagnetizingfieldisproportionaltotheareaofthecore.
Theeffectivefieldfromthecurrentisproportionaltothe

shouldbeproportionaltothesquarerootoftheappliedfield.
Inhigherfieldsitshouldapproachaconstantvaluemainly

longitudinalcorecollapsesforfieldsbelowacriticalfield,
Х
-

notresponsibleforthiscollapse,rather,itfavorstheexpan--
sionofthecentralcore,.
TheeffectivefieldfromthemagnetoelasticenergyThecalculatedresponseignoringmagnetostrictionisshownastheheavy
,thecentralcorecollapsesat



oftherightsignandmagnitudetoexplainthecollapseinislessthantheacdrivingfield.
(8),15April19960021-8979/96/79(8)/6529/3/$©1996AmericanInstituteofPhysics6529
effectsofthemagnetizationincrosssectionsawayfromthe2k1
␭ϭ,͑9͒
centralcrosssectionaretakenintoaccountbyassumingthat1003͑cϪc͒
theareaAofthecentralcorevariesasA(z)1112
ϭA(0)[1Ϫ(z/L)2]
Thisleadstominorchangesintheresponsecomparedto
2712
assumingthatthecentralcrosssectionappliestothewholeϭϪ͑Ϫ͒␭2ͩ2Ϫͪ͑͒


Thesimplest,butcrude,assumptionisthatthestrainsinNotethattheenergyislowestwhenthecorefillsthecross


notletthelatticeplanesbowanddoesnotallowforthe2ϭ
2inthecentralcrosssectionbeingMsrMs/3.
dependenceofthestrainonlogyand1/y,whichactuallyTheeffectivefieldcomesfromthedependenceofthe

Thesetermswillbeaddedbelowaftercarryingouttheuni-volume,

ץ-calculationtherearebuttwounknowns,␰and␨whichchar
H͑r͒ϭϪ͑a͒
Mmelץacterizethedisplacementsoftheatomsinboththecoreandeff
theframe,
ץ1
ϭϪͩ͑͒ͪ
amel2ץuϭy␰k,uϭx␰k,uϭz␨k͑2͒
y1x1z1Msr
␭2͑Ϫ͒
andthecorrespondingstrains:27100c11c121

Ms38ץץץ
eϭuϭk␰,eϭuϭk␰,eϭuϭk␨,
zz1Theeffectivefieldinthelimitofvanishingcore͑r2ϭ0͒isץyy13ץxx12ץ1
͑3͒
͑͒ϭϪ͑͒␭2͑Ϫ͒ϵϪ͑͒
Heff09/
ץץ1
,comparedtotheexperimentalcol-
uy0,melץuxץe4

theorderofmagnitudeisright,butinthiscrudemodelitis
ץץ1

e5uyuz0,4
-
plainsqualitativelythecomplexbehaviorseenexperimen-
ץץ1
,therearenarrowrangesof
zxfieldforwhichtherearetwostablesolutionsfortheareaofץxzץ26

͑͒͑͒
-
neticproperties.
fϭ2͕1͓͑␰2ϩ␨2͔͒ϩ͑␰2ϩ␨␰͒ϩ1͑␨Ϫ␰͖͒͑͒
a0k12c112c12235Whencalculationiscarriedoutforlessrestrictivestrain
fields,
andtheenergydensityinthecoreis222
ofHeffonrwillbejoinedbyaterminrlogrthatam-

acϭk2͕1͓c͑2␰2ϩ␨2͔͒ϩc͑␰2ϩ2␨␰͒ϩ2͑Ϫ␨ϩ␰͖͒.
01211123yieldsanenergydensitygivenby
͑6͒
32͑cϪc͒͑Ϫ2r2logr2ϩr4Ϫ1͒
Thetotalenergyperunitvolumeofthecentralcrosssectionϭͩ1112
amelMsHmelϩ
is2c4412c11
12
aϭaf͑Ϫr2ϩ1͒ϩacr2,͑7͒Ϫͩr2Ϫͪͪ͑13͒
mel003
ϭ
whererdi/distheratiooftheinner,2di,toouter,2d,andthecorrespondingeffectivefield
␰

␨Ϫ͒Ϫ2ϩ2Ϫ͒
andyieldsthestrains2͑c11c12͑logrr1
H͑r͒ϭ3Hͩ
effmelcϩ12c
11114411
␰ϭϪͩr2Ϫͪ,␨ϭͩr2Ϫͪ.
2͑cϪc͒3cϪc31

͑8͒3
Whenthecorefillsthecrosssection͑rϭ1͒,thestrain␨Theenergiesandeffectivefieldsforthetwoapproximate
͑͒␭͑͒͑͒

ϫϪ5␭
,givenbyfollowfromassumingthatthestraininatransversedomain
.,,,
yd2
Ϫ6logϩ3r2ͩϪϩ1ͪ
dy21
͑͒ϭϪ
gyϩϪ͒
c4412c112͑c11c12
1

3
Thefielddependenceofthemagnetizationisfoundpara-
metricallyfrom
ϭ2͑͒
MMsr22

2
rofthecoredomainforthesimpleargumentIandthemoredetailedand

-
͑͒͑͒C␴2͑cϪc͒
␲2ϩϪϩͩ1112
H4DMsrCIr3
thecriticalcurrentIc,forIc/2,andfor3Ic/,theexternaldcsus-rcϩ12c
ceptibilityiscontinuous,reversible,
1
ϫ͑logr2ϩ1͒Ϫr2ϩͪH,͑23͒
3mel
magnetizedinthexdirectiondependsonywhilemaintain-
-
ingthewallsbetweentransversedomainsat45°withrespectmel
-

cientfortheeffectivefieldfromthecurrentisCI
ϭ͒ϭ͒ϭ␥͑͒
uxxg͑yk1,uyyg͑yk1,/(4ͱ␲c).Thecoefficientfortheeffectivefieldfromthe
ϭϪ␴Ϫ&␴
ThesedisplacementsdonotpermitbowingoftheatomicwallenergyisC␴(41002110)/(dMs)or
ϭ␴
planes;amorecomplicatedcalculationshowsthisisnotim-C␴/(dMs)forwallenergiesperunitarea
␴ϭ)␴
-
lapseofthecoreatlowfields,elsewhereithasnegligible
.effectsץ
eϭkg͑y͒,eϭkͩyg͑y͒ϩg͑y͒ͪ,
-yEquations͑22͒and͑23͒accountforthedcmagneticreץ1211
sponseoftheironwhiskerwithfieldsandcurrentsalongthe
ץ1
ϭ␥-
e3k1,e4xk1g͑y,
ypendenceonr2,r,logr,and1/r,.,
Bythecalculusofvariations,treatmentoftheacsusceptibility,tobepublishedelsewhere,
takesintoaccounttheeffectsofeddycurrentsandtheinflu-
␳6
͑͒ϭϪϩ␴͑͒enceoffiniteamplitudeoftheacdrivingfieldatthecritical
gy2ϩlogy17
,accounting
forthespectacularpropertiesthataccompanytheexistence
whichintroducestheparameters␳and␴whichalongwith␨
ofacriticalpoint,withacriticalfieldandacriticalcurrent.
␴
Atthecriticalpointtheexternaldcsusceptibilityisinfinite,
-
continuous,andreversible.
scribedasbeforeby␰and␨,butnow␰isdeterminedby

␰ϭg(r)-
1thatthesimplepedagogicalargumentwasnotsufficient.
rametersthatminimizetheenergyare
3r2
␳ϭϪd2,͑18͒
cϩ12c
,,,inKineticsofOrderingand
GrowthatSurfaces,,NewYork,1990͒,
2Ϫ12
–341.
␴ϭϪϩ,͑19͒2
,,,,inMagnetismand
11124411StructureinSystemsofReducedDimension,,B.
andDieny,,,,NewYork,
1993͒,–193.
113J.-,,7006-8͑1994͒.
␨ϭ2Ϫ͑͒4͑͒
.-,,70311994.
c11c1235J.-,,SimonFraserUniversity,1994;
␳J.-,,5713͑1994͒.

␨,Jr.,MicromagneticsInterscience,NewYork,1963.
,,,-
,,4364͑1989͒.
.,,,