PhysicsLettersAll2(1993)236—242 PHYSICSLETTERS ANorth-Holland

Impurity effectson soliton dynamicsin planarferromagnets

StéphaneRocheandMichel Peyrard’PhysiqueNon-Linéaire:Ondeset StructuresCohérentes,UniversitédeBourgogne,6 BoulevardGabriel, 21000Dijon, France

Received30July 1992; revisedmanuscriptreceived7 October1992; acceptedfor publication20October1992Communicatedby A.R. Bishop

We investigatenumericallythedynamicsof solitonsin a ferromagneticspin chainandwe showthat thesine-Gordonapproxi-mationprovidesonly a poordescriptionof thesolitary excitationsin thepresenceof impurities. Dependingon theirenergyandthestrengthof theimpurity, solitonscanbe reflectedortransmitted.When theyarereflected,theycansufferabruptchangesinvelocity,which areassociatedto theswitchfrom onesolitonbranchto another.In somecasesthescatteringby an impuritycanexcitean internalmodeof thesoliton,which is ableto storesomeenergyandmodify theoutputof thescattering.

One-dimensionalmagneticchainsare interesting siderherethe ferromagneticcaseandshow that, insystemsbecause,while they are rather simple sys- thepresenceofimpurities,thesGdescriptioniseventemsfor which evolutionequationsare well known, more likely to providea very poor descriptionfirstthey exhibit very rich physicalproperties.In partic- because, due to the existence of other solitonular they havebeenlargely investigatedas goodcx- branches,the kink—impurity scatteringcanlead toamplesof systemshaving topological soliton exci- largeandabrupt velocity changeswhich cannotbetations. Antiferromagneticmaterialsoffer a definite accountedfor in the sGpicture,andsecondbecauseexperimentaladvantage[1] becausesolitons cor- impuritieswill enhancegreatlythe numberof non-respondto a ~t rotationof the spins,causinga flip- sG kinks in the system.The Hamiltonianof a spinping of the spinswhich is easily detectedin experi- chainwith nearestneighbor interactionsandplanarments. Neutron scattering investigations of the anisotropy,in an externalmagneticfield B~,issoliton dynamicsin these compoundsin the pres-enceof impuritieshaveexhibiteda crossoverfrom H J ~ Si S~+1+A ~ (Si )2

a ballistic to a diffusive behavior,drivenby the im-purity concentration,which is not yetexplained.Al- —g~i8B~~ S~”, (1)thoughthe theoreticaldescriptionof antiferromag-netic spin chainshas exhibited a large variety of withJ,A>0, andall notationsarestandard[8]. Theexcitations[2], themoretractablesine-Gordon(sG) z directionisthedirectionof thechainandtheeasy-approximationhasbeenusedto analyzethe impur- planeis thexyplane.In theclassicalapproximation,ity effects [3]. ThesG soliton providesalso an ap- the equationsof motion for the spin_componentsproximatedescriptionof the ferromagneticchains, tiv(y. z)=(l/S)Sx(Y ~ (whereS=..Js(s+l) is thebut in this casetoo a morecompletestudyhas re- modulusof the spin in units of h) arevealeda rich behavior [4,5]. Kumar [6] and Ma-gyariandThomas[7] havealreadyattractedatten- h&~’= SJ[ cT~’(a~1+ a~÷1) — a~’(o~i~+ a~,)]tion to the validity limits of the sG approximation — 2SAaY~Zfor the ferromagneticdomain walls. We shall con- 1

h’=SJ[a~(a~_1+a~+1)—a~’(a~_1+o~)]Presentaddress:EcoleNormaleSupérieuredeLyon, 46 Allée + 2SAti-v O.~+ gj~~B azd’Italie, 69364Lyon Cedex07, France.

236 0375-9601/93/s06.00© 1993 ElsevierSciencePublishersB.V. All rightsreserved.

Volume 172, number 4 PHYSICS LETTERS A 4 January 1993

fz&-f=SJ[a;(ay_, +ay+,)-a:(a;_1 +a?+,)]

-m4a?’ . (2)

By representing the spin field in spherical coordinates,

a,(z, t)= (cos 6, cos @,, cos 6, sin @,, sin f3,) ,

and taking the continuum limit, one can derive the following set of coupled partial differential equa- tions for the in-plane angle d and the out-of-plane deviation 13 [ 5,7],

@,cose=-eCC+(l-@$)sin8cosB+bsinecos@,

8, = @Cc cos 8-- 2e,& sin fL b sin @ . (3)

In these equations we have introduced the dimen- sionless quantities C= mz/a (where a is the lat- tice spacing), 7= (2AS/fi)t and b=gp,B&AS. The SC approximation is obtained at order e2 if a/a<, a/arNe (E-K 1) (which corresponds to the continuum approximation for permanent profile so- lutions), 8-e and b- c2. Equations (3) then reduce to

&-@,=bsin @, CL@, . (4)

Equations (4) show that in this approximation, the ferromagnetic solitary excitation is composed of a 2x kink in @ and a pulse in 8 which has an amplitude proportional to the soliton speed u,

&0=4 tan-‘exp[y&r-ur)] ,

t?,=-2&usech[yfi(<-ur)], (5)

wherey=(l-u ) ’ ‘I’. The static SC soliton is an ex- act solution of eqs. (3), but as pointed out by Ma- gyari and Thomas [ 71, it becomes unstable when b exceeds a critical value be(u) which decreases rap- idly as the kink velocity increases. Numerical sim- ulations [ 4,5], and a theoretical analysis using an ansatz able to describe large out-of-plane excitations [9] which can exceed ix, have exhibited other soliton-like excitations which are not described* by the SC approximation. Figure 1, obtained for b=O. 15, shows the energy-velocity curves for the solitary excitations in the ferromagnetic chain. It is useful for future discussion to separate the excita- tions with a positive out-of-plane deviation 8 (dashed line) and a negative 8 (full line). Let us discuss the properties of the soliton for the excitations having a

-0.5 0.0 0.5 velocity

Fig. 1. Plot of the energy-velocity relation for the soliton-like ex- citations in a ferromagnetic spin chain. The full line corresponds to solitons having a negative out-of-plane deviation 0 and the dashed line corresponds to solitons have a positive 8. The hori- zontal arrows indicate some possible switchings for the solitons scattered by impurities.

negative 6 (positive velocity for the SC approxi- mation). Figure 1 shows a first branch (I) from points A to B, for which the solution energy in- creases with its velocity. The energy-velocity rela- tionship along this branch is qualitatively similar to the one given by the sine-Gordon equation, there- fore this branch is henceforth called the sG-soliton branch. In the branches II, III and IV, significant de- viations from the SC soliton begin to be found and an increase in energy of the solitons results in a de- crease of the modulus of their velocity. When one follows the curve from A to D, the magnitude of the out-of-plane deviation 8 increases from 0 to $rc. Be- yond D the localized excitation is no longer a top- ological soliton as @ comes back to 0 after the ex- citation instead of switching to 2x. The excitations with a positive 8 are merely the reflection of the one discussed above by changing the sign of the velocity.

The existence of these various soliton branches, already noticed by Wysin et al. [ 41, is responsible for unusual behaviors in the presence of impurities which affect locally the coupling between the spins. If we drop the assumption that the exchange integral J is homogeneous along the chain and denote by Ji,i+ I

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Volume172, number4 PHYSICSLETTERSA 4 January1993

the couplingbetweensitesi and i+ 1, the equations (il)2(5)’’’’’ ~J’-

of motion (2) become

h&~’=SJ~~(a~a~_1+a~a~)

+SJ~1÷,(aj~o’~÷~+a~’÷1a~)—2SAa~s~, 15))

h&~=SJ,_1,1(a~a~1+a~’_~cr~)

JfI

h&~=SJ1_~,1(a”cr~_1+s~_ia~’)

+SJ1,~+1(o~’~’+~+a~÷(a~’)—gjiBB~a~’.(6)

We haveinvestigatedparticularlythecaseof a singlecouplinginhomogeneitybetweenthesitesk andk+ 1 I I

definedby ~ -m t~)) So 1(M)position

Jk,k.~-I =Jd,(bp

J1~÷1=J,ii=k. (7) 250

Equations (6) are integrated numerically with afourth order Runge—Kuttaschemewith a time step 200chosen to preserve the total energy of the spin chainto anaccuracyof about0.01%duringa completerun.The parameters of the model havebeenchosenso l5~that the kink width is large with respect to the lattice _____________________________________spacing (typically 20 lattice sites) to avoid discrete- ~ness effects. Typical parameters are s= ~, J= 13.6K, 1(5)

A =0.3 K and g~BB~=O.2K, which gives a value ofb = 0.15 situated in the middle of the stability range, -

0 ~ b ~ ~, of the static sG soliton [7]. Periodicboundaryconditionswereusedin all cases.Thekinks 4used as initial conditions have been obtained nu- .1mericallyby startingfrom a sG soliton with appro- )) 2)) .1(1 ~)) So 1)51

priateinitial velocity sothat it relaxesto the desired positionkink andusingaveragingprocesses to extract a so-lution which is then able to propagate in the perfect Fig. 2. Energydensityalongthechainversustime for a staticso-

liton situatedattime0 inthevicinity ofan impurity with J/Jd> I.latticewithout deformation.

Theenergydensityis shownby a grayscale darkerregionshay-Figure 2 shows the behaviorof a static soliton ing thehigherenergy.The black verticalline indicatestheposi-

placedat time 0 in the vicinity of an impurity for tionof theimpurity in thechain.(a) Staticsine-Gordonsoliton,which J/Jd> 1. The two possiblestatic solutions (b) staticsoliton with a non-zeroout-of-planeangle (solution

(pointsA andC in fig. 1) exhibit oppositebehav- correspondingto pointD in fig. I).iors. The sG soliton is attracted by the impurity andthen oscillates around the impurity site, while the [10]: for a local reduction of the coupling the im-non-sGsoliton (point C in fig. 1) is repelled.If the purity potentialhasa minimumat the impurity siteimpurity is suchthat J/Jd< 1, the behaviorsof the so that the sG soliton is attractedby the impurity.two solitons are exchanged.For the sO soliton, the The sametype of calculationcannotbe readilyex-numerical result is consistent with the impurity po- tended for the non-sOsoliton sincewe do not havetential calculated by the effective Lagrangian method an analytical solution in this case.

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Volume 172,number4 PHYSICSLETTERSA 4 January1993

For moving solitons,as the most interestingbe- thereforea low velocity. Figure 4 shows the energyhavior is obtainedfor J/Jd> 1, i.e. for a local re- densityalongthechainasafunctionof timefor a sOduction of thecoupling,weconfineourattentionto soliton of branchI (u= 0.12) interactingwith twothis case.Figure 3 summarizesthe effect of a local typesof defects.In case(a) (J/Jd= 2.0) thesolitoncoupling defect on solitons belonging initially to passesthrough the defect while in case (b) (J/branchesI, II or III (accordingto their initial energy Jd= 3.05) it is reflected. In both casesthe energyandout-of-planeangle).Thefirst importantresultisthat the output of the scattering is essentially con- (a)trolled by the energyof the soliton and not its ye- 200L_~“locity, i.e. solitons with the sameenergy but lying ondifferent branches,and thereforehaving different ‘‘ •

velocities,will show the same behavior. Figure 3 ,

showsthat thetwo-dimensionalspace(energy—J/J~) - “~

is dividedinto two regions,one inwhich thesolitons ~

are transmittedacrossthe defect,anotherin which ‘~‘~

they are reflected.As expected,whenJ/Jd—~ 1, the .~ 1(X)sizeof thetransmissionregionincreases,sincein the ,,,. ~

limit J/Jd= 1 thedefectvanishes.A resultwhichap-pearsmoresurprisingat a first glancewhenlookingat fig. 3 is thatsolitonswith a high energyare more ___________________________________likely to bereflectedthansolitonswith a low energy. ~..

Howeveronemustkeepin mind that, as shown in ~.r -

fig. 1, thevelocity of the solitonsis not a monotonic (.)‘. • I I

increasingfunction of their energy.Forinstancethe (1 20 40) ~) ~ IOUhigh energysolitons,which are reflectedby an im- positionpurity for J/Jd~1.2, are branchII or branchIII so- (b)litons which are closeto pointC in fig. 1 andhave

200~ ‘~“ ‘r. ~

1.100 ‘ I I I50~

1 075 Reflection 100

~:: T~ssion i:~1.000 0 20 41S~~ 80 1

I I Fig.4. Energydensityalongthechainversustimefor a sGsohton1.0 1.6 2.2 2.8 3.4 (branchI) with initial velocity 0.12 scatteredby an impurity.

-~/ Jd Theenergydensityis shownby a gray scaleasin fig. 2, andtheFig. 3. Behaviorof the soliton (i.e. reflectionor transmission) impurity is againindicatedby thelargeverticalblack line. (a)afterinteractionwith animpurity versustheenergyandvelocity “Weak impurity” J/Jd= 2.0. The soliton is transmitted. (b)oftheincomingsoliton. “Strongimpurity” J/Jd=3.05.Thesolitonis reflected.

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Volume172, number4 PHYSICSLETTERSA 4 January1993

densitystayslocalized aroundthe soliton after the (iiinteractionandananalysisof theresultsdoesnot.find L ‘‘‘‘‘‘‘‘

the emissionof magnons.Theenergy of the soliton F ,• -‘ ~. -‘

is conservedin the collision. In the caseof trans- 2(5) ‘

missionthis isattestedby a goodconservationofthesoliton speed.The caseof a reflection is much more -

interesting.All thesimulationsshow that thesign of ~° -‘

the out-of-planecomponent0 is conservedin the in-teraction. Although we haveno analyticalproofof .~

this result,which suggestsan underlyingconserva- 1(K) _____tion law, it hasimportantconsequenceson the dy- ‘

namicsof the solitons,becauseit meansthat in thediagramof fig. 1 an excitation mustalwaysstay on 5(0 _____________________________________

onetypeof line (full line or dashedline) evenwhen ~-

it interacts with impurities. It is interestingto notice Ithesameconservationof thesignofthe out-of-plane 0 ~ I I ,

componentwasfound in somekink—antikink scat- 0 20 40 hO 5(0 I(S). . . po~))I(~i1

teringsimulationswith initial velocitiesin a rangethatproducestotal reflection [4,51.Therefore, if asOsolitonof branchI is reflectedby an impurity, i.e. ,,,,,, ‘ I

changesthe signof its velocity while preservingitsenergy,it mustswitch to branchIII. Thisassociated 2(5)to an abrupt change in velocity which, in the ex- ~,

ampleshownin fig. 4, switchesfrom +0.12 to —0.47.The switch from branch I to branchIII is also ac- isocompaniedby a largechangein the shapeof the so-liton, which becomesmuch morelocalizedin space, ~with a larger energy density in the center. If the ~ ionbranchIII soliton generatedby the reflection inter-actsagainwith a similarimpurity, it is reflectedandswitchesbackto a branchI solitonwith low positive 50 ~‘

speedanda much broaderenergy distribution.The -.

switchingfrom one branch to anotherdue to theconservationof the sign of 0 is not restrictedto 50- 0 ‘i~. ..,,

litons originally in branchI. Figure 5 showsthebe- 0 20 4)0 1,0 50 00

havior of a slow branch III soliton (u= —0.15), positionwhich is transmittedacrossan impurity with ii Fig.5. Sameasfig. 4, but for asolitoninitially belongingto branch

~d= 1.23butreflectedifJ/Jd=l.3. In the lattercase III, with an initial velocity u=—0.15. (a) “Weak impurity” fiit is convertedinto a branchII soliton after reflec- Jd= 1.23.Thesolitonis transmittedandpreservesits speed.(b)

lion. In this region of the energy—velocitydiagram “Strong impurity” J/Jd=1.30. The soliton is reflectedandswitchesto branchII.

(around point C in fig. 1) the asymmetrybetweenthe velocitiesbetweenthe two branchesinvolved inthe switching (branchII andbranchIII) is not ashigh asfor the branchI—branch II switching dis- between—0.15 and +0.12).cussedbeforeandthereforethechangein theveloc- Besidesthis unusualchangein velocity at the re-ity of the soliton whenit is reflectedby an impurity flection, theswitchingof thesolitonsfrom onebranchis not asspectacularas in the first examplewe dis- to anotherin the presenceof impuritiescanbe ex-cussed,but it is still significant (velocity switching pectedto havemore fundamentalconsequences.In

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the analysisof the experimentsperformedon quasi- 200one-dimensionalmagneticcompoundswith impur -__________ities, sO solitons are generally assumed. Our resultsshow howeverthateven if thermallygeneratedso- [ - i.,. j~rig i .~i~’‘~

litons wereonly sOsolitons,in the presenceof im- ISO ~ 1!!~’puritiesstrongenoughto causereflectionofmostofthem, one would get an equilibrium distribution Itcontainingapproximatelyonehalfof sOsolitonsand ~one half of branchIII solitonswhich, for the sameenergy, havea very different speed.As a conse- I ‘ ~•- -~ —.

quence,thevelocity distributionofthesolitonswould ~.

becomedrasticallydifferent from a standardequi- 50 1 —.

librium distribution.For sG solitons,asthe energyE= 8~,nl/~/iJ~?increasesfor increasingvelocity 111’u, the thermaldistribution showsthe usualpeakat o .

low velocities.If abouthalfof thesesolitonsarere- 0 20 40 60 80 100

flectedby impuritiesandthereforeswitch to branch positionIII solitonswith the sameenergydistribution,asinfig. 4b the velocity distribution will split into two Fig. 6. Sameasfig. 4, but for a branchI soliton with an initial

speedof u=O.065 anda defectcharacterizedby J/Jd=4.5.Thepeaks,theformerpeakat low velocity corresponding impurity issituatedexactlyattheboundaryofthechainand,withto solitonswhich are still on the sO branch,andan- theperiodicboundaryconditions, it appearsat both endsof theother peak, centeredaroundu=0.5 which is asso- chainin the figure. Duetothe smallenergylossin thescatteringciatedtothelow energypart ofbranchIII. Although by the impurity, thebranchI soliton generatedafterthesecondthermodynamicstudieswould berequiredto assess reflectioncannotescapefrom the impurity attractivepotentialproperly the statisticalpropertiesof the nonlinear we

excitationsin the ferromagneticchainwith impuri-ties,this qualitativediscussionsuggeststhat impur- tractivepotentialof theimpurity andcollidesagainitiescanbeexpectedtogeneratea very distortedye- with the impurity. The processrepeatsa secondtime,locity distribution which couldexplain someof the but the figure showsthat the timeinterval betweenexperimentalobservationson magnetic materials two successivecollisionsof the branchI soliton withwith impurities, the impurity decreasesas the numberof collisions

In the resultspresentedabovewehavediscussed sincethe initial state increases.This confirmsthat,only examplesin which the energyof the soliton is at eachcollision,someenergy is lost by the soliton.conservedduring the‘scattering.Thisis howeveran We have not detectedthe emissionof magnonsidealizedcaseandin fact the collisionswith anim- (which is indeednot a proofthat they do not existpurity are neverperfectly elastic. In the casesdis- at all becausethey could havesuch a small ampli-cussedpreviouslythe energy losswasso small that tude that it is below the sensitivityof our observa-it could not be measuredin the simulations.When tion), but thereis anotherpossibilityfor the solitontheimpurity becomeseven“stronger”, i.e. for larger toloseenergy,it is toconvertitstranslationalenergyvaluesof J/Jd, the inelasticityof the scatteringcan into internal mode oscillations.The existenceof abecomesignificant. Figure 6 showsan example in localizedinternalmodeof ferromagneticsolitons,al-which the small energylossduring the scatteringis readynoticedby Wysin et al. [4] is easyto observesufficientto modify qualitativelythedynamicsof the by launchingin the chaina sO soliton with a speedsoliton. A slowbranchI soliton, reflectedafirst time largerthanthemaximumspeedofbranchI. It evolvesby an impuritywithJ/Jd= 4.5 is convertedtobranch intoanoscillatingbranchII or branchHI soliton.‘TheIII andthenbackto branchI afterasecondreflec- excitationof this localizedmodeby impurity scat-tion. But, after thetwo slightly inelasticreflections, teringis weak anddifficult to detectdirectly, but itthe branchI soliton is unableto escapefrom the at- canhaveimportantconsequenceson thesoliton dy-

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Volume (72, number4 PHYSICSLETTERSA 4 January1993

namicsbecauseits energy, although lost for the this behavioris interesting from the fundamentaltranslationalenergyof the kink, is still carriedby the point of view, becauseit showsthat magneticsoli-kinkandcanberestoredin favorablecircumstances, tonswith their internal degreeof freedommay ex-Figure7 showsan interestingexampleof sucha be- hibit veryrich properties,it isunlikely toplay anim-havior.A branchI soliton is scatteredby animpurity portant role in the statistical mechanics of thelargeenoughto causea reflection.Thentheresulting ferromagneticchain becauseit requiresa precisebranchIII soliton is able to passthrough the im- timing of the soliton—impurity interaction.purity. According to fig. 3, this changefrom reflec- The different simulationsof the scatteringof thelion to transmissionfor a given impurity corre- ferromagneticsolitonsby defectsdiscussedherepointspondsto a reductionof the soliton energy.It can out that, in the presenceof impurities, the sine-thusbe consideredasa manifestationof the inelas- Gordonapproximationis likely to fail toprovideanticity ofthescattering.But fig. 7 showsthataftertwo appropriatedescriptionof thesolitondynamics.Thetransmissionsthroughtheimpurity,thesamebranch exact nature of the impurity consideredis not es-HI soliton is later reflectedagain by the impurity, sential,providedthat it cancausea reflectionof thewhich atteststhat its energy hasrecovereda higher soliton, becauseit is relatedto the existenceof dif-value to reach again the reflection region in fig. 7. ferent soliton branches in the homogeneous chain.This behavior is understandable in terms of the in- We haveconsideredherea valueof b= 0.15 whichternalmode.If the energylost previouslywasstored is in themiddleof thestability domainof a staticsGinto the internal mode rather than dispersedinto soliton. Forlargervaluesof b thedeviationfrom themagnons,interactionwith the impurity canrestore sO behavioris expectedto be evenlarger.it to soliton translationalenergy.This phenomenon,which gives rise to resonantsoliton—antisolitonin- Wewould like to thankJ.P.Boucherforattractingteractionsin the ~4 or doublesine-Gordonmodels our attentionto the problemof magneticsolitons in[11] can explain the result shown in fig. 7. Although systemswith impuritiesandfor usefuldiscussions,

andA.C. Scott for a critical readingof the manu-script.Part of thiswork hasbeensupportedby the

200 ‘I’’’ I EEC contractSC1-0705.

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