Solid StateCommunications, Vol. 14,pp. 1153—1155,1974. PerganionPress. Printedin GreatBritain

DISTRIBUTIONOFMAGNON MODES IN DISORDEREDTWO DIMENSIONALHEISENBERGFERROAND ANTIFERROMAGNETS*

D.L. Huber

Departmentof Physics,Universityof Wisconsin,Madison,Wisconsin,U.S.A.

(Received17 January1974byM.F. Collins)

The numericalevaluationof thedistribution of magnonmodesin disorderedtwo dimensionalHeisenbergferro andantiferromagnetsis outlined.Calcu-lationsare carriedout for 30 X 30 arraysusingtechniquesdevelopedbyDeanandBacon.Detailedresultsare reportedfor anamorphousferro-magnethaving aGaussiandistributionof exchangeintegralsand for asubstitutionallydisorderedantiferromagnetresemblingK2Mn0~5Ni0~5F4.

IN A RECENTpaper’we indicatedhow numerical = (g~p.H4+

algorithmsdevelopedfor the analysisof phonon ~ ~-

— ~6Jj j+o~Sj~’j+o,‘-‘j+&,frequenciesin disorderedonedimensionalcrystallatticescouldbeappliedto the problemof determining whereg~andHJA are theg-factorandanisotropyfieldmagnonenergiesin disorderedmagneticchains.Sample associatedwith thefth site andJ

1,~+~is theexchangecalculationsof the densityof magnonmodeswere integralconnectingthejth sitewith its nearestcarriedout for two typesof systems;anamorphous neighborsites/+ &ferromagneticchaincharacterizedby a randomlyfluctuatingexchangeintegralanda substitutionally As pointedout in reference1 the normalmodedisorderedantiferromagneticchainresembling equationsfor the ferromagneticarrayare isomorphicCu~Cr1_~Cl2.Thepurposeof this noteis to report (providedc~is replacedby w

2) to the normalmodetheapplicationof similar techniquesto the problem equationsof anassemblyof harmonicoscillators.of determiningthe distributionof magnonmodesin Becauseof this correspondencewe canmakeuseofdisorderedtwo dimensionalHeisenbergmagnets. eigenvaluecountingproceduresdevelopedby DeanDetailedresultsare presentedfor two systems;an andBacon2in their analysisof phononfrequenciesamorphousferromagnethaving Gaussiandistribution in disorderedtwo dimensionallattices.of exchangeintegrals,andasubstitutionallydisorderedantiferromagnetresemblingK

2Mn05 Ni0~5F4. We havecarriedout calculationsof the distributionof modesin 30 X 30 squarelatticeswith free surface

As in reference1 thecalculationsare carriedout boundaryconditions.It wasassumedthatJI7 0,in the linear approximation,i.e., in the equationsof that themagnitudeof the spinwasthe sameat eachmotion for s the operatorS~is replacedby its site,and that theexchangeintegralbetweennearestexpectationvaluein the molecularfield groundstate, neighborsfollowed a Gaussiandistribution with mean(±~k.In the caseof ferromagnets,afterintroducing I Our resultsfor thethreecases((J — .T)

2)/2.P= 0,the normalizedoperators,U

1 = S7(S~)~’~,and 0.1,0.3,areshown in Fig.1, wheretheenergy,E, ispostulatinganharmonictime dependence,exp [—iwt}, measuredin unitsof4.weobtainthe equationsof motion(h= 1)

In the caseof no disorderthereis a peakin the* Researchsupportedby the NationalScience densityof statesat E = 1.0. When((J — J~)/2.T

2=Foundation 0.1 thepeakis shifteddownwardtoE = 0.75 and

1153

1154 DISTRIBUTIONOFMAGNON MODES Vol. 14, No. 11

00 ~~~~1~ theiris a tail in thedistribution extendingout to

E 2.5. For((J —J)2>/2J2= 0.3 thepeakis near— ______ . o E = 0. In additionthereare modeswith energiesin

______ the interval —0.3~ E<0 aswell ashigh frequency80 .0.1

2~Y’ modeswith energiesup toE= 3.0.~

2J~The datashownin Fig. I are to becomparedwith

60 a theoreticalcalculationof the densityof statesof anamorphoustwo dimensionalferromagnetwhichhas

43

beenreportedby Krey.~ His approach,which isequivalentto thesinglebond coherentpotential

40approximation(CPA) for a Gaussiandistribution ofexchangeintegrals,leadsto resultsfor thedensityofstateswhich arein excellentagreementwith our

20 findings.This is not entirely unexpectedsincecalcu-

1

lationsof thedensityof statesof anamorphousonedimensionalferromagnet,carriedoutin the CPA,arein excellentagreementwith the datafor theone

0.5 0 I 2 dimensionalferromagnetreportedin reference1 .~

E

The calculationof thedistribution of modesinFIG. 1. Distribution of magnonmodesin anamorphous antiferromagneticsystemsis slightly morecomplicated.ferromagneticlayer. A Gaussiandistribution of ex-changeintegralswaspostulatedwith ((J — 1)2>/212= The linearizedequationsof motion takethe form0,0.1,0.3.N(E)AEisthe numberof modesof a30X 3oarrayintheintervalbetweenE—0.1andE. (1)~~ = +Freesurfaceboundaryconditionsareassumedand + ~ (S

1S1+6)1’2U~+~ (2)

theenergyis measuredin unitsof 41S.wherenj = 0(1)if thejth spin is on theup (down)sublattice.Apart from the sign of the coefficient

(261) multiplying U~+5equation(2) is equivalentto the

normalmodeequationsof alattice coupledoscillators20 — K~Mn0~,Ni,•,F4 with masseswhich alternatein sign. TheDean—Bacon

-..-K,NnF4 (XI/2)formalismcanalsobeappliedto this problemit being

00 —o~o-K,NiF4 (Xl/2)necessaryonly to treatthe negativeandpositivefrequencymodeson anequalfooting(cf. reference1).

l~J 80I

Wehavecarriedouta seriesof calculationson a60

30 X 30 squareantiferromagneticarraywith parameterschosento resemblethoseof thecompound

40K2Mn~Ni1.~F4.We took(gpHA)~(= 0.59 cm

1,5

20 JNi—Ni = 77.9cm~,5&PIIA)Mr = 0.2cm’ 6 andJMn—Mn = 5.7 cm~.6The manganese—nickelexchange

1/2o ________________________— integral,JMn....Nj, was set equalto (JMn—MnJNi—Ni)0 00 200 300 Our resultsforx = 0.5 are shownin Fig.2. Also, for

E (cm~)comparison,wehaveplottedthe scaled(by a factor

FIG. 2. Distribution of magnonmodesin thedisordered of 1/2) distributionsfor 30 X 30 arrayscorrespondingcompoundK

2Mn0~5Ni0•5F4N(E)~.Eis thenumberof to K2MnF4 andK2NiF4. It is apparentthat thedis-modesof a 30 X 30 array in the interval between orderedsystemshowstwo distinctbands,an Mn-likeE — 10 cm

1 andE. The scaleddistributionscorre-spondingto K

2MnF4 andK2NiF4 are also shown.Free bandwith energiesextendingapproximately20 cm~surfaceboundaryconditionsareassumedin all three abovetheK2MnF4 maximumanda Ni-like bandwithcalculationsandJMnN1 is set equalto (JMfl.....MflJNINI)la. a peakin thevicinity of 260cm~’,whichis approximately

Vol. 14,No.11 DISTRIBUTION OFMAGNON MODES 1155

50 cm-i belowthe K2NiF4 maximum.In addition, ion structureis in thevicintiy of 200cm~.calculationson arrayscorrespondingto‘(2 Mn0~Ni0.75 F4 showstructurein thedensityof All of thecalculationswerecarriedouton astatesbetween50and70 cm~which is associated Univac 1108 computer.Theevaluationof theintegratedwith (comparatively)isolatedmanganeseions; in the densityof statesfor a 30 X 30 array tookapproximatelycaseof K2Mn0,5Ni0~F4 the correspondingsing~ elevensecondsperpoint.

REFERENCES

1. HUBERD.L.,Fhys.Rev. B 8,2124(1973).

2. DEAN P. andBACON M.D.,.&oc. R. Soc.A283,64(1965).

3. KREYU.,Int.J.Magnetism5,137(1973).

4. BOSES. andFOO E.N. Phys.Rev. B9, 347 (1974).

5. BIRGENEAURJ.,SKALYO J.,Jr. andSHIRANEG.,J.App!.Phys.41, 1303 (1970).

6. BREED D.J.,Physica37,35(1967).

L’evaluationnumériquedeladistribution desmodesdesmagnonsestesquissépourles ferro — et antiferromagnetiquesdeHeisenbergdésordonneéen deuxdimensions.Lescalculationssontaccomplipour les rablezuxde30 X 30 enutilisantdestechniquesdéveloppeéparDeanet Bacon.Lesresultatssontprésenteéendetailpourun ferromagnetiqueamorpheaveeun distributionGaussiend’integralesd’echangeet pourun antiferro-magnetiquesubstitutionellementdesordonnCequi ressembleaK2 Mn0.5Ni05 F4.