Solid StateCommunications,Vol.17,pp. 305—308,1975. PergamonPress. Printedin GreatBritain

EFFECTOF SINGLE ION ANISOTROPYON THE CRITICAL TEMPERATUREOF CLASSICALQUASI-ONE-DIMENSIONAL MAGNETS*

Al. McGurn, D.J. ScalapinoandY. Imryt

Departmentof Physics,Universityof California,SantaBarbara,California93106,U.S.A.

(Received20January 1975 byA.A. Maradudin)

Theeffectof single ion amsotropyenergyonthe three-dimensionalorderingtemperatureof a classicalquasi-one-dimensionalmagneticchainis estimatedusinga meanfield approximationfor the interchaincouplinganda classicalspinfield model for a chain.Numericalresultsarepresentedfor T(, asa func-tion of the intrachainandinterchainexchangeinteractionsand thesingleion anisotropy.

A NUMBER of Heisenbergexchangecoupledmagnetic Thestrongintrachainexchangecouplingresultsmaterialsexistwhich canbedescribedasquasi-one- in a significantshortrangeorderwhich developsindimensionalsystems.13In thesesystemsthespins the chainsbeforethemuchweakerinterchaincouplingalonga particularcrystaldirectionarestronglycoupled, cancausea transitionto a three-dimensionalorderedthusforming magneticchains.Thesechainsin turn state.Within a meanfield theoryapproach4’5we esti-arecoupledthroughweak interchainspininteractions, matetheeffectof the singleion anisotropytermonSuchsystemsoften canbedescribedby a Hamiltonian the three-dimensionalorderingtemperature,T~.Weof the form replaceoff chainspinsby a meanfield reducingthe

problemto thatof independentchainsdescribedby1C = — ~ E2JSn+im Sn~+J’Snm+8 Sn~

= — I [2JS~+~ S,,— D~S~~)2+ gpHn~~Sfl

_D(Sz~~)2+gpHSnmJ, (1) (2)wherewheren signifiesa lattice pointalonga chain,m is a

(two-dimensional)latticevectorperpendicularto the Hneff = (H + ~-~- (S~)’~.gji )

chains,and5 is a vectorconnectingnearestneighborWherefor a givenchainwehavetaken(S,~)asthe self-

siteson differentchains. Here theSnm are classicalspinsof unit magnitude,andJ and.1 are respectively consistentmeanfield valueof a spin on a neighboringtheintrachainandinterchainclassicalexchangecoup- chainat siten andz is the numberof nearestneighborling constantswith JI> J’I. Note that thesignsof J chains.andJ’ are not restricted;eitherinteractioncan be ferro-magneticor antiferromagnetic.Equation(1) alsoin- Within this approximationthetransitiontempera-cludesZeemanandsingle ion anisotropyinteractions. ture, T~,is given as the solutionto

___________ ~gp)2/2zIf’ = x2(T~) (3)

* Worksupportedby the U.S.Army ResearchOffice, for D <0 orDurham,North Carolina,U.S.A. (,gi)

2/2z If’ I = ~(T~) (4)

Onleave of absencefrom the Departmentof Physics for D> 0. Here ~(T~) and~(T~) arerespectivelyandAstronomy,Tel-Aviv University,Ramat-Aviv, the zerofield susceptibilitiesper spinfor fields in theTel-Aviv, Israel. zand in thex direction,of the one-dimensionalchain.

305

306 SINGLE JONANISOTROPYOF QUASI-ONE-DIMENSIONALMAGNETS Vol. 17,No.3

I I I I I ‘i,,,,,.

— —

~

0 I I I I I I0 B 16 24 32 40 ~‘O.l

101/2zIJ’I _______________________________________________

I_cFIG. l.PlotofkT~/-sJ4zIJJ’lvsIDl/2zIJ’IforD>0

(dashed) and D<0 (solid):

Thesesusceptibilitiesareunderstood to be ferromag-netic or staggeredfor ferromagneticor antiferromagnetic 0.I —chains respectively.

0.1As wearein the regionof significantshortrange o.oot 0.01 0.’

orderalongmechains,weapproximatex2(T) and zIJI/IJIx~(T)by their valuesfor the continuumspinfield FIG. 2. A log—losplot of kT~I fl vszIf’ fIJI formodel.

3’6Now equations(3)and(4)can besolved D = 0, equation(5), (dashed),planar(shortdashed),graphicallyfor T,~,usingthe numericalresultsfor the Ising (dot—dashed),andvariousratiosof 1)/I/I (solid).susceptibilitiesfoundIn reference6.

planarmodelThe results for kT~/’../4z jf.J’ vs DI/2zIJ’ are

plotted in Fig. 1. Notice that: (a) bothD>0 and = — ~ 2J(S~S~+1+ ~ (7)

D <0 curves steadily Increasefrom their D = 0 values for which we findof (2/3)~(b) thattheD >0 curve approaches a (2z If’ I”~kT~RJ4zIJJ’Ivalueof2~’;and(c)thatforD<0the kTCPI~Z~/IJI ~ 2 (8)increaseinkT~/~/4zIJJ’ is sharper than forD> 0. InFig. 2, kT~/lJjis plottedvs zIJ’/JI for variousratios which is identicalwith equation(6). ThusthepresenceD/IJI. Curvesof increasingID lie abovetheD = 0 of positivesingleion anisotropyenergyhaseffectivelycurves,whereagaintheeffect of D <0 anisotropyis reducedthe dimensionalityof theorder parametermuchmorepronounced, from threeto two in thekT~/lJI-+0 limit. Notethat

theexchangecouplingin equations(I) and(7) are theIn theregionzlJ’I/IJI 4 1 andkT~/2If 4 1 the same.A shift indexacan bedefinedby therelation

D = 0 curve is givenby5 kT~/IJl~ IJ’IJl~.In ourapproximationa= ~ for

both theisotropicandplanarmodels.For positiveD,we seefrom Fig. 2 thatalog—log plot of kT~/lJIvskT~/lJI (8 J’I~~— j’z .~-) . (5) zlJ’I/IJI appears roughlylike astraight line with a

In thepresenceof apositivesingleion anisotropy,we slopesmallerthan~, dueto thefact thatthe curvefmd, againusingthe spinfield results,6 interpolatesbetweenthe planarandisotropiccurves.(2z If’ l”~ (6) We havealsoplottedthe negativeanisotropyre-

kT~/JJIkT~,iJI-O If j suitsforD/IJI = —0.1,—0.3,—0.6and—2.0in

It isof interesttocomparethiswith thez ~ /IJI 4 1 Fig. 2. Theresultsfor theIsing modelandkT~f21 4 1 resultsfor theclassicaldiscrete = — ~2Jc~a~~

1 (9)

Vol. 17,No.3 SINGLE ION ANISOTROPYOF QUASI-ONE-DIMENSIONALMAGNETS 307

where8(x) = ñ(x)/a, n(x) is afield of unit magnitudeateachpoint anda is alattice constant(which we willtakeas unity). Note thatequation(10) describesferro-- —

magneticallycoupledspins.Thewall energyandlengthdoesnot dependon whetherthefield is ferro or anti-ferromagneticallycoupled.

We haveperformedaBloch wall calculationofthewall energyin achainof ourspinfield in whichthespinsat oppositeendsof the chainaretwisted

c. -. 180°to eachother from the stateof minimum energy.0 alignment.We haveobtainedthe 8(x) which minimizes

E[S(x)] subjecttothe aboveboundarycondition.ForD <0 the domainwall energyis 4~/IJDI andthedomain wall length is proportional to IJ/DI~awhereasfor D <0 the corresponding energy only depends onf. Theseresultssuggestthatfor large IDIikT~ andD <0 the spin field model should tend to a real fieldor Ising like model with arenormalizedcouplingcon-5stantJ’ = ‘,/IJDI andlengthrescalinga’aIf/DI~a.

0 100 000

IJI/zIJI Theseexpectationsareconsistentwith adetailednumericalstudyof the susceptibilityof the spinfield

FIG. 3. A log—logplot of kT~/IJI vs If liz If’ I for modelwith largenegativeD.D >0 andD <0 (solid) forvariousvaluesof D/kT~.Also shownare resultsfor D = 0, equation(5), (dashed),theplanarmodel(shortdashed)andthe Ising model The physicalreasonfor theselargeID I limits are(dot—dashed). the following. A positiveanisotropyenergyserves

only to restrict thespinsto aplane.In thisplanetheywherea~= ±1 arealsopresented.TheD <0 aniso- arefree to interactvia theexchangecouplinganddis-tropy resultsarefound toapproachan Ising modelin playa planarlimit askT~-÷0.For negativeanisotropy,apeculiarsensediscussedbelow, the spinfield formsaBloch-wall in orderto go from

an up spinpositioningto a downspin positioning. InWe notethatfor D <0 theanisotropyeffect is thewall, the spinspoint in intermediatedirections.

quite spectacular,changingits dependenceonz IJ’I/If I Large negative DI valuestendtoshortenthe lengthfromsquareroot to abehaviorwhich appearsnot to of thiswall while theinteractionJ supportsa largerbegivenby apowerlaw. In factonemightexpectas wall. Our solutionfor thewall minimizestheenergykT~/I/I -+0 the resultswould reduceto the corres- associatedwith thesetwo effectswithin thecontin-pondingIsing model,equation(9),just as thepositive uousspin model.For discretespinmodelsonemayanisotropyresultsreducedto aplanarlimit. That the still expecta simple Ising limit for D -÷— ~e,sincesituationis not that simplecan beeasilyseenfrom thewall in thiscasemight consistofjusttwo reversethetemperaturedependenceof thesusceptibilityfor spins.D <0 in thespinfield model,reference6.

Sincethe parameterwhich determinestheim-The failure of theD <0 resultsto becomesimple portanceof theanisotropyat the critical temperature

Ising askT~-+0 maybeunderstoodfrom asimple T~is the ratioD/kT~,wehaveplottedin Fig. 3 theBloch-wallcalculationof the energyto twist thespins ratio kT~IIll asafunction of IJI IzIJ’I for variousby 180°.For the spinfield model theenergyfunctional valuesof D/kT~.Wealso includedthe limiting casesis of theisotropic, Isingandplanarmodels.it is seenthat

E[S(x)I = f [~~2 + DSz2(x~j (10) for D/kT~> I the curvestendto theplanarlimit whilefor D/kT~<— 1 theytendto anIsing-ilke result.

308 SINGLE ION ANISOTROPYOF QUASI-ONE-DIMENSIONALMAGNETS Vol. 17,No.3

In conclusion,we haveestimatedtheeffect of approachestwo andforD/kT~<— I an Ising-like limitsingleion anisotropyon thecritical temperatureof with renormalizedcouplingconstantandlengthscalequasi-one-dimensionalmagneticsystemsby treating is achieved.We hopethattheseresultsmay be usefultheweakinterchaincouplingby a self-consistentmean for theinterpretationof experimentalresultson chain-field theoryapproach.4’5Within thisapproximation like magneticsystemswhichare neverexactly isotropic.7it is foundthatthe presenceof singleion anisotropyenergytendsto increasethethree-dimensionalorderingtemperaturefrom theD = 0 case.For D/kT~> I the Acknowledgement— We thankP.A. Montanoandeffective dimensionalityof theorder parameter P. Pincusfor usefuldiscussions.

REFERENCES

1. HUTCHINGSM.T., SHIRANEG~,BIRGENEAUR.J. andHOLT S.L.,Phys. Rev. B5, 1999 (1972).2. DE JONGH U. andMIEDEMA A.R., Adv. Phys. 23, 1(1974).

3. MCGURN A.R., MONTANO P.A. andSCALAPINO D.J.,Solid State Commun. (to be published).

4. STOUT J.W. andCHISHOLM R.C.,f. Chem.Phys. 36, 972,979 (1962).

5. SCALAPINOD.J., IMRY Y. andPINCUSP.,Phys. Rev. B (to be published).

6. MCGURN A.R. andSCALAPINOD.J. (submittedtoPhys. Rev.B)

7. Anisotropyenergiescanoften be on theorderof afew degreeswhile 3-D transitiontemperaturesfor extremequasi.l-Dcasesare alsoin andbelow thistemperaturerange.An example of a systemwith a planaranisotropyis C

8NiF3 (seeSTEINERM., KRUGERW. andBABEL D.,SolidState Commun. 9, 227 (1971)andalsorefer-ence3 above)while CrCI2 (reference4 above)andCoC12 2NC5H5 appearto bemore Ising-like [TAKEDAK.,MATSUKAWA S.andHASEDA T.,J. Phys. Soc. Japan 30,1330(1970)].