Solid StateCommunications,Vol.8, pp. 149—151,1970. PergamonPress. Printedin GreatBritain

THEORY OF INFRARED ABSORPTIONBY TWO MAGNON EXCITATIONS INTW0-DIMENSIONAL ANTIFERROMAGNETS

NaoshiSuzukiand Hiroshi Kamimura

Departmentof Physics, University of Tokyo, Tokyo, Japan

(Received12 December1969 by T. Muto)

The absorptioncoefficientspectradueto two magnonexcitationprocessesis calculatedfor two-dimensionalantiferromagnets.Onthe basisof the obtainedresults,predictionsaremadeasto twomagnonabsorptionsin K2NiF4 and Rb2MnF4. Featuresof magnonsidebandsin the optical spectraof K2NiF4 arealso discussed.

THE RECENTstudyof thequasi-elasticscattering ferromagnetconsistingof two sublatticesasof neutronon the group of compoundstypified by follows;K2NiF4 hasdefinitely shown that nearly genuinetwo-dimensionalantiferromagnetsdo exist in = ~ . S. ~- ~. Li~i S~.S +

nature.1 In particular,this experimenthas revealed

that long rangecorrelationsexist over thewide ~ >~J~iS~. S~temperaturerangeabovethe Neél temperatureTN. Thus,at this Stage,it is interestingto gei

8H~ES~-~- g/~L8H~~investigatevarious phenomenaconcerningthe -~

coIle~tivemagneticexcitation in two-dimensional where i and j run overthe spin-upand spin downantiferromagnets.

2In view of the absenceof any sublatticesin the layer, respectively,and Haoptical experimentson the collectiveexcitation is the anisotropyfield. Expressingthe Hamil-in two-dimensionalantiferromagnets,in this note tonian (1) in the magnonrepresentationand takingwe predict theoreticallythe featuresof the accountof only the nearestneighborexchangeabsorptionspectradueto two magnonexcitation J

1 and the next nearestneighborexchangeJ2,processesandof the magnonsidebandsin two- we obtain the following magnonHamiltonian:dimensionalantiferromagnets,with a hopethatthe presentreportwill stimulateexperimental H W~ (akak + ~kf

3k + 1) (2)studieson thesetopics. with

As an object of study we considera system = [g/~L~H + 4SJ1 4SJ2 + 2SJ2

of layerstructuredantiferromagneticcompoundssuchas K2NiF4 and Rb2MnF4. Namely, the system (cos aka. + COS ~, )]2 (3)consistsof layersof the squarelattice and in its — 16S

2J~cos2~kcos2£korderedstateeachlayerexhibits an antiferro- 2 2 vmagneticallyorderedspin arrangement.Neglecting The magnondensityof statesdefinedbythe exchangeinteractionbetweenlayersandfurther assumingthat the direction of the spins ~- @-) = ~‘ (w — wis perpendicularto the layer, we canwrite the is illustrated in Fig. la for the caseof J

1 >0spin Hamiltonianfor this two-dimensionalanti-

149

150 THEORY OF INFRARED ABSORPTION Vol. 8, No. 3

and J2 <0. In this case,the symmetrypoints With useof the Kubo formula5and from the

r, X and M in the Brillouin zoneof the present symmetryconsiderationfor ~ the absorptionsystemshownin Fig. lb correspondto the coefficientdueto thesimultaneousemissionofM

0, M~and M2 van Hove singularities3in the two magnonswith thewave vectork and —k is

wk versusk space,respectively, calculated,leadingto

32n2wS2Po2_____________ ________________ = 3*c~ coth(~~)(b) (a)

M cos2~ k~sin2 ~k~6(w — 2w) (7)U, __

________ k

for the electricfield vectorof the light E

parallelto the layer, wherewe haveassumedthatis non-zeroonly for the nearestneighborpafrs.

In eq. (7) ~ is the refractive index and P =

From D4h symmetryof the system,iç, mustbe in

layerand thus thetwo magnonabsorptionwillnot be observedfor the electric field vectorofSthe light perpendicularto the layer.

8 00 200 300320328MAGNON ENERGY (cnc’) ~ It is seenfrom eq. (7) that magnonswith k

near the point x are moststronglyexcited. TheFIG. 1. (a) Magnondensityof statesin K2NiF4 shapeof the two magnonabsorptionspectrumis

= 78 cm’ , = —2 cm~andgp-~H0= 0.59cm~. obtainedby inserting eq. (3) into eq. (7),The notationsa , w~and WM representmagnon and is schematicallyshown in Fig. 2. Althoughenergiesat the symmetrypoints F, X and M inthe Brillouin zone,which aregiven by two magnonabsorptionspectraof two-dimensionalw = [(g~~i~8H~+ 4SJ, )2 — (4SJ1)~II”

2, antiferromagnetshavenot beenobservedsofar,= g~

5H~+ 4SJ~— 4SJ2 and we can predict that two magnonabsorptionspectra= g/L~H0+ 4SJ1 — 8SJ2,respectively, of K2NIF4 andRb2MnF4will havea shapesuch

(b) Brillouin zoneof the two-dimensionalanti- asshownin Fig. 2. The expectedpeakfrequenciesferromagnetswith quadratic layers. The dimen- in thesespectracanbe estimatedfromsion of a unit cell is a. 2wx= 2(g~H

0+ 4SJ~— 4SJ2).SinceJ2 isvery smallcomparedwith J1 , we neglectJ2.

Now we are in a position to calculatethe Then,with useof J1 = 78 cm1 and

absorptioncoefficientfor electricdipole gj~H0= 0.59cm~’

2the peakfrequencyfortransitionassociatedwith two magnonexcitations. K

2NiF4 (TN = 97.1°K)is estimatedto be 624 cnc1

According to Tanabe,Moriya andSugano,4the and the low frequencyedgeof the spectrum,2wr,spin-dependentelectric dipole momentfor two appearsat 36 cm~.On the otherhand,formagnonexcitationsis written as Rb

2MnF4 (TN = 38.5°K,S = 5/2) its estimatedpeakfrequency,2w~,is 94 cni’ and its low

= Z7c~S, . S. (4~~ frequencyedgeliesat 10 cm~,wherewe have

Rewriting equation (4) in termsof magnon usedJ1 = 4.7 cm~deducedby Lines

6 andoperators,we have = 0.24cm~obtainedby Birgeneauel ai.7

P = S E17(k)(clk f3~— aZ ~), (5) Concerningmagnonsidebands,it will bek possibleto observethem in spectraof two-

where dimensionalantiferromagnets.For example,a

n(k) ~ 7T,~exp [_ ik. (R~— Ri)]. (6) magnonsidebandwill appearto the exciton linecorrespondingto the transition 3A

2 to1E within

On deriving eq. (5), we haveusedthe symmetry the t~e2 electronconfigurationof Ni24 ions inpropertyof ir(k), that is, 17(k) =

Vol. 8, No. 3 THEORY OF INFRARED ABSORPTION 151

K2NiF4, for the electric field vectorof light1’ • OK parallelto the layersand its bandshapewill be

I similar to that shown in Fig. 2 except that theI energyscaleis devidedby two and thus 2w is/ replacedby w. The estimatedseparationbetween

/ the peakfrequencyin the sidebandand the/ exciton line is 312 cm’, assumingJ2 = 0. Since/ the spin wave approximationholdsoverthe wider/ rangeof temperaturefor two-dimensionalanti-/ ferromagnetsthan for three-dimensionalones,

we expectthat the predictedfeaturesof two

magnonabsorptionspectraandmagnonsidebands2~ 2~2~ will be seenover the wide temperaturerange.

PHOTON ENERGYFIG. 2. A possibleshapeof two magnon Acknowledgements— We are grateful to Professorsabsorptionspectrain two-dimensionalanti- R. Kubo, Y. Uemura,S. Suganoand Y. Tomonoferromagnetswith quadraticlayersat T = 0 K. and Dr. K. Nakao for valuablediscussions.Sincethe point X correspondsto the saddlepoint, a sharppeakappearsat the frequencyof2w~in casewhereeffects of broadeningaresmall.

REFERENCES

1. BIRGENEAU R.J., GUGGENHEIM H.J. and SHIRANE G., Phys. Rev.Let:. 22, 720 (1969).

2. SKALYO J., Jr., SHIRANE G., BIRGENEAU R.J.andGUGGENHEIM H.J., to bepublished.

3. VAN HOVE L., Phys. Rev.89, 1189 (1953).

4. TANABE Y., MORIYA T. and SUGANO S., Phys. Rev.Let:. 15, 1023 (1965).

5. KUBO R., J. Phys.Soc.Japan.12, 570 (1957).

6. LINES M.E., Phys.Rev.164, 736 (1967).

7. BIRGENEAUR.J., GUGGENHEIMH.J. and SHIRANE G., to be published.

Die optischenAbsorptionsspektradurch2-Magnonenanregungenwerdenfür zwei-dimensionaleAntiferromagnetegerechnet.AufGrund der erhaltenenErgebnissewerdenVorhersagurigenüberdie2-Magnonenabsorptionenin K2 NiF4 und Rb2MnF4 gemacht.DieEigenschaftender Magnonseitenbanderin denoptischenSpektravon K2N1F4 werdenauchdiskutiert.