RPA-CPA theory for magnetism in disordered Heisenberg binary systems with long-range exchange integrals

Transcription

1 PHYSICAL REVIEW B 66, RPA-CPA theory for magnetsm n dsordered Hesenberg bnary systems wth ong-range exchange ntegras G. Bouzerar P. Bruno Max-Panck-Insttute für Mkrostrukturphysk, Wenberg 2, D-0620 Hae, Germany Receved 7 February 2002; pubshed 28 June 2002 We present a theory based on Green s-functon formasm to study magnetsm n dsordered Hesenberg systems wth ong-range exchange ntegras. Dsordered Green s functons are decouped wthn the Tyabcov scheme soved wth a coherent potenta approxmaton CPA method. The CPA method s the extenson of Backmann-Esterng-Beck approach to systems wth an envronmenta dsorder term whch uses cumuant summaton of the snge-ste noncrossng dagrams. The cruca pont s that we are abe to treat smutaneousy sef-consstenty the rom-phase approxmaton RPA CPA oops. It s shown that the summaton of the s-scatterng contrbuton can aways be performed anaytcay, whe the p,d, f contrbutons are dffcut to he n the case of ong-range coupng. To overcome ths dffcuty we propose provde a test of a smpfed treatment of these terms. In the case of the three-dmensona dsordered nearest-neghbor Hesenberg system, a good agreement between the smpfed treatment the fu cacuaton s acheved. Our theory aows us n partcuar to cacuate the Cure temperature, the spectra functons, the temperature dependence of the magnetzaton of each consttuent as a functon of concentraton of mpurty. Addtonay t s shown that a vrtua crysta treatment fas even at ow mpurty concentraton. DOI: 0.03/PhysRevB PACS numbers: 75.0.b, z, Cc, 7.0.w I. INTRODUCTION The coherent potenta approxmaton CPA s wdey used to study the effect of dsorder n crystas for revews see Refs. 2. The CPA was ntay deveoped ndependenty by Soven 3 Tayor 4 to study systems wth ony dagona dsorder. Usng a 22 formuaton, a generazaton to the presence of off-dagona dsorder was provded by Backman, Esterng, Berk BEB. 5,6 In these approaches the man dea s to repace the system by an effectve medum whch s determned by the condton that the averaged T matrx of a snge mpurty mmersed n the effectve medum s zero. An aternatve approach s based on cumuant expanson. 7,8 Ths atter method has the advantage that t can he the envronmenta dsorder term whch s characterstc of the Godstone s systems phonons, magnons. The proper treatment of the envronmenta dsorder term, by usng the cumuant expanson method, was used by Lage Stnchcombe, 9 who studed the duted Isng probem (S /2). Later, usng the 22 matrx method of Backman, Esterng, Berk, the method was extended by Whteaw 0 to the phonon probem. In ther cacuatons the coupng ocator are fxed quanttes restrcted to nearest-neghbor exchange coupngs. It s we known that magnetsm n cean ferromagnetc systems can be tacked wth Green sfuncton formasm usng Tyabcov decoupng procedure rom-phase approxmaton RPA. Ths method goes beyond a smpe mean fed snce t ncudes quantum fuctuatons. Addtonay, t fufs the Godstone Mermn- Wagner theorems whch s not the case of a mean-fed treatment. In the case of cean systems, combnng frstprncpe cacuatons to evauate the exchange ntegras RPA method, t was shown that one can provde a satsfactory Cure temperature for Co Fe,. whe, a smpe mean-fed cacuaton argey overestmates the Cure temperature. It s our objectve to provde n ths paper a generazaton of the RPA method to the dsordered systems. We show that by combnng n a sef-consstent manner the RPA method the CPA treatment of the dsorder we are abe to cacuate Cure temperature, magnetzaton of the dfferent consttuents, spectra weghts, etc. The CPA treatment s done n a smar way as done by Lage Stnchcombe by Whteaw. However, due to the Tyabcov decoupng scheme for the dsordered Green s functons, the ocators the effectve exchange ntegras are temperature dependent have to be determned sef-consstenty for a gven temperature. The paper s organzed as foows. In the frst secton we derve after the Tyabcov decoupng scheme the dsordered bnary aoy Green s functon whch ncudes dagona, offdagona, envronmenta dsorder. In Sec. II, we perform the cacuaton of the averaged Green s functons for the A respectvey B) atom. In Sec. III, by generazng Caen s formua we derve the equatons for the magnetzatons m A, m B, for the Cure temperature. In Sec. IV, we propose an aternatve smpfed treatment of the p,d,. scatterng contrbuton to the sef-energy to the case of system wth ong-range exchange coupng. Fnay n Sec. V we present some numerca resuts proceed to a test of our approxmaton of the sef-energy contrbuton of the hgher scatterng terms. II. DISORDERED GREEN S FUNCTION AND RPA DECOUPLING SCHEME We study the magnetsm n a bnary aoy A c B c ; A B can be ether magnetc ons or nonmagnetc. We denote ther spn, respectvey, S A S B. The tota Hamtonan reads /2002/66/04409/$ The Amercan Physca Socety

2 G. BOUZERAR AND P. BRUNO PHYSICAL REVIEW B 66, Ĥ j J j S S j D S z 2 B g S z, G j g j g G j g G j, 9 the J j D are rom varabes: J j J j J j wth the probabty P P j P c s the probabty that the ste s occuped by a atom (c concentraton of a atom. 2 Smary D D wth probabty P. The exchange ntegras are assumed to be ong range, our study s not restrcted to the nearest-neghbor Hesenberg mode. The second term whch descrbes ansotropy s ony reevant n the case of two-dmensona 2D systems to get a nonzero Cure temperature T c Mermn- Wagner theorem. However, n the case of 3D systems the contrbuton of ths term can be negected. We aso ncude the effect of an externa magnetc fed. Let us consder the foowng retarded Green s functon: G j tts t,s j 0, 2 denotes the statstca average at temperature T, Ô Z TreĤ Ô, ZTr(e Ĥ ). G j (t) s Fourer transform n Energy space s S ;S j G j Gj te t dt. Its equaton of moton reads G j 2m j S,H;S j, 5 m S z,orm m A respectvey m B )fare- spectvey B). After expng the second term on the rght sde of the equaty we obtan g BG j 2m j 3 4 J S z S S S z ;S j /2J /2J (m /m ) g denotes the ocator: g g A 0 respectvey g B 0 )fa respectvey B), g 0 E m m Eg B/2mD m m, 0 A or B. For convenence, we have aso ntroduced the reduced varabe E/2m; m denotes the averaged magnetzaton: m c m. The term whch s proportona to comes from the envronmenta dsorder term. Ths term s cruca to recover the Godstone mode requres to be treated very carefuy. We have ntroduced the coeffcent whch s n prncpe equa to, n order to foow the nfuence of the envronmenta dsorder term durng the cacuatons. Note aso that ths term appears because of RPA decoupng. If 0 Eq. 9 s anaogous to the propagator of an eectron n a dsordered medum wth on-ste potenta rom ong-range hoppng terms t off-dagona dsorder. In ths case the probem can be soved just wthn the BEB formasm. However, one shoud stress that the BEB formasm does not appy when the envronmenta term s present. Note aso that n our mode the ocator g 0,, are a temperature dependent, thus CPA RPA oops have to be treated smutaneousy n a sef-consstent manner. III. CUMULANT EXPANSION METHOD FOR THE AVERAGED GREEN S FUNCTIONS As t was done n Ref. 0, the basc dea s to wrte Eq. 9 as a ocator expanson n BEB manner. 5 We defne the rom varabe p : p f A s at ste or p 0 f s occuped by a B on. Therefore the ocator reads D S z S S S z. 6 g p g A 0 p g B 0 g A g B The next step conssts fo decoupng the hgher-order Green s functon. For the second term we use the stard Tyabcov decoupng 3 equvaent to RPA. The ast term due to ansotropy s somehow more compcated snce onste correatons are nvoved. Foowng the approach dscussed n Ref. 4 we adopt for ths term the Anderson-Caen decoupng scheme: 5 D S S z S z S 2D m, 2S S S 2 S z 2. After smpfcaton we fnd 7 8 Smary, p J AA p p J p p J p p J BB p. p J AA p p J, p p J,2 p p J BB p, 2 3 J, (m B /m A )J J,2 (m A /m B )J. The Green s functons are expressed n terms of a 22 matrx one gets for the equaton of moton

3 RPA-CPA THEORY FOR MAGNETISM IN DISORDERED... G j g A 0 0 g B j g A 0 0 g B m J AA m J m BB G AA mj BA J m J m G mj G mj G mj BB PHYSICAL REVIEW B 66, g 0 A 0 0 g B 0J, J AA J, p 0 0 J BB J,2 J BB p G j AA BA G j G j G BB. j 4 We have defned the varabes J, J, J BB. J BB The am s to exp ths expresson nto a product of the p factors, whch can then be averaged over dsorder by expng nto cumuants. For that purpose we separate out the factors ntroduce a new varabe by p c c A c). The dea s to separate out the vrtua crysta part, g g 0 A 0 0 g B 0 cg 0 A 0 0 cg 0. 5 B There s st the envronmenta term whch s more dffcut to he. As t was done by Lage Stnchcombe 9,by convertng nto k space the cacuatons become easer to perform. We defne the Fourer transform by G kk expk r expk r j G j. 6 j After some manpuaton one gets G kk G vc k kk G k vc c 0 0 c kk G k vc N q the 22 matrx V kq s defned by V kq J q AA J AA kq J, kq J q kq V kq G qk, J q J BB q J BB kq J kq the vrtua-crysta Green s functon G k vc, G k vc M 0 cm, the matrces M 0 M are M 0 g A g 0 B 7,2 8 9 J, 0 J k J BB k J BB20 M J k AA J AA J, J k J k J BB k J BB J,2. 2 Equaton 7 can be exped nto two subseres, G kk G () kk G (2) kk, 22 the subseres are, respectvey, G () kk G vc k kk N q G (2) kk G k vc kk G k vc V kk G k vc kk G k vc V kq G q vc kq qk 23 G vc N k V kq G vc q V qk G vc k kq qk q c c The averaged Green s functon s obtaned by averagng over products of by expng nto cumuants P (c). For nstance, k k2 P 2c N k k 2, 25 k k2 k3 P 3c N 2 k k 2 k 3, 26 k k2 k3 k4 P 4c k N 3 k 2 k 3 k 4 P 2 2c N k k 2 k 3 k 4 k k 3 k 2 k 4 k k 4 k 2 k The cumuants are systematcay obtaned by the generatng functon

4 G. BOUZERAR AND P. BRUNO PHYSICAL REVIEW B 66, gx,cncce x P c x!. 28 From ths equaton one gets P (c)c, P 2 (c)c(c), P 3 (c)c(c)(2c). In order to get a cosed form for the seres we have to make the usua CPA approxmaton whch conssts n keepng ony the dagrams wth no crossngs of externa nes. As t s was shown by Yonezawa 7 by Leath, 8 the sefconsstency requres a modfcaton of the sem-nvarants to be attrbuted to each vertex. In other words t means that the cumuants P (c) have to be repaced by a new set of coeffcents Q (c) whch satsfes the reaton Q cq 2 cxq 3 cx 2 c c x x c x, the modfed cumuants are m2! Q c m m!m!m! m c m In the snge-ste approxmaton, after averagng, one gets for the averaged 22 Green s-functon matrx Ḡ kk Ḡ k kk G k c 0 0 c k, 3 G kg k vc k. k denotes the sef-energy; t s gven by k Q 2 N q k Q 2 N q V kq G qv qk 32 Q 3 N 2 q,t V kq G qv qt G tv tk 33 V kq G qq 3 N 2 q,t V kq G qv qt G t. 34 The term k whch s very smar to the sef-energy s caed end correcton. 9 Note that, nsde the CPA oop, Eqs are the ony two equatons whch have to be soved sef-consstenty. To summarze, n Fg. we show a dagrammatc representaton of the prevous set of equatons. A. Evauaton of k It s convenent for the cacuatons to start by defnng q z r expqr. 35 FIG.. Dagrammatc representaton of the averaged Green s functon cacuated wthn the CPA oop. Ḡ s the tota averaged Green s functon, k s the sef-energy, k the end-correcton. The sum r runs over the th type of neghbors of the th she E from a gven ste 0 z s the tota number of neghbors n the she. Note that from now on w correspond to a summaton over the dfferent shes. Wth ths defnton t foows mmedatey that, J AA q J AA o z q. 36 We get a smar expresson for J BB (q) J (q). It s convenent to decompose the matrx V kq nto two terms, V kq V () kq V (2) kq, 37 V () kq V (), kq A D k q 38 V (2) kq V (2), kq D k q kàq. 39 A D are the foowng 2x2 matrces: A J AA o J o J o J o BB z, 40 D J o AA, J o 0 0 J BB o J o,2 z. 4 By usng the foowng very usefu property, 6 f f (r) sa functon whch s equvauated at each ste r of the th she E, then N q kàqfq k N q qfq. 42 By usng Eq. 42, we fnd sgnfcant smpfcatons n the cacuatons. Indeed, a the terms of the sum nvovng at

5 RPA-CPA THEORY FOR MAGNETISM IN DISORDERED... east one factor V (2) reduces to zero. Thus the end correcton term does not expcty depend on the envronmenta dsorder term. After cacuaton we fnay get k V (), k,0 Q 2 IQ 3 MQ 4 M 2 j F j. j 43 Lke V (),, F j sa22 matrx, M a N s N s matrx, each matrx eement M j sa22 matrx. N s denotes the number of consdered shes. V (), s gven n Eq. 38 F j M j are defned by PHYSICAL REVIEW B 66, In genera, the evauaton of the second term (2) (k) s much more compcated. One can get an anaytca form ony for smpe cases. For exampe, f the exchange ntegras are restrcted to ony nearest neghbors, the compete summaton of the sum can be performed by usng the space-group symmetry of the attce. 9,7 In the case of nearest-neghbor Hesenberg system one gets (2) k EC p 2kC d 2k2k 2, 5 F N q qg q 44 C p,d 2 Q 2 IQ 2 M p,d Q 3 M p,d D. 52 M j N q qg qv (),j q,0. 45 The sum n Eq. 43 s obtaned after dagonazaton of the 2N s 2N s matrx MP M dag P, Q 2 IQ 3 MQ 4 M 2 P c M dag Q IM dag P. 46 The functon c was prevousy defned n Eq. 29, c (M dag ) j c ( ) j are the egenvaues of M. Hence we get for the end correcton k V (), k,0 P c M dag Q IM dag P j F j. j 47 Let us now proceed further evauate the sef-energy k. B. Evauaton of k Usng the remarks made n the prevous secton, we fnd that the sef-energy can be wrtten k k () k (2), 48 () k respectvey (2) k ) s obtaned by repacng V k,q by V () k,q respectvey V (2) k,q ). Indeed we fnd that each term of the sum contanng both V () V (2) reduces to zero. After smpfcatons we obtan for () k, () k, j V k,0q IQ 2 MQ 3 M 2 j j k, 49 j (k) j (k)( 0 0 ). As prevousy done for the end correcton, usng the functon c (z) defned n Eq. 29 we obtan mmedatey Q IQ 2 MQ 3 M 2 P M dag P. 50 Note that we have ncuded n the sum the frst-order term dependng on c (Q ) whch comes from the vrtua crysta Green s functon G q vc. C p,d are evauated n the same way that t was done for () k (E) k (E). The matrces D, M p, M d are, respectvey, D,2 JAA J, 0 0 J BB z, 53 J M p 6 D G p, M d 4 D G d, G p(/n) q (2q)G (q) G d(/n) q (2q)2(q) 2 G (q). Note that the vrtua crysta approxmaton for k (2) (E) conssts of takng n Eq. 52 the frst term ony. Then t foows mmedatey that, C VCA p C VCA d c 2 D 56 whch substtuted n Eq. 5 eads to k (2),VCA EcD k Note that k (2),VCA s energy ndependent. It s aso mportant to stress that at the owest order the sef-consstency for (2) s not requred. Most of the ferromagnetc materas are of tnerant type, whch means that the exchange ntegras between dfferent ocazed magnetc ons are ong range drven by the poarzaton of the conducton eectrons gas as t s for the Ruderman-Ktte-Kasuya-Yosda RKKY mechansm. 8 Anaytcay, the generazaton of the prevous cacuatons to the more nterestng case J j are ong ranged s not an easy task. However, by truncatng the sum, the summaton can be performed numercay. It s mportant to note that (2) (k) s proportona to whch means that t orgnates ony from the envronmenta dsorder term, each term of the sum vanshes n the ong-waveength mt (2) (kä0)0. Ths mpes that even after truncaton of the sum at any order, the Godstone theorem remans fufed. Thus the ong-waveength magnons are aways treated prop

6 G. BOUZERAR AND P. BRUNO PHYSICAL REVIEW B 66, ery. Furthermore, snce (2) (k) corresponds to hgher-order scatterng terms (p,d, f, ) t s natura to expect that these terms shoud not affect the Cure temperature n a dramatc way. In other words, we expect that a truncaton of (2) (k) sum to the frst few terms shoud aready provde a good approxmaton of Cure temperature compared to the one obtaned wth the compete seres. However, t s cruca to consder at east the owest-order term the vrtua crysta contrbuton, otherwse even n the cean mt one woud not recover the correct resut the Godstone s theorem woud be voated. If we consder the ower approxmaton ( (2) 2 VCA, we get the expected resuts n the mt c0 c. It s not a pror cear whether such an approxmaton of (2) (k) to the owest order provdes satsfyng resuts for the Cure temperature at moderate mpurty concentraton. Such an approxmaton w be tested ater on. To concude ths secton, the compete averaged 22 Green s functon s obtaned after sovng sef-consstenty the set of Eqs wthn the CPA oop then usng Eqs to get k Ḡ k. However, as was aready mentoned n the ntroducton, the probem s not soved unt we are abe to cacuate the ocators g 0 the exchanged ntegras whch depend on the averaged magnetzaton m. The determnaton of m has to be done sefconsstenty n an addtona externa oop RPA. IV. MAGNETIZATION AND CURIE TEMPERATURE We assume that the averaged 22 Green s functon matrx Ḡ(k,E) s cacuated accordng to the prevous secton wthn the CPA oop. We show how from Ḡ (k,e), A or B we can get the mssng sef-consstent equatons RPA oop to get the temperature-dependent ocator g 0 the exchange ntegras. Ths w aow us to cacuate the eement-resoved magnetzatons m S z as a functon of temperature the Cure temperature. It was shown by Caen, n the case of a cean system pure A or B), that the magnetzaton can be expressed n the foowng way: 6 m S 2S 2S S 2S 2S, (/N) q (q) (q) s defned as q A q,e de e 2mE/kT, A q,e ImG q,e 60 s the spectra functon. Note aso that the Caen s approach to get the magnetzaton aows us to derve a ot of oca spn-spn correatons; they are ony expressed as a functon of. For nstance, S z 2 SSm 2 6 whch s needed to determne the ansotropy parameters gven n Eq. 8. In the case of cean systems, the normazed spectra functon A (q,e) s gven by A q,eeeq. 62 E(q)(q)/2m (q) denotes the magnon dsperson. In the case of a bnary or mutcomponent aoy ths formua can be generazed n the foowng way: A q,e ImḠ q,e c x, 63 c s the concentraton of the on we have for convenence ntroduced a T-dependent reduced varabe x m /m. Note that n the presence of mpurtes the spectra functon s no onger a functon, but because of the fnte magnary part of the sef-energy t w consst of peaks of fnte wdth wth a more or ess Lorentzan shape. In the case of bnary aoys we expect for a gven q two peaks, more generay n peaks for an n-component aoy. For a gven temperature the compete sef-consstency s obtaned by provdng good startng vaues for m, then performng the CPA oop whch provde Ḡ(k,E), fnay gong nto the RPA oop by usng Eqs. 58, 6, 63 one gets the new vaues of m (S z ) 2 whch are re-njected n the ocators g 0, the exchange ntegras,. Let us now show how to get the Cure temperature of a dsordered Hesenberg bnary aoy. We start by expng Eq. 59 n the mt T T C.e., m 0). We mmedatey get F N q kt C 2m F, 64 A q,e de. 65 E After expng Eq. 58 as a functon of / one obtans m S S 3 2m. kt C F 66 Snce the averaged magnetzaton m s defned by m c m, combnng the two prevous equatons one fnds for the Cure temperature k B T C 2 3 c S S F. 67 Equaton 67 s the RPA generazaton of the Cure temperature to a mutcomponent dsordered aoy. The prevous

7 RPA-CPA THEORY FOR MAGNETISM IN DISORDERED... PHYSICAL REVIEW B 66, FIG. 2. Cure temperature T C for dsordered nearest-neghbor Hesenberg ferromagnet as a functon of the mpurty concentraton c(a). The parameters are S A 2, S B 3, J AA 0.2, J BB 0.5. We have chosen three dfferent vaues for J. equaton provdes a drect measure of the weght w (/k B T C )c S (S )/F of each eement to the Cure temperature. V. NUMERICAL RESULTS In ths secton we provde an ustraton of the RPA-CPA theory a test for the approxmaton suggested above for the hgher-order scatterng contrbuton of the sef-energy. For smpcty, we consder the case of a 3D dsordered bnary aoy on a smpe cubc attce. Addtonay we restrct the exchange ntegras to nearest neghbor ony whch aows us to test the vadty of the approxmaton scheme suggested n Sec. III before estmatng 2. For further smpfcatons of the cacuatons we consder the case of a zero externa fed negect the ansotropy term whch s reasonabe for a 3D system. In Fg. 2, we have potted the Cure temperature as a functon of c obtaned wth the fu CPA treatment; the (2) part of the sef-energy s cacuated exacty fu summaton of the sum. Note that pure A respextvey B) corresponds to c respectvey c0). Dependng on the chosen set of parameters T C shows a mnmum J S A S B mn(j AA S A 2,J BB S B 2 ), a maxmum J S A S B max(j AA S A 2,J BB S B 2 ), or s monotonc mn(j AA S A 2, J BB S B 2 )J S A S B max(j AA S A 2,J BB S B 2 ). These three dfferent cases are shown n the fgure. As aready mentoned n Sec. III, t s dffcut to perform the fu summaton of (2) for the case of ong-range exchange ntegras whch s the case of many reastc nterestng systems, for exampe permaoy. As t was dscussed prevousy, the smpest approxmaton conssts of keepng ony the owest-order term of the sum vrtua crysta approxmaton. In the case of the nearest-neghbor Hesenberg system, (2) (2),VCA are respectvey gven n Eqs In Fg. 3 we have potted the Cure FIG. 3. Comparson between the Cure temperature cacuated as functon of the mpurty concentraton for a nearest-neghbor Hesenberg ferromagnet. a The fu CPA cacuaton, b approxmaton for (2) (2) VCA, c the vrtua crysta cacuaton. The chosen set of parameters s wrtten n the fgure. temperature cacuated wth a fu CPA treatment, the one performed wth the approxmaton dscussed prevousy, (2) (2) VCA, the one obtaned wth vrtua crysta approxmaton. In case, the averaged Green s functon s Ḡ k G k vc c c snce n VCA k 0. The comparson between the fu CPA the vrtua crysta approxmaton VCA shows that the Cure temperature dffers sgnfcanty. Even very cose to the cean mt the VCA appears to be napproprate, for nstance for c 0. we observe that T VCA C s about 35% arger than the fu CPA cacuated one. Note that the dsagreement s even more pronounced n the vcnty of c0 than c. Ths can be understood n the foowng way: snce J 3J BB.5J AA S A S B a substtuton of a B ste by an A ste cose to c0) ntroduces a change of energy wth respect to the pure case two tmes arger than a substtuton of a B ste by an A ste near c. As dscussed prevousy, t s nterestng to compare the Cure temperature the VCA s ony done on (2) (T 2,VCA c ). We observe a good agreement between the fu CPA cacuated T C T 2,VCA c, n the whoe range of concentraton; the agreement s even exceent for c0.6. A comparson between T VCA c T 2,VCA c n the vcnty of c0 c shows that the reason why the VCA approxmaton breaks down s essentay because of the crude approxmaton of the s part of the scatterng. Thus ths fgure vadates a smpe treatment of (2). It s aso expected that ncudng ony a few addtona terms of the sum w ead to an exceent agreement n the whoe range of concentraton. In Fg. 4 we show the temperature dependence of the eement-resoved magnetzatons. In order to demonstrate the versatty of our approach, we have chosen a set of param

8 G. BOUZERAR AND P. BRUNO PHYSICAL REVIEW B 66, FIG. 4. Magnetzatons m A,m B, averaged one c av c A m A c b m B as a functon of temperature. The spns are S A S b 3, the exchange coupngs are J AA.2, J BB 0.0, an antferromagnetc coupng between A B s taken J 0.5. The concentraton of A atoms s c A FIG. 5. Magnetc spectra densty (E)ImG (E)/x c as a functon of E. The contnuous ne corresponds to A the dashed ne to B for three dfferent concentraton of A: c 0.05, 0.5, The parameters are S A 2, S b 3, J AA 0.2, J BB 0.05, J 0.5, TT C. FIG. 6. Spectra functon S (q,e)(/)img (q,e) as a functon of E for dfferent momentum q qq(,,). The contnuous ne corresponds to A the dashed ne to B. The spns are S A 2 S b 3, the exchange coupngs are J AA 0.2, J BB 0.0, J 0.5, c A We have taken TT C. For carty of the pcture a sma magnary part 0. have been added. eters whch mmcs a ferrmagnetc behavor wth compensaton pont. Addtonay, the parameters are such that T C A T C B. Whe the temperature dependence of m A foows a stard behavor, m B (T) starts to strongy decrease even at ow temperature. For exampe, at T2.5, m A s reduced by ess that 20% whe m B 0.5 m B (0). As a resut of our choce of parameters we see that the averaged magnetzaton m av c A m A c B m B s nonmonotonc vanshes for an ntermedate temperature vaue compensaton pont. Its found that the functon (m B /m A )(T) decreases monotoncay wth temperature. As a resut, snce at T0, mb/m A S B /S A, thus f S B /S A c A /c B then m av w not have a compensaton pont. However, the condton that S B /S A c A /c B s not suffcent to get one, t s aso requred that m B /m A (T C )c A /c B. In Fg. 5 we now show the magnon spectra densty MSD (E)ImG (E)/x c as a functon of E. We consder three dfferent cases: amost cean A B a c, the ntermedate stuaton c A c B 0.5. In both Fgs. 5a c we observe that the MSD s very smar to the cean case. Ths s cearer n case c than a; t s easy to underst that when dopng A wth B the dfference n energy wth the undoped case s ony of order 0% J AA (S A ) J S A S B 0.9 whe when dopng B wth A the change s more drastc about 00%). To get a smar MSD to Fg. 5c for a weaky doped B sampe, one shoud take c In Fg. 6 we show the spectra functon S (q,e) as a functon of energy for dfferent vaues of the momentum q. Ths quantty s more nterestng that the ntegrated MSD snce t provdes drect nformaton about the eementary exctaton dspersons ther spectra weght. Addtonay t s drecty reated to neastc neutron-scatterng measurements. Let us now brefy dscuss Fg. 6. At precsey q0 momentum, n both S A,B, we observe two peak structures: a we defned peak 9 at E0, as expected snce our

9 RPA-CPA THEORY FOR MAGNETISM IN DISORDERED... theory fufs the Godstone theorem, a very broad one at ntermedate energy E. For ntermedate vaues of the momentum, t s dffcut to separate the peak one get a snge broad peak. We see ceary that the peaks are crossng each other at q(/2)(,,). Note that due to the dfferent spectra weght of the peaks to the coseness of ther ocaton, the snge peak-structure whch s observed at q/2 s ocated at dfferent energes for A B. From ths fgure we see aso that the dsperson of the second peak s amost fat E 2 (q), whe the Godstone mode E (q) Ref. 20 goes from E0 toe max 2 when movng n the (,,) drecton. VI. CONCLUSION In concuson, we have presented n ths paper a theory based on Green s-functon formasm to study magnetsm n PHYSICAL REVIEW B 66, dsordered Hesenberg systems wth ong-range exchange ntegras. The dsordered Green s functon are decouped wthn the Tyabcov procedure the dsorder dagona, off-dagona, envronmenta s treated wth a 22 modfed cumuant CPA approach. The cruca pont s that we are abe to treat smutaneousy sef-consstenty the RPA CPA oops. Our theory aows us n partcuar to cacuate Cure temperature, spectra functons, temperature dependence of the magnetzaton for each eement as a functon of concentraton of mpurty. Addtonay, we have proposed a smpfed treatment of the p,d, f. contrbuton of the sef-energy whch s dffcut to he n the case of ong-range exchange ntegras. The approxmaton was tested successfuy on 3D dsordered nearest-neghbor Hesenberg systems. Combned wth frst-prncpe cacuatons whch can provde the exchange ntegras, ths method appears to be very promsng to study magnetsm n dsordered systems. R.J. Eott, B.R. Leath, J.A. Krumhans, Rev. Mod. Phys. 46, F. Yonezawa K. Morgak, Supp. Prog. Theor. Phys. 53, P. Soven, Phys. Rev. 56, D.W. Tayor, Phys. Rev. 56, J.A. Backman, D.M. Esterng, N.F. Berk, Phys. Rev. B 4, A. Gons J.W. Gar, Phys. Rev. B 6, F. Yonezawa, Prog. Theor. Phys. 40, P.L. Leath, Phys. Rev. 7, E.J.S. Lage R.B. Stnchcombe, J. Phys. C 0, D.J. Whteaw, J. Phys. C 4, M. Pajda, J. Kudrnovský, I. Turek, V. Drcha, P. Bruno, Phys. Rev. Lett. 85, ; Phys. Rev. B 64, J j denotes the exchange ntegra between an atom of type at ste a atom at ste j, both mmersed n the effectve medum. Ths mpes that J j J j. 3 S.V. Tyabcov, Methods n Quantum Theory of Magnetsm Penum Press, New York, P. Fröbrch, P.J. Jensen, P.J. Kuntz, Eur. Phys. J. B 3, F.B. Anderson H.B. Caen, Phys. Rev. A 36, A H.B. Caen, Phys. Rev. 30, ; see aso H.B. Caen S. Shtrkman, Sod State Commun. 5, Y. Izyumov, Proc. Phys. Soc. Jpn. 87, A.A. Rudermann C. Ktte, Phys. Rev. 96, ; T. Kasuya, Prog. Theor. Phys. 6, 45956; K. Yosda, Phys. Rev. 06, For convenence a sma broadenng has been ntroduced to make the fgure easer for the reader. At exacty q0 we fnd that the peak at E0 sa peak consstent wth the fact that ImG (E 0)0. 20 As expected, anayzng cosey S (q,e) n the vcnty of q0, we fnd that E (q)dq