Ab initio determination of an extended Heisenberg Hamiltonian in CuO2 layers

a Xi :cond-ma /0010259 1 [cond-ma .s -el] 18 Oc 2000 Ab ini io de e mina ion o an ex ended Heisenbe g Hamil onian in CuO2laye s Ca men J. Calzado1and Jean-Paul Mal ieu Labo a oi e de Physique Quan ique. IRSAMC. Uni e si ´e Paul Saba ie , 31062 Toulouse, F ance. Abs ac Accu a e ab ini io calcula ions on embedded Cu4O12 squa e clus e s, agmen s o he La2CuO4la ice, con i m a alue o he nea es neighbo an i e omagne ic coupling (J=124 meV) p e iously ob ained om ab ini io calcula ions on bicen ic clus e s and in good ag ee- men wi h expe imen . These calcula ions p edic non negligible an i e omagne ic second- neighbo in e ac ion (J′=6.5 meV) and ou -spin cyclic exchange (K=14 meV), which may a ec he he modynamic and spec oscopic p ope ies o hese ma e ials. The dependence o he magne ic coupling on local la ice dis o ions has also been in es iga ed. Among hem he bes candida e o induce a spin-phonon e ec seems o be he mo emen o he Cu a oms, changing he Cu-Cu dis ance, o which he a ia ion o he nea es neighbo magne ic coupling wi h he Cu-O dis ance is ∆J/∆dCu−O∼1700 cm−1˚ A−1. 1 1On lea e om: Depa amen o de Qu´ımica F´ısica. Uni e sidad de Se illa. E-41012. Se illa. Spain. 1 1 In oduc ion Fi een yea s a e he disco e y o he high-Tcsupe conduc i i y in cup a es, nume ous heo- e ical and expe imen al s udies pay s ill a en ion o hese ma e ials and hei pa en undoped compounds in an a emp o explain hei elec onic p ope ies[1]. Rega ding he undoped ma e ials, he CuO2laye s, whe e supe conduc i i y akes place a e doping, can be seen as wo-dimensional spin la ices, whe e each Cu a om bea s an unpai ed elec on, which is an i- e omagne ically coupled wi h he nea es -neighbo s(NN). The alue o his NN coupling has been es ima ed om Raman sca e ing 128±6 meV[2, 3] and Neu on di ac ion expe imen s 134±5 meV[4, 5, 6], assuming a simple Heisenbe g Hamil onian, whe e only NN in e ac ions a e conside ed: H=JX <ij>,NN SiSj(J > 0) (1) whe e < ij > ep esen s a pai o NN si es. Howe e , his simple model does no sa is ac o ily ep oduce he whole Raman spec a o undoped cup a es [7, 8, 9, 10, 11, 12], and ex ended- Heisenbe g Hamil onians ha e been p oposed [13, 14, 15, 16]. The sophis ica ions in oduce some o he ollowing e ec s: - spin-phonon in e ac ions, - nex -nea es neighbo (NNN) magne ic couplings, J′, and - ou -spin cyclic (4SC) exchange, K. The spin-phonon coupling, i.e. he dependence o he magne ic coupling on he ib a ional dis o ions o he la ice, has been ecen ly in oked as possibly esponsible o he asymme y o he B1gpeak on he Raman spec a o cup a es [13, 17, 18, 19, 20]. The spin-phonon in e ac ion modi ies he magne ic coupling J h ough he dependence o he hopping in eg al ( pd) and he cha ge ans e ene gy (∆CT ) on he Cu-O dis ance [21, 22, 23]. A maximum con ibu ion o ±54 meV o he NN magne ic coupling coming om spin-phonon in e ac ions has been sugges ed [13], based on he linea dependence o Jon he Cu-O dis ance obse ed in he M2CuO4 amily [23] and he spin-wa e app oxima ion. Howe e , he calcula ions explici ly including he phonon- spin in e ac ion, using an adiaba ic app oxima ion o he phonons, [17], equie e un ealis ic alues o diso de o ep oduce he wid h and asymme y o he B1gpeak. I seems necessa y o in oduce addi ional e ms (as NNN coupling and 4SC exchange) o ep oduce he s uc u e o he Raman spec a [13]. The exis ence o he NNN magne ic coupling J′and he 4SC exchange Kcan be es ablished om a one-band Hubba d model [24, 25, 26, 27]. The NNN in e ac ions may be ei he a second o de e ec in o m o ∼ ′2/U, whe e ′is a second-neighbo hopping in eg al and Uis he classical on-si e Coulomb epusion, o ou h-o de e ec s scaling as ∼ 4/U3, whe e is he NN hopping in eg al. The 4SC exchange is a ou h-o de e m in he Hubba d model, in ol ing ci cula ion o he elec ons a ound he squa e and scaling as λ 4/U3, whe e λis a la ge nume ical ac o ( λ=40 [24, 25] o λ=80 [26, 27], depending on he o mal w i ing o he ope a o ), as shown in ea ly wo ks in quan um chemis y [24, 25] and solid s a e physics [26, 27]. Recen expe imen s ha e shown ha ou -spin cyclic exchange exis s in he wo dimensional solid 3He [28, 29, 30], in he 2D Wigne solid o elec ons o med in a Si in e sion laye [31] and in he bcc 3He [32, 33, 34]. As was p e iously shown, oxygen a oms play a c ucial ole in he spin exchange be ween Cu a oms in hese ma e ials [21, 22, 35, 36]. In his con ex , he one-band model is no su icien 2 o bea all he physics o such ma e ials and canno ix he a ios J′/J and K/J. The mul iple pa ame e s con ained in an ex ended Heisenbe g Hamil onian and he spin-phonon coupling canno be uni ocally ixed om he collec i e p ope ies o he ma e ial and, as a as possible, a p ejudiceless e alua ion o hem will be welcome. In he ecen pas , ab ini io quan um chemical calcula ions, using la ge basis se s and accu a e ea men o he elec onic co ela ion by means o ex ensi e con igu a ion in e ac ion (CI) expansions o he wa e unc ions, ha e been pe o med on bicen ic clus e s (Cu2O7and Cu2O11) [21, 22], p ope ly embedded in he Madelung ield o he in ini e c ys al, c ucial o co ec ly ep esen he elec onic s uc u e o hese sys ems[23]. These calcula ions p o ided sa is ac oy alues o J(138 meV) and o he i s -neighbo hopping in eg al o he hole-doped sys em ( =-0.55-0.57 eV). In bo h cases, he e alua ion o he e ec i e in e ac ion goes h ough he calcula ion o he spec um o he dime . An ex ension o his s a egy is p oposed he e which p o ides an e alua ion o J′and K om he calcula ion o he spec um o ou -Cu si es squa e embedded clus e s. By he way, he ans e abili y o he J alue om he wo-cen e o he ou -cen e clus e s will be e i ied. To es ima e he spin-phonon coupling, he bime allic clus e Cu2O7has been used, calcula ing he dependence o he single - iple sepa a ion on di e en geome y dis o ions. 2 Nex -nea es neighbo coupling and ou -spin cyclic exchange 2.1 S a egy o ex ac he e ec i e in e ac ions A squa e clus e con aining ou Cu a oms and hei nea es wel e in-plane oxygen a oms (a plaque e) will be used o ex ac hese pa ame e s (Figu e 1). Each Cu a om con ains an unpai ed elec on in an in-plane dx2-y2- ype o bi al. Fo such ame, he ou cen e - ou spin model space is spanned by six neu al de e minan s. I one calls a, b, c and d he magne ic o bi als, cen e ed in each Cu a om, he e a e wo kinds o de e minan s wi h Sz=0, he ully spin-al e nan de e minan s |a¯ bc ¯ d|and |¯ab¯cd|and ou pa ially- us a ed de e minan s |ab¯c¯ d|, |¯a¯ bcd|,|a¯ b¯cd|and |¯abc ¯ d|. The e ec i e Hamil onian spanned by such a model space can, in ull gene ali y, be w i en as: |a¯ bc ¯ d| |¯ab¯cd| |ab¯c¯ d| |¯a¯ bcd| |a¯ b¯cd| |¯abc ¯ d| −4h−g4g4h h h h −4h−g4h h h h −2h−2h′−g′ 4g′ 4h′h′ −2h−2h′−g′ 4h′h′ −2h−2h′−g′ 4g′ 4 −2h−2h′−g′ 4 whe e he ze o o ene gy is ha o he e omagne ic quin e s a e and he equi alences be ween di e en elemen s a e due o symme ic easons, imposed by he s uc u e o he plaque e. Fo ins ance, he elemen s ha¯ bc ¯ d|He |ab¯c¯ di ep esen s he exchange o he spins in band c. In he plaque e, his in e ac ion mus be equi alen o he exchange be ween aand d, ha is, he elemen ha¯ bc ¯ d|He |¯a¯ bcdi, and di e en om he h¯a¯ bcd|He |a¯ b¯cdielemen , which exchanges he spins on he diagonals. 3 The six eigens a es o his ma ix belong o di e en spin-space symme y i educible ep e- sen a ions. The spec um can be easily w i en om he basic pa ame e s as shown in Figu e 2. The e a e only ou ene gy-di e ences, and hen he ou pa ame e s can be uni ocally de ined. I we now pe o m an accu a e calcula ion o he six lowes eigens a es o his sys em, employing he bes ab ini io echniques, we will ob ain ou le el spacings which enable us o de e mine he ou desi ed e ec i e in e ac ions. 2.2 Ab ini io calcula ions As was p e iously men ioned, o es ima e he NNN and 4SC in e ac ions a squa e clus e con- aining ou Cu a oms and he i s wel e in-plane oxygen a oms has been conside ed, whe e all he a oms a e ea ed explici ly. The mos in e nal elec ons o he Cu a oms (1s22s2p63s2) ha e been eplaced by an e ec i e co e po en ial and he es o he elec ons (3p6d9 o Cu+2 and 1s22s2p6 o O−2, a o al o 156 elec ons) a e explici ly ea ed in he basis se s o iple-ze a quali y (double-ze a o O a oms) [37]. In o de o model he in ini e la ice, a well-es ablished app oach has been used, which consis s in eplacing he i s -shell o neighbo s (in-plane and ou -o -plane) by pseudopo en ials, which inco po a e bo h elec os a ic and exclusion e ec s o hese ions, and in conside ing he Madelung ield o he emo e a oms o he pe iodic la ice, acco ding o E jen’s echnique [38]. A es ic ed open-shell sel -consis en ield calcula ion (ROHF) o he quin e s a e has been ca ied ou , which de e mines he ou magne ic o bi als (a, b, c, d o hei symme y- adap ed combina ions co esponding o he i educible ep esen a ions a1g,b3uand euin he D4hsymme y g oup) (Figu e 3). These ou o bi als de ine a alence-space wi h one-elec on and one-o bi al pe si e, in one- o-one co espondence wi h he model spaces o he Heisenbe g Hamil onian o he pa en one-band Hubba d Hamil onian. The diagonaliza ion o he alence CI ma ix (CASCI), ha is, a ma ix wi h dimension 36 in he delocalized basis se , gi es a alue o -28 meV o he NN an i e omagne ic coupling, which is e y a om he expe imen al es ima ion. This e y limi ed CI only con ains he Ande son mechanism in he ba e one-band model. This le el o desc ip ion misses wo impo an phenomena, namely in e media e cha ge- ans e om he oxygen a oms o he Cu a oms and he dynamical pola iza ion e ec s o he in e nal elec ons and he su ounding a oms, which eac o he luc ua ion o he ield c ea ed by he ac i e elec ons. The ea men o hese e ec s equi es much la ge CI expansions. In o de o ake in o accoun he i s e ec , namely he hopping be ween oxygen and Cu a oms, i is c ucial o iden i y he doubly-occupied o bi als o he oxygen a oms which con- ibu e o his mechanism. They a e no necessa ily canonical o bi als, i.e. eigens a es o he Fock ope a o . The mos - ele an ligand-cen e ed o bi als will be ob ained as ene gy-di e ence dedica ed molecula o bi als [39]. These o bi als ha e been ob ained as ollow: 1.- F om he ou -elec ons in ou -o bi al ac i e space, a CI calcula ion has been pe o med, limi ed o he single exci a ions on he op o all he alence space de e minan s. 2.- The densi y ma ices, RSand RQ, o he lowes single and quin e s a es ha e been calcu- la ed. The exci a ion-ene gy dedica ed MOs a e he eigen ec o s o he di e ence o he densi y ma ices RS−RQ, es ic ed o he nea ly doubly occupied MOs. The eigen alues o his ma ix di e ence, called ’implica ion numbe s’, gi e a measu e o he pa icipa ion o he co esponding o bi al o he ene gy di e ence, hence o he an i e omange ic mechanism esponsible o he 4 lowe ing o he ene gy o he single s a e. The MOs o la ges implica ion numbe s a e essen- ially spanned by 2pa omic o bi als o he b idging oxygen a oms, as shown in Figu e 4. 3.- Now hese ou o bi als will be added o he magne ic ones o de ine an enla ged alence space in ol ing 12 elec ons in 8 MOs, co esponding o a wo-band Hubba d model since i includes bo h he 3d-like o bi al o he Cu a oms and 2po bi als o he b idging oxygen di ec ed along he Cu-O bonds, wi h op imized delocaliza ion ails on he ex e nal oxygen a oms. The e ec o he dynamical pola iza ion will be aken in o accoun by pe o ming all he single exci a ions on he op o his enla ged alence space. The esul ing CI ec o s a e expanded on a la ge space (∼5·106de e minan s). When applied o he dime ic clus e Cu2O7 he same s a egy p o ides a alue o J=128 meV, in good ag eemen wi h he expe imen al e alua ions and ou p e ious CI es ima es 138 meV[21, 22], which in ol ed dbasis unc ions on he b idging oxygen a oms (which had o be dele ed he e o make easible he calcula ions on he plaque e). The iden i ica ion o he ab ini io calcula ed spec um o he e ame ic clus e s wi h he expec ed spacings o Figu e 2 leads o he ollowing alues o he e ec i e in e ac ions: h= 60.22 meV ;h′= 5.01 meV g4= 7.00 meV ;g′ 4= 0.49 meV F om hese alues i is possible o es ablish he in e ac ions as w i en in he usual spin o mula ion o he ou -body ope a o [13, 14, 15]: H=X <ij>NN J(SiSj−1 4) + X <ij>NNN J′(SiSj−1 4) + +KX <ijkl> [(SiSj)(SkSl) + (SiSl)(SjSk)−(SiSk)(SjSl)−1 16] whe e he highe mul iple ene gy is ze o, Jco esponds o he NN in e ac ion, J′ o he NNN coupling and K o he ou -spin cyclic exchange. No ice ha he las e m p oduces he cyclic pe mu a ion o he ou spins on he plaque e plus o dina y wo-spins exchanges o all he pai s o spins o he plaque e including hose on he diagonals. W i en in he basis o he six Sz=0 de e minan s o he abcd con igu a ion, his Hamil onian has he ollowing o m: |a¯ bc ¯ d| |¯ab¯cd| |ab¯c¯ d| |¯a¯ bcd| |a¯ b¯cd| |¯abc ¯ d| −2J K/2J/2−K/8J/2−K/8J/2−K/8J/2−K/8 −2J J/2−K/8J/2−K/8J/2−K/8J/2−K/8 −J−J′0J′/2 + K/8J′/2 + K/8 −J−J′J′/2 + K/8J′/2 + K/8 −J−J′0 −J−J′ Iden i ying he wo ma ices and omi ing he negligible g′ 4 e m, one ob ains: K= 2g4→K= 14 meV J= 2h+K 4→J= 124 meV J′= 2h′−K 4→J′= 6.5meV 5 The alue o he NN an i e omagne ic coupling (J) is in good ag eemen wi h bo h he p e ious es ima ion on he dime (J=128 meV) and also wi h he expe imen al e alua ions (128±6 meV[2, 3] and 134±5 meV[4, 5, 6]). Ou es ima e o he NNN magne ic coupling (J′=6.5 meV) is in acco d wi h he limi o |J′| ≤9 meV, p oposed o his compound om Raman expe imen s[6]. Conce ning he ou -spin cyclic exchange, expe imen al e alua ions a e no a ailable and i is only possible o compa e wi h he K/J a ios used in some ecen nume ical simula ions o he abso p ion spec um. The he e-p esen ed alues o Kand Jgi e K/J ∼ 0.11, which is lowe han he alue o 0.25 assumed by Honda e al. [15] and han he alue o 0.30 aken by Lo enzana e al.[14] ( om an ea lie sugges ion by Schmid and Ku amo o [40]) bu la ge han he c i ical alue, (K/J)c=0.05 ±0.04, es ima ed by Sakai and Hasegawa [16] o he appea ance o a magne iza ion pla eau a hal he sa u a ion alue in he S=1 2 an i e omagne ic spin ladde s. The a io o he NNN and NN in e ac ions is J′/J=0.051, somewha la ge han he alue accep ed by Lo enzana e al. (J/J′=0.04) [14]. 3 Spin-phonon in e ac ions Addi ional calcula ions ha e been pe o med o e alua e he dependence o he magne ic coupling cons an on local geome ical dis o ions o he la ice. This e alua ion p oceeds h ough ab ini io calcula ions on bime allic clus e s using he same s a egy as in he p eceding sec ion (same basis se , same kind o op imiza ion o he molecula o bi als and same ype o Con igu a ion In e ac ion calcula ions). Fi e di e en local dis o ions ha e been conside ed, as shown in Figu e 5. Table 1 gi es hei co esponding o ce cons an s, associa ed equency and he de i a i e ∆J/∆dCu−O. Conce ning s ongly localized mo emen s, hese equencies a e di e en om he eal equencies o he la ice, bu o e an insigh on he so ness o he di e en mo ions. Among he dis o ions, he mo emen o he b idging oxygen a om along he Cu-Cu bond, leng hening one Cu-O bond and sho ening he o he one (mode 2), has a small o ce cons an (ω∼750 cm−1), bu does no a ec signi ican ly he J alue. The mo emen s ou o he Cu-Cu axis, ei he in-plane o along he caxis (modes 3, 4 and 5), induce s ong changes on J alue bu he o ce cons an and equencies a e la ge and, hen hese dis o ions do no seem o be esponsible o he dispe sion o J. The mo emen sho ening (o leng hening) he Cu-Cu bonds (mode 1) has bo h a signi ican impac on J(∆J/∆dCu−O=1700 cm−1·˚ A−1) and a low equency (ω∼800 cm−1). These alues should be compa ed wi h hose assumed in a ecen wo k[13], which akes ∆J/∆dCu−O=4350 cm−1·˚ A−1and in okes he expe imen al equencies 550 and 690 cm−1. Ou oughly calcula ed ib a ional equency is on line wi h he expe imen al one bu he calcula ed dependence o Jon he Cu-O dis ance is hal smalle ha he alue p e iously p oposed [13]. 4 Conclusions This wo k has e alua ed he ampli udes o he di e en in e ac ions gene ally in oked o explain he spec al ea u es o CuO2laye s which do no i wi h he simple Heisenbe g Hamil onian es ic ed o nea es neighbo coupling. Nei he he spin-phonon coupling no he nex -nea es neighbo magne ic in e ac ions no he ou -body cyclic e ec s a e negligible, hey appea o be o he o de o magni ude some imes assumed in nume ical simula ions o he collec i e e ec s. The he e-p esen ed ab ini io calcula ions a e ee om he simpli ica ions o a one-band o e en 6 o a wo-band model Hamil onian. We belie e ha he so-ob ained alues o he gene alized dis ance-dependen Heisenbe g Hamil onian a e eliable enough o dese e o be used in he e alua ion o he collec i e p ope ies o he ma e ial. Acknowledgemen s The au ho s a e indeb ed o he Eu opean Commission o he TMR ne wo k con ac ERBFMRX- CT96-0079, Quan um Chemis y o Exci ed S a es. C.J.C. acknowledges he inancial suppo h ough he TMR ac i i y ”Ma ie Cu ie esea ch aining g an s” G an No. HPMF-CT-1999- 00285 es ablished by he Eu opean Communi y. 7 Re e ences [1] J. O ens ein, A.J. Millis, Science 288, 468 (2000). [2] P. E. Sulewski, P.A. Fleu y, K.B. Lyons, S-W. Cheong, Z. Fisk Phys. Re . 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