Accurate ab initio determination of magnetic interactions and hopping integrals in La2-xSrxCuO4 systems

Accu a e ab ini io de e mina ion o magne ic in e ac ions and hopping in eg als in La 2−x S x CuO 4 sys ems Ca men J. Calzado, Ja ie F. Sanz, and Jean Paul Mal ieu Ci a ion: The Jou nal o Chemical Physics 112, 5158 (2000); doi: 10.1063/1.481093 View online: h p://dx.doi.o g/10.1063/1.481093 View Table o Con en s: h p://sci a ion.aip.o g/con en /aip/jou nal/jcp/112/11? e =pd co Published by he AIP Publishing A icles you may be in e es ed in Accu a e ab ini io po en ial ene gy cu e o O 2 . I. Non ela i is ic ull con igu a ion in e ac ion alence co ela ion by he co ela ion ene gy ex apola ion by in insic scaling me hod J. Chem. Phys. 132, 074109 (2010); 10.1063/1.3298373 Ab ini io po en ial ene gy su aces and nonadiaba ic in e ac ions in he H + + N O collision sys em J. Chem. Phys. 128, 124305 (2008); 10.1063/1.2894308 Magne ic and O bi al O de ing in Doped Mangani es AIP Con . P oc. 629, 203 (2002); 10.1063/1.1509146 Theo y o he pho oemission spec al unc ion in high T c cup a es AIP Con . P oc. 483, 130 (1999); 10.1063/1.59612 Ab ini io con igu a ion in e ac ion de e mina ion o he o e one ib a ions o me hyleneimine in he egion 2800–3200  cm −1 J. Chem. Phys. 107, 342 (1997); 10.1063/1.474395 Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 Accu a e ab ini io de e mina ion o magne ic in e ac ions and hopping in eg als in La2À x S x CuO4sys ems Ca men J. Calzadoa) Depa amen o de Quimica Fisica, Uni e sidad de Se illa, E-41012 Se illa, Spain and Labo a oi e de Physique Quan ique, IRSAMC, Uni esi e ´Paul Saba ie , F-31062 Toulouse Cedex, F ance Ja ie F. Sanz Depa amen o de Quimica Fisica, Uni e sidad de Se illa, E-41012 Se illa, Spain Jean Paul Mal ieu Labo a oi e de Physique Quan ique, IRSMAC, Uni e si e ´Paul Saba ie , F-31062 Toulouse Cedex, F ance 共Recei ed 13 July 1999; accep ed 14 Decembe 1999兲 The na u e o magne ic in e ac ions and elec on ans e p ocesses in La2⫺xS xCuO4sys ems a e s udied, by means o an ab ini io embedded clus e model app oach, using a di e ence dedica ed con igu a ion in e ac ion 共DDCI兲p ocedu e. Fo he undoped sys em, he c ucial ole played by he ligand o me al cha ge ans e 共LMCT兲con igu a ions in he magne ic p ocess makes necessa y he use o an enla ged DDCI space, which explici ly akes accoun o he elaxa ion o hese LMCT con igu a ions. This p ocedu e allows us o app oach he expe imen al magne ic coupling cons an alue. In hole-doped sys ems, he alue ob ained o he elec on ans e in eg al, , is o 0.54–0.57 eV. The ex a hole, cha ac e ized om he na u e and occupa ion o di e en ial na u al o bi als, has a s ong pcha ac e 共app oxima ely 50%兲and is essen ially localized in CuO2planes. These esul s a e in ag eemen wi h he expe imen al e idence abou hese kinds o compounds. Nei he he alue o no he na u e o he ex a hole a e se iously a ec ed by he op imiza ion o he o bi als used in he CI expansion. This sugges s ha a –Je ec i e Hamil onian is an adequa e model o s udy he elec onic p ope ies o hese sys ems. © 2000 Ame ican Ins i u e o Physics. 关S0021-9606共00兲30210-0兴 I. INTRODUCTION Al hough supe conduc i i y phenomena was de ec ed in 1911 by Kame lingh Onnes,1 he disco e y o he new high- empe a u e supe conduc ing ma e ials by Bedno z and Mu ¨lle 2in 1986 ga e a new impe us o he ield o supe con- duc i i y, s imula ing nume ous expe imen al and heo e ical wo ks. The p esence o coppe oxide planes wi h a uni cell CuO2is a common ea u e o he high c i ical empe a u e 共Tc兲supe conduc o s and i is belie ed ha supe conduc i - i y is closely ela ed wi h p ocesses which ake place wi hin hese planes. These ma e ials a e also cha ac e ized by he p esence o mixed alency, i.e., he supe conduc i i y is in- duced by doping wi h holes o elec ons, in such a way ha he o mal coppe oxida ion s a e is no wo.3 Unde s anding he beha io o hese s ongly co ela ed ma e ials equi es he use o model Hamil onians, which ex- plici ly ea he la ge Coulomb in e ac ions. Mean- ield ap- p oxima ions, as local densi y app oxima ion 共LDA兲calcula- ions, ex emely success ul a desc ibing many c ys alline solids, a e no ele an o such ma e ials. Fo ins ance, se - e al LDA calcula ions ha e ailed o e eal he an i e omag- ne ic na u e o La2CuO4.4 Ande son5was he i s o p opose a single d-band e ec- i e Hamil onian as an adequa e model o analyze he elec- onic s uc u e o cup a es. Howe e , he con o e sy was opened when di e en expe imen s showed he s ong oxy- gen cha ac e o he ex a hole in doped cup a es and he absence o e idences o mixed- alen coppe a oms.6These esul s ha e sugges ed he necessi y o in oduce he 2poxy- gen band in he model Hamil onian. These a e he g ounds o he h ee-band model o Eme y,7which pe mi s one o suc- cess ully in e p e expe imen al da a, such as x- ay pho o- emission spec a 共XPS兲and elec on ene gy loss spec a 共EELS兲spec a, especially in he high-ene gy limi .8A dis- ad an age is ha h ee-band Hamil onians equi e mo e pa- ame e s han a e accessible om expe imen s. The –Jmodel o Zhang and Rice9 ies o keep he ad an ages o bo h one- and h ee-band Hubba d models. I consis s o a single-band e ec i e Hamil onian, which im- plici ly accoun s o Cu 3dandO2phyb idiza ion, wi h simila esul s as a h ee-band model in he low-ene gy po - ion o he spec um.10,11 Indeed, Hybe sen e al.12 ha e shown ha he –Jwa e unc ions ha e an o e lap la ge han 90% wi h he eigen unc ions o a h ee-band Hamil- onian. The –JHamil onian only con ains wo pa ame e s: H⫽⫺ 兺 具 ij 典 ␴ 关ci ␴ ⫹共1⫺ni⫺ ␴ 兲共1⫺nj⫺ ␴ 兲cj ␴ ⫹h.c.兴 ⫹J兺 具 ij 典 共SiSj⫺1 4ninj兲,共1兲 whe e e e s o he hopping in eg al be ween adjacen iand jcen e s and Jis he e ec i e magne ic coupling be ween a兲Au ho o whom co espondence should be add essed. Elec onic mail: [email p o ec ed] JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 11 15 MARCH 2000 51580021-9606/2000/112(11)/5158/10/$17.00 © 2000 Ame ican Ins i u e o Physics Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 hem. The p esen a icle, which de elops he con en o a ecen ly epo ed le e ,13 shows ha hese pa ame e s can be accu a ely ob ained om high le el ab ini io quan um chemical calcula ions. The app oach goes h ough he con- side a ion o a ini e clus e , in ol ing a leas wo Cu a oms and he neighbo oxygen a oms, p ope ly embedded in he ield o he o he la ice ions. Ab ini io CI echniques may be applied o such a ini e clus e . Fi s he undoped sys ems a e conside ed, in which each coppe a om bea s an unpai ed elec on, and we de e mine he co esponding magne ic o - bi als by he SCF calcula ion o he uppe mul iple . In o de o de e mine he e ec i e magne ic coupling cons an be- ween he wo unpai ed elec ons, he ene gy di e ence be- ween he lowes single and iple s a es mus be accu a ely calcula ed. The de e mina ion o he exci a ion ene gies may ollow he logic o he so-called di e ence-dedica ed con- igu a ion in e ac ion 共DDCI兲,14 which gene a es he lis o de e minan s con ibu ing o he second-o de o pe u ba ion o he exci a ion ene gy. Fo a sys em wi h wo-elec on in wo o bi als 共aand b兲in he magne ic limi 共weak delocal- iza ion兲, he model space would consis o he wo neu al alence-bond de e minan s, 兩 ab ¯ 兩 and 兩 ba ¯ 兩 , and he co e- sponding lis 共DDCI-2 lis 兲in ol es de e minan s p esen ing up o wo inac i e o bi als only 共i.e., one hole and one pa - icle, o wo holes and wo pa icles兲. P e ious applica ions o ha s a egy o La2CuO4sys em had only eco e ed a alue o he magne ic coupling o ⫺95 meV,15 while he expe imen al alue is close o ⫺135 meV.16,17 The p esen wo k eexamines he physics o his p oblem and shows he necessi y o enla ge he lis o de e minan s in o de o allow he dynamical pola iza ion o he ligand→me al cha ge ans e con igu a ions, LMCT, o ake place. The enla ged CI spaces p oduce accu a e alues o he coupling cons an 共⫺140 meV兲. The sugges ion has led o a ecen calcula ion o he magne ic coupling in pe o ski es o Ni and Cu,18 which happens o be ex emely success ul. Then he hole-doped analog 共i.e., he La2⫺xS xCuO sys- em兲is conside ed. In o de o de e mine he hopping in e- g al , which delocalizes he hole共s兲on he la ice, we ha e conside ed he same clus e s as be o e (Cu2O7and Cu2O11) bu wi h one elec on less, and de e mined he ene gy di e - ence be ween he lowes double s a es o gand usymme y, he co esponding ene gy di e ence being conside ed as wice he hopping in eg al in he –Jmodel. The calcula ion o he ene gy di e ence ollows he same logic as abo e, i.e., is based on a di e ence dedica ed CI. The model space is he minimal one 共one elec on in wo ac i e molecula o bi als兲, and he CI lis again includes all de e minan s con ibu ing o he ene gy di e ence a he second o de o pe u ba ion.19 The calcula ion has been pe o med i s using he molecula o bi als o he iple s a e o he undoped clus e , in ol ing holes which a e, a he ze o-o de le el, essen ially localized on he Cu a oms. The CI expansion shows an impo an de- localiza ion o he holes on he oxygen a oms 共abou hal Cu and hal O兲, bu al eady p o ides a easonable alue o he hopping in eg al 共0.54–0.57 eV兲. Then he ac i e space, i.e., he gand uo bi als a e op imized by de e mina ion o ap- p oxima ed na u al o bi als. This elaxa ion ac ually esul s in a s ong mixing o he 3dCu o bi als wi h he 2po bi als o he su ounding o bi als, which become sligh ly dominan in he hole. Bu he esul ing alue o , calcula ed om hese op imized o bi als, emains he same. This con i ms he a- lidi y o he Zhang–Rice analysis and he possibili y o map- ping he local in e ac ions in an e ec i e Hamil onian o –J cha ac e , spanned by he 3dx2⫺y2Cu o bi als wi h app o- p ia e delocaliza ion ails, as discussed in he inal sec ion. II. CLUSTER MODELS AND ENVIRONMENT REPRESENTATION: COMPUTATIONAL DETAILS La2⫺xS xCuO4has a pe o ski e-like s uc u e and c ys- allizes in a body-cen e ed e agonal s uc u e. I consis s o CuO2laye s sepa a ed by wo LaO planes, which o m he cha ge ese oi . Cu a oms a e su ounded by co ne -sha ing oxygen oc ahed a.20 In he undoped ma e ial, neighbo ing oc ahed a along a 共110兲di ec ion a e sligh ly il ed in oppo- si e di ec ions wi h espec o he c-axis. Howe e , his dis- o ion will no be aken in o accoun o model he c ys al, and he undis o ed La1.85S 0.15CuO4s uc u e will be conside ed,21 expec ing ha o ho hombic dis o ions do no ha e signi ican e ec s on he and J alues. Fo he de e - mina ion o hese alues and he p ope ies o doped and undoped sys ems, bime allic clus e s we e used o model he sys em, including he nea es -neighbo oxygens, in such a way ha hey cons i u e wo- and h ee-dimensional clus e s (Cu2O7and Cu2O11, espec i ely, Fig. 1兲. The minimum clus e ep esen a ion, Cu2O, is an inadequa e model in bo h cases, and he co esponding esul s ob ained o Jand we e no included in his wo k. The clus e s a e su ounded by an a ay o poin cha ges, in o de o p o ide an adequa e ep- esen a ion o he c ys al Madelung po en ial. The cha ge alues a e ixed acco ding o he me hod o E jen.22 In he FIG. 1. Two- and h ee-dimensional clus e s used o model he La2⫺xS xCuO4sys em. 5159 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺ x S x CuO4 Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 clus e -poin cha ge in e ace, some o al ion po en ials 共TIPs兲a e included o ep esen he Cu⫹2and La⫹3ca ions nea es o he clus e a oms. They p e en an a i icial pola - iza ion o he clus e bounda y oxygen elec ons owa ds he posi i e poin cha ges. P e ious heo e ical s udies o Ma in23 and Wang e al.24 ha e shown he impo ance o including bo h poin cha ges and e ec i e po en ials o co - ec ly desc ibe he occupa ion o he 3do bi als, he ene gy o he Cu 4sand 4po bi als, and e en he Cu–O bond leng hs in geome y op imiza ion calcula ions. TIPs a e ep esen ed by la ge co e Hay–Wad e ec i e co e po en ials25 共ECPs兲wi h o mal cha ges 共⫹2 and ⫹3 o Cu and La, espec i ely兲. Fo me allic a oms, he ECPs p o- posed by S e ens, Basch, and K aus26 a e used o ep esen he en inne mos elec ons. Cu alence elec ons a e de- sc ibed using a (8s8p6d)/关4s4p3d兴basis se . Fo he cen- al oxygen a om o he clus e , an all-elec on basis se is employed. P elimina y calcula ions es ing he in luence o basis unc ion quali y ha e shown he necessi y o include pola iza ion unc ions, a leas in he cen al oxygen a om, in o de o ob ain he co ec ela i e s abiliza ion o single and iple s a es in he undoped sys ems. The basis se inally used o his a om is (10s5d1d) con ac ed o 关3s2p1d兴.27 Fo he es o he oxygen a oms, which do no con ibu e di ec ly o he spin and elec on ans e mechanisms, a (6s6p) con ac ed o 关2s2p兴basis se is used o he a- lence elec ons, while he co e 1sones a e desc ibed by an ECP.28 We ha e checked ha he pola izabili y o he oxy- gen dianions O⫺2in he la ice, calcula ed wi h his 2s2p basis is 1.3 Å3, in good ag eemen wi h he expe imen al da a 共0.5–3.2 Å3兲共Re . 29兲 o oxygen ions in me allic oxides. As a s a ing molecula o bi al se , he es ic ed open shell Ha ee–Fock 共ROHF兲wa e unc ion o he iple s a e o he undoped sys em is used. In he D2hsymme y poin g oup, his s a e is o B2usymme y and can be w i en as 3 ␾ u⫽ 兩 co egu 兩 . The wo ac i e o bi als gand ua e essen- ially he symme ic (a1g) and an isymme ic (b2u) combi- na ion o he dx2⫺y2o bi als o each o he wo Cu clus e a oms 共 espec i ely, 93% and 90% din cha ac e 兲. As is well known, and as p e iously no iced by Ma in23 in he case o La2CuO4, Ha ee–Fock app oxima ion p oduces an o e es- ima ion o he on-si e Coulomb epulsion (Ud) and an ex- cessi ely ionic ep esen a ion. We will show below ha con- igu a ion in e ac ion can co ec his de iciency and eco e he p ope me al–ligand co alency. III. UNDOPED SYSTEMS: MAGNETIC COUPLING CONSTANT DETERMINATION The La2CuO4is an an i e omagne ic insula o wi h an exchange cons an expe imen ally de e mined by means o neu on sca e ing 关J⫽⫺134⫾5 meV兴共Re . 16兲and Raman measu emen s 关J⫽⫺128⫾6 meV兴.17 In he undoped sys- em, each Cu⫹2ca ion con ains an unpai ed elec on essen- ially localized in a dx2⫺y2o bi al, acco ding o he e ag- onal ligand ield. Each Cu⫹2ion can be conside ed as a magne ic pa icle wi h spin S⫽1/2. The Heisenbe g–Di ac– Van Vleck Hamil onian30 is he mos used model o deal wi h he in e ac ion be ween magne ic pa icles. Fo wo pa - icles ha ing o al spin S, his Hamil onian has he ollowing exp ession: H ˆ⫽⫺JS ˆ1S ˆ2⫽⫺J关1 2共S ˆ1 ⫹S ˆ2 ⫺⫹S ˆ1 ⫺S ˆ2 ⫹兲⫹S ˆz1S ˆz2兴,共2兲 whe e Jis he magne ic coupling cons an , Siis he o al spin ope a o o cen e i,Si ⫹and Si ⫺a e he spin-up and spin- down ope a o s o pa icle i, and Szi is he ope a o o he z componen o spin o pa icle i. The eigen alues o he Heisenbe g Hamil onian a e exp essed as a unc ion o J,in such a way ha he di e ence be ween he lowes single and iple eigen alues gi es he alue o J: ES⫺ET⫽J.共3兲 Fo an i e omagne ic sys ems, he single is he g ound s a e and Jis nega i e. The p oblem lies in he accu a e de e mi- na ion o he ene gy alues o hese wo s a es. Fo a wo- cen e p oblem, ep esen ed by he localized dx2⫺y2mag- ne ic o bi al, aand b, hese wo s a es can be exp essed a a ze o-o de le el as ⌿T 0⫽1 &共 兩 ab ¯ 兩 ⫺ 兩 ba ¯ 兩 兲,⌿S 0⫽1 &共 兩 ab ¯ 兩 ⫹ 兩 ba ¯ 兩 兲.共4兲 Using delocalized magne ic o bi als g⫽(a⫹b)/&and u ⫽(a⫺b)/& hese s a es a e ⌿T 0⫽1 &共 兩 gu ¯ 兩 ⫺ 兩 ug ¯ 兩 兲,⌿S 0⫽1 &共 兩 gg ¯ 兩 ⫺ 兩 uu ¯ 兩 兲.共5兲 As has been p e iously es ablished,14,31 he ␾ 1⫽ 兩 ab ¯ 兩 and ␾ 2⫽ 兩 ba ¯ 兩 de e minan s cons i u e a model space, o e which i is possible o build an e ec i e 2⫻2 Hamil onian. A he second o de o he quasidegene a e pe u ba ion heo y 共QDPT兲, he o -diagonal elemen He o he ma ix ep e- sen a ion o his Hamil onian is equal o14,31 具 ␾ 1 兩 H ˆe 共2兲 兩 ␾ 2 典 ⫽ 具 ␾ 1 兩 H ˆ 兩 ␾ 2 典 ⫹兺 ␾ ␣ 苸S 具 ␾ 1 兩 H ˆ 兩 ␾ ␣ 典具 ␾ ␣ 兩 H ˆ 兩 ␾ 2 典 E1 0⫺E ␣ 0 ⫽J/2, 共6兲 whe e he de e minan s 兩 ␾ ␣ 典 in e ac simul aneously wi h bo h o he wo model space de e minan s, and hey a e he only ones which con ibu e o he ene gy di e ences ES ⫺ETa his le el. The con igu a ion in e ac ion ma ix buil o e his space is called DDCI-2.14 DDCI s ands o di e - ence dedica ed con igu a ion in e ac ion and he ‘‘2’’ indi- ca es ha his space includes exci a ions up o wo deg ees o eedom, i.e., consis s o all he de e minan s buil om he model space as exci a ions in ol ing wo inac i e holes o wo inac i e pa icles, o one inac i e hole and one inac i e pa icle. As p e iously desc ibed,14,32 he DDCI space is in- a ian unde he uni a y ans o ma ion o he ac i e o bi - als, and i is possible o gene a e he di e en ial space in he o iginal symme y o he clus e , ha is, using as ac i e o - bi als he delocalized ones and Eq. 共5兲as ze o-o de desc ip- ion o single and iple s a es. Bo h Cu2O7and Cu2O11 belong o he D2hsymme y poin g oup, and single and iple a e o A1gand B2usymme ies, espec i ely. The diagonaliza ion o he DDCI-2 ma ices gi es he co ec 5160 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Calzado, Sanz, and Mal ieu Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 sign o he Jcons an and oughly 80% o i s alue, as has been p e iously epo ed by Casano as e al.15 o La2CuO4 sys em and also compu ed in his wo k 共Table I兲.Asis shown in his able, he alues ob ained o he 2D and 3D clus e s a e p ac ically iden ical, in ag eemen wi h he ex- pe imen ally es ablished wo-dimensional na u e o magne- ism in his sys em.16,17 Since he di e ence be ween expe i- men al and heo e ical alues canno be associa ed o collec i e e ec s, as Illas e al.33 ha e ecen ly demons a ed, his unde es ima ion has o be ela ed o an inadequa e ea - men o elec onic co ela ion e ec s. Ac ually, he alue o he an i e omagne ic in e ac ion is qui e la ge, much la ge han in many binuclea sys ems o which DDCI-2 appea ed o be o su icien accu acy.31 We hough ha he disc epancy in he p esen case migh come om he la ge coe icien 共⬇0.1, see below兲o he ionic alence–bond s uc u e in he single s a e, which should be conside ed as belonging o he model space, now enla ged o in ol e beside ␾ 1and ␾ 2, ␾ 3⫽ 兩 aa ¯ 兩 and ␾ 4 ⫽ 兩 bb ¯ 兩 . ␾ 1and ␾ 2only di e by one o bi al om ␾ 3and ␾ 4 and in such a case and acco ding o Sla e ’s ules, he di e - ence dedica ed CI space has o be enla ged up o a DDCI-3 lis , in ol ing wo inac i e holes and one inac i e pa icle (2h–1p) and one inac i e hole and wo inac i e pa icles (1h–2p) con igu a ions. This lis inc eases he size o he CI ma ix by a ac o o 50–60, bu as shown in Table I, gi es a heo e ical es ima e which p ac ically ma ches he expe i- men al alue. In Table II he analyses o he single and iple wa e unc ions a e p esen ed, bo h o DDCI-2 and DDCI-3 spaces. Toge he wi h he comple e ac i e space 共CAS兲de e minan s, o he con ibu ions a e epo ed, co e- sponding o ligand o me al cha ge ans e con igu a ions, LMCT, whe e single exci a ions om sp ligand-like o bi als o he gand uac i e o bi als ake place 共schema ically ep- esen ed by Lg→dand Lu→d, whe e Lgand Lu ep esen , espec i ely, symme ic and an isymme ic combina ions o s and pligand basis unc ions, wi h mino con ibu ions o he d-me al unc ions, scheme I兲. SCHEME I. Two main e ec s can be dis inguished when DDCI-3 space is used: 共i兲 he inc ease o he coe 兩 gg ¯ 兩 /coe 兩 uu ¯ 兩 a io in he sin- gle s a e, 共ii兲 he appea ance o con ibu ions ex e nal o he CAS, bo h o single and iple s a es. The o me co esponds o an inc ease o he weigh o ionic alence–bond de e minan s, ␾ 3and ␾ 4in he single de- sc ip ion. The CAS p ojec o PCAS is de ined as PCAS⫽兺 i苸CAS 兩 ␾ i 典具 ␾ i 兩 .共7兲 Ope a ing o e 兩 ⌿S 典 and 兩 ⌿T 典 , we ob ain PCAS 兩 ⌿s 典 ⫽ ␣ 兩 gg ¯ 兩 ⫺ ␤ 兩 uu ¯ 兩 , 共8兲 PCAS 兩 ⌿T 典 ⫽ ␥ 关 兩 gu ¯ 兩 ⫺ 兩 ug ¯ 兩 兴. Assuming ha gand ua e composed only by he symme ic and an isymme ic linea combina ion o he localized dx2 ⫺y2o bi als, Eq. 共8兲 esul as PCAS 兩 ⌿s 典 ⫽ ␣ ⫹ ␤ & 冉 兩 ab ¯ 兩 ⫹ 兩 ba ¯ 兩 & 冊 ⫹ ␣ ⫺ ␤ & 冉 兩 aa ¯ 兩 ⫹ 兩 bb ¯ 兩 & 冊 , 共9兲 TABLE I. Exchange coupling cons an alues J, meV, ob ained wi h di e en DDCI spaces. Clus e Space n°de (T,S) O bi al se J共meV兲 Cu2O7 DDCI-2 7680,7687 HF ⫺96 IDDCI ⫺103 DDCI-2⫹Single 兵LMCT其12 454,12 461 HF ⫺135 DDCI-3 411 084,411 091 HF ⫺138 IDDCI ⫺142 Cu2O11 DDCI-2 11 772,11 779 HF ⫺99 IDDCI ⫺111 DDCI-2⫹Single 兵LMCT其22 586,22 593 HF ⫺131 DDCI-3 804 828,804 835 HF ⫺139 IDDCI ⫺145 Expe imen al alues: ⫺128⫾6meV a,⫺134⫾5meV b aRe e ence 17. bRe e ence 16. TABLE II. Single and iple wa e unc ions o DDCI-2 and DDCI-3 spaces in he canonical MO basis se . ROHF MO as ze o-o de desc ip ion. Only he con ibu ions equal o la ge han 0.05 a e epo ed. Clus e Con . DDCI-2 DDCI-3 TSTS Cu2O7gg ¯ ¯0.7703 ¯0.7380 uu ¯ ¯⫺0.6275 ¯⫺0.5717 gu ¯ ⫺0.7037 ¯⫺0.6629 ¯ ug ¯ 0.7037 ¯0.6629 ¯ Lu→d¯¯0.1152 0.1393 Lg→d¯¯0.0933 0.0721 Cu2O11 gg ¯ ¯0.7723 ¯0.7379 uu ¯ ¯⫺0.6246 ¯⫺0.5688 gu ¯ ⫺0.7037 ¯⫺0.6616 ¯ ug ¯ 0.7037 ¯0.6616 ¯ Lu→d¯¯0.1002 0.1217 Lg→d¯¯0.0862 0.0660 5161 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺ x S x CuO4 Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 PCAS 兩 ⌿T 典 ⫽& ␥ 冉 兩 ab ¯ 兩 ⫺ 兩 ba ¯ 兩 & 冊 . The alues o he coe icien s ( ␣ ⫹ ␤ )/&and ( ␣ ⫺ ␤ )/&, which conce n, espec i ely, he neu al and ionic alence– bond componen s, and & ␥ a e p esen ed in Table III. As is easy o check, he neu al/ionic a io in he single s a e di- minishes when he DDCI-3 space is used. The ques ion now is why he e ec o he ionic alence–bond single is en- hanced when he DDCI-3 space is used. The second e ec is he appea ance o con ibu ions as- socia ed o cha ge ans e p ocesses om o bi als cen e ed in sp combina ions o oxygen basis unc ions o o bi als 3dCu in cha ac e . Ac ually, Van Oos en e al.34 had ob- ained qui e accu a e esul s o exchange coupling in La2CuO4sys em 共⫺120 meV兲 om a low-dimensional non- o hogonal con igu a ion in e ac ion 共NOCI兲calcula ion. The me hod includes, in addi ion o he de e minan s ␾ 1, ␾ 2, ␾ 3, and ␾ 4, each o hese wi h a speci ically pola ized co e, he ligand→me al cha ge ans e de e minan s, in ol ing he o bi als o he b idging ligand: ␾ 5⫽...Cu⫹共↑↓兲⫺L⫹共↑兲⫺Cu⫹2共↓兲.., ␾ 6⫽...Cu⫹共↑↓兲⫺L⫹共↓兲⫺Cu⫹2共↑兲.., whe e L⫹is an O⫺ion (L⫽O⫽). These de e minan s appea in he DDCI-2 lis as one-hole s a es. Thei ole is c ucial o acili a e he Ande son mechanism. This mechanism is a second-o de e ec in h ough-space in e ac ions 共scheme II, M⫽Cu⫹2), SCHEME II. wi h he exchange coupling Jp opo ional o ⫺( 2/U), whe e Uis he exci a ion ene gy om ␾ 1 o ␾ 3and is he hopping in eg al be ween he wo magne ic o bi als, bu i becomes a ou h-o de e ec when he ligand is on he M–M axis, acco ding o he ollowing p ocess 共scheme III, M⫽Cu⫹2), SCHEME III. which in ol es ligand o me al hopping in eg als ( LM) and gi es a con ibu ion o he exchange p opo ional o ⫺( LM)4/(U⬘)2U, whe e U⬘and Ua e he exci a ion ene - gies om ␾ 1 o ␾ 5and ␾ 1 o ␾ 3, espec i ely. In his scheme, he coe icien s o he cha ge- ans e s a es and o he ionic o ms, ␾ 3, a e p opo ional o LM /U⬘and ( LM)2/UU⬘, espec i ely. The p ocess will be unde es ima ed i one does no in- oduce he speci ic dynamic epola iza ion occu ing on he LMCT con igu a ions. These epola iza ion e ec s a e b ough by he single exci a ions on he op o he de e mi- nan s ␾ 5, ␾ 6..., i.e., by 2h–1pde e minan s which do no belong o he DDCI-2 lis bu o he DDCI-3 one. This ex- plains why he LMCT de e minan s ca y signi ican weigh s in he wa e unc ion desc ip ions only when he DDCI lis is enla ged. The in e ac ion o hese de e minan s wi h hei single exci a ions, single 兵LMCT其, p oduces an impo an lowe ing o he e ec i e U⬘ene gies o he LMCT s a es and la ge con ibu ions o he single and iple wa e unc- ions. I is wo h while o no ice ha LMCT de e minan s a e also single exci a ions wi h espec o he ionic ␾ 3and ␾ 4 con igu a ions, in such a way ha he elaxa ion o he LMCT s a es p oduces a s abiliza ion o he ionic alence bond con igu a ion and an inc ease on i s weigh on he sin- gle desc ip ion. Reducing he e ec i e ene gy U⬘, he coe - icien s o he LMCT o ms and o he ionic de e min- an s, ␾ 3, inc ease 共 espec i ely, Lg→dand Lu→din Table II, and ( ␣ – ␤ )/&in Table III兲. The change o ( ␣ ⫺ ␤ )/& ⫽( LM)2/UU⬘by 20% can be ela ed o he change by 40% o he an i e omagne ic coupling, J⬃( LM)4/U(U⬘)2 ⫽( ␣ ⫺ ␤ )2/2U. F om a p ac ical poin o iew, i is impo an o no ice ha hese singly exci ed 兵LMCT其de e minan s a e no nu- me ous, since one o he inac i e holes is he Lgo Luo bi al o he ligands. The DDCI-2⫹single兵LMCT其spaces a e 40– 50- old smalle han DDCI-3 spaces bu , as is shown in Table I, he expe imen al alue o Jis also eco e ed. Wi h ega d o he na u e o he molecula o bi als Lg and Lu, Fig. 2 ep esen s he ROHF elec onic densi y maps o gand uac i e o bi als and Lgand Luo bi als o he Cu2O7clus e . As can be seen in his igu e, Lgand Luco - espond o he bonding combina ions o he Cu dx2⫺y2and sp oxygen o bi als 共scheme I兲, whe eas gand ua e he an i- bonding combina ions, wi h a s ong dcha ac e . The elec- onic dis ibu ion o he wo o bi al pai s 共gand Lg,uand Lu) a e qui e simila , making possible an op imal ligand– me al cha ge ans e in bo h hese cases. A. Dependence o he J alue on he molecula o bi al choice In o de o o e come he possible dependence o ou esul s on he s a ing molecula o bi al se , we ha e ob ained he S–Ta e age na u al o bi als by means o an i e a i e p ocedu e, p oposed by Ga cia e al.35 and implemen ed in he DDCI scheme. An a e age densi y ma ix o he single and iple s a es is buil : R ¯ ⫽(RS⫹RT)/2. A e he diago- naliza ion o his ma ix, a na u al o bi al se is ob ained. Wi h his new se , he DDCI p ocedu e is epea ed 共i e a i e dDDCI, IDDCI兲un il bo h he o bi als and he S–Tene gy TABLE III. Valence–bond p ojec ions o single and iple wa e unc ions. ROHF MO as ze o-o de desc ip ion. Clus e Coe . DDCI-2 DDCI-3 DDCI-2⫹single 兵LMCT其 Cu2O7& ␥ 0.995 0.938 0.986 ( ␣ ⫹ ␤ )/&0.988 0.926 0.975 ( ␣ ⫺ ␤ )/&0.101 0.117 0.119 Cu2O11 & ␥ 0.995 0.935 0.986 ( ␣ ⫹ ␤ )/&0.987 0.924 0.975 ( ␣ ⫺ ␤ )/&0.104 0.119 0.120 5162 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Calzado, Sanz, and Mal ieu Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 i e ence a e sel -consis en . Beside hei in e es o he de- e mina ion o op imal molecula o bi als in he ene gy di - e ence calcula ion, he densi y ma ices and he na u al o - bi als a e c ucial ins umen s in he analysis o he wa e unc ion and hey will be used o ha pu pose all h ough his wo k. Ac ually, he na u al o bi als and occupa ion num- be s make possible some concen a ion o in o ma ion e- ga ding wa e unc ions de eloped on housands o N-elec on con igu a ions. The alues o he exchange coupling ob ained wi h he con e ged a e age na u al o bi al se is p esen ed in Table I, o DDCI-2 and DDCI-3 spaces. The J alues ob ained wi h he canonical and na u al o bi al se s a e simila and only a sligh inc ease o he absolu e alue is obse ed when he na u al o bi al se is used. This e ec is la ge in he DDCI-2 space han in he DDCI-3. The IDDCI p ocedu e only p o- duces mino modi ica ions on he magne ic o bi als 共gand u ha e 92% and 89% o dcha ac e 兲. Figu es 3共a兲and 3共b兲 co espond, espec i ely, o he iple ROHF and na u al ID- DCI localized ac i e o bi al o Cu2O7, ob ained om a ␲ /4 o a ion o he symme y-adap ed gand uo bi als. The mino change in he slope o hese o bi als can be seen as a majo pa icipa ion o he poxygen o bi als in he na u al o bi als. IV. DOPED SYSTEMS: NATURE OF THE HOLES In La2CuO4, he subs i u ion o La⫹3ion by S ⫹2p o- duces a de ec o elec ons in he CuO2planes, i.e., in o- duces holes in he sys em. When he S concen a ion 共x兲 inc eases, di e en phase ansi ions ake place. Fo x ⬍0.05 he sys em is an an i e omagne ic insula o , while o 0.05⬍x⬍0.3 i becomes a supe conduc o . The maximum alue o he c i ical empe a u e (Tc⫽40 K) is obse ed a he op imal doping x⫽0.15. Me allic beha io is obse ed o S doping la ge han 0.3.36 The na u e and dis ibu ion o ca ie s de e mine he su- pe conduc i i y. This has mo i ed nume ous s udies, expe i- men al as well as heo e ical. Di e en expe imen al e i- dence has shown ha ca ie s in La2⫺xS xCuO4ha e a s ong 2poxygen cha ac e . Bo h XANES 共x- ay abso p ion nea edge s uc u e兲6and x- ay pho oemission spec a6ha e indi- ca ed ha he p edominan Cu a oms con igu a ion in doped coppe oxides is 3d,9wha e e x. The e is no e idence o mixed alency in hese sys ems. Also he 1soxygen EELS spec a sugges ha holes a e essen ially placed in he 2p oxygen band.6Se e al heo e ical wo ks, as he in e media e neglec o di e en ial o e lap 共INDO兲 esul s o Wang e al.24 o he ab ini io SCF/CI calcula ions epo ed by Ma in23 show ha he addi ional hole has a ound 50% p cha ac e . In acuum he ioniza ion po en ial o Cu⫹2ions is much la ge han ha o he O⫺2anions. In in ini e ionic la ices, howe e , his pic u e is no longe alid since he elec os a ic po en ial pushes he Cu dband abo e he O p band, in a way ha he highes pa ially occupied o bi als 共 he magne ic ac i e o bi als in undoped sys ems兲a e essen- ially Cu dx2⫺y2in cha ac e . This delocalized monoelec- onic pic u e would play in a o o a pa icipa ion o he d shell in he ioniza ion. Ou wo k will examine he ela i e O/Cu composi ion o he holes. FIG. 2. Isodensi y cu es o magne ic gand uo bi als, and he mos pa - icipa ing ligand o bi al Lgand Lu o Cu2O7clus e , plo ed on he xy-plane. 5163 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺ x S x CuO4 Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 As men ioned abo e, i is necessa y o include p ope ly he elec onic co ela ion in o de o adequa ely desc ibe he g ound s a e o he doped sys em. This will be he objec o he p esen sec ion. A. The wo s a e model Le us conside one coppe ion su ounded by ou oxy- gen a oms. Acco ding o he o iginal ideas o Zhang and Rice,9a hole a he oxygen can be in a symme ic o an i- symme ic s a e wi h espec o he cen al hole a he Cu ion. Holes in oxygen and Cu a oms will be coupled, esul ing in a spin single o iple s a e. These au ho s demons a ed ha he spin single s a e has he lowe ene gy, and assumed ha i is possible o wo k in his single subspace wi hou chang- ing he physics o he p oblem. In his con ex , he mo emen o he hole is equi alen o he displacemen o a Cu 1/2 spin and o a wo-cen e clus e , he p ocess can be summa ized in Scheme IV, SCHEME IV. which ep esen s wo degene a ed si ua ions, whe e an un- pai ed elec on 共o a hole兲oscilla es be ween wo equi alen posi ions. They can be exp essed by wo localized s a es: ␾ a 0⫹⫽ 兩 co e..a⬘ 兩 , ␾ b 0⫹⫽ 兩 co e..b⬘ 兩 .共10兲 Thei symme y-adap ed linea combina ions a e ⌿g 0⫹⫽1 &共 ␾ a 0⫹⫹ ␾ b 0⫹兲⫽ 兩 co e..g⬘ 兩 ; 共11兲 ⌿u 0⫹⫽1 &共 ␾ a 0⫹⫺ ␾ b 0⫹兲⫽ 兩 co e..u⬘ 兩 , whe e he o bi als a⬘and b⬘共 espec i ely, g⬘and u⬘) should esemble, o some ex en , hose o he magne ic p oblem. This pai o de e minan s de ines a new model space. As in he magne ic p oblem, a he second o de o he quasidegen- e a e pe u ba ion heo y, he o -diagonal elemen o he ma- ix ep esen a ion o he e ec i e Hamil onian buil o e his space is ela ed o he elec onic coupling cons an , :19 ⫽Hab e ⫽Hab 共0兲⫹兺 ␣ 具 ␾ a 0⫹ 兩 H ˆ 兩 ␾ ␣ 典具 ␾ ␣ 兩 H ˆ 兩 ␾ b 0⫹ 典 Ea 0⫺E ␣ 0.共12兲 Since ␾ a 0⫹and ␾ b 0⫹di e by only one spin-o bi al, i is easy o show ha he se o 兩 ␾ ␣ 典 de e minan s include all he exci a ions up o h ee deg ees o eedom 共DDCI-3 lis : single and double exci a ions excep hose which in ol e wo inac i e holes and wo inac i e pa icles兲. As in he magne ic p oblem, he a ia ional ea men o he DDCI lis in oduces highe o de co ec ions and a oids he inaccu acies due o he smallness o he denomina o s in he second-o de exp essions. Using symme y-adap ed con igu- a ions, is gi en by ⫽Eg⫺Eu 2,共13兲 whe e Egand Eua e he eigen alues o he ⌿g ⫹and ⌿u ⫹ s a es, which in he D2hpoin g oup a e o A1gand B2u symme y, espec i ely. Once he DDCI lis is buil , he di- agonaliza ion o he di e ence dedica ed CI ma ix gi es us he a ia ional es ima e o he hopping in eg al. In Table IV he alues ob ained o 2D and 3D clus e s a e collec ed. The size o he clus e does no a ec he alue o he ans e in eg al. This esul sugges s he wo- FIG. 3. Isodensi y cu es o ac i e localized o bi als in Cu2O7clus e , plo - ed on he xy-plane: 共a兲ROHF o bi al, 共b兲na u al IDDCI magne ic o bi al, 共c兲na u al IDDCI doped-sys em o bi al, and 共d兲localized di e en ial na u- al o bi al. 5164 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Calzado, Sanz, and Mal ieu Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17 dimensional cha ac e o he ca ie s in he doped coppe oxides. The alue o oscilla es be ween 0.53–0.55 eV in good ag eemen wi h he usually accep ed alue o 0.5 eV 共Re . 37兲共 o ins ance, 0.57–0.59 eV by Wang e al.,24 0.65 eV by Ma in,23 and 0.4 eV by Hybe sen e al.兲.12 In he basis o he canonical ROHF molecula o bi als, he p ojec ions in he CAS model space o he ⌿g ⫹and ⌿u ⫹ DDCI-3 wa e unc ions a e 0.77 and 0.76, espec i ely. This means ha only he 50% o he wa e unc ions can be ep- esen ed by he ex a hole localized in one o he Cu dx2 ⫺y2o bi als. The es o he con ibu ions comes om di - e en O 2p→Cu3dcha ge- ans e exci a ions. An appeal- ing poin is ha he mos pa icipa ing O 2pcen e ed o bi als co espond o he same Lgand Luo bi als which play an impo an ole in he magne ic p oblem. The e a e h ee kinds o de e minan s pa icipa ing in he g ound s a e wa e unc ion desc ip ion: 共i兲 he model space de e minan s, M–L–M⫹,M ⫹–L–M, whe e he e is an ex a hole in an ac i e localized o bi al: ␾ a 0⫹⫽Cu⫹2共↑兲⫺L共↑↓兲⫺Cu⫹3共.兲, ␾ b 0⫹⫽Cu⫹3共.兲⫺L共↑↓兲⫺Cu⫹2共↑兲; 共ii兲neu al cha ge- ans e exci a ions, M–L⫹–M, wi h an ex a hole in a ligand-cen e ed o bi al: ␾ 1 ⫹⫽Cu⫹2共↑兲⫺L⫹共↓兲⫺Cu⫹2共↑兲, ␾ 2 ⫹⫽Cu⫹2共↑兲⫺L⫹共↑兲⫺Cu⫹2共↓兲; 共iii兲ionic cha ge- ans e exci a ions, M⫹–L⫹–M⫺, M⫺–L⫹–M⫹, whe e he ex a hole is in an oxygen o bi al, and a me al–me al dA→dB ans e akes place simul aneously: ␾ 3 ⫹⫽Cu⫹3共.兲⫺L⫹共↑兲⫺Cu⫹1共↑↓兲, ␾ 4 ⫹⫽Cu⫹1共↑↓兲⫺L⫹共↑兲⫺Cu⫹3共.兲. The i s class 共 he model space兲in oduces he ex a hole in a3do bi al. Bu his ep esen s only 50% o he doped g ound s a e desc ip ion. Cha ge ans e exci a ions pu a hole in o bi als wi h s ong cha ac e in 2poxygen, inc eas- ing he elec onic densi y in 3dx2⫺y2, and he e o e dimin- ishing he 3dna u e o he hole 共sc eening he wo holes in 3do bi als兲. This e ec has ecei ed he name o elec onic pola on,23 in he sense ha an ex a hole in a coppe si e p oduces no a nuclea dis o ion bu an elec onic dis o ion o he neighbo oxygen, and a s ong p–d ehyb idiza ion occu s 共scheme V兲. In ac , he pcha ac e o ac i e o bi als has inc eased a e he hole doping. Fo ins ance, in he doped Cu2O7clus e he ac i e gand uo bi als ha e a 83% and 76% o dcha ac e , espec i ely, o be compa ed wi h he s a e-a e aged IDDCI na u al o bi als ob ained o he undoped sys em 共86% and 87%兲. Fo he Cu2O11 clus e , a simila e ec is obse ed in Scheme V. SCHEME V. B. Dependence o he alue on he molecula o bi al choice I should be no ed he e ha all he calcula ions so a ha e been ca ied ou using as ac i e o bi als he magne ic o bi als de e mined o J. This is in ag eemen wi h he ideas behind he –Jmodel, whe e a unique se o ac i e o bi als is used. Howe e , he e has been a lo o con o e sy abou he na u e o he hole. Ac ually, he e exis s unanimi y abou i s p edominan oxygen cha ac e . This means ha hole- adap ed o bi als, a⬘and b⬘, should be di e en han he o iginal aand bo bi als. Mo eo e , he possible dependence o he DDCI esul s on he ini ial o bi als has been p e i- ously men ioned. To o e come hese di icul ies, some DDCI-3 calcula ions ha e been pe o med i e a ing he na u- al o bi als o he doped p oblem, using he i e a i e di e - ence dedica ed CI 共IDDCI兲me hod. This p ocedu e does no change he alue o , as can be seen in Table IV. The mos impo an e ec o hese hole adap ed o bi als is he inc ease o he weigh o he CAS de e minan s in he ⌿g ⫹and ⌿u ⫹ desc ip ion 共 he coe icien s o he p ojec ions become 0.91– 0.90兲, i.e., he concen a ion o he wa e unc ions in an im- p o ed model space. The i e a ed gand uo bi als inco po a e by hemsel es he p–dhyb idiza ion, ha ing a 71% and 66% dcha ac e , espec i ely, o he doped g ound s a e o Cu2O7, as can be seen compa ing he ac i e magne ic and cha ge- ans e o bi als, aand a⬘关Figs. 3共b兲and 3共c兲兴. This explains he la ge con ibu ions o he CAS de e minan s in he wa e unc ion desc ip ion, inc easing he alidi y o he wo s a e model. C. Na u e o he hole The de e mina ion o he hole na u e in doped sys ems is no an easy ask. Two di e en elec onic a angemen s, neu- al 共undoped兲and ionic 共doped兲ones, based on di e en molecula o bi al basis se s, need o be compa ed. The ‘‘hole language,’’ usual in his ield, is no s aigh o wa d since i implici ly e e s o an ideal closed shell desc ip ion wi h d10 occupancy o Cu a oms. Mo eo e , he cha ge ans e p o- cesses, pa ially esponsible o hese co alen e ec s, a e unlike in he neu al and ionic sys ems. A new s a egy o o e come hese di icul ies has been de eloped. I consis s o building a di e en ial densi y ma- ix, in he common a omic basis se , which explici ly de- sc ibes he hole doping p ocess. I is a gene al p ocedu e, ha could be use ul o s udy o he phenomena, whe e s a es ela ed by hole doping a e in ol ed. The a omic-based den- si y ma ix o he neu al and ionic sys ems a e buil sepa- TABLE IV. Absolu e alues o he hopping in eg al, ob ained wi h DDCI-3 spaces and di e en MO se s. Clus e T iple ROHF 共eV兲IDDCI 共eV兲n°de (g,u) Cu2O70.553 0.575 255520, 255432 Cu2O11 0.532 0.544 526767, 526639 5165 J. Chem. Phys., Vol. 112, No. 11, 15 Ma ch 2000 Magne ic in e ac ion in La2⫺ x S x CuO4 Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.230.47 On: Thu, 27 Oc 2016 08:10:17