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
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[33] M. Roge , J.H. He he ing on, J.M. Del ieu, Re . Mod. Phys. 55, 1 (1983).
[34] M.C. C oss, D.S.Fishe , Re . Mod. Phys. 57, 881 (1985).
[35] R.L. Ma in, in Clus e s Models o Su ace and Bulk Phenomena, edi ed by G. Pacchioni
and P. Bagus (Plenum, New Yo k, 1992); R.L. Ma in , P.J. Hay J. Chem. Phys. 98, 8680
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(1996).
[36] J. Zaanen, G.A. Sawa zky, J.W. Allen, Phys. Re . Le . 55, 418 (1985).
[37] Fo me allic a oms, he ab ini io ela i is ic co e model po en ial p oposed by Ba andia ´an
(Z. Ba andia ´an, L. Seijo, Can. J. Chem. 70, 409 (1992)) has been used, whe e he Cu
alence elec ons a e desc ibed by a (9s6p6d)/[3s3p4d] basis se . Fo he oxygen a oms, an
all elec on basis se (10s5p) con ac ed o [3s2p] is employed (T.H. Dunning, J ., J. Chem.
Phys. 53, 2823 (1970); T.H. Dunning, J ., P.J. Hay, in Me hods o elec onic s uc u e
heo y, edi ed by H. F. Schae e III, Vol.2 (Plenum P ess 1977)).
[38] H.M. E jen, Phys. Re 39, 675 (1932).
[39] C.J. Calzado, J.P. Mal ieu, J. Cab e o, R. Caballol, submi ed.
[40] H. Schmid , Y. Ku amo o, Physica (Ams e dam) 167C, 263 (1990).
9
Cu OCu OO
O
O
O
O
mode 5
y
x
Cu OCu OO
O
O
O
O
mode 1
Cu OCu OO
O
O
O
O
mode 2
Cu OCu OO
O
O
O
O
mode 3
Cu OCu OO
O
O
O
O
z
mode 4
Figu e 5. Calzado and Mal ieu
16