Effective Hamiltonian for high-Tc ceramics, (…)CUmOn−x

ELSEVIER

Physica C 258 (1996) 30-40

PHYSICA ill5 --

Effective Hamiltonian for high-Tc ceramics, C.. • )CumOn - x George C. Asomba Department of Physics and Astronomy, University of Nigeria, NsukJul, Nigeria

Received 17 December 1993; revised manuscript received 30 June 1994

Abstract A new perspective on the Animalu nonlocal pairing model of high-Tc superconductivity is reported. currently stressing the possible interband (Cu3d-02p) charge transfer matrix (I). The mean-field Suhl-Matthias-Walker-type Hamiltonian obtained reveals hybrid and nonhybrid Cooper-pair wavefunctions. the latter manifesting anomalous splitting of quasiparticle energy. This result. presently interpreted as the Jahn-Teller (JT) effect. is briefly discussed in the context of data on the Cu-O bond length. In our brief Green's function analysis, an approximate t-dependence of the nonlocal pair potential (v,~. i ,;. j) and a semi-empirical expression for the JT stabilisation energy (Err), which we believe to be new. are presented. Numerical estimates of Err for YBa2CUJ06.9 compare well with photoemission data. Normal state energies obtained in the model are in excellent qualitative or quantitative agreement with the Mott-Hubbard picture. results from the Gutzwiller approximation of the Emery model and Ohkawa's breathing mode scheme for the copper ceramic oxides.

1. Introduction

P.w. Anderson first pointed out the proximity of high-Tc superconductors. ( ... )CumOn- x • to the Mott• Hubbard systems near the insulator-metal transition and proposed the resonance valence bond model based on an effective Hamiltonian. Baskaran et al. [1] later described these systems by the effective Hamiltonian H=-t L(I-niiT)C;;;,CJu(l-njii) (ij).(T

I +JL(S;, Sj - 4n;nj).

(1)

(ij)

where (ij) denotes nearest neighbour (nn) sites run• ning over the entire lattice. Ever since then, a number of authors [2-7] have employed several analytical ap• proaches to other interpretations of the Hamiltonian

(I), often as a t-J model. The roles of kinematic (", t) and exchange (f'V J) interactions have not been fully understood, however. and the emphasis on each varies. What appears to underline the different consid• erations is the uncertainty surrounding the anti ferro• magnetic (AFM) regime in the oxides: does AFM Of• dering exist only in the localised Mott-insulating (MI) state or does it extend to the itinerant superconducting phase where it may contribute to electron pairing as in the Kondo [8] system? Opinions and experiments differ on this vital issue. Kalinay [5], as some others, has nevertheless recognised the importance of kinematic interaction in electron pairing in the oxides, while emphasising that in spite of J making a subsidiary contribution to pairing. it (J) solely accounts for the d-wave symme• try revealed in the theory. However, the controversy [9] still rages over the true symmetry of pairing in

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G.C. AsombalPhysica C 258 (/996) 3CJ...40

high-Tc superconductors. One may, ipso facto, take the position favouring a freeze on J in the supercon• ducting state. The remnant of the Hamiltonian (I) is then of kinematic origin and includes immanently the restriction (localisation) on phase space available for electronic motion. The AFM ordering of this system consequently manifests only as a result of localisa• tion and strong correlations: as a precursor to the superconductivity. Recently, Animalu (AOEA) [10] proposed a non• local Cooper-pairing model of high-Tc superconduc• tivity in a two-band Hamiltonian formalism charac• terized by the breaking of the usual SU(2) symmetry via the mutation of the third (isospin) component of the underlying matrix, which manifests in the form

=

where T 1 - ''''IT)(,,,#I. Essential contents of the nonlocal effective model are the pseudopotential (fennionic projection operator) transfonnations: ct, C;(T

-+

et,

-+

Cil,

=(I -

nrr; )ctr, nrr;),

(3)

=Ciu( 1 -

for the electron/hole creation (annihilation) operator ct,( CilT) in the two-band Hubbard model [10] at half• filling (p, = 0), the latter of which is furthertruncated to simply yield

(4)

31

2. Effective kinematic model As implied above, we now expand Eq. (4). It yields

L +L

H=-

lijCtrCjlT

ijlT

lij(ctrCjucj;,CjiT

+ cfqcrr;ctrCjlT

iju

(5) The second tenn contains two types of tenn in the brackets. These are, in order, the two-band analogues of the so-called scattering and resonance broadening corrections to the Green's functions in the alloy anal• ogy of Hubbard 'III'. To the terms on the right-hand side (rhs) of Eq. (5), except the first, one applies Gor'kov-like factorization [14] such that when the c-number terms are ignored after adapting the paral• lel factorization [15] to the bosonic operators, QI ctr ctr, a2 =Crr;Cju, and a3 =cJ,;cjiT, arising from the third tenn in the bracket. one obtains. upon slightly adjusting the indices of t (i.e.• dropping some tenns):

=

H =-

L liiCtrcilT - L iu

tijctrCju

iju,i+ j

iu

-L

liJ(C I7i Cju)(cj;,CjiT}ctr Cfq

+ h.c.)

iju.i+j

+(

L

'Il(CiUCiU)ctrCj;,

iju,i+ j

where lij is the hopping integral. The index i(j) = d or p corresponds to the hybridized Cu 3d and 02p bands, thus incorporating Cu-O molecules in the su• perconductivity. Observe that Eq. (4) is the two-band version of the kinematic term of Baskaran et aI., in Eq. (4) of Rice et al. [ II] and in Refs. [4,12]. It is the objective of this paper to further inves• tigate the kinematic tenn and point out its possible new effects in the ceramic oxides, on the basis of Eq. ( 4). In a mean-field (MF) representation of the BCS type, following a procedure similar to the suggestion by AOEA but in the spirit of Hubbard 'III' [13], we presently stress the possible interband charge transfer (hopping) integral (tij(i :;. j) lii(jj»), opposite• spin pair correlations, and band hybridization.

»

-L

til (CiUCiu)(Cj;,CjU)ctrcfi;

+ h.c.).

iju,i+j

(6) The first term on the rhs of Eq. (6) is the intraband hopping energy, which we shall treat as not very differ• ent from the Fourier transform of the band kinetic en• ergy, the latter measured relative to the Fenni level EF. The second tenn is the band hybridization (mixing) term. The remaining tenns, deriving from anti parallel pairs (hybrid and nonhybrid) , are the interaction ener• gies which may lead to superconductivity. The above Eq. (6) arises from Eq. (5) if we guard against double counting in the summations. Our present decoupling scheme, which is familiar. is different from the Green's

32

G.c. AsombalPhysica C 258 (1996) 3D-40

function decoupling [16] which is not unique as the latter needed modification [12.13,17] to lead to the calculations of the Green's function corrections that yielded the correct strong coupling behaviour of the magnetic susceptibility in both the half-filled [ 12.17] and non-half-filled [12] limits of the Hubbard model. These results, which constitute figures of merit for such a decoupling and model. can be looked for in the present non local I-model for the ceramics. Gor'kov factorization (decoupling) [14], which was originally introduced in the singlet pairing BCS model of super• conductivity, was quite successful and has ever since been widely employed in the literature. It is therefore expected that the present application would lead to de• sirable new simplifications and valid physical results. In order to conveniently deal. later on, with the su• perconductivity. let us presently define. in Eq. (6). the potentials

\l;i = tii,

(7)

v,~

(8)

=-liiSi;.

and the corresponding Hermitian conjugates. where

(9) If \1;, and v,j are both binding, it then follows that. in the sense of the standard BCS gap expression for a single band. ( 10) all the terms on the rhs of Eq. (6), excepting the first two, are superconducting. especially if we set terms like ( 11)

This approximation may be justified by reasoning that. in a system where all electrons are in opposite• spin pairs, the expectation of the number of spins-up (spins-down), (n;;uCif»). is not different from the ex• pectation of the number of pairs. We may therefore. in the Bloch representation. write Eq. (6) as

H

= L E'k"Cj((TCiku "IT

+

L

",,,. ,.. ;

«\I;iSii

L

ijklT. i .. j

tijC~
+ V,jSJJ)c~TC~kl + h.c.)

+

L

«V;iSi;

ijkIT. i .. j

+ V;:Sjj)C~Tcj_k! + h.c.). (12)

where. in Eq. (6). the transformations have been made to arrive at Eq. (12): i=v"(-I).

tjj=Lexp(-ik.rJ )6jkt

(13)

j

eju

=L

exp( -ik . rj) . eftlT'

(14)

k

In the real oxide system. where each atomic site of Cu is coordinated to four nn 0 sites. a modification of Eqs. (13) and (14) to include the factor", N- t /2 • where N is the number of sites in the lattice. is easily made and one can account for possible multiple spins per atom, as may be considered. The hybrid pair wave_ function revealed in Eq. (l2L and which has earlier been emphasised [18], was not realised in the origi_ nal nonlocal pairing model [ 10]. but is indeed a kine_ matic resonance [13) effect resulting from nonlocal_ ity. Considering at present only one type of pair Con• duction.let us conveniently set in Eq. (12), consistent with our stress in this paper.

( 15) where ( 16)

which pertains to the nonhybrid pair wavefunction. and

.,

fJ. iik

=V;iSi; + \-Ii'Sjj.

associated with hybrid pair flow. One finally realises

H

=L

,kIT

6ikuC~ITCiklT

+ ~ L...t

•

-

L

;jktT ,i" j

+

+

(fJ.;ikC;kTCi-kl

t;jC~ITCjkIT

) + h.c..

( 17)

ijku. ioloj

The Hamiltonian (17) is not very different from the MF approximation of the Suhl-Matthias-Walker (SMW) model [19] referred to d and p bands. but now including band mixing. as in Refs. [10.20). and specifying the kinetic energies of electrons according to their spin orientations 0'(0'). where 0' =T for (j =1. and vice versa, as in Ref. [10]. In that case [19].

33

G.c. AsombalPhysica C 258 (1996) 30-40

both \1;; and V;j are binding. In the ceramic oxides, however, the potentials may not both be binding since, for fij(i j) attractive and binding. -tijSjj(i .;= j) may not be binding as Sjj is positive definite. For -f;,(i .;= j) binding, therefore, Eq. (16) reduces to

=

and one explicitly expresses this as

lk

(21)

= !«edkT - edk!) - (epk! - epkT»,

ii/c =! «epk! - epkT) - (edk! - edkT », -

Ak

I ='i(Addk -

(22)

-

(23)

Appk) ,

and the 4 x 4 matrices i'o, i's, and r are, respec• tively. the parity. chiral, and helicity operators [23]. Along with the four-component Nambu spinor wave• functions, (24)

(18)

This expression closely resembles the result given by AOEA [10] in his 'hadronics' formulation of the problem but appeared extrinsic to his MF approxima• tion in that theory. By writing in. in Eq. (18), the nonlocal pair po• tential V;j arising from phonon-aided and/or plasmon• assisted interband charge transfer and hybrid pair ex• pectation, one obtains (19) which has formal invariance with the BCS gap expres• sion (10). Contrastingly, however, the gap in the ith band now depends on the opposite-spin pair matrix in the jth band, and the binding potential is not the usual BCS localised (delta)-function approximation. In the sequel, we shall use a generalised [21] Nambu• Gor'kov [18,22] formalism for a brief investigation of the electronic states embodied in Eq. (17).

3. Generalised Nambu-Gor'kov Green's function analysis In the present paper we peep into the MI state by introducing an SMW-like symmetrisation of the MF Hamiltonian ( 17) in the generalised Nambu represen• tation. To this end, one equates to zero the symmetric (in energy) part of Eq. (17) and, in the absence of band hybridisation, takes only the antisymmetric por• tion of the Hamiltonian operator as approximating the MI state: (20) in which

and the Hermitian conjugate, Eqs. (20)-(24) lead to the Green's function, in its simplest form:

By the condition .

-

2

-2

-2 2

-2 -2

(26)

«IWn-tk) -l1k-Ak) -411kAk=O,

one arrives at the electronic energy spectrum:

Et± =~k ± (ilk -

2

+ A-2k ± 2ii/c A- k) 1{2 .

(27)

For exclusive d-subband ordering via antiferromag• netism, we may employ the energy conditions [ 10] : ep1«T = edkT

(28)

= -edk! = -U/2,

addk = -ll ppk

(29)

= -t,

where U is the Hubbard on-site potential. These lead. in Eqs. (21)-(23), to lk

= Tik = -U/4,

~k = -to

(30)

Finally, one realises, from Eq. (27), the MI state en• ergy spectra (31) or, in terms of some exchange constant I',

Et± =':fl';

E"k± = -U/2 ± I',

I'

= -to

Evidently. four energy branches manifest as though by Zeeman splitting of an energy band. We now specu• late that the energy spectrum (31) may be related to the charge transfer [24] regime. It will be recalled,

G.c. AsombalPhys;ca C 258 (/996) 30-40

34

however. that Walker and Ruijgrok [25] had. for in• teracting electrons in metals in the absence of an ex• ternal magnetic field. provided for a Zeeman interac• tion of electron spins with a fictitious magnetic field ( h) of wave vector q. We may. ordinarily. visualise Eq. (31) as characterising an anti ferromagnet with i' being the super-exchange integral. which is now inde• pendent of the Hubbard V-term since it (1') is solely determined by the interband charge transfer integral as though synonymous with it. Depending on the intrin• sic sign of t. observe that ferromagnetism may also be realised from Eq. (31) if i' is magnetic exchange. If. in Eq. (26). we rather take the 'perfect-square approximation', then .

-

2

-2

-2

(IWII-[k) -TJk-Ak~O.

(32)

This instead leads to two energy branches.

Ea ~ -V/4 ± ~(V2

+ 16t2 )1/2,

(33)

which. except for the sign (-) of the first term on the rhs which is positive by a slightly different rationali• sation. correspond to half the energy values obtained [ 10.26] for the ground state of the unhybridised two• band Hubbard model at half-filling and where the Hub• bard V-term is explicitly operative and accounted for a priori. This 'halving' of the electronic energy can be traced to the SMW-type symmetrisation, which prob• lem appears inherent in the procedure and will be dis• cussed in more detail in a future publication. Note, further, that in the form (33), for strong intraband (or on-site) correlation (t ~ V),

Ek+

~

-i.

2t 2

i=-U'

Ek- ~

-6

-4

-2

2

4

6

Fig. I. Excitation spectrum. Eq. (39). for Ek+ ~ O. ~k - l ,j == I. in units of 1 for convenience. for ( ... )CUI/lO._, superconductors showing the Jahn-Teller splitting of energy arising from interband charge transfer. (Points A and B labet new excited ground state energies. in place of possible X or Y a....~uming no splitting. Note that a single-band ground state at Y would only lead to a low Te • while at X it would lead to higher T, with the system mOre indisposed to pair-breaking influences such a.~ impurity scattering. This deduction is. essentially. based on hiiher expense in energy rather than on rigidity arising from pairing symmetry as by An• derson in Ref. 191. Observe that if e~(: 21) gets larger. the points A and B get higher up and Te increa.~es. making 1 the critical parameter for a possible room-temperature transition. in complete accord with our stress in the present I-model. The pa• rameter 21. which is the energy difference between the ground and first excited states of Cu 3d-O 2p hole (electron) ~pin-~inglets in the recent clu~ter model calculation of 21 vs. Tc by Ohta et al. I Mod. Phys. Len. B 5 (1991) 13151. here plays a role similar to that in the cluster model).

er -

(35)

-V/2 + J,

(34)

which illustrates the effect of a weak interband charge transfer (t ~ tii) as in most conventional systems ex• hibiting Heisenberg AFM ordering. and. as can be ob• served, the 'halving' subsists. We here suggest that the essential effect of doping is to alter the band energy conditions (28)-(29) by either a shift of the Fermi level or energy band(s), so that AFM and supercon• ducting phases are mutually precursor states of the ceramics. predicated on the doping level (x). The superconductivity may emanate from Eq. (17), as suggested above, under the band overlap and de• generacy condition

We now briefly investigate this possible superconduct_ ing state via the symmetric Hamiltonian, but including, as presaged by hybrid-pair expectation in Eq. (17), the hybridisation term, thus

(36) under condition (35). The corresponding Green's function is. in its simplified form: G,,(k, iw,,)

=(iw" + Atrl + ekr3 x(X-2ektrs)D~I,

where A2 - ek2 - t 2 • X -- ("tW" )2 - Uk

tr3Ys) (37)

G.c. Asnmba/Physica C 258 (/996) 30-40

(38) The condition

leads to the quasiparticle energy spectrum

EZ± =AZ + (ek ± 1)2.

(39)

This spectrum is plotted in Fig. I for Ek± ~ 0, 11k ,..., I, in units of I. It reveals an energy splitting of the other• wise single band BCS quasiparticle energy spectrum. This is rather unusual in conventional superconduc• tors. Such splitting is, however, well known [27] in electronically degenerate non-superconducting states of nonlinear molecular systems on account of electron• ~attice motion. The present anomalous energy splitting IS interpreted in a similar context of the so-called Jahn• Teller (JT) effect [28] arising from charge trans• fer, and observed as structural phase transitions in all known high-Tc superconductors [29]. Jahn and Teller were the first to point out that if a localised-orbital de• generacy is present and there is no other perturbation to remove this orbital situation, then there is a sponta• neous distortion of the interstice to some lower point symmetry that removes (reduces) the degeneracy. For a large concentration of JT ions (such as Cu 2+), elastic coupling between the distorted sites induces a cooperative distortion of the crystal to a lower space group. In this connection, van Vleck pointed out [30] that the normal vibrational modes that split the E I . g e ectronJc state are themselves two-fold degenerate: the one usually giving interstice tetragonal distortion, the other an orthorhombic distortion. These are indeed ~eali~ed [29] in the ceramic oxides by doping and, as IS eVIdent, as a charge transfer effect.

4, Comparison with experiment and other MF models To enable a simple assessment of the t-model and the underlying pairing approximation, let us, for the superconducting phase, approximately calculate theJT stabilisation energy. In a semi- empirical derivation, based on Eq. (39) and Fig. I, one obtains EJT simply as (40)

35

We believe that Eq. (40) is new. Observe further that t~is energy (EJT) vanishes, as expected, in conven• tional (lij ,..., 0, i oF j) superconductors but not in normal charge-transfer (Iii oF 0) materials. Numerical estimates of the stabilisation energy for YBa2Cu307-x (x=O.I, z(=2(n-x)/m) =4.6, Tc~88K, 1= 0.92eV (see Fig. 2) and 211/(k BTc ) = 3.5-8.0) from Eq. (40) are 0.91 eV and 0.89 eV, corresponding, re• sp~ctively, to the extremal gap-to-critical temperature ratIos [31] for the ceramics. The EJT values should be comparable to the bandwidth of Cu z + so as to allow the observed phase transitions, should we recall the guiding idea of Bednorz and Muller [29]. In the ex• band cited state of Cu 3d in the ceramics, the Ix2 is quoted as having a width of,..., 0.5 - 0.8 eV. This agrees well with the findings of angular-resolved pho• toemission [32], as, reasonably, with our present esti• mates (Err) based on the energy of itinerant charges and the superconductivity. An approximate z-dependence of the charge trans• fer integral is given in Fig. 2, and a rough evalua• tion of the hybrid-pair correlation function reveals the as having a three-halves power binding potential law I-dependence (Fig. 3). These simple results are of kinematic origin, as we originally set out to investi• gate, but were hitherto revealed by neither the earlier JT models [29,33,34] nor the I-J models [1-7] of high- Tc superconductivity. For the normal phase, we presently compare the energy results, Eqs. (33) and (34), arising in our 'pairing' model, with the antibonding conduction band energy in the d-p breathing mode scheme for the oxide ceramics of Ohkawa [35], given as

y;

V;i

Ec(k)

=!(eb+ 16t2 )1/2,

(41)

for arbitrary I and ea. In the above, in line with the assumption implicit in our model, we have set the planar structure (form) factor sin 2 (k x a/2)

+ sin2 (k ya/2) ~ I,

otherwise, appearing in the original expression (35). Trivially, Eq. (41) may be approximated as 41 2 Ec(k) ~ ~ea + -, ea

(42)

for I*.I « I .. which is essentially the same as Eq. (3) of that paper [35] for this limit, but at present for

36

C.c. A.wmba/Physica C 258 (/996) 30-40

3

.

>

.. 2

...

1"\ I~

I~

1';-

Ie I

10

20

30

40

so

60

z*

70

80

90

100

Fig. 2. Variation of interband charge transfer energy (I) with z·(= z/N(O). where z(= 2(n - x)/m) is the 'effective valence' of Cu in ( ... )CUIllOn- r compounds. (As for To vs. z in Ref. riO I, no plateau in t is realised within the present concept of z. The value of t for YBa2Cul06.lJ (z =4.6, bandstructure N(O) =0.40 (states/eV,Cu ion) by J. Yu et al. I Phys. Lett. A 122 (1987) 1981) is indicated by lhe broken line).

a negligible (or unitary) form factor. If, further, one identifies £G in that model with -U in our model (the parameter I perfonning similar roles in both cases). thecomparisonofEq. (41) withEq. (33) andofEq. (42) with Eq. (34) is straightforward. Both models. under these conditions, give similar Matt-Hubbard pictures, though the lower (bonding) Hubbard band is absent in Ohkawa's result and, as a consequence, the expected anti ferromagnetism of the U » I regime is not explicitly reproduced by that model. Ec(k) of Ref. [35] appears to reveal only the additive magnetic interaction energy, having lost the apparently isolated band reduced energy ~ -U/4. In spite of our not seek• ing to arrive at the conclusions reached by Ohkawa. it is evident from the above comparison that some un• derlying similarities do exist between the two models. Our MF model, however. appears better in its ability to predict the observed AFM order of the U » I regime in addition to the charge transfer state (U « 1), pos• sibly represented by Eq. (31), as envisaged [36] in

the oxide ceramics. As the Gutzwiller wave function [37] has the ad• vantage of taking into account the many-body charac_ ter of the problem, it is often believed that the corre_ sponding MF theory, usually arising from employing the Gutzwiller approximation (GA) [38], is more ap_ propriate than other MF models, such as our pairing approximation, for the description of strongly corre• lated 3d-electrons in the copper oxides. We, therefore, now place our BCS pairing approximation energy re• sults for the normal phase side by side with the corre• sponding results from the GA treatment of the Emery model by Hayn and Schumann (39). In order to ef• fect the comparison. we first point out the correspon• dence of E•. 2 ,11,1 and 4qVk in that model [39] with E/c:b U/2, I and C, respectively, in our model, where C(= 4qV/c) is the constant coefficient of 12 , as Vic is [39] only weakly dependent on the crystal vector k. In Eq. (5b) of Ref. [39]. therefore, one sets

G.c. A.wmbalPhysica C 258 (/996) 30-40 t(eV)

2

.,.

3

4

5

-2

.,

!!

:;:

....;>....

..

-4

*

;>

Fig. 3. Interhand charge tmnsfer energy (I) dependence of density of states-nonnalised pair intemction potential V·(: V;j/N(O), N(O) is the DOS ofCu) in ( .. . )CUI/IO.-x su• perconductors in the two-band model. (It ha~ a resemblance to the attractive portion of the cu-dependent effective pair interaction (upon change of origin) by Cui and Tsai I Physica B 169 ( 1991 ) 5771 for these systems. However, no asymptotic behaviour is pre• dicted here.)

(43)

(44) Formal agreement is again found between Eqs. (33) and (34) of our model and Eqs. (43) and (44), the lat• ter arising from the GA approach. We give in Fig. 4 the plots of the energy spectra, Eqs. (33), (43) and (44), for quantitative comparison. We observe that, while the Ek+ (Ek-) curves (for U » t) are asymptotic at large positive (negative) U values, the E2(E 1 ) curves are not. For negative C, the E2 (EI) curves exist only in the large positive or negative U, corresponding to the Mott-Hubbard regime; otherwise these curves show

37

little sensitivity to variations in C, quite in accord with Ref. [39]. There appears to be better agreement of the two models in the small ±U ranges, which should correspond to the broad-bandwidth regime where the magnitudes of the energies are more comparable. In spite of these slight differences, there is fair overall qualitative agreement between the two models com• pared here. In the U » t regime of Eq. (33) (bro• ken curves) there is excellent quantitative agreement between the GA of the Emery model and the present pairing approximation following Gor'kov. In addition, the prediction of antiferromagnetism, missed out in the Ohkawa model, is here recovered in the GA ap• proach, as by our pairing MF method. In assessing the comparisons made above, recall the 'halving' of energy, first pointed out in Section 3 of this paper. Although further results (for example, for thermo• dynamic properties) are needed to validate the present pairing approximate model, it is evident from the qual• itative and quantitative results and comparisons of this Section that the physics claimed in the present paper is not an artificial effect of the decoupling machinery. The t-model appears to be a truly realistic approximate model of high-Tc ceramic oxides, as it has indicated the potential to account for a large number of the un• usual features of these new compounds. Perhaps the greatest strength of the present model is that it needs no exotic analytic tools, as the simple approaches used in understanding the conventional DCS model are eas• ily adaptable to it.

5. Discussion, summary and conclusion We find it of further theoretical and experimental interest to discuss the Jahn-Teller effect in the copper oxides. This result, Eq. (39) and Fig. 1, we easily re• gard as the single most important revelation of the the• ory. This phenomenon has resulted from charge trans• fer. Charge transfer mechanisms, leading to charge compensating positive ion vacancies and vacancy ag• gregates, have been adduced [30,40] to explain the character of d 9 IT ions, isoelectronic with Cu 2+, in an octahedral complex or in irradiated alloy systems. In ( .. . )CumO.- x, the effect of charge transfer from (to) the Cu(2)-Oplanes to (from) the charge reser• voir layers (Cu( 1)0, BaO, BiO, ...) has been ex• amined [41] in YBa2Cu30J-x by analysis of neu-

a.c. AStJmbalPhysica C 258 (/996) 30-40

38

4.

C!l

-8·

-6·

_1-- -t-

-4.

6. I!J

U It unit~1

Pre'>!!nt Theor y: - - EK~ (arbitrary t and U I,

- - - - Ek~ (I:»

UI;

GA of Emery Mod,l, Ref,[ 391: E.:

E1 :

II

&

!

fJ

-4.

C (=4q Vk I:

0 1

o

•

4

•

c

8

Cl

[;J

-16

-e.

Fig. 4. Approximate energy spectra, Eq. (33), of the nonnal cuprate. for arbitrary CT energy (t) and for I » U. obtained in the present pailing '-model. compared with the results, Eqs. (43) and (44), of Ref. [391 in the Gutzwiller approximation (GA) of the Emery model. for different values of C{= 4qVd of that theory.

tron diffraction measurements of changes in the Cu-O bond lengths. Changes in Tc with the parameter x re• flected changes in Cu(2)-O( 1) distances, O( 1) being the apical ox.ygen. The Brown-Shannon [42] valence charge for Cu(2) has been calculated and found [41 ] to vary in the same manner as does Tc with x. Neu• tron crystal field spectroscopy was employed by Mesot et al. [43] to measure charge transfer energy spectra in ErBa2Cu306+y, and by using structural quantities from powder diffraction and a new parametrization model, taking account of 02p and rare-earth 4f hy• bridization, they calculated quantities of charges trans• ferred as a function of y in reasonable agreement with the crystallographic data of Cava et al. [44]. Bianconi

arrived at stereochemistry-based conclusions on the effects of ex.citonic charge transfer in YBa2Cu307 and BiCaSr2Cu20g using X-ray absorption spectroscopy (XAS) and extended X-ray absorption fine structure (EXAFS) measurements. In the former compound, the c-axis contraction as Tc goes from '" 50 K to ....., 90K with doping was interpreted as an increase in the so-called pseudo-JT effect [45], and, in the latter, measured Cu-O bond distances were in conformity with the reconstructed CU02 plane by Calestani et al, [46]. These changes in bond lengths, we suggest, give rise at some critical temperature, Tt, to observable co• operative JT distortion. This is usually a first-order change but one that, in the oxides, unusually coin_

a.c. Mamba/Physica cides with a nonnal-to-superconductor transition at a temperature Tc ( '" ~), in what may be construed as a cooperative or 'resonance' (13] effect. In summary, the basic picture of the present paper is that of a phase change in an otherwise localised-electron system, ( ... )Cu",On-x, char• acteri zed by ( I, x, T, ... ) , across some critical ity (te, Xc, Te ,·· '), to a collective-electron regime de• scribed by (I', x' , T', ... ) with the interband charge transfer matrix (t) as a crucial parameter. This is rem• iniscent of the idea of Goodenough (47], and many others. with regard to localised-versus-itinerant elec• trons in solids. What is interesting is that the present picture emerges from a Suhl-Matthias-Walker-type Hamiltonian of the BCS fonn, isomorphic with a dras• tically reduced two-band Hubbard (or I-J) model, while a simple matrix Green's function analysis has been conveniently and successfully invoked. We are then led to believe that the novel interpretation of the Hamiltonian (4) as an effective nonlocal I-model may have serious theoretical and experimental impli• cations for (and beyond) high-Tc superconductivity and its mechanism.

Acknowledgements I am indebted to the members of the Solid State Group, especially C.M.I. Okoye and C.N. Animalu, for useful discussions and encouragement. I am also grateful to M. Gulacsi, Y. Okabe, R. Gonczarek, J.P. Carbone. FJ. Ohkawa, Y. Ohta and others, too numer• ous to list here. for useful correspondence and encour• agement. I acknowledge A.O.E. Animalu for availing me of his paper, Ref. [10], and for discussions of and criticisms on this work as it progressed.

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