What can be learned about high-tc from local density theory

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Physics C 153-155 (1988) 1636-1641 North-Holland, Amsterdam

W H A T C A N B E L E A R N E D A B O U T H I G H Tc F R O M L O C A L D E N S I T Y T H E O R Y ?

J. Z A A N E N , O.JEPSEN, O . G U N N A R S S O N , A.T.PAXTON ,O.K.ANDERSEN

Max-Planck-lnstitut ffir

FestkSrperforschung,

D-7000 Stuttgart 80, F.R.G.

and A.SVANE

Institute of Physics, University of Aarhus, DK-8000 Aarhus C, Denmark

The significance of local density band structure results for high Tc compounds is critically discussed. It is pointed out that straightforward application of this method can be misleading because of the correlated nature of these materials. However, with LDA numbers can be derived for the parameters appearing in the models in which correlation is treated ezplicitely. In this way we arrive at the conclusion that despite a large U the high Tc materials should be viewed as itinerant materials. Turning to ground state properties, we show that the LDA /ails to describe the antiferromagnetism o/La2Cu04 and it is pointed out that this may be cured by the inclusion of self interaction corrections. Finally, using a proposed ozygen defect structure for YBa2CusOT-z and a simple tight binding model based on band structure calculations, we show that jumps and plateauz in the hole counts in the planes and chains occur as a f~nction of x. These features are perfectly correlated with the occurrence of the antiferromagnetic, as well as of the 60K and 90K superconducting phases.

1. I N T R O D U C T I O N

A fairly common a t t i t u d e among theorists is to re- ject results obtained for the high Tc superconducting oxides (HTSO) from b a n d s t r u c t u r e calculations, based on the local a p p r o x i m a t i o n to density functional theory (LDA). This is not groundless : it is expected t h a t the electronic structure of these Cu-oxides should bear a resemblance to t h a t of other 3d-oxides like NiO, CoO, etc. The LDA is notoriously in error for these materials. According to the LDA calculations of Terakura et al [1], the b a n d gap would be small (0.SeV) or non- existent, while these systems are in reality insulators with gap magnitudes ranging up to 4eV in N i O [2]. It is well established t h a t this discrepancy is due to the large Coulomb interactions between the 3d-electrons. As explained by Mott and H u b b a r d [3], if the energy cost (U) of the charge-fluctuation 3d'~+3d '~ --* 3d n+l + 3d '~-~ exceeds the b a n d w i d t h ( W ) , the single particle band picture breaks down completely. Nevertheless, in this p a p e r we shall indicate t h a t density functional calculations can be usefull in several respects.

Based on photoemission data, it has been shown recently ([4,5] and refs therein) t h a t the M o t t - H u b b a r d c o n c e p t indeed applies to the global electronic structure of these materials. In the simple Mott- H u b b a r d picture only the d-states are considered. In the real materials, however, also other charge degrees of freedom are present like the O2p- a n d transition metal 4s- states. In practice these can be quite i m p o r t a n t , also if one considers low energy properties. Experience

shows, t h a t LDA calculations are valuable tools in un- ravelling these complexities, because they can be used to derive numbers for the different p a r a m e t e r s enter- ing in the models used to interpret the photoemission data. In section 2 we will outline the picture as it has emerged for the 3d-oxides and illustrate the accuracy of t h e LDA-parameters. Applying this to the HTSO we show t h a t , despite the largeness of U, the d-electrons are p r o b a b l y not localized.

It is often thought t h a t despite the problems with the excitation spectra, LDA calculations give mean- ingfull results for ground state properties. In section 3 we address this issue by considering the antiferromagnetism of La2Cu04 and we will show t h a t according to the LDA this system would be far from being magnetic. One way o f improving the density functional is by including self-interaction corrections (SIC). Accord- ing to a realistic model calculation we find the promis- ing result t h a t upon including SIC an antiferromagnetic ground state is found with a spin polarization of the right order of magnitude.

Finally, in section 4 we show, using a proposed oxygen defect structure for YBa2Cu~07-~ and a simple tight binding model based on the LDA-band structure, t h a t sudden j u m p s in the hole-counts in the C u - O planes and chains occur as a function of non- stoichiom- etry, z. These j u m p s are perfectly correlated with the occurrence of the antiferromagnetic, as well as 6 0 K and 90K superconducting phases in this system. These findings s u p p o r t the view t h a t the hole count is the controlling p a r a m e t e r for the superconductivity.

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J. Zaanen et aL / High T c from local density theory 1637

2. HOW LOCALIZED A R E T H E d E L E C T R O N S IN T H E HTSO ?

With the combination photoemission - inverse photoemission one measures the probability of removing or adding a single electron at a given energy. In Fig.lb the result for N i O is shown [2] and compared with the LDA result [1] ( Fig.lc ). It is seen that these experimental data are not reproduced by the LDA, but also the relationship with the simple Mort-Hubbard picture is not clear. Instead, the spectral density can be described in fair detail (see Fig.lb [6]) using the Anderson impurity Hamiltonian [7] H = ~r0 + H1, -- .o4oo..o + + k~'t.o" w'~er Z U(ijlra)d]djd]dm, i f l m H1 = Z Vk'*m(dt=ck'== + h.c.). (1) knrr~ar

In H0 the electronic structure of the isolated d-shell, including crystal field splittings and the quasi-atomic Coulomb interactions, as well as the bands arising from the non-d states (first term) are gathered. H1 contains the hybridization between the d-states and the non-d bands. Upon switching off the hybridization, Fig.la is obtained [6]. The relevant non-d bands are the 02p valence- and Ni4s conduction bands, separated by a large gap. The location of the 3d-Hubbard bands with respect to these 'normal' bands is apparently different from what has been hypothesized i n the past [8 I. If doped, the electrons will go into the d V uPl~er Hubbard (UH) band. However, the multiplet-split d ~ lower Hub- bard (LH) band is below the b o t t o m of the 02p band, and it is far from obvious [9], that the holes can be described with an effective single band Hubbard Hamil- tonian. The gap-magnitude is given by the energy cost

(A)

of the charge-transfer fluctuation 3d '~ ~ 3d n-÷l + L , where L denotes a hole in the ligand band. Concern- ing the HTSO, if U < A the holes are in the LH band and (for instance) Anderson's resonating valence bond picture [10] would directly apply. If U > A, the holes would have 02p character, as proposed by Emery [11] and others [12]. Finally, if A becomes small compared to the hybridization, the d - electrons delocal- ize, independent of the magnitude of U [13] (mixed- valency) , and weak couph~ng ideas gain credibility.

From the preceding discussion it is obvious that knowledge of these parameters is an important ingre- dient in the search for the high T~ mechanism and the LDA has proved to be useful in this respect for other materials. In order to determine A or U one sets to zero the hopping between the relevant d-orbitals on a particular atom and the remainder of the crystal. By varying the number of d- electrons one obtains the desired quantities [14]. Applying essentially this pro- cedure to NiO, Norman and Freeman [15] obtained U = 7.9eV and A = 4.1eV. From the ~nterpreta- tion of the photoemission data ( Fig.la, b ) one finds U ~- 7.0eV and A _ 4.0eV [6]. Further, we notice that

~he A determined by 'excited state' LDA is very close to the A deduced from the ground state calculation ( Fig.lc ). Very recently it was shown by Gunnars- son et al [16], that also relevant information about the hybridization matrix elements can be extracted from LDA calculations in the framework of the linear muiBn tin orbital (LMTO) method [17]. Using this together with LDA values for A and U plus atomic F2 and F4 integrals, first principle results for the photoemission spectrum of M n in CdTe were obtained [16], being in surprisingly good agreement with experiment.

In Fig.2 we show the orbital composition of the LDA bands of paramagnetic La2Cu04 in the vicinity of the Fermi energy (E~), obtained from a ground state calculation using the L M T O method [17]. The most pro- minent feature is the band crossing E~t which is almost of pure d==_~= and in-plane oxygen ( O ) character. Mattheiss pointed out that this band is well described by a simple tight binding model where only the in plane z- or y 02p orbitals and the d==_y= Cu orbitals are taken into account [18]. This'band is unusually dis- persive ( see for instance Fig.lc ) and tight-binding fits to the L A P W bands [19], as well as first principle tight binding LMTO calculations [20] show that this is due to A ~ 0, while the transfer integrals are of essentially the same order of magnitude ( V _ 1.5 - 2.eV) as in (for instance) NiO.

' " ' a NiO - - U - - 3dg--3dTI ,,, 0-2p

i / / I

I

Ni-4s

b PES .~

I-PES

~,, L,/ ~. . ~P : i " ~ $ . Ni3d C

O-2p egt t2gi

2g Ni-4S

-I0 0 10

E N E R G Y A B O V E E F

leVl

F I G U R E 1

Spectral densities/or N i O . a: Theory ]or fully local- ized d - electrons, b: The experimental (inverse) pho- toemission spectra [~] compared with a result obtained from the Anderson impurity model [6]. c: LDA densi-

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1638 J. Zaanen et al. / High T c from local density theory

o.sJcu :,2.& l /c, i3z2., iK/co ,v\ .

0.0 i,i.i I.O, i " 05 o.o -o,5 - i,o

[1~'0] r [lOO] [1t01 F' [100] [lio] r" [lOO]

F I G U R E 2

LDA bands of La2CuO~ in the vicinity of EF in the folded (AF) Brillouin zone. The thickness of the lines

indicates the orbital character of the bands.

Using a 4[La2Cu04] supereeU and the L M T O - A S A m e t h o d [17] we calculated A and U in the LDA. For A we use the transition state technique [21]. Thus we have E(dX°L) - E ( d 9) ~_ e~ - ¢p, where ed and ep re- fer to the energies of the Cu 3d and O 2p orthogonal L M T O ' s , calculated for the 3d-occupancy 9.5. F r o m these calculations we deduced A = 0 -- leV, depend- ing on computational details. We again find, t h a t the ground- and excited state LDA A ' s are very similar.

The value of U was obtained from U = $ = E [ n d ] / $ n ~ ,

where the energy E is the energy of a n e u t r ~ syster~ with nd 3d-electrons. Note t h a t since the hopping from the 3d state is included explicitely in the Hamiltonian, the rather i m p o r t a n t renormalization effects on U from this hopping should not be included in the calculation o£ U. From the total energy we deduced U = 8eV.

In the valence b a n d photoemission spectra of the HTSO one always observes a feature located at 9 - 11 eV below E F [22,23,24]. According to the b a n d calculations there is no density of states in this region and this structure can be identified with the multiplets of the d 8 LH band [6]. Additional evidence is s u p p o r t e d by resonance photoemission where one finds the typ- ical and well understood resonance behaviour of a d s satellite [25]. This confirms t h a t U is as large as sug- gested by our calculation and it shows t h a t the LH b a n d is well removed from E F . Also our small A is confirmed by experiment. Several groups have ana- lyzed the core-XPS spectra, where the d - hole count in the ground state is measured, and they arrive at < na >-~ 0.5 [26], indicative of A < V [27]. It has also been argued t h a t the anomalous high energy scale for the (two dimensional) antiferromagnetism can be in- terpreted as an indication for small A / V [22]. Finally, the most direct evidence in favour of small A / V comes from inverse photoemission. In localized materials like NiO (Fig.lb) one finds always a distinct feature above E r corresponding to the narrow d '=+1 UH band. In the d a t a of the HTSO such a feature seems to be ab- sent and instead one finds instead a rather b r o a d b a n d around E F [23,28].

If our claim, i.e. large U/V and small A / V , turns out to be true this can have i m p o r t a n t consequences with respect to the n a t u r e of the superconductivity. At least on the level of the ' b a r e ' d-electrons one has to ac-

count for the charge degrees of freedom. However, one can also question the validity of the localized picture on the quasi-particle level. According to the T - m a t r i x [29] or Gutzwiller [30] pictures one expects an electronic mass renormalization of only -~ 1.5, if < n d > " 0.5. Of course, complications arise because of the antiferromagnetism found when the d==_~2-band is half filled, a n d the insulating n a t u r e of the antiferromagnets La2Cu04 and YBa2Cus06 is often p u t forward as evidence for having a local gap. However, one should keep in mind that the gap m a g n i t u d e (1.5 - 2.eV) is in fact quite small compared to the overall dispersional width ( ~ lO.eV, see Eq.(4)) of the plane band. One way to obtain more clarity about the locality of the gap could be by considering the electronic structure of the doped systems. Both for large and small A / V it is expected t h a t states, induced by the excess of holes, are found in the charge excitation gap. In the localized regime the energy scale of the gap itself (A) is much larger t h a n the interactions giving rise to self- localization (V2/A, V 2 / ( U - A) [9]). These hole states will therefore be in the neighborhood of the valence band edge (Fig.3b). However, if A < V these energy scales cannot be separated and therefore the new hole states can be located everywhere in the gap (Fig.3a). In this l a t t e r situation, Schrieffer's spin-bag concept [31] seems to be a more n a t u r a l 'generic' picture. We notice t h a t the EELS d a t a of Nficker et al [23] seem to favour this uniform gap filling.

3. LDA AND A N T I F E R R O M A G N E T I S M IN La2Ou04.

At half fiMng, the p l a n e band,has a near nesting in- stability for ~ ' = [ l l 0 ] ( F i g . 2 and Eq. (4)). The associ- ated breathing mode was i n s t r u m e n t a l in conventional t r a n s p o r t theory based on the electron-phonon interaction and the LDA for.La2_~SruCu04132 ]. This theory could account for Tc - 4 0 K , b u t in order to explain the t e m p e r a t u r e dependence of the resistivity above To, an unreasonable short mean free p a t h had to be assumed [33]. For the superconductivity of YBa2CusOz the theory failed completely [34].

a: Z~ < V ~ A P b : A >>V Ev d 10 I EF ENERGY A B O V E E F F I G U R E 3

Artist's impression of the filling of the gap by dopant induced states in (a) the mized.valent and (b) the lo-

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J. Zaanen et al. / High T c from local density theory 1639

The p r o p e r ground state of La~Ou04 is antiferro- magnetic with the above mentioned ~'-vector and a moment of ~_ 0.4pB. In the past, the spin-unrestricted LDA (LSDA) has been successful in yielding realistic magnetic moments for numerous transition-metal systems. However, the LSDA solution for L a ~ C u 0 4 is not spin polarized [20,35]. T h e size of the discrepancy is best seen from Fig.4 where we show the un- enhanced Xo 1 -- 2 # s H / m L D A and enhanced X -1 - 2 p B H / m L S D A inverse susceptibilties calculated by ap- plying an external staggered magnetic field H on the Cu sites. The sublattice magnetizations mLDA and mLgDA were obtained, respectively, without and with including the internal exchange-correlation field. From the susceptibilit~ curves we find the effective Stoner pa- rameter I = X~ TM - X -~. This is seen to be at least a factor 5 to small to account for the moment, although its m a g n i t u d e (-~ 0 . 7 - 1 . 0 ) is of the same order as com- monly accepted for Cu and N i metals. In these LMTO calculations the s - , p - , a n d d - o r b i t a l s were included on all atoms, as well as the La f-orbitals, and ex- treme care was taken in performing the Brillioun zone integration ( t e t r a h e d r o n m e t h o d with 300 irreducible k points). One possible source of error could be the spherically averaging of the spin density, b u t this can account for a factor of 2 at most.

In the LSDA an electron has an unphysical interaction with itself which may be explicitly subtracted. This gives rise to the so-called SIC LDA [36]. For m a n y properties these two density functionals have a com- parable accuracy [37]. In an application to the one- dimensional H u b b a r d model it was, however, found that the SIC gives a substantially better moment t h a n the LSDA [38]. It is therefore interesting to see if a similar improvement is obtained also for La2CuO4.

A full ab initio SIC calculation is difllcnlt to perform, since the one-particle solutions are not automatically orthogonal. To predict the outcome of such a calculation, we have studied a simple model of the C u 0 2 planes, including the Cu z 2 - y 2 o r b i t a l as well as the O 2p orbital pointing towards the nearest Cu neighbors. Mattheiss' value for the hopping integral V --- 1 . 6 e V between these orbitals was used. Each orbital was fur- thermore assumed to have an on-site Coulomb interaction. We used 'spectroscopic' values for these interac-

350 , , , , , 3 0 C

x

~

'-" 25o

_{x TT}

. i - ra 2 0 0 ~5 O0 i t t t

,.,.~ 0.2 0.3 0.,~ 0.5 0.6

m [lib/La2CuO 4] F I G U R E 4

Inverse ~taggered ~usceptibilitie8 vs. 8ublattice magne- tization according to LDA calculation8 for La2Cu04.

tions, i.e. 8eV (Cu) and 6 e V (O) [39]. The exchange: correlation energy was described in an analogous way to our earlier work [38], taking into account the C u and O charges and moments. The p a r a m e t e r s in the exchange-correlation energy expression were deduced from atomic and b a n d structure calculations to simu- late an ab initio SIC calculation as closely as possible. The orthogonality of the one-particle solutions was en- forced explicitly.

For this model SIC calculation We find an antiferromagnetic solution, with the moment 0.3 # ~ and with Cu hole count 0.5. In the LSDA one finds, as in the full L M T O calculation, t h a t the moment is very close to zero, while the small deviation from zero is due to the two-dimensional n a t u r e o f , t h e model. We con- clude t h a t the description of the antiferromagnetism in La~CuO4 is greatly improved in the SIC approximation compared with the LSDA approximation. 4. OXYGEN VACANCY S T R U C T U R E AND T H E

P H A S E D I A G R A M O F Y B a 2 C u s 0 7 - = .

An i m p o r t a n t theme in the theories about high Tc is the hole concentration in the perovskite planes. This is based on the phase d i a g r a m of La2_~SrwCu04. For y = 0 the df2_y2 b a n d is half filled, giving rise to antiferromagnetism. For y > 0 holes are introduced, which destroy the antiferromagnetic state and stabilize the superconducting state. It has even been claimed that Tc is directly proportional to the hole count in these materials [40]. In this light, the phase d i a g r a m of YBa~ Cus07-= as a function of the oxygen non- sto- ichiometry, z, is at first sight mysterious [41,42,43] (see top of Fig.5). For 0 < z < 0.2 the 90K superconductors are found. At a critTcal concentration z - 0.25, Tc decreases suddenly to 6 0 K , where it remains up to z -~ 0.5. Beyond z -~ 0.5, Tc gradually detoriates [41] and it has been established t h a t the z = 0.85 and x = 1 materials are antiferromagnets with high N e l l temperatures [43]. Considering the oxygen vacancies merely as donors of electrons, neither the A F state for z = 1, nor the sudden j u m p s between the different phases can be explained. However, it has recently been shown t h a t the vacancies are not d i s t r i b u t e d r a n d o m l y through the crystal, but instead form well defined defect structures [44]. The oxygen vacancies break the chains up in fragments. We will argue, using a simple tight binding model based on the LDA bands, t h a t these chain fragments act as the doping agents and t h a t in this system the same correlation exists between hole count and superconductivity as i n the £ a - c u p r a t e s .

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1640 J. Zaanen et aL / High T c from local density theory 100 , , , , , 500 80 / ,00 60 --.. : ~ 0 0 ~ ~-~ 40

\,

: 200 ,__z SC {: N: 20 ; 100 0 i i i t I 0 1.3 ~ _ _ ~ i lanes ,- 1.2 o , 1.1 ~'~.. ... 1.0 , , ~- ~-' . . . [ntoct 1.6 Chains .- 1.4 1 1.2 ~ 1.0 i J , , 0.0 0.2 0.4 0.6 0.8 1.0

X

F I G U R E 5

Phase diagram of Y B a 2 C u s O T - . a8 a f~nction of z [41,43] compared to the hole counts in planes and chains derived from the model discussed in the tezt.

have oxygen vacancies arranged with a regular spacing so as to satisfy t h e non-stochiometry. In the simplest case, this broken chain will t h e n comprise lengths of u CuO2-units connected by (v - 1) oxygen links, where

for 0 < z < 0.5 z = 1/(2u), and for 0.5 < z < 1

z = 0.5 + 1-/(2u). For values of x for which 1 / 0 z ) is not an integer, the broken chains will be m a d e up of two different lengths such t h a t the oxygen-vacancles are kept as far a p a r t as possible and form the simplest possible defect structure. This model agrees with the findings of Chaillot e~ al for z = 1/8, 1/2 and 7/8.

It is expected t h a t the LDA gives a reasonable picture of the ground state charge distribution. We have performed L M T O calculations [17] for Y2Ba4CusO,s = 2(YBa2Cus06.5). Per primitive cell there are 2 planes with 2 Cu02 units each, one intact chain, and one broken chain with v = 1. We find 2 d o u b l y degen- erate p l a n e - b a n d s which are very similar to the b a n d in La2Cu04 IFig.2) • T h e intact chain b a n d is also highly dispermve and can be very well described in the tight binding a p p r o x i m a t i o n as the antibonding b a n d obtained from the Cu dr=-. 2 orbital, the z-orbitals on the two oxygens in the e-c{irection, and the y-orbital on the oxygen along the chain. One finds for the dispersion of the a n t i b o n d i n g b a n d [46]

E ( k ) = - ~ - + )2 + 2 V 2 ( , ~ + 1 - cos~). (2)

Here - ~ < k < ~ is the b-projection of bk. The hopping integral from C u to an oxygen along the chain is

V, and to one of the oxygens in the c-direction, which are 5 per cent closer, is a V . A is as before the p - d level splitting. T h e p a r a m e t e r values A ~- 0eV, V = 2.4eV and a - 1.3 describe the antibonding LDA band. W i t h n holes in this band, kF = ~ ( 1 - n / 2 ) and EF = E(kF). According to the b a n d structure calculation the dis- persioniess a n t i b o n d i n g b a n d corresponding with the e m p t y chain is below E p , indicating t h a t this Cu is formally monovalent. The most convenient way to describe the electronic structure of a broken chain is to remove the linking oxygens, without changing the values of the parameters. In this approximation the states for a fragment of length v, with oxygens at positions y = 1 , . . . , ( u - 1) and with CuO2-units at = 1/2, ..., (u - 1/2) simply equal those states in the intact chain which have the Bloch factor eos(ky) and the wave vectors k - q~r/v with integer values of q from 0 to v - 1 T h e reason is t h a t these infinite-chain states have equal amplitudes at the z 2 - 92 orbitals at y = + l / 2 , v ~ 1/2, so t h a t they cannot couple via the missing O orbitals. The antibonding states of a fragment with length v thus have the discrete energy s p e c t r u m E(0), E(Ir/~), .., EQr(u - 1)/v), as indicated in figure 6. Starting with the empty chain, each time we a d d an oxygen, we create one additional antibonding level and two holes. As long as the level is above EF it will absorp the two holes and no doping occurs. The length vc of the longest non-donor fragment is therefore given by E(Tr/(~,~ + 1)) < EF <_ E(Tr/u¢). This fact, t h a t fragments less t h a n or equal to a critical one do not act as dopants, leads to the characteris- tic p l a t e a u x in the hole counts vs. oxygen vacancy concentration curves. Starting at z : 1, the plane b a n d is half-filied(npz = 1), exactly as in La2Cu04, and addition of oxygen does not change the hole count before the first fragment of length v~(1) + 1 occurs at z~(1) = 1 / 2 + 1 / ( 2 v ~ ( 1 ) ) . This is repeated beginning at z = 1 / 2 , giving rise to the 60K plateau, although the j u m p position ( z c ( 1 / 2 ) ) can in principle be different from z¢(1), because EF has shifted down. However, because it can be expected t h a t the change in E F is relatively small c o m p a r e d to the discrete level splittings, z~(1) and zc(1/2) will be similar (see Fig.5), in

LAI

÷__~

x~ 0 1 1

1

x - ~ x- 2 x= 3 x-I

F I G U R E 6

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J. Zaanen et al. /High T c from local density theory 1641

a s r e e m e n t w i t h e x p e r i m e n t . We notice t h a t t h e 6 0 K p l a t e a u in t h e d a t a i n d i c a t e d in Fig.5 e x t e n d s m e r e l y f r o m z ~ 0.40 to 0.25 which is u n d o u b t l y due to crys- tal imperfections. T h e d a t a of T o k u m o t o et a/~42], on t h e o t h e r h a n d , show this p l a t e a u to e x t e n d zrom z -~ 0.65 to 0.40.

P r o v i d e d we can use Eq.(2) b o t h for t h e hole d o n o r levels a n d t h e i n t a c t chain we can derive b o u n d s for t h e hole counts in t h e i n t a c t chain a n d planes in t h e 6 0 K m a t e r i a l given t h e l e n g t h of t h e 6 0 K plateau. We find for t h e hole count in t h e i n t a c t chain 4L < n ( 1 / 2 ) < (1 - L) - 1 , where L = 1 / 2 - z~(1/2) is t h e l e n g t h of t h e plateau. Using z c ( 1 / 2 ) - 0.25 we find 1 _< n ( 1 / 2 ) < 4 / 3 a n d for t h e hole c o u n t in t h e p l a n e ( a p t ( i / 2 ) = 3 / 2 - n ( 1 / 2 ) / 4 ) we find 7 / 6 < ztpl(1/2) < 5 / 4 . In o r d e r to be m o r e specific we h a v e to m a k e a s s u m p t i o n s a b o u t t h e q u a s i p a r t i c l e densities of states in t h e chain a n d p l a n e b a n d s [46], which c a n n o t be claimed to be known. As a m o d e l , we can t a k e t h e dispersion Eq.(2) for the chain b a n d and for t h e planes t h e tight b i n d i n g b a n d p r o p o s e d by M a t t h e i s s [18]

Ep~(k) = Ap~-~-+

( )2 +9.1,'~(2 -cosk~ --coskb).

(3)

where -Tr < ( k , , kb) < ~" a n d where Ap: gives t h e position of t h e plane b a n d w i t h respect to t h e chain b a n d . T h e h o p p i n g i n t e g r a l V as well as A h a v e b e e n t a k e n t h e s a m e as in t h e chain. In Fig.5 we c o m p a r e results for t h e hole- counts o b t a i n e d f r o m L D A p a r a m e t e r values (Apt --- A -- 0, ~ --- 1.3) w i t h a ' m o s t likely' result (Apl = A ---- 0 , a = 1). In p a r t i c u l a r , for t h e l a t t e r values t h e r e is a striking c o r r e l a t i o n between t h e hole counts in t h e planes a n d t h e chains and t h e o c c u r r e n c e of t h e SC a n d A F phases. It is obvious t h a t t h e effect on t h e hole counts of t h e i n c r e a s e d c - h o p p i n g s u g g e s t e d by t h e L D A calculations is essentially t h e s a m e as using t h e off-set Ap~ ~ V ( ~ - 2). T h i s p o i n t s to s o m e serious error in t h e L D A g r o u n d s t a t e charge distribution. It can be well i m a g i n e d t h a t C o u l o m b effects b e y o n d t h e L D A could b r i n g t h e hole counts back t o w a r d s those o b t a i n e d w i t h o u r second p a r a m e - ter choice. Finally, we a s s u m e d small A in o u r calculation. Strictly speaking our findings in this chapter are rather insensitive to the choice of A a n d they cannot be used to discriminate between the localized a n d itinerant pictures. At least as long as A takes the s a m e value in the chains a n d planes, increasing A will result in a rescaling of the bandwidth's with little effect on the positions where the j u m p s occur.

A C K N O W L E D G E M E N T S

We enjoyed s t i m u l a t i n g discussions w i t h Dr. A . M . Ole.~ a n d P r o f . ' s A. S i m o n a n d G.A. Sawatzky.

R E F E R E N C E S

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