Effective Hamiltonians for high-Tc superconductors

PhysicaC 152 (1988) 349-360 North-Holland, Amsterdam

EFFECTIVE HAMILTONIANS FOR HIGH-To SUPERCONDUCTORS E. SIGMUND

a

and K.W.H. STEVENS b

a InstitutfiirTheoretischePhysik, Teil3, UniversitdtStuttgart, Pfaffenwaldring57/V, D-7OOOFed. Rep. Germany b Department of Physics, University of Nottingham, Nottingham NG7 2RD, UK

Received 9 May 1988

Effective Hamiltonians are defined through an extension of an infinite order perturbation theory developed for magnetic insulators. Theycontain phonons, electron-phononand electron-electron interactions. It is reasoned that no divergences occureven if phonon energies match electronic energydifferences. A unitary transformation is used to decouple the phonons, which results in electron-electron interactions aving no isotope dependencies. The transformed Hamiltonians have low-lyingperiodic variational states composedof electron pairs, it is proposed that a better variational state will have pairs, without the periodicity.

1. Introduction In 1970 the book "Electron Paramagnetic Resonance of Transition Ions" [ 1 ] appeared. It was dedicated to the late Prof. J.H. Van Vleck, for virtually the whole interpretation of the properties of transition metal ions in insulating hosts had come from his original ideas. Over the years these had been further developed and in particlular the concept of spinHamiltonian had emerged as a convenient meeting point for theoreticans and experimentalists. Many examples had shown that it was possible to anticipate the form for the spin-Hamiltonian of a given system, confirm it experimentally and determine the parameters (the coefficients) of the various operator expressions in it. The theoretical problem then became more a matter of accounting for the parameter values than accounting for the occurrence of particular operator forms. Indeed this task is still not entirely completed, but perhaps what is not so often appreciated is that the ground rules have changed. It is generally straightforward to derive the spin-Hamiltonian form from a crystal field model. It is also straightforward to derive an identical form from a more general starting point in which no crystal field concepts are used. The major changes are then that the expressions for the spin-Hamiltonian parameters are not those of crystal field theory and that most of the various objections which from time to time have

been raised against crystal field theory have been circumvented. An extensive review of these more recent developments can be found in the work by Stevens [2] and a shorter version, with later references by Stevens [3 ]. The recent discovery of high-Tc copper-oxide type superconductivity by Bednorz and Mtiller [4] has produced a major upheaval in solid state physics. In particular the superconductivity had not been expected on theoretical grounds, despite the fact that the properties of transition metal ions as impurities in insulating hosts seemed to be well-understood. There was, of course, one major difference, that the spin-Hamiltonian forms do not usually allow for conductivity. However, the more recent ways of obtaining spin-Hamiltonians for insulating magnetic crystals are not, a priori, restricted to insulators, so it has seemed of interest to see whether these concepts can be used to produce "effective Hamiltonians" for these high-To materials which are, in some sense, the analogous of the spin-Hamiltonians of the insulators. An additional incentive came from the report that there was no isotope effect in the new superconductors [5] and the consequent assumption that this showed that phonons had no ~ole to play. It was already known from Jahn-Teller and similar studies oh insulators that in second order perturbation theory the electron-phonon interactions give rise to electron-electron interactions (expressed in

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E. Sigmund and K. g :ti. Stevens / E.[]bctive llamfltonians /br hi~,h-7~ superc,mductors

350

s p i n - s p i n f o r m ) which have no isotopic d e p e n d e n cies. Because o f the possible relevance o f this observation a shortened and incomplete version o f the theory now to be described has already been published [6]. Later work has shown that in fact there are small d e p e n d e n c e s [ 7 ], and that indeed they are not completely absent from the present theory (as was p o i n t e d out to us by M. W a g n e r ) . The discussion will be mainly concerned with modelling the two materials La2 , M , C u O 4 (M = Ba, Sr, etc., x~-0.15) and YBa2Cu3OT_,. ( y ~ 0 . 0 ..... 0.5). Other ceramic materials showing s u p e r c o n d u c t i v i t y are derivatives o f these, where all the elements other than Cu and O have been replaced by other elements. It seems to have been shown that the charge carriers responsible for the superconductivity are confined to the CuO4 planes (in the cases o f La2 , M , C u O 4 ) and to the CuO~ planes a n d / o r the CuO2 chains (in the cases o f YBaeCu307 ,.). The m e c h a n i s m leading to s u p e r c o n d u c t i v i t y seems to be the same for all the ceramic materials and therefore without loss o f generality we shall confine our discussion to models o f La: ,M,CuO4. In the d e v e l o p m e n t of the theory it will be useful to have three structures in mind. The first, which will be regarded as the p r o t o t y p e structure and which is not expected to be a superconductor, is regarded as m a d e up of layers o f Cu ~+ and O e ions, as illustrated in fig. 1. The layers are not electrically neutral, and it will be supposed that other ions are present, (e.g. La, Sr, Ba etc.) between the layers, to act as spacers and to p r o v i d e overall charge neutrality. The second structure is simply the first one with some of the electrons r e m o v e d ( d i s p l a y e d on to the spacer

/~,%

,,',L :

,; J• ~

J

ions) to produce, in a heuristic sense, a mixture of Cu :+ and Cu 3~ ions, with the oxygens remaining as O -~ . (This picture is substantially m o d i f i e d in due course.) The third structure is one which, in addition, has some o f the oxygen atoms removed. The theory, for each structure begins with a very general H a m i l t o n i a n , a p p r o p r i a t e for a many-electron many-nuclear structure, which is then re-expressed in a second quantised notation for the electrons, based on Wannier functions [ 8 ]. These are then m o d i f i e d to take account of the physical expectation that when the nucleus of an ion is displayed for a regular lattice site, as in a lattice vibration, the electrons on that ion follow i1 [ 9 ]. The net effect is shown to be equivalent to the form for an u n d i s p l a y e d set o f lattice points, with changed parameters. The next step is to set up a suitable perturbation formalism and it is at this stage that the treatments begin to differ, although the basic technique, which is due to Bloch [ 10], is c o m m o n . The p e r t u r b a t i o n expansion is of infinite order and produces an effective Hamiltonian, acting in a sub-space o f the u n p e r t u r b e d H a m i l t o n i a n , which gives the eigenvalues, but not the eigenstates, into which a degenerate set o f low-lying states o f the u n p e r t u r b e d Hamiltonian is split by the perturbation. For the first structure this can be written in typical s p i n - H a m i l Ionian form. For the other two structures this is not so, the differences between their forms p r i m a r i l y arising because the second structure possesses the s y m m e t r y properties of the fig. 1 plane whereas the third has no such s y m m e t r y properties because the oxygen a t o m s are assumed to have been r e m o v e d at r a n d o m . F r o m then on only the second and third structures are considered. They contain many different operators and now the emphasis shifts towards vibrational methods, a i m e d at showing that the effective H a m i l t o n i a n s can be expected to have low-lying ground states c o m p o s e d o f pairs of quasiparticles, much as in s t a n d a r d BCS theory [ 11 ].

, ]

: JI , 2. The Wannier functions

Fig. 1. T h e C u O p l a n e s . C o p p e r sites a r e d e n o t e d by c r o s s e s a n d o x y g e n sites by circles.

The usual way o f describing electronic m o t i o n in a crystal lattice is to introduce the concept o f a periodic potential and so arrive at a set o f conduction

E. Sigmund and K W.H. Stevens /Effective Hamiltoniansfor high-T,,superconductors bands. For each band Wannier functions can then be defined by setting

~(R,,) = ~

1

~ ~uke x p ( - i k . R n ) ,

where ~uk is a typical band wavefunction with propagation vector k and R is a specific site in the nth unit cell. The summation over k is over all the N vectors of the reduced Brillouin zone. For a given Rn there is a Wannier function for each band and for each band the functions for differing R~ are identical in shape but relatively displaced. Since the band wavefunctions are mutually orthogonal it follows that so are the Wannier functions. The lattice shown in fig. 1 has one Cu 2+ ion and two 0 2- ions in the unit cell, so Wannier functions constructed as above are to be associated with cells rather than with sites within cells, whereas for subsequent purposes it is desirable to have functions associated with sites. A first step is to consider the ~'k with k = 0 . Either there is no k = 0 function of another band which has the same energy, or there is degeneracy. In both cases the functions provide bases for irreducible representations of the point group of the lattice, and in constructing the ~(R,) for a particular band one should, if there is a choice of q/k=o, make it specific. The next point to note is that there is considerable ambiguity in the definition of a Wannier function because the phase of each q/k in the summation is quite arbitrary. A given k generates a "star" of k-vectors under the operations of the point group and it is therefore possible to choose the phases of the ~uk of each "star" so that q~(R~) transforms in exactly the same way as the chosen ~Uk=o.The symmetry of the ~(R,) is then explicitly determined. Some examples are given in fig. 2. Diagrams 2a and 2b have been chosen to illustrate that two bands of the same symmetry can produce two quite different looking Wannier functions. By taking suitable linear combinations of them new Wannier functions can be obtained, one which is more confined to the central Cu 2+ ion and the other which is more confined to the outer 0 2- ions. Diagram 2c has a different symmetry and is one of enough possibilities to ensure that there is scope to define Wannier functions which are largely confined to sites within the unit cell. These functions can be put into 1-1 correspondence with the atomic orbitals of the isolated ions, and given a

351

similar s, p, d, ... type of notation. It is, of course, only in the immediate vicinity of the site in the chosen unit cell that they have any similarity to atomic orbitals. They can also be put into l-1 correspondence with crystal states, and given a similar notation. The Wannier functions so obtained form a complete orthonormal set and so can be used in a second quantised formulation for electrons. The Hamiltonian is taken as

p2

H= Zz

p2

Zle2

+?

l,,,+e,_r,, ZlZse 2

e2

+½l ~EJ ]RI-t-QI-Rj-Qj] +½E i~J

Irz--rjl '

where lower case letters are used for electrons and upper case for nuclei. R/is the position vector of the lattice site nearest to t h e / t h nucleus, its actual position being R/q-Q/. P/is the momentum conjugate to Q/, so Pz and Qz are dynamical variables, the RI being constant vectors. The Hamiltonian therefore contains electron and nuclear kinetic and electrostatic potential energies. No magnetic types of interactions have been included, for simplicity, for they could be. The second quantised form is then:

Hsq Z/to "~1

"~A2 ,

where

p2 ZlZ]e 2 /"°= ~1 - ~ I +½ l~#J I R I + Q I - R j - Q , I I

'

(P2~IZIC2 "~l = .,E.~, a~, ~ -

) Rt+Qz-rl

a..

a*,o,a.~:

n2 ,¢Y2

and

f~2 ~ rtl o.,~n2o.2,.., an,,

an 3, an 4

rt3 ,n4

X a nlm a n2a2an4~2an3at , where a,,o, * etc. are creation operators for Wannier functions, with a~ being a spin index. The same Hamiltonian can be used for all three structures provided that suitable choices are made for the numbers of electrons and nuclei. In due course a perturbation treatment will be introduced and usually the unperturbed Hamiltonian is chosen to be, in some sense, as good as possible.

E. Sigmund and K. 14:1t. Stevens / E[/bcttve Hamiltonian~ tbr high-~ superconductors

352

Fig. 2. (a) A Wannier function ofx~'-y2 symmetry and of bonding character: (b of the same symmetry but of anti-bonding character: (c) an example of a different symmet~. However, the p e r t u r b a t i o n series is an expansion to infinite order so such a r e q u i r e m e n t is perhaps not necessary. Nevertheless, in case it is worthwhile to incorporate the physical r e q u i r e m e n t that when a nucleus moves it takes its W a n n i e r functions with it. To this end a new one-electron o r t h o n o r m a l set will be introduced, o b t a i n e d by applying a unitary transf o r m a t i o n to each W a n n i e r function. Thus a typical 0,, becomes 0,, , where 0,, = e x p ( - i S ) O , , ,

£,=

Z ia,,,l exp(iS)

- ~ IR,+Q,-rl

_

× exp(iS)la,,~ /" finial * att2o? and /

L=½

~ (,a .... a , : l { e x p i ( S , + S 2 ) } × -

e 2

-

it, - r 2 I

{exp - - i ( S l + S 2 ) } l a .... a,,4

with

X a,,,~, a n : a ~ a n 4 a 2 a n ~ , ~ k S= ~

•

{ ~ t ( R z ) p . Q z + p . Q z ~ t ( R , ) }.

3. The perturbation theory As in H, p is an electron m o m e n t u m operator, and Q~ is the d i s p l a c e m e n t o f the nucleus at position R~. The new o p e r a t o r is " # ( R D , which is an electronic operator, the projection o p e r a t o r o f all the W a n n i e r functions at site Rt [ 9 ]. S is chosen so that if all the Q~ are m a d e equal the t r a n s f o r m a t i o n is equivalent to a d i s p l a c e m e n t o f the whole lattice. The annihilation and creation operators for the ~ obey the same algebraic rules as those for the ~,, states so a new notation is not required. It follows that an alternative form for H~q is

The p e r t u r b a t i o n theory to be used is due to Bloch [10]. The first step is to define an u n p e r t u r b e d H a m i l t o n i a n Ho. It is to have a sequence o f wellspaced degenerate energy levels. Denoting its eigenvalues in ascending order by Eo, E,, etc. the eigenstates with energy Ei define a projection o p e r a t o r .~. Then Ho = ~,E,,~ and the p e r t u r b a t i o n is V= H - H~. The theory then shows that the eigenvalues of H which come from Eo, the lowest eigenvalue o f tf,, are those o f He, where

H =/,v + ~, + '~2 ,

H~=,~,

where

F o r each o f the three structures a different Ho is cho-

E,+V-

~

,~o ( E , - E , ) )

+...

;~,....

(1)

E. Sigmund and K. W.H. Stevens / Effective Hamiltonians for high-T,. superconductors

sen. Beginning with the first structure it is convenient to assume that the lattice is rigid. Then ~ is defined by the manifold of states in which all the 0 2and Cu 2+ ions are in their ground states. Thus all 0 2- will have closed shell structures, ( l s ) 2 (2s) 2 (2p) 6, where ls, 2s, ... denote the Wannier states which correspond to the isolated ion states. Similarly each Cu 2+ ion will be in its crystal field ground state, which is an x 2 - y 2 hole in a filled (3d) 1° configuration. Each such state has two-fold spin degeneracy, so ~o has 2 N degeneracy, where N is the number of Cu 2+ ions. Excited manifolds are obtained by moving electrons from occupied to unoccupied orbitals. The energies Ei are determined by evaluating the expectation values of H for each state in ~i. In general these expectation values have a distribution of values. Ei is therefore defined as the average expectation value. The expansion in He is an infinite series. Usually only the first few terms are examined in detail and then it is seen that the ~, operators on the left and right severely limit the operators which can occur. In the present case the only operators which can appear either leave the local ms unaltered or allow it to change by unity. As this is a property shared by spin operators He, to infinite order, can be written as an expression in local spin 1/2 operators. It is also apparent that if a given type of operator occurs in low order perturbation theory it is also likely to occur again in higher order. The effect is to renormalise its coefficient. If an operator does not occur in low order, when it does appear, in high order, it is likely to be small because of the high powers in energy differences in the denominator. The next step is to remove the restriction that the lattice is rigid, which immediately raises the question o f how to modify the unperturbed Hamiltonian. Since ~o contains no electronic operators it would seem, assuming that the QI are small, that it will, in some sense describe a spectrum o f lattice vibrations. It might therefore be expected that it should be included in the unperturbed Hamiltonian. But if this is done the unperturbed Hamiltonian no longer consists of a set of well-separated degenerate energy levels, for the phonon part produces a continuum o f levels. It therefore seems that Ao should be part o f the perturbation. Another difficulty then appears to arise, for the version of the Bloch theory given by Messiah [ 12 ] discusses H0+2V, where 2 is varied from 0 to 1. It

35 3

is assumed that for an infinitesimal 2 the eigenvalues coming from ~o do not overlap the eigenvalues coming from any other ~. But if V contains a harmonic oscillator this condition is not satisfied and it would therefore seem that ~o cannot be put into the perturbation, either. Fortunately there is another version of the perturbation theory in Bloch's original paper which avoids the above assumption and which actually gives a self-consistent expression for the effective Hamiltonian, that it is ~o VU, where U=

1

(Eo - H o )

(1-~)(V-UV)U

and U = ~0 if V= 0. U can be expanded in powers of V to give the infinite expansion ( 1 ). Thus Zo, and hence the whole of H can be used in V = H - H o . The ~, have to be redefined, so that a new ~ is the original one times the unit operator in the phonon subspace. It can be assumed that He does not contain any terms which are linear in the QI and which have scalar coefficients, for their presence would indicate an instability in the lattice, and it has been implicitly assumed that the underlying lattice structure for the vibrating version of fig. 1 is the equilibrium lattice. The main purpose of the above discussion has been to outline the perturbatio0 procedure in a relatively simple example. To use it for the second structure the electronic part of ~o has to describe a lattice in which all the 0 2- ions have closed shell ground states, ( N - r ) of the copper ions are Cu 2+ and r are Cu 3+. Since each Cu 2+ is at a site o f tetragonal symmetry it is convenient to introduce a creation operator, a*, for each ion, appropriate to a Wannier orbital of ( x 2 - y 2) symmetry. Cu 2+ is described by a+ or a_ acting on a (3d) 1° shell. Cu 3+ presents a problem because there is uncertainty over whether its lowest state has S = 1 or S = 0 . Since it is slightly easier to describe S = 0 this will be assumed for each Cu 3+ ion. The state is obtained by applying a+a_ to (3d) 1°. The electronic part of ~o has 2N--r(Nrr ) degeneracy, since all arangements of Cu 3+ sites are assumed to be possible. It is then straightforward to write down some of the operators which can be expected to occur in He, assuming that the Q~ are sufficiently small that power series expansions are valid. An example is

354

E. Sigmund and K. 14tH. Stevens /I-(l~'ctive Hamiltonians jor high-7~ superconduclors

p,~

Q,A,,Q,,

where A contains scalar and site n u m b e r operators o f the form ( 1 - a * + a + - a * _ _ a ). I f a site is occupied by a Cu 3+ ion with S = 0 the o p e r a t o r is equivalent to unity, whereas if the site is C u 2+ S = 1/2 it is zero. If these were the only operators in H~ they could be interpreted physically as allowing for the changes which occur in the lattice vibrational m o d e s on rearranging the Cu 3+ and Cu 2+ sites. A n o t h e r example is a,*~ra~ranramr, where m and n refer to different copper sites. This o p e r a t o r is only non-zero if both sites have Cu 2+ ions with m = 1/2. It can enter in first order from the electronic repulsion term, ~e2/Ir~j I- It can also arise in higher orders, in which case the coefficient ( p a r a m e t e r ) which goes with the o p e r a t o r form could be interpreted as describing a screened electrostatic interaction. Since there are no s p i n - d e p e n d e n t operators in H this o p e r a t o r will occur in a c o m b i n a t i o n a*.~,a%~a~.a

....

.

al a2

A

rather

similar

combination

is

.... a .... ano2a~,~a . . . . which describes an exchange

interaction between Cu 2+ ions at sites m and n. In first order its coefficient will be positive, corresponding to a ferromagnetic exchange interaction. In higher orders it is usually found, at least with insulators, that the corrections are o f negative sign and outweigh the first order c o n t r i b u t i o n to produce an overall antiferromagnetic interaction o f short range. In the present case the analogue is with insulators, rather than conductors, and the expectation is that the exchange will be o f antiferromagnetic character. Two further operators are o f particular interest, the first, o f type a % a .... is only non-zero if there is a Cu 2+ ion at site m and Cu 3+ at site n. Its effect is to interchange the two, a n d so can p r o v i d e a conduction mechanism. The o p e r a t o r can also occur in c o m b i n a t i o n with lattice displacements (e.g. Q~a*oa,,,~) which allow lattice v i b r a t i o n s to induce

charge movements (and vice versa). The second type ( in = n ) also involves lattice displacements, but with an electronic o p e r a t o r which preserves the charge at a site. Since the prototype lattice is regarded as in equilibrium the a s s u m p t i o n is that there is no force on the lattice unless the site is Cu 3+. So the interaction should be of forrn Q~(a,,+a*+ + a , , _ a * ). It is clear that many more complicated forms can appear, but unless there are some unforeseen effects most of the rest can be assumed to be small because they do not enter in low order p e r t u r b a t i o n theory. Several further observations can be made. It is imp o r t a n t to r e m e m b e r that each o p e r a t o r form is bordered to left and right by :~). It has been suggested that some o f the excitation energies in charge transfer processes (e.g. C u 3 + - O 2- changing to Cu -~+O ~- ) m a y fall in the range o f phonon quanta. Since the phonon energies have been put into t \ and so occur in the n u m e r a t o r s in the perturbation expansion, it seems possible that the expansion m a y not converge. On the other hand the expansion itself has been obtained from a self-consistent expression for U for which there is no reason to suppose that anything unusual will occur. It can therefore be expected that any a p p a r e n t divergences can be r e s u m m e d into a new non-divergent form. An example o f this kind has been described by Sigmund et al. [13], who showed that solitons were likely to result from such a resonant interaction. Such a possibility brings with it the question o f the meaning of an a* o p e r a t o r in an effective H a m i l t o n i a n . In the initial H it is an operator which places an electron in a W a n n i e r function. But as the theory is developed H changes its form while still containing a*, and, for example, the Coulomb interaction may become screened. It is then quite obvious that whatever a* is describing it is not an electron, and the simplest course is to regard it as describing a quasi-particle. Other investigators have used descriptions such as polaron, soliton, etc. To discuss the third structure all that is required is to omit the variables o f the absent 0 2- nuclei and m o d i f y the electronic part o f ~ appropriately. The s y m m e t r y is much reduced in consequence.

E. SigmundandK.W.H.Stevens/ EffectiveHamiltoniansfor high-Tosuperconductors

355

4. Electron-phonon decoupling There seems little immediate prospect of showing that He gives a description of a high-To superconductor because the analysis which leads to it is directed towards eigenvalues and presumably the criteria for superconductivity involve the nature of the eigenstates. Nevertheless in the early days of superconductivity the demonstration that the lowest eigenvalue of an effective Hamiltonian was well-separated from the next highest level was an important step forward. It is therefore of interest to examine He in a similar spirit, using variational methods. That is, an attempt will be made to guess wavefunctions which are such that the expectation values of He are particularly low. A first choice might be to use a wavefunction which is a direct product of a lattice vibrational part and an electronic part. This would not seem to be promising, because the vibrational part of He probably has a continuum of levels and the ground state is unlikely to be isolated. It, therefore, seemed more interesting to follow the ideas of Fr6hlich [14 ], which resulted in major progress in superconductivity theory. Any operator which is diagonal in site number operators commutes with ~o, and since on physical grounds it can be expected that lattice strains will be more strongly coupled to site number operators than to charge transfer operators it is of interest to attempt to decouple lattice strains from site number operators. There is, however, a difficulty over the elastic energy of the lattice, which is usually taken to be a quadratic form,

Q,(IIAIJ)Qj, LJ

where (IIA IJ) defines a positive definite matrix, A. If He is expanded in powers of the QI there will be terms of the above form, except that the (I[A IJ) will be functions of the site number operators, a form which expresses the physical requirement that the elastic energy depends on the detailed arrangement of Cu 2+ and Cu 3÷ sites. It therefore seems best to take the average of each matrix element of A, taken over all the states in ~0, and write the elastic energy as

Z Q,(II.41J)Qj+ ~ a,I ( A - A ) I J ) a J , where ./i is also a positive definite matrix if the lattice is stable for all Cu 2÷, Cu 3÷ configurations. There will also be terms linear in the Q~ of the form ZtQlVz, the v/are expressions containing scalars and site number operators. They express the physical requirement that when a Cu 2+ site changes to Cu 3÷ there are changes in the forces on all the other ions. To illustrate our procedure it is convenient to begin with a selection of terms from He, the others being added later. Suppose H;=~o

~ ~//+

2 Q~(IIdIJ)Q~+ 2 Qiv, ~0.

The transformation exp i (Zzu~Pt)/h changes H'e into

]

Z } - ~ , + Z (Q~+u~)(II.,iIJ)(Qj+uj)+ Z (Qt+ut)v~ ~o.

/~;=~o

The terms linear in QI disappear if

2~ (IlzilJ)ut+v~=O J

or

u1= -½Y, ( I I ( d ) - ' I J ) v j J

and

E. Sigmund and K. I4
356

/?;=& --l~j v] <, The first two terms in/t'~ contain only nuclear variables, whereas the third term contains only electronic operators. Were it not for the other terms in Hc it would seem that the electrons and phonons have been completely decoupled by a transformation which does not depend on the nuclear masses, in which case there would be no isotope effect. The transformation should, of course, be applied to all the operators in H~, and this is much easier if the above manipulations are repeated in a phonon representation. Therefore, set P~=(2Mt)'/2H, and Q~(2MD-~/2Z~. Then H'~ becomes ~o[ ~ H~ + ,.J~X'z"(4M'M')

,n+ ~XIv,(2M~)-,/~-],~,.

It can be diagonalised by an orthogonal transformation, 7". to become H~,-+2~-Z~-,

where

[HK, z~-,] =6h,h'.

K

Phonon creation and annihilation operators can then be introduced so that this part becomes

Y~h~Ka~a~. By the same procedure ~ Q~v~is changed into a form K

where v~ is an operator in site number operators. The decoupling transformation then takes the form I~ exp i ( u~aa-+ u 7,-a7,-) • K

Using exp i(ua-aK+ U~-a~-)C¢~,- exp --i(uKa~'+U*xa*X) =C~,.+iuh, H'~ becomes

fig2~,'(a*K+iuK) (aa---iul-) + Y~ [(aK--iul')Va + (a~,-+iu~)vT,-] K

K

and the terms linear in aT, and aa. disappear if --ifi~2KU~-+ V~ = 0 or

u~ = ivl,'/fi~K SO H'~ becomes Z fi~2~a~aK-- ~ ]VK]2/h~2K. K

K

The transformation applied to a ~ produces a*-VK/h(2~,, and to a it produces a-v*~/hF2x. All operators in tt~.

E. Sigm und and K. W.H. Stevens / Effective Hamiltonians for high-T,. superconductors

35 7

leave the total Ms unaltered, so starred and unstarred operators can be paired to conserve Ms. Further any a%a,,,, pair commutes with the transformation, so all that remains is to consider the transformation applied to a pair such as a,~+a~+ where m ¢ n , for exp iS(aTa2a~a4." ) exp ( i S ) = e x p iS(aTa2) e x p ( - i S ) exp (iS)(a~a4) e x p ( - i S ) . . . . Also, since the transformation is a product of exponentials it is sufficient to consider exp \

-h--~,~

j a * + a,+ e x p - \

hg2~

J"

In the left-hand exponential any am+*am+ terms in v~: or VKcan be replaced by unity and any a*+ a,+ by zero, so VKcan be written as VK(1, 0). On the right-hand side a * + am+ can be replaced by zero and a*+ a,+ by unity, so VKis written as VK(0, 1 ). a * + a,+ then commutes with both exponentials. Further, ifA and B are two commuting operators which also commute with a* and a, the relation exp 2(Ac~+Bc~*)=exp ()~Ba*) exp (LAc~) e x p ( 2 2 A B / 2 ) = e x p ( 2 A o ~ ) exp(ABc~*) e x p ( - 2 2 A B / 2 ) can be used to move all the operators to the left, so that exp \

h~2K

=expa~ xexp

j a * + a,+ exp - \-

~(0, 1)-~(1,0) h-QK

fv, (o, 1) vK(O,

~

j

exp --OlK hQK

o) vK(1, o)-2vK(O, 1)v (1, (fi~2r) 2

5. The variational ground state

The above transformation has changed the appearance of He to that of/4e without changing the eigenvalues. Nevertheless the choice of variational state in which each phonon mode is in its ground state is quite different using/te than it would be using He. Using/Te the transformed a * + a~÷ of the previous section simplifies considerably when averaged over the phonon ground state, giving 1-[ exp -- ½[ {v,~(0, 1 ) VK(O, 1 ) + V~(1,0) VK(1,0) K

- 2VK(0, 1 ) v,~( 1,0)} / (~Qp)2 ] Each of the factors in UK is less than or equal to unity, leading to the conclusion that all electron transfer terms are "quenched" (by factors which are not isotope independent). On the other hand all terms in He which are in site number operators are unaltered in/Te and there is no "quenching". These terms include the screened Coulomb and the exchange interactions. The quenching of all transfer terms poses

f a*+a~+ . something of a problem because these are the terms which allow electrons to hop from site to site and without them there would be no conduction. It therefore seems necessary to assume that although there is quenching it is not enough to remove the conduction. (There are processes in high order in the perturbation series which led to He which renormalise the low order transfer terms, and these may enhance the transfer processes. ) The procedure so far is very similar to ones which can be found in a number of previous papers, most of which are concerned with bipolarons (see Robaszkiewicz et al. [ 15 ] for references). In most of this work the transfer terms are further reduced by the use of another unitary transformation, with the result that each orbital contains either two or zero polarons. These authors have not been considering copper oxide superconductors, but had they been doing so it would seem that either each copper site would then be Cu 4+ or Cu E+ or that the Wannier functions which define the initial second quantised operators would be more like molecular orbitals and

358

E. Sigmund and 1<2~1~H, Stevens / Effective ltamiltontans fbr high- 7~ superconductors

the b i p o l a r o n would be the analogue o f a hydrogen molecule c o m p o s e d o f two Cu 3+ ions. Both of these possibilities have been excluded by our choice o f .~. Futher it is not obvious that such a t r a n s f o r m a t i o n can be justified when, as in the present case,/7~, operates in a subspace of closely spaced levels. The only example o f which we are aware in which this type of t r a n s f o r m a t i o n seems to have been used in the context o f copper oxide superconductors is in a very brief account by Ray [ 16 ] of a clearly much more extensive study. Because o f the possibility that the use o f such a t r a n s f o r m a t i o n m a y be difficult to justify variational m e t h o d s will be used instead. In choosing the electronic part o f the variational wavefunction a p r o b l e m already met in a different context again crops up, that the site n u m b e r operators have different values for the different states in • ~, so the quenching factors d e p e n d on the Cu 2+, Cu 3+ arrangements. As previously it would seem best to take the average over all states in :R), together with suitable correction terms. There are then terms in t']~. o f the form /

*

X .~T,,,,(a,,,+a,,+ + a * , _ a ..... ) + b . c . } ol.

~l

which, taken alone, would describe a conduction band. F o r the second structure its periodicity ensures that each state in the b a n d will be a Bloch function with a k-vector, provided, o f course, that the sample is o f infinite extent. If it is not, which is very likely, the periodicity will be absent and in m a y ways the structure will be similar to that o f our third structure. As this is not periodic it will be assumed that there is a b a n d o f states, described as a conduction band, with no concept o f k vector. F o r each structure a possible variational state would have each conduction b a n d filled to a F e r m i level. In so doing no significant notice is taken o f the exchange interactions and the e l e c t r o n - e l e c t r o n interactions which come from the two sources, C o u l o m b a n d e l e c t r o n p h o n o n interactions. A possible i m p r o v e m e n t might be to suppose that the antiferromagnetic exchange interactions produce two interpenetrating sub-lattices, one corresponding to m s = 1/2 and the other to m s = - 1/2, for this will go some way towards minimising the exchange part o f / t o . It would then be nessary to accept that it is the transfer terms between next-nearest neighbour sites which d e t e r m i n e the

conduction b a n d structure, for hopping between nearest neighbour sites is not possible without a spin reorientation. There would then be two identical and presumably narrow bands. They would be filled to a F e r m i level, the height of which is d e t e r m i n e d by the n u m b e r of Cu 3+ sites. If this arrangement is compared with the one in which the exchange is simply ignored, the b a n d width is there d e t e r m i n e d primarily by the nearest neighbour hopping and is therefore likely to be much greater. On filling this to a F e r m i level the energy associated with the conduction electrons will be less than for the case where the exchange is not ignored, and which choice is best will d e p e n d on how the lowering of energy by taking the exchange into account compares with the energy increase due to the reduced bandwidth. To obtain something like BCS superconductivity [ 11 ] some kind of electron pairing would seem to be necessary, to exploit the properties o f the e l e c t r o n electron interactions. Conventional BCS pairing uses a pair o f type t~a*k+ a*k - ) , where the state defining aT,+ is the time reversal o f the state defining a~ . Such a pairing is not possible for the third structure, nor for the second if the sample is o f finite size a n d / or the antiferromagnetic coupling case is the dominant one. The p r o b l e m o f how to construct C o o p e r pairs when there is no periodicity has been considered by Suhl et al. [17]. The fist step is to define creation and annihilation operators for the conduction band states and to express the e l e c t r o n - e l e c t r o n interaction in terms o f them. F o r convenience these operators will be given a numerical notation. The various operators in the e l e c t r o n - e l e c t r o n interaction are exa m i n e d in detail and one, say aT+ a* ~a4_ a3+. which has a negative coefficient, is selected. (aT+a* ) and a*3+ a*4- ) are called pairs. The remaining operators are then e x a m i n e d to see whether there is another pair (a'~+a~ ) which is such that a'~+a~_a~ at+ and a*a*~4_a~_.as+ both have negative coefficients. If they have a fourth pair, a*7+ a*~ is sought such that a*3+ a*4 _ a ~ _ a T + aT+a~_a~_aT+, and a'~+a'2 as_a7+ all have negative coefficients. This process is continued as far as possible. It m a y rapidly terminate, in which case a different initial a*a*aa is chosen, the object being to o b t a i n as large a linked set o f pairs as possible (actually an o p t i m u m set, not necessarily the largest, as will shortly becomes ap-

E. Sigmund and K. W.H. Stevens / EffectiveHamiltoniansfor high-Tosuperconductors parent). Suppose now that the linked set contains X pairs, and that 2X is much greater than the number of charge carrers, Y say, with Y even. Then from the set X, Y/2 pairs are selected and a many-electron state is defined by letting these Y/2 pairs act on the vacuum state. A large number of such many-electron states can be formed by different selections of the Y~ 2 pairs. These are then added together, all with possitive coefficients, to form a normalised many electron state. If now the expectation value o f / t e is evaluated it will contain diagonal type terms (in squares of the coefficients) of indeterminate signs together with off-diagonal type terms all of which are either negative or zero. Since the number of off-diagonal terms can be expected to be much larger than the number of diagonal terms a judicious choice of the positive coefficients can be expected to produce a particularly low mean energy. The network of pairs should be that which produces the minimum mean energy, rather than the one which has the most pairs in it. The copper oxide superconductors have a relatively small number of charge carriers, for each one is associated with a Cu 3+ site. Also, in a detailed examination of the electron-electron interaction induced by the phonons using a simple treatment of essentially the second structure, by Sigmund et al. [12], it was shown that there were a large number ofa~_ a*_ a_ a+ type terms with negative coefficients. This in itself is not sufficient for the Coulomb terms are of similar form and it is necessary for the negative coefficients of the one to outweigh the expected positive coefficients of other. The ~o operators which border all the second quantised forms are there to ensure that no copper sites are Cu ~+. They are, nevertheless a nuisance, and the above discussion of antiferromagnetic basic structure suggests a way of dispensing with them. Suppose that the second quantised notation is given a slightly different meaning and that am+* is replaced by a*, whenever the site m is on the A sub-lattice and that a*_ is replaced by b* whenever n is on the B sub-lattice. If then a vacuum state is defined as the state in which all Cu sites are Cu 3+ the ~o operators can be removed for there is now no way in which a site can become Cu ~+. kTeis then acting in a sub-space of ~o. The transfer operators are modified for transfer between the two sub-lattice are forbidden. Within

359

a given sub-lattice, such as the A sub-lattice, the transfer terms take the form

Z {Tmt, m2a*,amz +h.c.} and by a unitary transformation they can be brought into a canonical form,

Z eka~ak. The electron-electron terms can be expressed in terms of these new canonical operators, and this is advisable because there is then no scattering between conduction states by the transfer terms. It has seemed useful to go into the above description in some detail because there is now some scope for introducing a different prescription for choosing the A and B sub-lattices, to which the above arguments can be applied. Suppose then that the copper sites are arbitrarily allocated to A and B sub-lattice and the a * and b* operators are defined according to the above prescription. The effect is to make a different choice of sub-space of ~o, and having done so the ~o bordering operators can be removed. The transfer processes again operate within the separate sub-lattices and within each there will be a band structure, though no one described by k vectors. It will, unlike the antiferromagnetic case, contain some of the transfer processes between adjacent copper sites and so is likely to be broader. Against this some of the nearest neighbour antiferromagnetic exchange interactions have been removed, and the question therefore arises as to whether it is possible to choose A and B sub-lattices, with the optimum pairing, such that the ground state expectation value is below the best that can be achieved with A and B being the standard antiferromagnetic choice. Any broadening of the conduction band is favourable as far as the electron kinetic energy is concerned, but unfavourable as far as the exchange interaction is concerned. But beyond both of these considerations is the matter of optimimum pairing, which is now between electrons in different sub-lattices. Since with the antiferromagnetic choice it seems likely that a low mean energy state can be obtained using optimum pairing it would be surprising if an even lower state cannot be obtained by a more sophisticated choice of the A and B sub-lattices, again with optimum pairing. Any state formed in the above way will have a high degree of correlation between the electrons on the

360

E. Sigmund and K. 14iH. Stevens / E{lective Hamiltonians jbr high-7~ superconductors

two sub-lattices a n d it s e e m s n o t i m p r o b a b l e that if there is o n e such low-lying state there will be others o b t a i n e d by d i f f e r e n t c h o i c e s o f the A a n d B sub-lattices. I f so, there is t h e n the v a r i a t i o n a l g r o u n d stat w h i c h is c o n s i s t e n t with such an a r r a n g e m e n t has the d r a w b a c k that it m a k e s the c o n d u c t i o n take place by next-nearest n e i g h b o u r h o p p i n g , w h i c h is relatively u n f a v o u r a b l e energetically, e v e n t h o u g h there can be an isolated state o f BCS type. It therefore seems likely that there is s o m e o t h e r v a r i a t i o n a l state o f e v e n lower energy which, therefore, is n o t o f the c o n v e n t i o n a l a n t i f e r r o m a g n e t i c character. In fact e x p e r i m e n t a l e v i d e n c e is a c c u m u l a t i n g t h a t the c o p p e r - o x i d e c o m p o u n d s are e i t h e r a n t i f e r r o m a g n e t i c or s u p e r c o n ducting. It w o u l d be i n t e r e s t i n g to k n o w w h a t the c o n d u c t i o n b a n d w i d t h s are, for the t h e o r y w o u l d suggest that a n t i f e r r o m a g n e t i s m is to be a s s o c i a t e d with a n a r r o w b a n d w h e r e a s the s u p e r c o n d u c t i v i t y , e v e n a b o v e To, will be a s s o c i a t e d w i t h a w i d e r band. T h e w o r k by T o r r a n c e et al. [ 18 ] is p a r t i c u l a r l y interesting in this context, for it has b e e n s h o w n that there are a n u m b e r o f c o p p e r - o x y g e n c o m p o u n d s w h i c h are structurally s i m i l a r to the high-To ones, w h i c h are c o n d u c t o r s but not s u p e r c o n d u c t o r s d o w n to a b o u t 5 K. So far there is no i n f o r m a t i o n a b o u t the c o n d u c t i o n b a n d w i d t h s n o r a b o u t possible antiferromagnetism. In an earlier p a p e r [ 6 ] it s e e m e d that the e v i d e n c e was in f a v o u r o f Cu 2+, Cu 3+ a n d 0 2 sites, w i t h the Cu ~+ h a v i n g spin o f unity. W h e n the p r e s e n t w o r k was b e g u n it t h e n s e e m e d that C u 3+ was m o r e likely to h a v e S = 0. It n o w s e e m s t h a t o p i n i o n has m o v e d t o w a r d s h a v i n g all c o p p e r sites as Cu 2+ with the charge v a r i a t i o n s b e i n g on the o x y g e n sites so that s o m e are O 1- . It is a c o m p a r a t i v e l y s i m p l e m a t t e r to change the g r o u n d m a n i f o l d ( , ~ ) to w h a t e v e r charge a r r a n g e m e n t is t h o u g h t to be m o s t a p p r o p r i a t e and the d e t a i l e d d e v e l o p m e n t will p r o b a b l y t h e n follow s i m i l a r lines to t h o s e we h a v e d e s c r i b e d . O t h e r cases will be e x a m i n e d in d u e course. F o r all such cases it can be e x p e c t e d that the e f f e c t i v e H a m i l t o n i a n will

include Coulombic, exchange and phonon induced e l e c t r o n - e l e c t r o n i n t e r a c t i o n s s i m i l a r to those used here.

Acknowledgements We w o u l d like to t h a n k M. Wagner, F.W. Sheard, G.A. T o o m b s and R. H a u g for t h e i r interest and c o m m e n t s on s o m e o f the ideas p r e s e n t e d here.

References [ 1] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, O~ford. 1070/. [2] K.W.H. Stevens, Phys. Rep. 24C (1976) I. [3] K.W.H. Stevens, Magneto-Structural Correlations in Exchange Coupled Systems, eds. R.D. Willett, D. Gatteschi and O. Kahn (Reidel, Dordrecht, 1985) p. 105. [4] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 189. [5] B. Batlogg, R.J. Cava, A. Jayaraman, R.B. van Dover, G.~. Kourouklis, S. Sunshine, D.W. Murphy, LW. Rupp. H,S. Chcn, A. White, K.T. Short, A.M. Musjee and E.A. Rietman, Phys. Rev. Lett. 58 ( 1987l 2333. [6] E. Sigmund and K.W.H. Stevens, J. Phys. C.: Sol. St. Ph~s. 20 (1987) 6025. [7]T.A. Faltens, W.K. Ham, S.W. Keller, J. Leary, J.N. Michaels. A.M. Stacy and H-C. Loye, Phys. Rev. Lett. 59 (1987) 915. [8] O. Warnier. Phys. Rcv. 52 (1937) 191. [9] K.W.H. Stevens, J. Phys. C.: Sol. St. Phys. 6 (1973) 291. [ 10] C. Bloch, Nucl. Phys. 6 (1958) 329. [I I ] J. Bardeen, L.N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1975) 1175. [ 12] A Messiah, Quantum Mechaniscs. Vol. II (North-Holland. Amsterdam, 1962) p. 717. [13] E. Sigmund, R. Ruckh and K.W.H. Stevens, Physica B+(' 146 (1987) 312. [14] H. Fr(Shlich. Proc. Roy. Soc. A 215 (1952) 291. [ 15] S. Robaszkiewicz, R. Micnas and J. Ranniger, Phys. Rev. B36(1987) 180. [16] D.K. Ray, Phil. Mag. Lett. 55 (1987) 251. [17]H. Suhl, Low Temperature Physics, eds. C. DeWitt, B. Dreyfus and P.G. de Gennes (Gordon and Breach, New York, 1962 ) p. 233. [ 18 ] J.B. Torrance, Y. Tokura, A. Nazzal and S.S.P. Parkin, Phys. Rev. Lett. 60 (1988) 191.