Doped antiferromagnetic insulators : a model for high temperature superconductivity

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Doped antiferromagnetic insulators : a model for high temperature superconductivity

N.F. Mott

To cite this version:

N.F. Mott. Doped antiferromagnetic insulators : a model for high temperature superconductiv-

ity. Journal de Physique, 1989, 50 (18), pp.2811-2822. �10.1051/jphys:0198900500180281100�. �jpa-

00211104�

Doped antiferromagnetic insulators : ^a model for high temperature superconductivity

N. F. Mott

Cavendish Laboratory, Cambridge, ^G.B.

(Reçu ^{le 6 mars 1989,} accepté ^sous forme définitive ^{le 29 mai} 1989)

Résumé.

Une esquisse ^{de la nature} de la transition métal-isolant se produisant lorsque ^les composés antiferromagnétiques ^non métalliques LaVO3 ^et La2CuO4 ^sont dopés ^{avec du strontium}

ou du baryum ^est donnée. Il est suggéré, ^{en accord avec} d’autres auteurs, qu’un gaz dégénéré ^de polarons ^de spin ^{est formé.} Ce gaz, particulièrement ^{dans des} structures bidimensionnelles, _peut conduire à la formation de bipolarons, constituant des paires de bosons donnant naissance à la

supraconductivité ^{à haute} température ^{dans ce} composé ^{et d’autres} oxydes ^{du même} type.

Abstract.

An outline is given ^{of the nature} of the metal-insulator transition when the

antiferromagnetic non-metals LaVO3 ^or La2CuO4 ^are doped with strontium or barium. It is

suggested, following ^{other authors, that a} degenerate gas of spin polarons ^is formed, ^which particularly ⁱⁿ two-dimensional structures, ^can form bipolarons, and that these could be the boson

pairs which in the latter and similar oxides give ^{rise to} high temperature superconductivity.

Classification

Physics ^Abstracts

74.65

74.70H

71.30

1. Introduction.

The first of the high-temperature superconductors ^{to be} discovered was La2Cu04 doped ^with

strontium or barium (Bednorz and Müller [1]). This material is now known to be an

antiferromagnetic insulator with the magnetic ^{Cu2 + ions} arranged antiferromagnetically ^on

the CU02 planes ^{of the} layer ^{structure and a Néel} temperature ^{of c. 200 K. On} doping ^with

one of the divalent metals the ordered state of the moments rapidly disappears, ^{the Néel} temperature dropping ^{to zero,} and the material has the properties ^{of a} spin glass [2], ^the

moments having ^{fixed or} slowly varying orientations. Further doping ^{leads to a metal-}

insulator transition, and in the metallic state the material is superconducting.

The aim of this paper is ^to examine the nature of the metal-insulator transition in doped antiferromagnetic oxides, ^{and to} compare the properties ^{of the} superconductors with those of other oxides such as La, -.,Sr,,V03 which also show ^a metal-insulator transition. This material is not, ^{as far as is} known, ^a superconductor, ^{and does not have a} layer structure. We examine the possibility that the carriers in all these materials ^are spin polarons, and that the concept ^of

a degenerate gas of spin polarons ^{is a useful} approximation with which to describe their

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180281100

properties. Several authors (1) ^have suggested ^that polarons ^{can combine to form} bipolarons,

and that these are the boson pairs needed for superconductivity ; ^we shall examine this

hypothesis ^for spin polarons.

2. Metal-insulator transitions.

Two kinds of metal-insulator transition are relevant to this discussion. The first 1 call a Mott

transition, the second an Anderson transition.

1 define a Mott transition as one, in a crystalline system, ^{in which a} change of volume under pressure, ^a change ^of temperature ^{or of} composition ^{in an} alloy ^{leads to a} transition from ^an

antiferromagnetic ^or possibly RVB insulator to a metal, which may be antiferromagnetic ^or

may be ^« highly correlated ^» in the sense of Brinkman and Rice [7]. Such transitions ^are first order, ^{and are} necessarily accompanied by ^a change in volume and sometimes by ^a change ^of

structure. The treatment given by ^Mott [8] in 1949 is no longer relevant and following

Hubbard [9] ^we suppose that the transition ^occurs when two « Hubbard bands » overlap, ^so

that if their widths are Bl, B2, ^{and a} tight binding ^{treatment is} valid, the criterion should be

if the influence of any change ^{of structure or volume is} neglected. ^Here Bl, B2 ^{can be} equated

to 2 Ztl, ² Zt2 ^{where Z} is the coordination number and tl, t2 ^are the transfer integrals ^{for the}

two Hubbard bands. In general t2:> t1. U is the Hubbard intra-atomic energy (e2/Kr12>.

Even _so, equation (1) ^{is not exact, since when} long-range interaction is taken into account, ^all transitions of this kind are shown to be first order (Brinkman ^{and Rice} [10]). Typical examples ^{of these} transitions are observed in V203 under pressure, temperature ^{or on} addition of Ti203, and in the series Ni(Sl-xSex)2 (Wilson [11]).

The influence of disorder on Mott transitions has been widely discussed ; ^{it was} originally thought ^{that the} transition in doped ^{silicon was of this kind,} if the material is not

compensated. However, following calculations by Bhatt and Rice [12] and certain experimen-

tal data, ^{it now} appears (Mott [13]) that in the conduction band of many-valley materials the disorder induces a kind of self-compensation, ^so the transition is of Anderson type. ^Whether this is so for p-type ^{materials or} single-valley semi-conductors is not known.

Turning ^{now to Anderson} transitions, Anderson’s [14] paper of 1958 and research based ^on it showed that, ^{in the} approximation ^of non-interacting electrons, ^a degenerate electron gas in

a non-periodic field would undergo ^a second order transition to a non-conducting ^{state as the}

ratio of a disorder parameter Vo to ^the original band-width B is increased or as the mobility edge Ec passes through the Fermi energy EF. ^The scaling theory [15] of 1979 and much

experimental ^evidence showed that the zero-temperature conductivity ^increases linearly ^from

zero as Vo/B decreases from the critical value, ^there being ^no minimum metallic conductivity.

So if ^a is the conductivity ^{and x the} concentration,

with v = 1. Many-valley uncompensated semiconductors seem to be exceptions ^for n-type conductors ; for these the effects of long-range interactions are very important, and have been

(1) ^Edwards [3], Cyrot [4], ^Kamimura [5], Su and Chen [6], ^Emin [58], Kamimura et al. [59], ^de

Jongh [63].

studied extensively ^{in the} present decade ; they ^are probably responsible [16] ^{for the}

behaviour with v = 1

2

However calculations by Schreiber et al. [65] suggest that, ^for non-interacting electrons,

v 1.6.

The theory however is most directly applicable ^to compensated semiconductors, ^{where the}

lower Hubbard band is only partly filled. Work in Professor Friedel’s laboratory [17] ^{has been}

instrumental in showing that the transition takes place ^{in an} impurity band, ^{to which a} tight- binding model and the Anderson localization theory ^{can be} directly applied. ^Calculated

values of the electron concentration at which the transition _occurs, using the Anderson model

or that based on the Hubbard U, give very similar values for the critical concentration

here aH is the hydrogen ^{radius of a} donor. The reason is that the one depends ^on the ratio of B to U, the other of B to Vo, ^{but both occur in the} logarithm ^{of a} fairly large number. This is

probably why the well-known plot, reproduced ⁱⁿ figure 1, given by Edwards and Sienko [18]

shows a constant value of the quantity (3), namely ^0.26, though the transitions sampled may well be of both kinds.

Fig. ^1.

Plot of effective radius in equation (3) against log ne (Edwards and Sienko [18]).

As first pointed ^out by Alexander and Holcomb [19], ^{for a} value of n about three time nc the impurity band, in which the upper and lower Hubbard bands ^are already merged,

merges in its ^turn with the conduction band.

The non-conducting ^state for n .-- nc has also been extensively investigated, particularly ⁱⁿ

the ^« intermediate ^» range below the transition where the density ^{of states at the} Fermi energy remains finite and conduction at low enough temperatures ^is by variable-range hopping. ^The

material is not an amorphous ferromagnet, but rather similar to a spin-glass, ^{the moments} being frozen in random directions. It was suggested ^first by ^{de Gennes} [20] ^{and later} by ^the present ^author [21] ^{that an} electron with energy ^at the mobility edge may effect the orientation of the electrons on neighbouring sites, forming what is called a spin polaron ; ^if

the material were antiferromagnetic ^this might be necessary ^to give sufficient mobility

because an electron could not move to a nearest neighbour ^{site without} changing ^its spin

directions. For a spin glass, ^{on the other} hand, ^{a transfer} integral ^I involving ^{the whole} systems will, ^{on each atomic} site, ^{include wave} functions for both spin directions, though ^if

the number of sites within the polaron ^is large, I ^is likely ^{to be small,} giving ^a high ^effective

mass.

3. Metal-insulator transitions in antiferromagnetic insulators.

We have no reason to believe that ^a metal-insulator transition in LaV03 ^or La2Cu04 doped

with Sr or Ba is of Mott type. The former (see Fig. 2) ^{has been} extensively ^discussed by ^the present ^author [21] ^and by Doumerc et al. [22] ^{as an} Anderson transition but perhaps ^with

insufficient attention to the possible formation of spin polarons. By ^a spin polaron ^{we mean a}

situation in which a carrier is surrounded by ^a region in which the antiferromagnetic ^{order is}

broken down ; this may merge gradually ^{into an ordered} region. First, then, ^we give ^evidence

for the existence in at least one system ^of spin polarons. The clearest ^comes from the work of

Fig. ^2.

^{Variation of} the electrical conductivity ^of La, -,,SrV03 ^with 1 / T (Dougier ^and Hagenmuller

[57]).

Von Molnar and Penney [23] ^on Gd3 _ xVxS4. ^{Here V} stands for a gadolinium vacancy in the cubic structure, the concentration of vacancies being high, ^and through their random

positions producing Anderson localized states in the conduction band. Charge neutrality

ensures that x electrons per Gd ^{atom are} in the conduction band, forming ^a degenerate

electron gas ; ^at low temperatures ^the conductivity ^is low, tending ^{to zero with} temperature.

In a magnetic field, however, ^a transition to metallic behaviour takes place, ^{which seems of}

standard Anderson type ; this is shown in figure ^3. Magnetic ^{fields can cause} transitions in

doped semiconductors from metal to insulator ^or vice-versa, ^{as first} pointed ^out by Shapiro [24], by suppressing ^the quantum interference effect and thus increasing ^{0-, or} by shrinkage ^or

orbits in doped semiconductors and decreasing u. Here, however, ^the accepted explanation ^is

that the carriers form antiferromagnetic spin polarons ^{with the moments on the Gd} ions, ^and

this increases their effective mass and so allows Anderson localization. The system ^{then is} what Anderson has called a « Fermi glass ^»

^{that is a} Fermi distribution of states described

by ^{localized wave} functions. In the presence of ^a strong field, however, ^{the Gd moments are}

oriented parallel ^{to it, no} magnetic polarons ^{can form and so} the effective mass drops ; ^the mobility edge, initially deep down in the occupied ^{states, rises} through EF, leading ^{to an}

Anderson transition to metallic values of the conductivity.

Fig. ^3.

Dependence ^{of the} conductivity ^of Gd3 _ xVxS4 ^{in a} magnetic ^{field at T}

^{300 mK} (Von ^Molnar

and Penney [23]).

This work, then, shows that ^{a «} degenerate gas of spin polarons ^{» can} exist, ^{and can be an} acceptable approximation ^to the behaviour of a degenerate electron gas. It is interesting ^to

ask whether the same can be said for a degenerate gas of dielectric polarons. ^The present

author [25] ^has proposed ^{that the} slightly ^reduced crystalline ^materials SrTi03 ^and

KTa03 ^must be described in this way. The former shows metallic conductivity ^{if the} density ^of

carriers is greater ^{than 3} x10 18 CM- 3. According ^to equation (3), ^this implies ^a very large

value of the hydrogen ^radius.

The materials have very high static dielectric constants, ^{so we} deduce that ^K in (4) ^{must be the}

static dielectric constant. This can only ^{be so} if both the positive defects and the carriers are

able to polarize ^the medium, in other words that the carrier forms a (dielectric) polaron. ^Thus

the force between them will be e2/ K r2. ^In real space the polaron ^{must be} small, ^{to ensure that}

the force attracting ^{it to a} positive charge ^is e2/Kr2, ^not e2/ K o r2, ^{where K () is the} high frequency ^{and K} the static dielectric constant. At the same time the mass enhancement must not be too great ; Eagles [26] has estimated it as 7 - 10 me, ^so the polaron ^{should not be too}

small.

We now look further at the substance Lal _ xSrxV03. The strontium produces ^{a hole, which}

has been assumed until now to move in the vanadium d-band, and is thus an electronic

configuration 3d1 moving through ^the antiferromagnetic lattice of 3d2 states. If this is what is

formed, ^{it seems to us} highly likely ^{that a} spin polaron ^will result ; otherwise, ^as argued above, the carrier would need to move to next nearest neighbours, giving ^a very low mobility.

There remains the possibility however, ^{as seems to be the case for the} superconductors, ^that

the carrier is ^a hole in the oxygen p-band (see discussion below), ^{in which case this} argument would not be so strong. However, in either case the formation of a spin polaron ^seems possible ^with relatively large ^{effective mass,} of order 10 me.

For low concentrations of Sr, the carrier will be trapped by ^the negative charge produced by

the substitution for a Sr2 + ^{for a} La3 + ion.

As the concentration increases, ^the trapping will be of Anderson type ; EF ^{will increase,} Ec - EF diminish and a conventional Anderson transition occurs.

We have to ask, however, whether the transition takes place ^{in an} impurity band, ^or

whether the impurity and conduction band have merged. ^{We think it} unlikely ^that there is any

self-compensation here, ^as there appears ^to be in many-valley conduction bands. The

transition, if it took place ^{in an} impurity band, ^would therefore be of Mott type, taking place

when U - B. If U is too large ^{to allow} this, B ^{for an} impurity ^band being ^{small, it cannot occur}

until the bands have merged. It will then occur in the merged ^{band when} EF

Ec. ^{As the}

number of carriers increases, EF ^moves away from the band edge ^{as does the} mobility edge Ec, which results from increasing disorder. It is supposed ^that EF ^catches up with

Ec. ^If spin polarons ^{are formed,} EF ^and Ec ^{refer to the} degenerate gas of spin polarons.

There is an interesting difference between Lal _ xSrxV03 ^{and the} superconductor La2-,,Sr,,CU04. In the former [22], ^on adding ^{Sr, the Néel} temperature ^{does not} drop ^much

until the transition, ^{and then} antiferromagnetic ^order disappears. ^That Ec EF goes

continously ^{to zero} suggests ^an Anderson transition. The different behaviour of the Néel temperatures is doubtless a consequence of the ^two dimensional nature of the latter material.

Also in the vanadate at the transition the thermopower changes sign and becomes negative, suggesting ^a normal, ^{not a} p-type metal. In the insulating ^{state the} activation energy for conduction varies as (xc - x )1.8, ^{which is near} the index expected for classical percolation theory rather than the normal index v

1. Perhaps ^a rapidly changing ^{effective mass due to} spin polarons ^{could be} responsible, ^{if as we} suppose these ^are formed. There are several

unexplained ^features here, perhaps ^{also related to the} merging ^{of an} impurity ^{bad into a}

conduction band ; ^{the rate of} delocalization could be increased ^as Meff decreases.

In La2 - xSrxCu04, ^{on the other} hand, TN drops rapidly ^with increasing x with the formation

of a spin glass. ^{This seems to me} by ^{no means} surprising ; ^the spins ^are under the influence of

their neighbours ^{and the} trapped carriers, ^and TN ^should drop. ^{When it} disappears ^{a random} spin glass (localized ^random spins) ^remains (2) . ^But metallic behaviour must await the

merging ^of impurity with the valence bands for the spin polarons ; ^{when this} happens ^there

will be many sites for each carrier.

Many of the other high temperature superconductors ^{can, we} suggest, be described in the

same way. Thus YBa2Cu3O7 - x ^{should be} insulating ^{when x} =1/2 ^{while with} x > 1/2 holes are

2 2

introduced into the oxygen p band, ^so forming spin polarons with copper ions, ^{and a metal-}

insulator transition ^can take place.

The interesting ^{case of} Bal - xKxBi03 apparently ^{shows no} magnetic ^order [28] ; ^{it is a} superconductor when x > 0.25. We suggest ^{that the moments are} Bi4 + (it ^is interesting ^that they ^form Bi3 + and Bi5 + for x 0.25), and that for the insulating composition they ^{form a}

RVB state.

Anderson’s proposal [29] ^that insulating ^{RVB states} exist is verified by materials such as

TiBr3 ^{which are} insulators but show no antiferromagnetic ^{order but a small} paramagnetism nearly independent ^of temperature (Wilson ^{et al.} [30], ^{Maule et al.} [31]). Also calculations such as those of Kohmoto and Friedel [32] ^{and Gros} [33] suggest that the RVB state can be stable. In the metallic state, however, ^its significance is less clear (see § 5).

4. Bipolarons.

As already ^stated, photoemission observations suggest ^{that in} La2 - xSrxCu04 the carriers are

holes in the oxygen valence band, ^{or at} any ^rate in a band with strong oxygen 2p ^{content, as}

shown by X-ray absorption [34, 35]. ^{It seems to} the author improbable that there will be strong hybridisation ^{for the states} occupied by these holes. We consider hybridisation

between ^a 3d e. ^{state in the} Cu ions and the 2p oxygen, before the Hubbard U is introduced for the former ; this should be as in figure 4 ; the dotted lines represent ^the hybridised ^{states and} N(E) is finite at the Fermi energy. The Hubbard U will produce antiferromagnetism and open up ^a gap in the Cu d-band, ^{and the wave} functions here may have considerable p admixture. But the holes introduced by doping will, ^{after U is} introduced,

lie at P, where the 3d component is small. So in our view a relative position ^{of the two} bands,

which leads to strong hybrisation ^{for the d} functions, necessarily locates the holes in a part ^of the p-band ^where hybridisation is small. These we suggest ^form spin polarons with the Cu 3d moments, which combine into bipolarons ^{which are the} pairs needed for superconductivity.

Fig. ^4.

Showing hybridisation between the 2p oxygen band and that for motion of 3d8 through ^the

copper 3d9 states.

(2) See Maekawa et al. [60] « Motion of holes in magnetic insulators » who maintain that each hole

disturb ’TT’ 2 t / J spins, that t > J and so as little as one per ^cent of holes can disorder the spins.

We review first the evidence for the existence of dielectric bipolarons in certain other materials. One asks first whether under any circumstance ^two such polarons ^can combine,

because at large ^distances they ^must repel each other. Of course in a sense Cooper pairs ^are bipolarons, but the correlation length ^{is so} large that any repulsion is screened out. The clearest evidence for dielectric bipolarons in the literature comes from the work of Lakkis et al. [36] ^on Ti407 ; here every other Ti ion carries ^an electron, ^so every Ti3 + must have

neighbours ^{with the same} charge. Repulsion ^cannot prevent the formation of bipolarons

which give ^{over a} range of temperature ^{an activated} conductivity ^{but no} paramagnetism. ^The binding energy which forms these bipolarons ^is probably ^of homopolar type, ^{so in a sense it is}

a spin-dependent type, ^caused by ^the exchange ^of antiparallel spins. Other evidence does however suggest that isolated bipolarons ^can form ; ^{if so,} the interaction energy ^must be of the form shown in figure 5, with the attractive force of homopolar type. ^{In this} case, ^as with any alternative force localization will clearly ^{be more} probable ^{in two} dimensions than in three.

Fig. ^5.

Showing ^the potential energy between ^two polarons (dielectric ^or spin) ^if bipolarons ^{can form.}

We have however no direct evidence for the formation of spin bipolarons apart ^{from the} existence of superconductivity, with in the planes ^a small coherence length (-20 Â), ^and

even smaller across in planes (3).

Su and Chen [6] consider the formation of bipolarons from carriers with parallel ^and antiparallel spins, finding ^{that both can be} stable, particularly ^{in two} dimensional structures, because localization is always ^{easier in one or two} dimensions. Essential to ensure the validity

of such a model, ^{it has to} be shown that the spin interaction can overcome the Coulomb

repulsion, ^{as in} figure ^{5. This must occur, whether or not our} polaron ^{model is a} good one,

given ^that superconduction ^occurs with the observed small correlation length. The calculation of Su and Chen, suggesting ^a cigar shaped form for the polaron ^{in the} CU02 planes, ^{makes the} problem ^{of the} bipolaron almost one-dimensional and thus favours its stability ^{even more} strongly than another form would do.

Islam et al. [39] ^have attempted ^to calculate whether a spin bipolaron ^can form ; though they ^{cannot obtain a} positive binding energy, they consider it not impossible.

We next ask whether all the spin polarons, that it all the carriers, ^form bipolarons, ^or only ^a fraction, that is those in the upper part of the Fermi distribution. Another question is,

(3) Worthington ^{et al.} [37], Gallagher [38].

whether a bipolaron, moving through ^{the 3d9 states,} will suppress the antiferromagnetism,

since the influence on the spins ^{round a} bipolaron will be weaker than that round a polaron.

This point needs further investigation. Anderson’s model does involve, ^{as well as} bosons, fermions though ^{his, unlike ours,} carry ^no charge. ^We think however it is more likely ^{that the}

transition temperature ^{is that at} which the Bose gas becomes non-degenerate, and if this is _so,

according ^{to Prelovsek et al.} [64], the effective mass is in the range 20-40 me.

In the superconducting region the transition temperature Tc first rises with increasing

x and then drops. ^{If the} transition temperature ^{is that at which the} bipolaron dissociates, ^then

for concentrations near the Anderson transition the wave functions of any carrier according ^to

Mott [40] ^fluctuate strongly ^{within a} length e, which tends to infinity (4) ^as 1/(EF - E,,). ^{It is} likely that such fluctuations impede pair formation, and also because below EF ^{the wave}

functions remain localized. For high ^{values of x,} doubtless the pairs ^can impede each other’s formation.

5. Comparison with other models.

A degenerate gas of spin polarons ^{is a} theoretical model very similar ^to the highly ^correlated

electron gas introduced by Brinkman and Rice [10] ; in both models a small number of carriers _moves, causing ^the moments to resonate between their possible orientation. In both there is a mass enhancement. The quantity X (q, w ), ^of importance ^{for neutron} diffraction,

should we believe vary with q only when - hw - EF, ^where EF is the width of the band of

occupied ^{states reduced} by this interaction.

The model would suggest ^a smaller correlation length ^than actually observed. A detailed examination by Liang [41] in which the spin polaron ^is envisaged ^as causing ^local strains,

which in their turn are responsible ^{for the} scattering, suggests ^that coupling ^{should be}

between more distant sites. Liang describes his bipolarons ^as spin-phonon induced, ^{and some}

small isotope effect is to be expected. ^{Here the} spin produces ^a distortion, ^{which is} responsible ^for part of the cohesion. Fujimari [62] gives ^a discussion of various ways in which

a spin polaron ^can give cohesion. Calculations are needed to give quantitative values. The

binding energy may be much higher ^than kTc, ^so that above Tc ^{the carriers are still} bipolarons.

There is much evidence, ^to be reviewed elsewhere [65], that above Tc the carriers are very

heavy, ^as spin bipolarons might ^be.

6. Some relevant expérimental ^evidence.

This paper has taken the concept ^{of a} doped ^{« Mott »} insulator, and asked whether it can

produce superconductivity, and found that Cooper pairs ^with high binding energy, of order

kTN, ^are possible. This final section summarises some of the relevant experimental ^evidence.

It seems certain that pairs (Cooper pairs, ^holons, bipolarons ?) exist with charge 2 e ; ^the experimental evidence is summarized in the review by Bednorz and Müller [42].

Experiments ^{on neutron} scattering ^{and muon} spin rotation is stated to show that magnetism plays ^an essential role in the Cu02-based superconductors. Some of the evidence is given by Birgeneau ^{et al.} [43], ^{who state that for} L2-xSrxCu04 ^{the Cu 2+ moment is} independent ^of

x but the Cu 2+ spin-spin correlation function exhibits a dramatic x-dependence, being ^{of the}

order of the distance between the holes. This is in accord with the ideas expressed ^{in this}

paper. Greene et al. [44] show that the susceptibility ^above 7c ^{is enhanced} by ^electron correlations, ^{as we} expect. ^{A small} isotope effect is observed [45] ^{which must in our view}

(4) ^A parameter which tends to infinity ^{at a} second-order transition appears ^to be a general ^{feature of}

such transition, ^{as at} critical or Curie points.

accompany the formation of ^a spin polaron sharply localized in space. In the non-metallic state, in crystals ^of La2 - ySryCu1-xLi204 - ³ variable range hopping is observed [46] ; ^thus N (EF) ^{must be} finite, with Anderson localization at EF. Endo et al. [47], ^{in work on neutron} scattering, say that the large spin fluctuations give ^{credence to} the models in which the pairing

is magnetic ⁱⁿ origin, ^but according ^to Birgeneau et ^al. [48] ^{there is no} significant ^difference

between the superconducting and normal states. Mawdsley ^{et al.} [49] ^{consider that the} thermopower ^of YBa2CU307or- ^{is much} enhanced ; they suggest ^an enhancement of the effective mass by phonons ^{but in our view} spins ^{may be} equally responsible, ^{as in our model of} spin polarons.

Torrance et al. [50] have found that for La2-,,Sr,,CuO4 annealing in oxygen ^{to remove} oxygen vacancies leads ^{to a constant} value of Tc(36K) for x between 0.15 and 0.25 ; superconductivity disappears ^{at x}

^0.34 though ^{holes are} formed till x

0.4. Without

annealing they claim that increasing x beyond ^0.15 just adds oxygen vacancies rather than holes.

The material [51] YBa2CU307-y ^{seems to have no magnetic system,} if the copper is

envisaged ^{as a metallic} system made up of ^two Cu2 + and one Cu3 + , ^as might ^{be the case for} highly compressed Fe304(Wilson [52]). ^{It seems to us, however, that the Cu3 +} may be absent, being replaced by holes in the oxygen valance band. If so, this material may be classed with the others described here.

The newly discovered n-type superconductor [53, 54] La2 - xCexCu04 might fall into the

same pattern ^{with of} course no holes in the valence band and the stable [52] ^{Cu ion} (3d )1° forming ^a spin polaron ^{with the} surrounding ^3d9.

From the Zurich school, Takahashi and Zhang [55] ^{show, in} agreement ^{with our} postulate,

that a small concentration of holes in the Cu3d9 states on the quadratic ^{lattice can} destroy ^the antiferromagnetic order. On the other hand Zhang ^{and Rice} [56] consider that ^a hole in the oxygen p-band ^can combine with a Cu 3d9 ion to form ^a singlet, ^{which is a kind of} spin polaron. ^{We think} that, ^{whether a} singlet ^or triplet ^{is formed,} it will disturb the spin ^{of its} neighbours, forming ^a spin polaron ^and breaking down the AF order.

References

[1] BEDNORZ J. G. and MÜLLER K. A., Phys. ^{Rev. B 64} (1986) 189 ; ^{Rev. Mod.} Phys. ⁶⁰ (1988) ^585.

[2] BIRGENEAU R. J., KASTNER M. A. and AHARONY A., ^Z. Phys. ^{B 71} (1988) ^57.

[3] EDWARDS D. M. and MIYAZAKI M., ^J. Phys. ^{F. 17} (1987) ^L-311.

[4] ^CYROT M., Solid State Commun. 62 (1987) 82 ; ^{Nature 63} (1987) ^1015.

[5] ^{KAMIMURA H.,} MATSUMO S. and SAITO R., Solid State Commun. 66 (1988).

[6] SU W. P. and CHEN X. Y., Phys. ^{Rev. B 38} (1988) ^8879.

[7] BRINKMAN W. F. and RICE T. M., Phys. ^{Rev. D 2} (1970) ^4302.

[8] ^{MOTT N.} F., ^Proc. Phys. ^{Soc. A 62} (1949) ^416.

[9] ^{HUBBARD J.,} Proc. R. Soc. A 277 (1964) ^237.

[10] BRINKMANN W. F. and RICE T. M., Phys. ^{Rev. B 7} (1973) ^1508.

[11] WILSON J A. in : ^« The Metallic and Non-metallic States of Matter ^» (Taylor & Francis, London) 1985, p. 215.

[12] ^{BHATT R. N.} and RICE T. M., Phys. ^{Rev. B 33} (1988) ^1920.

[13] ^{MOTT N.} F., ^Philos. Mag. B ⁵⁸ (1988) ^{369 ; 1989 ibid} (in press).

[14] ^{ANDERSON P. W.} Phys. ^{Rev. 109} (1958) ^1492.

[15] ^ABRAHAMS E., ANDERSON P. W., LICCIARDELLO D. C. and RAMAKRISHNAN T. V., Phys. ^Rev.

Lett (1979) ^673.

[16] ^KAVEH M., ^Philos. Mag. B ⁵² (1985) ^45.

[17] ^{JÉROME D., RYTER C.,} SCHULZ H. J. and FRIEDEL J., ^Philos. Mag. B ⁵² (1985) ^403.

[18] ^{EDWARDS P. P.} and SIENKO M. J., Phys. ^{Rev. B 17} (1978) ^2575.

[19] ^{ALEXANDER M. N.} and HOLCOMB D. F., Rev. Mod. Phys. ⁴⁰ (1968) ^815.

[20] ^{DE GENNES P.} G., ^J. Phys. ^{Radium 23} (1962) ^630.

[21] ^{MOTT N. F., «} Metal-Insulator Transitions ^» (Taylor & Francis, London) ^1974.

[22] See for instance DOUMERC J. P., POUCHARD M. and HAGENMULLER P., The Metallic and Non- metallic States of Matter » (Taylor & Francis, London) ^1985, p. 287.

[23] VON MOLNAR S. and PENNEY T. in ^« Localization and Metal-Insulator Transitions _», (Mott Festschrift ³ (1985) p. 183, Institute of Amorphous Studies Press (Plenum ^Press, N.Y.) ;

for original ^{work see WASHBURN} S., ^{WEBB R.} A., ^{VON MOLNAR} S., HOLTSBERG F., FLOUQUET J.

and REMENJI G., Phys. ^{Rev. B 30} (1984) ^6224.

[24] ^{SHAPIRO B.,} Phys. Rev. B 25 (1982) ^426.

[25] ^{MOTT N. F., Adv.} Phys. ¹⁶ (1967) ^49.

[26] ^{EAGLES D.} M., Phys. ^{Rev. 178} (1969) ^668.

[27] ^{AHARONY A., BIRGENEAU R. J., CONIGLIO A.,} KASTNER M. A. and STANLEY H. E., Phys. ^Rev.

Lett. 60 (1988) ^1330.

[28] D’AMBRUMENAL N., Nature 335 (1988) ^113.

[29] ANDERSON P. W., Science 235 (1987) ^1196.

[30] WILSON J. A. MAULE C. H., STRANGE P. and TUTHILL J. N., J. Phys. ^{C 20} (1987) ^4159.

[31] ^{MAULE C. H.,} TOTHILL J. N., STRANGE P. and WILSON J. A., J. Phys. ^{C 21} (1988) ^2153.

[32] ^{KOHMOTO M. and FRIEDEL J.,} Phys. ^{Rev. B 38} (1988) ^7074.

[33] ^{GROS C.,} Phys. ^{Rev. B 38} (1988) ^931.

[34] ^FUJIMORI E., TAKAYAMA-MUROMACHI K., ^{UCHIDA Y. and} OKAIDO, Phys. ^{Rev. B 35} (1987) ^8814.

[35] ^{SHARMA D. D.,} Phys. ^{Rev. B 37} (1988) ^7448.

[36] ^{LAKKIS S., SCHLENKER C.,} CHAKRAVERTY B. K., ^{BUDER B.} and MAREZIO M., Phys. ^{Rev. B 14} (1976) ^1429.

[37] WORTHINGTON T. K., GALLAGHER W. J. and DINGER T. R., Phys. Rev. Lett. 5 (1987) ^180.

[38] GALLAGHER W. G., ^J. Appl. Phys. ⁶³ (1980) ^4217.

[39] ^{ISLAM M.} S., ^LESLIE M., ^TOMLINSON S. M. and CATLOW C. R. A., ^J. Phys. ^{C 21} (1988) ^L109.

[40] ^{MOTT N. F., Philos.} Mag. ^{B 49} (1984) L-75 ; MOTT N. F. and KAVEH M., ibid 52 (1985) ^177.

[41] ^{LIANG W. Y.,} Physica, ¹⁹⁸⁹ (in press).

[42] BEDNORZ J. G. and MÜLLER K. A., Rev. Mod. Phys. ⁶⁰ (1988) ^585.

[43] BIRGENEAU R. J., KASTNER M. A. and AHARONY A., ^Z. Phys. ^B (1988) ^57.

[44] ^{GREENE R. L. MALETTA H.,} PLUCKETT T. S., ^BEDNORZ J. G. and MÜLLER K. A., Solid State Commun. 63 (1987) ^379.

[45] ZUR LOYE H. C., ^{LEARY K.} J., KELLER S. W., FALTERS J. T. A., MICHAELS J. N. and STACY A. M., Science 238 (1987) ^1558.

[46] ^{KASTNER M. A.,} BIRGENAU R. J., ^{CHEN C.} Y., ^{CHIANG Y.} M., ^{GABBE D.} R., JENSSEN H. P., JUNK T., ^PETERS C. J., PICORE P. J., TINEKE T., THURSTON T. R. and TULLER H. L., Phys.

Rev. B 37 (1988) ^111.

[47] ^{ENDO Y., YAMADA K.,} BIRGENAU R. J., GABBE D. R., JENSSEN H. P., KASTNER M. A., ^PETERS C. J., ^{PICORE P.} J., THURSTON T. R., ^{TRINQUADA J.} M., ^SHIRANE G., ^HIDAKA Y., ODA M., ENOMOTA Y., SUZUKI M., ^MURAKAINI T., Phys. Rev. B 37 (1988) ^7443.

[48] BIRGENEAU R. J., ^ENDO Y., ^HIDAKI Y., ^KAKAURAE K., ^KASTNER M. A., ^MURAKAMI T., SHIRAM G., THURSTON T. A. and YAMADA K. (1989, preprint).

[49] ^MAWDSLEY A., ^{TRODAHL H.} J., ^TALLEN J., SARFATI J. and KAISER A. B., Nature 1989 (in press).

[50] TORRANCE J. B., ^{NUZZAL A.} I., ^BEZINGE A., ^{HUANG T.} C., PARKER S. S. P., Phys. ^{Rev. Lett. 61} (1980) ^1988.

[51] ^MASSIDDA S., ^YU J., FREEMAN A. J. and KOEBLING D. D., Phys. ^{Lett. A 122} (1987) ^198.

[52] ^{WILSON J. A.} Private Communication (Jan. 1989).

[53] ^{TOKURA Y.,} TAKAGI H. and UCHIDA S., Nature 337 (1989) ^345.

[54] ^{EMERY V.} J., Nature 337 (1989) ^308.

[55] TAKAHASHI Y. and ZHANG F. C., ^Z. Phys. ^B (in press).

[56] ZHANG F. C. and RICE T. M., Phys. ^{Rev. B 37} (1988) ^3759.

[57] DOUGIER P. and HAGENMÜLLER P. J., Solid State Chem. 15 (1975) ^158.

[58] ^EMIN D. and HILLERY M. S., Phys. ^{Rev. B 37} (1989) ^4060.

[59] ^KAMIMURA H., ^MATSUMO S. and SAIKO R. in ^« Mechanics of High Temperature Superconduc- tivity ^» Springer Series in Materials Science II, H. Kamimura and ^A. Oskiyama ^Eds. (1988)

p. 8.

[60] ^{MAEKAWA J., INOVE J.,} MIYAZAKI M. in ^« Mechanics of High Temperature Superconductivity ^» Springer Series in Materials Science II, H. Kamimura and A. Oskiyama ^Eds. (1988) p. 60.

[61] FUJIMARI A. in ^« Mechanics of High Temperature Superconductivity ^» Springer Series in Materials Science II, H. Kamimura and A. Oskiyama ^Eds. (1988) p. 187.

[62] ^{DE JONGH L.} J., Physica ^{C 152} (1988) ^171.

[63] ^{PRELOVSEK P.,} RICE T. M. and ZHANG F. C., ^J. Phys. ^{C 20} (1987) ^L229.

[64] ^{SCHREIBER M., KRAMER B.} and MACKINNON A., ^{to be} published ⁱⁿ Proceedings of Condensed Matter Conference of EPS (Budapest) ^1989.

[65] ^{MOTT N. F.,} communicated to Physica (1989).