Splitting or not splitting of the Van Hove singularity in the high T c superconductors

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Splitting or not splitting of the Van Hove singularity in the high T c superconductors

J.P. Pouget, Claudine Noguera, R. Moret

To cite this version:

J.P. Pouget, Claudine Noguera, R. Moret. Splitting or not splitting of the Van Hove sin- gularity in the high T c superconductors. Journal de Physique, 1988, 49 (3), pp.375-381.

�10.1051/jphys:01988004903037500�. �jpa-00210707�

375

Short Communication

Splitting ^{or not} splitting ^of the Van Hove singularity ^{in the} high Tc superconductors

J.P. Pouget(1), ^C. Noguera(2) ^{and R.} Moret(1)

(1) Laboratoire ^{de Physique des Solides, Associé au CNRS, Bâtiment 510,} Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France

(2)DPhG/SPAS, ^{CEN Saclay, 91191} Gif-sur-Yvette Cedex, ^France

(Regu le !1 décembre 1987, accepte ^{le 20} janvier 1988)

Résumé. 2014 La transition de phase ^{entre les} structures tétragonale ^et orthorhombique (T-O) ^observée

dans le composé supraconducteur à ^haute Tc,La2-xSrxCuO4-y, ^est analysée. ^On montre que le mou-

vement d’inclinaison des octaèdres CuO6 ^{constitue le} paramètre ^d’ordre primaire ^{de cette transition}

de phase et que la symétrie du cisaillement induit _par la déformation carré-rectangle ^du plan ^{de base}

des octaèdres ne lève _pas la dégénérescence de l’anomalie de Van Hove dans la densité d’états. Les

conséquences ^sur le mécanisme de la transition de phase ^{T-O et les} variations de la température ^de

transition supraconductrice, Tc, ^{avec le} remplissage ^{de bande sont} analysées. La transition T-O, ^ob-

servée dans l’autre série de supraconducteurs k ^haute Tc, YBa2CU3O6+03B4, ^est brièvement discutée du

point ^{de vue de la} symétrie.

Abstract.2014 The tetragonal-orthorhombic (T-O) structural phase transition observed in the high Tc superconductor La2-xSrxCuo4-y ^{is analysed.} It is shown that the tilt of the CUO6 ^{octahedra is the} primary ^order parameter ^{of this} phase transition and that the symmetry of the shear distortion induced

by ^the square ^to rectangle transformation of the _oxygen basal plane ^{of the} octahedra does not allow a

splitting of the Van Hove singularity ^{in the} density ^{of states.} Consequences ^on the mechanism of the T-O phase transition and the variations of the superconducting temperature Tc with the band filling ^are analysed. The T-O transition observed in the other series of high Tc superconductors, YBa2Cu3O6+03B4

is briefly discussed from the ^same symmetry point ^{of view.}

LE JOURNAL DE PHYSIQUE

J. Phys. ^{France 49} (1988) ^{375-381 MARS} 1988,

Classification

Physics ^Abstracts

74.70 - 61.50 K - 71.25

Introduction.

Since the _very beginning ^{of the} discovery

of the new high Tc superconductors L82-xSrx Cu04-y [1] ^and YBa2CU-306+,6 [2], questions

have been raised concerning ^the possible ^in-

terplay between the occurrence of _supercon-

ductivity and the existence of structural phase

transitions. In this respect, ^{the two com-}

pounds ^{seem to behave} quite differently. ^In

the La2-xSrxCU04-y ^{series at ambient pres-}

sure, the highest Tc ^{values are} observed in the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004903037500

376

orthorhombic phase [3, 4] ^{close to} the critical concentration zc- 0.2 for the occurrence of the

orthorhombic ^to tetragonal (4-T) ^{phase transi-}

tion [5, 6]. ^{However, there is strong evidence}

that superconductivity subsists in the T phase

: i) ^{at ambient pressure for} z>xc [3] ; ii) ^un-

der pressure for ^xxc (tic ^{goes on} increasing [7]

even when the 0 distortion has been suppressed by pressure [8]. The situation is less clear in the YBa2Cus06+6 ^{series due to the} variation of the _oxygen content with temperature [9] ^{but it}

seems that superconductivity is best observed in the _range of stoichiometry ^{where the 0 distor-}

tion occurs (b > 0.5 according ^{to Ref.} [10]). ^Yet

when iron is substituted for copper, supercon-

ductivity subsists while the 0 distortion seems

to disappear [11].

In the framework of the weak electron-elec- tron coupling limit, mechanisms have been _pro-

posed ^for La2-xSrxCu04-y [12-14, 4], ^{in which}

the stabilization of the 0 phase ^{is due to a low-} ering ^of the band electronic _energy coming ^from

a splitting ^{of the Van Hove} singularity ^{in the}

orthorhombic phase. ^Also it has been suggested

that the relative shift of the Fermi level, EF,

with respect ^{to this} split anomaly, ^when dop- ing ^is varied, ^is responsible for variations of the

density ^{of states at EF and as a} consequence, leads to variations of the superconducting ^tem- perature Tc.

In this brief report, ^we reconsider the rel-

evance of these models for La2-xSrxCU04-y ⁱⁿ

the light ^{of a} proper study ^{of the} symmetry

on the 0 phase. ^We emphasize ^the following points : (i) ^the tilt of the octahedra, rather than the shear deformation of the CU02 planes ^{is the} primary ^order parameter ^{of the T-0} transition ; (ii) this deformation _preserves equal ^{Cu-0 near} neighbour distances in all the directions of the Cu02 plane ; ^its symmetry is such that there is

no splitting ^of the Van Hove singularity ^{in the}

0 phase ^{and the} shape ^{of the} density ^{of states}

remains similar in the T and 0 phases ; (iii), ^on

the contrary, the 0 distortion of YBa2CuS 06+6,

which has a different symmetry ^{from the one ob-} served in La2-xSrxCuO,,_y ^creates inequivalent

Cu-0 near neighbour distances which induce a

splitting of the Van Hove singularity ^{in the den-} sity ^{of states} associated with the CU02 planes.

These conclusions raise a number of questions concerning the mechanism of the T-0 transi-

tion and the understanding ^{of the} high super-

conducting transition temperature.

1. The tetragonal-orthorhombic phase

transition in La -,,,Sr..,,CU04-y-

We have represented ⁱⁿ figures ^{la and lb}

the 2 dimensional (2D) ^{structure of the} Cu02 plane ^of La2-xSrxCU04-y rescpectively ^{in the}

T and 0 phases. ^{The 0} symmetry results both from a tilt, ^of angle a, around the. (110)T ^axis

of the octahedra and from an elastic deforma-

tion, e, of their basal plane [15-171 ; this latter

Fig. ^1.- Structure of the Cu02 layer ^{in the} tetragonal (a) and orthorhombic (b) phases ^of La2-a;Sra:Cu04-y, ^{and in} the orthorhombic pha-

se of YBa2Cu306+b (c). ^Squares and circles stand for the Cu and 0 atoms, respectively. ^In

(b) ^{hatched and open circles} correspond ^to oxy- gen atoms below and above the _copper plane, respectively. The C axis is perpendicular ^{to the} plane ^{of the} figure.

becomes a rectangle ^{with two} inequivalent ^near neighbour ^0-0 distances. In figure lb, two oxy- gen atoms (hatched circles) ^{lie below the plane}

and the two others (open circles) lie above. The Cu-0 distance (d ^{= 1.889 A} [15- 17] ) ^{remains un-}

changed through ^the transition and the 2D unit cell parameters ^{are defined} by :

in the tetragonal structure, ^and

in the orthorhombic structure. The shear dis- tortion angle ^{relative to the} tetragonal ^{axes is}

thus defined by :

The tilting mode of the octahedra leads ^to

a doubling ^{of the unit} cell, ^and structural stud- ies [16] have shown that it is associated with the _qt = (-!, ± .1, 0) ^{wave vectors.} The elastic shear corresponds ^{of course to} the uniform wave

vector q, ⁼ (0, 0, 0) . ^{It is thus easy to show} by symmetry analysis ^{that the} corresponding ^order parameters ^are coupled ^at the lowest order by ^an

invariant of. the form 8 a2 (8 ^{and a belong re-} spectively ^{to the} T7 (k14 ⁼ qs) ^{and f3} (k13 ⁼ qt)

irreductible representations ^{of the} D4h ^tetrago-

nal space group in Kovalev’s notation [18]).

We first emphasize ^{that the} staggered ^{tilt of}

the octahedra is the primary ^order parameter ^of the T-0 transition, because the criticality ^{of its}

qt ^{wave vector} is demonstrated by ^the appear-

ance of superstructure reflections below TTO

and by structural fluctuations which diverge ^at TTO. ^The experimental evidences for these lat- ters are displayed ⁱⁿ figure 2, obtained from the

X-ray ^diffuse intensity I(q) around the positions

of the extra peaks ⁱⁿ Lal.8gSro.12 CU04-y (Ref.

[5]) : ^{both the} susceptibility X(qt) ^associated

with the order parameter ^a and the coherence

lengths along ^{the 3} directions ao,bo,co ^defined

in figure Ib, ^{tend to} diverge ^at TTO- ^{215K. The} progressive softening ^{of a} phonon ^{branch at the}

reduced wave vector qt , when the temperature approaches TTO= ^{423K in} La2Cu04-y ^{gives the}

same information [2oj .

Because of the coupling Oa 2 ^{between 8}

and a, 8 behaves like a secondary ^order pa- rameter of the T-0 transition. A standard free _energy development including this linear -

quadratic coupling shows, ^{in the mean field} ap-

proximation, ^{that B must} behave like a2 below

TTO, at ^{least for} temperatures ^{close to} TTO [21].

We have plotted ⁱⁿ figure ^{3 the} temperature ^de- pendence ^{of the} X-ray intensity ^{of a satellite re-} flection, ^{which is} proportional ^{to the} square of the order parameter a, together with 8 values

Fig. ^2.- Structural fluctuations in Lal.ggSro.12 CU04-y ^{above TTO.} Temperature dependence

of (a) the inverse susceptibility associated with the tilt order parameter x-1 (qt) (as given by ^the

inverse X-ray ^diffuse intensity ^I (qt ) ^corrected by

the thermal population ^factor T ) ; (b) ^{the in-}

verse correlation lengths ^{in the} ao, bo ^and co

directions (deduced ^{from the half width at half}

maximum of the diffuse scattering ^corrected by

the experimental resolution). ^{As indicated in}

reference [5], ^a domain-size effect limits the cor-

relation lengths ^below TTO - ^215K.

Fig. ^3.- Temperature dependence ^{of the} super- structure reflection intensity, proportional ^to a2,

for the two equivalent ^domains (open ^{and filled} circles) ⁱⁿ Lal.g8Sro.12CU04-y ^{and of the shear}

distortion 8 (crosses), ^as defined in equation (2),

(adapted ^{from Ref.} [5]).

378

determined on the same single crystal. Although

more points ^{would be} required ^{to obtain a} quan- titative relationship, ^it is clear that 0 varies in

a way close to a2 in all the temperature range.

An additional consequence of such a coupling ^is

that the susceptibility associated with the shear order parameter ⁸ (inverse ^{elastic constant} C661,

defined with respect ^{to the} tetragonal frame)

must present, ^{in the mean field} approximation,

a sudden increase at TTO [21]. ^The observation of a decrease of the shear moduli in polycrys-

talline Lal.g,5Sro.1.5CU04-y ^{at TTO, 180K} [22]

is in agreement ^{with such a} prediction, ^{but fur-}

ther confirmation should be searched on single crystals.

Finally, ^{we want to} emphasize ^{that the tem-} perature dependence ^{of 8 is} mainly ^{related to the}

increase of e below TTO. Actually ^from equa- tions (1) ^and (2), ^a development ^{of 9 to lowest}

order in e and a gives :

From the data of Lal.8SSrO.lSCU04-y ^{at 10}

K z ^{= 0.0098} A, Q ^{= 0.057 rd} _[171) ^{the first}

term is about 4.5 times larger ^than a2/2. ^{0 is}

thus roughly proportional ^{to z.} It is worth not-

ing that, ^even in the absence of the deformation

e of the octahedra basal plane, ^a shear distor- tion 9 N -Q2/2 would result from the tilt of octahedra. However its sign ^{would be} opposite

to the observed one. This means that there is

a specific mechanism for that elastic distortion which remains to be understood, in addition to the one driving ^the instability of tilt of the oc-

tahedra.

We summarize the main conclusions of this section : (i) the tilt of the octahedra drives the T-0 transitions ; (ii) ^a distortion c of the basal

plane of the octahedra accompanies this tilt with

e - a2; (iii) ^{in the} orthorhombic phase, ^as repre- sented in figure lb, all the Cu-0 distances in the basal plane ^of the octahedra remain unchanged

when the angle 4 - Cu - 0 deviates from 90° .

2. Consequences for the electronic struc- ture of La2-.,Sr., ,,,CU04-y-

The electronic structure of La2-,,Sr,,

CU04-y ^{has been a matter} of discussion due

to conflicting views about the relative value of electron-electron repulsions ^{ver8u8 band} width, U/t. ^{Our aim here is to} point ^{out some} proper- ties of the one electron band structure in the 0

phase ^{which have been} clearly misunderstood,

and which could provide ^a proper starting point

either for the elaboration of a theory ^of super-

conductivity ^{in the} hypothesis ^{of low} U/t ^val-

ues, ^or for further introduction of the electron- electron interactions in the hypothesis ^of high

U/t.

It is widely recognized ^{that in} La2-zSrzCu04-y ^the highest partly filled band is built through hybridization ^{of the} dx2_y2 ^or-

bitals of the _copper atoms with the _pz and py orbitals of _oxygen [12, 231. In the absence of Sr substitution and 0 deficiency, ^{this band is}

half filled. Neglecting ^any interlayer coupling,

and within nearest-neighbour tight-binding ap-

proximation, the Fermi surface is a square : it

displays ^four equivalent ^saddle points (A ^{to D in}

Fig. 4a) ^{on the boundary of the square Brillouin}

zone. They correspond ^{to a} unique logarithmic

Van Hove singularity ^{in the} density ^{of states.}

It has been argued that in the 0 phase, ^the

Brillouin zone is no longer ^a square but rather a

rectangle (Fig. 4c). Because of the suppression

of the diagonal ^mirror plane symmetry ^{of the} lattice, the saddle points ^become inequivalent,

two of them (B ^and D) ^{occur at} less than half-

filling ^{and the} remaining ^two (A ^and C) ^{at more}

than half filling. ^{As a} consequence the Van Hove

singularity ^{in the} density ^{of states} splits ^{in two} parts, symmetric ^with respect ^{to half} filling.

Actually ^{this is not what occurs in the 0} phase ^of La2-zSrzCu04-y because the above

splitting ^would require ^the geometry ^schema- tized in figure ^{lc with two different Cu-0 dis-}

tances (d1 ^and d2), yielding ^{to two} inequiva-

lent Cu-0 hopping integrals. ^We have stressed in the previous section that on the contrary, the four Cu-0 near neighbour distances remain

equal ^{and even} unchanged, compared ^{to those}

of the T phase. The actual elastic distortion,

shown in figure lb, keeps (ao,co) ^and (bo,co)

as mirror planes. The Brillouin zone (neglect- ing ^the doubling ^of the unit cell [25]) ^becomes

a truncated rhombus and the Fermi surface for half filling ^{is a} rectangle (Fig. 4b). Although

this surface is distorted from the original square

one, its four corners remain equivalent by ^the operations ^of symmetry ^{of the} lattice. This im-

plies that in the 0 phase ^of L82-:.:SrxCuO’-If’

the density ^{of states} presents ^a single ^{Van Hove} singularity ^{for half} filling in close resemblance with that of the T phase. Nevertheless, ^{the dis-}

tortion shown on figure lb induces an effective

weakening ^{of the} original ^orbital hybridization

(d:’:2_1f2 ^{with poor} py) ^{together with new hy-}

bridizations (dx2_!f2 ^with pz...) . ^{This will prob-}

ably change details of the band structure (band width,...), ^{but the} general shape will subsist.

Fig. ^4.- 2D Brillouin zone and Fermi surface

(for ^a half-filled band) associated with a Cu02 layer ^{for the} tetragonal (a) and orthorhombic

(b) ^{phases in} La2-xSrxCu04-y. A, B, ^{C and}

D are the 4 symmetry equivalent ^saddle points

which create the Van Hove singularity. ^{In the}

case of YBa2Cu30g+b the orthorhombic distor- tion splits ^this degeneracy ^and gives ^{rise to 2} pairs ^of points ^which correspond ^{to less} (B, D)

and more (A, C) ^{than half filling} (c). ^{For clar-}

ity ^reasons the Brillouin zone (BZ) depicted ⁱⁿ (b) ^ignores the actual doubling of the unit cell which would make the BZ coincide exactly ^with

the Fermi surface ABCD.

The mechanism of the T-0 phase ^transition

which was previously attributed [12-14, 4] ^{to a}

lowering of the electronic _energy due to the split- ting of the Van Hove singularity ^{has thus to be}

considered. At the same time, it has been _sug-

gested [12, 14, 4] ^{that this splitting is} responsible

for the occurrence of the maximum of the su-

perconducting transition temperature Tc ^{in the} vicinity ^{of z - 0.15. The basic} idea is that for such a doping the Fermi level position EF ^is just

at the _energy of the lowest Van Hove singularity.

In a standard interpretation ^{of the} superconduc- tivity, Tc ^{has to} be maximum there, ^{due to a}

maximum of n (EF ). Actually, ^the picture ^which

emerges from. the previous discussion is rather

that’, ^as doping increases in La2-sSrsCu04-1f’

the band gets gradually depleted from half fill-

ing, ^so that the Fermi level is further and further apart from the Van Hove singularity. ^{This does}

not demonstrate that a classical interpretation

of superconductivity ^{is not} valid, but rather that the role attributed to the splitting ^{of the Van}

Hove singularity ^to explain ^{both the} dependence Tc (x) ^{and the origin} of the T-0 transition does not apply ^to L82-sSr:eCu04-1f.

3. Tetragonal ^to Orthorhombic phase

transition in YBa2CusOa+6 .

The compound YBaZCu30g+b ^also pre- sents a tetragonal ^to orthorhombic phase ^tran-

sition when 6 exceeds a critical value 6c (dc ~

0.5 in Refs. [9,10]). According to current inter-

pretation, 6 ^{is the} oxygen ^content in the _copper

planes ^{which are} sandwiched between the BaO

planes. ^We will call them the Cu06 planes by

contrast with the two CU02 planes which lie be- tween the yttrium ^atoms and the BaO planes.

In the orthorhombic phase, ^the occupancies ^of

the _oxygen sites 04 ^and O5, along ^{the b and a}

directions in the Cu06 plane become different.

The symmetry breaking in the site _occupancy

can thus be characterized by ^{the order} parame- ter :

where P (0i) ^{is the mean Oi site} occupancy. As 6 increases above 8e, r¡ grows until finally ^reach- ing ^a saturation for 6 ⁼ 1 where almost all the

Ob ^{sites are} empty [24]. ^The preferential ^occu-

pancy of the 04 ^{sites is} accompanied by ^{an elas-}

tic distortion of the tetragonal unit cell : elon-

gation along b and contraction along ^{a. The}

orthorhombic deformation, ^{of order} parameter

e ⁼ 2 (bo-ao) / (bo+ao), is illustrated in figure

lc for a CU02 plane. ^Note that, contrary ^to

the La2-,Sr,Cu04-y ^{case, the ao and bo axes}

380

remain parallel ^{to the} tetragonal ^ones. Finally, symmetry ^reasons impose ^{a linear} coupling ^be-

tween the two order parameters ^{e and tl.}

We first emphasize that, ^{with the same sim-} plified tight binding model of the Cu02 planes

as-the one used in La2-a:Sra;Cu04-y, ^{there ex-}

ists a splitting of the Van Hove anomaly ^associ-

ated with the Cu02 planes in the orthorhom- bic phase ^of YBa2 Cu3 Cg+a . Figure ^{lc shows}

that the orthorhombic distortion gives ^{rise to}

two inequivalent distances between the _copper and _oxygen near neighbour atoms, ^{and thus to}

a 2D rectangular ^{Brillouin zone with two} pairs

of inequivalent ^saddle points (Fig. 4c). ^This

is precisely ^the situation, improperly ^attributed

to L82-xSrzCU04-y, ^{where it was thought that}

the lowering ^of electronic, energy, due ^to the

splitting of the Van Hove singularity ^could help stabilizing the orthorhombic distortion, ^{if the} filling ^of the band is not too far from half filling.

Such a filling ^is realized in YBa2Cu306+6

where 6 is close _{to zero,} since there are almost

no oxygen ^atoms in the Cu06 planes. ^{Yet no te-} tragonal ^to orthorhombic distortion is observed,

even at low temperature [19]. ^{The lowering of}

the electronic _energy by itself is thus insufficient to stabilize the orthorhombic phase [26] ^{:’ this}

may be due either to a broadening ^{of the Van}

Hove singularities, smearing fine details of the

splitting, ^{or to other not} yet recognized ^reasons.

When the T-0 transition _occurs, the _pre- vious analysis demonstrates that the density ^of

states associated with the Cu02 planes ^should present ^two split ^{Van Hove} singularities. ^But

the evaluation of the gain of electronic _energy for larger ^{values of} 6, requires ^a difficult deter- mination of the Fermi level : the electronic lev- els of the Cu06 planes and the bands of the

Cu02 planes ^are in chemical equilibrium and,

as 6 varies, ^{there is a} change ^{in their} filling

but also in their relative positions ^{due to} crys- tal field effects. Therefore, ^{it is not} possible

to conclude about a possible electronic origin

of the T-0 transition. Considering ^the present available experimental data, ^{the most} likely hy- pothesis about the T-0 transition remains an

order-disorder mechanism on the _oxygen sub- lattice based on competition between effective oxygen-oxygen interactions and entropy ^effects.

Yet, it is clear that future efforts should be put

on the understanding of the electronic structure of YBa2Cu306+6 for 6 > 6c , where both _super-

conductivity ^{and T-0} ordering ^{are observed.}

.

Acknowledgements.

Very stimulating discussions with S. Barisic and J. Friedel are recognized.

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CAPPONI, J.J., CHAILLOUT, C., HEWAT, A.W., LEJAY, P., MARAZIO, M., NGUYEN, N., RAVEAU, B., SOUBEYROUX, J.L., THOLENCE, ^{J.L. and} TOURNIER, R., ^Eu- rophys. ^{Lett. 3} (1987) ^{1301 ;}

BENO, M.A., SODERHOLM, L., ^CAPONE II, D.W., HINKS, D.G., JORGENSEN, J.D., GRACE, J.M., SCHULLER, I.K., SEGRE,

C.U. and ZHANG, K., Appl. Phys. ^Lett.

51 (1987) ^6.

[25] ^In the actual reduced zone scheme, the ^Bril-

louin zone coincides with the Fermi surface at half filling, ^ABCD (Fig. 4b). ^{But, within}

the approximations mentionned in the text, the folding of the bands is not accompanied by ^the opening ^{of a} gap ; ^{as a} consequence, the extended and reduced zone schemes are

equivalent ^to describe the electronic struc- ture.

[26] ^{This statement may also apply to}

La2CuO4 (J. Friedel, private ^communi-

cation).