Pressure dependence of Tc and electron phonon interaction in MBa2Cu3O7 systems

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Pressure dependence of Tc and electron phonon interaction in MBa2Cu3O7 systems

J.M. Besson

To cite this version:

J.M. Besson. Pressure dependence of Tc and electron phonon interaction in MBa2Cu3O7 systems.

Journal de Physique, 1989, 50 (12), pp.1433-1443. �10.1051/jphys:0198900500120143300�. �jpa-

00211006�

Pressure dependence ^of Tc and electron phonon interaction in

MBa2Cu3O7 systems

J. M. Besson

Physique des Milieux Condensés (*), Université Pierre et Marie Curie, ^T. 13, ^E. 4, ^{4 Place} Jussieu, ⁷⁵²⁵² Paris, France

(Reçu ^{le 21 mars} 1989, accepté le 21 avril 1989)

Résumé.

Les coefficients de pression ^des températures critiques Tc, dans la série

MBa2Cu3O7-y ^{sont analysés (M}

^{yttrium ou un} élément des terres rares). En utilisant le modèle BCS à deux dimensions, ^on _peut relier les variations de Tc ^avec y ^{et avec} la pression hydrostatique, la diminution de y étant équivalente ^{à une} surpression interne. Les modes de

compression symétrique Ag ^{du barium sont} responsables de l’interaction électron-phonon, ^{et leur}

variation sous pression ^est comparée ^au comportement de modes similaires dans d’autres structures à basse dimensionnalité (1-D ^et 2-D). ^{Un mécanisme est} proposé pour expliquer

l’interaction électron-phonon, ^mécanisme qui contribuerait au couplage électron-électron.

Abstract.

Existing data for the pressure derivatives of critical temperatures Tc ^{in the}

MBa2Cu3O7-y ^{family are} analyzed (M

^{Y, or rare} earth). Using ^a two-dimensional BCS model, the electron-phonon coupling parameter ^{is shown to be} responsible for the increase of

Tc, which varies with y in the ^same way ^as under an equivalent internal pressure. The anharmonic barium symmetrical stretching ^{mode is} compared with similar phonons in other 1-D or 2-D quasi-

low-dimensional structures. A possible mechanism for large electron-phonon interaction and electron-electron pairing ^is proposed.

Classification

Physics ^Abstracts

74.20F

62.50 - 71.38

Introduction.

High pressure measurements have been used since the discovery ^of high 7c superconductors

to try ^to understand their mechanism, especially ^after large pressure coefficients [1] ^for 7c ^{were shown to occur in those} crystals. Attempts ^to relate the observed pressure

dependence ^{of the critical} temperature ^to existing theories have only ^been moderately

successful and a very complete discussion of the subject [2] ^has recently ^been published.

Here, the available data on the pressure dependence ^{of the critical} temperature ^{of the}

MBa2CU307 -y ^{series are examined (M}

Y, Eu, Gd, ^{Sm and} Yb). ^{The other series are not}

(*) Physique des Milieux Condensés is associated with the C.N.R.S.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120143300

discussed, ^{for lack of} enough experimental ^{data. It} will thereon be designated ^{as MBCO for} brevity, ^{unless otherwise} specified.

The first section is devoted to a critical selection of experimental values for the pressure variation of Tc. ^{The variation of the} electron-phonon interaction in MBCO under pressure is then extracted by ^{use of the} 2-D BCS formalism [3-5]. ^This indicates that the large increase, under _pressure, of the interaction of electrons with the Ba2 + symmetric stretching ^mode along ^{z is} expected, ^and quantitatively ^{fits the} observed behavior [6] ^of corresponding ^modes (Cs+) ^{in other} quasi-low-dimensional systems (CsCdBr3). ^{In the} conclusion a coupling

mechanism for electrons is proposed, through ^the Ba2 + Ag ^ion vibration, which may induce

large charge transfer in the ^z direction, âs in other two-dimensional structures [7].

Analysis ^of high pressure results.

A systematic comparison ^of dTc/dP ^has recently ^{been done} [2] ^{for the four main classes of} high 7c supraconductors :

A clear trend is that dTc/dP is smaller in the high 7c crystals. ^{Below 90 or 100 K} nevertheless, ^data points ^{scatter all over the} dTc/dP - 7c plane especially ^when plotted ^as

d log Tc/d log ^{V vs.} Tc. ^{In the} present analysis ^{we will} keep ^{to the} following ^rules.

i) Definition of the « critical temperature » Tc : ^a number of definitions have been

proposed ^for Tc : ^zero resistance

onset temperature (Tco)

midpoint of resistance drop (Tcm)

^{and others.} In any case, under pressure, these quantities may well behave in opposite

directions. For instance [8] ⁱⁿ YBa2Cu307 - y ^{samples, the zero} resistance temperature decreases under _pressure, Tco ^{increases and} 7cm ^{is almost} constant... This flattening ^{out of the} resistivity drop under pressure may indeed in some cases be a bona fide physical ^effect yet ^to be explained, but could also be related to pressure gradients, crystal inhomogeneities,

internal strain (and stress) variations, ^{etc... The most desirable data} (zero resistance) ^are very few. Onset T co values under pressure ^{are more numerous} but subject ^to uncertainties due to

variable and optimistic definitions of the authors, especially ^{on the} high temperature ^side.

Besides, they ^{are the most sensitive to} pressure-induced inhomogeneities. Values for

7cm ^are comparatively more numerous and are used here as the best compromise. These shall be referred hereafter simply ^as Te

midpoint of resistance drop.

ii) ^When comparing Tc’s ^under pressure, only [dTc/dP ]p

^{o shall be used,} that is the initial

slope, ^not the evolution of the slope ^under pressure, in the few cases when this is known. This restriction is related to the reasons quoted ^above (i) : Especially ^{in the} higher pressure range

(P >_ ¹⁰ GPa ) 7c(P) measurements are not done under hydrostatic conditions and have not

yet been shown to be reversible, ^{in the} published literature.

iii) Only the MBCO series will be discussed here. Too few data exist over a large 7c range in the La (SrBa) ^{CuO and} (TlBi)(CaSr) ^CuO compounds. ^{Studies on} crystals ^where

Cu is substituted [9] ^with Fe, ^{Ni or Zn must not be included} here. This would be quite

misleading : here, only ^the dependence ^of 7c ^on electron-phonon interaction will be

discussed, ^{not the} electronic contribution which is surely ^affected by substitution of other elements onto copper sites.

iv) ^For analogous reasons one data point [10] concerning Yo.4Bao.6Cu03 ^{cannot be retained}

for comparison. Barium, there, ^{is not in the same} configuration ^{as in} MBa2CU307 and this is of

prime importance ^{for the} present discussion.

v) The usable data points ^{which are retained} after this selection [11-16] ^are given ^{in the}

table I. They ^{are also} plotted ⁱⁿ figure ^la. Actually ^we could have used a few more for the

high 7c region (- 90 K) ^but they ^{all fall in the same area with} small, positive ^{or, in a few cases,} negative ^{values and this does not} modify ^the shape ^{of the curve in} figure ^{1. Before} proceeding

to the next section, ^{let us note that the} low-T, points (n° ^{5, 6 and} 10) ^{are the} oxygen-deficient samples (y - ^0.5 ). Their pressure coefficient is high ^{and their} c-parameter ^is larger. ^The dependence ^of the c-axis and 7c ^on O-deficiency is well documented [17] ^{and shows} strong correlation in both orthorhombic superconducting phases, ^{above and} below y

0.3.

Table I. - Data points ^{used in} this discussion. T, ^is the critical temperature ^{at 50 %} of ^the

resistance drop. ^Number of ^the samples ^{are recalled in} figures ^{1 a and 2} for identification.

COMPARISON WITH 2-D BCS. - These results will now be examined in the light ^{of the 2-D} BCS theory [3].

À depends [3, 4], among others, ^{on the} electron-phonon interaction, the relevant phonon frequency, and the number [5] ^of adjacent CU02 sheets in the compound.

To ^{is an} ad hoc characteristic temperature ^{which accounts both} for the width of the 2-D electronic density ^{of states} singularity, and for the equivalent Debye temperature ^for electron-phonon interaction. To - ^{600 K is a} value that fits well with experiments ^{and was}

used for figure ^1b. Actually, any To between 400 K and 900 K, may be used without changing

the conclusions.

In the present discussion, ^the variation under pressure of 7c will be related to À through equation (1) and will be shown to depend largely upon the variation of the electron-phonon

interaction. The variation of the phonon frequency ^{will be} neglected, since it is a second order

Fig. 1.

a) Left-hand ordinates : dTc/dP ^vs. T, from the table. Figures correspond ^to the first column.

Full line is related to the curve in figure ² by equation (1). b) Right-hand scale : dashed line is A from

equation (1).

contribution (10 ^{to 20 % at} most) ^as compared ^{to the} large (over ^{one order of} magnitude)

variation of the electron-phonon interaction. Since all compounds studied here have the same

number of adjacent Cu02 ^{sheets, no} scaling ^factor [5] need be taken into account either.

Curve (la) ^{is now} plotted in different coordinates in figure 2, ^{that is} using ^{À 2 in} place ^of 7c through equation (1), ^{in order to take the} square ^{of the} electron-phonon interaction as the

appropriate ^function of pressure. We thus plot :

using :

Integration ^of (2) gives :

F is any function that goes well through ^{the data} points and the inverse of which is readily integrable. ^It has, for the time being, ^no physical meaning. Integration ^can also be done

numerically. ^{This is shown in} figure 3, with pressure being ^defined only within the arbitrary integration ^{constant C.}

What does this mathematical procedure mean ? First of all, hydrostatic pressure brings

about a known variation of Tc, ^{that is} d7c/dP (experiments). ^{Each observed} sample, ^{with its}

characteristic Tc (abscissae scale) ^{is then} assigned ^an equivalent intemal pressure. This is caused in the present ^case by ^variable oxygen ^content but should not be related explicitly ^to

the oxygen deficiency, ^{or, for that matter to} the known [17] variation of the crystal ^axes

through ^{some «} experimental ^{» bulk} modulus. The reason for this is that long-range ^forces

Fig. ^2.

Function F in equation (2) : obtained from figure 1 with ordinates and abscissae transformed

by equations (3) ^and (1) respectively. Full line is the slope ^{of the curve in} figure ^3.

Fig. ^3.

Variation of the expected electron-phonon interaction with extemal pressure ^or intemal pressure in MBCO’s, represented by ^variation of À 2 vs. pressure + integration ^constant (C ⁱⁿ Eq. (4)).

(Coulomb interaction for instance) ^do bring ^{in an} analogue ^{of an intemal} pressure but with

tensorial properties which may be quite different from hydrostatic pressure. For example, ⁱⁿ

layered compounds, ^{the Coulomb} glue ^is largely ^a uniaxial stress, ^{not a} uniaxial pressure, and

not a purely hydrostatic pressure either, ^as regards ^{the actual} crystal deformation. In the case

of molecular crystals ^{in the} generalized ^{sense, that is 0-D, 1-D or 2-D} structures which is the

case here, ^{there are} obvious differences in the behavior under pressure of properties ^{linked to}

the intramolecular space ^or intermolecular space. The latter which is ^more compressible ^at

low pressures may actually become stiffer than the former under compression [7, 18]. ^Thus,

other properties ^{of the} crystal ^{will not} vary in the ^same way ^as Tc does under internal _pressure.

Notably, ^{the c-axis does decrease in} length ^under increasing ^internal pressure when y decreases and this variation can be plotted ^{as a} function of Te, forgetting ^the intermediate _y variable which actually ^causes the effect. The decrease of c with increasing Tc ^is quite ^smooth

with a slope dTc,/d log ^{c - 6 000} K and shows no strong discontinuity ^at the orthorhombic orthorhombic phase transition at y

0.3. But if one attempts ^to relate it to an intemal pressure through ^a bulk modulus of some - 170 GPa which is the average value for this class of compounds, ^{one finds an} intemal pressure scale which is different from that derived from

Te. ^Another consequence ^{of this} separate behavior of inter- and intramolecular regions ⁱⁿ crystals ^{with two or more} kinds of chemical bonds is that the so-called ^« Grüneisen constants » have little physical significance ^{and are} usually not constant. They ^{can be used} only ^for isotropic crystals ^{with one} type of chemical bonds (interatomic potentials). Notably a quantity

such as : d log Tc/d log ^V (V ^the volume) ^{will not} be used here.

In summary, the meaning ^{of the} integration ^{done here on} À 2, ^{on which} Te depends ^{is : The}

variation of Te ^{is due to an} equivalent intemal pressure which brings ^{about a} compression ^of

some region ^{of the} crystal ^{structure which} is, ^{at the} present stage ^{of the} argument, undetermined. This region ^is responsible ^{for the} electron-phonon interaction which is one

step among the steps leading ^to superconductivity. ^{It need not be the same} region ^where current-carrying regions ^are located. As a matter of fact, ^it probably ^{is not} here, ^{as will be} proposed in the conclusion.

Discussion.

At the present time, ^no experimental data exist on the evolution of the electron-phonon

interaction in superconducting perovskites under pressure. It is thus necessary ^to rely ^{on what}

is know in other low-dimensionality systems ^{to see} if any reasonable physical analogy ^{can be}

drawn. In this section a comparison will be made first with the quasi ^1-D CsCdBr3 perovskite,

then with a bona fide 2-D layered crystal : GaSe under high pressure.

a) COMPARISON WITH CsCdBr3. - This 1-D structure is made up of rigid ionicovalent chains of (CdBr6 )- ^octahedra separated by chains of softer Cs+ ions. Under pressure the Cs+ ions do indeed compress [6] ^{as their rare} gas analogue, ^{that is xenon,} which has the same

closed electronic shells. At ambient, ^{the Coulomb} glue between the CdBr6 ^{and the}

Cs+ chains is weak enough ^{to leave} ample space ^to the cesium atoms in their rhom- bododecahedral bromine cages : Phonons involving ^{movement of Cs atoms normal to the c-} axis, ^{have a} large pressure coefficient and weak Raman intensity. ^{Above - 5} GPa, ^{the outer}

electrons of Cs+ come into contact with the bromine shells and react in the same way ^{as rare} gas ^atoms do under pressure : they ^become highly incompressible in the hard sphere regime [19, 20]. Thus, ^{in the} low-pressure (van ^der Waals) regime, Cs vibrations are more or less

harmonic, electron shells and core moving together. ^{Above some 10 GPa, on the} contrary, the electronic shell is blocked and the phonon mode induces strong charge transfer, ^which

means high electron-phonon interaction. This cross-over from the low- to high- pressure

regimes ^is clearly ^seen in the pressure dependence ^{of the} frequency w ^{of the} Ea 29 ^{mode in}

CsCdBr3 ^{which involves} the motion of Cs : d log w /dP decreases from - 10-1 GPa-1 at ambient down to - 1 x 10- 2 GPa-1 ^at 10 GPa. At the same time, ^{the Raman} intensity ^{of this}

mode increases by ^{an order of} magnitude ^{relative to the total} intensity of all other modes.

Now the Raman intensity ^{in a} transparent crystal ^is proportional ^{to the Raman cross section}

which is :

a, (3 : intermediate states

i, ^{s :} incident and scattered photons

HER, HEp : electron-radiation and electron-phonon hamiltonians. Far away from any resonance, which is the case here, ^{the Raman} intensity ^{will be} proportional ^{to the} square ^of the electron-phonon interaction (HEP ), ^{and thus to} À 2, ^{which is the reason} why ^this parameter

was chosen for figure ^3.

In figure 4, ^the calculated values of À 2 of figure ^{3 for} MBa2Cu307 - y ^are superposed ^with

the Raman intensity values of the Cs mode (Ea2 g ) ^{in CsCdBr3 taken from figure} 9 in reference

[6]. ^After scaling ^the ordinate scale by ^an arbitrary factor, ^and adjusting ^{C in} equation (3) ^that

is shifting the pressure abscissae, the fit is obvious. Nevertheless the shape of the full line in

figure 2 may be varied by ^{a sizeable amount to still} go through ^the experimental points ^and

the apparent quality of the fit in figure 4 may be ^{no more} than a coincidence. Thus it is

presented ^{here as a} semiquantitative trend, ^{not a} precise comparison. ^{For this,} obviously, experimental values of the Raman tensor and frequency ^{of the Ba} vibrations in MBCO under pressure should be used. Experimental ^data [21] ^{are available on} other modes but not on this

low-frequency [22] (~ 120 cm-1 ) phonon. The barium vibration under scrutiny ^{is the} Ba(z) symmetrical stretching mode which will compress the Cu(2) 0(3) planes ^{in the} same manner as the Ea’ 29 cesium vibrations compress the ionicovalent chains in CsCdBr3. ^Before pursuing

the discussion, ^indirect experimental ^{evidence of the} possible evolution of this mode under

Fig. ^4.

Comparison of the variation under pressure of electron-phonon interaction which would accouent for dTc/dP ^{in MBCO from 2-D BCS} (full line) with observed electron-phonon interaction in a

1-D crystal under pressure. Circles : experimental points ^of figure 9 in reference [6] ^{for Raman} intensity

of the E2 g mode of Cs in CsCdBr3 (Ir) ^vs. pressure. Full line : ^curve No 3, adjusted in ordinates by ^a

scaling ^{factor to} give the best fit with the experimental points.

pressure may be found in its variation with y, ^as regards : i) ^{Raman cross} section and

ii) frequency ^variation.

i) ^{Raman cross section.} - Only ^{the Raman} intensity of this mode is given in reference [23]

for _y

0. For y

0.75 which is

unfortunately

tetragonal, ^it disappears. Thus, ^{the fact} that the Ba 118 cm-1 line is indeed not seen in the latter, ^as expected ^{from the} previous

discussion does not give very strong support ^{to it. All that can be} said is : it does not contradict it.

ii) ^Raman frequency variation.

Here again ^the y-dependence ^{has to be used. The} phonon density ^{of states from neutron} scattering experiments [24] ^{shows a} positive shift of the

phonon frequencies, of about 20 % for y decreasing ^{from 1 to} 0 in the region ^of

-

14 meV, ^at (120 cm-1 ) ^which corresponds ^{to the w} (k) ^{branch of the} Ba (z ) ^{mode. Here} again, ^the expected shift would be of - 20 % by ^{reference to} the pressure behavior of

E2a g ^{in CsCdBr3} which is the right ^{order of} magnitude, but the existence of the tetragonal-->

orthorhombic transition again ^{casts some} doubt upon the validity ^{of this} comparison.

To conclude this part of the discussion :

i) the increase under pressure of the electron phonon interaction which is required by ^2-D

BCS theory ^{to account for the} dTc/dP ^{values in} MBCO, and for the y-induced overpressure effect on Tc ^is quantitatively ^{identical to} the observed increase under pressure in CsCdBr3 ^of

the interaction of electrons with the Cs mode which compress the chains normal ^to the c-axis ; ii) the effect of barium Ag vibrations on the Cu(2) 0(2,3) plane in MBCO could be

analogous ^to that of Cs atoms on CdBr6 octahedra in compressed CsCdBr3. Both involve motions of rigid ^filled shells, analogous ^{to those of xenon under the} simplifying assumption

that Cs and Ba be fully ^ionized which, admittedly, ^is only ^an approximation. ^Under high

pressure for CsCdBr3, ^{or for} MBa2CU307 (y

0), ^{at ambient,} these shells are blocked and the motion of the Ba or Cs atoms induces large charge ^{transfer to} and from the (CdBr6)

chains in the first case, to and from the CU02 planes ^{in the} second ;

iii) indirect indication that this behavior is indeed similar to that of MBCO can be derived from the fact that the Ba mode is absent for y

1, and that the frequency shift of this mode between y

0 and y

1 is that which would be expected ^{for the} corresponding equivalent

pressure, but it is ^not possible ^{to be more} specific because of the orthorhombic-tetragonal phase transition which occurs around y - 0.6.

b) COMPARISON WITH OTHER 2-D STRUCTURES. - We noted before, ^{that the} Ag ^{mode of}

the Ba 2, ion should induce large charge transfer when its outer electronic shells were pushed against ^the CU02 planes ^{and thus cause} large interaction of electrons with this phonon. ^Now

the eigenvectors pattern ^of this mode is analogous ^{to the} homopolar layer breathing ^modes

which occur in layered crystals ^{with two} sheets per layer, ^{in indium or} gallium chalcogenides.

These vibrations alternatively ^involve symmetrical compression (dilation) ^{of the van der}

Waals gap and dilation (compression) ^{of the} layers.

An analogous ^{situation occurs in MBCO} where, ^{in each} layer, ^two Cu(2) 0(3) ^{sheets are}

linked by ^the Ba2 + and Cu(l) 0(1,4) chains. The interlayer ^van der Waals gap in GaSe

corresponds ^{to the} quasi-empty ^{rare earth} planes ^{in MBCO.}

In GaSe, ^a remarkable electron-phonon interaction occurs between the carriers at the energy gap level and this particular phonon. ^This interaction is large enough ^to completely

dominate that of all other modes for electrons and holes ^at the gap energy. This is well documented in a number of papers and for instance, ^the temperature dependence ^{of the}

energy gap [25], ^{or of the} exciton linewidth [26] ^{can be} entirely explained by ^the interaction of

charge carriers, only ^{with the} layer breathing mode which causes large charge ^transfer

between the intra- and interlayer spaces and thus large electron-phonon interaction.

Although ^{there is no} direct relation between the band structures of GaSe and MBCO, ^the

same dominant role must also be played in the latter by the barium Ag mode which must indeed induce symmetrical breathing ^{of the} layers.

A second effect of this particular ^{mode can be} proposed. ^{In the} previous paragraph, ^we

discussed the oscillatory charge ^{transfer and/or induced} dipole ^{of the} Ag ^{mode. We will now}

show that it might also induce a static local strain in the lattice along the z-axis. It has been noted before [27] ^{that the Ba 2, ions move} inside the CuO _{cages in} a double well or symmetry- breaking potential ^{and have} large amplitudes. ^{Due to the local} asymmetry ^{of the} site, ^the restoring ^{force when Ba moves} towards the rigid Cu(2) 0(2,3) planes ^{will be} larger ^{than when}

it moves in the other direction which is less densely settled. This means that the amplitudes

would be asymmetrical which will be compensated by ^an average ^movement of the center of

gravity away from the Cu(l) planes. ^{This is} only the classical rectification effect in the response of ^a non-linear systems ^{to a sine wave. The two Ba atoms} in the unit cell will thus on

the average ^come closer together ^{under the} eigenvectors ^{of the} Ag ^{phonon. What happens}

then to the Cu(l) planes under the motion of this phonon ^{is not} clear. On the average they might ^{as well be} pushed ^out into the Y-R.E. gap by ^the Ba2 + electronic shells, ^{or be} pulled in, by the decrease of the Ba-Ba distance. _{In any} case the important point is that the

Ag ^{barium mode decreases} the Ba-Ba distance and is thus analogous ^to hydrostatic pressure in that is squeezes electrons ^out of the inner part ^{of the} layer. ^{This is a} general ^case in molecular

crystals, that is solids where different chemical bond speciés coexist : pressure transfers

charge density ^{from the} strongest ^to the weakest bonds. Thus, ^as regards charge transfer, ^this

mode is similar to a static compression, irrespective ^{of its} frequency ^{or wave vector. Now in}

other 2-D structures, ^{such as} GaSe, ^{we have} recently ^shown [7] ^that high pressure, through charge ^{transfer out of the} intralayer space, brings ^{about a} very large ^increase (dell /dP - ¹ GPa-1 ) ^{of the} component of the static dielectric constant tensor ( Ep ) parallel ^to

the c-axis. This increase is due to depletion ^{of the} intralayer charge density which in the case

of GaSe, spills ^{out into} the vdW gap leaving B 1. about constant. In the case of MBCO, ^this

would mean that the Ba mode propagating ^{- for} argument’s ^sake

perpendicular ^{to z would}

be accompanied by ^a static local overpressure and thus ^a larger value of the component ^{of the} dielectric constant parallel ^{to c. This} would decrease the Coulomb repulsion ^{for electrons in} adjacent layers, with similar _x, y coordinates, which interact through ell. This Coulomb

repulsion minimum will of course occur at the same location as the charge density ^maximum brought ^{about in or close to the} Cu(2) planes by expulsion of electrons from the inner part ^of the layer ^{and both} singularities will, ^of necessity, follow the barium phonon ^k-vector.

Conclusion.

The last section of the discussion is obviously highly speculative ^{and the} hand-waving

arguments presented there should be verified by ^more quantitative calculations. Nevertheless,

it is interesting, ⁱⁿ conclusion, ^{to check a} few of the consequences of the proposed ^scheme against experimental ^evidence.

i) ^The proposed mechanisms are specific ^{to the} crystal structures of MBCO compounds,

not of La1 _ x (BaSr )xCu204 crystals. ^The only ^common point is the succession of La 3+ double

layers with 2-D CuO octahedra. This justifies the fact that they ^{were not} compared ^{with the}

MBCO’s. They ^are similar but not identical. Among others, they ^{could not be} compared ^with

the same A(P) ^scale.

ii) ^{With the} present model and reference [17] ^an increase of 7c ^{from 50 to 90 K} through

oxygen concentration variation would correspond (Figs. ^{2 and} 3) ^to about 15 GPa equivalent

pressure, ^whereas by varying y, ^a compression of the c-axis of less than 1 % is observed for this 7c variation. Even with an isotropic ^{bulk modulus of 200} GPa, ^this corresponds ^to only

6 GPa, ^{not 15 GPa. This} illustrates what we said at the beginning : oxygen concentration is

important ^{in a small} region around the chains where it locally ^{increases the internal}

overpressure ^over the Ba 21 ions but on the whole does not much increase in the Y-Re gap ^or in the Cu(2) layers ^where conduction presumably ^takes place, ^{but not the} pressure-dependent coupling mechanism which is proposed ^here.

iii) ^{We have} here drawn an analogy between MBCO and two-dimensional crystals ^{with two}

atoms (or ^molecular subunits) ^{in the} layer. The structural analogy ^{was utilized} only ^{to discuss}

its impact ^{on the Ba} Ag ^{mode, and intra- to inter- charge} transfer. A variety ^{of other} phenomena ^are predictable ^{of course,} in relation to this, ⁱⁿ particular ^on the 0 and Cu vibrations. Among others, ^the intralayer shear modes should have either a small

dw /dP ^{or even a} negative ^one, through charge depletion. Interestingly enough ^{such a mode}

has been observed [21] ⁱⁿ YBa2CU307 ^{but not} yet explained. ^{It is} tempting ^to assign ^{is to a}

Bg ^{mode involving for instance 0(1) which} should decrease in frequency with pressure and

possibly ^{increase in} intensity because of the Ba2 + ions displacement, which is the reported

behavior [21].

iv) ^For mathematical simplicity only, ^{the curve of} figure ^{2 was} restricted to positive ^values

at high ^À 2. Actually, above 90 K it goes ^to slightly negative values which means that the A 2(P ) ^{curve in} figures 3 and 4 decreases slowly ^above 7c - 90 K. Thus the observed maximum [2] ^of Te with pressure in MBCO can very easily be reconciled with the present scheme : The pressure-induced charge transfer towards the 0(2) layers ^{will at a} given point

start pushing the Fermi level away from the logarithmic ^maximum of the 2-D density ^{of states,}

and thus decrease Tc, ^as actually ^observed. Nevertheless, ^{this is a second order effect in the} analysis presented here : The largest contributions to the shift of Tc with pressure and oxygen content can be assigned solely ^{to variable} electron-phonon interaction.

v) ^{The model used here} also shows antiferroelectric order to be a natural consequence ^of the structure of 2-D perovskites, ^{but this will not be} discussed here [28]. ^{For the} present, ^let

us note that for large electron-phonon interaction, the mirror symmetry ^{of the} Ag ^mode

eigenvectors ^{is not} compatible with the inversion symmetry ^of perfect antiferroelectric order.

At most, antiferroelectric fluctuations would be allowed to remain in material with electron-

phonon interaction large enough ^{to lead to} superconductivity. Interestingly enough, ^{the same}

conclusion has been reached, recently [29], by ^{a different} argument, ^{that is} by showing ^the supraconducting ^and antiferroelectric instabilities to be mutually ^excluded by symmetry.

Here, it is the interaction mediator (Ag ^{phonon) which breaks} antiferromagnetic ^{order. The} important point ^{to note is that the ln2} parquet equations ^used in reference [29] for their 2-D

model, ^are exactly equivalent ^to equation 1 for the interaction.

Acknowledgments

This _paper heavily ^{relies on the} data, references, ^and discussions of reference [2]. ^{1 thank}

Rinke J. Winjgaarden ^for early communications of his manuscript ^which actually ^{was the}

basis for this note. It is a pleasure ^to acknowledge ^the help ^{of Julien Bok for fruitful}

discussions and suggestions ^{on this} subject, ^{and of} Jacques Friedel for pointing ^out the shift of

the Fermi level under _pressure.

References

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