Electron mechanism for the tilting transition in La2-x SrxCuO4

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Electron mechanism for the tilting transition in La2-x SrxCuO4

S. Barisić, I. Batistić

To cite this version:

S. Barisić, I. Batistić. Electron mechanism for the tilting transition in La2-x SrxCuO4. Journal de

Physique, 1988, 49 (2), pp.153-158. �10.1051/jphys:01988004902015300�. �jpa-00210680�

Electron mechanism for the tilting transition in La2-xSrxCuO4

S. Bar~si0107 and I. Batisti0107*

Department ^of Physics, Faculty ^of Science, ^P.O.B. 162, ^Marulicev trg 19, ⁴¹⁰⁰¹ Zagreb, Croatia, Yugoslavia

*Institute of Physics ^{of the} University, ^P.O.B. 304, Bijeni010Dka c.46, ⁴¹⁰⁰¹ Zagreb, Croatia, Yugoslavia

(Regu ^{le 25 novembre} 1987, accepti ^{le 21 dicembre} 1987)

Résumé.2014 Nous montrons que la déformation qui ^fait onduler les plans CuO2 ^est couplée ^{aux électrons}

par l’intermédiaire de la variation des éléments de matrice Umklapp ^dans l’interaction électron-électron.

Ces derniers ne sont importants que pour de faibles concentrations de Sr, ^{si les} interactions, supposées

alors répulsives, ^sont suffisamments petites. ^Le premier ^{terme dans la} contribution à l’énergie ^de

déformation est quadratique ^en déformation et diminue de façon critique ^la fréquence ^{des modes}

d’ondulation lorque ^la température décroît. La résistivité, ^bien qu’augmentée par Umklapp, ^{est continue}

à travers la transition à la phase ondulée. Notre modèle ressemble quelque peu à celui développé précédemment pour les supraconducteurs organiques. Nous discutons aussi brièvement les propriétés

du modèle qui concernent la supraconductivité ^à Tc ^élevée.

Abstract.- It is shown that the tilting deformation is couplet ^to the electrons through ^the tilting

induced variation of the Umklapp ^{matrix elements. The latter are} important only ^{for small concen-}

trations of Sr if the electron-electron interaction, (which ^{are assumed to be} repulsive), ^{are reasonably}

small. The leading ^{term in the} resulting deformation _energy is quadratic ^{in the} tilting deformation and leads to the critical devease of the tilting ^mode frequency ^{when the} temperature is lowered. The

resistivity, although ^enhanced by Umklapps, ^is continuous through ^the tilting transition. The model bears some ressemblence to that developed previously ^for 4kF deformations in organic superconductors.

Some consequences of the model for the high Tc superconductivity ^are briefly ^discussed.

Classification

Physics ^Abstracts

64.00 - 74.20

Recent measurements [1,2] of the lattice

properties have shown that the orthorhombic ^to

tetragonal transition, ^at T

423 K in La2Cu04,

is driven by ^the (0.5, 0.5,0) ^{tilting mode 1/, de-}

picted [3] ^{in figure 1. Actually} it turned out

[1,2] ^that f/2 "-I (T - T)2Ø "-I E where e is the

(0,0,0) shear of the basal Cu02 plane. c appears therefore only ^{as a} secondary ^order parameter induced by ^the tilting q, through ^the e1/2 ^Lan-

dau term. This conclusion is corraborated by

the study [2] ^{of the} pretransitinal softening ^{of the} phonon ^{modes in} monocrystals ^of La2Cu04. ^The

described result is important provided ^{that the}

instability ^{of the} tilting mode reflects the _prop- erties of the electron system. ^The understanding

of the tilting transition as the one which occurs

at the highest temperature may then shed some

light ^{on the nature of other} transitions, [4,5] ^in-

cluding superconductivity [6].

There are some indications that the elec- trons are the source of the tilting instability.

First, ^{T is of the same order of} magnitude ^{as the}

transition temperature of electronic transitions to the antiferromagnetic [4,5] (AF) ^{and not so}

far from the superconducting [6] ^{Tc. Second the}

phonon softening is rather well localized [1] ⁱⁿ

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004902015300

154

the Brillouin _zone, suggesting large correlation

lengths. Third, ^{T is} strongly dependent [1,7,8] ^on

the doping ^{x in} e.g. La2-a?Sra:Cu04. ^The tilting disappears ^{at x ;ze 0.2.} On the other hand it ^was

recognized [9,10] ^{very early that ’1 is not cou-}

pled linearly ^{to the band} electrons, and therefore any kind of straightforward Peierls tansition is ruled out. This fact even led to the suggestions

that the tilting ^{is not} electronically driven, ^dis- regarding ^its strong dependence ^on doping.

Fig. ^{1.- Tilted} Cu06 octahedron within the _cage of La. The arrows indicate the shits of the _oxygen atoms.

Here we wish to discuss a mechanism of cou-

pling ^the tilt 17 ^{to the} electrons, which, although

nonlinear in _’1, qualitatively explains all available observations [ 2,7,8]. ^Actually it will be argued

that the Umklapp matrix elements between elec- trons depend appreciably ^on q 2. These matrix elements are important ^{for the} nearly half-filled band (pure La2-xCu04) but their effect disap-

pers rapidly ^upon doping.

Our starting point ^{is the} tight-binding ^band

model for layered ^oxides [13]. ^In this model the bands are parametrized in therms of the electron site energies Ed ^and Ep ^{on the Cu and 0 sites} respectively, Ed - Ep

Apd > 0, ^{and the} overlap integral t between those sites. The corresponding density ^{of states} is shown in figure ^{2. Note that} Apd ^{appears as a gap} between the two sub-bands

and, ^{in a} sense, makes the _upper sub-band half- filled. The band width of each sub-band is

where 6

Apd/4t. ^{It is} usually ^believed [12] ^that

6 is small, ^{61. The} corresponding ^{band struc-}

ture is exhibited in figure ^{3. This} figure ^shows

that the Fermi surface can be divided in the van

Hove (vH) singularities, where the electron veloc-

ity vanishes, ^{and the} quasi one-dimensional (ld)

sections in between. The electron which corre-

spond ^to those latter states are quite ^{mobile and} propagate along ^the diagonals ^{of the} Cu02 plane.

Fig. ^{2.- Densities of states of the lower and} upper sub-bands for 6

Apd/4t

^{0.2. The} gap between sub-bands is equal ^to Apd ^{= Ed -} Ep.

Let us next turn to interactions. Bearing ⁱⁿ

mind that the _pure system ^{exbibits AF} ordering

[4,5] ^{we choose} repulsive Coulomb interaction.

The latter are convenienly parametrized ^{in terms}

of the Hubbard matrix elements Ud ^and Up ^for

the Cu and 0 sites respectively and the intersite matrix element V, ^{while the} long range Coulomb forces can be taken as screened out [13]. ^{In the}

band representation ^of figures ^{2 and 3 the inter-}

action matrix elements are then given ^{as linear}

combinations of the Ud, Up ^{and V.} However, ^not

all of these combinations are important. ^{In the}

weak coupling ^limit (V, Up,dW) ^{only the matrix}

elements which belong ^{to the} Hamiltonian HB ^of the _upper sub-band _may play ^an essential role.

Among the intraband Umklapp matrix elements

we need only those involved in electron-hole di- agrams, which _appear in the parquet ^{or ladder} approximations. Taking ^{k’s of} Uu (ki, k2, k3, k4),

with k1 ⁺ k2

k3 + k4 ^{+ G} in the notations of

figure ^{3 to the} representative ^{terms are} (ignoring

V for simplicity)

where

with 6

Apd/4t. ^{The Umklapp} (2a) ^relating

the vH electrons depends only ^on Ud, ^since Up

is removed by interference effects. In contrast to that the Umklapp (2b) ^which couples ^the quasi

ld electrons depends ^{on both} Ud ^and Up. ^Unlike

the Umklapp (2a) ^{the Umklapp} (2b) ^{may even}

vanish for special ^of Ud ^and Up ^{and in particu-}

lar for Ud

2Up ^{at 6}

^{0 when the gap} Agap

between the sub-bands in figure ² is closed. Post-

poning ^the description ^{of the} physical meaning ^of

this cancellation let us turn now to the evalua- tion of the dependence ^{of UU on 11. To this end}

we note that both Umklapps (2) ^are linear in 6 for 6 1.

Fig. ^{3.- Brillouin zone of the upper} sub-band. The

capitals A, B, ^C and D indicate the regions ^{of the vH} singularities, ^{while letters a,} b, c, and d correspond

to the flat 1D-parts ^{of the Fermi surface.}

Apparently ^both Umklapps (2) depend onq either through Apd ^{or through t. The symmetry} requires ^this dependence ^{to be} quadratic ^{in both}

cases. Indeed, considering Apd, ^the tilt induced rotation of the Cu04 octahedron (Fig. 1) ^{in the}

crystal field of the La 3+ _cage decreases the site energy Rep ^{of the in plane 0’8 which move to-}

wards Lag+. The effect can be evaluated using

the generalized Evjen-Frank method, [14]

where for simplicity ^{we have} disregarded ^{the dif-}

ference between the lattice constants c and a.

The corresponding variation of t can be found

by expanding ^t in the conventional _way. Insert-

ing those variations in equation (3) and then in

equation (2) ^{the Umklapp} matrix elements (2a,b)

can easily be found at 6 1 as

with i

A, ^a respectively. UU ^{is in} general large

due to the large prefactor of t7 2 ⁱⁿ equation (4).

In addition Ulu ^is always positive. ^Even UlU ^is

of the right sign ^{in the} large part ^{of the} parame- ter space (Ud, Up, 6) . !7u > 0 is essential for the

understanding ^{of the} tilting transition, ^{as will be} argued ^now.

The harmonic deformation _energy which

corresponds ^to equation (5) ^{can be} conveniently

determined under two assumptions. First, ^the Born-Oppenheimer ^adiabatic approximation ^will

be assumed to hold when t7 is varied. Sec-

ond, UU will be considered as a relevant _pa- rameter, i.e. that it controls the singular ^be-

haviour of diffe ent correlators which define the average value HB of the electron intra-band Hamiltonian HB . ^The development of those sin-

gularities decreases (fIB) and the increase of UU

can then only decrease HB . In other words a (HB) lauu ^0. The ’1 induced variation of the electron

_energy HB ^{is thus negative pro-}

vided that UU > 0. As a result the frequency Co

of the tilting ^{mode is decreased} according ^to

where 1;;0 ^{is its bare value.}

The Umklapp ^{term is relevant} only ^when

the correlator [15] ^{which multiples it in} HB

is itself singular. ^{When this} singularity ^{is domi-}

nant its contribution to (fIB) ^can be evaluated

156

in two steps [16]. ^{In the first} step ^the Umklapp

part ^of H$ is treated in second order perturba-

tion theory. ^The resulting convolution xiO) ^{of the}

two singular electron-hole bubbles is next renor-

malized [15-17] ^to k4 by inserting ^the leading logarithmic corrections, which involve all inter- actions and the Umklapps (2) ⁱⁿ particular. R4

describes the pd valence fluctuations in the limit 6 1 when the Umklapps (2) ^are linear i 6. This

can be understood on noting ^that Apd ^{enters the} original Hamiltonian as the external staggering

field which is linearly coupled ^to the site den-

sity np. ^In the corresponding ^band picture ^the

external 6 is linearly coupled ^to R4 when , the Umklapps (2) ^are linear in 6 (8 1) ^i.e. R4

is the susceptibility ^{to 6. In} addition, ^{the val-}

ues of the Umklapps (2) ^{at 6}

^{0 add to this 8}

an internal, interaction-induced staggering ^field.

As discussed in connection with equations (2)

and (3) this field is in general finite, (i.e. ^it

can vanish only ⁱⁿ equations (2b) ^{at Ud}

2Up).

The softening ^{of the} tilting frequency (6) ^reflects

then directly the increased susceptibility R4 ^of

the system ^{to the} (external ^and internal) stag- gering field, ^{i.e. the} pd valence fluctuations are

enhanced. As the effective charge ^{of the 0 site}

is increased the 0 ions are attracted towards the

positive La3+ sites and the tilting ^occurs.

The experiments show that the dependence of w(q) ^on [1,2] T, q and [1,2,7,8] ^{x is apprecia-}

ble but not dramatic, which is consistent with the weak coupling approach used here. Actually,

whatever perturbation theory (parquet ^{or lad-} der) ^is used in the evaluation of a (HOB) l,9Uu

the associated high temperature length ^scale

should be given by elementary diagrams, namely f

a.W /T. Smaller scales can occur only ^be-

low some electronic phase transition or crossover.

The bare e ^describes reasonably well the width of the observed anomaly ^{ak N 10-1} a-, ^and

the concentration zc which suppresses the tilt-

ing, zc = ^a ç-l ~ ^10-1. Indeed the doping ^{x in-}

troduces the holes in the _upper subband of figure

2. The resulting shift of the Fermi energy sepa- rates the normal and Umklapp contributions of the electron-hole diagrams. ^In addition, ^both log

and (log)2 contributions of respectively, ^the quasi

ld parts ^{and the van Hove} singularities ^{are de-} pleted. ^{As a} result the Umklapps ^{cease to be rel-}

evant. The effect occurs on the scale AkF = ç-l.

Note that the electron redistribution fre- quency ^can by found in the same way ^as C. ^{It is}

equal ^{to T. Since T} > Wo ^this justifies ^{the use of}

the adiabatic approximation ⁱⁿ deriving equa- tion (6).

It should be further pointed ^{out that the}

electronic properties ^are predicted ^to be continu-

ous through ^the tilting transition. This is conve-

niently illustrated with the help ^of X4, ^{but holds}

also in the more general ^{case of} equation (6). ^Ac-

cording ^to equation (6) ^{R4 is finite at T. Below} T _only the _average values of umklapps (2) ^{are af-}

fected by ^the development ^of (q) . ^{To the second}

order this effect is given by equation (5). ^The

variation of Umklapps ^{is in} principle ^reflected

in all correlation functions including i4. ^How-

ever the effect is smoothed out when UU ^{is fi-}

nite in equation (5). ^In particular only ^{a weak}

crossover effect is expected ^{to occur at} T in the

resistivity, ⁱⁿ good agreement with observations

[2]. ^X4 should continue to increase below T head-

ing towards the Mott metal to insulator transi- tion in fIB. The actual evaluation of i4 ^{or of}

the other correlators in (fIB) ^{requires at least}

the solution of the parquet problem. ^{The latter}

was discussed [18,19] ^for layered oxides, retaining only the contributions of the van Hove singular-

ities. In particular ^this amounts to disregarding

x4al ^{in x4.} via) ^{was not} explicitly considered al-

though ^{the low} temperature enhancement of the renormalized Umklapp (2a) indicates that xi A)

diverges ^{at low T.} Unfortunately ^{it was assumed}

[18,19] ^that Apd ^{» t} (i.e. cos 4(p -- ^{1 in} Eq.

(2a)), the limit in which _ri is only weakly ^cou- pled ^to the electrons Uu 0 ^{0 in equation (5)}

and subsequently). ^The many-body analysis ^of

the UbA) =1= ^{Ud case} is therefore desirable.

It is also difficult to include the contribution of the quasi ld electrons. However if these elec- trons can be treated separately from those in vH

singularities (which ^{is not} obvious) ^the problem

is solvable again and, ⁱⁿ particular, the correla- tion function xia) ^{can be found} [15]. ^{In addi-}

tion it is instructive to discuss the ld problem

in some detail. Keeping ^thus only the ld contri- bution equation (2b) ^{it can be easily seen that}

the latter corresponds [20] ^to the contribution of

an alternating chain of d and _p sites with site

energies Ed

Ed, Ep

Ep, ^{Coulomb energies} Ud

Ud ^and CJp2Up ^{and the overlap f}

^{ý2t. For}

Apd ^small (b 1 ) ^{and !7d}

Up ^{= U this model}

maps ^on that for the Bechgaard organic ^metals

[21]. ^{Indeed in this case equation} (2b) ^{reduces to}

u - !7.Apd/2.B/2.f characteristic [22] ^{of those}

materials. For Apd :0 ^{0 and} Ud :A Up ^{the model}

describes the hypothetical miwed stack conduc- tors. It is interesting ^{to note with} layered ^oxides

in mind, ^{that even when} [12] Apd

⁰ tiba) ^of

equation (2b) (with Up =, 2Up) ^{is finite} provided

that CTd :A Up. Already in the weak coupling ^limit Ud, Up W the chain is dimerized on the level of interactions (uba) i- 0) ^even when there is

no doubling of the unit cell in the band struc- ture (ape

0) . ^{The behaviour of the correla-}

tion function X41 is controlled by u ^whatever

the structure of UU ^{in terms of Ud,} Up ^and Apd.

However only for s 1 does the enhancement of

xia) ^{lead to a charge} distribution on the _p and

s sites which differs appreciably from the band values.

In spite ^{of this} analogy between the layered

oxides and organic metals the enhancement of

xia) in the latter case is accompanied by ^the temperature dependent dimerization _{al - a2} of the pd distances rather than by tilting. ^{In the} Bechgaard ^salts Apd ^{is the} bond rather than the site _gap. The dimerization of distances adds a

term linear in a1 - a2 to this dimerization _gap.

This contrasts with equation (4) where the de-

pendence ^on deformation is quadratic. ^{As a re-}

sults a (small) temperature ^or composition ^de- pendent dimerization is always present ^{in Bech-} gaard ^salts [17]. ^A similar dimerization in lay-

ered oxides would change ^the gap according ^to LB2

A2 ^{+ A2 where Aa} is linear in _{al - a2.} As

long ^{as the site} gap Apd ^{is large with respect to}

the bond _gap Od ^{this is} analogous ^to equation

(4). However the _q

0 dimerization mode in

layered oxides has a bare frequency ^much larger

than wo ⁱⁿ equation (6) and it is therefore rea-

sonable that this mode remains stable at T.

The enhancement of the renormalized Cou- lomb interactions inthe _pure x - 0 La2Cu04

goes presumably ^{into the} screening [24,18] ^{in the}

metallic x > 0.1 case, analogous ^to that intro- duced in the conventional theory ^of supercon-

ductivity. ^The interactions mediated by ^{the low} frequency phonons [10] ^or other excitations _may then become important. ^{Thus the} present ^de-

scription ^{of the} tilting transition in terms of the Coulomb interactions does not necessarily imply

that the Coulomb interactions alone explain ^the high Tc superconductivity, although such models have been proposed ^for Bechgaard ^salts [21]. ^{It is}

merely sugested ^{here that the} phase diagram ^ob-

served in layered oxides for x small, analogous

in _many respects ^{to that of} Bechgaard salts, ^can

be reasonably explained ^{in terms} of moderate ^or

even weak Coulomb interactions.

Acknowledgements.

We are particularly indepted ^{to J.P.} Pouget

for communicating ^us this results prior ^to pub- lication, ^{and or an} important discussion. Useful discussions with A. Bjelis ^{and E. Tutis are} grate- fully acknowledged. ^{The work was} supported ⁱⁿ part by ^{the Yu-US} collaboration _program DOE JF 738.

References

[1] MORET, R., POUGET, ^{J.P. and} COLIN, G., Europhys. ^{Lett. 4} (1987) ^365.

[2] BIRGENEAU, R.J., CHEN, C.Y., GABBE, D.R., JENSSEN, H.P., KASTNER, M.A., PETTERS, C.J., PICONE, P.J., ^TINEKE THIO, THURSTON, T.R., TULLER, H.L., AXE, J.D., BONI, ^{P. and} SHIRANE, G., Phys.

Rev. Lett. ⁵⁹ (1987) ^1329.

[3] ^{GANGULY, P. and RAO, C.N.R., J. Solid}

State Chem. 53 (1984) ^193-216.

[4] VAKNIN, D., SINHA, S.K., MONCTON, D.E., JOHNSTON, D.C., NEWSOM, J.M., ^SAFIN- YA, ^{C.R. and KING} Jr., H.E., Phys. ^Rev.

Lett. 58 (1987) ^2802.

[5] ^MITSUDA, S., SHIRANE, G., SINHA, S.K., JOHNSTON, D.C., ALVAREZ, M.S., VAKNIN,

D. and MONCTON, D.E., Phys. ^{Rev. B36}

(1987) ^822.

FRELTOFT, T., FISCHER, J.E., SHIRANE, G., MONCTON, D.E., SINHA, S.K., VAKNIN, D., RAMEIKA, J.P., COOPER, ^{A.S. and} HARSHMAN, Phys. ^{Rev. B36} (1987) ^826.

[6] ^{BERDNORZ, J.G. and} MULLER, K.A., ^Z.

Phys. ^B36 (1986) ^189.

[7] FUJITA, T., AOKI, Y., MAENO, Y.,

SAKURAI, J., FUKUBA, ^{H. and} FUJII, H.,

Jpn. ^J. Appl. Phys. ²⁶ (1987) ^L-368.

158

[8] FLEMING, R.M., BATLOGG, B., CARA, ^R.J.

and RIETMAN, E.A., Phys. ^{Rev. B35}

(1987) ^7191.

[9] EMERY, V.J., unpublished.

[10] ^{BARISIC, S., BATISTIC, I. and FRIEDEL, J.,}

Europhys. ^{Lett. 3} (1987) ^{1231 and refer-}

ences therein.

[11] JORGENSEN, J.D., SCHUTTLER, H.B., HINKS, D.G., ^CAPONE II, D.W., HANG, K.Z., BRODSKY, ^{M.B. and} SCALAPINO, D.J., Phys. Rev. Lett. 58 (1987) ^1024.

[12] MATTHEISS, L.F., Phys. Rev. Lett. 58

(1987) ^1028.

[13] BARISIC, S., ^J. Phys.France ⁴⁴ (1983) ^183.

BARISIC, S. and E. TUTIS, ^J. Phys. ^C19

(1986) ^6303.

[14] KITTEL, C., Introduction to Solid State

Physics, (John Wiley ^and Sons, ^{Inc. New}

York London) ^1986.

[15] EMERY, V.J., Phys. ^{Rev. Lett. 37} (1976)

107, ^{and in} Highly Conducting ^One-Dimen-

sional Solids, ^{Ed. J.T.} Devreese, ^R.P.

Evrard and V.E. van Doren (Plenum, ^New York) ^1979.

[16] LEE, P.A., RICE, ^{T.M. and} KLEMM, R.A., Phys. ^{Rev. B15} (1977) ^2984.

[17] EMERY, V.J., BRUINSMA, ^{R. and} BARISIC, S., Phys. Rev. Lett. 48 (1982) ^1039.

[18] DZYALOSHINSKII, I.E., ^Zh. Eksp. ^{Fiz. 93}

(1987) ^1487.

[19] SCHULZ, H.J., Europhys. ^{Lett. 4} (1987)

609.

[20] ^{LUTHER, A., High Tc} Conference in Trieste, July ^4. -8, ^1987.

[21] ^{For a recent review see C.} Bourdonnais in Low-Dimensional Conductors and Super- conductors, ^edited by D. Jerome and L.G.

Carron (Plenum, ^New York, ^{NATO ASI}

1986).

[22] ^BARISIC, S., BRAZOVSKII, S.A., ^{in Recent} Developments in Condensed Matter Physics,

Ed. J.T. Devreese (Plenum, ^New York)

1981, ^{Vol. 1.}

[23] PINTSCHOVIUS, L., BASSAT, J.M., ODIER, P., GERVAIS, F., HENNION, ^{B. and REI-}

CHARDT, W., ^{to be} published.

[24] BYCHKOV, Yu.A., GOR’KOV, L.P., ^DZYALO SHINSKII, I.E., ^Zh. Eksp. ^{Teor. Fiz. 50} (196) ^738.

[25] ^GARLAND, J.W., Phys. ^{Rev. Lett. 11}

(1963) ^114.