Possible role of non-bonding Pπ oxygen orbitals by orbital Kondo effect in high-Tc superconductivity

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0038-1098/8953.00+.00 Pergamon Press p l c

Solld State Communications, Vol. 70, No, 4. pp. 439-442, 1989.

P r i n t e d in Great B r i t a i n .

POSSIBLE ROLE OF NON-BONDING P~ OXYGEN ORBITALS BY ORBITAL KONDO EFFECT IN HIGH-Tc SUPERCONDU~ A. Zawadowski* Department of Physics, University of Illinois, 1110 W. Green St., Urbana, IL 61801, USA (Received 11 December 1988; in revised form, 6 February 1989 by E. Burstein)

The possible role of the p~ oxygen orbitals in high temperature superconductivity is discussed by considering those scattering processes between conduction electrons in which one px orbital occurs in the intermediate states. The responsible coupling is the offdiagonal Coulomb coupling between atomic and bond charges, which is enhanced by Kondo-type correlation effects of orbital origin. Considering the 1:2:3 compound the role of the bridging oxygens between the planes and chains is discussed.

logarithmic approximation is not fully justified, that serves as a good guideline to concentrate on the largest contributions. Such logarithmic corrections remind one to the spin Kondo problem, but here the pseudo-particle representing the spin is replaced by the heavy parking orbital. Considering electron-electron scattering the two basic diagrams are depicted in Fig. 1 where in the intermediate states there are a heavy and a light particle. The integration over the momentum of the light particle is carried out by integrating first in the direction perpendicular to the Fermi surface and then over the Fermi surface element dS/vF where vF is the actual Fermi velocity. Replacing the scattering potential V by its value at the Fermi surface the contribution obtained is

Since the discovery of high temperature superconductivityl it has become the most vigorously studied subject in condensed matter physics. Although most physicists are inclined to accept that the new superconductivity is BCS type, the mechanism responsible for it has remsined unclear.2 Among the mechanisms that have already been proposed those are the most attractive which are based on the Coulomb interaction between the electrons. One of the most challenging versions of such theories is the excitonic mechanism proposed by Varma, Schmitt-Rink and Abrahams. 3 However, one may have the impression, that it is hard to develop such a theory in which the on-site Coulomb repulsion is overwhelmed by the induced excitonie interaction without large enhancements Iike logarithmic corrections. The present paper is devoted to study of the

(1) possible role of the non-bonding p~r orbitals of oxygens as possible intermediate states with particular emphasis on the bridging oxygen connecting the planes and chains in the 1:2:3 compounds. As Kondo-type logarithmic corrections of orbital origin are considered to enhance the coupling, thus the induced coupling is very sensitive on the values of the initial parameters, therefore, instead of calculating the transition temperature "re directly, the range of the parameters will be determined where high Tc superconductivity may occur. That situation is similar to the spin Kondo problem. Recently one- and two-dimensional models have been proposed4,5 which can be regarded as a modified version of the excitonic mechanism. In these theories, considering the two-conduction-electron scattering channels such two .particle intermediate states occur in which one of the particles ~s a conduction electron, the other one is, however, a localized, weakly hybridizing heavy "parking" orbital with energy ell nearby the Fermi energy. Thus, the energy width D and the distance eh of these orbitals measured from the Fermi energy eF are taken smaller than 15% of the band width D of the conduction electrons. In such cases, the summation over the intermediate states results ha logarithmic corrections proportional to J~n (D/Max {eh, A}). Even ffthe

where d S and VF belong to momentum k and it is assumed that the energy variable co and temperature T are small (T, Ic01 < ell, A). For the sake of simplicity, the heavy band is taken completely above or below the Fermi energy. If the vertices V depend only on the momentum transfer (e.g.

Vk'lk'h, ~'2"-~-~ V(k 2 - k)

), then the two terms cancel in

expression (1).

lm

1.

*Permanent address: Central Research Institute for Physics, 1525 Budapest, P.O.B. 114, HUNGARY 439

~

l

The two basic electron-electron scattering diagrams are depicted in time ordered representation. The heavy and light lines represent the heavy "parking" and conduction electrons, respectively. The wavy lines stand for the interaction t .

It-

440

NON-BONDING

Vol. 70, No. 4

P ~ OXYGEN ORBITALS

(¢)

In order to avoid that cancellation, the non-commutative nature of the spin dependent Kondo coupling is substituted by the off-diagonal Coulomb matrix element 2

f,'0

I

(2) 3.

which appears in the Hamiltonian Hr involving heavy and fight particle annihilation operators h , , a a n d a n,8,o with spin Gas

Hi'=t'

•

h:,oa:,8,~an,8,-oan,8,o

n,8,a

(3)

where ¢' and $ are atomic wave functions, n and (n,8) are different site indexes related to the heavy and light particles, and 8 refers to the nearest neighbors of the heavy particle at site n. In the case of the Hamiltonian H i the formfactor appearing in the Fourier wansform prevents the cancellation in expression (1) and plays the role of the spin operators characteristic in the Kondo problem, t is known as the Coulomb interaction between an atomic charge and a bond charge formed by the overlap S of two orbitals on neighboring sites (see Fig. 2). The basic process responsible for superconductivity is, that fTom a double occupied fight site one particle (electron or hole) moves to the "parking" orbital while the other fight particle is delocalized in the fight band (see Fig. 3). The final process is when the heavy particle moves back to the light b~ind and meets the other one on one of the sites neighboring the "parking" orbital. The electrons involved have opposite spins. The screening of the particle on the "parking" orbital by light particle has a crucial role as it leads to the enhancement of the coupling { as it is demonstrated in Refs. 4 and 5. The enhancement factor is (D/Max {A, E:h}) 2PoUwhere Po is the conduction electron density of the fight particle for one spin direction (see the definition later) and U is the Coulomb repulsion. This enhancement shows similarities to the anisotropic Kondo problem. During the screening process the screening cloud keeps a long time memory of which site was the startingdouble occupied site. It turns out from the following calculation that the particle fxom the "parking" orbital returns with larger amplitude to the original site to meet the other member of the starting pair than to any other site. That phenomena shows the formation of a ground state highly correlated in the real space.

The conduction electron orbitalsare represented schematically, which are accompanied by a heavy orbital (hatched). In order to show the processes responsible for the attractive interaction the starting and the final processes are represented by the black and white arrows indicating the electron hoppings. Between these hopping processes the fight electron is delocalized, while the heavy one stays at one given site. The 1" and $ symbols in the brackets stand for electrons with up and down spins.

The possible role of other orbitals like d = = and p~ 3zon the Cu and O atoms in the CuO2 plane has a~.a~y been raised by Webe~5 and by Fr~man 7 and Goddard8 and their collaborators, respectively. As it is shown in Ref. 5 the d q.= 2 orbital in the CuO2 plane cannot play the role of the "p~r'l~Jng"orbitalat least as far as the commonly accepted tight binding picture is used for the CuO2 plane ignoring the O-O overlap, namely, the generated interaction just vanishes. Furthermore, the aaz 2.r=orbital may not be narrow enough either. The cubic BiO materials do not have such orbitals,9 but they have non-bonding p~ oxygen orbitals. The following discussion is restricted to the 1:2:3 materials. There are tm orbitals in the CuO2 plane and on the bridging 0(4) between the planes and the chains (see for notation 0(4) Ref. 1(3). The first ones form a considerably broader band as they have six next-neighbors oxygens with non-zero overlap (see Fig. 4). According to the band structure calculations 11 there is a flat lax band associated with 0(4) in a distance <0.2 eV below the Fermi level with a comparable width for YBa2Cu307 which moves further away for YBa2Cu30611. The existance of such a band is supported by the measurement of the Hall coefficient12. The px~t-O(4) orbital has off-diagonal matrix with two 0(2) oxygens in the

plone

Cu(I)[.,~ O ( I ) otomic charge 2.

K-bond chorge

The orbitals of two atoms are shown. The area of the bond charge is hatched and the atomic ch~ge is also indicated. The Coulomb interaction between the bond and atomic charges is the off-diagonal Coulomb i~i-ua{ .

4.

The CuO2 plain and the CuO chain with the bridging oxygen 0(4) between them are depicted scbemaficaily for YBa2Cu307-8 compound. Some of the p a and I~ oxygen orbitalsare also represented and some of the possible hopping processes between them are indicated with double arrows.

Vol.

70,

441

NON-BONDING Px OXYGEN ORBITALS

No. 4

plane, which are indicatedby double arrows, while pyx has four (Fig. 4.). These orbitals have the proper symmetry to provide attraction in the plane but pxx may be more relevant than py~. In the following, only the bonding orbitals of the Cu02 plane and the px-O(4) orbitals are kept. Using the simplified fight binding band structure only one of the two broad bands is considered in which the Fermi level lies. For the broad bands (+) the dispersion curve is .2

2 .1/2

co+(k) = + (tx.~+ ty,~')

P- = 1,0 whether 8 is parallel or perpendicular to 8c¢ the (~-axis. Furthermore

where

F-'-"

=

62

f~

88. ( 2 = ) % I vF (s)

exo fiak~ (8-" -

(9)

and

(4)

where ta,l~ = 2t sin(ka t ) , t is the hopping matrix element between dx2.# and po orbitals (a = x,y) and a is the lattice constant. If the Bloch state annihilation operator in the broad band is C~,o, then the site annihilation operators for the

and

2f VF(S) dsl2

P°= (:~ )2

dxz_# and pct-s (ct = x,y) orbitals are expressed in terms of Ck,cras -, , 1 x-, ik"an 1 ~ an,o=a:~z.,e c~,~ and an,~'a=~2.je k

k

(11)

ik'a(n+l~)ta~ z -~-' c~, o I~J

where the Cu sites are labelled by n = (nx, ny) integers and the oxygen site (~, 8 ) i s a(nx + 21-8x, ny + 21 By)where

is

(5)

where VF(S) and k s are the Fermi velocity and momentum at 1 the Fermi surface element dS,*~,~s= *_~fis = - ~ t VJIc0d

the unite vector directed to the oxygen neighbors, and a = x,y is parallel to ~. The annihilation operators h ~,a,o for the pct~ orbitals of O(4). Below the site ~ will be labelled by n also. The Coulomb interaction between the particles in the plane and the "parking" orbitals can be written in terms of the occupation number operators r~ and rt~.~ for the plane and n~,h,a for the 0(4) as

x

F~= -F~ ffi -F.~ holds, therefore, the second term in the last bracket in Eq. (8) drops out- Thus the sum can be replaced by A = 4An{F~"~+ F~_~ > 0 where the factor 4 is a

He= ~_~n;,h(Ud nn+ Upnn,~) n

where a(= x,y depending whether ~ is parallel to the x- or yaxis. Finally, A2 = IFsl 2 = (c0s/2t) 2 as a result of a straightforward calculation. Furthermore, as t~,~= -ta,.~ the identity

(6)

where Ud and U p are the two Coulomb energies

(Up

~ Ud). Hc Can be expressed in terms of c-operators -by using Eqs. (5) and of an effectiveinteractionbetween hand c-particleswhich is associatedwith a form factor

consequence of four possible ~ vectors for the two "parking" orbitals. A can be calculated numerically and it is of the order of unity at not very near the band edge. Thus, the final expression for the induced interaction is

Q----LA[ 2U

U d + Up [cos( 1 (kx- k'x )a) tx,~tx,~'co~. (x<--->y)]

o MaX{eh,A}]~°*°

(12)

(7) where this term can be approximated by an average value U and orj:(k) at a the Fermi surface is denoted by coS: (In case of a non-separable form factor the vertex Eq. for ~-can not be solved exactly, see Ref. 5.) The effective interaction between light electrons can be calculated by taking into account the vertex corrections for diagrams shown in Fig. 1. The final result obtained in Ref. 5 is for the generated interaction between Cooper pairs is

o }4upoA, g= 2U max{~h,A}

T_, .~ P.~. Pr~.F --(' =~_-F --F-) 5,8'

the parameter i can be estimated as i = S U ~r~ where S is the overlap between pn-O(4) and pg-O(2) orbitals, where 1/2 reflect the distance between the bond charge and the O(2) atom. S can be estimated on the basis of Ref. 12 as S = -0.08, thus i = 0.1 • U. As the induced interaction must compete with the on-site Coulomb interaction to get superconductivity, therefore, one can estimate the order ofmag-

(8)

442

NON-BONDING P~ OXYGEN ORBITALS

nitude of the factor [D/Max {Eh, A}]

4poU

around 200-300.

Using for D/Max {ere A} - 8, poU must be somewhat less than unity. This estimation provides a reasonable range for poU where superconductivity may occur. According to this calculation it does not seem to be unrealistic that the smallness of the overlap integral can be compensated by the large vertex correction. The most important feature of the present theory is that the superconducting coupling is very sensitive on the product poU. Larger induced coupling can be obtained in those cases where the density of state is large and the form factors are essential. A rough estimation gives that the largest coupling can be obtained where the lower (upper band) is about 30% occupied (unoccupied). The distance between the CuO'2 plane and the bridging 0(4) plays also an important role in two ways: (i) the coupling t is very sensitive on that distance, (ii) the Coulomb coupling U is less sensitive, but it has a large effect being in the exponent in Eq. (12). The Cu(2) - 0(4) distance can be changed by alloying mainly the chains. According to the experiments 13 on Co doped materials the superconductivity disappears as that dismnce is enlarged by 4%. A change b~t2/U~ 2.5 • 10-2 in the Coulomb coupling can account for the reduction of the induced el-el coupling g by 15% which change may be big enough to make impossible the compensation of the on-site Coulomb repulsion in the plane. Such experimental facts 13, however, may be explained by the change in the charge transfer, as well. The oxygen doping may result (i) in occuring a spin-wave-superconducting phase transition as it is demonstrated for a similar one-dimensional model in Ref. 5. (ii) in moving the narrow px band associated with 0(4) away from the Fermi level with decreasing oxygen concentration in YBa2Cu307.¢, thus the superconducting transition temperature must be reduced essentiallyl2,13 (see Eq. 12). It is interesting to note, that a similar role of the heavy band was discussed by J. Ruvalds in the framework of a theory 14, where the interaction is mediated by acoustic plasmons. In the later theory, however, the interband transitions are omitted.

Vol. 70, No. 4

Finally, it is important to emphasize that not the entire parking orbital band must be flat, it is enough that a flat part of it is nearby, above or below the Fermi energy. Furthermore, it must be mentioned, that in the present theory the on-site Coulomb interaction in the CuO2 plane has not been taken into account. It is easy to show that the perturbative treatment of that does not lead to leading logarithmic coneetions, thus the results obtained remain valid. That will be changed, however, for larger on-site interaction, due to the superexchange between the formed Cu moments, which is not treated here as it appears beyond the logarithmic approximation. The superexchange can be responsible for antiferromagentism. The superconductivity can compete with that only in that range of clopping, where the induced coupling is large thus where the upper band contains a considerable amount of holes as is discussed earlier. The conclusion of the present paper is that the px orbitals of oxygens may play as intcrmexliate states an important role in inducing attractive interaction between electrons. The responsible off-diagonal Coulomb coupling is essentially enhanced by correlation effects of orbital, Kondotype origin. The range of coupling parameters required to explain the high superconducting transition temperature are reasonable, but Tc depends very sensitively on those. After the manuscript was completed a preprint by Z. Tesanovic, A. B. Bishop, R. L. Martin and K. A. Mtiller was received where detailed analysis of the role of the nonbonding pg orbit is provided. Note added on December 21, 1988: I have just learned that Zlatko Tesanovic, A. R. Bishop and Richard Martin in preprint LA-UR-88-2534 made the suggestion that the non-bonding p~ oxygen orbita/s on the chains contribute to develop attractive interaction between electrons. Acknowledgement--The author is greatly indebted to D. Pines, J. Bardeen, and A. J. Leggett for their hospitality and valuable discussions at the University of Illinois, and benefited from participating in, and was supported by, the summer programs at Los Alamos National Laboratory and Aspen Center for Physics. The discussions with many physicists, especially with E. Abrahams, D. K. Campbell, L. H. Weber made particular influence on the present work. The work was also partially supported by NSF DRM85-24101.

References 1.

2. 3. 4. 5. 6. 7. 8.

9.

J. G. Bednorz and K. A. Mtiller, Zcitschrift flit Physik B 64, 189 (1986). See for a review: T. M. Rice, Zeitschrift far Physik B 67, 141 (1987). C. M. Varma, S. Schmitt-Rink, E. Abrahams, Solid State Communications 62, 681 (1987). A. Zawadowski, Physical Review B 39 ..... (1989). A. Zawadowski, Proceedings of the Nobel Symposia on "Physics of Low Dimensional Physics", 1988, Gr~ftavallen. Physica Scripta 39, ... (1989). W. Weber, Zeitschrift ~ r Physik B 70, 323 (1988). Jaejun Yu, S. Massida, A. J. Freeman, D. D.Koeling, Physics Letters A 122, 198 (1987). Y. Guo, J.-M. Langlois, W. A. Goddard HI, Science 239, 896 (1988). B. Batlogg, R. J. Cava, L. W. Rupp, Jr., A. M. Mujsce, J. J. Krajewski, J. P. Remeika, W. F. Peck, Jr., A. S. Cooper, and G. P. Espinosa, Physical Review Letters .6!, 1670 (1988).

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14.

W . I . F . David, et al., Nature 327, 310 (1987). S. Massida, Jaejun Yu, A. J. Freeman, and D. D. Koelling, Physics Letters A, 122, 198 (1987); Key Taeck Park, Kiyoyuki Terakura, Tamio Oguchi, Akira Yanase and Minoru Ikeda, Journal of the Physical Society of Japan, ~ 3445 (1988). Z.Z. Wang, J. Clayhold, N. P. Ong, J. M. Tarrascon, L. H. Greene, W. R. McKinnon and G. W. Hull, Phys. Rev. B 36, 7222 (1987). P.C. Miceli, J. M. Tarascon, L. H. Greene, P. Barboux, F. J. Rote//a, and J. D. Jorgensen. Physical Review B 32, 5932 (1988). The data for Cu(2) - 0(4) distance was provided by L. H. Gr~ne (private communication); R. J. Cava, et al., Physica C 153-155. 560 (1988). J. Ruvalds, Phys. Rev. B. ~ 8869 (1987); Y. Ishii and J. Ruvalds, Phys. Rev. B. (to be published).