High-Tc superconductivity by pairing of resonant delocalized holes

~Solld

S t a t e Communications, P r i n t e d i n Great B r i t a i n .

Vol. 70,

No.

3. pp. 283-286, 1989.

0038-1098/8953.00+.00 Pergamon P r e s s plc

HIGH-T c SUPERCONDUCTIVITY BY PAIRING OF RESONANT DELOCALIZED HOLES To Tsang Department of Physics, Howard University, Washington, D. C. 20059, U. S. A. (Received 10 December 1988 by W. Y. Kuan)

When CuO 2 sheets in the (123)- or (214)-compounds are oxidized, the holes would delocalize to a limited extent. The resonant square state would form from three electrons with parallel spins located on a square of four Cu atoms. This state is not susceptible to phonon scattering. There is antiferromagnetic pairing between these states with opposite spins. The theoretical results agree quite well with the experimental data.

unpaired d x 2 v 2 electron with energy much higher (Sy ~ 2 eV) than the other four pairs of electrons. 12 The system is a ~ott-Hubbard solid (where the energ~ band theories are not applicable) with one d x 2 y 2 electron per Cu-atom due to the Hubbard U-parameter. 13-15 In the following, our discussions will be limited to this d electron only. The nearest neighbor d-electron spins are arranged in upand down-spin (~ and p) pairs for the GS because of the antiferromagnetic 14-16 (AFM) superexchange interaction. In the GS, the infinite sheets have checkerboard patterns of ~ and spins. The required energy 2J for the flip of one spin pair (from antiparallel to parallel spins) is estimated 17 to be 0.036 eV. Starting from the un-oxidized GS, we will now consider the oxidation process (creation of a hole). The delectron with spin ~ is removed from site a at the position (i,j) (i-th row and j-th column) in the xy-plane. The energy of this localized hole state ( L H S ) w i l l be denoted as E L. It may be seen that th~ energy of this state may be lowered considerably by delocalization and resonance. We will also consider three other sites, b,c and d, at the positions (i+l,j),(i÷l,j+l) and (i,j+l). The sites a,b,c,d form a square in the xy-plane. In the LHS, the spins are ~ ,~ and ~ for the b, c and d-sites, while the a-site is empty. By flipping the ~ - s p i n at the c-site to ~-spin, we get the parallelspin state (PSS) with the following determinant wave function,

Discoveries 1-4 of superconductivity (SC)in La2_xSrxCu04 (214-compounds), RBa2Cu307_ x (the 123-compounds), and their derivatives have sparked an enormous world-wide surge of activity~ Nevertheless, the pairing mechanism and the nature of the SC are still not well understood at present° D-7~ The common features of these SC structures are sheets of Cu-0 bonds forming a network of squares. 8'9 The interactions are relatively weak between the sheets° (The out-of-plane Cu-O bond lengths of 2.4~ are considerably longer tha~ the in-plane Cu-0 bond length of 1o9A) Some of the Cu ÷2 have been oxidized to Cu +3 (the holes)° In our present work, we have proposed a specific mechanism for this oxidation process and the pairing of the holes. The mechanism has some interesting features. The holes are delocalized and stabilized by resonance. However, the delocalization is limited to four Cu-sites. The hole structure and the pairing appear to be specific for these sheet structures° Our theoretical results are also consistent with the experimental data (such as the short coherence length 6 and the absence of the isotope effect10'11)° Furthermore, there are considerable differences (as well as similarities) from the conventional Cooper pairing and BCS theory° For the un-oxidized ground state (GS) of an infinite sheet of CuO 2 squares in the xy-plane, all Cu atoms are in the +2 valency (d 9) state° For Cu +2 with square-planar coordination,it is only necessary to consider the 283

284

PAIRING OF RESONANT DELOCALIZED HOLES

(6-½)

Sb(rl)~l

~c(rl)~l

~d(rl)~l 1

@b(r2)~2

~c(r2)~2

~d(r2)~21

(r3)~3

@c(r3)~3

@d(r3)~3J

thus it is necessary to flip two spins to et the PSS. On diagonalizing the 6x6genergy matrix, the lowest eigenvalue is,

ERR=EL- 2.41 t +16J

where #b is the metal dx2_y 2 orbital (with small admixtures of ligand orbitale) on site b, r I is the position of electr0n I, etc. This wave function may be factorized into ~(bcd)~l~2~3, where ~1~2~3 is the spin part, and ~(bcd) is the spatial part and is given by, I lSb(rl)~c(rl ) (6"~)l~b(r 2)

~d(rl) I

(1)

~c(r2 ) ~d(r2) I

|@b(r3 ) ~c(r3 ) ~d(r3)J

t 0 t

t

0

t)

Q t 0

t Q t

0 t Q

where t = < ~ a l H l @ b >

(4)

Within the likely range 4 < ( t / J ) < 1 9 , then ERS would be lower than either E L or ERR. Thus the LHS would delocalize to the RS state. However, there would be no further delocalization. The phonons (lattice vibrations) would change the bond lengths, hence there would be fluctuations in the transfer integrals. The non-diagonal matrix elements in equation (2) may be replaced by t+~t i, where i=1,2,3,4. After diagonalization, the lowest eigenvalue is~ ERS =~L -2t+8J- [< (~ti) 2>/(2t)J

The PSS energy is higher than the LHS energy by 8J because of the spin flip at c-site involves four Cu nearest neighbors° We will denote the PSS energy as Q=EL+8J. Although Q is higher than E L , the PSS energy can be lowered considerably through resonance. Since the a-site is empty, there are four equivalent spatial functions, ~b(bcd), $(cda), ~(dab) and $(abc). Using these four functions as the basis functions, the matrix for the Hamiltonian is (Q

Vol. 70, No. 3

(2)

is the effective

transfer integral. 13-15 Typically, t is about an order of magnitude larger than J. The lowest eigenvalue of the matrix isa ERS =Q- 2t =EL- 2t +8J. (3 ) where RS stands for "resonant square". ERS is expected to be considerably lower (byN1 eV) than EL since tN10J. Thus there would be delocalization of the holes. The RS state of the hole is specific for the two-dimensional Cu02 sheet structure. Two characteristics (limited delocalization and phonon immunity) of the RS state may be intimately related to the SC. We will first consider the expansion of the RS state of the hole to the resonant rectangle (RR) with six sites° There are five electrons for the RR,

[5)

where we have used (~t~)=0,<~ti~tj>=0_ for i~j, and (~ti2> is independent of i. The energy ERS is actually lowered by the fluctuations due to phonons. Hence the RS state is immune to phonons. There are certain similarities between the RS states and the Cooper pairs. Both are stabilized by the phonon interactions. We will now discuss the condensations of RS pairs with opposite spins for the (123)-compounds with high hole concentrations. An approximate selfconsistent (molecular field) Ising model calculation will be given for this condensation which is somewhat similar to the AFM phase transition. For the square lattice, the coordination number is z=4. However, two types of coordinations (at side and at corner) may occur for the RS as shown in Fig. 1. The probabilities are 1/3 and 2/3 for these coordinations. For side coordination as shown in Fig. I(A), the energies are +2j and -2j for parallel and antiparallel spins, where j=(3/4)2j since the average number of electrons is 3/4 on each site of the RS. The corner coordination is shown in Figs. I(B) and 1(C) for the upper left corner. Another neighboring RS is shown in dashed lines. It is impossible to have antiparallel spins (AFM ordering) between all RS shown in Figs. I(B) and 1(C). Upper left corner coordination blocks out AFM on two sides (the upper Side and left side). 0nly two sides would be available for AFM. The average spin of the RS will be denoted by S. The weights would be (1/3) 4 for four side coordinations, 4(1/3)3(2/3) for three side and one corner coordinationL When the central

PAIRING OF RESONANT DELOCALIZEDHOLES

Vol. 70, No. 3 !

I-IN (a)

I-I

i

"-"

Tc=93°K, the experimental value is mole of the (123)-

~Cp=(O.77±O.17)Rper

[:]

(b)

(c)

Fig. I. (A) Side coordination between the central RS (shaded square) and the neighboring RS (unshaded square). (B), (C) Corner coordinations. Other neighboring RS are shown as squares with dashed lines.

RS has up spin, the energies are 8iS and 4jS respectively. For two corner coordinations toward one side, the weight is 4(1/3)2(2/3) 2 , and the energy is 2jS. For other coordinations, the energy is zero. When the central RS spin is up, the average energy is,

(1/3)4(8jS)+4(1/3)3(2/3)(4jS) +4(1/3)2(2/3)2(2jS) = 8iS/9 Similarly, the energy is -8jS/9 when the central RS has down spin. The self-consistent condition is S=tanh(8jS/9kT) where k is the Boltzmann constant and T is the temperature. The transition temperature T c is given by the condition 8j/(9kTc)=l.

Using

j=(3/4)2j and the theoretical value J=205°K of Guo etal. 17, we get T =102°K. This result agrees quite well c with the experimental value of 93°K for the (123)-compounds. In our pairing model, the specific heat anomaly ~ C p has the standard value of 1.5R per gram mole of holes, where R is the gas constant. For one mole of RBa2Cu307_ x, there are 2 moles of Cu atoms in the sheets. The maximum number of RS is 2(1/4) mole since each RS occupies 4 Cu-sites in the sheet. The theoretical value of ~ C p is 0.75R per mole of the compound. The experimental value 18 is ~ C J T c = 2 3 ~ 5 mJ/°K 2 per mole of Cu, or

~C~Tc=69±15

285

per mole of the (123)-compound. Using

r--'1

mJ/°K 2

compound. There is very good agreement between the theoretical and experimental values. The dimension of the RS is ~4~. The dimension of the RS hole pair is ~10~, and is comparable to the observed coherence lengths of the (123)compounds. These distances are considerably shorter than the dimensions of Cooper pairs (~I04 ~). In contrast to the Cooper pairs, the phonons are no t d i r e c t l y involved either in the formation of the RS or the spin-pairing of the RS (although both the RS and Cooper pairs are stabilized by the phonons)o Thus it is expected that there would be no isotope effect on the T c of the SC oxides. This is consistent with the experimental observations° 10'11 In the RS state, the hole is distributed over four Cu-siteso Instead of one Cu +3 site, there are four Cu +2'25 sites. The valencies of Cu in compounds may be identified through the photoelectron spectra (XPS). On comparison with the XPS of Cu +1 and Cu +2 compounds19,there have been great difficulties in identifying the Cu +3 species in the XPS of the (123)-compounds 20. Again, the experimental results (the possible absence of Cu +3 species) are consistent with the RS state. In summary, it has been proposed that the hole state in Cu02 sheets would delocalize to the parallel-spin RS, but no further. This limited delocalization is specific for the Cu02 sheets. The RS pairs are formed by AFM coupling. There are good agreements with various experimental results,such as T c, the specific heat anomaly, the absence of isotope effects and Cu +3 species, the short coherence lengths, etc. A c k n o w l e d g e m e n t - - T h e partial financial support by National Aeronautics and Space Administration grant NAG5-156 is gratefully acknowledged.

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58, 9o8 (1987)o 4. Z. X. Zhao etal.,Kexue Tongbao 32, 412 (1987).

286

PAIRING OF RESONANT DELOCALIZED HOLES

5. T.M. Rice, Z. Phys. B67, 141 (1987). 6. T.H. Geballe and J.K. Hulm, Science 239, 367 (1988). 7. S.A. Wolf and V.Z. Kresin, Novel Superconductivity. Plenum Press, New Y o r k (1987). 8. Q.W. Yan etal.,Phys. Rev. B~6, 5599 (1987). 9. W.I.P. David etal., Nature 327, 310 (1987). 10. B. Batlo~g etal., Phys. Rev. Lett. 58, 2333 (1987). 11o L.C. Bourne etal., Phys. Rev. Lett. 58, 2337 (1987). 12. F. J. Adrian, Phys.Rev. B37, 2326 (1988). 13. D. Adler, Solid State Physics 21, 1 (1968). 14o P.W. Anderson, Solid State Physics 14, 99 (1963).

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15. P.W. Anderson, in Magnetism, edited by G.T. Rado and H.Suhl, Vol. I, p. 25. Academic Press, New York(1963). 16o P.J. Hay, J.C. Thibeault and R. Hoffmann, J. Amer. Chem. Soc. 97, 4884 (1975). 17o Y. Guo, J.M. Langlois and Wo A. Goddard, Science 239, 896 (1988). 18. C. Ayache etal., Solid State Commun. 64, 247 (1987). 19. L. Yin, ToTsang and I.Adler, Proceedings of Seventh Lunar Science Conference (1976), p. 891. 20. D.D. Sarma and C.N.R. Rao, Solid State Commun. 65, 47(1988).