J . Phys. Chem. 1989, 93, 4414-4422
4414
FEATURE ARTICLE Physical Chemistry of High-Temperature Oxide Superconductorst T. V. Ramakrishnant and C. N. R. Rao* Indian Institute of Science, Bangalore 560 012, India (Received: November 21, 1988)
The discovery of high-temperature superconductivity in metal cuprates has ushered in a new era in the chemistry and physics of solids. At the same time, it has also highlighted our inadequate knowledge of the electronic structures of metal oxides. In this Feature Article, we briefly discuss the structure and properties of the different families of oxide superconductors discovered since 1986, paying specific attention to the states of copper and oxygen in the cuprates. We list those experimental observations related to the superconducting and normal states that have to be explained by theoretical models and present a n overview of the current theoretical models. We conclude by indicating possible future directions.
1. Introduction Metal oxides, particularly those of transition elements, possessing a variety of structures and properties have been intensively studied in recent years. Specially noteworthy is the phenomenal range of electronic and magnetic properties exhibited by metal ~xides.l-~Thus, oxides with diverse magnetic properties ranging from ferromagnetism (e.g., C r 0 2 , Lao,5Sro,sMn03)to antiferromagnetism (e.g., NiO, LaCrO,) are well-known. Many oxides possess two or more orientation states or domains that can be switched from one state to another through the application of one or more appropriate forces; ferroelectricity (e.g., BaTiO,, KNbO,) and ferroelasticity (e.g., CaAI,, Si208) are phenomena where spontaneous electric polarization or spontaneous strain is switched by the application of an electric field or a mechanical stress. There are oxides with metallic properties (e.g., LaNiO,, Reo,) at one end and highly insulating behavior (e.g., BaTiO,, Al,O,) at the other; then there are oxides that traverse both these regimes with change of temperature, pressure or composition (e.g., V203, Lal-,Sr,CoO,). Interesting electronic properties arising from charge density wave transitions (e.g., Nal,WO,), valence ordering ( e g , Fe304),and defect ordering (e.g., Ca2Fe205)are also known; however, no discovery, however fascinating, has created as much excitement as that of high-temperature superconductivity4 in the cuprates. Superconductivity in metal oxides itself has been known for some time in oxides such as NbO and K,W03 exhibiting T:s in the 1-4 K range; the highest T, in oxides before 1987 was 13 K exhibited by BaPbl-xBi,035 and Lil+xTi2-x046 The new oxide superconductors have pushed the transition temperature up to 125 K . The discovery of high T, superconductivity in oxides has highlighted our inadequate knowledge of the electronic behavior of oxide materials. It is generally believed that the unusual properties of metal oxides which distinguish them from metallic elements and alloys, covalent semiconductors, and ionic insulators arise from several factors: (i) Oxides formed by d-block elements have narrow electronic bands, because of the small overlap between the metal d and oxygen p states. The bandwidths are typically of the order of 1 or 2 eV (rather than 5-15 eV as in most metals). (ii) Electron correlation effects play an important role, obviously helped by the narrow electronic bands. The local electronic structure can be described in terms of atomic-like states (e.g., Nil+(dio),Ni2+(d9), and Ni3+(ds)for Ni in NiO) as in the Heitler-London limit. (iii) The polarizability of oxygen is of importance. That is, the divalent
-
'Contribution No. 576 from the Solid State & Structural Chemistry Unit. *Department of Physics. 'To whom correspondence should be addressed at the Solid State & Structural Chemistry Unit.
0022-3654/89/2093-44l4$0l.50/0
oxide ion, 02-, does not exactly describe the state of oxygen. Configurations such as 01-would have to be included, especially in the solid state. This gives rise to lattice and electronic distortions (polaronic and bipolaronic effects). Species such as 01-,which are actually oxygen holes (with a ps configuration instead of the could be mobile and correlated. filled p6 configuration of 02-), (iv) Many oxides are not truly three-dimensional, but exhibit a low-dimensional feature. For example, La2Ni04with the K,NiF4 structure is two-dimensional compared to the analogous LaNiO,, which is a three-dimensional perovskite. Because of these and other unique features, it has been difficult to make much progress in working out proper theoretical models for complex oxides. The high T, oxide superconductors are complex copper oxides possessing low-dimensional features as well as highly covalent Cu-0 bonds. The difficulty in adequately describing such systems in terms of theory is therefore obvious. Structure-property relations in the high-temperature superconducting cuprates have been adequately reviewed in the past few In this article, we shall first describe the salient features of the high T, cuprates, paying some attention to the states of copper and oxygen which play a crucial role in determining the electronic properties of cuprates. We shall examine models for the electronic structure of cuprates and highlight the important experimental observations that models have to explain. We shall then attempt to briefly present the theoretical scenario today. Finally, we speculate on experimental and theoretical efforts needed to obtain further insight into phenomenon of high-temperature superconductivity and the materials exhibiting it.
2. Superconducting Cuprates High-temperature superconducting cuprates discovered since 1986 belong to five families: (i) La2-,M,Cu0, (M = Ca, Sr, or ( I ) Rao, C. N. R.; Subbarao, G. V. Phys. Status Solidi A 1970, I , 597. (2) Goodenough, J. B. Prog. Solid State Chem. 1971, 5, 149. (3) Rao, C. N. R.; Gopalakrishnan, .I.New Direcfions in Solid State Chemistry; Cambridge University Press: Cambridge, U.K., 1986. (4) Bednorz, J. G.; Muller, K. A. Z . Phys. B 1986, 64, 189. (5) Sleight, A. W.; Gillson, J. L.; Bierstedt, P. E. Solid State Commun. 1915, 17, 27. (6) Johnston, D. C.; Prakash, H.; Zachariasen, W. H.; Viswanathan, R. Mater. Res. Bull. 1973, 8, 777. (7) Nelson, D. L.; Whittingham, M. S.;George, T. F., Eds. Chemistry of High-Temperature Superconductors; ACS Symposium Series 35 1; American Chemical Society: Washington, DC, 1987. (8) Rao, C. N. R. J . Solid Sfate Chem. 1988, 7 4 , 147. (9) Rao, C. N. R., Ed. Chemistry of Oxide Superconductors; Blackwell: Oxford, 1988. (10) Muller, J.; Olsen, J. L., Eds. Proceedings of the Interlaken M2HTSC Conference; Physicu C 1988, 153-155, 1-1774. ( I 1) Rao, C. N. R., Ed. Chemical and Structural Aspects of High Temperature Superconductors; Progress in High-Temperature Superconductivity 7 ; World Scientific: Singapore, 1988.
0 1989 American Chemical Society
Feature Article
The Journal
Physical Chemistry, Vol. 93, No. 1I, I989 4415
C b
Sr.Bi
cu CaSr cu Sr,BI
cu2 04 0 . BI
.*
0 . Sr,BI
cu 1 /
@. '
(0)
cu, Ca .Sr
05 (b)
(C)
Figure 1. Unit cells of (a) La2$3rX(Bax)CuO4,(b) YBa2Cu307, and (c) Bi2(Ca,Sr)3C~208 (schematic).
Ba) of the' K2NiF4 s t r u c t ~ r e ; ~(ii) ~ ' the ~ LnBa2Cu307 (123) system,8*16where Ln = Y, La, Nd, Sm, Eu, Gd, Dy, Ho, Er, Tm, or Yb; (iii) the Bi2(Ca,Sr)n+lC~n02n+4 system;' 1~17-20 (iv) the T12Ca,-] B a 2 C ~ , 0 2 n + 4~ y s t e m ; ~ I * ~ and ' - ~ ~ (v) the More recently, T1Ca,,Ba2Cu,0h+3 (n = 1,2,3,4) an oxide without Cu, Bal,K,Bi03, has been found to exhibit2* a Tc of -30 K. In Figure 1 we show the structures of the first three families of cuprates in order to illustrate certain commonalities. The structures of the T1 cuprates are quite similar to those of the Bi cuprate shown in Figure 1. All these cuprates contain defect perovskite layers and all but the 123 compounds possess rock-salt-type oxide layers. All the cuprates contain two-dimensional C u - 0 sheets, and in addition 123 compounds contain
(12) Uchida, S.;Takagi, H.; Kitazawa, K.; Tanaka, S. Jpn. J. Appl. Phys. 1987, 26, L1. (13) Chu, C. W.; Hor, D. H.; Meng, R. L.; Gao, L.; Huang, Z. J.; Wang, Y. Q. Phys. Rev. Lett. 1987, 58, 405. (14) Cava, R. J.; van Dover, R. B.; Batlogg, B.; Reitman, E. A. Phys. Rev. Lett. 1987, 58, 408. (15) Ganguly, P.; Mohan Ram, R. A.; Sreedhar, K.; Rao, C. N. R. Solid State Commun. 1987, 62, 807. (16) Wu, M. K.; Ashburn, J. R.; Torng, C. J.; Hor, P. H.; Meng, R. L.; Gao, L.; Huang, Z. J.; Wang, Y.Q.; Chu, C. W. Phys. Rev. Lett. 1987,58, 908. (17) Maeda, H.; Tanaka, Y.;Fukutomi, M.; Asano, T. Jpn. J. Appl. Phys. 1987,27, L209. (1 8) Subramanian, M. A.; Torardi, C. C.; Calabrese, J. C.; Gopalakrishnan, J.; Morrissey, J. J.; Askew, T. R.; Flippen, R. B.; Chowdhry, U.; Sleight, A. W. Science 1988,239, 1015. (19) Hervieu, M.; Michel, C.; Domenges, B.; Laliguant, Y.;Lebail, A.; Feray, G.; Raveau, B. Mod. Phys. Lett. B 1988, 2, 491, 835. (20) Rao, C. N. R.; Ganapathi, L.; Vijayaraghavan, R.; Rangarao, G.; Murthy, K.; Mohanram, R. A. Physica C 1988, 156, 827. (21) Sheng, Z. Z.; Hermann, A. M. Nature 1988,332, 5 5 , 138. (22) Torardi, C. C.; Subramanian, M. A.; Calabrese, J. C.; Gopalakrishnan, J.; Morrissey, K. J.; Askew, T. R.; Flippen, R. B.; Chowdhry, U.; Sleight, A. W. Science 1988, 240, 631. (23) Ganguli, A. K.; Nanjundaswamy, K. S.; Subbanna, G. N.; Umarji, A. M.; Bhat, S. V.; Rao, C. N. R. Solid State Commun. 1988, 67, 39. (24) Maignan, A.; Michel, C.; Hervieu, M.; Martin, C.; Groult, D.; Raveau, B. Mod. Phys. Lett. B 1988, 2, 681. (25) Ganguli, A. K.; Subbanna, G. N.; Rao, C. N. R. Physica C 1988,156, 181. (26) Hervieu, M.; Maignan, A.; Martin, C.; Michel, C.; Provost, J.; Raveau, B. J. Solid State Chem. 1988, 75, 212. (27) Parkin, S. S. P.; Lee, V. Y.;Nazzal, A. I.; Savoy, R.; Beyers, R.; La Placa, S. J. Phys. Rev. Lett. 1988, 61, 750. (28) Cava, R. J.; Batlogg, B.; Krajewski, J. J.; Farrow, R.; Rupp, L. W.; White, A. E.; Short, K.; Peck, W. F.; Kometani, T. Nature 1988,332,814.
r
-
I
-
I 0.2
I
d
I 0.4
-
r
1 ,1 .
Figure 2. Variation of T, of Y B a 2 C ~ 3 0 7 with 4 6. Shaded portion includes data from magnetic measurements from different laboratories. Points are based on measurements from this laboratory.
one-dimensional Cu-0 chains. The coordination of Cu is essentially square planar, and the Cu-0 bond distance is around 1.9 A, indicative of high covalency. We shall now briefly review some of the important features of the different cup rate^,^^ highlighting those observations that are seminal to our understanding the phenomenon as well as the system. Oxides of the La2-,M,Cu04 family are ordinarily tetragonal and become orthorhombic near 180 K.30 The T i s are in the 25-40 K range (at an optimal value of x ) depending on the M ion. Substitution of Ca by Ni, Zn, and such ions adversely affects superconductivity; La can be replaced by Eu, Gd, and other rare earth ions up to 5% without badly affecting the superconductivity. In the 123 compounds, the Ln ion has little effect on the T,, but the Tc is markedly dependent on the oxygen stoichiometry 6. In the case of YBa2C~307-b, T, is nearly constant (-90 K) up to 6 = 0.2 but drops to a constant value of -55 K between 6 of 0.2 and 0.4 (Figure 2); further increase in 6 drastically lowers the (29) The reviews listed earlier (ref 7-1 1) provide most of the useful references to the original literature. We shall cite some of the additional references where necessary: We would like to be excused for any errors of judgement in our choice of references. (30) Day, P.; Resseinsky, M.; Prassides, K.; David, W. I. F.; Moze, 0.; Soper, A. J. Phys. C Solid State 1987, 20, L429.
4416
The Journal of Physical Chemistry, Vol. 93, No. 11. 1989
T, until it becomes nonsuperconducting when 6 = 0.6.31 The structure is orthorhombic over the entire 6 range of 0.0-0.60 but becomes tetragonal when 6 2 0.60. It is not clear whether there is any special structural feature that is associated with the 55 K T, plateau (Figure 2). We have recently found that the b parameter of 90 K T, orthorhombic structure is always equal to c/3, but not so in the orthorhombic structure showing 55 K T,. That is b = (c/3) only when the chains are highly populated. In the 123 compounds, ordered orthorhombic structures with 90 K T, are found only when the 0 1 oxygen in the Cu-0 chains are highly populated and ordered; the distribution of the chain oxygen between the 01 and 0 5 sites gives rise to disordered Equal population of orthorhombic structures with low T,‘S.~~ the 01 and 0 5 sites, just as complete depletion of the 0 1 oxygen, gives rise to tetragonal structures. The chains, however, do not seem to play a determining role in the high T, behavior. YBazC~307-6 samples with 6 = 0.1-0.6 can be oxidized to the 6 = 0.0 phase, but the diffusion of oxygen is a highly activated process. It is believed that formal mixed valence of Cu is essential for superconductivity of these cuprates. Yet we find superconductivity in Y B a 2 C ~ 3 0 6(, 5T , 45 K) which is expected to contain Cu only in the 2+ state. It seems likely that intergrowth of the O7 and the O6 phases is present in the compositions. If so, we by should be able to detect the presence of Cu2+in YBa2C~306.5 EPR spectroscopy. Well-annealed samples with 6 0.5 do not show evidence for Cu2+ in EPR and other measurements. LnBa2Cu307may be considered to be the x = 1 member of the more general Ln3-xBa3+xC~6014+d family. A common occurrence in the 336 and 123 compounds is the exchange between the Ln and Ba sites specially when Ln is a large rare-earth ion such as La. Such an exchange does not occur in the YBa2Cu307 because of the small size of yttrium. The Ba ion in YBazCu307 can, however, be replaced by La to advantage, rendering the oxygen less labile. Generally speaking, the smallest rare-earth ion tolerated by the 123 orthorhombic structure is Yb. Lu does not form the 123 compound, but superconducting Y I - x L ~ x B a z C ~ 3where 0 7 , the weighted average radius of the rare earth corresponds to Yb, can be made. Orthorhombicity was considered to be a necessary criterion for high T, in the 123 compounds for some time, but tetragonal YBazCu307where Co, Fe, or Ga partly substitute for Cu have since been found to show high T, behavior. These tetragonal oxides also exhibit orthorhombic microdomains. Substitution of Cu by Ni and Zn affects the oxygen stoichiometry and lowers the Tc. Orthorhombic 123 compounds show extensive twinning, created during their formation from the high-temperature tetragonal structures. Across the twin boundaries there is a 90’ rotation of the a and b axes. There is no discernible change in oxygen stoichiometry across the twins. While twins may play a crucial role in determining properties such as the critical current density of 123 compounds, they are not the cause of superconductivity. Thus, orthorhombic PrBa2Cu307,which is not superconducting, shows twins. The absence of superconductivity in the 123 compounds of Ce, Pr, and Tb has been suspected to lie in the bivalency (3+ and 4+) of these lanthanide ions, but the exact cause is not yet understood. M e m b e r s of t h e B i z ( C a , S r ) n + l C ~ n 0 2 n +and 4 T12Can-IBa2C~n02n+4 series have similar structures and contain two Bi, TI-0 type rock-salt layers. The bismuth cuprates show modulation in the structure and a van der Waals gap of -3.2 A between the Bi-O layers. Members of the T1Ca,lBazCu,0z,,+3 have a single TI-0 type rock-salt layer. The Bi, Ca, and Sr sites in the Bi cuprates are interchangeable, and the compositions
-
(31) Johnston, D. C.; Jacobson, A. J.; Newsam, J. M.; Lewandinski, J. T.; Goshorn, D. P.; Xie, D.; Yelon, W. B. ACS Symp. Ser. 1987, No. 351, Chapter 14. (32) (a) Beech, F.; Miraglia, S.; Sontoro, A,; Roth, R. S . Phys. Reu. B 1987,35,8778. (b) Jorgensen, J. D.; Beno, M. A.; Hinks, D. G.; Soderholm, L.; Volin, K. J.; Hitlerman, R. L.; Grace, J. D.; Schuller, I . K.; Segre, C. U.; Zhang, K.; Kleefisch, M. S . Phys. Reu. B 1987, 35, 7915.
Ramakrishnan and Rao
o T12Ca-Ba-Cu 0
T I : Ca-Ea-Cu
A Bi
Ca-Sr-Cu
I
1
1
1
2
3
n
Figure 3. Variation of T, in bismuth and thallium cuprates with the number of Cu-0 sheets, n. Broken lines indicate trends observed recently when n > 3.
(Bi:Ca:Sr ratios) are never exactly 2122 or 2223 as described by the general formula Bi2(Ca,Sr)n+lC~n02n+4. Most samples tend to be Ca rich. Bi can be partly substituted by Pb (up to -25%), and this generally favors the formation of better monophasic compositions with slightly enhanced T,’s. In the Bi2(Ca,Sr)n+lC~nOZn+4 series, the first three members with c parameters of -25,31, and 38 8, have been characterized, the T,‘s being 60 f 20, 85 f 5, and 107 f 3 K, respectively. The phasic purity of these cuprates, however, remains a problem. The n = 2 member is most stable, but the composition and properties of the n = 1 member have been elusive, showing the onset of superconductivity at 60 f 20 K in some of the preparations. In the T12Ca,lBazCu,0M4 series, the n = 1,2, and 3 members (c parameters 23, 29, and 36 A) show T,’s of 80, 110, and 125 K, respectively. The n = 2 and 3 members of the T1Ca,lBazCu,0M3 series show Tc’sof 90 and 1 15 K, respectively, and these are lower than most of the corresponding members of the T12series. In the T1 cuprates, just as in the Bi cuprates, we see a progressive increase in T, as well as in the c parameter with the number of Cu-0 sheets only up to n = 3; when n > 3, the thallium cuprates do not seem to show a further increase in T, (Figure 3). An ultramicrostructural feature common to the Bi and TI cuprates is the presence of inter growth^.^^ It has not yet been possible to exploit intergrowth structures to improve superconducting properties. It has not been possible to prepare well-defined superconducting compositions of the TI-Ca-Sr-Cu-0 system with the general formula Tll,z(Ca,Sr),IC~,0Zn+3,4.These members are, however, ~ progressive stabilized by partial substitution of T1 by p b . ” ~ ~Thus, increase in x in Tll-,PbXCaSrzCuzO, increases the T, up to 90 K when x = 0.5;x = 0.3 has a T,of 70 K. Tl,,5Pb0,SCaSr2C~307 has a T, of 120 K. Progressive substitution of Ca by Y in T1Ca,IBa2C~nOZn+3 and Biz(Ca,Sr)n+lC~nOZn+4 lowers the T,, until superconductivity is lost on complete r e ~ l a c e m e n t . ~ ~
3. Important Properties of Superconducting Cuprates Some of the important properties of the superconducting state are the critical field, the critical current, the magnetic penetration depth, and the coherence length. The H,,(O) and H,,(O) of YBa2Cu307parallel to the c axis are close to 1 and 120 T, respectively; the magnetic penetration depth is -900 A.37 The coherence length (size of the Cooper pair) is 10-30 8, in the ab (33) Rao, C. N. R.; Thomas, J. M. Acc. Chem. Res. 1985, 18, 113. (34) Ganguli, A. K.;Nanjundaswamy, K. S.; Rao, C. N. R. Physica C 1988, 156, 788.
(35) Subramanian, M. A.; Torardi, C. C.; Gopalakrishnan, J.; Gai, P. L.; Calabrese, J. C.; Askew, T. R.; Flippen, R. B.; Sleight, A. W.,1988, 242, 249. (36) Ganguli, A. K.; Nagarajan, R.; Nanjundaswamy, K. S.; Rao, C. N. R. Mare?. Res. Bull. 1989, 24, 103. (37) Umezawa, A.; Crabtree, G . W.; Liu, J. 2.Physica C 1988, 153-55, 1461.
Feature Article plane of YBa2Cu307as well as other cuprates but only about 2-4 A along the perpendicular to the plane.37 The meaning of such a small coherence length is not clear. Anisotropy is also found in the magnetic and electrical properties of YBa2Cu307and the other cuprates. The critical current of a superconductor should be at least lo5 A cm-2 for magnet and other applications at the operating temperatures. There has been considerable divergence of results reported on the 123 compounds with regard to the critical current. While values of lo5 A cm-2 or higher have been reported in films and single grain materials38of YBazCu3O7,it has not been possible to obtain good, reproducible samples because of the presence of grain boundaries and weak flux pinning. The situation is the same with TI and Bi cuprates, though there is a report by Kowak et of relatively high critical currents in polycrystalline thin films of T12CaBa2Cu20y. For all practical purposes no I 8 0 isotope effect has been observed in Y B a z C ~ 3 0 7 , 3implying 9 that the conventional phonon mechanism may not be valid here. The magnitude of the superconducting gap has been a matter of debate. Different measurements (IR absorption, point-contact tunneling spectroscopy, etc.) give different results, but many seem to cluster around 2A 2: 3-4(kBT,).40 There are also reports of larger gaps or no gap. In conventional superconductors, there is a specific heat discontinuity at T,, proportional to the electronic specific heat at that temperature. It is not clear whether such a discontinuity is present in the oxide superconductors. Experimental observations are consistent with either a discontinuity or with a rise in specific heat over a narrow temperature range presaging a weak Mike anomaly. The latter would be due to fluctuation effects associated with the rather short superconducting coherence length. In the superconducting cuprates, the copper ions are EPR silent. However, they show intense nonresonant absorption of microw a v e ~ . The ~ ~ presence of Cuz+ in YBaZC~307-6 is clearly evidenced in magnetic measurements in the nonsuperconducting YBa2Cu306, which is antiferromagnetic ( T N = 450 K). LnBa2Cu307compounds with Ln = Gd, Dy, etc., show magnetism at low temperatures due to the Ln ion. X-ray absorption spectroscopy, electron spectroscopy, and related techniques have thrown light on the electronic structures of the cuprates. The d-d correlation energy is the largest relevant energy in these oxides. Copper is in a mixed-valent state, corresponding to the d-orbital occupation of 9.2-9.5 (we shall discuss this aspect as well as the state of oxygen in the next section). Curiously, the density of states near the Fermi energy in La2Cu04 and La2-xMxCu04both are small, as revealed by photoemission spectra. Does this imply that the states near the Fermi energy make very poor quasiparticles? An angle-resolved photoemission study of Bi2CaSr2Cu20842 has shown that there are p-like bands near and below EF,with a very small width (-0.5 eV). It appears that (ep - td) is close to 1-3 eV and the pd mixing integral to is in the range of 1-2 eV.
-
4. States of Copper and Oxygen in Cuprate Superconductors The state of copper in the cuprate superconductors is of seminal importance to the mechanism of superconductivity in these materials. In stoichiometric La2Cu04,Cu is present in the 2+ state; this would not be the case in La2-,M,CuO4 or YBa2Cu307. It has been generally believed that the presence of Cu in the mix(38) See Malozemoff, A. P. Physica C 1988, 153-55, 1049, for a survey. Kowak, J. F.; Venturini, E. L.; Banghman, R. J.; Morosin, B.; Ginley, D. Physica C 1988, 156, 103. (39) (a) Batlogg, B.; Cava, R. J.; Jayaraman, A,; van Dover, R. B.; Kourouklis, G. A,; Sunshine, S.; Murphy, D. W.; Rupp, L. W.; Chen, H. S.; White, A.; Short, K. T. Mujsce, A. M.; Rietman, E. A. Phys. Rev. Leu. 1987, 58, 2333. (b) Bourne, L. C.; Crommie, M. F.; Zettl, A,; zur Loye, H.-C.; Keller, S. W.; Leary, K. L.; Stacy, A. M.; Chang, K. J.; Cohen, M. L.; Morris, D. E. Phys. Rev. Lett. 1987, 58, 2337. (40) Van Bentum, P. J. M.; et al. Physica C 1988, 153-55, 1718. (41) (a) Bhat, S.V.; Ganguly, P.; Ramakrishnan, T. V.; Rao, C. N. R. J . Phys. C: Solid Srare 1987, 20, L559. (b) Portis, A. M.; Blazey, K. W.; Muller, K. A.; Bednorz, J. G. Europhys. Lett. 1988, 5 , 487. (42) Takahashi, T.; Matsuyama, H.; Yoshida, H. K.; Okabe, Y.; Hosoya, S.;Seki, K.; Fujimoto, H.; Sato, M.; Inokuchi, H. Nature 1988, 334, 691.
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989 4417 ed-valent (2+, 3+) state, in such "doped" oxides, is essential to explain the magnetic and superconducting properties. Oxygen excess in these materials is expected to create holes on Cu in the form of Cu3+. However, X-ray absorption measurements show no evidence for the presence of Cu3+in YBa2Cu307. Instead, very indicate the presence of copper in the 1+ state and holes on oxygen, Ol-. This is corroborated by electron energy loss spectroscopy which shows an s-p transition of o ~ y g e n . ~X-ray ~ , ~ photoelectron spectroscopy and Auger spectroscopy also show that, for all practical purposes, there is no Cu3+ in YBazCu3O7and also in Bi2(Ca,Sr)3Cu208+6.There is, however, definitive evidence for the presence of the Cul+ state. The Cu(2p) feature at 933 eV due to dIoL-' d10L-2final states and the d8 multiplet ('G state) in the Auger spectrum arise from the presence of CUI' species in the ground Evidence for the dimerization of oxygen holes giving rise to peroxo-type species, O?-, has been presented based on O( 1s) spectra.45 It is believed by many workers today that mobile oxygen holes are responsible for the superconductivity of the cuprates. In the Bi and TI cuprates, the Bi-0 and T1-0 layers seem to donate such holes. The-question then arises as to the nature of such holes or specifically as to which orbitals of oxygen are involved in forming the holes. Chakraverty et a!l7 among many others propose that the ds-9 orbital of copper (in the Cu-0 sheets) overlaps with the pu orbital of oxygen (formed by a combination of px and py orbitals), forming a broad Cu-0 band consistent with high covalency of the Cu-0 bands. They also propose that the holes are in the pu state (within the Cu-0 band) and that they are more favored by the dI0-state CUI+ ions than by Cu2+ (d9) ions. We suggest that a 01-Cul+-O'- state (which we have designated as a peroxiton) is energetically favored compared to a hole bipolaron, O'--Cu2+-01-. The presence of holes in pu also explains the absence of antiferromagnetism in the cuprates. Guo, Langlois, and G ~ d d a r don ,~~ the basis of cluster calculations, suggest that oxidation of Cu beyond Cuz+creates oxygen pI holes bridging two Cu2+sites. The pr holes are ferromagnetically coupled to adjacent Cuz+ d electrons, and hopping of the pr holes in the Cu-0 sheets from site to site would then be responsible for the conductivity. The pI of these workers is not the pr orbital but a pxy orbital of oxygen. It is not clear that these will form narrow bands distinctly separated from the broad Cu-0 band involving d,z-,,z (Cu) and oxygen pu orbitals. It is likely that this pI will lie within the broad Cu-0 band. Cluster calculations may tend to overlocalize the holes; the high mobility required of the holes becomes possible when they are in the broad hybridized Cu-0 band. It is, therefore, difficult to distinguish the pr holes of CuO et al. from the hole model proposed from this laboratory. If we consider that the d2-9 (Cu) orbital overlaps with the px orbital of oxygen, then it is not at all certain that the p,, of oxygen will be well separated from the Cu-0 band. Holes in the pI orbitals of oxygen have been proposed by a few other workers as well, and an examination of the models reveals that they are not genuine a orbitals, but are actually u orbitals. Measurements of the orientation dependence of the O(1s) absorption edge as well as of electron spectroscopic studies of single crystals of YBa2Cu,07 and Bi2CaSr2Cu208establish that there are no holes at the Fermi level on oxygen orbitals perpendicular to the CuOz The O(2p) hole state seems to have at least 95% pxy character, where x and y are parallel to the C u 0 2 plane. It should be noted that oxygen hole pairing has been suggested by many theoretician^,^^,^^
+
(43) Bianconi, A.; De Sourtis, M.; Flank, A. M.; Fontaine, A,; Lagarde, P.; Marcelli, A.; Yoshida, H. K.; Kotani, A. Physica C 1988, 153-155, 1760. (44) Fuggle, J. C.; Weijs, P. J. W.; Schorl, R.; Sawatzky, G. A,; Fink, J.; Nucker, N.; Durham, P. J.; Temmerman, W. M. Phys. Rev. B 1988.37, 123. (45) Sarma, D. D.; Ganguly, P.; Sreedhar, K.; Rao, C. N. R. Phys. Rev. B 1987,836, 2371. (46) Sarma, D.D.; Rao, C. N . R. Solid State Commun. 1988, 65, 47. (47) Chakraverty, B. K.; Sarma, D. D.; Rao, C. N . R. Physica C 1988, 156, 413. (48) Guo, Y.; Langlois, J.; Goddard, W. A., 111 Science 1988, 239, 896. (49) Fink, J.; Nucker, N.; Romberg, H.; Fuggle, J. C. Unpublished results. (50) Himpsel, F. J.; Chandrashekhar, G. V.; McLean, A. B.; Shafer, M. W. Phys. Rev. B 1988, 38, 11946.
4418
The Journal of Physical Chemistry. Vol. 93, No. 1 I , IF'89
p;
PI
Ramakrishnan and Rao
p;
Figure 4. (a) d orbitals of copper. The x and y coordinates are in the a b plane, and z is along the c axis. (b) p orbitals of oxygen, with the same
set of axes as in (a). but it is not entirely certain that the peroxo-type species observed in the O( Is) spectra correspond to the paired holes conceived by them. We can relate the presence of 01-holes to the average charge p of (Cu--O)p since p is known to be related to the superconducting properties of the cup rate^.^^ A finite positive value of p (say, 0.2, which seems to delineate superconductors and insulators) can U pr'"(x*-y*)" only result from (Cuz+-O1-)'+ in combination with (CU~+--O~-)~ and (Cui+-O1-)ospecies since there is likely to be no (Cu3+-OZ+)'+ Figure 5. Ligand field split levels of Cu 3d in a distorted octahedral species. The relative importance of ( C U ~ + - O ~and - ) ~(CUI+-O'-)~ coordination analogous to that found in the cuprates (from ref 54). could determine whether a particular cuprate is a superconductor or an insulator. The importance of oxygen holes is also under( B ) Hubbard Model. The Hubbard model is described by the scored by the discovery of relatively high T, (-30 K) in Hamiltonian Bal,K,Bi03.28 Here, Bi3+-01--type species could be important. H = x ( t d - p)dia+dlo xt'ij(di,+dj, dj,+di,) U&ifnii I P ij i 5. Electronic Structure of Cuprates where the terms represent dxz+ electron energy (e,, - p) with the Copper has a distorted octahedral coordination in the cuprates, chemical potential p, intersite hopping t'i, and intraatomic corwith four square-planar oxygen ions and two more weakly bonded relation energy U, respectively. In the oxide superconductors, the oxygens perpendicular to the plane. From a valence count, Cu average number of d + z electrons per site is ( I - 6), where 6 can is seen to exist as a mixture of CuZ+(d9),Cu'+(dlo), and Cu3+(d8) be changed by alloying, e.g., Sr addition in Laz-xSr,Cu04 (in this in order of decreasing stability; correspondingly, oxygen is in mixed case, x N 6), or by oxygen stoichiometry, e.g., in YBa2Cu307-y 02-(p6) and Oi-(p5) states. Electronic structure models with these (where 6 E 0.2-0.3 for y E 0). For 6 = 0 and strong correlation geometrical and configurational features range from strong to ( U >> 4 t 9 , each spin 1 / 2 electron is localized to its site, and the weak crystal field limits and vary greatly in the level of detail system is an insulator (Mott insulator) due to correlation, with considered necessary, not to mention the energy parameters. an excitation gap ( U - 4 t 9 . The system has only spin degrees ( A ) Strong Crystal Field Limit. This approach in its conof freedom at low energies, with antiferromagnetic coupling J ventional ligand field forms4 would be natural if the crystal field ( t {j/b')between nearest-neighbor spins. For low hole densities splitting A C F were much larger than the correlation energy (or 6 with respect to the half-filled Mott limit, the system is an interconfiguration energy difference) U. This is most probably unusual, strongly correlated metal. There have been suggesnot the case in the cuprates, the relevant numbers being AcF t i o n that ~ ~the ~ ground ~ ~ ~state is actually a superconductor. Soon 2 eV and U 7 eV. One is thus closer to the weak crystal field after the discovery of superconductivity in Laz-,Sr,CuO4, Anlimit U >> ACF. Despite this, the strong C F approach is useful d e r ~ o nargued ~ ~ , ~on~ the basis of the above kind of crystal field for its global perspective and because of the simple, explicit picture that a one-band Hubbard model is appropriate for these correlated one-band model it leads to. systems and proposed a novel ground state related to the low spin Parts a and b of Figure 4 show the Cu d and 0 p orbitals, of Cu and the low dimensionality of the cuprates (section respectively, prior to hybridization. Their semicovalent admixture 7C). for td > cp leads to the level structure shown in Figure 5 . These ( C ) Weak Crystal Field or Strong Correlation Limit. If, as levels broaden into bands in the solid. For the Cu2+, 02-conpointed out earlier, the correlation energy U is much larger than figuration, the d9-y like subband is half-full. If this subband (of both the charge-transfer energy ( t d - tp) and the crystal field width 4t3 is well separated from others, one has, effectively a splitting AcF (or equivalently, p-d hybridization to), the d states one-band model. This band is not fully occupied, since the deare close to the atomic limit; the relevant single-site d configuviation from Cuz+,02-leads to fewer than one hole per unit cell. rations are d9 (one hole, in the d2-9 orbital), d'O (no hole, energy The large energy cost of having two d,~-~zelectrons per Cu site ledl). and ds (two d holes, extra energy U with respect to two means that this band is strongly correlated. If U is much larger separate holes). The p hole states are taken to be weakly corthan 4t', but much smaller than the intersubband splitting or the related. The p and d hole levels broaden into a band because of crystal field kF, a strongly correlated single-band Hubbard model p d hybridization to. This is the simplest two-band model, with is appropriate. Now, as mentioned earlier, the latter is not true; a strongly correlated d-like band and a weakly correlated p-like However, because a more realistic Le., one actually has U >> kF. band hybridizing with it. two-band model leads, in some regimes, to a Hubbard-like picture The mode15135sdescribes an insulator for one hole per unit cell, which is the simplest model for correlations in solids, we discuss if the top of the strongly correlated d hole band (similar to the it below. lower Hubbard band in the strongly correlated Hubbard model)
+
+
+
-
-
-
(51) Emery, V. J . Phys. Rev. Lett. 1987, 58, 2794. (52) Hirsch, J. E. Phys. Reu. Lett. 1987, 59, 228. ( 5 3 ) Takura, Y.; Torrance, J. B.; Huang, T. C.; Nazzal, A . I. Phys. Reo. B 1988, 38, 7156.
(55) Valls, 0. T.; Tesanovic, Z. Phys. Reu. Lett. 1984, 53, 1497. (56) Ramakrishnan, T. V. In Metallic and Nonmetallic Slates of Matter; Rao, C. N. R., Edwards, P. P., Eds.; Taylor and Francis: London, 1985; p
(54) Anderson, P. W. Proceedings of the Enrico Fermi Summer School, Frontiers and Borderlines in Many Particle Physics, Varena; North-Holland: Amsterdam, 1988.
16. (57) Anderson, P. W . Science 1987, 235, 1196. (58) Zhang, F. C. Rice, T. M. Phys. Reo. B 1987, 37, 3759
The Journal of Physical Chemistry, Vol. 93, No. 11, I989 4419
Feature Article is separated from the bottom of the p band by a gap; Le., if (ep""" - tdmax), the charge transfer or conductivity gap is greater than zero. Each unit cell has a d-like hole with spin The d hole spins are coupled antiferromagnetically to each other, the coupling J being mediated by Cu-0 hybridization to (and of fourth order in t o ) . Adding more holes into the p derived bands makes the system metallic. The carriers are p like or oxygen holes. There is an antiferromagnetic superexchange coupling between a d hole spin and the hybridizing p hole spin (located essentially on the same unit cell as the d hole). This is of second order in to. This coupling leads to a singlet, so that at low p hole densities every added hole exists and moves about as a spinless singlet. In this limit, a one-band strongly correlated Hubbard model again describes the system; Le., one has a low density of "holes" (actually composite singlets), the remaining sites being occupied by spin objects.ss The Hamiltonian of the two-band model is given by
H= Ctddiu+diu + C~pP/u+P/u + UCnitniJ+ CC(tidiu+P/u + h.c.1 IP
1.u
I
1,u
Is
where the first two terms are d and p hole energies, the third term is the large correlation energy, U, and the last term is the overlap between a d state and a p hole Wannier state mode essentially constructed out of an appropriate combination of px, p,, orbitals surrounding the former. ( D ) Realistic Complications. In the simplest of the two-band models described above, other d- and p-like bands, as well as interactions other than u d d have been left out. It is possible that they are essential, say, for high-temperature superconductivity! We therefore briefly mention these realistic complications and outline reasons for their possible importance. There are, in addition to the states included, four d states (namely, dr2, d,, dyr, dxz)as well as the two p states (p, and the nonbonding px, p,, combination) per unit cell. In a lattice, due to d-p as well as p-p hybridization, these mix and broaden into bands, the former set lying above the lower Hubbard dX2-,,2hole band and the latter set above the bonding px, p,, band considered or overlapping with it. These higher lying states could be responsible for various polarization processes (e.g., excitonic processes); Le., virtual transitions to them could mediate interaction between holes. Specific possibilities have been discussed by a number of authors.47-49*s9*60 At least two Coulomb interaction derived terms, namely have been ignored. The former tends to Udpndn: and Uppnlon/-u promote, for a fixed total hole number, charge transfer or np, n d difference. This could be crucial for excitonic superconductivity mechanisms.6'*62 The Up, term is generally believed to be not crucial, since for low p hole density, the holes can always arrange themselves to minimize this repulsion and still remain quite mobile. Overlap matrix elements other than vd,pw = to are also expected to be sizable. They can change the shape and width of bands; there is no strictly nonbonding set of levels, so that bands made out of these have a nonzero width also. ( E ) Electronic Structure Calculations. We have described above two simple models. The basic energy and interaction parameters entering these models or more realistic ones are not determined. Since the nature of the system described (metal, insulator, antiferromagnetic, superconductor) does depend on the parameter values, it is obviously necessary to know them and their variation with doping and with change of chemical species. An experimental method that provides such information is electron spectroscopy (section 4). Here we briefly describe attempts at first principles computation of electronic structure. Electronic structure calculations, which are generally based on a one-electron approximation, are believed to be qualitatively (59) Johnson, K. H.; McHenry, M. E.; Couterman, C.; Collins, A,; Donovan, M. M.; OHandley, R. Kalonji, G. Physica C 1988, 153-155, 1165. (60) Weber, W. Z . Phys. B 1987, 70, 323. (61) Varma, C. M.; Schmitt-Rink, S.; Abraham, E. Solid State Commun. 1987, 62, 681. Abraham, E. Physica C 1988, 153-55, 1622. (62) Hirsch, J. E. Physica C 1988, 153-155, 549.
c.;
inadequate in these highly correlated systems. For example, in such calculations, the insulating state of LazCu04can be ascribed only to a spin density wave gap; this should disappear above the spin density wave transition temperature Tsow (or T N 220 K) and with it the insulating state. Since this does not happen, the system is not a one-electron-band insulator, but a strong correlation insulator. It is believed, however, that self-consistent one-electron theory is sufficiently accurate for energetics and also for electron densities (or potentials). From the former, quantities such as correlation and charge-transfer energies can be calculated as appropriate derivative^.^^ For example, Schluter, Hybersten, and Christenand udp to be lo, 6 , and 4 ev, respectively. sed3 estimate Udd, upp, The charge-transfer energy (eP - td) N 2 eV. From densities and potentials, band parameters (e.g., overlap integrals) can be found. The p hole bandwidth is about 4 eV.64 A complementary approach is to solve numerically the quantum problem for a small cluster of Cu and 0 atoms, without further approximations.6s This would provide one with an estimate of molecular parameters relevant to the solid, including even the Cooper pairing mechanism, since the small pair size inferred in various ways makes such estimates reasonably real is ti^.^^^^^ In between these two limits lie calculations of a single correlated Cu d level hybridizing with an oxygen p band. This kind of approach has been very useful for NiO and rare-earth solids (e.g., mixed-valent systems) especially for interpreting photoemission. An interesting example is the work of Eskes and Sawatzky66on the problem of two holes hybridizing with the oxygen p band. They find results similar to those of Zhang and RiceSSfor the two-band model. Sarma and Taraphder6' find, for cd > cp (inverted order with respect to that commonly believed), that a careful one-hole calculation leads to a split-off low-energy magnetic hole state and other behavior consistent with experiment.
-
6. Observations Models Must IExplain There is consideraMe divergence of emphasis among workers about what needs to be explained. Many believe that these materials are like other metals which become superconducting; only the transition temperature is higher so that we need to find a new, stronger, pairing mechanism. However, the proximity of these systems to an insulating magnetic state means that matters are much more interesting. There are a number of properties in the normal and superconducting phases, each of which would be considered curious, and many of which are common to oxides. Taken together, they are quite striking. We briefly describe some characteristics of cuprates in both the superconducting and normal phases. ( A ) Superconducting State. The basic experimental features have already been mentioned in section 3. Briefly, they are the following: (i) The very short coherence 1en th or the Cooper pair size (-20 A along the phase and about 3 perpendicular to the plane). (ii) Specific heat at temperatures well below T,. The electronic specific heat is not exponentially small, but changes as P,where a seems to vary from 1 to 2. This means either that the gap vanishes at some points on the Fermi surface or that there are other low-lying excitations.6s While most measurements show a linear specific heat, the coefficient varies widely, for the same composition; it is possible therefore that this is an extrinsic effect due to impurity phases. Further, no such term is seen in the Bi compounds. (iii) The nonexponential dependence of N M R relaxation rate l/T1 on temperature, again indicating the presence of low-lying excitation^.^^ (iv) Resistive and heat capacity be-
1
(63) Schliiter, M.; Hybertsen, M. S.; Christensen, N. E. Physica C 1988, 153-155, 1217. (64) Key, T. P.; Terakura, K.; Oguchi, T.; Yanse, A,; Ikeda, M. J. Phys. SOC.Jpn., in press. (65) Fujimori, A. Phys. Rev. B 1987, 35, 8814. (66) Eskes, H.; Sawatzky, G. A. Phys. Reu. Left. 1988, 61, 1415. (67) Sarma, D. D.; Taraphder, A,, to be published. (68) See, for example, the review of Kitazawa, K.; Takogi, H.; Kishio, K.; Hasegawa, T.; Uchida, S.; Tajima, S.; Tanaka, S.; Fueki, K. Physica C 1988, 153-155, 9.
4420
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989
havior near (and above) T,, suggesting that order parameter fluctuations are significant. (v) Many properties, e.g., the energy gap, the temperature dependence of Hc2, the size of the flux quantum, all suggesting Cooper pairing with a BCS-like superconducting state. ( B ) Normal State: ( i ) Magnetic Properties. The “parent” substances (undoped starting points) of cuprate superconductors are antiferromagnetic; neutron scattering reveals large in-plane magnetic correlation lengths =200 A above T N and very stiff spin-wave excitations. The latter are also probed by Raman scattering which shows a broad inelastic structure not simply reconciled with a conventional two-magnon spectrum as is possible in some other antiferromagnets. High-quality data70 on large single crystals of La,,&ro I C U Oshow ~ that the magnetic correlation lengths are still sizable and that there is a quasielastic peak incommensurate with the lattice by an amount expected from the Fermi surface for this doping. (ii) Optical Properties. There is a sharp low-frequency conductivity peak (Aw 5 0.05 eV) which is temperature dependent and has a low oscillator strength. Drude fits of this, measured plasma frequencies, and other data require substantial mass enhancement; ( m * / m ) = 10 is indicated. There is a broad peak at w N 0.05 eV in cr ( w ) with a long tail going over a 2-eV range, and another peak (presumably interband) at -4 eV.71 The low-energy (hw < 2 eV) conductivity is very non-Drude-like. (iii) Density of States. A large number of point contact experiments lead to a (tunneling) density of quasiparticle states which shows a gaplike structure at low energies ( h u g A = 4kBT,) but, surprisingly, a linear dependence on energy beyond this, extending in many cases over 1 eV or ~ 0 . It~ has 3 been ~ ~ argued that this is extrinsic, i.e., due to charging effects at the tip. A more interesting possibility is that it is intrinsic; the system is not like a normal metal with a nearly constant density of states near the Fermi energy. (io) Transport Properties. The most striking single feature is the resistivity, which for relatively clean samples is linear in temperature from T, to several hundred degrees kelvin, with a ~ linear behavior is observed in highly slope of order 1 C I R / K . ~This doped nonsuperconducting La2-,Sr,Cu04 for x 2 0.3, as also for low T , systems of this family. Thus it cannot be attributed to some medium-temperature effect involving phonon density of states; presumably a process other than electron-phonon scattering is the cause of resistivity. Large negative magnetoresistance and anomalous small thermopower whose sign changes with temperature have been reported. The Hall coefficient is positive, indicating transport by holes. In some systems, e.g., La2-,Sr,Cu04, the inferred hole density is nearly equal to x ; in others, e.g., YBa2Cu3O7,the Hall constant changes by a factor of 2 over a 200 K temperature range above Tc. (0) Non-Korringa Relaxation. N M R data (1 / T I ) of Cu indicate that the relaxation processes in the plane and chain are different; the former does not depend linearly on temperature as expected for Cu2+ exchange coupled to hole carriers but seem independent of T.69 It is thus clear that members of this family of substances are neither standard antiferromagnetic insulators nor standard metals.
7. Theoretical Models In a large class of theories, the main attempt is to find a mechanism for Cooper pairing, the assumption being that otherwise, these systems are like ordinary metals. Since, as we have seen above, this is not quite true, a deeper understanding of the properties of strongly correlated metals is needed. The most (69) Physicn C 1988, 153-155, 75-97 (panel discussion on resonance effects). (70) Shirane, G.; Endoh, Y.; Birgeneau, R. J.; Kastner, M. A,; Hidaka, Y.; Oda, M.; Suzuki, M.; Murakami, T. Phys. Reu. L e t f . 1987, 59, 1613. (71) Thomas, G.A.; Orenstein, J.; Rapkine, D. H.; Cappizzi, M.; Millis, A . J.; Bhatt, R. N.; Schneemeyer, L. F.; Waszezak, J. V . Phys. Rec. Lett. 1988, 61, 1317. (72) Gurvitch, M.: Fiory, A . T. Phys. Rec. Lett. 1987, 5 9 , 1337.
Ramakrishnan and Rao detailed and original attempt of this kind is the resonating valence bond (RVB) theory due to Anderson and c o - w ~ r k e r s . ~ In ~ *this ~’ section, we outline various strong pairing proposals and the RVB theory and conclude with a qualitative discussion of alternative strong correlation approaches. ( A ) Phonon Mechanisms. A few authors argue for a phonon mechanism, which in the present case requires both strong elec3-10 is needed) and rather stiff tron-phonon coupling (X phonons. There is no direct evidence for either. On the contrary, from the saturation of resistivity with temperature in metals which 0.5-1.0, and nonsaturation of resistivity here, it can have X be argued that X 5 0.3.72Also the dependence of T , on oxygen isotopic mass is very weak,39much weaker than what is expected if the oxygen atom vibrations cause pair formation. A different scenario involving the lattice centrally is the formation of bipolarons and their Bose c ~ n d e n s a t i o n . ~This ~ is possible here since oxygen ions are highly polarizable. A problem with small bipolaron models is their exponentially large effective mass which correspondingly reduces the Bose condensation temperature or T,. There is at present no credible theory for a dense gas of intermediate-sized bipolarons and no experimental evidence of a local fluctuating lattice distortion related to them. ( B ) Electronic Mechanisms. The relatively large T , suggests that electronic excitations with an energy scale of the order 1 eV are involved in mediating the attraction between carriers. Several possibilities discussed over the years have been proposed again in the present context. (i) Spin Fluctuations. There is strong antiferromagnetic coupling between copper spins in the cuprates, so that as carriers (holes in the p, d, or hybridized bands) move, spins are disturbed. This could lead to pairing of carriers oia exchange of antiferromagnetic spin fluctuations (with an energy scale -0.1-0.3 eV). Similar scenarios have been discussed in the context of heavy fermion supercond~ctivity.~~ They lead to a d-like pair wave function and relatively weak coupling superconductivity. There is no strong evidence for or against d wave pairing in the systems. (ii) Spin Bipolaron or Spin Bag. In two dimensions, with nearest-neighbor hopping on a square lattice, the normal metallic state is unstable against spin-spin density wave formation for even an infinitesimal U. The spin density wave amplitude A ( x ) is different around a carrier (spin polaron). It turns out that, on adding another carrier, it is energetically favorable for the two to be together and deform A ( x ) locally. This is the “spin bag”.7s This does not require really a long-range SDW, but only shortrange spin correlations. Again, since the SDW energy scale is of the order 0.2 eV, T,’s of the observed size are possible. (iii) Excitonic Mechanisms. In the oxide superconductors, several possibilities for attraction mediated by electronic polarization have been proposed. Cu-0 charge-transfer excitations induced by the p-d Coulomb term Un,nd, nonbonding p excited states, and higher lying d orbitals are all possible candidates. A general difficulty with these mechanisms is that the effective electron interaction is required to be attractive over sizable length and time scales, while we know it to be repulsive at long distances and times. Since there is no separation of energy scales as for phonons, these two requirements are difficult to reconcile. (C)Resonating Valence Bond Model. A n d e r s ~ n ~has ~ . ~argued ’ that the spin of the d9 configuration and the two dimensionality of the Cu-0 layers make for a novel non-Niel magnetic ground state (the RVB state) with characteristic elementary excitations. This original proposal has generated a large amount of theoretical work. Several unusual normal state properties can be rationalized using RVB-based ideas. Despite all this, the problem is difficult enough so that the theory cannot be considered as firmly established. We outline some of the basic ideas and predictions. In three dimensions, the exchange interaction J leads to NCel order,
-
-
(73) Chakravarty, B. K. J . Phys. Letr. 1979, 40, L-99. Alexandrov. A,; Ramringer, J . Phys. Rev. B 1981, 32, 1796. (74) Miyake, K.: Schmitt-Rink, S.; Varma, C. M. Phys. Rec. B 1986, 34, 6554. (75) Schrieffer, J. R.; Wen, X. G.; Zhang, S . C. Phys. Rec. Lett. 1988, 60, 944.
The Journal of Physical Chemistry, Vol. 93, No. 1 1 , 1989 4421
Feature Article for all spins including S = Le., for the half-filled strongly correlated Hubbard band. In one dimension there is no Ntel order, but the exactly known ground-state wave function is a superposition of singlet pairs (valence bonds) formed out of a spin and its first, third, fifth, ... neighbors. The system Yresonates”(in the Pauling or Kekult sense) between an infinite number of mutually nonorthogonal states or configurations, each consisting of a particular set of singlet pairs. This state is stabilized both by resonance (or overlap) and because of singlet formation. Anderson argues that in two dimensions, for a spin system the ground state is an RVB and not a Ntel state. Numerical calculations are inconclusive, since energy differences are small. The question may be irrelevant for metallic systems which have a finite concentration of holes (with respect to one d state per site). In this case, a Ntel state may be dynamically “annealed” out by the motion of holes, and the resulting quantum spin liquid may indeed be approximated by an RVB-like state. However, no detailed description of such a state is available. The elementary excitations with respect to this ground state are either a wrong spin (an isolated unpaired spin with its companion far away) or a site without any spin when an electron is removed from the system. The former is called a spinon, and the latter, a holon. The questions of their statistics, the extent to which they can be regarded as defined and noninteracting, and their energy dispersion are still not settled. One intuitively reasonable scenario is that spinons are charge-neutral fermions with a nonzero density of states and that holons are charged bosons. They interact with each other, not particularly weakly. Presumably both excitations have a bandwidth of J . Calculations based on the above simple picture of spinons and holons lead to a tunneling density of states linear in energy, a electrical resistivity linear in temperature, and a linear (in T ) specific heat in the superconducting regime, though there is some disagreement as to the results.76 What is the nature of superconductivity in the RVB model? Baskaran, Zhou, and Anderson7’ argue that singlet pairs present in the RVB ground state are “performed” Cooper pairs with binding energy of J and size of the order of interatomic spacing. On introducing holes, this system can transport charge and is a superconductor. Another possible scenario is hole boson condensation. Among other matters this is not consistent (at a simple level) with flux quantization in units of (2e)-’. Because of this, hole pairing via interlayer hopping t , has been suggested as a possible mechanism for superconducti~ity.~~ A pair of holons moving in a plane exchange a spinon so that a pair of holonspinon composites, each of which is the real hole, tunnel coherently to the next layer. Using the c-axis resistivity to estimate t l , reasonable estimates of Tc are obtained. ( D ) Correlated Electron Models. The realization that d electrons in the cuprates are strongly correlated has led to renewed interest in the basic physics of such systems, e.g., the one- and two-band models discussed above. Some of the relevant questions are the nature of the ground state in the electronically commensurate, insulating case (Le., one spin 1/2 d hole per planar Cu site) and the nature of the low extra hole density system. Considerable work on the one d hole per site system (half-filled strongly correlated Hubbard band) has shown that, in two dimensions, most probably the ground state is a Ntel antiferromagnet with reduced sublattice magnetizati~n.’~*~~ For example, in the work of DasguptaSo~*’ a one-parameter variational wave function is chosen such that when the parameter cos 0 has one limiting value (1 or 0), the ground state is the Niel state, while when cos 0 = 1/2’12, the wave function describes a nearestneighbor RVB state, i.e., one with equal-phase nearest-neighbor
-
(76) Kallin, C.; Berlinksy, A . J. Phys. Reo. Lett. 1988, 60, 2556. Anderson, P. W.; Zou, Z. Phys. Rev. Lett. 1988, 60, 2557. (77) Baskaran, G.; Zou, A.; Anderson, P. W . Solid Stare Commun. 1987, 63, 973. (78) Wheatley, J . M.; Hsu,T. C.; Anderson, P. W. Phys. Rev. B 1988,37, 5897. (79) Roger, J. D.; Young, A . P. Phys. Rev. B 1988, 37, 5978. (80) Dasgupta, C. Phys. Rev. E 1989, 39, 3786. (81) Ramakrishnan, T.V. Physica C 1988, 153-155, 555.
singlets superposed. The ground-state energy is minimized numerically by using an exact enumeration of configurations for a 20-spin system. The minimum occurs for cos 0 = 0.83, corresponding to a reduced sublattice magnetization (MJM?) = 0.70. However, non-nearest-neighbor RVB states have lower energy, and it is clear from simulations on large systems that the Ntel and RVB state energies are very close in energy. The question of the exact ground state for the spin system is of academic interest for superconductors, since the latter have a nonzero hole density 6. For a one-band Hubbard model, it has been argueds1that a moving hole wipes out spin memory so that the spin system equilibrates with an effective temperature T N 621, where 6 is the hole density. Thus any antiferromagnetic (AF) order is rapidly suppressed, and a quantum spin liquid with short-range A F correlations results. This paramagnetic Fermi liquid is in principle quite different from the RVB with its spinon and holon excitations. There has been considerable work on the mobility of holes in the AF background.81*82The consensus is that because of coupling to spin fluctuations, the hole mass is considerably enhanced; i.e., ( m * / m ) ( t / J ) . The hole is a spin polaron. For superconductivity, the question of interest is whether the carriers attract each other. There are some computer simulations suggesting that they do. Here, the difference between a one-band and a two-band model is apparent. In the latter, as mentioned earlier, there is a range of parameters for which the extra p hole forms a local singlet with a d hole; this composite object is the Hubbard “hole”. If two such objects are placed next to each other, they will attract, via processes in which intermediate states (such as two p holes being in the same unit cell) are involved.81 This kind of intrinsic attraction does not make sense in a literal one-band Hubbard model. Whether this attraction is crucial or not remains to be seen. The regime of greatest interest, namely, one where 6 is sizable (-0.2-0.4), is the least studied. A simple Fermi liquid model has been discussed by Kotliar, Lee, and Read,83who find that the system is a d-wave superconductor due to exchange of spin fluctuations, much like in heavy fermion models.74
8. Future Directions Despite the intense activity generated by the discovery of superconductivity in cuprates, understanding of both the compounds and their superconductivity remains elusive. There are several reasons for this. First, it is not easy to prepare reproducible samples that are well characterized with respect to stoichiometry, oxygen content, and microstructure. This has caused difficulties in obtaining reliable data on good samples. High-quality single-crystal measurements of many properties (e.g., resistivity, Cu NMR, Hc2)have gradually become available in recent months. The picture emerging is of a class of materials that are more unusual in their normal state than in their superconducting state! Attention is therefore shifting toward the basic electronic nature of oxides. Here, surprisingly few analytical results are available. For example, the phase diagram of the Hubbard model in the (U/zt),6 plane (at T = 0) is not precisely known, in three or in two dimensions. At 6 = 0 or half-filling, there is a metal-insulator transition (in three dimensions at least) as U >, zt. This is the Mott transition, beyond which there is a gap in the charge excitation spectrum and the low-lying states involve only spin fluctuations. There is no detailed theory yet of this transition and no picture of the low-lying excitations in this regime, as well as of what doping (6), coupling to the lattice, and temperature do. In two dimensions, there is a spin density wave (SDW) transition for the smallest value of U. What is the nature of the competition between a SDW and the Mott transition? How is it affected by correlation, doping and temperature? In the (6,u) plane, for very large U, and very small 6, a giant ferromagnetic (82) Kane, C. L.; Lee, P. A,; Read, N. Phys. Reu. B, in press. Schraimann, B. I.; Siggia, E. D. Phys. Reu. Letf. 1988, 60, 740. Schmitt-Rink, S.;Varma, C. M.; Ruckenstein, A. E. Phys. Rev. Lett. 1988, 60, 2793. (83) Kotliar, G . ; Lee, P. A.; Read, N. Physica C 1988, 153-155, 538.
J . Phys. Chem. 1989, 93, 4422-4429
4422
polaron ground state has been argued for by Nagaoka. Does it cross over to an antiferromagnet for smaller U and larger 6, or to a spin liquid, or both? Is the spin liquid similar in nature to that described by the RVB picture, or is it just a Fermi liquid with unusual parameters, or something else again? In which part of the parameter space is superconductivity favored? These are some of the questions needing answers. Regarding two-band models, even less is known. Here, the basic parameters are (cp - e,,), 6 , and the d-p interaction U,, ( U is assumed large always). In the context of the mixed-valent semiconductor SmS, as (eP - td) decreases, a first-order transition to a low excitation gap, mixed-valent semiconductor occurs. Does this continue to be true for the spin cuprates? What is the effect of doping on the phenomena in either case? Is Udp crucial for superconductivity? For negative charge-transfer energy ep t d , the Anderson lattice model is used for magnetic moments in metals and for heavy fermions. Is there a smooth crossover in properties as a function of charge transfer? It is quite likely that exploring these questions seriously will lead to a better understanding of the rich variety of electronic phenomena in transition-metal oxides. It may also point the way to new classes of systems which, because of optimal pairing, are really high-temperature superconductors. Given the kind of the experimental and theoretical attention the oxides are getting, this does not seem an unrealistic prospect. Many strategies for the synthesis of new families of high T, superconductors seem possible. In the oxide family alone, there
are many possibilities. One possibility is to look for oxides with low 02--cation charge-transfer energy (e.g., layered nickelates), favoring formation of oxygen holes. One need not restrict oneself to two-dimension,al oxides alone. The discovery of Bal-,KXBiO3 ( T , 30 K) suggests the possibility of other three-dimensional oxides exhibiting high T,. Ba1-,K,BiO3 itself is interesting, showing a large isotope effect,E4but no static magnetic orders5 (unlike in the two-dimensional cuprates).
-
Note Added in Proof. There have been a few developments of interest in the past few months. Among them, mention must be made of the discovery (i) of a series of Pb cupratesS6of the type Pb2Cal,Ln~r2Cu308,containing CUI+in high proportion^,^^ (ii) of the electron superconductorE8Nd2,Ce,Cu04, and (iii) of a series of cupratesg9of the type T1Cal,Ln,Sr2Cu207+6 showing electron or hole superconductivity depending on x. (84) Hinks, D. G.; Richards, D. R.; Dabrowski, B.; Marx, D. T.; Mitchell, A . W. Nature 1988, 335, 419. (85) Uemura, Y. J.; Sternlieb, B. J.; Cox, D. E.; Brewer, J. H.; Kadono,
R.; Kempton, J. R.; Kiefl, R. F.; Kreitzman, S. R.; Luke, G.M.; Mulhern, P.; Riseman, T.;Williams. D. L.: Kossler. W. J.: Yu.X. H.: Stronach. C. E.; Subramanian, M. A.; Gopalakrishnan, J.; Sleight, A. W. Nature 1988, 335, 151 (86) (87) (88) (89)
Cava, R. J.; et al. Nature 1988, 336, 21 1. Rao, C. N . R.; et al. Phys. Reu., in press. Tokura, Y . ;Takagi, H.; Uchida, S. Nature 1989, 337, 345. Rao, C. N. R.; Ganguli, A. K.; Vijayaraghavan, R. Phys. Reo. B, in
press.
ARTICLES Bond Orders from ab Initio Calculations and a Test of the Principle of Bond Order Conservation G . Lendvay Central Research Institute for Chemistry, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary (Received: April 27, 1988: In Final Form: October 31, 1988)
Bond orders and valence indexes of atoms were calculated from ab initio wave functions using the definition suggested by Mayer. Changes of these quantities were investigated under conditions occurring in chemical reactions. Calculations were performed to reveal the dependence of bond orders on bond length. The results are compared with Pauling’s bond order-bond length relation. Bond orders were also calculated in triatomic metathesis reactions. Bond orders are able to describe the similarity of the transition state of reactants or products. Along the minimum-energy path the principle of conservation of bond order is valid to a good approximation. The obtained correlations are applied to a bond energy-bond order type model to estimate trends in reaction series.
I. Introduction Methods of computational quantum chemistry are able to produce physical properties of molecules with increasing accuracy and reliability. It is not possible however to calculate simple qualitative properties like valence of atoms or the bond order between two atoms in a molecule. The quantity calculable from a b initio wave functions which is often related to the strength of bonds in molecules is Mulliken’s overlap population CpEA~vEBPauSpu. but i t is not close to the integer bond order values expected. The recent formulation of valence indexes of atoms and bond orders between atoms in molecules suggested by M a ~ e r ’ can - ~ bridge the gap between the “overdetailed” infor0022-3654/89/2093-4422$01.50/0
mation supplied in the form of large matrices by theoretical chemistry and the need of chemists for qualitative information. Mayer’s method has been successfully applied to a large number of stable molecule^,^^^-^ and a preliminary study of bond orders ( I ) Mayer, I. Chem. Phys. Lett. 1983, 97, 210; 1985, 117, 396 (addendum). (2) Mayer, 1. Int. J . Quantum Chem. 1986, 29, 73. (3) Mayer, I. Int. J . Quantum Chem. 1986, 29, 477. (4) Baker, J . Theor. Chim. Acta 1985, 68, 221. (5) yillar, H. 0.; Dupuis, M. Chem. Phys. Lett. 1987, 142, 59. (6) Angyin, J. G.;Pokier, R. A. Manuscript in preparation. (7) Mayer, I. In Modelling of Structure and Properties of Molecules; Maksif, Z. B., Ed.; Ellis Harwood: Chichester, 1987; p 145.
‘01989 American Chemical Society
