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Valence bond order and superconductivity
Journal of Molecular Structure (Theo&em), 231 (1991) 335-355 Elsevier Science Publishers B.V., Amsterdam
335
VALENCE BOND ORDER AND SUPERCONDUCTIVITY
ERIK W. KVAM Superconductor
Ventures, Inc., 232 East 82nd Street, New York, NY 10028 (U.S.A.)
(Received 1 October 1990)
ABSTRACT The three-dimensional interstitially localized valence bond structure leads to a model of real space pairing of superconducting electrons in a three-dimensional lattice. The valence bond ground states are coupled to the superconducting ground states through a topologically synchronous transformation. Superconductivity depends critically on the three-dimensional isotropy of the valence bond structure within the three-dimensional unit cell.
INTRODUCTION
Since the discovery of superconductivity by Kamerlingh Onnes in 1911, and the discovery of high T, superconductivity in the La-Ba-Cu-0 system by Bednorz and Miiller [ 1] in 1986, no superconducting mechanism has been elucidated which is common to low T, and high T, superconductors. The present model describes a three-dimensional isotropically ordered valence bond structure which may be common to all superconducting materials. The model focuses on YBa&u30, (Y-Ba-Cu-0) and (Nd,Ce)&uO, (NdCe-Cu-0) as representative of hole-type and electron-type layered cuprate superconductors, on (Ba,K)BiO, (Ba-K-Bi-0) as representative of the cubic bismuthate superconductors, and on NbaGe (Nb-Ge) as representative of the Al5 superconductors. The model could be extended to the metallic element superconductors in which T, has been shown to depend on optimum interatomic distances in hole-type superconductors [ 21. The likelihood of a common s-wave pairing superconducting mechanism in both high T, (higher than 40 K) and low T, (lower than 40 K) superconductors has been shown by recent experiments in which the electron pairs maintained phase coherence across an Nb-Y-Ba-Cu-0 junction [ 31. Heretofore, the actual structures giving rise to superconductivity have not been adequately described or understood. The lack of such understanding has proved a serious obstacle to achieving room temperature superconductivity because the search for new superconductors has been a trial-and-error process, guided primarily by variation of known superconducting materials.
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Elsevier Science Publishers B.V.
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The topological model offers a simple method for predicting whether a material will superconduct, based on the valence bond structure. The interstitial valence bond structure of a superconducting material can be specified by quantum chemical calculations using “generalized valence bond” theory to describe bonding structures and energies in metallic atom microclusters [ 451. The interstitial valence bond structure can explain why the lengthening of the Cu (2)0 (1) apical bond is identified with the loss of superconductivity in Y-Ba-Cu0. VALENCE BOND ORDER OF SUPERCONDUCTORS
The working hypothesis of the present model is that superconducting materials possess an isotropically ordered three-dimensional interstitial valence bond structure. This structure has a space-filling tetrahedral-octahedral geometry [ 61. The three-dimensional isotropy [ 71 of the structure within the superconducting layers is crucial for superconductivity, and the loss of such isotropy is identified with the loss of superconductivity. Generalized valence bond (GVB) theory systematically accounts for electron correlation effects by variationally optimizing each valence electron and hole orbital wavefunction during the formation of the bond to minimize energy and to obtain an accurate structural description of the molecule [ 51. Conceptually, this procedure treats the two-center one-electron or one-hole valence bond as the wavefunction of singly occupied orbitals which can compress and tense to describe a dynamically stable energy equilibrium, i.e. a valence bond, between two or more atoms. Optimum bonding involves singly occupied valence electron orbitals localized through interstitial regions between the ionic cores [ 81. Interstitial valence electron localization, according to GVB principles, minimizes Pauli repulsion and is consistent with the Lewis-Langmuir octet rule by which each atom shares valence electron orbital pairs to achieve a closed-shell configuration [9]. Localized valence electron and hole orbitals describe a “bent” bond, as distinguished from a and o bonds in cases of double, triple and conjugated bonds [lo]. The GVB theory is unlike mean-field theory which describes a variety of ad hoc electron orbital hybridizations, depending on the chemical bonding environment of such orbitals. The application of empirical rules [4] derived from GVB theory yields the following description of interstitially localized valence bonds in the copper oxide and Al5 superconductors. GVB hybridized orbitals can share a valence electron pair in a covalent bond (two singly occupied orbitals) or dative bond (one doubly occupied orbital and one unoccupied orbital). The ionic cores and valence electrons in a copper oxide superconductor tend to occupy vertices in a tetrahedral-octahedral structure (the “L net” ) in which
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the valence electron and hole orbitals describe the ‘bent” covalent bonds which connect such vertices in the superconducting state. The vertices in a second tetrahedral-octahedral structure (the “K net”), maximally interstitial to the L net, describe the superconducting orbital structure. In the non-superconducting state, the Cu3d9 valence electron localizes on the K net with a resulting loss of K-net and L-net isotropy and superconductivity. The K net and L net are interpenetrating face-centered cubic (f.c.c.) structures [ 281. Such structures are commonly recognized as highest density lowest energy structures in three dimensions in cluster studies [11,12], which may explain why superconductivity is most commonly found at low temperatures (below 10 K) in f.c.c. metallic elements. Vertices and lines are used in Fig. 1 and in the other figures to simplify the description of the overall symmetry of the valence electrons and valence bonds. Such vertices and lines symbolize the spatial regions in which the valence electrons and orbitals are localized, and do not imply that such electrons are orbitals are actually located at fixed points or along fixed lines in real space. (a)
(b)
;i
F
(cl
/@
& Ba
\
?
cuti
1
O(4)
0
o
Localized
Valence
Delocalized (Superconducting)
Bond
State
Antibonding
State
Fig. 1. YBa&u,O+ (a) interstitially localized covalent bonds, (b) complete interstitially localized valence bond structure (“L net”); (c ) delocalized (superconducting) orbital structure (“K net”).
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The upper and lower portions of Fig. 1(a) show the alternate covalent bent bonds formed by the Cu (3 ) 3dg orbitals and the 0 (2 ) 2p and 0 (3 ) 2p orbitals in the Cu(2)-0 (2)-O(3) layer the “Cu(2) layer”) of Y-Ba-Cu-0, and by the 0 (2 ) 2p and 0 (3 ) 2p orbitals and the Ba5d orbitals. The valence bond structure of the Cu (2 ) layer of YBa-Cu-0 could apply equally to the Cu-0 layers of the (La,Sr ) -, (Bi,Sr ) - and (Tl,Ba) -based copper oxide superconductors. The central portion of Fig. 1 (a) shows the alternate covalent bonds formed by the Cu(1)3dgorbitalsandtheO(4)2porbitalsoftheCu(1)-O(4)-O(5) layer (the “Cu ( 1) layer” ) . The covalent bonds in Fig. 1 (a) describe part of the three-dimensional tetrahedral-octahedral structural termed the L net. The complete L net, including non-bent bonds within the Cu-0 layers and vectors correlating interstitial valence electrons, is shown in Fig. 1 (b ). Figure 1 (c ) shows the interpenetrating K-net structure for Y-Ba-Cu-0. (a)
(b)
(cl
Nd
0
0
Fig. 2. (Nd,Ce)&uO,:
j_ocalited
Valence
Bond state
Delocalized Antibondim (superconducting) state
(a) interstitially
localized covalent bonds, (b) complete interstitially
lo-
calizedvalence bond structure (“L net”); (c) delocalized (superconducting) orbital structure (“K net” 1.
Figure 2 (a) shows the alternate covalent bonds formed by the Cu3dg orbitals and the 02p orbitals, and by the NdSd/Ce3d orbitals and 02p orbitals in the T’ phase of Nd-Ce-Cu-0. Figure 2(b) shows the complete L net for the T’ phase. Figure 2 (c) shows the K-net structure for this compound Figure 3 (a) shows the complete set of alternate non-bent bonds formed by the Bi6p orbitals, (Ba,K) 5d orbitals and the 02p orbitals in the cubic Ba-KBi-0 superconductor. Figure 3 (b) shows the K-net structure for this compound. Figure 4 (a) shows the alternate covalent bonds formed by the Nb4p orbitals in Nb-Ge, an Al5 intermetallic superconductor. Figure 4(b) shows the complete L net for Nb-Ge. Figure 4 (c) shows the K-net structure for this compound. Resonating-GVB calculations by Guo et al. [ 131of a five-coordinate CI_I~O~~ half-octahedral cluster and a four-coordinate Cu301,, planar cluster in Y-BaCu-0, and of a six-coordinate CuzO,, octahedral cluster in Laz_,Sr,CuO, (LaSr-Cu-0), yield optimum localized wavefunctions having nine electrons in d orbitals on each copper. Oxidation of the Cu2+ ( dg) did not lead to Cu3+ (d8) but to a singly occupied 02~71 orbital spin-coupled to the singly occupied copper d orbitals and localized interstitially to two Cu2+ sites. (a)
(b)
0
Delocalized Antibonding (Superconducting) State
Fig. 3. (Ba,K)BiO,: (a) interstitially localized valence bond structure (“L net”); (b) delocalized (superconducting) orbital structure (“K net”).
340 (a)
(b)
(c)
0
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o
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Delocalized
0
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Valence
0
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O
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Antibonding
(Superconducting)
State
Fig. 4. Nb3Ge: (a) interstitially localized covalent bonds; (b) complete interstitially localized valence bond structure (“L net”); (c) delocalized (superconducting) orbital structure (“K net”).
In the ionized CuZOl, cluster, Guo et al. find that the planar 02pa orbital is oxidized when x = 0, but that the apical 02~~ orbital is preferentially oxidized when x=0.3, which suggests that the 02pa orbital is localized interstitially to the planar and apical oxygen atoms in the superconducting state. In the ionized planar Cu,O,, cluster, 02p bonding orbitals and singly occupied Cu3dg orbitals are shown oriented toward regions corresponding to L-net vertices interstitial to the Cu-0 and Ba-0 layers, and resonant singly and doubly occupied 02p non-bonding and/or antibonding orbitals are shown oriented toward K-net positions with the suggestion that holes in such orbitals are high mobility charge carriers. MSXcr calculations for CuOI planar clusters by Sarma and co-workers suggest that the non-spin-paired Cu3dg electron bonds interstitially with an 03s orbital in a 4A1, non-magnetic highest occupied molecular orbital (HOMO) ground state, just below an antiferromagnetic and antibonding lowest unoccupied molecular orbital (LUMO) of 3B,, [ 141.They suggest that the stability of the 4A,, orbital depends critically on the overlap (dimerization) of 03s orbitals and the symmetry of the O-O distances, and that a slight increase in such distances pushes the cluster into the antiferromagnetic 3&, state in which the unpaired 3dg electron is localized to the Cu site and not interstitially. Like
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Guo et al., they suggest that charge fluctuation is not due to Cu3d electrons but to 02p electrons which resonate between 02- and O’- species. X-ray absorption spectra and X-ray photoemission spectra of La-Sr-Cu-0 and Y-Ba-Cu-0 consistently show Cu3d/02p hybridization of localized Cu3d/ 02p bonding orbitals and Cu3d/02p covalent bonding in the superconducting state [ 15-181. These spectra consistently show that the ground state consists of either 2p3dg ( Cu2+) or a mixture of 2p3dg and 2p3d” (Cu’+ ) configurations, with little or no 2p3ds ( Cu3+ ). Similar results are reported for Tl,CaBa,Cu,O, (Tl-Ca-Ba-Cu-0) and Bi,CaSr,Cu,O, (Bi-Ca-Sr-Cu-0) [ 161. Bianconi’s results [ 171 suggest that in the hybridized 2p3dg and ligand 2p3d1’ configuration, the hole fluctuates between the Cu site and the 0 site in the pda covalent bond (consistent with Sleight’s view [ 19 ] ), and that the Cu3d metal orbital is localized while the hybridized 02p ligand orbital is delocalized, consistent with Hall effect findings [ 18,201 that the charge carriers are holes in Y-Ba-Cu-0. Chakraverty et al. [ 161 propose that bipolaronic (Cu”’ ) or peroxitonic (Cul+ ) dimerization of nearest-neighbor 01- yields Cu3dg and Cu3dl” interstitially localized covalent bonds, and real space pairing of holes, above the antibonding oxygen level, which are the charge carriers. Van Bruggen [21] proposed a similar model in which the monovalent Cu3d’O ground state induces low density of states 02p valence band holes as charge carriers. Chakraverty’s 01s core level spectroscopy results point to dimerization of 01- holes. The results of Sarma et al. [ 151 provide evidence of oxygen dimerization below T,, lending support to the peroxitonic model. The interstitial valence bond model is consistent with the covalency found in both low T, and high T, superconductors [ 221. The covalent bonds in YBa-Cu-0 are strongly localized, with evidence of delocalization of the bonding orbitals above T, based on a jump in the heat capacity at T, [ 231. Interstitial valence bonds in copper oxide superconductors are consistent with suggestions that the Cu( 1) layer in Y-Ba-Cu-0 acts as a reservoir of electronic charge, and that the mechanism for superconductivity in the layered copper oxide superconductors employs a non-homogeneous charge distribution [ 24-261. In the Y-Ba-Cu-0 superconductor, the average valence on the orbitals interstitial to the Cu (2) and Ba-0 layers is roughly + 0.25 [ 7,241. In Nd-CeCu-0, the average charge on the orbitals interstitial to the Cu-0 and (Nd,Ce) layer is roughly -0.13. The interstitial valence bond model describes the holedoped and electron-doped valence electron structures, which have been shown by Hall measurements to correspond to superconducting states in which the charge carriers are holes [ 201 and electrons [ 271 respectively. Messmer and Murphy [28] propose that the GVB theory can describe a superconducting ground state which is a coherent superposition of bonding structures (degenerate states) which exhibit synchronous spatial alternation of valence bonds (a form of “resonance”). Messmer and Murphy suggest that
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highly correlated valence electron orbitals localized at interstitial Wigner lattice sites can completely describe a superconducting wavefunction.
Cu3dg LOCALIZATION IN COPPER OXIDE SUPERCONDUCTORS
Localization of the Cu3dg valence electron may play a key role in determining the Cu (2)-O (1) distance and the overall isotropicity of the L-net and the K-net structures. In Y-Ba-Cu-0 and the other copper oxide superconductors, each Cu (1) and Cu (2) atom has one uncoupled 3dg valence electron [ 171. In the superconducting state of Y-Ba-Cu-0, the Cu (1) 3dg orbital localizes through an L-net vertex adjacent to the Cu (1) layer, as shown in Fig. 5 (a). (b)
(a)
0 @O
0
Localized
Valence
0
0
@O
Bond State
Fig. 5. (a) Superconducting YBa&us07: alternate Cu(2)-0 (1)-O (2)-O(3)-Ba covalent bent bonds comprising 3dg2p bonding orbitals interstitially localized through L-net vertices, and alternate Cu( 1)-O (1)-O (4)-Ba-Cu( 1) bent covalent bonds comprising 3dg2p and 3ds2p bonding orbitals interstitially localized through L-net vertices and the vacant O(5) site. (b) Normal YBa,Cus07: Cu(2)-O( 1) non-bent (ionic) valence bonds comprised 3ds2p bonding orbit&, interstitially localized through K-net vertices.
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The Cu (2) 3dg orbital localizes through an L-net vertex adjacent to the Cu (2) layer, as shown in Fig. 5 (a). In the superconducting state shown in Fig. 5 (a), the Cu ( 1) 3dg orbital forms a bent covalent bond with an 0 (1) 2p orbital, 0 (4) 2p orbital or Ba5d orbital. Alternatively, two such Cu ( 1) 3dg orbitals can spin-couple through a vacant 0 (5) site to describe a strongly correlated covalently bound Cu3d’ ( Cu3+ ) state [ 171, equivalent to a singlet-coupled hole pair localized at the 0 (5) site [ 26,291. (Tranquada et al. [ 261 suggest that the number of holes is maximized because the Cu (1)-O (4) chain acts as an electron reservoir. Tokura et al. [ 291 conclude that holes associated with the average charge in the Cu( 1)-O(4) plane are localized and provide an insulating reservoir of charge.) De Jongh [ 301 suggests that Cu1+/Cu2+ “ dimerization” in the Cu ( 1) chain is plausible. A hole pair localized through the O(5) site may screen the interaction between the localized Cu( 1)3dg valence electrons and maintain K-net and L-net isotropy within the Cu (1) layer. Such a hole pair repels oxygen from the 0 (5) site to create the observed Cu( 1)-O (4) chain structure [31]. Hole pair delocalization from the 0 (5) site at temperatures above about 55-60 K may produce the superconducting phase transition reported at this temperature [ 251. In the superconducting state, the Cu(2)3dg orbital localizes through an Lnet vertex, as shown in Fig. 5 (a). In the normal non-superconducting state, the Cu(2)3dg orbital localizes through a K-net vertex, as shown in Fig. 5(b). Messmer and Murphy [ 281 suggest that single valence electron localization on an f.c.c. lattice (a K net) interpenetrating a second f.c.c. lattice (an L net) of ion cores prevents superconductivity in the alkali and noble metals (Cu, Ag, Au). In the superconducting state shown in Fig. 5 (a), the Cu ( 2)3dg orbital forms a covalent bent bond with an 0 (2) 2p orbital, 0 (3) 2p orbital, or Ba5d orbital [ 321. (Using Rietveld refinement of structural models from pulsed neutron powder diffraction data, Kwei et al. [ 321 find no changes in Cu (2)-O (1) lengths at T,, but do find large-amplitude motions involving the Ba2+ ions and the CuO, planes. ) In the normal non-superconducting state of Y-Ba-Cu-0 shown in Fig. 5 (b) , the Cu (2) 3dg orbital forms a weak non-bent (ionic) bond with the 0 ( 1) 2p orbital. The Cu (2 ) -0 ( 1) bond is reported [ 33 ] to be significantly weaker than the other Cu-0 bonds in Y-Ba-Cu-0. The weakness and ionicity of the Cu (2 )-0 ( 1) bond is shown by its extended (Jahn-Teller effect ) length of about 2.3 A in orthorhombic Y-Ba-Cu-0 (about 2.5 A in tetragonal Y-BaCu-0), compparedwith the covalent Cu (2)-O (2) and Cu (2)-O (3) bond lengths of about 1.9 A [34]. Oudet hypothesizes that loss of superconductivity at T, is caused by thermally induced disorder of the covalent bonds involving the Cu3dg valence electrons [ 351, which is consistent with the model of valence bond order hypothesized herein.
344 PROPERTIES OF THE VALENCE BOND STRUCTURE
Localization of the Cu(2)3dg single valence electron at a K-net vertex instead of at an L-net vertex creates short weak non-bent ionic Cu (2)-O (1) bonds between atoms at longer interatomic distances, instead of long strong bent covalent Cu(2)-0 (1) bonds between atoms at shorter interatomic distances. The ionic Cu3dg valence bonds create K-net and L-net anisotropy within the Cu-0 layers, reflective of a mixed ionic/covalent bond structure in which the L-net ionic cores are not equidistant from one another as they are in an isotropic covalent bond structure. The length of the Cu (2)-O (1) covalent/ionic bond, as shown by the interatomic distances reported for Y-Ba-Cu-0 [ 341, shows L-net isotropy below T,, and L-net anisotropy above T,. For Y-Ba-Cu-0, the critical temperature is reported to increase as the Cu (2 ) -0 ( 1) interatomic distance decreases from about 2.46 A (tetragonal) to about 2.3 A (orthorhombic), indicating greater covalent bonding and less ionic bonding by the Cu(2)3dg single valence electron [ 361. The T, is found to depend on the Cu (2)-O (1) interatomic distance, and not on the orthorhombic-to-tetragonal transition per se [ 25,371. More striking is the change in interatomic distances reported for Tl-BaCa-Cu-0 [ 381. In the thallium-based superconductor, the critical temperature is greatest (125 K) where the Tl-Ba-Ca-Cu-0 compound having three Cu-0 layers has a Cu( 2)-O (1) interatomic distance of about 2.0 A, and is less (112 K) where the Tl-Ba-Ca-Cu-0 compound having two Cu-0 layers has a Cu (2)0 (1) interatomic distance of about 2.7 A. Like the Cu (2)-O (1) bond in YBa-Cu-0, the apical Cu-0 bond in Tl-Ba-Ca-Cu-0 is reported to be weaker than the planar Cu-0 bonds [39], and may be inferred to lengthen as T, decreases [ 401. Similar results have been reported for Bi-Sr-Ca-Cu-0 [ 411. In the electron-doped T’ phase of Nd-Ce-Cu-0, Murayama et al. [42] find that T, does not change with pressure as it does in the copper oxide superconductors. They concluded that changes in the apical Cu-0 distance must affect T, because the electron-doped superconductor lacks apical oxygen atoms. Sharp changes in bond lengths are reported to coincide with T, in this material [ 431. LaGraff et al. [ 441 report that annealing Y-Ba-Cu-0 in fluorine gas shortens the Cu (2 ) -0 ( 1) bond because fluorine ions intercalate at 0 (5 ) sites and reduce Ba-Ba repulsion. The critical temperature is not significantly affected by the fluorine anneal which suggests that Cu (1 )3dg valence electrons are able to covalently spin-pair through the fluorine anion at the 0 (5) site. The interstitial valence bond structure produces the oxygen stoichiometry which plays a crucial role in the high T, superconductivity of Y-Ba-Cu-0. At oxygen concentrations below 6.5, the Cu (1) atom loses its octahedral coordination with the O(4) atoms, and its freed 3d valence electrons bond aniso-
345
tropically with the remaining 0 (4) and 0 ( 1) atoms, destroying L-net isotropy [ 451. The T, is found to depend, inter alia, on the isotropy of the Cu (1)-O (1) interatomic distances [ 451. Localization of the Cu(2)3dg single valence electron at a K-net vertex, ionically bonding the Cu(2) and 0( 1) atoms of Y-Ba-Cu-0, may produce the antiferromagnetism reported for LazCuO, [46] and Y-Ba-Cu-0 [47] in the normal non-superconducting state. In the normal state, the spin of the localized Cu (2) 3dg single valence electron is free to align alternately, i.e. antiferromagnetically, with the spins of Cu(2)3dg valence electrons localized at nearest-neighbor K-net vertices inside the Cu (2) layer. A similar effect may explain the static magnetic order detected in the Nd-Ce-Cu-0 electron-superconductor series [48]. Similarly, Chakraverty’s model [ 161, based on Cu1+/Cu2+ disproportionation, yields a description of antiferromagnetism in the excited (ionic) state. Epiotis [49] has developed a high T, superconductivity model in which the interstitial orbitals are in the two-dimensional Cu-0 plane, and are not interstitial to the Cu-0 and (Ba,Sr)-0 planes. If the interstitial orbitals in EpiOtis’s model are ordered three-dimensionally and interstitially to the Cu-0 and (Ba,Sr)-0 planes, the stage is set for “meshing”, in the manner shown in Fig. 7, of the delocalized Cu3d electron pairs and localized 02p lone pairs hypothesized by Epiotis. The tetrahedral-octahedral character of the L net suggests that tetrahedrally hybridized sp3 valence electrons of carbon compounds will permit application of the present model in organic or polymeric superconductors with very high critical temperatures [ 28,501. ANISOTROPY IN COPPER OXIDE SUPERCONDUCTORS
The (Ln,Ca,Y)-0 interatomic distances range from 2.4 to 2.5 A in the copper oxide superconductors [ 34,38,51,52 1. Such distances are significantly shorter than, and anisotropic to, the average (Ba,Sr)-0 and O-O distances of 2.8-3.0 A within such superconductors. The L net and K net interstitial to the Cu-0 and Ln-0 layers is anisotropic to the L net and K net interstitial to the Cu-0 and (Ba,Sr)-0 layers, as shown in Fig. 1. Such K-net and L-net anisotropy could explain observed anisotropies in electrical resistivity [ 531, critical current density [ 541, upper critical field [ 551 and vortex structure [ 561 of Y-Ba-Cu-0. Superconducting charge carriers do not flow synchronously through the anisotropic K net interstitial to the Ln-0 and Cu-0 layers, as described below. That such charge carriers flow at all along the c axis in single-crystal Y-Ba-Cu-0 shows that yttrium atoms are occasionally located (owing to defects ) approximately 2.8 A from the oxygen atoms of the Cu(2) layer, and describe a K net and an L net interstitial to the Ln-0
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and Cu-0 layers which is isotropic to the K net and L net interstitial to the Cu-0 and (Sr,Ba)-0 layers. Twinning, as observed in high Z’, superconductors [57], is caused by ionic core defects at L-net vertices and disrupts L-net and K-net isotropy with a deleterious effect on superconductivity. The Cu-0 layering up to three layers preserves L-net and K-net isotropy at enhanced T, because the yttrium, lanthanide or calcium atom stabilizes the oxygen sublattice against such defects in the Y-Ba-Cu-0, Bi-Sr-Ca-Cu-0 and Tl-Ba-Ca-Cu-0 superconductors 1561. The isotropy of the K-net and L-net structures is the key to three-dimensional flow of charge carriers, as discussed below. The anisotropy of such structures produces observations of anisotropy, and of non-linear resistivity above Z’, [ 591, suggestive of three-dimensional superconductivity. SUPERCONDUCTING ORBITAL STRUCTURE
The superconducting orbital structure is a cuboctahedral-octahedral structure (the “K net”) described by vertices maximally interstitial to the L-net vertices occupied by ionic cores and interstitially localized hybridized valence electrons and holes. The K net represents a superconducting ground state in real space coupled to the L-net localized valence bond ground state by the topological transformation shown in Figs. 6 and 7. Figures 6 (a) and 7 (a) correspond to the L net. Figures 6 (e) and 7 (e) correspond to the K net. The ground state of the L net in Fig. 7 (a) is quantum mechanically coupled to the ground state of the K net in Fig. 7 (e ) . The symmetry of this coupling is modeled by the topological transformation shown in Fig. 7. McAdon and Goddard [ 41 point out that T,, symmetry, including the D2,, subgroup, is preserved in a transformation like that shown in Fig. 7. The L net may behave as the polarized medium in Bardeen-Cooper-Schieffer (BCS ) theory [ 601. When all the L-net vertices are occupied by ionic cores (as in the f.c.c. metals and the cubic bismuthates), the K net and L net interact, through the topological transformation shown in Figs. 6 and 7, to fit the qualitative BCS picture of phonon pairing as follows. A superconducting free electron passing through a K-net vertex momentarily attracts ionic cores located at second nearest neighbor L-net vertices. The attraction of ionic cores at such vertices corresponds to the topological transformation from Fig. 6 (a) towards Fig. 6(b). The resulting phonon excitation furnishes the attractive interaction responsible for Cooper pairing of the first superconducting electron with a second superconducting electron on the K net. The collective excitation of the ionic core and valence bond lattice is shown by the topological transformation of the space-filling L net from Fig. 7(a) to Fig. 7 (b). As the second superconducting electron approaches the K-net vertex, the L-net ionic cores relax or repel from the configurations shown in Figs. 6 (b ) and 7 (b) back toward those in Figs. 6 (a) and 7 (a), respectively.
341
(d)
(a)
IY
(e)
(b)
(cl
X Fig. 6. Synchronous topological transformation: (a) cuboctahedral phase (L net); (b) icosahedral phase; (c ) inter-icosahedral phase; (d) quasi-icosahedral phase; (e 1 octahedral phase (K net).
When some of the L-net vertices are occupied by valence bond pairs, the symmetric correlation between both the ionic core and valence bond electron states and the superconducting states is modeled in Fig. 7. The L net represents a valence bond structure with minimal Pauli repulsion, and the K net represents a superconducting ortibal structure maximally interstitial to the L net and minimally (possibly negative [ 611) Coulomb repulsive relative to the L net. Superconducting charge carriers are hypothesized to scatter synchronously along the vectors connecting nearest-neighbor K-net vertices. The K net models a scattering matrix for pairwise species. When some of the L-net vertices are occupied by valence bond pairs, phonon excitations of the L net could be augmented by excitonic [ 50,621, plasmonic [ 631, bipolaronic [ 301 or peroxitonic [ 161 excitations in the weak coupling limit [ 641.
(a)
Cd)
(b)
(c)
Fig. 7. Space-filling synchronous topological transformation: (a) cuboctahedral phase; (b) icosahedral phase; (c) inter-icoeahedral phase; (d) quasi-icosahedral phase; (e) octahedral phase.
Synchronous spin interactions between superconducting charge carriers passing through second-nearest-neighbor K-net vertices may be mediated by ionic cores and bonding orbitals localized through L-net vertices maximally interstitial to such K-net vertices. The topological symmetry of such spin interactions may be modeled by the topological transformation from that shown in Fig. 7 (d) to that in Fig. 7 (e ). The topological transformation shown in Fig. 7 depends on the isotropy of the K net and L net. The region interstitial to the Ln-0 and Cu-0 layers does not participate in the superconducting flow because the K net and L net in this region are anisotropic to the K net and L net interstitial to the Cu-0 and (Ba,Sr)-0 layers. If the states available in the isotropic K-net region are lower
349
in energy than those in the anisotropic K-net region, the charge carriers will prefer the isotropic K-net region. In the absence of an applied electromagnetic field, the K-net vertices topologically describe a set of empty interstitial electronic states below the energy gap, consistent with the reported [65] energy gap in the copper oxide superconductors. The distance between nearest-neighbor K-net vertices in the copper oxide superconductors is about 1.4 A and may approximate a superconducting coherence length comparable to calculated c axis superconducting coherence lengths of about 1.5-2 A by Oh et al. [ 661, and of about 1 A by Deutscher [ 671 and Kresin and Wolf [68]. King [69] hypothesized that a porous skeletal bonding topology can describe the valence bond structure of high T, superconductors, and suggested that the skeletal bonding manifold may relate to the short mean free path and small BCS coherence length (l-2 A) in such superconductors [ 66,681. Phonon interactions sufficient for Cooper pairing are found in neither the high T, copper oxide superconductors [ 701, nor the Al5 superconductors [ 711. However, phonon energies sufficient for Cooper pairing have been identified in Ba-K-Bi-0 [ 721. Figure 3 (a) shows that the bonding structure of Ba-KBi-0 is non-bent (ionic), and is not characterized by localized interstitial valence electrons as in the copper oxide and Al5 superconductors. The Ba-K-Bi-0 bonding structure shown in Fig. 3 (a) is a complete tetrahedral-octahedral (f.c.c.) L net, which may resemble the bond structure of the f.c.c. lead crystal in which phonon coupling has been shown by isotope effect studies [60]. Although the L-net bonding geometry is the same in the phononcoupled and non-phonon-coupled superconductors, phonon effects may be identified with superconductivity only if all the L-net vertices are occupied by ionic core states, as in lead and Ba-K-Bi-0. PROPERTIES OF THE SUPERCONDUCTING
ORBITAL STRUCTURE
The trajectory in the (x,y ) plane of a vertex (A-+A’ ) of the transformation shown in Fig. 6 is elliptical and is described by the equation 2x2-2xy+2y2=1
(1)
where each edge of each triangular surface shown in Fig. 6 has a length of unity, all the vertices lie in the (xy), (x,2) or (y,z) planes, and the volumetric centers of the polyhedra shown in Fig. 6 lie at the origin. The combined trajectories of all the vertices in the (x,y ) plane are shown in Fig. 8. The coordinates of vertex A-A’ at each stage of the transformation in Fig. 6 are shown in Table 1. The trajectories of the other vertices shown in Fig. 6 are given by the following equations:
350
2x2+2xy+2y2=1
(2)
2x2~2xz+2z2=1
(3)
2y2?2yz+2z2=1
(4) A simple description of the topological transformation shown in Fig. 7 could be obtained by a rotation of + x/6 and a translation of +-l/J6 of each triangular surface (180” domain wall) shown in Fig. 7. The topological transformation shown in Fig. 7 could model the topological symmetry of quasi-particle excitations thought by Anderson et al. [ 731 and by Kivelson et al. [ 741 to cause Bose condensation in the strong coupling theories of superconductivity [ 751. Each triangular surface shown in Figs. 6 (a) and 6 (e) and Figs. 7 (a) and 7 (e) rotates by + n/6 and translates by -t l/,/6 within a cylinder whose axis Y .
- _____ . . _-_
.---.
_-._._.____...
_ .._.. _.._.
---._-
--..
_-if__.____
.-____ -.._-
Fig. 8. Trajectory in the (x,y) plane of vertex A+A’ in the synchronous topological transformation.
351 TABLE 1 Transformation
coordinates
Fig.
Polyhedron
x
Y
z
6(a)
Cuboctahedron Icosahedron Inter-icosahedron Quasi-icosahedron Octahedron
l/J2 0+J51/4 JW3 U+J5)/4 l/J2
l/J2 l/2 l/J6 (-1+J5v4 0
0 0 0 0 0
6(b) 6(c) 6(d) 6(e)
passes through the center of the surface and the volumetric center of each octahedron and cuboctahedron shown in Figs. 6 (a) and 6 (e ) and Figs. 7 (a) and 7 (e) . Each triangular surface shown in Fig. 7 displays either a right-handed or left-handed helical motion. Each such motion describes a topical symmetry related to the topical symmetries of all other triangular surfaces by the topological transformation of Fig. 7. Figures 7 (b) and 7 (d) show that the topological transformation of Fig. 7 contains icosahedrally symmetric phases [ 761 based on one of the two possible helical motions (left- or right-handed) of each triangular surface [ 771. (Szasz and co-workers [ 761 have proposed that an icosahedral Jones zone/Fermi surface interaction causes icosahedral distortion or “dynamical frustration” of the oxygen lattice.) There exist two geometrically distinct combinations of such phases based on the two possible helical motions of each triangular surface. The two geometrically distinct combinations yield geometrically equivalent phases like that shown in Fig. 7 (e). The triangular surfaces in Fig. 7 resemble “odd rings”, composed of odd numbers of interatomic bonds, which Rivier hypothesizes are punctured by topologically stable Zz “odd-line” defects (x1 (SO (3) ) = Z,) in metallic and covalent glasses [ 771. Dynamic odd-line defects lead to viscous relaxation in glass, consistent with observations of giant flux creep in the copper oxide superconductors [ 781. The K nets within two superconducting grains may couple through K-net vertices common to each grain [ 791. Decoupling of such K nets may describe a metastable electronic liquid phase [80] or a superconducting glass phase consistent with findings by several researchers (see for example ref. 81) that copper oxide superconductors exhibit metastable glass-like behavior under conditions of applied magnetic field and zero-field cooling. Such decoupling may explain the giant flux creep observed in the copper oxide superconductors 1731. Rivier shows that the energy gap in amorphous silicon can be described by a topological formulation in which the two ground states are characterized by bonding and antibonding states [ 821. The energy gap in the superconductors
352
may be described by a similar procedure in which the two ground states are described by the L net of valence bond states and the K net of antibonding (conduction) states. Applying Bloch’s theorem, the free energy of the superconductor will be a periodic function of the flux triggering the gauge transformation [ 831. If the flux lives on the triangular surfaces shown in Fig. 7, the flux may be quantized by the + x/6 rotation and + l/J6 translation along the odd-line core puncturing each such surface to depict qualitatively the vortices and flux lattice of a type-II superconductor [ 771. Gammel and co-workers [84] and Dolan et al. [56] observed hexatic flux quanta vortices in Y-Ba-Cu-0, and stated that such vortices are comparable to the triangular array of vortices in conventional type-II superconductors under similar conditions. The observed hexatic flux pattern may show that such flux is correlated through the centers of the hexagonally correlated triangular surfaces in Fig. 7. The synchronous motion of such surfaces, as shown in Fig. 7, suggests a “pumping” action responsible for the Meissner effect. Observations by Dolan et al. [56] of “undulations” in chains of anisotropic flux vortices in Y-Ba-Cu-0 may also provide evidence for the topological transformation shown in Fig. 7. CONCLUSION
The present model describes an interstitial valence bond structure and a superconducting structure interstitial to the valence bond structure. Superconductivity depends critically on K-net and L-net isotropy. In the superconducting state, the L-net interstitial valence bond structure can produce the following effects characteristic of superconductivity in copper oxide superconductors: (1) photoemission spectroscopy evidence of Cu3d/02p hybridization, Cu-0 covalent bonding and oxygen dimerization, (2) Hall resistivity of electron vs. hole carriers, (3) oxygen stoichiometry and Cu-0 chains in Y-Ba-Cu-0, (4) Jahn-Teller distortion of Cu-0 half-octahedra, (5) antiferromagnetic and ferromagnetic normal states, (6) three-dimensional anisotropies in critical current, resistivity, upper critical field and vortex structure measurements, (7) microstructural twinning, and (8) increased T,with multilayer structures. The K-net superconducting orbital structure is maximally interstitial to the L-net valence bond structure and can produce the following effects characteristic of superconductivity in copper oxide superconductors: (1) a superconducting energy gap, (2 ) Cooper pairing, (3) short superconducting coherence lengths, (4) an isotope effect, (4) intergrain coupling, (5) metastable glasslike behavior and (6 ) hexatic type-11 flux quanta. The topological model may aid the discovery and design of new higher T,
353
superconducting materials with valence bond structures conforming to the L net and a superconducting orbital structure conforming to the K net.
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335
VALENCE BOND ORDER AND SUPERCONDUCTIVITY
ERIK W. KVAM Superconductor
Ventures, Inc., 232 East 82nd Street, New York, NY 10028 (U.S.A.)
(Received 1 October 1990)
ABSTRACT The three-dimensional interstitially localized valence bond structure leads to a model of real space pairing of superconducting electrons in a three-dimensional lattice. The valence bond ground states are coupled to the superconducting ground states through a topologically synchronous transformation. Superconductivity depends critically on the three-dimensional isotropy of the valence bond structure within the three-dimensional unit cell.
INTRODUCTION
Since the discovery of superconductivity by Kamerlingh Onnes in 1911, and the discovery of high T, superconductivity in the La-Ba-Cu-0 system by Bednorz and Miiller [ 1] in 1986, no superconducting mechanism has been elucidated which is common to low T, and high T, superconductors. The present model describes a three-dimensional isotropically ordered valence bond structure which may be common to all superconducting materials. The model focuses on YBa&u30, (Y-Ba-Cu-0) and (Nd,Ce)&uO, (NdCe-Cu-0) as representative of hole-type and electron-type layered cuprate superconductors, on (Ba,K)BiO, (Ba-K-Bi-0) as representative of the cubic bismuthate superconductors, and on NbaGe (Nb-Ge) as representative of the Al5 superconductors. The model could be extended to the metallic element superconductors in which T, has been shown to depend on optimum interatomic distances in hole-type superconductors [ 21. The likelihood of a common s-wave pairing superconducting mechanism in both high T, (higher than 40 K) and low T, (lower than 40 K) superconductors has been shown by recent experiments in which the electron pairs maintained phase coherence across an Nb-Y-Ba-Cu-0 junction [ 31. Heretofore, the actual structures giving rise to superconductivity have not been adequately described or understood. The lack of such understanding has proved a serious obstacle to achieving room temperature superconductivity because the search for new superconductors has been a trial-and-error process, guided primarily by variation of known superconducting materials.
0166-1260/91/$03.50
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Elsevier Science Publishers B.V.
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The topological model offers a simple method for predicting whether a material will superconduct, based on the valence bond structure. The interstitial valence bond structure of a superconducting material can be specified by quantum chemical calculations using “generalized valence bond” theory to describe bonding structures and energies in metallic atom microclusters [ 451. The interstitial valence bond structure can explain why the lengthening of the Cu (2)0 (1) apical bond is identified with the loss of superconductivity in Y-Ba-Cu0. VALENCE BOND ORDER OF SUPERCONDUCTORS
The working hypothesis of the present model is that superconducting materials possess an isotropically ordered three-dimensional interstitial valence bond structure. This structure has a space-filling tetrahedral-octahedral geometry [ 61. The three-dimensional isotropy [ 71 of the structure within the superconducting layers is crucial for superconductivity, and the loss of such isotropy is identified with the loss of superconductivity. Generalized valence bond (GVB) theory systematically accounts for electron correlation effects by variationally optimizing each valence electron and hole orbital wavefunction during the formation of the bond to minimize energy and to obtain an accurate structural description of the molecule [ 51. Conceptually, this procedure treats the two-center one-electron or one-hole valence bond as the wavefunction of singly occupied orbitals which can compress and tense to describe a dynamically stable energy equilibrium, i.e. a valence bond, between two or more atoms. Optimum bonding involves singly occupied valence electron orbitals localized through interstitial regions between the ionic cores [ 81. Interstitial valence electron localization, according to GVB principles, minimizes Pauli repulsion and is consistent with the Lewis-Langmuir octet rule by which each atom shares valence electron orbital pairs to achieve a closed-shell configuration [9]. Localized valence electron and hole orbitals describe a “bent” bond, as distinguished from a and o bonds in cases of double, triple and conjugated bonds [lo]. The GVB theory is unlike mean-field theory which describes a variety of ad hoc electron orbital hybridizations, depending on the chemical bonding environment of such orbitals. The application of empirical rules [4] derived from GVB theory yields the following description of interstitially localized valence bonds in the copper oxide and Al5 superconductors. GVB hybridized orbitals can share a valence electron pair in a covalent bond (two singly occupied orbitals) or dative bond (one doubly occupied orbital and one unoccupied orbital). The ionic cores and valence electrons in a copper oxide superconductor tend to occupy vertices in a tetrahedral-octahedral structure (the “L net” ) in which
337
the valence electron and hole orbitals describe the ‘bent” covalent bonds which connect such vertices in the superconducting state. The vertices in a second tetrahedral-octahedral structure (the “K net”), maximally interstitial to the L net, describe the superconducting orbital structure. In the non-superconducting state, the Cu3d9 valence electron localizes on the K net with a resulting loss of K-net and L-net isotropy and superconductivity. The K net and L net are interpenetrating face-centered cubic (f.c.c.) structures [ 281. Such structures are commonly recognized as highest density lowest energy structures in three dimensions in cluster studies [11,12], which may explain why superconductivity is most commonly found at low temperatures (below 10 K) in f.c.c. metallic elements. Vertices and lines are used in Fig. 1 and in the other figures to simplify the description of the overall symmetry of the valence electrons and valence bonds. Such vertices and lines symbolize the spatial regions in which the valence electrons and orbitals are localized, and do not imply that such electrons are orbitals are actually located at fixed points or along fixed lines in real space. (a)
(b)
;i
F
(cl
/@
& Ba
\
?
cuti
1
O(4)
0
o
Localized
Valence
Delocalized (Superconducting)
Bond
State
Antibonding
State
Fig. 1. YBa&u,O+ (a) interstitially localized covalent bonds, (b) complete interstitially localized valence bond structure (“L net”); (c ) delocalized (superconducting) orbital structure (“K net”).
338
The upper and lower portions of Fig. 1(a) show the alternate covalent bent bonds formed by the Cu (3 ) 3dg orbitals and the 0 (2 ) 2p and 0 (3 ) 2p orbitals in the Cu(2)-0 (2)-O(3) layer the “Cu(2) layer”) of Y-Ba-Cu-0, and by the 0 (2 ) 2p and 0 (3 ) 2p orbitals and the Ba5d orbitals. The valence bond structure of the Cu (2 ) layer of YBa-Cu-0 could apply equally to the Cu-0 layers of the (La,Sr ) -, (Bi,Sr ) - and (Tl,Ba) -based copper oxide superconductors. The central portion of Fig. 1 (a) shows the alternate covalent bonds formed by the Cu(1)3dgorbitalsandtheO(4)2porbitalsoftheCu(1)-O(4)-O(5) layer (the “Cu ( 1) layer” ) . The covalent bonds in Fig. 1 (a) describe part of the three-dimensional tetrahedral-octahedral structural termed the L net. The complete L net, including non-bent bonds within the Cu-0 layers and vectors correlating interstitial valence electrons, is shown in Fig. 1 (b ). Figure 1 (c ) shows the interpenetrating K-net structure for Y-Ba-Cu-0. (a)
(b)
(cl
Nd
0
0
Fig. 2. (Nd,Ce)&uO,:
j_ocalited
Valence
Bond state
Delocalized Antibondim (superconducting) state
(a) interstitially
localized covalent bonds, (b) complete interstitially
lo-
calizedvalence bond structure (“L net”); (c) delocalized (superconducting) orbital structure (“K net” 1.
Figure 2 (a) shows the alternate covalent bonds formed by the Cu3dg orbitals and the 02p orbitals, and by the NdSd/Ce3d orbitals and 02p orbitals in the T’ phase of Nd-Ce-Cu-0. Figure 2(b) shows the complete L net for the T’ phase. Figure 2 (c) shows the K-net structure for this compound Figure 3 (a) shows the complete set of alternate non-bent bonds formed by the Bi6p orbitals, (Ba,K) 5d orbitals and the 02p orbitals in the cubic Ba-KBi-0 superconductor. Figure 3 (b) shows the K-net structure for this compound. Figure 4 (a) shows the alternate covalent bonds formed by the Nb4p orbitals in Nb-Ge, an Al5 intermetallic superconductor. Figure 4(b) shows the complete L net for Nb-Ge. Figure 4 (c) shows the K-net structure for this compound. Resonating-GVB calculations by Guo et al. [ 131of a five-coordinate CI_I~O~~ half-octahedral cluster and a four-coordinate Cu301,, planar cluster in Y-BaCu-0, and of a six-coordinate CuzO,, octahedral cluster in Laz_,Sr,CuO, (LaSr-Cu-0), yield optimum localized wavefunctions having nine electrons in d orbitals on each copper. Oxidation of the Cu2+ ( dg) did not lead to Cu3+ (d8) but to a singly occupied 02~71 orbital spin-coupled to the singly occupied copper d orbitals and localized interstitially to two Cu2+ sites. (a)
(b)
0
Delocalized Antibonding (Superconducting) State
Fig. 3. (Ba,K)BiO,: (a) interstitially localized valence bond structure (“L net”); (b) delocalized (superconducting) orbital structure (“K net”).
340 (a)
(b)
(c)
0
0
0
0
0 0
0 0
0
0 0 0
0
0
o
Localized
Delocalized
0
0
Valence
0
Bond
O
0
State
Antibonding
(Superconducting)
State
Fig. 4. Nb3Ge: (a) interstitially localized covalent bonds; (b) complete interstitially localized valence bond structure (“L net”); (c) delocalized (superconducting) orbital structure (“K net”).
In the ionized CuZOl, cluster, Guo et al. find that the planar 02pa orbital is oxidized when x = 0, but that the apical 02~~ orbital is preferentially oxidized when x=0.3, which suggests that the 02pa orbital is localized interstitially to the planar and apical oxygen atoms in the superconducting state. In the ionized planar Cu,O,, cluster, 02p bonding orbitals and singly occupied Cu3dg orbitals are shown oriented toward regions corresponding to L-net vertices interstitial to the Cu-0 and Ba-0 layers, and resonant singly and doubly occupied 02p non-bonding and/or antibonding orbitals are shown oriented toward K-net positions with the suggestion that holes in such orbitals are high mobility charge carriers. MSXcr calculations for CuOI planar clusters by Sarma and co-workers suggest that the non-spin-paired Cu3dg electron bonds interstitially with an 03s orbital in a 4A1, non-magnetic highest occupied molecular orbital (HOMO) ground state, just below an antiferromagnetic and antibonding lowest unoccupied molecular orbital (LUMO) of 3B,, [ 141.They suggest that the stability of the 4A,, orbital depends critically on the overlap (dimerization) of 03s orbitals and the symmetry of the O-O distances, and that a slight increase in such distances pushes the cluster into the antiferromagnetic 3&, state in which the unpaired 3dg electron is localized to the Cu site and not interstitially. Like
341
Guo et al., they suggest that charge fluctuation is not due to Cu3d electrons but to 02p electrons which resonate between 02- and O’- species. X-ray absorption spectra and X-ray photoemission spectra of La-Sr-Cu-0 and Y-Ba-Cu-0 consistently show Cu3d/02p hybridization of localized Cu3d/ 02p bonding orbitals and Cu3d/02p covalent bonding in the superconducting state [ 15-181. These spectra consistently show that the ground state consists of either 2p3dg ( Cu2+) or a mixture of 2p3dg and 2p3d” (Cu’+ ) configurations, with little or no 2p3ds ( Cu3+ ). Similar results are reported for Tl,CaBa,Cu,O, (Tl-Ca-Ba-Cu-0) and Bi,CaSr,Cu,O, (Bi-Ca-Sr-Cu-0) [ 161. Bianconi’s results [ 171 suggest that in the hybridized 2p3dg and ligand 2p3d1’ configuration, the hole fluctuates between the Cu site and the 0 site in the pda covalent bond (consistent with Sleight’s view [ 19 ] ), and that the Cu3d metal orbital is localized while the hybridized 02p ligand orbital is delocalized, consistent with Hall effect findings [ 18,201 that the charge carriers are holes in Y-Ba-Cu-0. Chakraverty et al. [ 161 propose that bipolaronic (Cu”’ ) or peroxitonic (Cul+ ) dimerization of nearest-neighbor 01- yields Cu3dg and Cu3dl” interstitially localized covalent bonds, and real space pairing of holes, above the antibonding oxygen level, which are the charge carriers. Van Bruggen [21] proposed a similar model in which the monovalent Cu3d’O ground state induces low density of states 02p valence band holes as charge carriers. Chakraverty’s 01s core level spectroscopy results point to dimerization of 01- holes. The results of Sarma et al. [ 151 provide evidence of oxygen dimerization below T,, lending support to the peroxitonic model. The interstitial valence bond model is consistent with the covalency found in both low T, and high T, superconductors [ 221. The covalent bonds in YBa-Cu-0 are strongly localized, with evidence of delocalization of the bonding orbitals above T, based on a jump in the heat capacity at T, [ 231. Interstitial valence bonds in copper oxide superconductors are consistent with suggestions that the Cu( 1) layer in Y-Ba-Cu-0 acts as a reservoir of electronic charge, and that the mechanism for superconductivity in the layered copper oxide superconductors employs a non-homogeneous charge distribution [ 24-261. In the Y-Ba-Cu-0 superconductor, the average valence on the orbitals interstitial to the Cu (2) and Ba-0 layers is roughly + 0.25 [ 7,241. In Nd-CeCu-0, the average charge on the orbitals interstitial to the Cu-0 and (Nd,Ce) layer is roughly -0.13. The interstitial valence bond model describes the holedoped and electron-doped valence electron structures, which have been shown by Hall measurements to correspond to superconducting states in which the charge carriers are holes [ 201 and electrons [ 271 respectively. Messmer and Murphy [28] propose that the GVB theory can describe a superconducting ground state which is a coherent superposition of bonding structures (degenerate states) which exhibit synchronous spatial alternation of valence bonds (a form of “resonance”). Messmer and Murphy suggest that
342
highly correlated valence electron orbitals localized at interstitial Wigner lattice sites can completely describe a superconducting wavefunction.
Cu3dg LOCALIZATION IN COPPER OXIDE SUPERCONDUCTORS
Localization of the Cu3dg valence electron may play a key role in determining the Cu (2)-O (1) distance and the overall isotropicity of the L-net and the K-net structures. In Y-Ba-Cu-0 and the other copper oxide superconductors, each Cu (1) and Cu (2) atom has one uncoupled 3dg valence electron [ 171. In the superconducting state of Y-Ba-Cu-0, the Cu (1) 3dg orbital localizes through an L-net vertex adjacent to the Cu (1) layer, as shown in Fig. 5 (a). (b)
(a)
0 @O
0
Localized
Valence
0
0
@O
Bond State
Fig. 5. (a) Superconducting YBa&us07: alternate Cu(2)-0 (1)-O (2)-O(3)-Ba covalent bent bonds comprising 3dg2p bonding orbitals interstitially localized through L-net vertices, and alternate Cu( 1)-O (1)-O (4)-Ba-Cu( 1) bent covalent bonds comprising 3dg2p and 3ds2p bonding orbitals interstitially localized through L-net vertices and the vacant O(5) site. (b) Normal YBa,Cus07: Cu(2)-O( 1) non-bent (ionic) valence bonds comprised 3ds2p bonding orbit&, interstitially localized through K-net vertices.
343
The Cu (2) 3dg orbital localizes through an L-net vertex adjacent to the Cu (2) layer, as shown in Fig. 5 (a). In the superconducting state shown in Fig. 5 (a), the Cu ( 1) 3dg orbital forms a bent covalent bond with an 0 (1) 2p orbital, 0 (4) 2p orbital or Ba5d orbital. Alternatively, two such Cu ( 1) 3dg orbitals can spin-couple through a vacant 0 (5) site to describe a strongly correlated covalently bound Cu3d’ ( Cu3+ ) state [ 171, equivalent to a singlet-coupled hole pair localized at the 0 (5) site [ 26,291. (Tranquada et al. [ 261 suggest that the number of holes is maximized because the Cu (1)-O (4) chain acts as an electron reservoir. Tokura et al. [ 291 conclude that holes associated with the average charge in the Cu( 1)-O(4) plane are localized and provide an insulating reservoir of charge.) De Jongh [ 301 suggests that Cu1+/Cu2+ “ dimerization” in the Cu ( 1) chain is plausible. A hole pair localized through the O(5) site may screen the interaction between the localized Cu( 1)3dg valence electrons and maintain K-net and L-net isotropy within the Cu (1) layer. Such a hole pair repels oxygen from the 0 (5) site to create the observed Cu( 1)-O (4) chain structure [31]. Hole pair delocalization from the 0 (5) site at temperatures above about 55-60 K may produce the superconducting phase transition reported at this temperature [ 251. In the superconducting state, the Cu(2)3dg orbital localizes through an Lnet vertex, as shown in Fig. 5 (a). In the normal non-superconducting state, the Cu(2)3dg orbital localizes through a K-net vertex, as shown in Fig. 5(b). Messmer and Murphy [ 281 suggest that single valence electron localization on an f.c.c. lattice (a K net) interpenetrating a second f.c.c. lattice (an L net) of ion cores prevents superconductivity in the alkali and noble metals (Cu, Ag, Au). In the superconducting state shown in Fig. 5 (a), the Cu ( 2)3dg orbital forms a covalent bent bond with an 0 (2) 2p orbital, 0 (3) 2p orbital, or Ba5d orbital [ 321. (Using Rietveld refinement of structural models from pulsed neutron powder diffraction data, Kwei et al. [ 321 find no changes in Cu (2)-O (1) lengths at T,, but do find large-amplitude motions involving the Ba2+ ions and the CuO, planes. ) In the normal non-superconducting state of Y-Ba-Cu-0 shown in Fig. 5 (b) , the Cu (2) 3dg orbital forms a weak non-bent (ionic) bond with the 0 ( 1) 2p orbital. The Cu (2 ) -0 ( 1) bond is reported [ 33 ] to be significantly weaker than the other Cu-0 bonds in Y-Ba-Cu-0. The weakness and ionicity of the Cu (2 )-0 ( 1) bond is shown by its extended (Jahn-Teller effect ) length of about 2.3 A in orthorhombic Y-Ba-Cu-0 (about 2.5 A in tetragonal Y-BaCu-0), compparedwith the covalent Cu (2)-O (2) and Cu (2)-O (3) bond lengths of about 1.9 A [34]. Oudet hypothesizes that loss of superconductivity at T, is caused by thermally induced disorder of the covalent bonds involving the Cu3dg valence electrons [ 351, which is consistent with the model of valence bond order hypothesized herein.
344 PROPERTIES OF THE VALENCE BOND STRUCTURE
Localization of the Cu(2)3dg single valence electron at a K-net vertex instead of at an L-net vertex creates short weak non-bent ionic Cu (2)-O (1) bonds between atoms at longer interatomic distances, instead of long strong bent covalent Cu(2)-0 (1) bonds between atoms at shorter interatomic distances. The ionic Cu3dg valence bonds create K-net and L-net anisotropy within the Cu-0 layers, reflective of a mixed ionic/covalent bond structure in which the L-net ionic cores are not equidistant from one another as they are in an isotropic covalent bond structure. The length of the Cu (2)-O (1) covalent/ionic bond, as shown by the interatomic distances reported for Y-Ba-Cu-0 [ 341, shows L-net isotropy below T,, and L-net anisotropy above T,. For Y-Ba-Cu-0, the critical temperature is reported to increase as the Cu (2 ) -0 ( 1) interatomic distance decreases from about 2.46 A (tetragonal) to about 2.3 A (orthorhombic), indicating greater covalent bonding and less ionic bonding by the Cu(2)3dg single valence electron [ 361. The T, is found to depend on the Cu (2)-O (1) interatomic distance, and not on the orthorhombic-to-tetragonal transition per se [ 25,371. More striking is the change in interatomic distances reported for Tl-BaCa-Cu-0 [ 381. In the thallium-based superconductor, the critical temperature is greatest (125 K) where the Tl-Ba-Ca-Cu-0 compound having three Cu-0 layers has a Cu( 2)-O (1) interatomic distance of about 2.0 A, and is less (112 K) where the Tl-Ba-Ca-Cu-0 compound having two Cu-0 layers has a Cu (2)0 (1) interatomic distance of about 2.7 A. Like the Cu (2)-O (1) bond in YBa-Cu-0, the apical Cu-0 bond in Tl-Ba-Ca-Cu-0 is reported to be weaker than the planar Cu-0 bonds [39], and may be inferred to lengthen as T, decreases [ 401. Similar results have been reported for Bi-Sr-Ca-Cu-0 [ 411. In the electron-doped T’ phase of Nd-Ce-Cu-0, Murayama et al. [42] find that T, does not change with pressure as it does in the copper oxide superconductors. They concluded that changes in the apical Cu-0 distance must affect T, because the electron-doped superconductor lacks apical oxygen atoms. Sharp changes in bond lengths are reported to coincide with T, in this material [ 431. LaGraff et al. [ 441 report that annealing Y-Ba-Cu-0 in fluorine gas shortens the Cu (2 ) -0 ( 1) bond because fluorine ions intercalate at 0 (5 ) sites and reduce Ba-Ba repulsion. The critical temperature is not significantly affected by the fluorine anneal which suggests that Cu (1 )3dg valence electrons are able to covalently spin-pair through the fluorine anion at the 0 (5) site. The interstitial valence bond structure produces the oxygen stoichiometry which plays a crucial role in the high T, superconductivity of Y-Ba-Cu-0. At oxygen concentrations below 6.5, the Cu (1) atom loses its octahedral coordination with the O(4) atoms, and its freed 3d valence electrons bond aniso-
345
tropically with the remaining 0 (4) and 0 ( 1) atoms, destroying L-net isotropy [ 451. The T, is found to depend, inter alia, on the isotropy of the Cu (1)-O (1) interatomic distances [ 451. Localization of the Cu(2)3dg single valence electron at a K-net vertex, ionically bonding the Cu(2) and 0( 1) atoms of Y-Ba-Cu-0, may produce the antiferromagnetism reported for LazCuO, [46] and Y-Ba-Cu-0 [47] in the normal non-superconducting state. In the normal state, the spin of the localized Cu (2) 3dg single valence electron is free to align alternately, i.e. antiferromagnetically, with the spins of Cu(2)3dg valence electrons localized at nearest-neighbor K-net vertices inside the Cu (2) layer. A similar effect may explain the static magnetic order detected in the Nd-Ce-Cu-0 electron-superconductor series [48]. Similarly, Chakraverty’s model [ 161, based on Cu1+/Cu2+ disproportionation, yields a description of antiferromagnetism in the excited (ionic) state. Epiotis [49] has developed a high T, superconductivity model in which the interstitial orbitals are in the two-dimensional Cu-0 plane, and are not interstitial to the Cu-0 and (Ba,Sr)-0 planes. If the interstitial orbitals in EpiOtis’s model are ordered three-dimensionally and interstitially to the Cu-0 and (Ba,Sr)-0 planes, the stage is set for “meshing”, in the manner shown in Fig. 7, of the delocalized Cu3d electron pairs and localized 02p lone pairs hypothesized by Epiotis. The tetrahedral-octahedral character of the L net suggests that tetrahedrally hybridized sp3 valence electrons of carbon compounds will permit application of the present model in organic or polymeric superconductors with very high critical temperatures [ 28,501. ANISOTROPY IN COPPER OXIDE SUPERCONDUCTORS
The (Ln,Ca,Y)-0 interatomic distances range from 2.4 to 2.5 A in the copper oxide superconductors [ 34,38,51,52 1. Such distances are significantly shorter than, and anisotropic to, the average (Ba,Sr)-0 and O-O distances of 2.8-3.0 A within such superconductors. The L net and K net interstitial to the Cu-0 and Ln-0 layers is anisotropic to the L net and K net interstitial to the Cu-0 and (Ba,Sr)-0 layers, as shown in Fig. 1. Such K-net and L-net anisotropy could explain observed anisotropies in electrical resistivity [ 531, critical current density [ 541, upper critical field [ 551 and vortex structure [ 561 of Y-Ba-Cu-0. Superconducting charge carriers do not flow synchronously through the anisotropic K net interstitial to the Ln-0 and Cu-0 layers, as described below. That such charge carriers flow at all along the c axis in single-crystal Y-Ba-Cu-0 shows that yttrium atoms are occasionally located (owing to defects ) approximately 2.8 A from the oxygen atoms of the Cu(2) layer, and describe a K net and an L net interstitial to the Ln-0
346
and Cu-0 layers which is isotropic to the K net and L net interstitial to the Cu-0 and (Sr,Ba)-0 layers. Twinning, as observed in high Z’, superconductors [57], is caused by ionic core defects at L-net vertices and disrupts L-net and K-net isotropy with a deleterious effect on superconductivity. The Cu-0 layering up to three layers preserves L-net and K-net isotropy at enhanced T, because the yttrium, lanthanide or calcium atom stabilizes the oxygen sublattice against such defects in the Y-Ba-Cu-0, Bi-Sr-Ca-Cu-0 and Tl-Ba-Ca-Cu-0 superconductors 1561. The isotropy of the K-net and L-net structures is the key to three-dimensional flow of charge carriers, as discussed below. The anisotropy of such structures produces observations of anisotropy, and of non-linear resistivity above Z’, [ 591, suggestive of three-dimensional superconductivity. SUPERCONDUCTING ORBITAL STRUCTURE
The superconducting orbital structure is a cuboctahedral-octahedral structure (the “K net”) described by vertices maximally interstitial to the L-net vertices occupied by ionic cores and interstitially localized hybridized valence electrons and holes. The K net represents a superconducting ground state in real space coupled to the L-net localized valence bond ground state by the topological transformation shown in Figs. 6 and 7. Figures 6 (a) and 7 (a) correspond to the L net. Figures 6 (e) and 7 (e) correspond to the K net. The ground state of the L net in Fig. 7 (a) is quantum mechanically coupled to the ground state of the K net in Fig. 7 (e ) . The symmetry of this coupling is modeled by the topological transformation shown in Fig. 7. McAdon and Goddard [ 41 point out that T,, symmetry, including the D2,, subgroup, is preserved in a transformation like that shown in Fig. 7. The L net may behave as the polarized medium in Bardeen-Cooper-Schieffer (BCS ) theory [ 601. When all the L-net vertices are occupied by ionic cores (as in the f.c.c. metals and the cubic bismuthates), the K net and L net interact, through the topological transformation shown in Figs. 6 and 7, to fit the qualitative BCS picture of phonon pairing as follows. A superconducting free electron passing through a K-net vertex momentarily attracts ionic cores located at second nearest neighbor L-net vertices. The attraction of ionic cores at such vertices corresponds to the topological transformation from Fig. 6 (a) towards Fig. 6(b). The resulting phonon excitation furnishes the attractive interaction responsible for Cooper pairing of the first superconducting electron with a second superconducting electron on the K net. The collective excitation of the ionic core and valence bond lattice is shown by the topological transformation of the space-filling L net from Fig. 7(a) to Fig. 7 (b). As the second superconducting electron approaches the K-net vertex, the L-net ionic cores relax or repel from the configurations shown in Figs. 6 (b ) and 7 (b) back toward those in Figs. 6 (a) and 7 (a), respectively.
341
(d)
(a)
IY
(e)
(b)
(cl
X Fig. 6. Synchronous topological transformation: (a) cuboctahedral phase (L net); (b) icosahedral phase; (c ) inter-icosahedral phase; (d) quasi-icosahedral phase; (e 1 octahedral phase (K net).
When some of the L-net vertices are occupied by valence bond pairs, the symmetric correlation between both the ionic core and valence bond electron states and the superconducting states is modeled in Fig. 7. The L net represents a valence bond structure with minimal Pauli repulsion, and the K net represents a superconducting ortibal structure maximally interstitial to the L net and minimally (possibly negative [ 611) Coulomb repulsive relative to the L net. Superconducting charge carriers are hypothesized to scatter synchronously along the vectors connecting nearest-neighbor K-net vertices. The K net models a scattering matrix for pairwise species. When some of the L-net vertices are occupied by valence bond pairs, phonon excitations of the L net could be augmented by excitonic [ 50,621, plasmonic [ 631, bipolaronic [ 301 or peroxitonic [ 161 excitations in the weak coupling limit [ 641.
(a)
Cd)
(b)
(c)
Fig. 7. Space-filling synchronous topological transformation: (a) cuboctahedral phase; (b) icosahedral phase; (c) inter-icoeahedral phase; (d) quasi-icosahedral phase; (e) octahedral phase.
Synchronous spin interactions between superconducting charge carriers passing through second-nearest-neighbor K-net vertices may be mediated by ionic cores and bonding orbitals localized through L-net vertices maximally interstitial to such K-net vertices. The topological symmetry of such spin interactions may be modeled by the topological transformation from that shown in Fig. 7 (d) to that in Fig. 7 (e ). The topological transformation shown in Fig. 7 depends on the isotropy of the K net and L net. The region interstitial to the Ln-0 and Cu-0 layers does not participate in the superconducting flow because the K net and L net in this region are anisotropic to the K net and L net interstitial to the Cu-0 and (Ba,Sr)-0 layers. If the states available in the isotropic K-net region are lower
349
in energy than those in the anisotropic K-net region, the charge carriers will prefer the isotropic K-net region. In the absence of an applied electromagnetic field, the K-net vertices topologically describe a set of empty interstitial electronic states below the energy gap, consistent with the reported [65] energy gap in the copper oxide superconductors. The distance between nearest-neighbor K-net vertices in the copper oxide superconductors is about 1.4 A and may approximate a superconducting coherence length comparable to calculated c axis superconducting coherence lengths of about 1.5-2 A by Oh et al. [ 661, and of about 1 A by Deutscher [ 671 and Kresin and Wolf [68]. King [69] hypothesized that a porous skeletal bonding topology can describe the valence bond structure of high T, superconductors, and suggested that the skeletal bonding manifold may relate to the short mean free path and small BCS coherence length (l-2 A) in such superconductors [ 66,681. Phonon interactions sufficient for Cooper pairing are found in neither the high T, copper oxide superconductors [ 701, nor the Al5 superconductors [ 711. However, phonon energies sufficient for Cooper pairing have been identified in Ba-K-Bi-0 [ 721. Figure 3 (a) shows that the bonding structure of Ba-KBi-0 is non-bent (ionic), and is not characterized by localized interstitial valence electrons as in the copper oxide and Al5 superconductors. The Ba-K-Bi-0 bonding structure shown in Fig. 3 (a) is a complete tetrahedral-octahedral (f.c.c.) L net, which may resemble the bond structure of the f.c.c. lead crystal in which phonon coupling has been shown by isotope effect studies [60]. Although the L-net bonding geometry is the same in the phononcoupled and non-phonon-coupled superconductors, phonon effects may be identified with superconductivity only if all the L-net vertices are occupied by ionic core states, as in lead and Ba-K-Bi-0. PROPERTIES OF THE SUPERCONDUCTING
ORBITAL STRUCTURE
The trajectory in the (x,y ) plane of a vertex (A-+A’ ) of the transformation shown in Fig. 6 is elliptical and is described by the equation 2x2-2xy+2y2=1
(1)
where each edge of each triangular surface shown in Fig. 6 has a length of unity, all the vertices lie in the (xy), (x,2) or (y,z) planes, and the volumetric centers of the polyhedra shown in Fig. 6 lie at the origin. The combined trajectories of all the vertices in the (x,y ) plane are shown in Fig. 8. The coordinates of vertex A-A’ at each stage of the transformation in Fig. 6 are shown in Table 1. The trajectories of the other vertices shown in Fig. 6 are given by the following equations:
350
2x2+2xy+2y2=1
(2)
2x2~2xz+2z2=1
(3)
2y2?2yz+2z2=1
(4) A simple description of the topological transformation shown in Fig. 7 could be obtained by a rotation of + x/6 and a translation of +-l/J6 of each triangular surface (180” domain wall) shown in Fig. 7. The topological transformation shown in Fig. 7 could model the topological symmetry of quasi-particle excitations thought by Anderson et al. [ 731 and by Kivelson et al. [ 741 to cause Bose condensation in the strong coupling theories of superconductivity [ 751. Each triangular surface shown in Figs. 6 (a) and 6 (e) and Figs. 7 (a) and 7 (e) rotates by + n/6 and translates by -t l/,/6 within a cylinder whose axis Y .
- _____ . . _-_
.---.
_-._._.____...
_ .._.. _.._.
---._-
--..
_-if__.____
.-____ -.._-
Fig. 8. Trajectory in the (x,y) plane of vertex A+A’ in the synchronous topological transformation.
351 TABLE 1 Transformation
coordinates
Fig.
Polyhedron
x
Y
z
6(a)
Cuboctahedron Icosahedron Inter-icosahedron Quasi-icosahedron Octahedron
l/J2 0+J51/4 JW3 U+J5)/4 l/J2
l/J2 l/2 l/J6 (-1+J5v4 0
0 0 0 0 0
6(b) 6(c) 6(d) 6(e)
passes through the center of the surface and the volumetric center of each octahedron and cuboctahedron shown in Figs. 6 (a) and 6 (e ) and Figs. 7 (a) and 7 (e) . Each triangular surface shown in Fig. 7 displays either a right-handed or left-handed helical motion. Each such motion describes a topical symmetry related to the topical symmetries of all other triangular surfaces by the topological transformation of Fig. 7. Figures 7 (b) and 7 (d) show that the topological transformation of Fig. 7 contains icosahedrally symmetric phases [ 761 based on one of the two possible helical motions (left- or right-handed) of each triangular surface [ 771. (Szasz and co-workers [ 761 have proposed that an icosahedral Jones zone/Fermi surface interaction causes icosahedral distortion or “dynamical frustration” of the oxygen lattice.) There exist two geometrically distinct combinations of such phases based on the two possible helical motions of each triangular surface. The two geometrically distinct combinations yield geometrically equivalent phases like that shown in Fig. 7 (e). The triangular surfaces in Fig. 7 resemble “odd rings”, composed of odd numbers of interatomic bonds, which Rivier hypothesizes are punctured by topologically stable Zz “odd-line” defects (x1 (SO (3) ) = Z,) in metallic and covalent glasses [ 771. Dynamic odd-line defects lead to viscous relaxation in glass, consistent with observations of giant flux creep in the copper oxide superconductors [ 781. The K nets within two superconducting grains may couple through K-net vertices common to each grain [ 791. Decoupling of such K nets may describe a metastable electronic liquid phase [80] or a superconducting glass phase consistent with findings by several researchers (see for example ref. 81) that copper oxide superconductors exhibit metastable glass-like behavior under conditions of applied magnetic field and zero-field cooling. Such decoupling may explain the giant flux creep observed in the copper oxide superconductors 1731. Rivier shows that the energy gap in amorphous silicon can be described by a topological formulation in which the two ground states are characterized by bonding and antibonding states [ 821. The energy gap in the superconductors
352
may be described by a similar procedure in which the two ground states are described by the L net of valence bond states and the K net of antibonding (conduction) states. Applying Bloch’s theorem, the free energy of the superconductor will be a periodic function of the flux triggering the gauge transformation [ 831. If the flux lives on the triangular surfaces shown in Fig. 7, the flux may be quantized by the + x/6 rotation and + l/J6 translation along the odd-line core puncturing each such surface to depict qualitatively the vortices and flux lattice of a type-II superconductor [ 771. Gammel and co-workers [84] and Dolan et al. [56] observed hexatic flux quanta vortices in Y-Ba-Cu-0, and stated that such vortices are comparable to the triangular array of vortices in conventional type-II superconductors under similar conditions. The observed hexatic flux pattern may show that such flux is correlated through the centers of the hexagonally correlated triangular surfaces in Fig. 7. The synchronous motion of such surfaces, as shown in Fig. 7, suggests a “pumping” action responsible for the Meissner effect. Observations by Dolan et al. [56] of “undulations” in chains of anisotropic flux vortices in Y-Ba-Cu-0 may also provide evidence for the topological transformation shown in Fig. 7. CONCLUSION
The present model describes an interstitial valence bond structure and a superconducting structure interstitial to the valence bond structure. Superconductivity depends critically on K-net and L-net isotropy. In the superconducting state, the L-net interstitial valence bond structure can produce the following effects characteristic of superconductivity in copper oxide superconductors: (1) photoemission spectroscopy evidence of Cu3d/02p hybridization, Cu-0 covalent bonding and oxygen dimerization, (2) Hall resistivity of electron vs. hole carriers, (3) oxygen stoichiometry and Cu-0 chains in Y-Ba-Cu-0, (4) Jahn-Teller distortion of Cu-0 half-octahedra, (5) antiferromagnetic and ferromagnetic normal states, (6) three-dimensional anisotropies in critical current, resistivity, upper critical field and vortex structure measurements, (7) microstructural twinning, and (8) increased T,with multilayer structures. The K-net superconducting orbital structure is maximally interstitial to the L-net valence bond structure and can produce the following effects characteristic of superconductivity in copper oxide superconductors: (1) a superconducting energy gap, (2 ) Cooper pairing, (3) short superconducting coherence lengths, (4) an isotope effect, (4) intergrain coupling, (5) metastable glasslike behavior and (6 ) hexatic type-11 flux quanta. The topological model may aid the discovery and design of new higher T,
353
superconducting materials with valence bond structures conforming to the L net and a superconducting orbital structure conforming to the K net.
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