New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Percolative theories of strongly disordered ceramic hightemperature superconductors

Contributed by J. C. Phillips, November 18, 2009 (sent for review October 31, 2009)
Abstract
Optimally doped ceramic superconductors (cuprates, pnictides, etc.) exhibit transition temperatures T _{c} much larger than strongly coupled metallic superconductors like Pb (T _{c} = 7.2 K, E _{g}/kT _{c} = 4.5) and exhibit many universal features that appear to contradict the Bardeen, Cooper, and Schrieffer theory of superconductivity based on attractive electronphonon pairing interactions. These complex materials are strongly disordered and contain several competing nanophases that cannot be described effectively by parameterized Hamiltonian models, yet their phase diagrams also exhibit many universal features in both the normal and superconductive states. Here we review the rapidly growing body of experimental results that suggest that these anomalously universal features are the result of marginal stabilities of the ceramic electronic and lattice structures. These dual marginal stabilities favor both electronic percolation of a dopant network and rigidity percolation of the deformed lattice network. This “double percolation” model has previously explained many features of the normalstate transport properties of these materials and is the only theory that has successfully predicted strict lowest upper bounds for T _{c} in the cuprate and pnictide families. Here it is extended to include Coulomb correlations and percolative band narrowing, as well as an angular energy gap equation, which rationalizes angularly averaged gap/T _{c} ratios, and shows that these are similar to those of conventional strongly coupled superconductors.
All known microscopic theories of metallic superconductors are based on the isotropic Bardeen, Cooper, and Schrieffer (BCS) model of Cooper pairs formed by attractive electronphonon interactions that overwhelm repulsive Coulomb interactions, yet many features of electronphonon interactions in the BCS theory that are confirmed in metals appear to be weak or absent in the infrared spectra of ceramic layered high temperature superconductors (HTSC) (1). The ceramics often exhibit multiple phases and must be doped to be not only superconductive but even metallic. By contrast layered undoped MgB_{2} with T _{c} ∼ 40 K is an “ideal” example of a metallic BCS crystalline superconductor where T _{c} can be calculated quite accurately (2). When MgB_{2} is substitutionally doped with ∼10% Al (on the Mg sites) or C (on the B sites), T _{c} decreases drastically to ∼10 K (3). Many authors have invoked residual spin interactions in the doped state, left over from the magnetic undoped states, as a source of this difference (1). However, it has been known for a long time that electronspin interactions (for instance, with magnetic impurities) drastically suppress T _{c} by breaking S = 0 Cooper pairs (4), and more generally that the insulating antiferromagnetic (AF) and metallic Cooper pair fourparticle plaquette channels are completely independent (5). While ceramics often exhibit multiple phases, recently the data have shifted overwhelmingly in favor of phonon exchange as the only attractive interaction correlated with T _{c} and responsible for ceramic HTSC (6, 7).
An Ockham’s razor, semiclassical alternative to abandoning electronphonon interactions instead abandons translational invariance and the continuum approximation and supposes that currents are carried by discrete molecular wave packets that percolate coherently from dopant to dopant along filamentary interlayer paths, if necessary avoiding entirely insulating or weakly metallic regions associated with AF or strong JahnTeller distortions that destroy large parts of Fermi lines. So far this “zigzag” interlayer percolative model (ZZIP) (8) has not been used to calculate either normalstate or superconductive properties from first principles. However, if one is familiar with the filamentary principles associated with selforganized glassy networks (9, 10), one can use homology arguments to describe material properties (such as T _{c}) very economically. In this way not only was a master function defining strict least upper bounds on T _{c} established (11), but this master function also very successfully predicted for the complex oxyhalide Ba_{2}Ca_{2}Cu_{3}O_{6}F_{2} (apical F) to be 78 K, in excellent agreement with the highest T _{c} value for Ba_{2}Ca_{3}Cu_{4}O_{8}F_{2} reported later, 76 K (12). As shown in Fig. 1, also gives (13) a very good account of the largest T _{c}s in noncuprate ceramic HTSC, where 〈R〉 is the number of chemical bonds/atom. It is clear that the connectivity and stability of ionic host lattice networks are marginal for 〈R〉 = 2, where the largest T _{c} is reached. Topological network considerations have explained phase diagrams of superionic conductors (14). Percolative rehydroxylation is characteristic of ceramics and can be used to date them (15).
Quantum Percolation and AntiHermitian Interactions
In the ZZIP model the mechanism of quantum percolation was pictured semiclassically, in terms of discrete molecular wave packets that percolate coherently from dopant to dopant. Here we will outline a primitive Hamiltonian matrix model that could eventually be used for numerical quantum simulations. Because T _{c} correlates well with shifts δω _{LO} in the longitudinal optical phonon frequency near the (100) Brillouin zone boundary (11), one anticipates that the spectra of the localized dopant states are broadened by δE _{1} and their centers displaced from E _{F} by an amount δE _{2}, with both δE _{1} and δE _{2} of order ω _{LO}. Neglecting the δE _{1} broadening, these states can be represented by a twostate, twodopant model Hamiltonian with and (in units of δE _{2}). The critical offdiagonal matrix elements actually reflect dopantdopant interactions mediated by filamentary wave packets propagating in the metallic planes (CuO_{2} planes for the cuprates). The key feature of those interactions has been identified in very extensive studies of electron pair interactions in small molecules.
Manyelectron Coulomb interactions reduce Fermion bandwidths, for instance, by screening ionsingle electron interactions, and many methods have been proposed for treating these screening effects formally (16). However, classical percolation is already a complex process, and to describe it simply one can turn to recent studies of twoelectron excited states of manyelectron atoms and molecules, without doing configuration interaction, by using an antiHermitian contracted Schrodinger equation (ACSE) to study twobody charge density matrices (17). The ACSE is an optimized reduction of the enormously complex problem of quantum configuration interaction. In principle the twobody charge density matrices are related to the three and fourbody charge density matrices, which in turn are related to still higherorder charge density matrices. Moreover, because interference effects are not included, classical correlation functions in general do not correspond to actual Fermionic wave function solutions of the Schrodinger equation itself. Nevertheless, it has been found in many examples that restricting the higher order corrections to the leading antiHermitian thirdorder term yields excellent results, at least an order of magnitude better than obtained from the full (Hermitian + antiHermitian) thirdorder term (17). Classically speaking, the antiHermitian interaction corresponds to damping of the pair interaction by a third particle; apparently this damping erases most of the higher order Hermitian interactions, which is a great simplification.
Returning to our twodopant Hamiltonian, we parameterize and as A + Bi, where A is the Hermitian part and Bi is the antiHermitian part (still in units of δE _{2}). According to Mazziotti (17), A ≪ B, and we set A to zero. Now the eigenvalues of our twostate model are given by E ^{2} = 1  B^{2}, and the bandnarrowing effect is obvious without further calculation for B < 1. For B > 1, the localized dopant states become unstable, and one can say that a percolative metalinsulator transition occurs at B = 1.
The main features of HTSC phase diagrams are evident from this simple twostate model. For B ≪ 1, low dopant density x and δE _{1} disorder have produced an insulating dopant impurity band. For moderate x, dopantdopant spacing is reduced, the filamentary Coulomb fluctuations become significant, and 0 < 1  B(x) ≪ 1. The broad impurity density of states peak at E _{F} narrows, and when it becomes narrow enough, Cooper pairs can form in this band, leading to HTSC. For B(x) > 1, a normal metallic state arises, filamentary/dopant localization ceases, and the impurity band peak in N(E) for E near E _{F} is lost.
Percolative Angular Gap Equation
The foregoing model sketches dopant percolation in an isotropic context. In actuality, the dopants are embedded in a crystalline host, where there is present at higher temperatures a strongly anisotropic dwave pseudogap associated with nearest neighbor CuO interactions in the CuO_{2} plane. This pseudogap stabilizes the host lattice, and it itself is the result of rigidity percolation. The latter is the origin of the phenomenologically successful model of shown in Fig. 1. Electronic charges must percolate through this marginally stable pseudogap maze.
In a double charge/rigidity percolation model there are two global factors that should be considered: the planar anisotropy of the energy gap E _{g}(φ) and the transition probability P(φ) that the gap will percolate at a given azimuthal angle φ. Quantum percolation (8) must occur coherently on each percolative filament, so for quantum percolation one should regard P(φ) as a probability amplitude, with characteristic analytic properties. Currents will be carried on each filament by wave packets composed of Bloch waves. The wave packets will have average energies E and average velocities v. In ARPES experiments these wave packets are projected on the momenta k of high energy photoemitted electrons, and the percolative supercurrents are separated from insulating pseudogap states (probably associated with JahnTeller distortions) by defining Fermi arcs. When this projective kspace method of analysis is used, it is often difficult to separate the pseudogap and the superconductive gap states, except in underdoped samples where the latter is much smaller than the former (18, 19). [At optimal doping, as in (7), the pseudogap and the superconductive gap appear to be nearly the same.] However, even after E _{g}(φ) has been assigned a dwave angular dependence , to establish a quantum percolation model one must still determine the probability amplitude P(φ) that the gap will percolate phasecoherently at a given angle φ. The function P(φ) can be regarded heuristically as akin to an interdopant transfer coupling, similar to the transfer couplings T used in phenomenological descriptions of the Josephson effect.
Here P(φ) is assigned a simple functional dependence for all cuprates, the basic idea being that metallic and superconductive wave packets will percolate most effectively if they avoid the φ = 0 CuCu nearest neighbor {10} directions where the pseudogap E _{p}(φ) and superconductive gap E _{g}(φ) are strongest (20). These effective percolation directions are the φ = π/4 next nearest neighbor {11} directions where electronphonon interactions are weakest, thus P(φ) should be complementary to E _{g}(φ), for example, P(φ) = sin^{n}2φ, where n should be optimal.
The model is tested by studying the detailed electronic structure of Fermienergy dopant resonances (type B, predicted in Sec. IX of ref. 11) as revealed by recent largescale scanning tunneling microscope (STM) experiments (21). As expected, the resonant amplitude of the superconductive gap is greatly reduced in the φ = 0 nearest neighbor {10} directions where the pseudogap E _{p}(φ) is strongest. However, only a modest reduction is found in the φ = π/4 next nearest neighbor {11} directions where electronphonon interactions are weakest, which is exactly what one expects from quantum percolation of superconductive wave packets (see Fig. 2). Although this experiment does not display either E _{p,g}(φ) or P(φ) fully, it does confirm qualitatively their complementary character.
The gap equation is obtained by coherently averaging over φ to obtain E _{g} = I _{2}/I _{1}, where I _{2} = ∫dφE _{g}(φ)P(φ) and I _{1} = ∫dφP(φ) with 0 ≤ φ ≤ π/4. We find . Unambiguous values of E _{g} and are not obtainable from optical data (1), which primarily show a polaronic midinfrared data band (22). These ceramic midinfrared data merely reinforce the large qualitative difference between layered ceramics (La_{2x}Sr_{x}CuO_{4}) and cubic Ba_{1x}K_{x}BiO_{3} known from their phase diagrams. The former has a parabolic (nearly semicircular) T _{c}(x), while the latter has a triangular T _{c}(x), with a maximum T _{c} at the metalinsulator transition and a normal BCStype infrared spectrum with E _{g}/kT _{c} = 3.2 (23), and a metalinsulator transition at x = 0.3 in Ba_{1x}K_{x}BiO_{3}, which has been interpreted in terms of bipolaron formation (24).
Because of the selforganized internal structure, dopant configurations can adapt to thermal fluctuations, and the dissipative scattering (25) characteristic of large gap/T _{c} ratios (Pb, T _{c} = 7.2 K, E _{g}/kT _{c} = 4.5) is absent from Ba_{1x}K_{x}BiO_{3} (E _{g}/kT _{c} = 3.2, weak coupling). However, in a ceramic HTSC, Boyer et al. (19) found , or E _{g}/kT _{c} = (10.6 ± 2.9)/3I_{1} = 4.4 ± 1.3 if one chooses n = 2. The fit with n = 2 can be justified topologically: Suppose each time a wave packet leaves an interlayer dopant to percolate in a metallic plane, it adds a sin 2φ factor to P(φ). Then to “reset” itself in an equivalent state, it will have to return to another (statistically equivalent) interlayer dopant, serially experiencing a second factor sin 2φ. Thus n = 2 in P(φ) could be an intrinsic feature of interlayer effects in the ZZIP model. This value of E _{g}/kT _{c} = 4.4 is similar to that of Pb.
It appears that the layered structure of ceramic HTSC has led to an entanglement of dopant and JahnTeller distortions that is much more complex than is apparent from the data on cubic Ba_{1x}K_{x}BiO_{3}. This entanglement is imaged in the STM study (Fig. 2) of a rare layered ceramic Fermienergy pinning resonance (21), which is probably an apical oxygen vacancy. This resonance is stable and observable by STM, whereas most Fermienergy pinning interstitial O dopant resonances are too fragile to be imaged with an meV potential. It may be that the spatial and spectral entanglement of dopantcentered superconductive gaps with pseudogaps in ceramic superconductors often renders them intrinsically inseparable.
Footprints of ZZIP
The doubly percolative, dopantcentered ZZIP network has left many large footprints (12). Thus extended xray absorption fine structure (EXAFS) studies of lattice distortions (26) have revealed a third subT _{c} phase transition (in addition to the JahnTeller transition at T ^{∗} and the superconductive transition at T _{c}) shown in Fig. 3. This transition at T = T _{OP} can be explained as a glass transition of the ZZIP network (27); it is especially interesting because, unlike the other two transitions, it cannot be modeled with a Landau order parameter, and it is completely inexplicable with meanfield toy model Hamiltonians (Hubbard and tJ models). (It could, however, be compared to formation of CysCys disulfide bonds in proteins.)
The planar resistivity ϱ _{ab}(T) is most linear in T at and near optimal doping (12, 28). The dopant networkforming interactions, although individually weak, are cumulatively very strong because the dielectric energy gained by screening internal electric fields increases as the network conductivity increases due to dopant selforganization (12), and this explains the maximization of the Tlinearity of ϱ _{ab}(T) near optimal doping (12, 28). The superconductive transition itself can be narrowed by annealing (29). The classic zerofield work of Ando et al. (28) has recently been extended to high magnetic fields with spectacular results (30). Ando et al. (28) found that there is a T > 100 K regime of ϱ _{ab}(x,T) that is linear in T at x = x _{c} = 0.16 in zero field, which coincides with , which also occurs in La_{2x}Sr_{x}CuO_{4} at x = x _{c}. Cooper et al. (30) found the much simpler La_{2x}Sr_{x}CuO_{4} pattern shown in Fig. 4, where the linear regime has been shifted to . This highfield linear regime extends at fixed x vertically all the way from T = 0 right up to the highest Ts studied (200 K). The vertical linear T regime is shaped like an hourglass, wide at T = 0 and T = 200 K, and narrow at T ∼ 100 K. This narrowing is due to the formation of pseudogap (JahnTeller distorted) islands, which first percolate at T ^{∗} ∼ 100 K and then stabilize between T ^{∗} and T _{c} (see Fig. 3). The low temperature linearity requires a reformation of the nanodomain network into a doubly percolative network, with two kinds of dopants (11).
A combined description of refs. 28 and 30 follows from thinking of the hardwired ZZIP network as a combination of irrotational filamentary open and solenoidal vortex closed components (11). In zero field (26) the competition between these two topological components is relatively simple (independent of x) for T > T ^{∗} = 100 K, but below 100 K it becomes more complex as pseudogap and normal metallic nanodomain walls compete in partitioning CuO_{2} planes. Application of a high magnetic field in overdoped samples (x > x _{c}) leaves only the solenoidal vortex closed components, which absorb some dopant electronic kinetic energy in purely static vortex motion. This orbital reorganization effect both shifts x _{c} from 0.16 in the zerofield case to in high fields and is also responsible for fixing the T linear behavior of sliding vortices at this x = 0.185(5) value up to 200 K, giving the striking hourglass structure shown in Fig. 4.
Conclusions
We have sketched a twodopant model Hamiltonian that shows that the onset of Fermionic charge percolation can be represented by an antiHermitian dopantdopant transfer interaction. By establishing an anisotropic percolative gap equation and identifying a new subgap glassy phase transition, ZZIP builds a bridge between theories of isotropic metallic and anisotropic ceramic superconductivity. Percolation shows up even in the cubic perovskites SrTiO_{3} (31) and SrRuO_{3} (32). Although SrTiO_{3} (STO) is not a high temperature superconductor, it is a ceramic that exhibits superconductivity with at very low carrier densities n _{3D} ∼ (10^{18}–10^{20})/cm^{3}, in both chemically doped bulk and electricfield induced surface accumulation layers (31). The coincidence of the chemically doped and electricfield induced surface values could be accidental, but there is a natural percolative explanation for this coincidence: is reached when the percolative filaments begin to merge, just as in the EXAFS data for HTSC (26, 27).
There is also evidence for percolative conductivity at high temperatures in SrRuO_{3} (SRO): According to Herranz et al., “On the other hand, at temperatures above the ferromagnetic transition (150 K), the temperature dependence is linear, but remarkably it is found that the slope of ρ(T) increases as disorder increases. This latter result is radically dissimilar to what is observed in other complex electronic systems, where disorder commonly induces a reduction of the slope” (32). Because the percolative network is fragile, it can be disrupted thermally, and its thermal fragility will increase with increasing disorder, as measured by the resistivity minimum temperature.
Overall it is difficult to see how a successful theory of HTSC can avoid recognizing its percolative nature. In fact, when proper curvilinear cuprate percolative basis states are forcibly projected onto Bloch plane wave states, it becomes difficult to separate the pseudogap from the superconductive gap. This causes Fouriertransformed scanning tunneling evidence for the superconductive transition to weaken or disappear, but with one important exception. This is the difference between the slopes of the temperature dependences of Fourierderived q scattering from k regions to k + q regions that have the same or opposite signs of the dwave order parameter (Fig. 4 of ref. 33). This is the Fouriertransformed qualitative analogue of the complementary angular behavior (21) of E _{p,g}(φ(k)) and P(φ(q)) discussed above analytically in connection with the percolative gap equation.
Footnotes
 ↵ ^{1}To whom correspondence should be addressed. Email: jcphillips8{at}comcast.net.

Author contributions: J.C.P. designed research, performed research, and wrote the paper.

The author declares no conflict of interest.
References
 ↵
 ↵
 ↵
 ↵
 Phillips JC
 ↵
 Dzyaloshinskii IE,
 Larkin AI
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Shimizu S,
 Mukuda H,
 Kitaoka Y,
 et al.
 ↵
 ↵
 ↵
 Wilson MA,
 et al.
 ↵
 ↵
 ↵
 McElroy K,
 et al.
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Zhang CJ,
 Oyanagi H
 ↵
 Phillips JC
 ↵
 ↵
 ↵
 Cooper RA,
 Wang Y,
 Vignolle B,
 et al.
 ↵
 ↵
 ↵
 Lee J,
 Fujita K,
 Schmidt AR,
 et al.
Citation Manager Formats
Sign up for Article Alerts
Article Classifications
 Physical Sciences
 Physics