In lightly hole-doped cuprates, scanning tunneling microscopy reveals bond-centered electronic domains four lattice constants wide, organized as a central hole-rich ladder flanked by hole-poor ladders. Deutscher and de Gennes (2007) proposed that these domains arise from local hole pairing: when two doped holes occupy a short Cu–O–Cu or O–Cu–O–Cu–O segment, the Cu–O bonds contract and strongly enhance the oxygen–copper transfer integral, binding the pair once a simple criterion on that transfer integral, an on-cluster two-hole energy, and a band parameter is satisfied; pair formation coupled to a charge density wave then generates the observed 4a-periodic ladder pattern, superconductivity emerges by pair propagation along the hole-rich ladders, and right-angle inter-ladder coupling is conjectured to produce the macroscopic d-wave state. The authors themselves flagged the mechanism as tentative. The open problem is to determine whether the mechanism operates in real lightly doped cuprates — through realistic material parameters for the pairing criterion, a controlled treatment of the strictly one-dimensional, strongly disordered ladders and their couplings, and experimental discrimination — and whether it quantitatively accounts for the observed domain patterns, the ladder-localized gap scale of a few tens of meV, the Fermi arcs, and the macroscopic d-wave coherence.

Background

The cuprate high-temperature superconductors are layered copper oxides whose parent compounds are antiferromagnetic Mott insulators — insulators in which conduction is blocked not by filled atomic bands but by the repulsion between electrons localized on the copper sites. Introducing holes (positive charge carriers) into the copper-oxide CuO2 planes — the square lattice of copper atoms bridged by oxygen atoms on which superconductivity occurs — produces superconductivity at transition temperatures far above those of conventional superconductors. The lightly doped (underdoped) regime, between the insulator and the composition of maximal transition temperature, hosts the most debated phenomena of the field: the pseudogap (a partial suppression of low-energy electronic states appearing well above the superconducting transition temperature), Fermi arcs (arc-shaped fragments of a Fermi surface — the closed contour of electron-occupied momenta — where a full closed contour is never completed), and short-range charge order (spatial modulations of the hole density that persist only over a few lattice spacings).

A key experimental observation came from scanning tunneling microscopy/spectroscopy (STM/STS), which measures the tunneling current from a sharp tip at atomic resolution as a function of position and voltage. Kohsaka and co-workers (Science, 2007) applied tunneling-asymmetry imaging — mapping the difference between the conductance for injecting and for extracting electrons, itself a signature of strong correlations — to two lightly hole-doped cuprates, Ca1.88Na0.12CuO2Cl2 and dysprosium-substituted Bi2Sr2CaCu2O8+δ. They found intense spatial variations located primarily on the planar oxygen sites, arranged in a bond-centered pattern (centered on the Cu–O chemical bonds rather than on atomic sites), without long-range order but organized into unidirectional domains four lattice constants (4a) wide, with the two possible domain orientations, at right angles to each other, distributed randomly over the surface. Within each domain a central ladder — a strip two lattice spacings wide, centered on a column of oxygen atoms and its two flanking copper columns — is hole-rich, while the neighboring ladders are hole-poor; superconducting-gap-like spectral features sit on the central ladder. The established theoretical frame for such inhomogeneity was the stripe picture — self-organized lines of high hole density separating antiferromagnetic regions (Zaanen and Gunnarsson, 1989) — generalized to electronic liquid-crystal phases of a doped Mott insulator, that is, partially ordered electronic states intermediate between a uniform metal and a fully ordered lattice, of which stripes are one example (Kivelson, Fradkin and Emery, 1998) — in which the lattice plays no primary role. Independent evidence for nanoscale charge inhomogeneity existed from nuclear quadrupole resonance — a spectroscopy that measures the electric-field gradient at nuclear sites and is therefore sensitive to the local charge environment (Ofer et al., 2006) — and from X-ray signatures of spatially modulated Mottness — position-dependent strength of the correlation-driven insulation — in a related compound (Abbamonte et al., 2005).

Against this background, Deutscher and de Gennes (C. R. Physique, 2007) — the last article of de Gennes, completed by Deutscher after de Gennes's death and published as-is by the editorial committee, which wrote that the future will tell whether its suggestions are relevant — proposed a spatial, lattice-based interpretation. Their postulate: when each of the two copper atoms of a Cu–O–Cu segment carries a doped hole, the Cu–O distances contract, and because the transfer integral (electronic hopping matrix element) between an oxygen and a neighboring copper site, tOCut_{\mathrm{OCu}}, involves the overlap of copper 3d and oxygen 2p orbitals twice, it is extremely sensitive to the bond length and increases strongly upon contraction. They estimated the binding of a two-hole spin singlet (the total-spin-zero two-particle state, the channel relevant for superconductivity) on the minimal three-site cluster with a variational ansatz — a trial wavefunction built from four candidate singlet configurations — where UU denotes the large unperturbed energy of the configurations with both holes on the cluster. The eigenvalue condition is

ϵ(ϵ−U)=4 tOCu2\epsilon(\epsilon - U) = 4\,t_{\mathrm{OCu}}^{2}

which for large UU gives the pair energy ϵpair=−4 tOCu2/U\epsilon_{\mathrm{pair}} = -4\,t_{\mathrm{OCu}}^{2}/U. Comparing with two independent holes at the bottom of the band, with half band width ϵ0=4t0\epsilon_{0} = 4t_{0}, the bound pair wins when

tOCu>(U t0/2)1/2.t_{\mathrm{OCu}} > \left(U\,t_{0}/2\right)^{1/2}.

Recognizing that doped holes prefer oxygen sites (the high ionization potential of Cu++ disfavoring copper), they proposed an alternative five-site cluster, an oxygen-centered O–Cu–O–Cu–O segment with the two holes on the edge oxygens, whose energy gain from admixture of intermediate configurations is likewise proportional to tOCu2t_{\mathrm{OCu}}^{2}.

The mesoscale picture runs as follows. Pair formation, bond contraction (dimerization), and a charge density wave — a periodic modulation of the hole density — are coupled and reinforce each other, so a hole-rich contracted central bond depletes its neighbors, generating alternating hole-rich and hole-poor ladders with period 4a, as observed. Pairs cannot propagate along the contracted segment's axis but can in the perpendicular direction, making the central ladders conducting and potentially superconducting. At the roughly 10% hole doping of the experiments, a domain needs about twenty copper sites to hold one hole pair, so the slightly elongated domains observed contain a few pairs; if the holes concentrate on the central ladder, its local density is near optimal doping, so a superconducting gap of a few tens of meV is expected there — as observed — while the edge ladders stay near the antiferromagnetic composition. Because the ladder pattern is highly disordered and breaks translational and rotational symmetry, well-defined quasiparticles (long-lived particle-like excitations with a sharp momentum) are not guaranteed; the authors argue that any coherence should survive only for momenta at 45 degrees to the ladder axes, consistent with the Fermi arcs — arc-shaped remnants of a Fermi surface seen by angle-resolved photoemission spectroscopy (ARPES) in lightly doped cuprates, where a full closed Fermi contour is never completed. Macroscopic superconductivity requires coupling between ladders oriented at right angles, which the authors conjecture produces the observed gap anisotropy, with the strongest gap along the ladder axes, and possibly the d-wave symmetry of the macroscopic superconducting order parameter — the experimentally established fact that the cuprate superconducting gap changes sign between the two perpendicular Cu–O bond directions.

One constraint from standard theory bears directly on this mechanism. Since the work of Zhang and Rice (1988), the accepted picture of hole-doped cuprates is that doped holes reside primarily on the oxygen sites, forming spin singlets with the copper spins (Zhang–Rice singlets). This disfavors the copper-centered three-site variant of the cluster mechanism and directs attention to the oxygen-centered five-site variant — a qualification the source paper itself introduced. The authors stressed that the whole idea is very tentative, and the quantitative pairing criterion above has not been evaluated with material-specific parameters in the literature cited here.

Problem Statement

Determine whether the Deutscher–de Gennes local-pair mechanism actually operates in real lightly doped cuprates and quantitatively accounts for the observed emergence of superconductivity there. Four linked results jointly decide this single objective:

(1) Parameter criterion. Determine realistic values of the oxygen–copper transfer integral tOCut_{\mathrm{OCu}} — including its enhancement by hole-induced Cu–O bond contraction — of the on-cluster two-hole energy UU, and of the band parameter t0t_{0} (half band width ϵ0=4t0\epsilon_{0}=4t_{0}), for candidate lightly doped cuprates such as the studied Ca1.88Na0.12CuO2Cl2 and Bi-based compounds, and establish whether the pairing criterion

tOCu>(U t0/2)1/2t_{\mathrm{OCu}} > \left(U\,t_{0}/2\right)^{1/2}

holds — in particular for the oxygen-centered five-site variant selected by the Zhang–Rice preference of doped holes for oxygen sites — with uncertainties small enough for a robust verdict on whether local hole pairs form.

(2) Ladder-localized superconductivity. Determine how the strictly one-dimensional character and the strong disorder of the hole-rich central ladders affect the magnitude and robustness of a superconducting gap on individual ladders (nominally of a few tens of meV), and whether coherent quasiparticles can form on them at all.

(3) Emergent macroscopic order. Determine whether the weak, spatially inhomogeneous inter-ladder couplings — especially between ladders at right angles — can establish long-range phase coherence, and whether they quantitatively yield the observed macroscopic d-wave order parameter and gap anisotropy.

(4) Experimental linkage. Test experimentally the proposed causal link between local pair formation and (i) the 4a-periodic bond-centered domain patterns, (ii) the ladder-localized gap scale, and (iii) the directional quasiparticle coherence that leads to the Fermi arcs.

The question is decided by establishing or refuting the mechanism; (1) is the microscopic foundation, (2) and (3) are the mesoscale consequences that connect it to the observed phenomenology, and (4) is the empirical arbiter of the whole chain.

References

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