Origin of the anomalously large superconducting gap ratio $2\Delta(0)/k_BT_c\approx 6$ in cuprates with an extended saddle-point singularity (Abrikosov, Campuzano and Gofron, 1993)

A 1993 angle-resolved photoemission study by Abrikosov, Campuzano and Gofron reported that the CuO $_2$ -plane band of the cuprate superconductors YBa $_2$ Cu $_3$ O $_{6.9}$ and YBa $_2$ Cu $_4$ O $_8$ is nearly flat along the $\Gamma$ -Y direction of the Brillouin zone over roughly half the zone - an 'extended' saddle-point (van Hove) singularity - with the saddle point only an estimated 5-20 meV below the Fermi level. Modeling this spectrum by a dispersion quadratic in $k_x$ and piecewise quadratic in $k_y$ (a huge effective mass over a finite momentum range) yields a density of states diverging as $1/\sqrt{\epsilon-\epsilon_0}$ , stronger than the logarithmic divergence of an ordinary two-dimensional saddle point. Within BCS theory with a momentum-independent s-wave pairing interaction, the authors solved the gap equation analytically in both regimes $T_c\ll\mu-\epsilon_0$ and $T_c\gg\mu-\epsilon_0$ , obtaining $2\Delta(0)/k_BT_c=3.52$ and $3.795$ respectively - both far below the experimentally observed $\approx 6$ - and left open the origin of this anomaly, suggesting strong-coupling effects, alternative gap symmetries, or more sophisticated pairing interactions as possible resolutions, with auxiliary topics noted in connection with the work (the imprecise Fermi-level offset $\mu-\epsilon_0$ , the unanalyzed isotope effect, the deeper CuO $_2$ -plane band, and the normal-state consequences of the spectrum). Later work refuted parts of the scenario - the YBa $_2$ Cu $_3$ O $_{6.9}$ near-Fermi-level peak was reinterpreted as a surface state, $d_{x^2-y^2}$ gap symmetry was established, and Migdal-Eliashberg strong-coupling analysis showed that even a strong extended singularity cannot reproduce the observed $T_c$ enhancements and isotope-effect reductions - but the core anomaly, a microscopic account of $2\Delta(0)/k_BT_c\approx 6$ together with high $T_c$ and a near-vanishing isotope effect in these band structures, remains without a generally accepted explanation.

Background

Superconductivity - the lossless flow of electrical current below a critical temperature $T_c$ - occurs in the copper-oxide (cuprate) ceramics at temperatures up to about 130 K at ambient pressure, far above those of conventional superconductors. In the cuprates YBa $_2$ Cu $_3$ O $_{6.9}$ (hereafter Y123, $T_c\approx 90$ K) and YBa $_2$ Cu $_4$ O $_8$ (Y124), superconductivity arises in the CuO $_2$ planes (the copper-oxide sheets that host the superconducting electrons). In BCS (Bardeen-Cooper-Schrieffer) theory, electrons form Cooper pairs through an effective attractive interaction, opening a gap $\Delta$ in the electronic excitation spectrum at the Fermi level. A celebrated universal prediction is the ratio of twice the zero-temperature gap to $k_BT_c$ : $2\Delta(0)/k_BT_c=3.53$ for an isotropic s-wave gap (one with the same magnitude in all momentum directions) in the weak-coupling limit, independent of material parameters. The cuprates violate this badly: tunneling and photoemission measurements give $2\Delta(0)/k_BT_c\approx 6$ in optimally doped Y123 and Bi $_2$ Sr $_2$ CaCu $_2$ O $_8$ (here 'doping' is the mobile-carrier concentration set by chemical substitution, and 'optimal' the value at which $T_c$ is maximal), and up to $\sim 8$ in some compounds. The oxygen isotope effect - the shift of $T_c$ upon replacing $^{16}$ O by $^{18}$ O, conventionally parametrized as $T_c\propto M^{-\alpha}$ with $\alpha=0.5$ in simple phonon-mediated superconductors - is also anomalous: $\alpha$ is nearly zero at optimal doping in YBCO (the yttrium-barium-copper-oxide family, i.e. Y123 and Y124).

One candidate explanation available in 1993 involved van Hove singularities. A saddle point of the band dispersion $\epsilon(\mathbf{k})$ - a point where the gradient of $\epsilon$ vanishes but the curvature has opposite signs along the two momentum directions - produces a singularity in the electronic density of states (DOS). In two dimensions an ordinary saddle point gives a logarithmic divergence, $\nu(\epsilon)\propto\ln(4E/|\epsilon-\epsilon_0|)$ , where $\epsilon_0$ is the saddle energy and $E$ a band cutoff. If the Fermi level $\mu$ lies within reach of $\epsilon_0$ , the pairing kernel is effectively strengthened and $T_c$ is enhanced. Angle-resolved photoemission spectroscopy (ARPES) - which measures the electronic dispersion directly by ejecting electrons with photons and resolving their energy and emission angle - had found in early studies of the cuprates a narrow peak just below the Fermi level whose position barely moved along certain momentum cuts, suggesting an unusually flat band.

Abrikosov, Campuzano and Gofron (1993) reported ARPES measurements on Y123 and Y124 showing that the CuO $_2$ -plane band is nearly dispersionless along the $\Gamma$ -Y direction (from the zone center $\Gamma$ to the zone-edge point Y) over about half the Brillouin zone, with a saddle point at Y at a binding energy below the roughly 30 meV instrumental resolution, estimated from fits at about 10 meV below the Fermi level (uncertainty range 5-20 meV) - an 'extended' saddle point, in which the band is flat over a finite region of momentum space rather than only in the immediate vicinity of a quadratic extremum. They modeled the dispersion as quadratic in $k_x$ (mass $m_x$ ) and piecewise quadratic in $k_y$ - a very large mass $M$ over a range $k_y<k_{y0}$ and a much smaller mass $m_y$ beyond, with $M\gg m_y$ - and derived the DOS in the intermediate energy window $\frac{k_{y0}^2}{2M}\ll\epsilon-\epsilon_0\ll\frac{k_{y0}^2}{2m_y}$ :

\nu(\epsilon)=\frac{k_{y0}}{(2\pi)^2 d}\sqrt{\frac{m_x}{2}}\,\frac{1}{\sqrt{\epsilon-\epsilon_0}},

per spin, with $d$ the interlayer spacing - a square-root divergence, stronger than the logarithm of an ordinary saddle point.

Within BCS theory with a momentum-independent s-wave point interaction of strength $g$ , they solved the gap equation analytically in the two regimes set by the offset $\mu-\epsilon_0$ . For an ordinary logarithmic saddle point one already finds $2\Delta(0)/k_BT_c=3.52$ ; for the extended saddle point, in the far regime $T_c\ll\mu-\epsilon_0$ the ratio is again $2\Delta(0)/k_BT_c=2\pi/\gamma=3.52$ (with $\gamma$ the exponential of Euler's constant) and $T_c=\frac{8\gamma}{\pi}(\mu-\epsilon_0)e^{-1/\lambda}$ with $\lambda=\frac{g\sqrt{2m_x}\,k_{y0}}{(2\pi)^2 d\sqrt{\mu-\epsilon_0}}$ ; in the near regime $T_c\gg\mu-\epsilon_0$ the ratio rises only to $2\Delta(0)/k_BT_c=3.795$ , and $T_c=(1.346)^2\frac{g^2\,2m_x k_{y0}^2}{(2\pi)^4 d^2}$ becomes independent of $\mu-\epsilon_0$ and of the boson energy cutoff, so the isotope effect vanishes in this limit and $T_c$ of order 100 K follows for couplings of order 0.1. Thus, even with the stronger square-root divergence, the extended saddle point within the s-wave point-interaction model leaves the gap ratio at essentially its BCS value - far below the observed $\approx 6$ .

The paper therefore left the origin of the large ratio $2\Delta(0)/k_BT_c\approx 6$ unresolved, suggesting that its explanation may require more sophisticated pairing mechanisms, strong-coupling effects, or alternative gap symmetries. Auxiliary topics discussed in connection with the work include pinning down $\mu-\epsilon_0$ experimentally (it selects the theoretical regime), analyzing the isotope effect, treating the deeper CuO $_2$ -plane band seen at higher binding energy, and computing the normal-state properties implied by the extended-saddle-point spectrum. Subsequent developments - the establishment of $d_{x^2-y^2}$ gap symmetry, the reinterpretation of the Y123 near-Fermi-level peak as a surface state, and quantitative bounds on what a DOS singularity alone can achieve - reshaped this question without closing it.

Problem Statement

Determine the microscopic origin of the anomalously large superconducting gap ratio $2\Delta(0)/k_BT_c\approx 6$ (computed from the maximum, antinodal gap), documented in optimally doped YBa $_2$ Cu $_3$ O $_{6.9}$ and Bi $_2$ Sr $_2$ CaCu $_2$ O $_8$ (and up to $\sim 8$ in some cuprates), taken jointly with the high transition temperatures ( $T_c\approx 80$ - $90$ K) and the near-vanishing oxygen-isotope exponent at optimal doping, given that ARPES finds the CuO $_2$ -plane band nearly flat along the $\Gamma$ -Y direction - an extended saddle-point (van Hove) singularity close to the Fermi level, solidly established in YBa $_2$ Cu $_4$ O $_8$ , while for bulk YBa $_2$ Cu $_3$ O $_{6.9}$ the near- $E_F$ peak was later reinterpreted as a surface state - and that the BCS treatment of this spectrum with a momentum-independent s-wave point interaction yields $2\Delta(0)/k_BT_c=3.52$ - $3.795$ in both regimes $T_c\ll\mu-\epsilon_0$ and $T_c\gg\mu-\epsilon_0$ . Concretely: identify the pairing physics that reproduces all three observables simultaneously in a single quantitative account grounded in the measured band structure. Auxiliary topics discussed in connection with the source work - the bulk Fermi-level offset $\mu-\epsilon_0$ (estimated at 5-20 meV at the time, which decides the applicable theoretical regime; for Y123 the original near-Fermi-level peak was later reinterpreted as a surface state, so the bulk offset remains to be established), the possible role of the deeper CuO $_2$ -plane band, and the normal-state consequences of the extended-saddle-point spectrum - are background context for this account, not joint completion requirements.

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