Quantitative confrontation of Abrikosov's extended saddle-point, weak-screening phonon model of cuprate superconductivity with real materials

A. A. Abrikosov's 2000 theory of high-Tc cuprate superconductivity combines extended saddle-point singularities in the electronic spectrum, a weakly screened (small-momentum) phonon-mediated attraction $V(k)=g[\kappa^2/(k^2+\kappa^2)]^n$ with a phenomenological exponent $n>1$, and a small short-range repulsion $U$. The theory accounts semi-quantitatively for the high transition temperatures, the suppressed oxygen and negative copper isotope effects, the coexistence of even (s-wave-like) and odd (d-wave-like) order-parameter solutions, the inelastic-neutron-scattering peak, and the positive curvature of the upper critical field -- but it leaves five quantitative confrontations with real materials unresolved: whether the even/odd selection inequality $U>g[\kappa^2/(2p_0^2)]^n$ is satisfied in actual cuprates and dopings; under what experimental conditions the predicted impurity-driven odd-to-even transition occurs; whether the assumed weak-screening interaction form (the exponent $n$ and screening length $\kappa$) holds in real materials; what the coupling constant $g_2$ and frequency $\omega_{02}$ of the low-frequency copper phonons invoked in the negative copper isotope effect are, and which phonon modes scatter rather than pair; and whether the realistic inter-saddle hopping $\eta$ places any material in the asymptotic regime where $H_{c2}$ is strongly enhanced. None of the five parts has been resolved in the later literature. The framework lost mainstream attention without a dedicated quantitative refutation or confirmation, while adjacent results (phase-sensitive establishment of d-wave pairing, isotope-effect theory in other frameworks, disorder studies) address the surrounding physics from different premises. The problem remains open.

Background

Discovered in 1986, the high-temperature superconducting (high-$T_c$) cuprates are layered oxides whose superconductivity resides in weakly coupled CuO$_2$ planes. By the early 1990s, angle-resolved photoemission spectroscopy (ARPES -- a technique that measures how the electron energy depends on momentum) had found that in several cuprates the quasiparticle dispersion $\epsilon(k_x,k_y)$ contains unusually flat, quasi-one-dimensional regions near the Fermi level at the saddle points $(\pi,0)$ and $(0,\pi)$ of the square Brillouin zone. Abrikosov called this an "extended saddle point singularity": the density of states behaves as $(\epsilon-\epsilon_0)^{-1/2}$, a much stronger divergence than the logarithmic one of an ordinary two-dimensional van Hove singularity (a momentum where the electron group velocity vanishes so that the density of states is non-analytic).

In a series of papers culminating in the 2000 source paper, Abrikosov proposed a phonon-based theory built on three ingredients. First, the extended saddle point singularities enhance the density of states and modify the BCS (Bardeen-Cooper-Schrieffer) pairing integral, so that $T_c$ can be high and, when the Fermi energy lies within a characteristic phonon frequency of the singularity, nearly independent of that frequency -- explaining the almost vanishing oxygen isotope effect at optimal doping (the isotope exponent is $\alpha=-\partial\ln T_c/\partial\ln M$ for an ionic mass $M$). Second, "bad screening": cuprates are intermediate between metals and ionic crystals, so the Coulomb-based interactions are screened only at large distances, and the phonon-mediated attraction takes the small-momentum form $V(k)=g[\kappa^2/(k^2+\kappa^2)]^n$, where $\kappa$ is the inverse screening radius (small compared with the Fermi momentum $p_0$) and the exponent $n>1$ is treated phenomenologically. Because only small momenta are exchanged, the order parameter becomes strongly anisotropic, with maxima at the saddle regions, and may be even (s-wave-like) or odd (d-wave-like) under a $\pi/2$ rotation in the plane. Third, a small short-range repulsion of spin-fluctuation or Hund origin, with amplitude $U$, competes with the phonon attraction.

The model then confronts experiment along five routes. To first approximation the even and odd solutions are degenerate; a thermodynamic selection follows from the sign of a small overlap parameter $\lambda_2$: the odd (d-wave-like) solution wins only if $U>g[\kappa^2/(2p_0^2)]^n$, so which state a given cuprate realizes depends on microscopic magnitudes that the model does not establish. Large-angle impurity scattering suppresses the odd state's $T_c$ (in the standard Abrikosov-Gor'kov way, i.e. pair-breaking that drives $T_c$ to zero linearly in the scattering rate) and, the model predicts, can drive a transition into the even state; electron-bombardment experiments on YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) by the groups of Ginsberg, Giapintzakis and Kirk suppressed $T_c$ strongly without observing such a transition in the accessible temperature interval. The negative isotope effect upon $^{63}$Cu--$^{65}$Cu substitution in underdoped YBCO (discovered by Franck's group) is attributed to low-frequency Cu phonons near $150\,\mathrm{cm}^{-1}$ that act as short-range (neutral-defect-like) scatterers with coupling $g_2$ and frequency $\omega_{02}$, giving $\alpha\approx-6.326\,g_2^2\nu(\omega_{02}/2\pi T_c)^2$ with $\nu$ the density of states; all phonons contribute to the attraction, so scattering and pairing roles must be discriminated mode by mode. Finally, a two-component Ginzburg-Landau functional for the order parameters $\Psi_a,\Psi_b$ of the two saddle vicinities, coupled by a hopping term $\alpha\eta|\Psi_a+\Psi_b|^2$ whose dimensionless strength $\eta=|\lambda_2|/(\lambda_1+|\lambda_2|)^2$ is built from the same overlap parameters $\lambda_1,\lambda_2$ that decide the even/odd selection, predicts $H_{c2}\approx(4mc\alpha/e)|\tau|(1+|\tau|/\eta)$ (with $\tau=(T-T_c)/T_c$ and $\alpha$ here a Ginzburg-Landau coefficient, not the isotope exponent), i.e. an anomalously large upper critical field with positive curvature for sufficiently small $\eta$ -- although including the paramagnetic (Pauli) pair-breaking effect was already known by the author to spoil the agreement at low temperatures.

Independent of this model, phase-sensitive experiments (junction and tricrystal interferometry, reviewed by Tsuei and Kirtley) established $d_{x^2-y^2}$ pairing in the hole-doped cuprates, and strong-coupling (Eliashberg) analyses by Radtke and Norman found that band-structure enhancement from extended saddle points alone cannot account for high $T_c$. The quantitative questions the model left open -- whether its parameters for real materials satisfy its selection rules and whether its distinctive predictions actually occur -- are the subject of this problem.

Problem Statement

Complete, quantitatively, the confrontation of Abrikosov's extended-saddle-point, weak-screening phonon model of cuprate superconductivity with real materials, by resolving the following five linked questions. The model's equations and parameter definitions are fixed by the source paper (Abrikosov, Physica C 341-348, 97 (2000); companion paper Physica C 336, 163 (2000)); all five parts concern that one model and jointly determine whether it is quantitatively viable for the cuprates it claims to describe.

(1) Parameter determination and even/odd selection. Determine, by experiment or microscopic calculation for specific cuprates at specified dopings, the short-range repulsion amplitude $U$, the phonon coupling strength $g$, the inverse screening length $\kappa$, the Fermi momentum $p_0$, and the exponent $n$, with uncertainties, and decide whether the selection inequality $U > g\left[\frac{\kappa^{2}}{2p_{0}^{2}}\right]^{n}$ is satisfied -- in which case the thermodynamic equilibrium superconducting state is the odd (d-wave-like) solution -- or whether the even (s-wave-like) solution is favored, and for which materials and dopings each case holds.

(2) Impurity-driven odd-to-even transition. Establish the experimental conditions -- disorder strength, defect type (large-angle versus small-angle scatterers), temperature interval, and material/doping -- under which the predicted impurity-driven transition from the odd to the even superconducting state occurs, or establish with stated sensitivity that it does not occur. (The original electron-bombardment experiments on YBCO suppressed $T_c$ below 12 K without observing it.)

(3) Form of the effective interaction. Validate or falsify the assumed weak-screening form of the phonon-mediated interaction by directly characterizing the momentum dependence of the effective electron-electron interaction in real cuprates -- specifically the exponent $n$ and the screening length $\kappa$ entering $V(k)=g[\kappa^2/(k^2+\kappa^2)]^n$ with $n>1$ and $\kappa\ll p_0$ -- by measurement or microscopic calculation.

(4) Copper isotope effect. Determine the coupling constant $g_2$ and frequency $\omega_{02}$ of the low-frequency copper phonons (candidates near $150\,\mathrm{cm}^{-1}$) invoked in the model's explanation of the negative copper isotope effect in underdoped YBCO, and discriminate which phonon modes act as short-range scatterers -- suppressing the odd state and producing the isotope shift through $\alpha\approx-6.326\,g_2^2\nu(\omega_{02}/2\pi T_c)^2$ -- and which contribute to the pairing attraction.

(5) Inter-saddle hopping and the upper critical field. Measure or compute realistic values of the dimensionless inter-saddle hopping parameter $\eta=|\lambda_2|/(\lambda_1+|\lambda_2|)^2$ (in the source paper the hopping term of the Ginzburg-Landau functional is $\alpha\eta|\Psi_a+\Psi_b|^2$, with $\lambda_1,\lambda_2$ the overlap parameters of the even/odd selection) for actual cuprates, and establish whether the asymptotic regime in which the two-component Ginzburg-Landau result $H_{c2} \approx \frac{4mc\alpha}{e}\,|\tau|\left(1+\frac{|\tau|}{\eta}\right)$ (with $\tau=(T-T_c)/T_c$; here $\alpha$ is the Ginzburg-Landau coefficient of the source paper, not the isotope exponent) yields a strongly enhanced $H_{c2}$ with positive curvature is ever reached, including the interplay with paramagnetic (Pauli) pair breaking at low temperatures.

The common scientific objective is a decision on the model's quantitative viability: whether the microscopic parameters of real cuprates place them where the model says they must be for its explanations (high $T_c$, isotope effects, order-parameter symmetry, impurity sensitivity, $H_{c2}$ behavior) to hold.

References

1. A. A. Abrikosov, "Theory of high-Tc superconducting cuprates based on experimental evidence", Physica C 341-348, 97-102 (2000). DOI: 10.1016/S0921-4534(00)00399-3, https://doi.org/10.1016/S0921-4534(00)00399-3; arXiv:cond-mat/9912394, https://arxiv.org/abs/cond-mat/9912394 (source paper)
2. A. A. Abrikosov, "Theory of the negative isotope effect on copper in underdoped YBCO", Physica C 336, 163-169 (2000). DOI: 10.1016/S0921-4534(00)00280-x, https://doi.org/10.1016/S0921-4534(00)00280-x
3. R. J. Radtke and M. R. Norman, "Relation of extended Van Hove singularities to high-temperature superconductivity within strong-coupling theory", Phys. Rev. B 50, 9554-9560 (1994). DOI: 10.1103/PhysRevB.50.9554, https://doi.org/10.1103/PhysRevB.50.9554; arXiv:cond-mat/9404081, https://arxiv.org/abs/cond-mat/9404081
4. C. C. Tsuei and J. R. Kirtley, "Pairing symmetry in cuprate superconductors", Rev. Mod. Phys. 72, 969-1016 (2000). DOI: 10.1103/RevModPhys.72.969, https://doi.org/10.1103/RevModPhys.72.969
5. J. G. Storey, J. L. Tallon and G. V. M. Williams, "Saddle-point van Hove singularity and the phase diagram of high-Tc cuprates", Phys. Rev. B 76, 174522 (2007). DOI: 10.1103/PhysRevB.76.174522, https://doi.org/10.1103/PhysRevB.76.174522; arXiv:0707.2239, https://arxiv.org/abs/0707.2239
6. A. A. Kordyuk, "Electronic band structure of optimal superconductors: From cuprates to ferropnictides and back again", Low Temp. Phys. 44, 477-486 (2018). DOI: 10.1063/1.5037550, https://doi.org/10.1063/1.5037550; arXiv:1803.01487, https://arxiv.org/abs/1803.01487
7. J. Giapintzakis, D. M. Ginsberg, M. A. Kirk and S. Ockers, "Testing models of the symmetry of the superconducting pairing state by low-temperature electron irradiation of an untwinned single crystal of YBa2Cu3O7-delta", Phys. Rev. B 50, 15967-15973 (1994). DOI: 10.1103/PhysRevB.50.15967, https://doi.org/10.1103/PhysRevB.50.15967
8. D. M. Ginsberg, J. Giapintzakis and M. A. Kirk, "Evidence of order-parameter symmetry and of a normal metallic state in electron-irradiated YBa2Cu3O7-delta (YBCO)", Czech. J. Phys. 46, 1203-1204 (1996). DOI: 10.1007/BF02562716, https://doi.org/10.1007/BF02562716
9. M. Covington, M. Aprili, E. Paraoanu, L. H. Greene, F. Xu, J. Zhu and C. A. Mirkin, "Observation of surface-induced broken time-reversal symmetry in YBa2Cu3O7 tunnel junctions", Phys. Rev. Lett. 79, 277-280 (1997). DOI: 10.1103/PhysRevLett.79.277, https://doi.org/10.1103/PhysRevLett.79.277
10. A. V. Antonov, A. I. El'kina, V. K. Vasiliev, M. A. Galin, D. V. Masterov, A. N. Mikhaylov, S. V. Morozov, S. A. Pavlov, A. E. Parafin, D. I. Tetelbaum, S. S. Ustavschikov, P. A. Yunin and D. A. Savinov, "Experimental Observation of s-Component of Superconducting Pairing in Thin Disordered HTSC Films Based on YBCO", Phys. Solid State 62, 1598-1603 (2020). DOI: 10.1134/S1063783420090036, https://doi.org/10.1134/S1063783420090036
11. J. P. Franck, "Experimental studies of the isotope effect in high Tc superconductors", Physica C 282-287, 198-201 (1997). DOI: 10.1016/S0921-4534(97)00196-2, https://doi.org/10.1016/S0921-4534(97)00196-2
12. J. P. Franck and D. D. Lawrie, "The copper isotope effect in oxygen-deficient YBa2Cu3O7-x", J. Supercond. 8, 591-594 (1995). DOI: 10.1007/BF00727433, https://doi.org/10.1007/BF00727433
13. D. E. Morris, A. P. B. Sinha, V. Kirtikar and A. V. Inyushkin, "Copper isotope shifts in Pr and La substituted 123 and 124 cuprate superconductors; comparison with oxygen isotope shifts", Physica C 298, 203-208 (1998). DOI: 10.1016/S0921-4534(97)01730-9, https://doi.org/10.1016/S0921-4534(97)01730-9
14. D. Shimada and N. Tsuda, "On the isotope effect in Tc for cuprate superconductors", Physica C 371, 52-56 (2002). DOI: 10.1016/S0921-4534(01)00965-0, https://doi.org/10.1016/S0921-4534(01)00965-0
15. G. Sangiovanni and O. Gunnarsson, "Isotope effect in the pseudogap state of high-temperature copper oxide superconductors", Phys. Rev. B 84, 100505 (2011). DOI: 10.1103/PhysRevB.84.100505, https://doi.org/10.1103/PhysRevB.84.100505; arXiv:1104.0940, https://arxiv.org/abs/1104.0940
16. A. Greco and R. Zeyher, "Isotope effect on the superconducting critical temperature of cuprates in the presence of charge order", Supercond. Sci. Technol. 29, 015002 (2016). DOI: 10.1088/0953-2048/29/1/015002, https://doi.org/10.1088/0953-2048/29/1/015002; arXiv:1511.05096, https://arxiv.org/abs/1511.05096
17. Gan Sun, Qing-Geng Yang, Da Wang and Qiang-Hua Wang, "Anomalous isotope effect in d-wave superconductors on the square lattice", Phys. Rev. B 109, L180508 (2024). DOI: 10.1103/PhysRevB.109.L180508, https://doi.org/10.1103/PhysRevB.109.L180508; arXiv:2405.16138, https://arxiv.org/abs/2405.16138
18. N. F. Q. Yuan and L. Fu, "Classification of critical points in energy bands based on topology, scaling, and symmetry", Phys. Rev. B 101, 125120 (2020). DOI: 10.1103/PhysRevB.101.125120, https://doi.org/10.1103/PhysRevB.101.125120
19. A. A. Abrikosov, "Bad screening and applicability of Migdal's theorem to layered cuprates", Physica C 424, 149-152 (2005). DOI: 10.1016/j.physc.2005.05.006, https://doi.org/10.1016/j.physc.2005.05.006
20. V. I. Belyavsky and Yu. V. Kopaev, "Pair Fermi contour and repulsion-induced superconductivity in cuprates", Phys. Rev. B 67, 024513 (2003). DOI: 10.1103/PhysRevB.67.024513, https://doi.org/10.1103/PhysRevB.67.024513
21. K. Kamiya, T. Masui, S. Tajima, H. Bando and Y. Aiura, "Absence of the 1/8 anomaly in the oxygen isotope effect of YBa2Cu3Oy", Phys. Rev. B 89, 060505 (2014). DOI: 10.1103/PhysRevB.89.060505, https://doi.org/10.1103/PhysRevB.89.060505