Relating the optically extracted electron-boson coupling function to the critical temperature of high-$T_c$ cuprates: d-wave projection with pair-breaking and the underdoped pseudogap regime

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Infrared optical conductivity of the cuprate superconductors can be inverted, via strong-coupling (memory-function/Eliashberg) theory, to yield a momentum-averaged electron-boson 'glue' spectrum $\tilde{\Pi}(\omega)$, which van Heumen et al. (Phys. Rev. B 79, 184512 (2009)) showed consists of a robust 50–60 meV peak plus a high-energy continuum and correlates with the doping trend of the superconducting transition temperature $T_c$. The quantitative link between this optically measured glue and $T_c$ is, however, not established: treating the full $\tilde{\Pi}(\omega)$ as pair-forming yields only upper bounds on $T_c$, because the projection of a momentum-averaged glue onto the d-wave pairing channel — including pair-breaking contributions — is not known, and the Fermi-liquid-based inversion fails in the strongly underdoped, pseudogap regime. Later work has narrowed the microscopic-identity question (the glue is dominantly electronic) and improved inversion methodology, but the rigorous glue-to-$T_c$ mapping from optical data remains open. The problem is to determine this mapping and extend it across the cuprate phase diagram, thereby establishing whether the spectroscopically measured bosonic spectrum quantitatively accounts for high-temperature superconductivity.

Background

Cuprate high-$T_c$ superconductors have a superconducting gap with $d$-wave symmetry (the gap changes sign between the Cu–O bond directions and the diagonals of the CuO$_2$-plane Brillouin zone), and in conventional Eliashberg strong-coupling theory superconductivity arises from electrons exchanging a bosonic excitation — the 'glue'. The extended Drude formalism rewrites the measured infrared optical conductivity $\sigma(\omega)$ in terms of a memory function (optical self-energy) $\hat{M}(\omega)$, and in a Fermi-liquid normal state the frequency-dependent part of $\hat{M}(\omega)$ is a thermal average of the electron-boson coupling function $\tilde{\Pi}(\omega)$ (the cuprate analogue of the phonon spectral function $\alpha^2F(\omega)$ of conventional superconductivity). Because optical conductivity integrates over the whole Brillouin zone, the extracted $\tilde{\Pi}(\omega)$ is a momentum average, not a momentum-resolved interaction. van Heumen, Muhlethaler, Kuzmenko, Eisaki, Meevasana, Greven and van der Marel (Phys. Rev. B 79, 184512 (2009); arXiv:0904.1223) inverted the optical spectra of ten cuprates of the bismuth- and mercury-based families Bi-2201, Bi-2212, Bi-2223 and Hg-1201 (compounds with one, two, three and one CuO$_2$ planes per unit cell, respectively), from strongly underdoped to overdoped and found a robust peak at 60 ± 3 meV (resolvable as one or more modes between 45 and 75 meV) plus a broad continuum extending to roughly 0.3 eV in the highest-$T_c$ samples. Feeding the full spectrum into the Eliashberg $T_c$ equation reproduced the doping trend of $T_c$, but only as an upper bound, for two acknowledged reasons. First, for a $d$-wave superconductor the momentum structure of the interaction is essential: as shown for spin-fluctuation pairing by Millis, Sachdev and Varma (Phys. Rev. B 37, 4975 (1988)), low-frequency bosonic fluctuations act as pair breakers in anisotropic (sign-changing) gap states, and the authors state explicitly that 'it is not possible to separate pair-breaking from pair-forming contributions to $\tilde{\Pi}(\omega)$ in an unambiguous way', so the full measured glue was simply assumed to pair favorably, making the calculated $T_c$'s overestimates. Notably, for the most overdoped Bi-2212 sample the 50–60 meV peak alone gives $T_c < 20$ K while the continuum above 100 meV gives about 160 K, implying that high-energy — hence purely electronic — modes carry an essential part of the pairing strength. Second, the formalism rests on a Landau Fermi-liquid description of the normal state: the internal-consistency test succeeded for all samples except the most strongly underdoped Bi-2201, where the fit produces an unphysical low-frequency peak in $\tilde{\Pi}(\omega)$, plausibly because the pseudogap (a partial depletion of the electronic density of states that opens well above $T_c$ in underdoped cuprates) and other non-Fermi-liquid effects violate the assumptions of the inversion. Subsequent momentum-resolved probes sharpened the difficulty rather than removing it: electronic Raman scattering, whose $B_{1g}$ and $B_{2g}$ polarization geometries weight antinodal and nodal Fermi-surface regions respectively, yields glue spectra that differ between channels and from the optical average (Muschler et al., J. Phys.: Condens. Matter 22, 375702 (2010); Fanfarillo et al., J. Phys.: Condens. Matter 28, 065701 (2016)), showing directly that a momentum-averaged glue does not uniquely encode a momentum-resolved pairing interaction; and laser-based angle-resolved photoemission (ARPES) self-energy inversion by Bok et al. (Sci. Adv. 2, e1501329 (2016)) determined the attractive interaction in the $d$-wave channel and a nearly angle-independent repulsive component near optimal doping of Bi-2212 — a momentum-resolved determination, but from photoemission rather than from the optical spectrum, and over a limited doping range. Ultrafast optical experiments (Giannetti et al., Nat. Commun. 2, 353 (2011); Dal Conte et al., Science 335, 1600 (2012); Dal Conte et al., Nat. Phys. 11, 421 (2015)) established that the retarded interaction in optimally doped cuprates builds up on a ~16 fs timescale, faster than any lattice vibration, so the glue is dominantly electronic — narrowing, but not settling, the microscopic identity (spin fluctuations, orbital-current fluctuations, or other electronic modes) of the continuum. Theoretical analysis (Chubukov and Schmalian, Phys. Rev. B 101, 180510(R) (2020)) further showed that peaks in the single-particle self-energy near the gap frequency can arise from soft thermal fluctuations for essentially any boson and cancel in the total measured self-energy, complicating the inference of a glue from spectroscopy.

Problem Statement

The problem is to establish, quantitatively, the relation between the electron-boson coupling function $\tilde{\Pi}(\omega)$ extracted from the infrared optical conductivity of high-$T_c$ cuprates and their superconducting critical temperature $T_c$ — the title question of van Heumen et al. (2009) — which currently yields only doping-trend agreement and upper bounds on $T_c$. A complete answer must jointly resolve, as parts of this single objective: (i) the rigorous projection of the experimentally determined, momentum-averaged $\tilde{\Pi}(\omega)$ onto the d-wave pairing channel, separating pair-forming from pair-breaking contributions (in the sense of Millis–Sachdev–Varma), so that $T_c$ is computed rather than overestimated — or a demonstration, with explicit constructions, that the momentum-averaged optical data do not determine the d-wave channel and must be supplemented by momentum-resolved input (Raman, ARPES, neutron), in which case the combined-procedure mapping must be given; (ii) extension of the strong-coupling inversion and $T_c$ analysis into the strongly underdoped, pseudogap regime where the Fermi-liquid-based formalism currently fails (the unphysical low-frequency structure in underdoped Bi-2201), including whatever vertex corrections (interaction corrections to the optical response beyond the leading-order memory-function approximation) or pseudogap modifications of the electronic structure are required; and (iii) within this mapping, identification of which parts of the measured spectrum — the 45–75 meV peak region and the high-energy continuum above ~100 meV — actually contribute to pairing, and with what microscopic modes (spin, orbital-current, or other electronic fluctuations). Success means that the measured spectra, processed through the established mapping, reproduce the observed $T_c(p)$ — the superconducting transition temperature as a function of hole doping $p$ of the CuO$_2$ planes — across the cuprate phase diagram within experimental uncertainties, or that the obstruction to doing so is precisely characterized and the combined-procedure mapping is given.

References

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