Ultrafast optical spectroscopy of optimally electron-doped La2−xCexCuO4±δ\mathrm{La}_{2-x}\mathrm{Ce}_x\mathrm{CuO}_{4\pm\delta} (LCCO, x≈0.10x \approx 0.10, Tc≈25T_c \approx 25 K) decomposed the electron–boson spectral function into a phononic part and an electronic part Πs(Ω)\Pi_s(\Omega) whose weight below ∼65\sim 65 meV grows strongly below T∗≈60T^* \approx 60 K, and concluded, via an extended McMillan estimate, that two-dimensional antiferromagnetic spin fluctuations can fully account for superconducting pairing. That conclusion rests on indirect thermodynamic and temperature-dependence arguments rather than on momentum-resolved spin spectroscopy tied to the same samples. The open problem is to complete this glue identification: quantitatively connect measured spin spectra to the optically extracted Πs(Ω)\Pi_s(\Omega), λs(T)\lambda_s(T), and TcT_c on matched samples with microscopic anisotropic theory; identify the electronic excitations persisting above T∗T^* (spectral weight above ∼65\sim 65 meV); determine the long-timescale competing order causing a nanosecond relaxation sign change at T†T^{\dagger}, provisionally attributed to three-dimensional antiferromagnetism; and delimit possible cooperative participation of phonons. No resolution of these questions is known in the literature to date.

Background

Superconductivity in the copper oxides (cuprates) is generally understood to arise from an effective interaction between charge carriers and bosonic excitations — either lattice vibrations (phonons) or collective electronic modes such as spin fluctuations. Identifying this 'pairing glue' and its strength is a central open question of high-temperature superconductivity. Most experimental effort has concerned hole-doped cuprates; the present problem lives on the electron-doped side, in La2−xCexCuO4±δ\mathrm{La}_{2-x}\mathrm{Ce}_x\mathrm{CuO}_{4\pm\delta} (LCCO), where substituting Ce4+\mathrm{Ce}^{4+} for La3+\mathrm{La}^{3+} near x≈0.10x \approx 0.10 adds electrons to the CuO2\mathrm{CuO}_2 planes and superconductivity appears below Tc≈25T_c \approx 25 K. Electron-doped cuprates sit close to antiferromagnetic (AFM) order in their temperature–doping phase diagram, lack the pronounced pseudogap complexity of hole-doped compounds (see Armitage, Fournier and Greene, Rev. Mod. Phys. 82, 2421 (2010), for a review), and exhibit a family-wide scaling between anomalous normal-state magnetization and TcT_c (Jin et al., Nature 476, 73 (2011)) — features that make them a comparatively clean testbed for the spin-fluctuation pairing hypothesis. The strength of the electron–boson interaction is encoded in the bosonic spectral function: α2F(Ω)\alpha^2 F(\Omega) for phonons and its spin-channel analog I2χ(Ω)I^2\chi(\Omega), with χ\chi the dynamical spin susceptibility. Ultrafast optical spectroscopy can disentangle these contributions: a femtosecond pump pulse drives the sample out of equilibrium and the transient reflectivity ΔR(t)/R\Delta R(t)/R is recorded. Fitting the energy flow among electrons, hot phonons, the lattice, and electronic bosonic excitations with such an 'effective temperature model' (an extension of the standard two-temperature model of electron–phonon relaxation) yields the total bosonic spectrum split into phononic and electronic parts; the strategy was demonstrated on a hole-doped cuprate by Dal Conte et al., Science 335, 1600 (2012). Wang et al., Phys. Rev. B 97, 155157 (2018), applied this to a set of optimally doped LCCO thin films with tuned oxygen content δ\delta. They found: (i) the electronic spectrum Πs(Ω)\Pi_s(\Omega) carries a specific heat below 0.018 of the electronic one, certifying electronic (non-phononic) origin; (ii) its weight below ∼65\sim 65 meV — the upper limit of cuprate phonon energies, where different bosons overlap — grows strongly on cooling below T∗≈60T^* \approx 60 K, with the dimensionless coupling λs=2∫Πs(Ω) dΩ/Ω\lambda_s = 2\int \Pi_s(\Omega)\, d\Omega/\Omega increasing sharply; (iii) using the extended McMillan formula, i.e. the Allen–Dynes strong-coupling estimate of TcT_c from a coupling constant and a logarithmic moment of the spectrum (Allen and Dynes, Phys. Rev. B 12, 905 (1975)), the spin channel at 30 K yields a maximum Tc≈34T_c \approx 34 K (above the measured 25.1 K), whereas phonons with λp≈0.47\lambda_p \approx 0.47 alone give at most ≈15\approx 15 K; (iv) by comparison with angular-magnetoresistance evidence for the onset of two-dimensional (2D) AFM correlations at similar temperatures in LCCO (Jin et al., Phys. Rev. B 80, 012501 (2009)), Πs\Pi_s below 65 meV was attributed to 2D AFM spin fluctuations; and (v) in the nanosecond tail of ΔR(t)/R\Delta R(t)/R, a long-lived relaxation component changes sign at a temperature T†T^{\dagger} (≈16\approx 16 K in one sample), anti-correlated with TcT_c across oxygen-tuned samples, suggesting a competing order coexisting with superconductivity — with three-dimensional (3D) AFM, established as a competing ground state in electron-doped Nd1.85Ce0.15CuO4\mathrm{Nd}_{1.85}\mathrm{Ce}_{0.15}\mathrm{CuO}_4 (Kang et al., Nature 423, 522 (2003)), a natural candidate. Two probes used for such follow-ups are inelastic neutron scattering (INS), which measures the dynamical spin susceptibility at finite momentum in bulk samples, and resonant inelastic x-ray scattering (RIXS) at the Cu L edge, which measures charge and spin excitations versus in-plane momentum in small samples; 'paramagnons' are the heavily damped spin excitations of a metal in the vicinity of AFM order. The spin-glue conclusion of Wang et al., however, rests on indirect evidence — a thermodynamic bound, temperature dependences, and consistency with transport — and the authors themselves state that they cannot clarify the origin of the electronic excitations above T∗T^* and that they cannot exclude cooperative participation of phonons or other excitations.

Problem Statement

The open problem is to complete the microscopic identification of the pairing glue in optimally electron-doped La2−xCexCuO4±δ\mathrm{La}_{2-x}\mathrm{Ce}_x\mathrm{CuO}_{4\pm\delta} that the ultrafast optical extraction of Wang et al. (Phys. Rev. B 97, 155157 (2018)) began. Its components jointly support one objective — a closed, quantitative account of which bosonic excitations pair the electrons in this electron-doped cuprate, how strongly, and how a competing order modulates it: (1) Direct spin-character test: measure the momentum- and temperature-resolved spin spectrum (2D AFM spin fluctuations/paramagnons) of LCCO at matched doping (x≈0.10x \approx 0.10) and comparable oxygen content and TcT_c, from above T∗≈60T^* \approx 60 K down through TcT_c, e.g., by inelastic neutron scattering or RIXS, and test whether it quantitatively reproduces the optically extracted electronic spectrum Πs(Ω)\Pi_s(\Omega) — its sub-65-meV weight, its growth below T∗T^*, and the coupling λs(T)\lambda_s(T) — thereby directly confirming or refuting the assignment of the pairing glue to 2D AFM spin fluctuations. (2) Quantitative pairing sufficiency: replace the phonon-derived extended McMillan estimate with a microscopic strong-coupling treatment appropriate to an anisotropic unconventional superconductor with a dx2−y2d_{x^2-y^2} gap (the gap changes sign under a 90-degree in-plane rotation), taking the measured momentum-dependent spin spectrum as input, and determine whether it reproduces the measured Tc≈25T_c \approx 25 K, and whether the phonon channel (λp≈0.47\lambda_p \approx 0.47) or other excitations must participate cooperatively. (3) Origin of the above-T∗T^* excitations: determine the microscopic origin of the electronic spectral weight that persists above T∗T^* and is distributed above ∼65\sim 65 meV — e.g., residual short-range spin correlations, charge excitations, or quantum-critical fluctuations (fluctuations of a nearby continuous zero-temperature phase transition). (4) The T†T^{\dagger} competing order: identify the long-timescale order underlying the nanosecond relaxation sign change at T†T^{\dagger} and its anti-correlation with TcT_c across oxygen-tuned samples, testing the 3D antiferromagnetism candidate against alternatives. A complete answer requires all four components, each with the following completion standard: (1) must supply a momentum- and temperature-resolved spin spectrum for LCCO at matched doping — either an executed measurement of it or a fully specified, sourced measurement route that reaches it — together with its quantitative comparison to the optically extracted Πs(Ω)\Pi_s(\Omega) and λs(T)\lambda_s(T); (2) must supply a quantitative microscopic strong-coupling calculation that reproduces the measured Tc≈25T_c \approx 25 K to within the uncertainty of the reported value; (3) and (4) must each supply a sourced, testable identification — of the origin of the above-T∗T^* weight and of the order underlying the T†T^{\dagger} sign change, respectively. No component, and no proper subset of components, resolves the task on its own.

References

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