Leggett's "midinfrared" scenario for cuprate superconductivity: the unfinished program of self-consistent theory and decisive experimental tests

A. J. Leggett (Proc. Natl. Acad. Sci. USA 96, 8365 (1999)) conjectured that superconductivity in the cuprates is driven by the saving, through the improved screening that accompanies Cooper-pair formation, of the part of the Coulomb energy associated with long in-plane wavelengths ($q \lesssim d^{-1} \approx 0.3$ Å$^{-1}$) and midinfrared frequencies (0.1–2 eV). The scenario semi-quantitatively explains the layering law $\Delta T_c(n) \propto 1 - 1/n$ of Ca-spaced homologous series and predicts a drop of order 10% in the midinfrared electron-energy-loss (EELS) cross-section below $T_c$. The source paper itself enumerates what remains to be done: a self-consistent gap equation (the standard Eliashberg technique being inadequate in the relevant non-weakly-coupled regime), an account of the origin of the normal-state midinfrared peak, detailed differential optical and Raman calculations, the Sr/Ba-spaced bilayer question, universality of the ideal homologous series, the effect of relaxing the nonessential assumptions, the differential EELS "make-or-break" test, and the quantification of intermultilayer Coulomb contributions. Subsequent work includes sum-rule bounds on the Coulomb energy (Turlakov–Leggett 2003), an experiment on Sr-spaced bilayer films (Di Luccio et al. 2003), a microscopic Umklapp-based study (Lee 2015), and temperature-dependent ellipsometry (Levallois et al. 2016) reporting a small-$q$ Coulomb-energy change comparable to the condensation energy but with a doping-dependent sign, obtained by extrapolating the optical response to finite $q$. No self-consistent midinfrared gap equation has been derived, the decisive differential EELS measurement has not been performed (momentum-resolved EELS in the 0.1–2 eV window across $T_c$ exists — Mitrano et al. 2018 — but does not implement the required differential test), the origin of the midinfrared peak remains debated, and the promised differential Raman calculations have not appeared. The problem — completing the theory and the decisive tests of the midinfrared scenario — remains open.

Background

The superconducting copper oxides (cuprates), discovered in 1986, lose all electrical resistance at transition temperatures up to about 135 K, and the microscopic mechanism of their superconductivity is still unresolved. Their crystal structure is built from copper-oxide (CuO2) planes separated by insulating charge-reservoir layers. In many families a group of $n$ adjacent planes (a "multilayer") is separated from the next group by Ca layers ("Ca-spaced" compounds, e.g. the 2-Tl and Hg homologous series); $T_c$ systematically rises from $n = 1$ to $n = 3$ in these series. In the standard theory of superconductivity (BCS–Eliashberg), pairing is mediated by the exchange of a bosonic "glue" (usually lattice vibrations) treated as a weak, retarded interaction, and a self-consistent "gap equation" determines the superconducting energy gap; whether this framework applies to the cuprates is a central open question.

Two experimental facts motivate the midinfrared scenario. First, the optical response of the cuprate normal state generically contains a broad "midinfrared" (MIR) peak in the loss function $L(\omega) = -\mathrm{Im}\,\epsilon(\omega)^{-1}$ — the quantity that measures how the material dissipates energy through charge-density fluctuations, accessible both optically (at wave vector $q \approx 0$, e.g. by ellipsometry) and by electron energy-loss spectroscopy (EELS), which probes the same loss function at finite momentum transfer $q$. This peak extends from roughly 0.1 eV to 1–2 eV and its microscopic origin is debated. Second, thermal-difference optical measurements (Holcomb et al., Phys. Rev. B 53, 6734 (1996)) had shown sub-percent reflectance changes at the superconducting transition.

A. J. Leggett (J. Phys. Chem. Solids 59, 1729 (1998); Phys. Rev. Lett. 83, 392 (1999)) observed that a substantial amount of Coulomb energy is locked up in the cuprates at long in-plane wavelengths and midinfrared frequencies, and proposed — the "midinfrared scenario" (Leggett, Proc. Natl. Acad. Sci. USA 96, 8365 (1999)) — that the driving force of superconductivity is the saving of this energy: Cooper-pair formation improves the screening of the small-$q$ Coulomb interaction and thereby lowers the expectation value of the Coulomb energy $\langle V_c \rangle$. The relevant region is $q \lesssim d^{-1} \approx 0.3$ Å$^{-1}$, much smaller than the scale $q_o \sim 1$ Å$^{-1}$ of normal-state in-plane behavior (with $d \approx 3.5$ Å the spacing of adjacent CuO2 planes within a multilayer). Writing $K(q,\omega)$ for the complex two-dimensional polarization response of the CuO2 planes from which the c-axis dielectric response is built, the scenario tracks the fractional change $\eta(q,\omega) = \Delta K/K$ on entering the superconducting state. Because this region lies far below the two-dimensional Thomas–Fermi wavevector (the scale separating weakly from strongly screened Coulomb coupling), Leggett argued that the standard Eliashberg technique is "almost certainly inadequate" there.

The scenario yields the layering law $\Delta T_c(n) = T_c(n) - T_c(1) = \mathrm{const} \cdot (1 - 1/n)$ for "ideal homologous series" (series whose single-plane properties are independent of the plane and of $n$), in reasonable agreement with the 2-Tl and Hg families. Its "make-or-break" prediction is a decrease, of order $0.1\,\epsilon_{\mathrm{sc}}^{-1}$ (with $\epsilon_{\mathrm{sc}}$ the effective dielectric constant of the ionic background), i.e. of order 10%, in the EELS cross-section of the MIR region on entering the superconducting state; in the source's own words, a reliable experimental upper bound substantially below 10% on the change of the (transmission) spectrum, coupled with a direct verification of the source's estimate for the differential condensation energy $\delta^{(2)}E_{\mathrm{cond}}$ (with $E_{\mathrm{cond}}$ the $T = 0$ superconducting condensation energy per CuO2 unit and $\delta^{(2)}E_{\mathrm{cond}}$ its difference $E_{\mathrm{cond}}^{(2)} - E_{\mathrm{cond}}^{(1)}$ between the $n = 2$ and $n = 1$ members of a homologous series), would cast serious doubt on the hypothesis, and one below about 1% would "definitively refute" it. The quantitative predictions are derived under an explicit list of simplifying assumptions, labeled B.1–B.7 in the source and called nonessential in the sense that relaxing them is expected to change the quantitative predictions but not the qualitative conclusions; the ones referenced below are: B.1 — the screened interconduction-electron Coulomb interaction is approximated using a constant, frequency-independent background (ionic-core) dielectric constant $\epsilon_{\mathrm{sc}}$; B.2 — Coulomb interactions between electrons in different multilayers are neglected; B.3 — the matrix of single-plane polarization responses coupling the planes within a multilayer is approximated by its diagonal part; B.4 — quantities are expanded to lowest order in the relative increment of $T_c$ between successive layer numbers; B.5 — the $q$-dependence of the single-plane polarization response $K_o(q,\omega)$ and of $\eta$ is neglected for $qd \lesssim 1$; B.6 — superconducting-state changes are expanded to lowest order in $\eta$; and B.7 — the ratio $\zeta = K_o/(d\,\epsilon_{\mathrm{sc}})$ is treated as large compared with unity. The predicted optical reflectance changes are roughly an order of magnitude smaller. Later theoretical work clarified the bookkeeping: the partial Coulomb energy at wave vector $q$ is a thermally weighted frequency integral of the loss function, and Umklapp processes (electron scattering that conserves momentum only up to a reciprocal-lattice vector) are essential to the sum rules that bound it (Turlakov and Leggett, Phys. Rev. B 67, 094517 (2003)). The condensation energy — the free-energy difference between the normal and superconducting states at zero temperature — is the benchmark against which any claimed energy saving must be measured.

The scenario deliberately takes the normal-state MIR peak as an unexplained input and derives no pairing gap; the 1999 paper closes by listing what is needed to turn the conjecture into a testable theory and to test it decisively.

Problem Statement

Establish or refute Leggett's midinfrared scenario as a quantitative mechanism of cuprate superconductivity, along the lines its source explicitly identifies as unresolved. The common objective of all components is to determine whether the saving of long-wavelength midinfrared Coulomb energy, through the improved screening that accompanies Cooper-pair formation, is the principal driver of superconductivity in the cuprates.

(1) Theory: derive a self-consistent gap equation (or an equivalent non-perturbative, self-consistent treatment of screening-driven pair formation) within the MIR scenario; the standard Eliashberg technique is expected to fail because the energy saving is concentrated at $q \ll q_o$, where the Coulomb interaction is not weakly coupled.

(2) Input ingredient: account for the origin of the normal-state midinfrared peak itself, which the scenario must currently take as an input, or quantify how the scenario's conclusions depend on the candidate origins.

(3) Predictions: compute in detail the differential (superconducting-minus-normal-state) optical and Raman properties predicted by the scenario, for quantitative comparison with experiment.

(4) Layering structure: complete the treatment of the intramultilayer tunnelling Hamiltonian $\hat{H}_{\mathrm{intra}}$ (the term of the model Hamiltonian describing tunnelling between adjacent CuO2 planes within a multilayer) and of the absence of superconductivity in bulk bilayers spaced entirely with Sr or Ba — including the confrontation with subsequent experiments on Sr-spaced bilayer films.

(5) Universality: establish experimentally and theoretically the degree of universality of the 'ideal homologous series' concept, in particular testing the source's stated belief that the apparent agreement of the layering law for the Bi and 1-Tl (single-thallium-layer) homologous series is "a pure coincidence" because assumption B.4 fails badly for these series.

(6) Robustness: quantify the effect of relaxing the nonessential assumptions B.1–B.7 on the quantitative predictions, in particular B.1 — the constant, frequency-independent background dielectric constant $\epsilon_{\mathrm{sc}}$, which the source calls the most problematic of the nonessential assumptions, since the background (ionic-core) dielectric constant is likely to be material- and position-dependent and quite likely anisotropic — as well as the assumed $q$-independence of $K_o$ and $\eta$ for $qd \lesssim 1$ and the largeness of $\zeta = K_o/(d\,\epsilon_{\mathrm{sc}})$.

(7) Decisive experiment: perform a differential EELS measurement (transmission or reflection geometry) with sufficient sensitivity to detect the predicted order-10% change of the midinfrared cross-section across the superconducting transition, together with higher-precision, lower-temperature optical measurements; interpret the outcome against the source's own criteria — a reliable upper bound substantially below 10% on the change of the midinfrared spectrum, coupled with a direct verification of the source's estimate for the differential condensation energy $\delta^{(2)}E_{\mathrm{cond}}$, would cast serious doubt on the scenario, and one below about 1% would definitively refute it.

(8) Intermultilayer Coulomb energy: quantify the role of intermultilayer Coulomb interactions when assumption B.2 is relaxed, and their contribution to the condensation energy in single-plane and multilayer compounds.

The common objective of components (1)–(8) — the unresolved program the source identifies, together with the supporting investigations it names — is to determine whether the saving of long-wavelength midinfrared Coulomb energy, through the improved screening that accompanies Cooper-pair formation, is the principal driver of superconductivity in the cuprates: operationally, whether or not it accounts for the largest share of the condensation energy. On the source's own criteria, the scenario is refuted if a reliable experimental upper bound below about 1% on the change of the midinfrared spectrum across the superconducting transition is established; a reliable upper bound substantially below 10%, together with a direct verification of the source's estimate for the differential condensation energy $\delta^{(2)}E_{\mathrm{cond}}$, would cast serious doubt on the scenario but would not by itself settle the question.

References

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