Microscopic origin of the normal-state pseudogap in the c-axis optical conductivity of underdoped YBa2Cu3Ox: confirming or excluding resonant-defect tunneling against competing mechanisms

In oxygen-deficient (underdoped) YBa2Cu3Ox, the optical conductivity perpendicular to the copper-oxide planes, $\sigma_c(\omega)$, shows a normal-state 'pseudogap': a flat, nearly frequency-independent background with a gaplike depression below roughly 400-500 cm^-1 and a maximum whose position and height do not depend on temperature. A. A. Abrikosov (1996) proposed that this pseudogap is a threshold effect of resonant tunneling between the CuO2 bilayers via localized levels formed at broken CuO-chain fragments in the barrier, fitting the then-available YBa2Cu3O6.7 data, but the fit required unknown inputs: the distribution $f(E_j)$ of resonant levels and the in-plane scattering rate $1/\tau(\omega,T)$, and the analysis left superconducting-fluctuation, resonating-valence-bond (spin-charge separation), and preformed-pair mechanisms unexcluded. Abrikosov's own follow-ups extended but did not confirm the model; Ca/Pr-substitution experiments (as summarized in Timusk's 2003 infrared review) showed the c-axis spectra depend on plane doping rather than chain oxygen filling, disfavoring chain-fragment resonant-center models; the mainstream reviews recast the question in density-of-states and spectral-weight frameworks; and later calculations pursued the competing RVB branch without decisive discrimination. The question of the microscopic mechanism of the c-axis optical pseudogap remains open.

Background

YBa2Cu3Ox (YBCO) is a layered cuprate high-temperature superconductor. Its structure consists of superconducting CuO2 planes (stacked as bilayers) separated along the crystallographic c axis by 'blocking' layers containing one-dimensional CuO chains; the oxygen content x controls the number of holes (missing electrons acting as mobile positive carriers) introduced into the CuO2 planes, with x near 7 (fully oxygenated) giving optimal doping and x below roughly 6.9 giving underdoping. The optical conductivity $\sigma(\omega)$, measured by far-infrared reflectance and converted to conductivity via a Kramers-Kronig analysis (an exact analytic transform between the real and imaginary parts of a causal response function), is strongly anisotropic: for light polarized in the planes the response is metallic and Drude-like (a Lorentzian centered at zero frequency), whereas for polarization along the c axis the conductivity of underdoped samples is small and nearly frequency independent, indicating incoherent, hopping-type interplane transport. Frequencies below are quoted in spectroscopic wavenumber units, cm^-1 (1 cm^-1 corresponds to about 0.124 meV).

In 1993-1995, C. C. Homes and coworkers observed that in underdoped YBCO the c-axis conductivity develops, already far above the superconducting transition temperature $T_c$, a gaplike depression at low frequencies: $\sigma_c(\omega)$ is suppressed below roughly 400-500 cm^-1, rises to a maximum, and the position and height of this structure do not change appreciably with temperature up to room temperature. The depression is essentially absent in fully oxygenated YBa2Cu3O6.95 and weak in the chain compound YBa2Cu4O8. This is one facet of the broader 'pseudogap' phenomenon of underdoped cuprates (seen also in the nuclear magnetic resonance (NMR) Knight shift, a resonance measure of the local spin susceptibility; in angle-resolved photoemission spectroscopy (ARPES); and in tunneling), whose microscopic origin was and remains contested.

A. A. Abrikosov had earlier proposed that c-axis charge transport in YBCO proceeds by resonant tunneling: an electron tunnels between adjacent CuO2 bilayers not directly but through a localized impurity state in the barrier, and the tunneling amplitude is resonantly enhanced when the level energy $E_j$ lies near the Fermi energy. In oxygen-deficient samples, broken CuO-chain fragments between the bilayers were posited to supply a density of such resonant centers. Extending this model to finite frequency, Abrikosov (1996) showed that $\sigma_c(\omega)$ acquires a threshold-like pseudogap: with a flat distribution of resonant levels in a stripe $(E-\varepsilon, E+\varepsilon)$ and an in-plane scattering time $\tau$ (the lifetime of a carrier against in-plane collisions, with inelastic width $1/\tau$), integration of the tunneling expression gives $\sigma_c(\omega)$ proportional to $(T/\tau)\,\omega_0^{-3}$ times a logarithmic factor that turns on only when the frequency $\omega_0$ exceeds the lower stripe edge $E-\varepsilon$. Because the threshold is set by the resonant-level spectrum, its location and height are temperature independent, reproducing the observed phenomenology; a fit to the YBa2Cu3O6.7 ($T_c=63$ K) data of Homes et al. gave $E-\varepsilon\approx 271$ cm^-1 at 70 K, consistent with static-resistivity estimates. The derivation, however, assumes the separation condition $1/\tau\ll\omega_0$ (so that the two resonant peaks of the tunneling spectrum are well resolved), and the fit degraded at 110 K and at low frequencies.

The 1996 paper explicitly acknowledged the open inputs of the model: the true distribution $f(E_j)$ of resonant levels is unknown (a flat stripe was assumed for analytic simplicity, and the fine 'wiggling' of the experimental curve was tentatively attributed to an uneven distribution), and the paper proposed to extract that distribution by differentiating the measured frequency-integrated conductivity at low temperature in the normal state; the frequency- and temperature-dependent in-plane scattering rate $1/\tau(\omega,T)$ had 'not been definitely established', so that microscopic tunneling parameters cannot be unambiguously extracted from $\sigma_c(\omega)$ alone unless $\sigma_{ab}(\omega)$ is measured simultaneously on the same sample; extracting $f(E_j)$ at low temperature is obstructed by superconductivity, which restricts the normal-state window to samples with low $T_c$ near the metal-insulator transition; and the alternative mechanisms of the pseudogap were not excluded. Those alternatives, as named in the source, are: superconducting fluctuations above $T_c$ (which could produce a shallow depression); the resonating-valence-bond (RVB) idea of spin-charge separation, in which the electron fractionalizes into a neutral spinon and a charged holon and the c-axis current requires their recombination, favored by Uchida and coworkers from c-axis spectra of La2-xSrxCuO4; and preformed pairs, local pairing of electrons well above $T_c$, whose double charge would suppress interplane tunneling.

The question is embedded in a broader electrodynamical problem. Later work established that the c-axis superfluid density $\rho_{s,c}$ (the zero-frequency spectral weight of the superconducting condensate along the c axis) obeys a scaling relation $\rho_{s,c}\propto\sigma_{\mathrm{dc}}T_c$, and that the superconducting-state spectral-weight redistribution in the c-axis response violates the simple Ferrell-Glover-Tinkham sum rule (a bookkeeping of optical spectral weight that must be conserved when a gap opens), implications for interlayer coherence and kinetic energy that were debated by L. B. Ioffe and A. J. Millis and by D. N. Basov and coworkers. Whether the normal-state pseudogap in $\sigma_c(\omega)$ is an interlayer-tunneling threshold effect, as in the resonant-level model, or a direct image of a density-of-states gap of the CuO2 planes, remains the discriminating question.

Problem Statement

What is the microscopic origin of the normal-state pseudogap - the threshold-like low-frequency suppression of $\sigma_c(\omega)$ below roughly 400-500 cm^-1 whose position and height are temperature independent - in the c-axis optical conductivity of oxygen-deficient YBa2Cu3Ox? In particular, the following clarifying considerations bear on the answer: (i) can the resonant-tunneling mechanism - interplane transport through localized levels formed at broken CuO-chain fragments in the barrier, with the threshold fixed by the lower edge $E-\varepsilon$ of the resonant-level spectrum - be confirmed or quantitatively excluded as the explanation of $\sigma_c(\omega,T)$ across the underdoped doping range; (ii) can the phenomenological inputs the model requires be determined independently, namely the distribution function $f(E_j)$ of resonant levels and the frequency- and temperature-dependent in-plane scattering rate $1/\tau(\omega,T)$; (iii) what is the domain of validity of the threshold approximation $1/\tau\ll\omega_0$ (the requirement that the in-plane scattering width $1/\tau$ be small compared with the photon frequency $\omega_0$, so that the two resonant peaks of the tunneling spectrum are well resolved) on which the analytic form rests, given the observed degradation of the fits at higher temperatures and low frequencies; and (iv) which of the competing mechanisms - superconducting fluctuations above $T_c$, resonating-valence-bond spin-charge separation (in which the electron fractionalizes into a neutral spinon and a charged holon that must recombine for c-axis current to flow), preformed pairs (locally paired electrons above $T_c$ whose double charge suppresses interplane pair tunneling), or a density-of-states gap of the CuO2 planes imaged by incoherent interlayer transport - actually produces the pseudogap? The candidate mechanisms are constrained by the full c-axis phenomenology - temperature independence of the threshold, absence of the pseudogap in fully oxygenated samples, dependence on plane doping rather than chain oxygen filling under Ca/Pr substitution, sensitivity to in-plane Zn disorder, and the spectral-weight and sum-rule constraints.

References

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