Impurity-driven odd-to-even parity transition of the superconducting order parameter in YBa2Cu3O7−δ: completing the quantitative theory and its low-temperature test

In A. A. Abrikosov's extended-saddle-point model of cuprate superconductivity, an order parameter that is odd under a $\pi/2$ rotation of the CuO $_2$ plane ( $d$ -like) in the clean limit is strongly pair-broken by nonmagnetic impurities, while the critical temperature of the even-parity ( $s$ -like) phase is nearly impurity-independent; the two curves must therefore cross, and the 1996 calculation (Phys. Rev. B 53, R8910) predicted an impurity-driven transition to the even phase, with a kink and a long low-temperature tail in $T_c$ versus impurity concentration. That calculation was deliberately restricted to a tetragonal band structure, isotropic scatterers, and scattering confined to the extended-saddle-point regions of the Fermi surface; the predicted kink lay beyond the 12 K lower limit of the underlying electron-irradiation experiment on YBa $_2$ Cu $_3$ O $_{7-\delta}$ . This problem asks for the completion of that theory — the orthorhombic calculation for YBa $_2$ Cu $_3$ O $_{7-\delta}$ , inclusion of the scattering channels that transfer quasiparticles between the singular and nonsingular parts of the Fermi surface (restoring a slow decay of the even-phase $T_{ce}$ ), and constrained interaction parameters $\lambda_1$ and $\lambda_2$ — and for its confrontation with $T_c$ -versus-defect-concentration measurements below 12 K, to decide whether the impurity-driven parity change actually occurs. None of these items has been carried through in the published literature: no orthorhombic or multichannel calculation within the model exists, and the one experiment reaching $T_c\to0$ in electron-irradiated YBa $_2$ Cu $_3$ O $_{7-\delta}$ (2003) saw no kink or long tail and reported deviations from pair-breaking behavior, without testing the parity question within the model.

Background

In a superconductor, the Cooper-pair condensate is described by an order parameter whose symmetry in momentum space can be isotropic (conventional $s$ -wave) or anisotropic, changing sign over the Fermi surface. The high-critical-temperature ( $T_c$ ) copper oxides discovered in the late 1980s, chief among them YBa $_2$ Cu $_3$ O $_{7-\delta}$ (YBCO), posed the question of which symmetry their order parameter has; through the mid-1990s an $s$ -like versus $d$ -like debate was open.

One theoretical framework of that period was A. A. Abrikosov's extended-saddle-point model (developed in Phys. Rev. B 52, R15738 (1995) and related papers). It assumes that the band structure of the CuO $_2$ planes contains extended saddle points — Fermi-surface locations where the quasiparticle dispersion is nearly flat along an entire direction, so that the density of electronic states is strongly enhanced in the 'a' and 'b' vicinities of those points, the neighborhoods of the saddle points in which the model's pairing interaction is concentrated. The effective interaction consists of a strong long-range (phonon-mediated) attraction acting within these singular regions, parametrized by a coupling $\lambda_1$ , plus a weak short-range repulsion parametrized by $\lambda_2$ . In this model the natural classification of the order parameter is its parity under a $\pi/2$ rotation of the CuO $_2$ plane: an 'odd' order parameter changes sign (associated with $d$ -like symmetry, e.g. $d_{x^2-y^2}$ ), an 'even' one is invariant ( $s$ -like). The clean-limit criterion of the model favors the odd phase for the highest- $T_c$ compounds (YBCO, Bi-based cuprates) and the even phase for lower- $T_c$ , especially electron-doped ones such as Nd $_{2-x}$ Ce $_x$ CuO $_4$ .

Impurities act very differently on the two classes of order parameter. In an isotropic $s$ -wave superconductor, nonmagnetic impurities leave $T_c$ essentially unchanged; for anisotropic order parameters with nonzero angular momentum $L\neq0$ , nonmagnetic scattering mixes states on which the gap has opposite signs and suppresses $T_c$ according to the classical pair-breaking formula (first derived for magnetic impurities in isotropic superconductors and generalized in the 1960s to anisotropic, nonmagnetic scattering): $T_c$ falls with the scattering rate $\tau^{-1}$ (proportional to the residual resistivity) and vanishes at a critical $\tau_c^{-1}$ . An experiment by the Ginsberg–Kirk–Giapintzakis group provided the impetus: 400 keV electron bombardment of YBCO displaces oxygen atoms, creating dipolar defects that scatter quasiparticles at large angles, and the observed $T_c$ suppression fitted the pair-breaking prediction for an odd order parameter (published as Czech. J. Phys. 46 (1996)).

The source paper (A. A. Abrikosov, Phys. Rev. B 53, R8910 (1996)) then computed, within this model, how impurities affect the competition between the two parities. For a tetragonal crystal, isotropic scatterers, and scattering restricted to the singular regions, the odd-phase $T_c$ follows the pair-breaking curve toward zero at $\tau_c^{-1}=(\pi/4\gamma)T_{c0}^{\rm clean}$ (with $\gamma\approx1.78$ ), while the even-phase transition temperature

$T_{ce}=\frac{8m_1\gamma}{\pi}\exp\left[-(\lambda_1-|\lambda_2|)^{-1}\right]$

(with $m_1$ the electronic cutoff energy, about 30 meV from photoemission) is independent of impurity concentration in this approximation. Since the odd-phase $T_c$ reaches zero at a finite impurity content while $T_{ce}$ stays finite, the curves must cross: beyond the crossing the metal should undergo a transition into the even phase, and the measured $T_c$ versus impurity concentration should display a kink at the crossing followed by a long, flat tail (the even-phase $T_{ce}$ being impurity-independent in this approximation).

The paper itself stated what was missing. YBCO is orthorhombic, while the calculation was tetragonal (the orthorhombic case was deferred to a subsequent paper). Scattering into the nonsingular parts of the Fermi surface — regions of much lower density of states — was neglected; the more accurate formulas of the earlier Physica C 244, 243 (1995) paper show that including those channels makes $T_{ce}$ decay slowly with impurity concentration, reaching zero only asymptotically. The scatterers were treated as isotropic, whereas the actual defects are dipolar and strongly anisotropic. The couplings were only indirectly constrained: from the photoemission cutoff and the measured $T_c$ values one obtains $|\lambda_2|\approx0.075$ and $|\lambda_2|/\lambda_1\approx0.21$ , compatible with the assumed hierarchy $|\lambda_2|\ll\lambda_1$ but hedged. Finally, the kink and tail fell outside the experimental window: the irradiation data reached only $T>12$ K, while the estimated transition temperature lies 'in the degree range, or slightly less' — the paper itself calls observing the odd-even transition 'a crucial check of the validity of our concept'.

Problem Statement

Determine, within the extended-saddle-point model of cuprate superconductivity, whether nonmagnetic impurities drive a transition from the odd-parity ( $d$ -like) to the even-parity ( $s$ -like) superconducting phase in YBa $_2$ Cu $_3$ O $_{7-\delta}$ , and make the determination quantitative and experimentally testable. Concretely, complete the calculation of Phys. Rev. B 53, R8910 (1996) by:

extending it from the tetragonal to the orthorhombic Fermi surface of YBa $_2$ Cu $_3$ O $_{7-\delta}$ ;
including the scattering channels that transfer quasiparticles from the extended-saddle-point regions to the nonsingular parts of the Fermi surface, thereby replacing the impurity-independent even-phase $T_{ce}$ of the original approximation with its correct, slowly decaying dependence on impurity concentration;
either determining the interaction parameters $\lambda_1$ and $\lambda_2$ from microscopic calculation or independent measurement, or quantifying how sensitively the predictions depend on them and on the assumed hierarchy $|\lambda_2|\ll\lambda_1$ ; and
confronting the resulting prediction for $T_c$ as a function of impurity concentration — the presence or absence of a kink and a long low-temperature tail marking the odd-to-even transition — with measurements on YBa $_2$ Cu $_3$ O $_{7-\delta}$ extended below the 12 K limit of the original electron-irradiation data into the few-kelvin range where the model places the transition, with the data analyzed within the extended-saddle-point model.

Items (1)-(3) are the theory side of one objective: a quantitatively reliable prediction of the $T_c$ versus impurity-concentration curves of both parities for the real orthorhombic material; item (4) is the experimental test of that single prediction. All four items are required to resolve the task, and they are exhaustive: the source calculation's isotropic-scatterer treatment is retained, and its replacement by anisotropic (dipolar) defect scattering is not part of the task. Item (4) requires confronting the prediction with $T_c$ -versus-defect-content measurements on YBa $_2$ Cu $_3$ O $_{7-\delta}$ below the 12 K limit of the original electron-irradiation data, into the few-kelvin range where the model places the transition, with the data analyzed within the extended-saddle-point model. The answer must settle whether the impurity-driven parity change survives the completed model and whether it is observed.

References

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