Microscopic validity of coherent resonant tunneling through localized interlayer states in high-$T_c$ cuprates: decoherence, eigenvalue scatter, finite-range potentials, and disorder

In layered cuprate superconductors such as $\mathrm{YBa_2Cu_3O_x}$ and $\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$, the out-of-plane ($c$-axis) resistivity is anomalously large and grows as doping decreases. A. A. Abrikosov (1995-1997) proposed that this arises from resonant tunneling between $\mathrm{CuO_2}$ planes through localized states created by oxygen defects, and in Phys. Rev. B 55, 11735 (1997) verified the mechanism's two key assumptions on exactly solvable one- and two-center models, while explicitly leaving open whether the required coherence of the tunneling amplitudes survives in real materials. The open problem is the microscopic, quantitative determination of whether coherent resonant tunneling through ensembles of localized centers survives in real cuprate interlayers: environmental decoherence mechanisms were never quantified; the crossover criterion for the tolerable spread of center energies was never computed; the ad hoc ultraviolet cutoff standing in for the localized potential was never replaced by a finite-range microscopic potential; and the heuristic substitution of coherent amplitude sums by a center density was never replaced by statistical averaging over disordered center positions and energies. None of these items has been resolved in the literature to date.

Background

The cuprate high-$T_c$ (high superconducting transition temperature $T_c$) superconductors are layered oxides in which superconductivity resides in the $\mathrm{CuO_2}$ planes; transport perpendicular to the planes (along the crystallographic $c$ axis) is far worse than in-plane transport, with resistivity $\rho_c$ up to $10^5$ times larger than the in-plane value and semiconducting-like temperature dependence in the underdoped regime (oxygen or carrier concentrations below those at which $T_c$ is maximal). Abrikosov proposed that $c$-axis transport proceeds not by direct tunneling through the insulating interlayer but by resonant tunneling: charge carriers cross via localized states (bound at oxygen defects in the chains or interlayer oxides) whose energy nearly matches the carrier energy, so that the transmission through two barriers in series can exceed the direct single-barrier transmission by orders of magnitude. In this mechanism the tunneling amplitude through each localized center adds coherently across centers, so $\rho_c$ is controlled by how many centers satisfy the resonance condition rather than by the bare barrier opacity. He used this to explain the measured dependence of $\rho_c$ on oxygen content $x$ and on temperature in $\mathrm{YBa_2Cu_3O_x}$ (Phys. Rev. B 52, R7026 (1995)) and the pseudogap (a suppression of low-frequency spectral weight relative to a normal metal) in the $c$-axis optical conductivity (Phys. Rev. B 54, 12003 (1996)).

The 1997 paper (Phys. Rev. B 55, 11735 (1997)) tests the theory's two load-bearing assumptions on simple models. First, geometric resonance: in a one-dimensional rectangular barrier of height $U$ and width $d$ (the interlayer spacing) containing an attractive delta-function center (a point-like potential whose strength is measured by a parameter $\beta$) at position $z_0$, resonant transmission occurs when $2\alpha=\beta$, where $\alpha=[2m(U-E_0)]^{1/2}/\hbar$ is the inverse decay length of the wavefunction under the barrier at the electron energy $E_0$ at which resonance is achieved; the center may be displaced from the barrier midpoint by up to $1/\alpha$ (about $2\ \mathrm{\AA}$ for $U-E_0=1$ eV), which tolerates the roughly $1\ \mathrm{\AA}$ offset of localized states in the BiO layers of $\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$, so resonance survives realistic geometric disorder. Second, coherence: in a three-dimensional two-center model in the median plane, a strict delta-function potential in 3D supports no bound states, so the momentum integral was cut off at $k=K$ as a stand-in for a finite-range potential; coherent amplitude addition requires the transverse separation $\rho_0$ of the two centers to satisfy $\rho_0\ll\sqrt{d/\alpha}$, and for many centers the coherent-sum bracket was replaced heuristically by the center density, giving roughly one center per area $d/\alpha$ (about $60\ \mathrm{\AA^2}$ for $\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$ with $d=15\ \mathrm{\AA}$ and $1/\alpha=4\ \mathrm{\AA}$). The theory's basic small parameter is $e^{-\alpha d}\ll1$ (about 0.024 for those parameters), and near resonance the principal-value part of the level shift and the finite resonance width were neglected (the resonance was replaced by a delta function).

The source program itself flags what remains unproven (Phys. Usp. 41, 605 (1998), Sections 2-3 and Appendices I-II, which reproduce the 1997 models): the remaining open question is the coherence of the resonant states, which may be violated by the environment or by the distance between centers; the scatter of center eigenvalues is acknowledged without a crossover criterion; the cutoff $K$ is acknowledged as ad hoc, with the finite range and shape of realistic localized potentials left unexplored; and the many-center treatment is a heuristic density substitution with no statistical averaging over disordered center positions or energies and no interactions between centers beyond simple amplitude addition. The author's subsequent work (Phys. Rev. B 57, 7488 (1998); Phys. Usp. 41, 605 (1998)) analyzed only the incoherence caused by energy mismatch and large separation, with the coherence interval - the energy window within which centers of nearly equal energy act coherently - of order $\Omega\sim(U-E_0)e^{-\alpha d}$.

Problem Statement

Determine, microscopically and quantitatively, whether coherent resonant tunneling through ensembles of localized interlayer states survives in real high-$T_c$ cuprates, in the setting of Abrikosov's 1997 models (rectangular interlayer barrier with localized centers, resonance condition $2\alpha=\beta$, basic small parameter $e^{-\alpha d}\ll1$). Concretely: (1) determine and quantify the effect of environmental factors on the coherence required for amplitude addition (candidate mechanisms include electron-phonon and electron-electron interactions and fluctuations), and compare that effect with the resonant linewidth/coherence interval $\Omega\sim(U-E_0)e^{-\alpha d}$; (2) compute the crossover criterion for how large a spread of the localized-state eigenvalues can be tolerated by the resonant coherent contribution, beyond the order-of-magnitude coherence window $\Omega\sim(U-E_0)e^{-\alpha d}$ - including the quantitative suppression law for spreads of order $\Omega$; (3) replace the ad hoc ultraviolet cutoff $K$ of the three-dimensional treatment with a finite-range microscopic localized potential and establish how the results depend on its range and shape; and (4) treat the disorder in the spatial arrangement and energetic distribution of the centers, and the interactions between centers beyond simple additive amplitudes, by an explicit statistical average rather than the heuristic substitution of the coherent-sum bracket by the center density. These four items are the linked sub-issues of one scientific objective: establishing (or excluding) the microscopic validity of the coherence assumption on which the resonant-tunneling explanation of anomalous $c$-axis transport in cuprates rests. Establishing the microscopic validity requires all four to hold, whereas a demonstrated failure of any single one excludes it.

References

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