In a 2002 Physical Review B brief report, A. A. Abrikosov proposed that the out-of-plane (c-axis) charge transport of 214-family cuprates such as La2−xSrxCuO4 — where doping is done by substituting La3+ with Sr2+ or Ba2+ rather than by adding oxygen — proceeds by resonant tunneling through the two symmetric La(Sr) layers lying between adjacent CuO2 planes. In a one-dimensional model with delta-function potentials he found two resonances at β/2α≈1±e−2αa\beta/2\alpha \approx 1 \pm e^{-2\alpha a} whose transparency reaches order unity when the interplane distance dd exceeds 4a4a (aa being the distance of each layer from the median plane), a condition satisfied by La214 crystallography, and he explicitly deferred the detailed, material-specific theory. The open problem is the construction of that theory: a three-dimensional, finite-range treatment of two-center resonant tunneling in 214 cuprates and its testable predictions for interlayer conductivity.

Background

High-temperature superconducting cuprates are layered crystals whose key electronic units are the CuO2 planes; conductivity along the crystallographic c axis (perpendicular to the planes) is far weaker than within the planes and is typically described as tunneling through the blocking layers between adjacent CuO2 planes. The '214 family' comprises La2−xCuO4-based compounds (so named for the La2CuO4 stoichiometry), e.g. La2−xSrxCuO4 and La2−xBaxCuO4, in which doping is achieved not by adding oxygen but by substituting La3+ with divalent Sr2+ or Ba2+; the substituent atoms sit in two La(Sr) layers located between neighboring CuO2 planes, off the median plane. Near the doping level x=1/8x = 1/8 these compounds show a pronounced dip of the superconducting transition temperature TcT_c, first reported by Moodenbaugh et al. (1988). Resonant tunneling denotes the enhancement of the tunneling probability (the 'transparency' of the barrier) to values of order unity — instead of the exponentially small direct-tunneling probability e−αde^{-\alpha d}, with dd the barrier thickness and α=2m(U−E)\alpha=\sqrt{2m(U-E)} for barrier height UU and particle energy EE — when the barrier contains a localized bound state whose energy matches the tunneling particle's energy. Abrikosov had earlier proposed (Phys. Usp. 41, 605 (1998); Physica C 317–318, 154 (1999)) that c-axis transport in oxygen-doped cuprates such as YBa2Cu3O7−δ and Bi2Sr2CaCu2O8+δ proceeds by resonant tunneling via localized states at oxygen atoms in or near the median plane, a picture he used to explain the pseudogap state (a partial suppression of low-energy electronic excitations in the normal state of underdoped cuprates). That model does not apply to the 214 family, because the dopant sites are not in the median plane. Abrikosov's 2002 paper therefore asked whether two symmetric but otherwise arbitrary localized centers — the two La(Sr) layers at positions z=±az=\pm a between CuO2 planes separated by dd — can act as a resonant chain. He solved the one-dimensional Schrödinger equation with delta-function potentials of strength β\beta at z=±az=\pm a, assuming small direct tunneling (e−αd≪1e^{-\alpha d}\ll 1) and aa of the same order as dd. The exact result contains two resonances at β/2α≈1±e−2αa\beta/2\alpha \approx 1 \pm e^{-2\alpha a}. If d>4ad>4a (the two centers closer than half the barrier thickness), the transparency at each resonance is TR=4(E/U)(1−E/U)T_R = 4(E/U)(1-E/U), i.e. of order unity, with a deep minimum between the two maxima — a split-peak energy dependence; if 2a<d<4a2a<d<4a, the maxima merge into one and the transparency is exponentially small, ∝e−2α(4a−d)\propto e^{-2\alpha(4a-d)}, yet still exponentially larger than direct tunneling. Using the crystallographic data of Radaelli et al. (1994) for La2−xSrxCuO4 (x = 0.2, 295 K) — nearest CuO2-plane spacing d=6.623d = 6.623 Å and CuO2-to-La(Sr) distance 1.84531.8453 Å, giving 4a=5.8664a = 5.866 Å <d< d — he concluded that the La214 structure falls in the favorable, order-unity-transparency regime. The author stressed, however, that the model is one-dimensional with strict delta-function potentials and should be regarded only as a 'useful hint'; that the goal was to check the possibility of the mechanism 'and not a detailed theory, which is very hard to construct because of many unknown factors' (e.g. the strength parameter β\beta and the interlayer coupling coefficient); that the mechanism is 'rather delicate and depends on many factors'; and that a more detailed model might explain the TcT_c dip at x=0.125x=0.125 if regular La/Sr(Ba) ordering there destroyed the resonant chains. It should be noted that the mainstream account of the x=1/8x = 1/8 anomaly is instead charge/spin stripe order — the self-organization of doped holes into line-like regions separating antiferromagnetically ordered domains, as detected by neutron scattering (Tranquada et al., Nature 375, 561 (1995)) — a competing framework within which Abrikosov's suggestion was not developed.

Problem Statement

Construct the quantitative, material-specific theory of resonant tunneling through the two symmetric La(Sr) layers between adjacent CuO2 planes in 214-family cuprates that Abrikosov's 2002 one-dimensional delta-function model deliberately omitted, with the single overarching objective of deciding whether the two-center resonant-tunneling mechanism is quantitatively viable for c-axis charge transport in these compounds. The jointly required results are: (i) compute the tunneling transparency of the two-center barrier in a three-dimensional model with finite-range short-range potentials at the crystallographically located La/Sr sites, and determine how the resonance conditions β/2α≈1±e−2αa\beta/2\alpha \approx 1 \pm e^{-2\alpha a}, the resonance amplitudes, and the criterion d>4ad > 4a are modified relative to the one-dimensional result (for La2−xSrxCuO4, d=6.623d = 6.623 Å and 4a=5.8664a = 5.866 Å); and (ii) derive the resulting interlayer (c-axis) conductivity with its doping and temperature dependence and confront it with measured c-axis transport of 214 compounds. The material-specific inputs — the effective potential and bound-state energy at the La/Sr site and the interlayer coupling coefficient — are currently unknown; these inputs must either be derived microscopically, be constrained by independent experiment, or their uncertainty must be shown to leave the conclusions robust. A quantitative demonstration that the mechanism cannot operate in realistic 214 material parameters also answers the question, in the negative. The paper's speculative suggestion that a more detailed model might explain the TcT_c dip at x=1/8x=1/8 in La2−xBaxCuO4 through regular La/Sr(Ba) ordering destroying the resonant chains, and the fate of the exponentially small splitting under disorder and broadening, are background motivation and possible applications, not required completion conditions.

References

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