Is the linear c-axis magnetoresistance of underdoped YBa$_2$Cu$_3$O$_{6+δ}$ caused by quantum-limit suppression of resonant-tunneling coherence?

The c-axis (interplane) resistivity $ρ_c(H)$ of underdoped YBa$_2$Cu$_3$O$_{6+δ}$ ($T_c ≈ 60$ K, magnetic field $H∥c$) exhibits a large, positive magnetoresistance that is strictly linear in $H$ up to the highest fields applied in the original pulsed-field measurements ($\approx 55$ T) and quasilinear at high field in later measurements up to 60 T, with a slope that grows strongly on cooling. A. A. Abrikosov (Phys. Rev. B 61, 5928 (2000)) proposed that this linearity arises from Landau quantization of the in-plane carrier motion, which suppresses quantum coherence between resonant tunneling centers located in the median planes between CuO$_2$ layers, and derived $ρ(H) = ρ_{coh}^{(0)}[1 + (1 − ρ_{coh}^{(0)}/ρ_{in}^{(0)})·qeH/(cn_i)]$ in the limit $eH/ℏcn_i ≪ 1$ under the quantum-limit condition $exp(−eHd/αc) ≪ 1$. Abrikosov himself flagged two corrections to be resolved before comparison with experiment: the uncomputed field dependence of the activation energy $E−μ$ entering the zero-field coherent and incoherent resistivities (which 'could drastically change the magnetoresistance, particularly at low temperatures'); and the superconducting-fluctuation background near $T_c$; the averaging $ρ_0^2 → q/n_i$ standing in for a microscopic treatment of resonant-center disorder statistics is presented in the source as a step of the derivation rather than as a flagged open issue. High-field measurements a decade later (Vignolle et al., 2012) confirmed the strong low-temperature c-axis magnetoresistance as a normal-state property but reinterpreted it as semiclassical two-band orbital magnetoresistance from Fermi-surface reconstruction, without testing the resonant-tunneling mechanism; and the zero-field resonant-tunneling theory was later microscopically refined (Gutman and Maslov, 2007–2008) but never extended to finite field. The mechanism question — including the field-dependent activation energy and the fluctuation interplay that the source itself flagged as corrections to be resolved — remains open.

Background

YBa$_2$Cu$_3$O$_{6+δ}$ (YBCO) is a layered cuprate high-temperature superconductor: superconducting CuO$_2$ planes are stacked along the crystallographic $c$ axis and separated by charge-reservoir layers, so the electrical resistivity is strongly anisotropic. 'Underdoped' means the hole concentration in the CuO$_2$ planes is below the value at which the superconducting transition temperature $T_c$ is maximal; in this regime in-plane transport is metallic-like while $c$-axis (interplane) transport is insulating-like, and a 'pseudogap' suppresses the electronic density of states at the Fermi level $μ$. Because superconductivity masks the normal state at low fields, strong magnetic fields applied parallel to the $c$ axis are the standard tool for suppressing superconductivity and exposing the underlying normal-state transport; the magnetoresistance (MR) is the field-induced change of resistivity.

In pulsed fields up to roughly 55 T, the $c$-axis resistivity $ρ_c(H)$ of underdoped YBCO crystals with $T_c ≈ 60$ K displays a large, positive MR that is strictly linear in $H$ over the whole measured range, with a slope that grows strongly on cooling (Balakirev et al., 2000); later high-field studies extended the measurements to 60 T and found the high-field MR quasilinear (Vignolle et al., 2012); at smaller fields the MR is quadratic in $H$, as classical orbital MR predicts. A linear-in-field MR of fixed sign over tens of tesla is unusual: Abrikosov's quantum linear magnetoresistance — linear because only the lowest Landau level participates — had been derived for semimetals, not for layered superconductors carrying current parallel to the field.

Abrikosov (2000) proposed an explanation within the resonant-tunneling model of $c$-axis transport that he had developed in a 1999 review. In this picture, $c$-axis conduction in underdoped cuprates is dominated by two-hop tunneling via localized resonant centers — impurity states associated with oxygen atoms — located in the median plane between adjacent CuO$_2$ planes a distance $d ≈ 12$ Å apart. A center whose bound-state energy $E$ lies near the Fermi energy $μ$ greatly enhances tunneling; the barrier height $U$ above $E$ enters the inverse decay length $α = √(2m(U−E))$ (the source uses units with $ℏ=1$ except where common units are explicitly noted). Tunneling through two centers of equal binding energy can be coherent (tunneling amplitudes add; in-plane momentum approximately conserved) or incoherent (probabilities add; momentum not conserved). At zero field the two-center tunneling probability is $W_2 = 2W_1[1 + exp(−αρ_0^2/2d)]$, where $ρ_0$ is the in-plane separation of the two centers. Abrikosov then included Landau quantization: for $H∥c$ the in-plane motion is quantized into Landau levels, and in the quantum-limit regime $exp(−eHd/αc) ≪ 1$ only the lowest level contributes, giving $W_2 = 2W_1[1 + exp(−(eHρ_0^2/4c)·coth(eHd/2αc))]$ — the field suppresses the coherent part while leaving the incoherent part intact. Averaging over many centers through the phenomenological substitution $ρ_0^2 → q/n_i$ (with $q ∼ 1$ and $n_i$ the areal concentration of resonant centers) yields an interpolation formula for the conductivity,

$σ(H) = σ_{coh}^{(0)}·exp(−qeH/(cn_i)) + σ_{in}^{(0)}·[1 − exp(−qeH/(cn_i))],$

where $σ_{coh}^{(0)}$ and $σ_{in}^{(0)}$ are the zero-field coherent and incoherent contributions. In the weak-field limit $eH/ℏcn_i ≪ 1$ (written in common units, as in the source, which otherwise assumes $ℏ=1$) the resistivity becomes linear in field:

$ρ(H) = ρ_{coh}^{(0)}·[1 + (1 − ρ_{coh}^{(0)}/ρ_{in}^{(0)})·qeH/(cn_i)].$

The mechanism requires shallow resonant states ($U−E < 0.1$ eV for the quantum-limit condition to hold at $H ∼ 30$ T) and $n_i ≳ 10^{13}$ cm$^{−2}$ (atomic-scale concentrations above roughly 1%); the source judges both conditions definitely satisfied for YBCO not too close to the metal-insulator transition.

Abrikosov explicitly flagged two corrections to be considered before comparing this result with experiment: (i) the linear law 'may be incomplete' because the activation energy $E−μ$ entering $ρ_{coh}^{(0)}$ and $ρ_{in}^{(0)}$ can itself depend on $H$ — a correction that 'could drastically change the magnetoresistance, particularly at low temperatures'; and (ii) the measured data were taken rather close to $T_c$, where superconducting fluctuations — whose conductivity in layered superconductors in perpendicular magnetic field was computed by Dorin et al. (1993) — have their own strong $H$-dependence, so comparison requires high fields and temperatures far enough above $T_c$. The averaging $ρ_0^2 → q/n_i$ is itself phenomenological, without a microscopic treatment of the statistics of center positions, the distribution of binding energies, or the temperature dependence of the centers' occupation, but the source presents it as a step of the derivation rather than as a flagged open issue.

Modern context: quantum-oscillation measurements (periodic oscillations of the resistivity in $1/H$ produced as Landau levels successively cross the Fermi level; their frequencies map the Fermi-surface geometry) and Hall measurements established that below the onset temperature of charge order — a static modulation of the electronic charge within the CuO$_2$ planes — the Fermi surface of underdoped YBCO is reconstructed into small electron and hole pockets. In that framework a strong semiclassical two-band orbital MR is expected, and Vignolle et al. (2012) fitted the strong low-temperature $ρ_c(H)$ of underdoped YBCO up to 60 T with the two-band form $ρ_{c0} + aB^2/(1 + bB^2)$. This is the main competing interpretation of the linear c-axis MR, mechanistically distinct from Abrikosov's quantum-limit resonant-tunneling proposal.

Problem Statement

Decide whether Abrikosov's resonant-tunneling quantum-limit mechanism is the correct explanation of the large, positive, linear-in-$H$ $c$-axis magnetoresistance of underdoped YBa$_2$Cu$_3$O$_{6+δ}$ for $H∥c$ — by establishing, refining, or falsifying the mechanism through the gaps the source paper itself identified. Within the resonant-center model of $c$-axis transport (localized centers in the median planes between CuO$_2$ layers, coherent plus incoherent two-center tunneling), the corrections the source paper itself flags as still unresolved are: (1) the field dependence of the activation energy $E−μ$ entering the zero-field resistivities $ρ_{coh}^{(0)}$ and $ρ_{in}^{(0)}$ — the correction the source calls the most important — and whether the linear law $ρ(H) = ρ_{coh}^{(0)}·[1 + (1 − ρ_{coh}^{(0)}/ρ_{in}^{(0)})·qeH/(cn_i)]$ survives once this dependence is included, particularly at low temperatures; and (2) the superconducting-fluctuation contribution to $ρ_c(H,T)$ near $T_c$ and its interplay with the tunneling magnetoresistance, so that theory and experiment can be compared in the regime of high fields and temperatures sufficiently far above $T_c$. The mechanism is subject to the source's stated admissibility conditions: shallow resonant states $U−E < 0.1$ eV, areal density $n_i ≳ 10^{13}$ cm$^{−2}$, and the quantum-limit condition $exp(−eHd/αc) ≪ 1$ at fields of order 30–60 T — conditions the source itself judges satisfied for YBCO not too close to the metal-insulator transition. The mechanism is tenable only if it accounts quantitatively for the observed $ρ_c(H,T)$ within these admissibility conditions and is distinguishable from the competing two-band orbital-magnetoresistance account; otherwise it is refuted by showing that no admissible parameters reconcile it with experiment.

References

1. A. A. Abrikosov, 'Linear c-axis magnetoresistance in underdoped YBa2Cu3O6+δ', Physical Review B 61, 5928–5929 (2000). DOI: 10.1103/PhysRevB.61.5928, https://doi.org/10.1103/PhysRevB.61.5928 (source paper).
2. F. F. Balakirev, Yoichi Ando, A. Passner, J. B. Betts, L. F. Schneemeyer, Kouji Segawa, G. S. Boebinger, 'Large, linear c-axis magnetoresistance in YBa2Cu3O7−δ', Physica C 341–348, 1877–1878 (2000). DOI: 10.1016/S0921-4534(00)01236-3, https://doi.org/10.1016/S0921-4534(00)01236-3 (the motivating experiment, cited as unpublished in the source paper).
3. A. A. Abrikosov, 'Resonant tunneling in high-Tc superconductors (review)', Physica C 317–318, 154–174 (1999). DOI: 10.1016/S0921-4534(99)00056-8, https://doi.org/10.1016/S0921-4534(99)00056-8 (the review establishing the resonant-tunneling model of c-axis transport; cited as Ref. 3 of the source paper).
4. V. V. Dorin, R. A. Klemm, A. A. Varlamov, A. I. Buzdin, D. V. Livanov, 'Fluctuation conductivity of layered superconductors in a perpendicular magnetic field', Physical Review B 48, 12951–12965 (1993). DOI: 10.1103/PhysRevB.48.12951, https://doi.org/10.1103/PhysRevB.48.12951 (cited as Ref. 4 of the source paper for the superconducting-fluctuation contribution).
5. A. A. Abrikosov, 'Resonant tunneling through two symmetric centers', Physical Review B 66, 212501 (2002). DOI: 10.1103/PhysRevB.66.212501, https://doi.org/10.1103/PhysRevB.66.212501 (Abrikosov's own follow-up extending resonant tunneling to 214-type cuprates at zero field; does not revisit the magnetoresistance mechanism).
6. B. Vignolle, B. J. Ramshaw, James Day, David LeBoeuf, Stéphane Lepault, Ruixing Liang, W. N. Hardy, D. A. Bonn, Louis Taillefer, Cyril Proust, 'Coherent c-axis transport in the underdoped cuprate superconductor YBa2Cu3Oy', Physical Review B 85, 224524 (2012). DOI: 10.1103/PhysRevB.85.224524, https://doi.org/10.1103/PhysRevB.85.224524, arXiv:1107.5422.
7. T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R. Okazaki, Y. Matsuda, 'Field-induced quantum critical route to a Fermi liquid in high-temperature superconductors', Proceedings of the National Academy of Sciences USA 105, 7120–7123 (2008). DOI: 10.1073/pnas.0712292105, https://doi.org/10.1073/pnas.0712292105, arXiv:0805.2215.
8. T. Shibauchi, L. Krusin-Elbaum, M. Hasegawa, Y. Kasahara, R. Okazaki, Y. Matsuda, 'Evidence for field-induced quantum critical route to a Fermi liquid in overdoped Tl2Ba2CuO6+x', Journal of Physics: Conference Series 150, 052234 (2009). DOI: 10.1088/1742-6596/150/5/052234, https://doi.org/10.1088/1742-6596/150/5/052234.
9. L. Krusin-Elbaum, T. Shibauchi, Y. Kasahara, R. Okazaki, Y. Matsuda, R. D. McDonald, C. H. Mielke, M. Hasegawa, 'Interlayer magnetotransport in the overdoped cuprate Tl2Ba2CuO6+x: Quantum critical point and its downslide in an applied magnetic field', Physical Review B 82, 144530 (2010). DOI: 10.1103/PhysRevB.82.144530, https://doi.org/10.1103/PhysRevB.82.144530.
10. D. B. Gutman, D. L. Maslov, 'Anomalous c-axis transport in layered metals', Physical Review Letters 99, 196602 (2007). DOI: 10.1103/PhysRevLett.99.196602, https://doi.org/10.1103/PhysRevLett.99.196602, arXiv:0704.0613.
11. D. B. Gutman, D. L. Maslov, 'Boson-assisted tunneling in layered metals', Physical Review B 77, 035115 (2008). DOI: 10.1103/PhysRevB.77.035115, https://doi.org/10.1103/PhysRevB.77.035115.
12. A. J. Schofield, J. R. Cooper, 'Quasilinear magnetoresistance in an almost two-dimensional band structure', Physical Review B 62, 10779–10784 (2000). DOI: 10.1103/PhysRevB.62.10779, https://doi.org/10.1103/PhysRevB.62.10779 (a competing semiclassical account of quasilinear magnetoresistance, cited alongside the source paper in Shibauchi et al. 2008).
13. B. S. Tewari, A. Dhyani, Ajay, 'Influence of inter cell resonant tunneling on the out-of-plane electronic transport behavior in layered high Tc cuprates', The European Physical Journal B 66, 67–74 (2008). DOI: 10.1140/epjb/e2008-00375-6, https://doi.org/10.1140/epjb/e2008-00375-6 (zero-field resonant-tunneling model of c-axis transport).