Unresolved validation program for Abrikosov's single-Landau-band quantum magnetoresistance mechanism in layered semimetals

A. A. Abrikosov (Phys. Rev. B 60, 4231 (1999)) proposed that the large, nonsaturating, linear-in-field magnetoresistance (MR) of layered semimetals — motivated by the layered rare-earth diantimonides and by graphite — arises in the extreme quantum limit, where only the lowest Landau band of a quasi-two-dimensional graphitelike spectrum is occupied and non-Born impurity scattering gives $\rho_{xx}\propto H\,N_i/n_0^2$. The paper itself identified open questions it did not resolve and suggested experiments it did not perform: Hall determination of the excess-carrier density $n_0$ to test the central condition $n_0>N_i$ and the predicted temperature dependence, with the explicit falsification branch that an exactly half-filled band ($n_0=0$, no linear-in-field Hall effect) "would mean that our model is wrong"; the conflict between single-Landau-band conductivity and the observed Shubnikov–de Haas oscillations, whose origin (bulk, leads, or magnetic breakdown) had to be clarified; approximations whose robustness the paper acknowledged but did not establish — the non-Born scattering rate's dependence on the impurity model and screening, with order-unity prefactors; and isolation of the electronic mechanism from magnetic effects using nonmagnetic graphitelike realizations such as controlled intercalated graphite. The validation-or-falsification of the mechanism remains open.

Background

Linear, nonsaturating magnetoresistance (MR) — resistivity growing strictly proportionally to magnetic field $H$ without saturation at high field — has been known in graphite and related layered and narrow-gap materials for roughly a century, and its microscopic explanation became a long-standing problem. A. A. Abrikosov proposed a quantum route. In a gapless semimetal whose carriers have a linear, relativistic-like (Dirac or Weyl) dispersion, a strong magnetic field can reach the extreme quantum limit (EQL), in which Landau quantization (the discretization of closed cyclotron orbits into highly degenerate Landau levels) leaves only the lowest Landau level occupied. His 1998 theory for three-dimensional gapless semiconductors ('quantum magnetoresistance', Phys. Rev. B 58, 2788) showed that, with impurity scattering treated beyond the Born approximation — resumming the ladder series of repeated impurity scatterings, which is necessary because the huge Landau-level degeneracy amplifies scattering — the EQL yields $\rho_{xx}\propto H$.

The 1999 paper, 'Quantum magnetoresistance of layered semimetals' (A. A. Abrikosov, Phys. Rev. B 60, 4231 (1999)), adapted this mechanism to a quasi-two-dimensional semimetal: a graphitelike linear in-plane dispersion $v(\sigma_x p_x+\sigma_y p_y)$ with weak interlayer hopping $-t\cos(p_z d)$ (interlayer bandwidth $2t$, interlayer spacing $d$). It was motivated by the linear MR measured by Bud'ko, Canfield, Mielke, and Lacerda in the layered rare-earth diantimonides and by graphite. When the Landau splitting exceeds both $k_B T$ and $2t$, only the $n=0$ Landau band is occupied; the model then fixes the Hall conductivity $\sigma_{xy}=e c n_0/H$ in terms of the excess carrier density $n_0$ (the small imbalance between electron and hole densities), and gives a longitudinal conductivity set by non-Born scattering from $N_i$ strongly screened, effectively pointlike impurities, leading to an in-plane resistivity essentially linear in field,

$$\rho_{xx}=\frac{H\,N_i}{\pi e c\,n_0^{2}}\;F(\theta,m),$$

where $F$ is a dimensionless factor depending on $\theta=T/t$ and $m=\mu/t$ ($\mu$ the chemical potential, itself fixed by $n_0$ and $H$ through the filling of the lowest Landau band). $F$ decreases with increasing temperature, and matching the observed decrease of MR with increasing temperature selects the regime $n_0>N_i$, directly testable through the Hall constant $R=1/(e c n_0)$. The paper acknowledged open issues on which the mechanism stands or falls. First, the Shubnikov–de Haas (SdH) oscillations — periodic-in-$1/H$ resistivity oscillations produced as Landau levels pass through the Fermi level — reported in the diantimonide data have a period implying a large Fermi surface of order one electron per atom, in tension with the small-carrier-density, single-band EQL scenario; the paper proposed the oscillations might instead originate in the metallic measurement leads or in magnetic breakdown (semiclassical tunneling between nearby orbits across a small band gap). Second, the transport derivation retains only the dominant term of the non-Born series, leaving an order-unity (about 1/4) uncertainty in the scattering rate $1/2\tau$, and assumes short-range impurities justified by an estimate that screening is strong — an assumption whose failure would change the result. Third, the candidate materials contain magnetic rare-earth ions and can undergo magnetic or structural phase transitions that modify MR, so electronic and magnetic contributions must be separated, for which nonmagnetic realizations of the graphitelike spectrum (clean graphite with controlled intercalation and doping) were proposed. A further complication, established by later work, is that classical mechanisms produce nearly indistinguishable linear MR: spatial mobility inhomogeneity (Parish and Littlewood) and guiding-center diffusion in a smooth random potential (Song, Refael, and Lee). These later mechanisms — none of which appears in the 1999 paper — form part of the interpretive context of modern tests, in which demonstrating linearity alone is not decisive; the paper's own open questions, however, concern the tests and internal consistency of the QMR mechanism itself.

Problem Statement

Determine whether the large, nonsaturating, linear-in-field magnetoresistance observed in the layered rare-earth diantimonides — and, as proposed, in graphitelike layered semimetals such as graphite, preferably intercalated — arises from the single-Landau-band quantum magnetoresistance (QMR) mechanism of A. A. Abrikosov (Phys. Rev. B 60, 4231 (1999)), by resolving the open questions that the paper itself identified:

(1) Determine the excess-carrier density $n_0$ of the layered rare-earth diantimonides and of graphitelike realizations such as intercalated graphite from the Hall response — the paper proposes Hall measurements as the decisive test, noting that $R=1/(e c n_0)$ holds at high fields and that the Hall constant becomes field-dependent once the $n=1$ band is populated — together with an independent determination of the impurity concentration $N_i$, and test the mechanism's central consistency condition $n_0>N_i$ and the predicted temperature dependence of $\rho_{xx}=H\,N_i/(\pi e c\,n_0^{2})$ (up to a dimensionless factor of $T/t$ and $\mu/t$; in the regime $\sigma_{xx}\ll\sigma_{xy}$) against experiment, including the prediction that the magnetoresistance decreases with increasing temperature only in this impurity-sparse regime, as observed. The paper's falsification branch: in an exactly half-filled band ($n_0=0$) there is no linear-in-field Hall effect, which "would mean that our model is wrong".

(2) Resolve the conflict, stated by the paper, between single-Landau-band conductivity at high fields and the Shubnikov–de Haas oscillations observed in the diantimonide magnetoresistance (whose period implies a large Fermi surface of order one electron per atom): establish whether the oscillations arise from the sample bulk, from the metallic measurement leads, or from magnetic breakdown, and — given the bulk quantum oscillations established in modern samples — whether resolved bulk Shubnikov–de Haas oscillations can coexist with single-Landau-band extreme-quantum-limit occupancy, or whether the assumed spectrum is incorrect.

(3) Establish the robustness of the non-Born resummation result for the scattering rate $1/2\tau$ — an open question arising from approximations the paper explicitly acknowledges (its evaluation "depends on the assumption about $U$", the strongly screened pointlike-impurity model, and an order-unity truncation error, "in the imaginary part the mistake is 1/4") — against alternative impurity models (pointlike versus long-range versus smooth disorder) and against less effective screening, and quantify the prefactor uncertainty left by the dominant-term truncation of the ladder series.

(4) Isolate the purely electronic QMR mechanism from magnetic effects — magnetic rare-earth moments and field- or temperature-driven magnetic phase transitions that modify the magnetoresistance — for which the paper proposes nonmagnetic realizations of the graphitelike spectrum, such as clean graphite with controlled intercalation and doping ("There are no magnetic atoms, and hence no magnetic phase transitions"), using clean and doped samples to compare the predicted temperature dependence.

The common scientific objective uniting these items is the confirmation or refutation of the layered-semimetal single-band QMR mechanism as the explanation of the observed linear magnetoresistance in these systems. The items are separate open diagnostics rather than a jointly required program: each supplies a single-item falsification rule (absence of the predicted linear-in-field Hall response refutes the model; bulk quantum oscillations irreconcilable with single-band occupancy would mean the assumed spectrum is incorrect), while confirmation requires the consistency of all the mechanism's load-bearing assumptions (excess-carrier dominance over impurities, single-band occupancy, the disorder model underlying the scattering rate, and the electronic rather than magnetic origin of the effect).

References

1. A. A. Abrikosov, 'Quantum magnetoresistance of layered semimetals', Physical Review B 60, 4231-4234 (1999). DOI: 10.1103/PhysRevB.60.4231, https://doi.org/10.1103/PhysRevB.60.4231
2. A. A. Abrikosov, 'Quantum magnetoresistance', Physical Review B 58, 2788-2794 (1998). DOI: 10.1103/PhysRevB.58.2788, https://doi.org/10.1103/PhysRevB.58.2788
3. A. A. Abrikosov, 'Quantum linear magnetoresistance', Europhysics Letters 49, 789-793 (2000). DOI: 10.1209/epl/i2000-00220-2, https://doi.org/10.1209/epl/i2000-00220-2
4. A. A. Abrikosov, 'Quantum linear magnetoresistance; solution of an old mystery', Journal of Physics A: Mathematical and General 36, 9119-9131 (2003). DOI: 10.1088/0305-4470/36/35/301, https://doi.org/10.1088/0305-4470/36/35/301
5. S. L. Bud'ko, P. C. Canfield, C. H. Mielke, A. H. Lacerda, 'Anisotropic magnetic properties of light rare-earth diantimonides', Physical Review B 57, 13624-13638 (1998). DOI: 10.1103/PhysRevB.57.13624, https://doi.org/10.1103/PhysRevB.57.13624 (experimental motivation of the 1999 paper)
6. M. M. Parish, P. B. Littlewood, 'Non-saturating magnetoresistance in heavily disordered semiconductors', Nature 426, 162-165 (2003). DOI: 10.1038/nature02073, https://doi.org/10.1038/nature02073
7. Justin C. W. Song, Gil Refael, Patrick A. Lee, 'Linear magnetoresistance in metals: Guiding center diffusion in a smooth random potential', Physical Review B 92, 180204 (2015). DOI: 10.1103/PhysRevB.92.180204, https://doi.org/10.1103/PhysRevB.92.180204
8. R. G. Goodrich, D. Browne, R. Kurtz, D. P. Young, J. F. DiTusa, P. W. Adams, D. Hall, 'de Haas-van Alphen measurements of the electronic structure of LaSb2', Physical Review B 69, 125114 (2004). DOI: 10.1103/PhysRevB.69.125114, https://doi.org/10.1103/PhysRevB.69.125114
9. Igor A. Luk'yanchuk, Yakov Kopelevich, 'Phase Analysis of Quantum Oscillations in Graphite', Physical Review Letters 93, 166402 (2004). DOI: 10.1103/PhysRevLett.93.166402, https://doi.org/10.1103/PhysRevLett.93.166402
10. H. Kempa, P. Esquinazi, Y. Kopelevich, 'Integer quantum Hall effect in graphite', Solid State Communications 138, 118-122 (2006). DOI: 10.1016/j.ssc.2006.02.020, https://doi.org/10.1016/j.ssc.2006.02.020
11. Z. Q. Li, S.-W. Tsai, W. J. Padilla, S. V. Dordevic, K. S. Burch, Y. J. Wang, D. N. Basov, 'Infrared probe of the anomalous magnetotransport of highly oriented pyrolytic graphite in the extreme quantum limit', Physical Review B 74, 195404 (2006). DOI: 10.1103/PhysRevB.74.195404, https://doi.org/10.1103/PhysRevB.74.195404
12. Yuanbo Zhang, Joshua P. Small, Michael E. S. Amori, Philip Kim, 'Electric Field Modulation of Galvanomagnetic Properties of Mesoscopic Graphite', Physical Review Letters 94, 176803 (2005). DOI: 10.1103/PhysRevLett.94.176803, https://doi.org/10.1103/PhysRevLett.94.176803
13. Adam L. Friedman, Joseph L. Tedesco, Paul M. Campbell, James C. Culbertson, Edward Aifer, F. Keith Perkins, Rachael L. Myers-Ward, Jennifer K. Hite, Charles R. Eddy, Glenn G. Jernigan, D. Kurt Gaskill, 'Quantum Linear Magnetoresistance in Multilayer Epitaxial Graphene', Nano Letters 10, 3962-3965 (2010). DOI: 10.1021/nl101797d, https://doi.org/10.1021/nl101797d
14. Iryna Ovsiienko, Lyudmila Matzui, Igor Berkutov, Il'gar Mirzoiev, Tetyana Len, Yuriy Prylutskyy, Oleksandr Prokopov, Uwe Ritter, 'Magnetoresistance of graphite intercalated with cobalt', Journal of Materials Science 53, 716-726 (2017). DOI: 10.1007/s10853-017-1511-x, https://doi.org/10.1007/s10853-017-1511-x
15. Mitsuaki Owada, Yudai Awashima, Yuki Fuseya, 'Corrections to the magnetoresistance formula for semimetals with Dirac electrons: the Boltzmann equation approach validated by the Kubo formula', Journal of Physics: Condensed Matter 30, 445601 (2018). DOI: 10.1088/1361-648X/aae03c, https://doi.org/10.1088/1361-648X/aae03c
16. Alexander Kazantsev, Alexey Berdyugin, Andre Geim, Alessandro Principi, 'On the origin of Abrikosov's quantum linear magnetoresistance', arXiv:2208.06273 (2022). arXiv ID: 2208.06273, https://arxiv.org/abs/2208.06273
17. Ya. I. Rodionov, K. I. Kugel, B. A. Aronzon, 'Quantum magnetoresistance of Weyl semimetals with strong Coulomb disorder', Physical Review B 107, 155120 (2023). DOI: 10.1103/PhysRevB.107.155120, https://doi.org/10.1103/PhysRevB.107.155120
18. Na Xin, James Lourembam, Piranavan Kumaravadivel, A. E. Kazantsev, Zefei Wu, Ciaran Mullan, Julien Barrier, Alexandra A. Geim, I. V. Grigorieva, A. Mishchenko, A. Principi, V. I. Fal'ko, L. A. Ponomarenko, A. K. Geim, Alexey I. Berdyugin, 'Giant magnetoresistance of Dirac plasma in high-mobility graphene', Nature 616, 270-274 (2023). DOI: 10.1038/s41586-023-05807-0, https://doi.org/10.1038/s41586-023-05807-0
19. A. T. Lonchakov, S. B. Bobin, 'Quantum linear magnetoresistance and magnetic-field-induced metal-insulator transition in the Weyl semimetal candidate HgSe', Journal of Applied Physics 135, 235703 (2024). DOI: 10.1063/5.0213546, https://doi.org/10.1063/5.0213546
20. Yuxi Qiao, Fuyi Wang, Huaiqiang Wang, Zicheng Tao, Zhicheng Jiang, Zhengtai Liu, Soohyun Cho, Fayuan Zhang, Qingkai Meng, Wei Xia, Yichen Yang, Zhe Huang, Jishan Liu, Zhonghao Liu, Zengwei Zhu, Shan Qiao, Yanfeng Guo, Haijun Zhang, Dawei Shen, 'Experimental observation of multiple topological nodal structure in LaSb2', Science China Physics, Mechanics & Astronomy 67, 247411 (2024). DOI: 10.1007/s11433-023-2320-7, https://doi.org/10.1007/s11433-023-2320-7 (preprint: arXiv:2208.10437, https://arxiv.org/abs/2208.10437)
21. Xian-Peng Zhang, Yan-Qing Feng, Haiwen Liu, Yugui Yao, 'Magnetoresistance from decoherence', arXiv:2604.17672 (2026). arXiv ID: 2604.17672, https://arxiv.org/abs/2604.17672