- Letter
- Open Access
Disorder-mediated linear and nonlinear magnetotransport in the charge-density-wave material
Phys. Rev. Research 7, L042056 – Published 8 December, 2025
DOI: https://doi.org/10.1103/1zzj-lcf1
Abstract
We report disorder-mediated first-order linear and higher-order nonlinear (magneto-)transport of (TNS) in the charge-density-wave (CDW) regime. CDW transition temperature () and carrier density are proportional and inversely proportional to residual resistance ratio of samples, respectively. Such relation helps to understand the unique CDW order therein. High- TNS exhibits negative first-harmonic magnetoresistance () under a magnetic field () parallel to the direction of alternating current (), which may arise from the anomalous velocity induced by the Berry curvature of three-dimensional topological bands near the Fermi level. As drops, a positive-to-negative transition is observed with decreasing perpendicular , which is likely due to the contribution of Zeeman effect on current pathways in the disordered system. Moreover, interestingly, the second-harmonic nonlinear signals are suppressed, while the third-harmonic signals are significant and sensitive to both and . Such observations, together with scaling analysis, suggest the quantum geometry quadrupole at play and the modulation of disorder on third-order nonlinearity. Our results pave an avenue for tailoring distinct-order magnetoresistive phases in disordered topological materials.
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References (62)
- J. L. Olsen, Electron Transport in Metals (Interscience, New York, 1962).
- A. B. Pippard, Magnetoresistance in Metals (Cambridge University Press, Cambridge, UK, 1989).
- J. Heremans, Solid-state magnetic field sensors and applications, J. Phys. D 26, 1149 (1993).
- X. L. Wang, Y. Du, S. X. Dou, and C. Zhang, Room temperature giant and linear magnetoresistance in topological insulator nanosheets, Phys. Rev. Lett. 108, 266806 (2012).
- M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N.-P. Ong, and R. J. Cava, Large, non-saturating magnetoresistance in , Nature (London) 514, 205 (2014).
- J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, Evidence for the chiral anomaly in the Dirac semimetal , Science 350, 413 (2015).
- T. Liang, Q. Gibson, M. N. Ali, M. H. Liu, R. J. Cava, and N. P. Ong, Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal , Nat. Mater. 14, 280 (2015).
- X. C. Huang, L. X. Zhao, Y. J. Long, P. P. Wang, D. Chen, Z. H. Yang, H. Liang, M. Q. Xue, H. M. Weng, Z. Fang, X. Dai, and G. Chen, Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs, Phys. Rev. X 5, 031023 (2015).
- L. R. Thoutam, Y. L. Wang, Z. L. Xiao, S. Das, A. Luican-Mayer, R. Divan, G. W. Crabtree, and W. K. Kwok, Temperature-dependent three-dimensional anisotropy of the magnetoresistance in , Phys. Rev. Lett. 115, 046602 (2015).
- C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, Z. Liu, Y. Chen, W. Schnelle, H. Borrmann, Y. Grin, C. Felser, and B. Yan, Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP, Nat. Phys. 11, 645 (2015).
- N. Kumar, Y. Sun, N. Xu, K. Manna, M. Y. Yao, V. Süss, I. Leermakers, O. Young, T. Förster, M. Schmidt, H. Borrmann, B. H. Yan, U. Zeitler, M. Shi, C. Felser, and C. Shekhar, Extremely high magnetoresistance and conductivity in the type-II Weyl semimetals and , Nat. Commun. 8, 1642 (2017).
- A. Narayanan, M. D. Watson, S. F. Blake, Y. L. Chen, D. Prabhakaran, B. Yan, N. Bruyant, L. Drigo, C. Felser, T. Kong, P. C. Canfield, and A. I. Coldea, Linear magnetoresistance caused by mobility fluctuations in -doped , Phys. Rev. Lett. 114, 117201 (2015).
- O. Breunig, Z. W. Wang, A. A. Taskin, J. Lux, A. Rosch, and Y. Ando, Gigantic negative magnetoresistance in the bulk of a disordered topological insulator, Nat. Commun. 8, 15545 (2017).
- D. Zhang, W. Jiang, H. Yun, O. J. Benally, T. Peterson, Z. Cresswell, Y. Fan, Y. Lv, G. Yu, J. G. Barriocanal, P. W. Swatek, K. A. Mkhoyan, T. Low, and J.-P. Wang, Robust negative longitudinal magnetoresistance and spin-orbit torque in sputtered and topological semimetal, Nat. Commun. 14, 4151 (2023).
- S. Lai, H. Y. Liu, Z. W. Zhang, J. Z. Zhao, X. L. Feng, N. Z. Wang, C. L. Tang, Y. D. Liu, K. S. Novoselov, S. Y. A. Yang, and W. B. Gao, Third-order nonlinear Hall effect induced by the Berry-connection polarizability tensor, Nat. Nanotechnol. 16, 869 (2021).
- C. Wang, R.-C. Xiao, H. Liu, Z. Zhang, S. Lai, C. Zhu, H. Cai, N. Wang, S. Chen, Y. Deng, Z. Liu, S. A. Yang, and & W.-B. Gao, Room-temperature third-order nonlinear Hall effect in Weyl semimetal , Natl. Sci. Rev. 9, nwac020 (2022).
- A. Y. Gao, Y. F. Liu, J. X. Qiu, B. Ghosh, T. V. Trevisan, Y. Onishi, C. W. Hu, T. M. Qian, H.-J. Tien, S. W. Chen, M. G. Huang, D. Bérubé, H. C. Li, C. Tzschaschel, Z. Sun T.Dinh, S. C. Ho, S. W. Lien, B. Singh, K. Watanabe, T. Taniguchi, D. C. Bell, H. Lin, T. R. Chang, C.-H. R. Du, A. Bansil, L. Fu, N. Ni, P. P. Orth, Q. Ma, and S. Y. Xu, Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure, Science 381, 181 (2023).
- N. Z. Wang, D. Kaplan, Z. W. Zhang, T. Holder, N. Cao, A.-F. Wang, X. Y. Zhou, F. F. Zhou, Z. Z. Jiang, C. S. Zhang, S. H. Ru, H. B. Cai, K. Watanabe, T. Taniguchi, B. H. Yan, and W.-B. Gao, Quantum metric-induced nonlinear transport in a topological antiferromagnet, Nature (London) 621, 487 (2023).
- T.-Y. Zhao, A.-Q. Wang, X.-G. Ye, X.-Y. Liu, X. Liao, and Z.-M. Liao, Gate-tunable Berry curvature dipole polarizability in Dirac semimetal , Phys. Rev. Lett. 131, 186302 (2023).
- P. He, H. Isobe, G. K. W. Koon, J. Y. Tan, J. Hu, J. Li, N. Nagaosa, and J. Shen, Third-order nonlinear Hall effect in a quantum Hall system, Nat. Nanotechnol. 19, 1460 (2024).
- H. Li, C. P. Zhang, C. J. Zhou, C. Ma, X. Lei, Z. J. Jin, H. T. He, B. K. Li, K. T. Law, and J. N. Wang, Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator , Nat. Commun. 15, 7779 (2024).
- X.-Y. Liu, A.-Q. Wang, D. Li, T.-Y. Zhao, X. Liao, and Z.-M. Liao, Giant third-order nonlinearity induced by the quantum metric quadrupole in few-layer , Phys. Rev. Lett. 134, 026305 (2025).
- H. Yu, X. Li, Y.-Q. Bie, L. Yan, L. Zhou, P. Yu, and G. Yang, Quantum metric third-order nonlinear Hall effect in a non-centrosymmetric ferromagnet, Nat. Commun. 16, 7698 (2025).
- Z. Z. Du, C. M. Wang, S. Li, H.-Z. Lu, and X. C. Xie, Disorder-induced nonlinear Hall effect with time-reversal symmetry, Nat. Commun. 10, 3047 (2019).
- H. Liu, J. Zhao, Y.-X. Huang, X. Feng, C. Xiao, W. Wu, S. Lai, W.-B. Gao, and S. A. Yang, Berry connection polarizability tensor and third-order Hall effect, Phys. Rev. B 105, 045118 (2022).
- C.-P. Zhang, X.-J. Gao, Y.-M. Xie, H. C. Po, and K. T. Law, Higher-order nonlinear anomalous Hall effects induced by Berry curvature multipoles, Phys. Rev. B 107, 115142 (2023).
- D. Mandal, S. Sarkar, K. Das, and A. Agarwal, Quantum geometry induced third-order nonlinear transport responses, Phys. Rev. B 110, 195131 (2024).
- S. A. Sunshine and J. A. Ibers, Synthesis, structure, and transport properties of tantalum nickel selenide () and tantalum platinum selenide , Inorg. Chem. 25, 4355 (1986).
- R. M. Fleming, S. A. Sunshine, C. H. Chen, L. F. Schneemeyer, and J. V. Waszczak, Defect-inhibited incommensurate distortion in , Phys. Rev. B 42, 4954 (1990).
- Z. X. Dai, C. G. Slough, W. W. McNairy, and R. V. Coleman, Charge-density-wave formation in studied by scanning tunneling microscopy, Phys. Rev. Lett. 69, 1769 (1992).
- A. Spijkerman, A. Meetsma, J. L. de Boer, Y. Gao, and S. van Smaalen, Modulated structure of in its incommensurate charge-density-wave state at 16 K, Phys. Rev. B 52, 3892 (1995).
- J. Lüdecke, M. Schneider, and S. van Smaalen, Independent and distortions in the incommensurately modulated low-temperature structure of , J. Solid State Chem. 153, 152 (2000).
- J. Lüdecke, E. Riedl, M. Dierl, K. Hosseini, and S. V. Smaalen, Influence of niobium on the charge-density-wave transition of , Phys. Rev. B 62, 7057 (2000).
- J. M. He, Y. R. Zhang, L. B. Wen, Y. S. Yang, J. Y. Liu, Y. S. Wu, H. L. Lian, H. Xing, S. Wang, Z. Q. Mao, and Y. Liu, Band dependence of charge density wave in quasi-one-dimensional probed by orbital magnetoresistance, Appl. Phys. Lett. 111, 052405 (2017).
- J. M. He, L. B. Wen, Y. S. Wu, J. Y. Liu, G. X. Tang, Y. S. Yang, H. Xing, Z. Q. Mao, H. Sun, and Y. Liu, Angle-dependent magnetoresistance as a sensitive probe of the charge density wave in quasi-one-dimensional semimetal , Appl. Phys. Lett. 113, 192401 (2018).
- H. P. Chen and Z. T. Sun, Phonon dispersion and electronic structure in quasi-one-dimensional layered single crystal, J. Mater. Chem. C 9, 5162 (2021).
- L. X. Liu, X. Luo, X. C. Yang, Y. Li, W. Q. Tu, M. Cheng, T. Y. Wang, N. Zhou, R. R. Zhang, X. G. Zhu, W. H. Song, W. J. Lu, J. S. Chen, and Y. P. Su, Origin of the anomalous electrical transports in quasi-one-dimensional , J. Phys. Chem. Lett. 14, 10736 (2023).
- M. D. Watson, A. Louat, C. Cacho, S. Choi, Y. H. Lee, M. Neumann, and G. Kim, Spectral signatures of a unique charge density wave in , Nat. Commun. 14, 3388 (2023).
- C. Yao, L. B. Wen, Y. Zhao, Y. L. Zhu, Y. Wang, G. X. Tang, X. X. Yan, J. M. Zhang, Y. X. Huang, H. Sun, Z. Q. Mao, Y. P. Qi, Y. Zheng, and H. Xing, Electric field effect on the charge density wave in the quasi-one-dimensional semimetal , Phys. Rev. B 110, 165136 (2024).
- P. Xiao, X. Sun, Y. Chen, Y. Han, Z.-M. Yu, W. J. Liu, X. Li, W. D. Xiao, and Y. G. Yao, Robust edge states of quasi-1D material and applications in saturable absorbers, Nano Lett. 24, 10402 (2024).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/1zzj-lcf1 for more theoretical calculations and transport data of TNS, which includes Refs. [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56].
- G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
- G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium, Phys. Rev. B 49, 14251 (1994).
- P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
- J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation, Phys. Rev. B 46, 6671 (1992).
- J. C. Gao, Q. S. Wu, C. Persson, and Z. J. Wang, Irvsp: To obtain irreducible representations of electronic states in the VASP, Comput. Phys. Commun. 261, 107760 (2021).
- I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001).
- N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
- J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys. 32, 37 (1964).
- J. Ge, T. Luo, Z. Lin, J. Shi, Y. Liu, P. Wang, Y. Zhang, W. Duan, and J. Wang, Magnetic moments induced by atomic vacancies in transition metal dichalcogenide flakes, Adv. Mater. 33, 2005465 (2021).
- Q. Cheng, G. Xian, Y. Huang, H. Guo, L. Pan, H. Zhou, J. Wang, S. Lv, C. Shen, X. Lin, H. Chen, Y. Li, H. Yang, and H.-J. Gao, Three-dimensional weak localization and negative magnetoresistance in high-quality crystals, Sci. China Mater. 66, 2393 (2023).
- F. Arnold, C. Shekhar, S. C. Wu, Y. Sun, R. D. dos Reis, N. Kumar, M. Naumann, M. O. Ajeesh, M. Schmidt, A. G. Grushin, J. H. Bardarson, M. Baenitz, D. Sokolov, H. Borrmann, M. Nicklas, C. Felser, E. Hassinger, and B. Yan, Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP, Nat. Commun. 7, 11615 (2016).
- X. Dai, Z. Z. Du, and H. Z. Lu, Negative magnetoresistance without chiral anomaly in topological insulators, Phys. Rev. Lett. 119, 166601 (2017).
- B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. J. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature (London) 547, 298 (2017).
- L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007).
- H.-Z. Lu and S.-Q. Shen, Quantum transport in topological semimetals under magnetic fields, Front. Phys. 12, 127201 (2017).
- T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Catalogue of topological electronic materials, Nature (London) 566, 475 (2019).
- M. G. Vergniory, B. J. Wieder, L. Elcoro, S. S. P. Parkin, C. Felser, B. A. Bernevig, and N. Regnault, All topological bands of all nonmagnetic stoichiometric materials, Science 376, eabg9094 (2022).
- X. K. Zhang, L. Gao, H. H. Yu, Q. L. Liao, Z. Kang, Z. Zhang, and Y. Zhang, Single-atom vacancy doping in two-dimensional transition metal dichalcogenides, Acc. Mater. Res. 2, 655 (2021).
- J. Wang, H. Li, C. Chang, K. He, J. S. Lee, H. Lu, Y. Sun, X. Ma, N. Samarth, S. Shen, Q. Xue, M. Xie, and M. H. W. Chan, Anomalous anisotropic magnetoresistance in topological insulator films, Nano Res. 5, 739 (2012).
- J. Xu, M. K. Ma, M. Sultanov, Z. L. Xiao, Y.-L. Wang, D. F. Jin, Y. Y. Lyu, W. Zhang, L. N. Pfeiffer, K. W. West, K. W. Baldwin, M. Shayegan, and W. K. Kwok, Negative longitudinal magnetoresistance in gallium arsenide quantum wells, Nat. Commun. 10, 287 (2019).
- X. Sun et al., Dataset for “Disorder-mediated linear and nonlinear magnetotransport in charge-density-wave ”, Fig- share, Dataset (2025), https://doi.org/10.6084/m9.figshare.29586974.v1.