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Theory and simulations of critical temperatures in CrI3 and other 2D materials: easy-axis magnetic order and easy-plane Kosterlitz-Thouless transitions

Abstract

The recent observations of ferromagnetic order in several two-dimensional (2D) materials have generated an enormous interest in the physical mechanisms underlying 2D magnetism. In the present Prospective Article, we show that Density Functional Theory combined with either classical Monte Carlo simulations or renormalized spin-wave theory can predict Curie temperatures for ferromagnetic insulators that are in quantitative agreement with experiments. The case of materials with in-plane anisotropy is then discussed, and it is argued that finite size effects may lead to observable magnetic order in macroscopic samples even if long range magnetic order is forbidden by the Mermin–Wagner theorem.

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References

  1. 1.

    N.D. Mermin and H. Wagner: Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models. Phys. Rev. Lett 17, 1133 (1966).

    CAS  Article  Google Scholar 

  2. 2.

    B. Huang, G. Clark, E. Navarro-Moratalla, D.R. Klein, R. Cheng, K.L. Seyler, D. Zhong, E. Schmidgall, M.A. McGuire, D.H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu: Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270 (2017).

    CAS  Article  Google Scholar 

  3. 3.

    Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A.F. May, W. Wu, D.H. Cobden, J.-H. Chu, and X. Xu: Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2. Nat. Mater 17, 778 (2018).

    CAS  Article  Google Scholar 

  4. 4.

    D.J. O’Hara, T. Zhu, A.H. Trout, A.S. Ahmed, Y.K. Luo, C.H. Lee, M.R. Brenner, S. Rajan, J.A. Gupta, D.W. McComb, and R.K. Kawakami: Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit. Nano Lett 18, 3125 (2018).

    Article  CAS  Google Scholar 

  5. 5.

    M. Bonilla, S. Kolekar, Y. Ma, H.C. Diaz, V. Kalappattil, R. Das, T. Eggers, H.R. Gutierrez, M.-H. Phan, and M. Batzill: Strong room-temperature fer-romagnetism in VSe2 monolayers on van der Waals substrates. Nat. Nanotechnol 13, 289 (2018).

    CAS  Article  Google Scholar 

  6. 6.

    Z.-L. Liu, X. Wu, Y. Shao, J. Qi, Y. Cao, L. Huang, C. Liu, J.-O. Wang, Q. Zheng, Z.-L. Zhu, K. Ibrahim, Y.-L. Wang, and H.-J. Gao: Epitaxially grown monolayer VSe2: an air-stable magnetic two-dimensional material with low work function at edges. Sci. Bull 63, 419 (2018).

    Article  CAS  Google Scholar 

  7. 7.

    J.-U. Lee, S. Lee, J.H. Ryoo, S. Kang, T.Y. Kim, P. Kim, C.-H. Park, J.-G. Park, and H. Cheong: Ising-Type Magnetic Ordering in Atomically Thin FePS3. Nano Lett 16, 7433 (2016).

    CAS  Article  Google Scholar 

  8. 8.

    K. Yosida: Theory of Magnetism (Springer Berlin Heidelberg, 1996).

    Book  Google Scholar 

  9. 9.

    C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z.Q. Qiu, R.J. Cava, S.G. Louie, J. Xia, and X. Zhang: Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546, 265 (2017).

    CAS  Article  Google Scholar 

  10. 10.

    J.L. Lado and J. Fernández-Rossier: On the origin of magnetic anisotropy in two dimensional CrI3. 2D Mater 4, 035002 (2017).

    Article  CAS  Google Scholar 

  11. 11.

    D. Torelli and T. Olsen: Calculating critical temperatures for ferromagnetic order in two-dimensional materials. 2D Mater 6, 015028 (2018).

    Article  CAS  Google Scholar 

  12. 12.

    S. Sarikurt, Y. Kadioglu, F. Ersan, E. Vatansever, OÜ Aktürk, Y. Yüksel, Ü Akıncı, and E. Aktürk: Electronic and magnetic properties of monolayer α-RuCl3: a first-principles and Monte Carlo study. Phys. Chem. Chem. Phys 20, 997 (2018).

    CAS  Article  Google Scholar 

  13. 13.

    C. Yasuda, S. Todo, K. Hukushima, F. Alet, M. Keller, M. Troyer, and H. Takayama: Néel Temperature of Quasi-Low-Dimensional Heisenberg Antiferromagnets. Phys. Rev. Lett 94, 217201 (2005).

    CAS  Article  Google Scholar 

  14. 14.

    K. Kim, S.Y. Lim, J.-U. Lee, S. Lee, T.Y. Kim, K. Park, G.S. Jeon, C.-H. Park, J.-G. Park, and H. Cheong: Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS3. Nat. Commun 10, 345 (2019).

    Article  CAS  Google Scholar 

  15. 15.

    D. Torelli, K.S. Thygesen, and T. Olsen: High throughput computational screening for 2D ferromagnetic materials: the critical role of anisotropy and local correlations. 2D Mater 6, 045018 (2019).

    CAS  Article  Google Scholar 

  16. 16.

    P. Chen, W.W. Pai, Y.-H. Chan, V. Madhavan, M.Y. Chou, S.-K. Mo, A.-V. Fedorov, and T.-C. Chiang: Unique Gap Structure and Symmetry of the Charge Density Wave in Single-Layer VSe2. Phys. Rev. Lett 121, 196402 (2018).

    CAS  Article  Google Scholar 

  17. 17.

    A.O. Fumega and V. Pardo: Absence of ferromagnetism in VSe2 caused by its charge density wave phase, arXiv:1804. 07102 (2018).

    Google Scholar 

  18. 18.

    G. Duvjir, B.K. Choi, I. Jang, S. Ulstrup, S. Kang, T. Thi Ly, S. Kim, Y.H. Choi, C. Jozwiak, A. Bostwick, E. Rotenberg, J.-G. Park, R. Sankar, K.-S. Kim, J. Kim, and Y.J. Chang: Emergence of a Metal-Insulator Transition and High-Temperature Charge-Density Waves in VSe2 at the Monolayer Limit. Nano Lett 18, 5432 (2018).

    CAS  Article  Google Scholar 

  19. 19.

    P.K.J. Wong, W. Zhang, F. Bussolotti, X. Yin, T.S. Herng, L. Zhang, Y.L. Huang, G. Vinai, S. Krishnamurthi, D.W. Bukhvalov, Y.J. Zheng, R. Chua, A.T. N’Diaye, S.A. Morton, C. Yang, K. Ou Yang, P. Torelli, W. Chen, K.E.J. Goh, J. Ding, M. Lin, G. Brocks, M.P. de Jong, A.H. Castro Neto, and A.T. S. Wee: Evidence of Spin Frustration in a Vanadium Diselenide Monolayer Magnet. Adv. Mater 31, 1901185 (2019).

    Article  CAS  Google Scholar 

  20. 20.

    P.C.W. Holdsworth and S.T. Bramwell: Magnetization: A characteristic of the Kosterlitz-Thouless-Berezinskii transition. Phys. Rev. B 49, 8811 (1994).

    Article  Google Scholar 

  21. 21.

    P.W. Anderson: Antiferromagnetism. Theory of Superexchange Interaction. Phys. Rev. 79, 350 (1950).

    Article  Google Scholar 

  22. 22.

    P.W. Anderson: New Approach to the Theory of Superexchange Interactions. Phys. Rev. 115, 2 (1959).

    CAS  Article  Google Scholar 

  23. 23.

    C. Xu, J. Feng, H. Xiang, and L. Bellaiche: Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI3 and CrGeTe3 monolayers npj Comput. Mater. 4, 57 (2018).

    Article  CAS  Google Scholar 

  24. 24.

    A. Banerjee, C.A. Bridges, J.-Q. Yan, A.A. Aczel, L. Li, M.B. Stone, G.E. Granroth, M.D. Lumsden, Y. Yiu, J. Knolle, S. Bhattacharjee, D.L. Kovrizhin, R. Moessner, D.A. Tennant, D.G. Mandrus, and S.E. Nagler: Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733 (2016).

    CAS  Article  Google Scholar 

  25. 25.

    M. Heide, G. Bihlmayer, and S. Blügel: Describing Dzyaloshinskii-Moriya spirals from first principles. Phys. B Condens. Matter 404, 2678 (2009).

    CAS  Article  Google Scholar 

  26. 26.

    T. Koretsune, T. Kikuchi, and R. Arita: First-Principles Evaluation of the Dzyaloshinskii-Moriya Interaction. J. Phys. Soc. Jpn. 87, 041011 (2018).

    Article  Google Scholar 

  27. 27.

    J. Liu, M. Shi, J. Lu, and M.P. Anantram: Analysis of electrical-field-dependent Dzyaloshinskii-Moriya interaction and magnetocrystalline anisotropy in a two-dimensional ferromagnetic monolayer. Phys. Rev. B 97, 8 (2018).

    Google Scholar 

  28. 28.

    O. Besbes, S. Nikolaev, N. Meskini, and I. Solovyev: Microscopic origin of ferromagnetism in the trihalides CrCl3 and CrI3. Phys. Rev. B 99, 104432 (2019).

    CAS  Article  Google Scholar 

  29. 29.

    K. Wang, S. Nikolaev, W. Ren, and I. Solovyev: Giant contribution of the ligand states to the magnetic properties of the Cr2Ge2Te6 monolayer. Phys. Chem. Chem. Phys. 21, 9597 (2019).

    CAS  Article  Google Scholar 

  30. 30.

    N. Marzari, A.A. Mostofi, J.R. Yates, I. Souza, and D. Vanderbilt: Maximally localized Wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419 (2012).

    CAS  Article  Google Scholar 

  31. 31.

    A. Görling: Symmetry in density-functional theory. Phys. Rev. A 47, 2783 (1993).

    Article  Google Scholar 

  32. 32.

    A. Görling: Proper Treatment of Symmetries and Excited States in a Computationally Tractable Kohn-Sham Method. Phys. Rev. Lett. 85, 4229 (2000).

    Article  Google Scholar 

  33. 33.

    H. Xiang, C. Lee, H.-J. Koo, X. Gong, and M.-H. Whangbo: Magnetic properties and energy-mapping analysis. Dalt. Trans. 42, 823 (2013).

    CAS  Article  Google Scholar 

  34. 34.

    A. Jacobsson, C. Etz, M. Ležaic, B. Sanyal, and S. Blügel: Frozen Magnon Calculations Beyond the Long Wavelength Approximation, arXiv:1702. 00599 (2017).

    Google Scholar 

  35. 35.

    D. Ködderitzsch, W. Hergert, W.M. Temmerman, Z. Szotek, A. Ernst, and H. Winter: Exchange interactions in NiO and at the NiO(100) surface. Phys. Rev. B 66, 064434 (2002).

    Article  CAS  Google Scholar 

  36. 36.

    M. Pajda, J. Kudrnovský, I. Turek, V. Drchal, and P. Bruno: Ab initio calculations of exchange interactions, spin-wave stiffness constants, and Curie temperatures of Fe, Co, and Ni. Phys. Rev. B 64, 174402 (2001).

    Article  CAS  Google Scholar 

  37. 37.

    T. Olsen: Assessing the performance of the random phase approximation for exchange and superexchange coupling constants in magnetic crystalline solids. Phys. Rev. B 96,125143 (2017).

    Article  Google Scholar 

  38. 38.

    S.K. Bose and J. Kudrnovský: Exchange interactions and Curie temperatures in Cr-based alloys in the zinc blende structure: Volume- and composition-dependence from first-principles calculations. Phys. Rev. B 81, 054446 (2010).

    Article  CAS  Google Scholar 

  39. 39.

    J. Enkovaara, C. Rostgaard, J.J. Mortensen, J. Chen, M. Dutak, L. Ferrighi, J. Gavnholt, C. Glinsvad, V. Haikola, H.A. Hansen, H.H. Kristoffersen, M. Kuisma, A.H. Larsen, L. Lehtovaara, M. Ljungberg, O. Lopez-Acevedo, P. G. Moses, J. Ojanen, T. Olsen, V. Petzold, N.A. Romero, J. Stausholm-Møller, M. Strange, G.A. Tritsaris, M. Vanin, M. Walter, B. Hammer, H. Häkkinen, G.K.H. Madsen, R.M. Nieminen, J.K. Nørskov, M. Puska, T.T. Rantala, J. Schiøtz, K.S. Thygesen, and K.W. Jacobsen: Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method. J. Phys. Condens. Matter 22, 253202 (2010).

    CAS  Article  Google Scholar 

  40. 40.

    T. Olsen: Designing in-plane heterostructures of quantum spin Hall insulators from first principles: 1T’-MoS2 with adsorbates. Phys. Rev. B 94, 235106 (2016).

    Article  Google Scholar 

  41. 41.

    A.H. Larsen, J.J. Mortensen, J. Blomqvist, I.E. Castelli, R. Christensen, M. Dutak, J. Friis, M.N. Groves, B. Hammer, C. Hargus, E.D. Hermes, P.C. Jennings, P.B. Jensen, J. Kermode, J.R. Kitchin, E.L. Kolsbjerg, J. Kubal, K. Kaasbjerg, S. Lysgaard, J.B. Maronsson, T. Maxson, T. Olsen, L. Pastewka, A. Peterson, C. Rostgaard, J. Schiøtz, O. Schütt, M. Strange, K.S. Thygesen, T. Vegge, L. Vilhelmsen, M. Walter, Z. Zeng, and K.W. Jacobsen: The atomic simulation environment—a Python library for working with atoms. J. Phys. Condens. Matter 29, 273002 (2017).

    Article  Google Scholar 

  42. 42.

    S.V. Tyablikov: Methods in the Quantum Theory of Magnetism (Springer US, Boston, MA, 1967).

    Book  Google Scholar 

  43. 43.

    S. Haastrup, M. Strange, M. Pandey, T. Deilmann, P.S. Schmidt, N.F. Hinsche, M.N. Gjerding, D. Torelli, P.M. Larsen, A.C. Riis-Jensen, J. Gath, K.W. Jacobsen, J.J. Mortensen, T. Olsen, and K.S. Thygesen: The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals. 2D Mater. 5, 42002 (2018).

    CAS  Article  Google Scholar 

  44. 44.

    S. Haastrup, M. Strange, M. Pandey, T. Deilmann, P.S. Schmidt, N.F. Hinsche, M.N. Gjerding, D. Torelli, P.M. Larsen, A.C. Riis-Jensen, J. Gath, K.W. Jacobsen, J.J. Mortensen, T. Olsen, and K.S. Thygesen: Reply to comment on The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals’. 2D Mater. 6, 048002 (2019).

    CAS  Article  Google Scholar 

  45. 45.

    R. Allmann and R. Hinek: The introduction of structure types into the Inorganic Crystal Structure Database ICSD. Acta Crystallogr. A 63, 412 (2007).

    CAS  Article  Google Scholar 

  46. 46.

    S. Gražulis, A. Daškevic, A. Merkys, D. Chateigner, L. Lutterotti, M. Quirós, N.R. Serebryanaya, P. Moeck, R.T. Downs, and A. Le Bail: Crystallography Open Database (COD): an open-access collection of crystal structures and platform for world-wide collaboration. Nucleic Acids Res. 40, D420 (2012).

    Article  CAS  Google Scholar 

  47. 47.

    N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I.E. Castelli, A. Cepellotti, G. Pizzi, and N. Marzari: Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246 (2018).

    CAS  Article  Google Scholar 

  48. 48.

    P.M. Larsen, M. Pandey, M. Strange, and K.W. Jacobsen: Definition of a scoring parameter to identify low-dimensional materials components. Phys. Rev. Mater. 3, 034003 (2019).

    CAS  Article  Google Scholar 

  49. 49.

    J.V. José, L.P. Kadanoff, S. Kirkpatrick, and D.R. Nelson: Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B 16, 1217 (1977).

    Article  Google Scholar 

  50. 50.

    X. Wang, K. Du, Y.Y. Fredrik Liu, P. Hu, J. Zhang, Q. Zhang, M.H.S. Owen, X. Lu, C.K. Gan, P. Sengupta, C. Kloc, and Q. Xiong: Raman spectroscopy of atomically thin two-dimensional magnetic iron phosphorus trisulfide (FePS3) crystals. 2D Mater. 3, 031009 (2016).

    Article  CAS  Google Scholar 

  51. 51.

    A.R. Wildes, V. Simonet, E. Ressouche, G.J. McIntyre, M. Avdeev, E. Suard, S.A.J. Kimber, D. Lançon, G. Pepe, B. Moubaraki, and T.J. Hicks: Magnetic structure of the quasi-two-dimensional antiferromagnet NiPS3. Phys. Rev. B 92, 224408 (2015).

    Article  CAS  Google Scholar 

  52. 52.

    A.L. Chernyshev and M.E. Zhitomirsky: Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities. Phys. Rev. B 79, 144416 (2009).

    Article  CAS  Google Scholar 

  53. 53.

    P.A. Maksimov, Z. Zhu, S.R. White, and A.L. Chernyshev: Anisotropic-Exchange Magnets on a Triangular Lattice: Spin Waves, Accidental Degeneracies, and Dual Spin Liquids. Phys. Rev. X 9, 021017 (2019).

    CAS  Google Scholar 

  54. 54.

    B. Huang, G. Clark, E. Navarro-Moratalla, D.R. Klein, R. Cheng, K.L. Seyler, D. Zhong, E. Schmidgall, M.A. McGuire, D.H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu: Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270 (2017).

    CAS  Article  Google Scholar 

  55. 55.

    N. Sivadas, S. Okamoto, X. Xu, C.J. Fennie, and D. Xiao: Stacking-Dependent Magnetism in Bilayer CrI3. Nano Lett. 18, 7658 (2018).

    CAS  Article  Google Scholar 

  56. 56.

    P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z.-Y. Lu, and W. Ji: Stacking tunable interlayer magnetism in bilayer CrI3. Phys. Rev. B 99, 144401 (2019).

    CAS  Article  Google Scholar 

  57. 57.

    C. Cardoso, D. Soriano, N.A. García-Martínez, and J. Fernández-Rossier: Van der Waals Spin Valves. Phys. Rev. Lett. 121, 67701 (2018).

    CAS  Article  Google Scholar 

  58. 58.

    T. Song, X. Cai, M.W.-Y. Tu, X. Zhang, B. Huang, N.P. Wilson, K.L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, M.A. McGuire, D.H. Cobden, D. Xiao, W. Yao, and X. Xu: Giant tunneling magnetoresistance in spin-filter van der Waals heterostructures. Science 360, 1214 (2018).

    CAS  Article  Google Scholar 

  59. 59.

    Z. Wang, I. Gutiérrez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamoglu, E. Giannini, and A.F. Morpurgo: Very large tunneling magnetoresistance in layered magnetic semiconductor CrI3. Nat. Commun. 9, 2516 (2018).

    Article  CAS  Google Scholar 

  60. 60.

    T. Song, M.W.-Y. Tu, C. Carnahan, X. Cai, T. Taniguchi, K. Watanabe, M.A. McGuire, D.H. Cobden, D. Xiao, W. Yao, and X. Xu: Voltage Control of a van der Waals Spin-Filter Magnetic Tunnel Junction. Nano Lett. 19, 915 (2019).

    Article  CAS  Google Scholar 

  61. 61.

    B. Huang, G. Clark, D.R. Klein, D. MacNeill, E. Navarro-Moratalla, K.L. Seyler, N. Wilson, M.A. McGuire, D.H. Cobden, D. Xiao, W. Yao, P. Jarillo-Herrero, and X. Xu: Electrical control of 2D magnetism in bilayer CrI3. Nat. Nanotechnol. 13, 544 (2018).

    CAS  Article  Google Scholar 

  62. 62.

    S. Jiang, L. Li, Z. Wang, K.F. Mak, and J. Shan: Controlling magnetism in 2D CrI3 by electrostatic doping. Nat. Nanotechnol. 13, 549 (2018).

    CAS  Article  Google Scholar 

  63. 63.

    E. Suárez Morell, A. León, R.H. Miwa, and P. Vargas: Control of magnetism in bilayer CrI3 by an external electric field. 2D Mater. 6, 025020 (2019).

    Article  CAS  Google Scholar 

  64. 64.

    J. Liu, M. Shi, P. Mo, and J. Lu: Electrical-field-induced magnetic Skyrmion ground state in a two-dimensional chromium tri-iodide ferromagnetic monolayer. AIP Adv. 8, 055316 (2018).

    Article  CAS  Google Scholar 

  65. 65.

    A. Mook, J. Henk, and I. Mertig: Edge states in topological magnon insulators. Phys. Rev. B 90, 024412 (2014).

    Article  CAS  Google Scholar 

  66. 66.

    S.S. Pershoguba, S. Banerjee, J.C. Lashley, J. Park, H. Ågren, G. Aeppli, and A.V. Balatsky: Dirac Magnons in Honeycomb Ferromagnets. Phys. Rev. X 8, 011010 (2018).

    Google Scholar 

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Olsen, T. Theory and simulations of critical temperatures in CrI3 and other 2D materials: easy-axis magnetic order and easy-plane Kosterlitz-Thouless transitions. MRS Communications 9, 1142–1150 (2019). https://doi.org/10.1557/mrc.2019.117

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