Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-29T14:41:44.175Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  12 September 2019

Thomas Olsen*
Affiliation:
Computational Atomic-Scale Materials Design, Department of Physics, Technical University of Denmark, 2800 Kgs, Lyngby, Denmark
*
Address all correspondence to Thomas Olsen at tolsen@fysik.dtu.dk
Get access

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.

Type
Prospective Articles
Copyright
Copyright © Materials Research Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Mermin, N.D. and Wagner, H.: Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models. Phys. Rev. Lett 17, 1133 (1966).CrossRefGoogle Scholar
2.Huang, B., Clark, G., Navarro-Moratalla, E., Klein, D.R., Cheng, R., Seyler, K.L., Zhong, D., Schmidgall, E., McGuire, M.A., Cobden, D.H., Yao, W., Xiao, D., Jarillo-Herrero, P., and Xu, X.: Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270 (2017).CrossRefGoogle Scholar
3.Fei, Z., Huang, B., Malinowski, P., Wang, W., Song, T., Sanchez, J., Yao, W., Xiao, D., Zhu, X., May, A.F., Wu, W., Cobden, D.H., Chu, J.-H., and Xu, X.: Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2. Nat. Mater 17, 778 (2018).10.1038/s41563-018-0149-7CrossRefGoogle ScholarPubMed
4.O'Hara, D.J., Zhu, T., Trout, A.H., Ahmed, A.S., Luo, Y.K., Lee, C.H., Brenner, M.R., Rajan, S., Gupta, J.A., McComb, D.W., and Kawakami, R.K.: Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit. Nano Lett 18, 3125 (2018).10.1021/acs.nanolett.8b00683CrossRefGoogle ScholarPubMed
5.Bonilla, M., Kolekar, S., Ma, Y., Diaz, H.C., Kalappattil, V., Das, R., Eggers, T., Gutierrez, H.R., Phan, M.-H., and Batzill, M.: Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substrates. Nat. Nanotechnol 13, 289 (2018).CrossRefGoogle ScholarPubMed
6.Liu, Z.-L., Wu, X., Shao, Y., Qi, J., Cao, Y., Huang, L., Liu, C., Wang, J.-O., Zheng, Q., Zhu, Z.-L., Ibrahim, K., Wang, Y.-L., and Gao, H.-J.: Epitaxially grown monolayer VSe2: an air-stable magnetic two-dimensional material with low work function at edges. Sci. Bull 63, 419 (2018).CrossRefGoogle Scholar
7.Lee, J.-U., Lee, S., Ryoo, J.H., Kang, S., Kim, T.Y., Kim, P., Park, C.-H., Park, J.-G., and Cheong, H.: Ising-Type Magnetic Ordering in Atomically Thin FePS3. Nano Lett 16, 7433 (2016).CrossRefGoogle ScholarPubMed
8.Yosida, K.: Theory of Magnetism (Springer Berlin Heidelberg, 1996).CrossRefGoogle Scholar
9.Gong, C., Li, L., Li, Z., Ji, H., Stern, A., Xia, Y., Cao, T., Bao, W., Wang, C., Wang, Y., Qiu, Z.Q., Cava, R.J., Louie, S.G., Xia, J., and Zhang, X.: Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546, 265 (2017).CrossRefGoogle ScholarPubMed
10.Lado, J.L. and Fernández-Rossier, J.: On the origin of magnetic anisotropy in two dimensional CrI3. 2D Mater 4, 035002 (2017).CrossRefGoogle Scholar
11.Torelli, D. and Olsen, T.: Calculating critical temperatures for ferromagnetic order in two-dimensional materials. 2D Mater 6, 015028 (2018).CrossRefGoogle Scholar
12.Sarikurt, S., Kadioglu, Y., Ersan, F., Vatansever, E., Aktürk, , Yüksel, Y., Akıncı, Ü, and Aktürk, E.: Electronic and magnetic properties of monolayer α-RuCl3: a first-principles and Monte Carlo study. Phys. Chem. Chem. Phys 20, 997 (2018).CrossRefGoogle ScholarPubMed
13.Yasuda, C., Todo, S., Hukushima, K., Alet, F., Keller, M., Troyer, M., and Takayama, H.: Néel Temperature of Quasi-Low-Dimensional Heisenberg Antiferromagnets. Phys. Rev. Lett 94, 217201 (2005).CrossRefGoogle ScholarPubMed
14.Kim, K., Lim, S.Y., Lee, J.-U., Lee, S., Kim, T.Y., Park, K., Jeon, G.S., Park, C.-H., Park, J.-G., and Cheong, H.: Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS3. Nat. Commun 10, 345 (2019).CrossRefGoogle ScholarPubMed
15.Torelli, D., Thygesen, K.S., and Olsen, T.: High throughput computational screening for 2D ferromagnetic materials: the critical role of anisotropy and local correlations. 2D Mater 6, 045018 (2019).CrossRefGoogle Scholar
16.Chen, P., Pai, W.W., Chan, Y.-H., Madhavan, V., Chou, M.Y., Mo, S.-K., Fedorov, A.-V., and Chiang, T.-C.: Unique Gap Structure and Symmetry of the Charge Density Wave in Single-Layer VSe2. Phys. Rev. Lett 121, 196402 (2018).CrossRefGoogle Scholar
17.Fumega, A.O. and Pardo, V.: Absence of ferromagnetism in VSe2 caused by its charge density wave phase, arXiv:1804. 07102 (2018).CrossRefGoogle Scholar
18.Duvjir, G., Choi, B.K., Jang, I., Ulstrup, S., Kang, S., Thi Ly, T., Kim, S., Choi, Y.H., Jozwiak, C., Bostwick, A., Rotenberg, E., Park, J.-G., Sankar, R., Kim, K.-S., Kim, J., and Chang, Y.J.: Emergence of a Metal-Insulator Transition and High-Temperature Charge-Density Waves in VSe2 at the Monolayer Limit. Nano Lett 18, 5432 (2018).CrossRefGoogle ScholarPubMed
19.Wong, P.K.J., Zhang, W., Bussolotti, F., Yin, X., Herng, T.S., Zhang, L., Huang, Y.L., Vinai, G., Krishnamurthi, S., Bukhvalov, D.W., Zheng, Y.J., Chua, R., N'Diaye, A.T., Morton, S.A., Yang, C., Ou Yang, K., Torelli, P., Chen, W., Goh, K.E.J., Ding, J., Lin, M., Brocks, G., de Jong, M.P., Castro Neto, A.H., and Wee, A.T.S.: Evidence of Spin Frustration in a Vanadium Diselenide Monolayer Magnet. Adv. Mater 31, 1901185 (2019).CrossRefGoogle Scholar
20.Holdsworth, P.C.W. and Bramwell, S.T.: Magnetization: A characteristic of the Kosterlitz-Thouless-Berezinskii transition. Phys. Rev. B 49, 8811 (1994).Google Scholar
21.Anderson, P.W.: Antiferromagnetism. Theory of Superexchange Interaction. Phys. Rev. 79, 350 (1950).CrossRefGoogle Scholar
22.Anderson, P.W.: New Approach to the Theory of Superexchange Interactions. Phys. Rev. 115, 2 (1959).CrossRefGoogle Scholar
23.Xu, C., Feng, J., Xiang, H., and Bellaiche, L.: Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic CrI3 and CrGeTe3 monolayers. npj Comput. Mater. 4, 57 (2018).CrossRefGoogle Scholar
24.Banerjee, A., Bridges, C.A., Yan, J.-Q., Aczel, A.A., Li, L., Stone, M.B., Granroth, G.E., Lumsden, M.D., Yiu, Y., Knolle, J., Bhattacharjee, S., Kovrizhin, D.L., Moessner, R., Tennant, D.A., Mandrus, D.G., and Nagler, S.E.: Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733 (2016).CrossRefGoogle Scholar
25.Heide, M., Bihlmayer, G., and Blügel, S.: Describing Dzyaloshinskii-Moriya spirals from first principles. Phys. B Condens. Matter 404, 2678 (2009).10.1016/j.physb.2009.06.070CrossRefGoogle Scholar
26.Koretsune, T., Kikuchi, T., and Arita, R.: First-Principles Evaluation of the Dzyaloshinskii-Moriya Interaction. J. Phys. Soc. Jpn. 87, 041011 (2018).CrossRefGoogle Scholar
27.Liu, J., Shi, M., Lu, J., and Anantram, M.P.: 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.Besbes, O., Nikolaev, S., Meskini, N., and Solovyev, I.: Microscopic origin of ferromagnetism in the trihalides CrCl3 and CrI3. Phys. Rev. B 99, 104432 (2019).CrossRefGoogle Scholar
29.Wang, K., Nikolaev, S., Ren, W., and Solovyev, I.: Giant contribution of the ligand states to the magnetic properties of the Cr2Ge2Te6 monolayer. Phys. Chem. Chem. Phys. 21, 9597 (2019).10.1039/C9CP01034CCrossRefGoogle ScholarPubMed
30.Marzari, N., Mostofi, A.A., Yates, J.R., Souza, I., and Vanderbilt, D.: Maximally localized Wannier functions: Theory and applications. Rev. Mod. Phys. 84, 1419 (2012).CrossRefGoogle Scholar
31.Görling, A.: Symmetry in density-functional theory. Phys. Rev. A 47, 2783 (1993).CrossRefGoogle ScholarPubMed
32.Görling, A.: Proper Treatment of Symmetries and Excited States in a Computationally Tractable Kohn-Sham Method. Phys. Rev. Lett. 85, 4229 (2000).CrossRefGoogle Scholar
33.Xiang, H., Lee, C., Koo, H.-J., Gong, X., and Whangbo, M.-H.: Magnetic properties and energy-mapping analysis. Dalt. Trans. 42, 823 (2013).CrossRefGoogle ScholarPubMed
34.Jacobsson, A., Etz, C., Ležaić, M., Sanyal, B., and Blügel, S.: Frozen Magnon Calculations Beyond the Long Wavelength Approximation, arXiv:1702. 00599 (2017).Google Scholar
35.Ködderitzsch, D., Hergert, W., Temmerman, W.M., Szotek, Z., Ernst, A., and Winter, H.: Exchange interactions in NiO and at the NiO(100) surface. Phys. Rev. B 66, 064434 (2002).CrossRefGoogle Scholar
36.Pajda, M., Kudrnovský, J., Turek, I., Drchal, V., and Bruno, P.: Ab initio calculations of exchange interactions, spin-wave stiffness constants, and Curie temperatures of Fe, Co, and Ni. Phys. Rev. B 64, 174402 (2001).CrossRefGoogle Scholar
37.Olsen, T.: Assessing the performance of the random phase approximation for exchange and superexchange coupling constants in magnetic crystalline solids. Phys. Rev. B 96, 125143 (2017).CrossRefGoogle Scholar
38.Bose, S.K. and Kudrnovský, J.: 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).CrossRefGoogle Scholar
39.Enkovaara, J., Rostgaard, C., Mortensen, J.J., Chen, J., Dułak, M., Ferrighi, L., Gavnholt, J., Glinsvad, C., Haikola, V., Hansen, H.A., Kristoffersen, H.H., Kuisma, M., Larsen, A.H., Lehtovaara, L., Ljungberg, M., Lopez-Acevedo, O., Moses, P.G., Ojanen, J., Olsen, T., Petzold, V., Romero, N.A., Stausholm-Møller, J., Strange, M., Tritsaris, G.A., Vanin, M., Walter, M., Hammer, B., Häkkinen, H., Madsen, G.K.H., Nieminen, R.M., Nørskov, J.K., Puska, M., Rantala, T.T., Schiøtz, J., Thygesen, K.S., and Jacobsen, K.W.: Electronic structure calculations with GPAW: a real-space implementation of the projector augmented-wave method. J. Phys. Condens. Matter 22, 253202 (2010).CrossRefGoogle ScholarPubMed
40.Olsen, T.: Designing in-plane heterostructures of quantum spin Hall insulators from first principles: 1T'-MoS2 with adsorbates. Phys. Rev. B 94, 235106 (2016).CrossRefGoogle Scholar
41.Larsen, A.H., Mortensen, J.J., Blomqvist, J., Castelli, I.E., Christensen, R., Dułak, M., Friis, J., Groves, M.N., Hammer, B., Hargus, C., Hermes, E.D., Jennings, P.C., Jensen, P.B., Kermode, J., Kitchin, J.R., Kolsbjerg, E.L., Kubal, J., Kaasbjerg, K., Lysgaard, S., Maronsson, J.B., Maxson, T., Olsen, T., Pastewka, L., Peterson, A., Rostgaard, C., Schiøtz, J., Schütt, O., Strange, M., Thygesen, K.S., Vegge, T., Vilhelmsen, L., Walter, M., Zeng, Z., and Jacobsen, K.W.: The atomic simulation environment—a Python library for working with atoms. J. Phys. Condens. Matter 29, 273002 (2017).CrossRefGoogle Scholar
42.Tyablikov, S.V.: Methods in the Quantum Theory of Magnetism (Springer US, Boston, MA, 1967).CrossRefGoogle Scholar
43.Haastrup, S., Strange, M., Pandey, M., Deilmann, T., Schmidt, P.S., Hinsche, N.F., Gjerding, M.N., Torelli, D., Larsen, P.M., Riis-Jensen, A.C., Gath, J., Jacobsen, K.W., Mortensen, J.J., Olsen, T., and Thygesen, K.S.: The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals. 2D Mater. 5, 42002 (2018).CrossRefGoogle Scholar
44.Haastrup, S., Strange, M., Pandey, M., Deilmann, T., Schmidt, P.S., Hinsche, N.F., Gjerding, M.N., Torelli, D., Larsen, P.M., Riis-Jensen, A.C., Gath, J., Jacobsen, K.W., Mortensen, J.J., Olsen, T., and Thygesen, K.S.: Reply to comment on ‘The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals’. 2D Mater. 6, 048002 (2019).CrossRefGoogle Scholar
45.Allmann, R. and Hinek, R.: The introduction of structure types into the Inorganic Crystal Structure Database ICSD. Acta Crystallogr. A 63, 412 (2007).CrossRefGoogle ScholarPubMed
46.Gražulis, S., Daškevič, A., Merkys, A., Chateigner, D., Lutterotti, L., Quirós, M., Serebryanaya, N.R., Moeck, P., Downs, R.T., and Le Bail, A.: Crystallography Open Database ({COD}): an open-access collection of crystal structures and platform for world-wide collaboration. Nucleic Acids Res. 40, D420 (2012).CrossRefGoogle ScholarPubMed
47.Mounet, N., Gibertini, M., Schwaller, P., Campi, D., Merkys, A., Marrazzo, A., Sohier, T., Castelli, I.E., Cepellotti, A., Pizzi, G., and Marzari, N.: Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds. Nat. Nanotechnol. 13, 246 (2018).CrossRefGoogle ScholarPubMed
48.Larsen, P.M., Pandey, M., Strange, M., and Jacobsen, K.W.: Definition of a scoring parameter to identify low-dimensional materials components. Phys. Rev. Mater. 3, 034003 (2019).CrossRefGoogle Scholar
49.José, J.V., Kadanoff, L.P., Kirkpatrick, S., and Nelson, D.R.: Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B 16, 1217 (1977).CrossRefGoogle Scholar
50.Wang, X., Du, K., Fredrik Liu, Y.Y., Hu, P., Zhang, J., Zhang, Q., Owen, M.H.S., Lu, X., Gan, C.K., Sengupta, P., Kloc, C., and Xiong, Q.: Raman spectroscopy of atomically thin two-dimensional magnetic iron phosphorus trisulfide (FePS3) crystals. 2D Mater. 3, 031009 (2016).CrossRefGoogle Scholar
51.Wildes, A.R., Simonet, V., Ressouche, E., McIntyre, G.J., Avdeev, M., Suard, E., Kimber, S.A.J., Lançon, D., Pepe, G., Moubaraki, B., and Hicks, T.J.: Magnetic structure of the quasi-two-dimensional antiferromagnet NiPS3. Phys. Rev. B 92, 224408 (2015).CrossRefGoogle Scholar
52.Chernyshev, A.L. and Zhitomirsky, M.E.: Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities. Phys. Rev. B 79, 144416 (2009).CrossRefGoogle Scholar
53.Maksimov, P.A., Zhu, Z., White, S.R., and Chernyshev, A.L.: Anisotropic-Exchange Magnets on a Triangular Lattice: Spin Waves, Accidental Degeneracies, and Dual Spin Liquids. Phys. Rev. X 9, 021017 (2019).Google Scholar
54.Huang, B., Clark, G., Navarro-Moratalla, E., Klein, D.R., Cheng, R., Seyler, K.L., Zhong, D., Schmidgall, E., McGuire, M.A., Cobden, D.H., Yao, W., Xiao, D., Jarillo-Herrero, P., and Xu, X.: Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270 (2017).CrossRefGoogle Scholar
55.Sivadas, N., Okamoto, S., Xu, X., Fennie, C.J., and Xiao, D.: Stacking-Dependent Magnetism in Bilayer CrI3. Nano Lett. 18, 7658 (2018).CrossRefGoogle ScholarPubMed
56.Jiang, P., Wang, C., Chen, D., Zhong, Z., Yuan, Z., Lu, Z.-Y., and Ji, W.: Stacking tunable interlayer magnetism in bilayer CrI3. Phys. Rev. B 99, 144401 (2019).CrossRefGoogle Scholar
57.Cardoso, C., Soriano, D., García-Martínez, N.A., and Fernández-Rossier, J.: Van der Waals Spin Valves. Phys. Rev. Lett. 121, 67701 (2018).CrossRefGoogle Scholar
58.Song, T., Cai, X., Tu, M.W.-Y., Zhang, X., Huang, B., Wilson, N.P., Seyler, K.L., Zhu, L., Taniguchi, T., Watanabe, K., McGuire, M.A., Cobden, D.H., Xiao, D., Yao, W., and Xu, X.: Giant tunneling magnetoresistance in spin-filter van der Waals heterostructures. Science 360, 1214 (2018).CrossRefGoogle ScholarPubMed
59.Wang, Z., Gutiérrez-Lezama, I., Ubrig, N., Kroner, M., Gibertini, M., Taniguchi, T., Watanabe, K., Imamoğlu, A., Giannini, E., and Morpurgo, A.F.: Very large tunneling magnetoresistance in layered magnetic semiconductor CrI3. Nat. Commun. 9, 2516 (2018).CrossRefGoogle ScholarPubMed
60.Song, T., Tu, M.W.-Y., Carnahan, C., Cai, X., Taniguchi, T., Watanabe, K., McGuire, M.A., Cobden, D.H., Xiao, D., Yao, W., and Xu, X.: Voltage Control of a van der Waals Spin-Filter Magnetic Tunnel Junction. Nano Lett. 19, 915 (2019).CrossRefGoogle Scholar
61.Huang, B., Clark, G., Klein, D.R., MacNeill, D., Navarro-Moratalla, E., Seyler, K.L., Wilson, N., McGuire, M.A., Cobden, D.H., Xiao, D., Yao, W., Jarillo-Herrero, P., and Xu, X.: Electrical control of 2D magnetism in bilayer CrI3. Nat. Nanotechnol. 13, 544 (2018).CrossRefGoogle ScholarPubMed
62.Jiang, S., Li, L., Wang, Z., Mak, K.F., and Shan, J.: Controlling magnetism in 2D CrI3 by electrostatic doping. Nat. Nanotechnol. 13, 549 (2018).CrossRefGoogle ScholarPubMed
63.Suárez Morell, E., León, A., Miwa, R.H., and Vargas, P.: Control of magnetism in bilayer CrI3 by an external electric field. 2D Mater. 6, 025020 (2019).CrossRefGoogle Scholar
64.Liu, J., Shi, M., Mo, P., and Lu, J.: Electrical-field-induced magnetic Skyrmion ground state in a two-dimensional chromium tri-iodide ferromagnetic monolayer. AIP Adv. 8, 055316 (2018).CrossRefGoogle Scholar
65.Mook, A., Henk, J., and Mertig, I.: Edge states in topological magnon insulators. Phys. Rev. B 90, 024412 (2014).CrossRefGoogle Scholar
66.Pershoguba, S.S., Banerjee, S., Lashley, J.C., Park, J., Ågren, H., Aeppli, G., and Balatsky, A.V.: Dirac Magnons in Honeycomb Ferromagnets. Phys. Rev. X 8, 011010 (2018).Google Scholar