Microscopic mechanism of electrostatic gate control of the metamagnetic transition and linear magnetoelectric response in bilayer CrI$_3$

In dual-gated bilayer chromium triiodide (CrI$_3$), a layered antiferromagnetic insulator, electrostatic gating tunes the critical field of the field-driven metamagnetic (layered-antiferromagnet-to-ferromagnet) transition by up to about 30%, with the shift tracking the nominal carrier density $n$ rather than the vertical displacement field $D$, and the two time-reversed layered-antiferromagnetic states acquire a magneto-optical signal linear in gate voltage with opposite signs. The originating experiments established this phenomenology but left its microscopic origin open. Eight years of follow-up work has measured the doping dependences of the interlayer exchange and anisotropy fields and produced several mutually different theoretical mechanisms (magnetic polarons, $d$-orbital electron counting, band-edge superexchange, interlayer-hopping modulation, interfacial charge transfer), but no quantitative microscopic account reproduces the observed dependences, and the actual layer-resolved carrier distribution under gating remains unmeasured. The problem is to establish this mechanism and the accompanying quantitative electronic-structure and many-body description, together with the coupling constants of the spin-layer-locking-mediated linear magnetoelectric response.

Background

Two-dimensional (2D) van der Waals (vdW) magnets are atomically thin crystals that retain intrinsic magnetic order; the archetype, CrI$_3$, was shown in 2017 to be a ferromagnetic insulator in the monolayer limit and a layered antiferromagnet in the bilayer (Gong et al. 2017; B. Huang et al. 2017). In bilayer CrI$_3$, each monolayer is a ferromagnetic sheet with out-of-plane magnetization below a Néel temperature (the ordering temperature of an antiferromagnet) of about 45 K, and the two layers are coupled antiferromagnetically, so the net magnetization vanishes in the ideal case; an applied perpendicular magnetic field drives a metamagnetic transition, a spin-flip from the layered antiferromagnetic (AFM) state to a ferromagnetic (FM) state at a critical field of roughly 0.8–1 T.

In 2018, Huang, Clark, Klein and co-workers (DOI 10.1038/s41565-018-0121-3) placed bilayer CrI$_3$ in a dual-gate field-effect device, in which a top and a bottom gate allow the nominal carrier density $n$ and the vertical electric displacement field $D$ to be varied independently (both estimated from a parallel-plate capacitor model). Using magneto-optical probes of the out-of-plane magnetization — the magneto-optical Kerr effect (MOKE, the rotation of polarization of light reflected from a magnetized surface) and reflectance magnetic circular dichroism (RMCD, the difference in absorption of left- and right-circularly polarized light) — they showed that gate voltage shifts the metamagnetic critical field by up to about 30%, and that the AFM–FM phase boundary in the $(n, D)$ plane runs nearly parallel to the constant-$n$ contours, so that the shift is dominated by the carrier density rather than by the electric field. This is striking because CrI$_3$ is an insulator, so the densities involved (of order $10^{13}\,\mathrm{cm}^{-2}$, about 0.1 electron per Cr ion) should only weakly perturb a band picture, yet the interlayer exchange changes qualitatively. In the companion experiments, Jiang, Li, Wang, Mak and Shan (DOI 10.1038/s41565-018-0135-x) extracted the interlayer exchange $J_\perp$ as a function of doping and found it changes sign (AFM to FM) at a critical density of roughly $2\times10^{13}\,\mathrm{cm}^{-2}$, while a naive picture of band filling accounts for only a few percent of the observed changes in saturation magnetization; Jiang, Shan and Mak (DOI 10.1038/s41563-018-0040-6) showed conversely that a pure displacement field can also switch the magnetic state via a linear magnetoelectric effect (a coupling between electric field and magnetization). Whether the same gate control extends to the Néel temperature of CrI$_3$ itself is not addressed by these experiments.

At zero magnetic field, Huang et al. further observed a MOKE signal linear in gate voltage whose slope has opposite signs for the two energetically nearly degenerate time-reversed AFM states (spins of the top and bottom layer interchanged). The proposed explanation invokes spin-layer locking: in the layered AFM, the spin orientation of each layer is locked to its layer pseudospin (the label of the geometric top or bottom layer), so that a gate-induced layer polarization (unequal carrier density in the two layers) is accompanied by a net magnetization that reverses sign between the two AFM states. This argument, which builds on the earlier prediction by Sivadas, Okamoto and Xiao (DOI 10.1103/PhysRevLett.117.267203) of a gate-controllable Kerr effect in collinear layered antiferromagnets, is qualitative: no microscopic coupling constants were derived. The same experiments also noted a remnant (non-cancelling) magnetization in van-der-Waals-assembled bilayers, attributed to possible out-of-plane symmetry breaking during fabrication, and that the capacitor-model doping estimate is an oversimplification, leaving the absolute carrier densities and their distribution across the two layers undetermined.

Problem Statement

Establish the microscopic mechanism by which electrostatic gating controls magnetism in bilayer CrI$_3$, as observed by Huang et al. and the companion 2018 experiments. Concretely: (1) identify and quantify the electronic pathway — modification of the orbital occupation of the Cr $d$ states, of the interlayer superexchange (the exchange coupling transmitted virtually through the bridging iodine ligands rather than by direct interlayer hopping) or of the interlayer hopping itself, of the magnetic anisotropy, or of many-body effects specific to a dilutely doped insulating 2D antiferromagnet, such as magnetic polarons (a doped carrier self-trapped in a ferromagnetic droplet within the antiferromagnetic matrix) — by which carrier densities of order $10^{13}\,\mathrm{cm}^{-2}$ shift the metamagnetic critical field and change the sign of the interlayer exchange at a critical density of roughly $2\times10^{13}\,\mathrm{cm}^{-2}$, explaining why naive band filling underestimates the observed changes by about an order of magnitude; (2) determine experimentally the actual carrier density and its distribution across the two layers under gating, replacing the parallel-plate capacitor estimate; (3) derive the microscopic coupling constants responsible for the zero-field linear magnetoelectric response, in which gate-induced layer polarization, through spin-layer locking, produces net magnetizations of opposite sign in the two time-reversed layered-antiferromagnetic states, and settle the relative roles of interlayer-hopping quenching, layer-resolved band-structure changes, and disorder or fabrication-induced symmetry breaking; and (4) identify the origin of the remnant net magnetization in fabricated bilayers, so that the symmetry-breaking channel can be separated from the intrinsic gate response. Parts (1) and (3) are jointly posed, in that the electronic pathway and the linear-magnetoelectric coupling constants are to be described by one quantitative microscopic theory whose predictions reproduce, with a single consistent parameter set, the measured doping dependences of the interlayer exchange and anisotropy fields and the gate-dependent magneto-optical response. Parts (2) and (4) are open experimental determinations that would test and constrain such a theory.

References

1. Bevin Huang, Genevieve Clark, Dahlia R. Klein, David MacNeill, Efrén Navarro-Moratalla, Kyle L. Seyler, Nathan Wilson, Michael A. McGuire, David H. Cobden, Di Xiao, Wang Yao, Pablo Jarillo-Herrero, Xiaodong Xu. Electrical control of 2D magnetism in bilayer CrI3. Nature Nanotechnology 13, 544–548 (2018). DOI: 10.1038/s41565-018-0121-3, https://doi.org/10.1038/s41565-018-0121-3 (preprint arXiv:1802.06979, https://arxiv.org/abs/1802.06979)
2. Shengwei Jiang, Lizhong Li, Zefang Wang, Kin Fai Mak, Jie Shan. Controlling magnetism in 2D CrI3 by electrostatic doping. Nature Nanotechnology 13, 549–553 (2018). DOI: 10.1038/s41565-018-0135-x, https://doi.org/10.1038/s41565-018-0135-x (preprint arXiv:1802.07355, https://arxiv.org/abs/1802.07355)
3. Shengwei Jiang, Jie Shan, Kin Fai Mak. Electric-field switching of two-dimensional van der Waals magnets. Nature Materials 17, 406–410 (2018). DOI: 10.1038/s41563-018-0040-6, https://doi.org/10.1038/s41563-018-0040-6 (preprint arXiv:1803.07272, https://arxiv.org/abs/1803.07272)
4. Tiancheng Song, Matisse Wei-Yuan Tu, Caitlin Carnahan, Xinghan Cai, Takashi Taniguchi, Kenji Watanabe, Michael A. McGuire, David H. Cobden, Di Xiao, Wang Yao, Xiaodong Xu. Voltage control of a van der Waals spin-filter magnetic tunnel junction. Nano Letters 19, 915–920 (2019). DOI: 10.1021/acs.nanolett.8b04160, https://doi.org/10.1021/acs.nanolett.8b04160 (preprint arXiv:1807.05285, https://arxiv.org/abs/1807.05285)
5. Xiao-Xiao Zhang, Lizhong Li, Daniel Weber, Joshua Goldberger, Kin Fai Mak, Jie Shan. Gate-tunable spin waves in antiferromagnetic atomic bilayers. Nature Materials 19, 838–842 (2020). DOI: 10.1038/s41563-020-0713-9, https://doi.org/10.1038/s41563-020-0713-9 (preprint arXiv:2001.04044, https://arxiv.org/abs/2001.04044)
6. D. Soriano, M. I. Katsnelson. Magnetic polaron and antiferromagnetic-ferromagnetic transition in doped bilayer CrI3. Physical Review B 101, 041402 (2020). DOI: 10.1103/PhysRevB.101.041402, https://doi.org/10.1103/PhysRevB.101.041402 (preprint arXiv:1912.05280, https://arxiv.org/abs/1912.05280, posted under the preprint title 'Magnetic polaron and antiferro-ferromagnetic transition in doped bilayer CrI3')
7. Jiewen Xiao, Binghai Yan. An electron-counting rule to determine the interlayer magnetic coupling of the van der Waals materials. 2D Materials 7, 045010 (2020). DOI: 10.1088/2053-1583/ab9ea4, https://doi.org/10.1088/2053-1583/ab9ea4 (preprint arXiv:2003.09942, https://arxiv.org/abs/2003.09942)
8. Chao Lei, Bheema L. Chittari, Kentaro Nomura, Nepal Banerjee, Jeil Jung, Allan H. MacDonald. Magnetoelectric response of antiferromagnetic CrI3 bilayers. Nano Letters 21, 1948–1954 (2021). DOI: 10.1021/acs.nanolett.0c04242, https://doi.org/10.1021/acs.nanolett.0c04242 (preprint arXiv:1902.06418, https://arxiv.org/abs/1902.06418, posted under the preprint title 'Magnetoelectric Response of Antiferromagnetic Van der Waals Bilayers')
9. Yan Lu, Haonan Wang, Li Wang, Li Yang. Mechanism of carrier doping induced magnetic phase transitions in two-dimensional materials. Physical Review B 106, 205403 (2022). DOI: 10.1103/PhysRevB.106.205403, https://doi.org/10.1103/PhysRevB.106.205403 (preprint arXiv:2212.10386, https://arxiv.org/abs/2212.10386)
10. Guanghui Cheng, Mohammad Mushfiqur Rahman, Zhiping He, Andres Llacsahuanga Allcca, Avinash Rustagi, Kirstine Aggerbeck Stampe, Yanglin Zhu, Shaohua Yan, Shangjie Tian, Zhiqiang Mao, Hechang Lei, Kenji Watanabe, Takashi Taniguchi, Pramey Upadhyaya, Yong P. Chen. Emergence of electric-field-tunable interfacial ferromagnetism in 2D antiferromagnet heterostructures. Nature Communications 13, 7348 (2022). DOI: 10.1038/s41467-022-34812-6, https://doi.org/10.1038/s41467-022-34812-6 (preprint arXiv:2203.13051, https://arxiv.org/abs/2203.13051)
11. Guanghui Cheng, Mohammad Mushfiqur Rahman, Andres Llacsahuanga Allcca, Avinash Rustagi, Xingtao Liu, Lina Liu, Lei Fu, Yanglin Zhu, Zhiqiang Mao, Kenji Watanabe, Takashi Taniguchi, Pramey Upadhyaya, Yong P. Chen. Electrically tunable moiré magnetism in twisted double bilayers of chromium triiodide. Nature Electronics 6, 434–442 (2023). DOI: 10.1038/s41928-023-00978-0, https://doi.org/10.1038/s41928-023-00978-0 (preprint arXiv:2204.03837, https://arxiv.org/abs/2204.03837)
12. B. Liu, J. S. Feng, H. J. Xiang, Z. Dai, Zhi-Xin Guo. Efficient electric field control of the magnetic phase in bilayer magnets via interlayer hopping modulation. Physical Review B 111, L140401 (2025). DOI: 10.1103/PhysRevB.111.L140401, https://doi.org/10.1103/PhysRevB.111.L140401
13. Srdjan Stavrić, Paolo Barone, Silvia Picozzi. Delving into the anisotropic interlayer exchange in bilayer CrI3. 2D Materials 11, 015020 (2024). DOI: 10.1088/2053-1583/ad1313, https://doi.org/10.1088/2053-1583/ad1313 (preprint arXiv:2305.16142, https://arxiv.org/abs/2305.16142)
14. Ivan Verzhbitskiy, Goki Eda. Electrostatic control of magnetism: Emergent opportunities with van der Waals materials. Applied Physics Letters 121, 060501 (2022). DOI: 10.1063/5.0107329, https://doi.org/10.1063/5.0107329
15. Ping Wang, Fuzhuo Lian, Renjun Du, Xiaofan Cai, Song Bao, Yaqing Han, Jingkuan Xiao, Kenji Watanabe, Takashi Taniguchi, Jinsheng Wen, Hongxin Yang, Alexander S. Mayorov, Lei Wang, Geliang Yu. Gate control of 2D magnetism in three- and four-layers CrI3/graphene heterostructures. Applied Physics Letters 124, 012406 (2024). DOI: 10.1063/5.0178916, https://doi.org/10.1063/5.0178916 (preprint arXiv:2404.18061, https://arxiv.org/abs/2404.18061, posted under the preprint title 'Gate control of 2D magnetism in tri- and four-layers CrI3/graphene heterostructures')
16. Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern, Yang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan Wang, Z. Q. Qiu, R. J. Cava, Steven G. Louie, Jing Xia, Xiang Zhang. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546, 265–269 (2017). DOI: 10.1038/nature22060, https://doi.org/10.1038/nature22060
17. Bevin Huang, Genevieve Clark, Efrén Navarro-Moratalla, Dahlia R. Klein, Ran Cheng, Kyle L. Seyler, Ding Zhong, Emma Schmidgall, Michael A. McGuire, David H. Cobden, Wang Yao, Di Xiao, Pablo Jarillo-Herrero, Xiaodong Xu. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017). DOI: 10.1038/nature22391, https://doi.org/10.1038/nature22391
18. Nikhil Sivadas, Satoshi Okamoto, Di Xiao. Gate-controllable magneto-optic Kerr effect in layered collinear antiferromagnets. Physical Review Letters 117, 267203 (2016). DOI: 10.1103/PhysRevLett.117.267203, https://doi.org/10.1103/PhysRevLett.117.267203