Topological magnon gap engineering in van der Waals ${\mathrm{CrI}}_{3}$ ferromagnets

Topological magnon gap engineering in van der Waals ${CrI}_{3}$ ferromagnets

Verena Brehm, Pawel Sobieszczyk, Jostein N. Kløgetvedt, Richard F. L. Evans, Elton J. G. Santos, and Alireza Qaiumzadeh

Phys. Rev. B 109, 174425 – Published 14 May 2024

Abstract

The microscopic origin of the topological magnon band gap in ${CrI}_{3}$ ferromagnets has been a subject of controversy for years since two main models with distinct characteristics, i.e., Dzyaloshinskii-Moriya (DM) and Kitaev, provided possible explanations with different outcome implications. Here, we investigate the angular magnetic field dependence of the magnon gap of ${CrI}_{3}$ by elucidating what main contributions play a major role in its generation. We implement stochastic atomistic spin-dynamics simulations to compare the impact of these two spin interactions on the magnon spectra. We observe three distinct magnetic field dependencies between these two gap opening mechanisms. First, we demonstrate that the Kitaev-induced magnon gap is influenced by both the direction and amplitude of the applied magnetic field, while the DM-induced gap is solely affected by the magnetic field direction. Second, the position of the Dirac cones within the Kitaev-induced magnon gap shifts in response to changes in the magnetic field direction, whereas they remain unaffected by the magnetic field direction in the DM-induced gap scenario. Third, we find a direct-indirect magnon band gap transition in the Kitaev model by varying the applied magnetic field direction. These differences may distinguish the origin of topological magnon gaps in ${CrI}_{3}$ and other van der Waals magnetic layers. Our findings pave the way for exploration and engineering topological gaps in van der Waals materials.

Received 15 December 2023
Revised 8 March 2024
Accepted 15 April 2024
Corrected 22 May 2024

DOI:https://doi.org/10.1103/PhysRevB.109.174425

Physics Subject Headings (PhySH)

Magnetism Spin dynamics Topological materials

2-dimensional systems

Condensed Matter, Materials & Applied Physics

Corrections

22 May 2024

Correction: The second sentence of Sec. V contained an error in wording and has been fixed. The omission of an acknowledgment statement and a support statement in the Acknowledgments have been fixed.

Authors & Affiliations

Verena Brehm ^1,*, Pawel Sobieszczyk ², Jostein N. Kløgetvedt ¹, Richard F. L. Evans ³, Elton J. G. Santos^4,5,6, and Alireza Qaiumzadeh ¹

¹Center for Quantum Spintronics, Norwegian University of Science and Technology, 7034 Trondheim, Norway
²Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krakow, Poland
³School of Physics, Engineering and Technology, University of York, York, YO10 5DD, United Kingdom
⁴Institute for Condensed Matter Physics and Complex Systems, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, Untied Kingdom
⁵Higgs Centre for Theoretical Physics, The University of Edinburgh, EH9 3FD, United Kingdom
⁶Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastián, Basque Country, Spain

^*verena.j.brehm@ntnu.no

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Vol. 109, Iss. 17 — 1 May 2024

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Figure 1
Magnon dispersion of single-wave theory (black dashed lines). The parameters for (a)the Kitaev model and (b) the DM model can be found in the Supplemental Material [45]. Note that, in the Dzyaloshinskii-Moriya (DM) model, only first-nearest-neighbor (NN) exchange is included in the analytical model, while up to third NNs are included in the atomistic simulation. This leads to a stretching of the low-energy band and a compression of the high-energy band [5] but does not affect the size and position of the topological Dirac gap at the $K$ and $K^{'}$ points.
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Figure 2
Analytic comparison of the positions of the Dirac-like cones in the magnon dispersion relation at out-of-plane (OOP, left column) and in-plane (IP, right columns) orientation with constant-energy cuts. In the Kitaev ( $κ$ ) model, the Dirac-like cones are at the $K$ points in (a)the OOP configuration but migrate into (b) and (c) the IP configuration. In contrast, (d)–(f) in the Dzyaloshinskii-Moriya (DM) model with a tilted DM vector, the cones remain at the $K$ points. Here, the low-energy band $σ = -$ is shown, and the color map scales from blue (low energy) to red and yellow (high energy). The magnetic field strength is $4.5 T$ [19].
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Figure 3
Impact of Dzyaloshinskii-Moriya (DM) interaction on the magnon dispersion relation in the out-of-plane magnetization configuration. (a) Tilting the DM vector only impacts the edge of the Brillouin zone (BZ). The gap size is read out and presented below. (b) When the relative angle $θ_{DM}$ between the DM vector and the magnetization direction increases from parallel $θ_{DM} = 0^{\circ}$ (yellow) to orthogonal $θ_{DM} = 90^{\circ}$ (blue), the gap at the $K$ symmetry point $Δ_{K}$ closes. The numerical results shown as points agree with the analytical result $Δ_{K}^{DM} (θ_{DM}) = Δ_{K}^{DM} (0) cos θ_{DM}$ , drawn with the black line, where $Δ_{K}^{DM} (0) = 9 \sqrt{3} A$ .
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Figure 4
Impact of the tilting angle $θ_{m}$ on dispersion, from out-of-plane (OOP, $θ_{m} = 0$ ) over in-plane (IP, $θ_{m} = π / 2$ ) to negative OOP ( $θ_{m} = π$ ). Solid lines are obtained analytically within linear spin-wave theory and in agreement with numerical data, obtained from atomistic simulations, for selected angles, shown with squares for the Dzyaloshinskii-Moriya (DM) model and diamonds for the Kitaev model. (a) Gap at the $K$ point $Δ_{K}$ for the Kitaev (green dashed) and DM (black solid) models dependent on the magnetization direction $θ_{m}$ . The DM direction is fixed at $54^{\circ}$ . (b) Displacement of the Dirac-like cones at $k_{DC}^{σ}$ from the $K$ point in the acoustic ( $σ = -$ ) and optic ( $σ = +$ ) branches for the Kitaev (green dashed lines) and DM (black and gray solid lines, both remaining zero)models. Characteristic angles indicated by vertical lines. (c) Constant-energy cut in the Kitaev model at $θ_{m} = 35^{\circ}$ showing the migration of the Dirac gap. For tilting the magnetization, a magnetic field of $4.5 T$ is applied [19].
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Physical Review B

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Topological magnon gap engineering in van der Waals ${CrI}_{3}$ ferromagnets

Verena Brehm, Pawel Sobieszczyk, Jostein N. Kløgetvedt, Richard F. L. Evans, Elton J. G. Santos, and Alireza Qaiumzadeh

Phys. Rev. B 109, 174425 – Published 14 May 2024

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Physical Review B

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Topological magnon gap engineering in van der Waals CrI3 ferromagnets

Verena Brehm, Pawel Sobieszczyk, Jostein N. Kløgetvedt, Richard F. L. Evans, Elton J. G. Santos, and Alireza Qaiumzadeh

Phys. Rev. B 109, 174425 – Published 14 May 2024

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Topological magnon gap engineering in van der Waals ${CrI}_{3}$ ferromagnets