OOMMF extension: Dzyaloshinskii-Moriya interaction for crystallographic class D_2d

David Cortés-Ortuño^1,2, Marijan Beg^1,3, Martin Lang¹, Vanessa Nehruji^1,4, Ryan A. Pepper¹, and Hans Fangohr^1,5,6

¹ Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, United Kingdom
² Department of Earth Sciences, Utrecht University, 3584 CD Utrecht, The Netherlands
³ Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, United Kingdom
⁴ Department of Physics, University of Durham, Durham DH1 3LE, United Kingdom
⁵ Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
⁶ Center for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany

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About

Different Dzyaloshinskii-Moriya interaction forms can be expressed using different combinations of Lifshitz invariants, depending on the crystallographic class [1]. D_2d crystallographic class has the following energy density:

$\LARGE w_x = D \left( \mathcal{L}_{zx}^{(y)} + \mathcal{L}_{yx}^{(z)} \right) = D \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial y} \times \hat{y} - \frac{\partial \mathbf{m}}{\partial z} \times \hat{z} \right)$ $\LARGE w_y = D \left( \mathcal{L}_{xy}^{(z)} + \mathcal{L}_{zy}^{(x)} \right) = D \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial z} \times \hat{z} - \frac{\partial \mathbf{m}}{\partial x} \times \hat{x} \right)$ $\LARGE w_z = D \left( \mathcal{L}_{yz}^{(x)} + \mathcal{L}_{xz}^{(y)} \right) = D \mathbf{m} \cdot \left( \frac{\partial \mathbf{m}}{\partial x} \times \hat{x} - \frac{\partial \mathbf{m}}{\partial y} \times \hat{y} \right)$

where $D [\text{J}/\text{m}^{2}]$ is the DMI constant and $\mathbf{m} = \mathbf{M}/M_\text{s}$ is the normalised magnetisation field.

Periodic boundary conditions

This OOMMF extension supports periodic boundary conditions. It works with both Oxs_RectangularMesh and Oxs_PeriodicRectangularMesh.

Installation

Copy DMI_D2d_[xyz].cc and DMI_D2d_[xyz].h files from the src directory in this repository into the app/oxs/local OOMMF directory.
Recompile OOMMF, usually by running

$ tclsh oommf.tcl pimake distclean && tclsh oommf.tcl pimake upgrade && tclsh oommf.tcl pimake

Examples

We give examples in .mif files for the relaxation resulting in an isolated skyrmion state in a confined cuboid (with no periodic boundary conditions) and relaxation resulting in a skyrmion lattice in a unit cell of an infinite system (with periodic boundary conditions). Examples demonstrate the use of all three terms (with appropriately adjusted geometry).

Other crystallographic classes

DMI extensions for other crystallographic classes are C_nv and T(O).

Support

If you require support, have questions, want to report a bug or suggest an improvement, please raise an issue in ubermag/help repository.

Contributions

All contributions are welcome, however small they are. If you would like to contribute, please fork the repository and create a pull request. If you are unsure how to contribute, please contact us by raising an issue in ubermag/help repository, and we will help you get started and assist you on the way.

How to cite

David Cortés-Ortuño et al. Proposal for a micromagnetic standard problem for materials with Dzyaloshinskii–Moriya interaction. New J. Phys. 20, 113015 (2018).
David Cortés-Ortuño, Marijan Beg, Martin Lang, Vanessa Nehruji, Ryan A. Pepper, and Hans Fangohr. OOMMF extension: Dzyaloshinskii-Moriya interaction for the crystallographic class D2d. DOI: 10.5281/zenodo.1196451 (2018).

License

Licensed under the BSD 3-Clause "New" or "Revised" License. For details, please refer to the LICENSE file.

Acknowledgements

OpenDreamKit – Horizon 2020 European Research Infrastructure project (676541)
EPSRC Programme Grant on Skyrmionics (EP/N032128/1)

References

[1] A. Bogdanov and D. Yablonskii. Thermodynamically stable "vortices" in magnetically ordered crystals. The mixed state of magnets. Zh. Eksp. Teor. Fiz 95, 178 (1989).

joommf/oommf-extension-dmi-d2d

OOMMF extension: Dzyaloshinskii-Moriya interaction for crystallographic class D2d