# Dzyaloshinskii–Moriya-like interaction in ferroelectrics and antiferroelectrics

## Abstract

The Dzyaloshinskii–Moriya interaction (DMI) between two magnetic moments mi and mj is of the form $${\rm{D}}_{ij} \cdot ({\rm{m}}_i \times {\rm{m}}_j)$$. It originates from spin–orbit coupling, and is at the heart of fascinating phenomena involving non-collinear magnetism, such as magnetic topological defects (for example, skyrmions) as well as spin–orbit torques and magnetically driven ferroelectricity, that are of significant fundamental and technological interest. In sharp contrast, its electric counterpart, which is an electric DMI characterized by its $${{\bf{D}}}_{ij}^{\prime}$$ strength and describing an interaction between two polar displacements ui and uj, has rarely been considered, despite the striking possibility that it could also generate new features associated with non-collinear patterns of electric dipoles. Here we report first-principles simulations combined with group theoretical symmetry analysis which not only demonstrate that electric DMI does exist and has a one-to-one correspondence with its magnetic analogue, but also reveals a physical source for it. These findings can be used to explain and/or design phenomena of possible technological importance in ferroelectrics and multiferroics.

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## Data availability

All the data (for example, raw data and Matplotlib-based scripts for analysing and visualizing the data) supporting the present work are available from the corresponding author upon request. Please note that our data figures were prepared with the use of some Matplotlib-based scripts (for example, with some mathematical processes such as post-processing of the data or fitting inside the scripts). Consequently, we prefer to share our raw data as well as the scripts to interested readers on request, so that we can help them in case of need. We do not upload our data and scripts because the latter may depend on the version of Python.

## Code availability

The VASP code for the numerical simulations in this work can be found at https://www.vasp.at; the code L-INVARIANT can be found at https://github.com/PaulChern/LINVARIANT/; the Mathematica software is available at https://www.wolfram.com/mathematica; the Matplotlib is available at https://matplotlib.org; other codes and scripts can be obtained on request from the corresponding author.

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## Acknowledgements

H.J.Z and L.B. thank the Department of Energy, Office of Basic Energy Sciences, under award number DESC0002220 for the DFT simulations. P.C. and S.P. acknowledge the Office of Naval Research under grant number N00014-17-1-2818 for symmetry analysis. The simulations based on DFT were done using the Arkansas High Performance Computing Center.

## Author information

Authors

### Contributions

L.B. and H.J.Z. conceived the work. H.J.Z. performed the DFT simulations. H.J.Z., P.C., S.P. and S.A. carried out symmetry analysis. All authors participated in the discussion and preparation of this work.

### Corresponding authors

Correspondence to Peng Chen or Laurent Bellaiche.

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### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Figs. 1-7, Discussion I–VII, Tables I–VI and refs. 1–22.

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Zhao, H.J., Chen, P., Prosandeev, S. et al. Dzyaloshinskii–Moriya-like interaction in ferroelectrics and antiferroelectrics. Nat. Mater. (2020). https://doi.org/10.1038/s41563-020-00821-3