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  • Technical review
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First-principles calculations for Dzyaloshinskii–Moriya interaction

Nature Reviews Physics volume 5, pages 43–61 (2023)Cite this article

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Abstract

Understanding the magnetic interactions is of fundamental importance in condensed-matter physics as well as in the application of spintronic devices. In the past 13 years, the Dzyaloshinskii–Moriya interaction (DMI) has attracted increasing attention, as it can trigger topological chiral magnetism, such as magnetic skyrmions, which is of particular interest from both fundamental and applied points of view. First-principles calculations have played an essential role in figuring out the microscopic properties of DMI and helped to search for the materials with strong DMI. In this Technical Review, we present a comprehensive and systematic survey of the first-principles-calculations methods for DMI, along with an overview of the first-principles calculations of the DMI properties of typical material systems and the DMI-induced magnetic phenomena.

Key points

  • The Dzyaloshinskii–Moriya interaction (DMI) is a kind of antisymmetric exchange coupling that arises as a consequence of the spin–orbit coupling in the magnetic system with broken inversion symmetry.

  • First-principles calculations of DMI can be based on the methods of mapping of total energies or mapping of energy derivatives.

  • First-principles calculations with spin–orbit coupling for the noncollinear magnetic structure is the critical step to obtaining the DMI parameter.

  • DMI resides in many magnetic bulk and thin-film systems with broken inversion symmetry.

  • DMI favours twisted spin pairs, leading to the formation of many chiral topological magnetic structures, such as skyrmions, and has a strong impact on the spin dynamics.

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Fig. 1: Spin configurations for the total-energy mapping method of Dzyaloshinskii–Moriya interaction calculations.
Fig. 2: Dzyaloshinskii–Moriya interaction properties of bulk materials.
Fig. 3: Dzyaloshinskii–Moriya interaction properties of magnetic thin-film heterostructures.
Fig. 4: Dzyaloshinskii–Moriya interaction properties of 2D magnets.
Fig. 5: Illustration of Dzyaloshinskii–Moriya-interaction-induced phenomena.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (grant nos. 11874059 and 12174405); Key Research Program of Frontier Sciences, CAS (grant no. ZDBS-LY-7021), Pioneer and Leading Goose R&D Program of Zhejiang Province (grant no. 2022C01053), Ningbo Key Scientific and Technological Project (grant no. 2021000215), Zhejiang Provincial Natural Science Foundation (grant no. LR19A040002) and Beijing National Laboratory for Condensed Matter Physics (grant no. 2021000123).

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Authors and Affiliations

  1. Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, China

    Hongxin Yang, Jinghua Liang & Qirui Cui

  2. School of Physics, Nanjing University, Nanjing, China

    Hongxin Yang

  3. Center of Materials Science and Optoelectronics Engnieering, University of Chinese Academy of Sciences, Beijing, China

    Hongxin Yang

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  1. Hongxin Yang
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Contributions

H.Y. conceived the Technical Review. H.Y., J.L. and Q.C. cowrote the manuscript.

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Correspondence to Hongxin Yang.

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Nature Reviews Physics thanks Yan Zhou, Wenbo Mi and another anonymous reviewer for their contribution to the peer review of this work.

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Glossary

Berry curvature

The Berry curvature Ω encodes how the eigenstate \(|n({\bf{R}})\rangle \) evolves as a local function of parameter R and is defined as \(\varOmega ={\nabla }_{{\bf{R}}}\times {{\bf{A}}}_{n}\) with \({{\bf{A}}}_{n}=i\langle n({\bf{R}})| \frac{\partial }{\partial {\bf{R}}}| n({\bf{R}})\rangle \).

Bloch-type spiral

The spin spiral can be classified as Bloch-type spiral, in which the spins rotate in a helical way around an axis that is parallel or antiparallel to the propagation direction.

Fert–Levy model

In the magnetic metallic system, the DMI between two magnetic atoms can be mediated by a third heavy atom. The resultant DMI parameter depends on the SOC strength of the heavy atom and the relative orientation of the three atoms involved.

Lifshitz invariants

In Landau’s theory of phase transitions, the free-energy expansion with respect to the order parameter η can include terms of the form ηi∂kηj, in which ηi is a component of η. Invariant polynomials in the expansion that contain terms of this form are called Lifshitz invariants.

Néel-type spiral

The spin spiral can be classified as Néel-type spiral, in which the spins rotate in a cycloidal way around an axis that is perpendicular to the propagation direction.

Rashba SOC

The Rashba effect is a momentum-dependent splitting of spin bands in two-dimensional condensed matter systems caused by SOC. The Rashba SOC Hamiltonian reads as \({H}_{{\rm{R}}}={\alpha }_{{\rm{R}}}(\widehat{{\bf{z}}}\times {\bf{k}})\cdot {\boldsymbol{\sigma }}\), in which αR characterizes the strength of Rashba SOC and σ is the vector of the Pauli matrices.

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Yang, H., Liang, J. & Cui, Q. First-principles calculations for Dzyaloshinskii–Moriya interaction. Nat Rev Phys 5, 43–61 (2023). https://doi.org/10.1038/s42254-022-00529-0

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