Dzyaloshinskii-Moriya Interaction

• Dzyaloshinskii-Moriya Interaction is an antisymmetric exchange mechanism arising in systems with broken inversion symmetry and significant spin–orbit coupling.
• It stabilizes chiral magnetic textures such as cycloidal spirals and skyrmions, with practical demonstrations in metallic zigzag chains and thin film heterostructures.
• Advanced ab initio methods and spin-spiral calculations enable precise quantification of DMI, guiding materials engineering for robust spintronic applications.

The Dzyaloshinskii-Moriya interaction (DMI) is an antisymmetric exchange interaction between localized or itinerant magnetic moments that emerges in systems with broken inversion symmetry and sizeable spin–orbit coupling (SOC). DMI favors non-collinear spin configurations, stabilizes chiral magnetic textures such as cycloidal spirals and skyrmions, and underpins a broad class of phenomena in condensed matter physics and spintronics. At the atomistic level, the DMI energy for two neighboring spins is defined as $$E_{\text{DMI}} = \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j)$$, where $\mathbf{D}_{ij}$ is the DMI vector whose magnitude and orientation are dictated by the geometry, SOC, and the electronic structure. Below, the fundamental principles, computational approaches, physical effects, materials engineering strategies, and extensions of DMI are reviewed.

1. Fundamental Mechanisms and Theoretical Formulation

DMI arises when the inversion symmetry between two magnetic sites is broken, and there is a finite SOC. Historically formulated for localized spins by Dzyaloshinskii and Moriya, the generic Hamiltonian is

$$\mathcal{H}_{ij} = - J_{ij}\,\mathbf{S}_i \cdot \mathbf{S}_j + \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j) + \mathbf{S}_i \cdot \mathcal{A}_{ij} \cdot \mathbf{S}_j,$$

with $J_{ij}$ the isotropic Heisenberg exchange and $\mathcal{A}_{ij}$ a symmetric anisotropic exchange tensor. The DMI term is present when the midpoint between sites $i$ and $j$ lacks inversion symmetry, and $\mathbf{D}_{ij}$ points according to cross-product rules derived from the local chemistry and lattice.

In systems with itinerant electrons, DMI can be rigorously derived using multiple-scattering theory and Green's function formulations. The antisymmetric exchange coupling encompasses both spin currents, generated through noncollinear spin textures even in the absence of SOC, and charge currents. The general expressions for DMI capture its dependence on band structure, hybridization, and the presence of noncollinearity: $$\mathbf{D}_{ij} = \frac{2}{\pi} \text{Re} \int d\epsilon \, \text{tr} \left\{ \Delta_i G_{ij}^{(00)} \Delta_j \mathbf{G}_{ji}^{(1)} - \Delta_i G_{ij}^{(01)} \Delta_j \mathbf{G}_{ji}^{(0)} \right\},$$ where $G_{ij}^{(00)}$ and $G_{ij}^{(01)}$ are components of the Green's function even/odd under site exchange, and $\Delta_i$ is the exchange splitting (Cardias et al., 2020, Cardias et al., 2020).

2. Microscopic Origin and Model Hamiltonians

The microscopic origins of DMI are rooted in the interplay between SOC, orbital hybridization, and inversion symmetry breaking. In metallic zigzag chains with alternating 3$d$ and heavy 5$d$ atoms (Fe–Pt, Co–Pt, Fe–Ir, etc.), FLAPW-DFT calculations and minimal tight-binding models reveal that both spin canting and structural inversion asymmetry (e.g., in a zigzag geometry) are essential for a finite DMI. Hybridization parameters between relevant $d$ orbitals ($t_1$, $t_2$), the magnitude and sign of SOC, and the energy separation between occupied and unoccupied spin states govern the effective $\mathbf{D}$ vector (Kashid et al., 2014). For small spin-spiral vectors $q$, the DMI energy correction behaves linearly: $E_{\text{DM}}(q) \approx D q,$ with $D$ set by symmetry (e.g., fixed along the $z$-axis for certain low-symmetry geometries).

A compact model Hamiltonian encapsulating spin–noncollinearity, hybridization, and SOC,

$$H = \begin{pmatrix} E_A - \tfrac{i}{2}I_m\cos\varphi & \cdots \ \vdots & \cdots \end{pmatrix},$$

reinforces that both off-diagonal hybridization ($t_2$) and canting angle $\varphi$ are minimal requirements for DMI (Kashid et al., 2014).

At the electronic structure level, DMI correlates strongly with the anisotropy of the magnetic orbital moment and the intra-atomic dipole moment $m_p$, both ultimately tied to the symmetry and distribution of $d$-orbital electron occupation. Asymmetric occupation, enhanced at lower temperatures or by specific 5$d$/3$d$ combinations, leads to stronger DMI (Kim et al., 2017).

3. First-Principles Evaluation and Band Structure Effects

Accurate computation of DMI in real materials is feasible by several ab initio strategies:

1. Spin-spiral total energy calculations: Compute the total energy of a helical (spin-spiral) configuration $n(\mathbf{r}) = [\cos(qz), \sin(qz), 0]$ to extract the linear $q$-dependence $E(q) = Jq^2 - Dq$ and deduce $D = \partial_q E(q)|_{q\to 0}$ (Koretsune et al., 2018).
2. Spin gauge/spin current formalism: Map to a frame aligned locally with the magnetization, uncovering a link between DMI and equilibrium spin current. The DMI vector is

$$D_{\mu}^{\alpha} = \sum_l j_{s, l}^{\perp, \alpha},$$

i.e., only the transverse equilibrium spin current contributes (Koretsune et al., 2018). This approach directly connects DMI to band anticrossings and allows for evaluation from uniform-state properties.

1. Perturbative expansion (RKKY extension): Treat the exchange coupling as a perturbation, leading to an effective DMI expressed via derivatives of the non-interacting spin susceptibility,

$$D_{\mu}^{\alpha} = \frac{J_e^2}{2} \lim_{q\to 0} \frac{\partial \chi_0^\gamma(q,0)}{i\partial q^\mu}$$

This highlights the role of Fermi surface geometry, electron density, and the presence of anticrossing points. Sign changes of DMI often accompany changes in Fermi level position (Koretsune et al., 2018).

All approaches confirm that features such as band crossings and strength of SOC (e.g., in FeGe, Mn$_{1-x}$Fe$_x$Ge, Fe$_{1-x}$Co$_x$Ge) crucially govern the magnitude and sign of the DM coefficient.

4. Non-Relativistic and Spin-Current-Induced DMI

DMI is not solely a relativistic (SOC-induced) phenomenon. In noncollinear magnets—even without SOC—a finite spin-current exists due to the underlying magnetic texture, and this current can mediate a sizable DMI (Cardias et al., 2020, Cardias et al., 2020). The general spin Hamiltonian retains a $$\mathbf{D}_{ij}\cdot(\mathbf{m}_i \times \mathbf{m}_j)$$ term where the DM vector's direction and magnitude may be tailored by the configuration (e.g., the canting angle on a triangular motif as in Mn$_3$Sn). In Kagome lattices and trimer clusters, the DMI can be primarily attributed to these spin-current channels, sometimes dominating over relativistic (SOC) mechanisms.

This generalization broadens the theoretical description of DMI:

• In collinear configurations, SOC is required for DMI.
• In noncollinear configurations, spontaneous spin or charge currents can generate DMI even if SOC is negligible.
• Spin-current-induced DMI components are enhanced in systems with strong structure-driven noncollinearity and frustrated exchange interactions.

5. DMI in Practical Systems: Magnitude, Sign, and Chirality

The magnitude and sign of DMI are highly sensitive to:

• Hybridization between magnetic 3$d$ and nonmagnetic 5$d$ orbitals,
• The electronic structure near the Fermi energy,
• Geometry-induced inversion asymmetry.

For 3$d$–5$d$ zigzag chains, first-principles data shows that, for instance, Ir and Au promote DMI of one sign, Pt of the opposite sign, due to rigid-band shifts with different 5$d$ elements (Kashid et al., 2014).

When the DMI is strong enough to compete with Heisenberg exchange and magnetocrystalline anisotropy, chiral spirals form as ground states. The sign of $D$ dictates the sense (left- or right-handedness) of the spiral and can be switched by minor modifications in composition or lattice structure.

6. Extensions: Interlayer, Bulk, and Multipolar DMI

Beyond conventional interfacial DMI:

• Interlayer DMI couples magnetic layers (e.g., in FM/NM/FM trilayers) via nonmagnetic spacers through conduction electron–mediated three-site interactions. The interlayer DMI can stabilize 3D spirals and is described by generalized Levy–Fert models (Vedmedenko et al., 2018).
• Bulk DMI in centrosymmetric layers emerges due to "hidden" long-range inversion asymmetry arising from strong SOC and orbital hybridization, even in otherwise centrosymmetric, composition-uniform materials. Here, the DMI strength increases with magnetic thickness, contrasting with the interfacial case (Zhu et al., 2022).
• Multipolar DMI: In complex multi-orbital systems (e.g., 5$d^1$ ions in perovskites), DMI can act not only between dipoles but also quadrupoles and octupoles when suitable symmetry and SOC are present. This expands the spectrum of chiral order parameters and low-energy excitations (Hosoi et al., 2018).

7. Experimental Characterization and Materials Control

Experimentally, DMI can be quantified via:

• Domain-wall (DW) motion and spin-wave (SW) nonreciprocity measurements, shown to yield mutually consistent values for DMI strength if controlled for artifacts (Kim et al., 2018).
• Brillouin light scattering, ferromagnetic resonance linewidth broadening (which is broadened proportionally to $D^2$ due to increased two-magnon scattering channels) (Oh et al., 2017).
• X-ray magnetic circular dichroism and detailed orbital-resolved spectroscopies, which evidence strong correlations between DMI and orbital anisotropies (Kim et al., 2017).

Strategies to control and engineer DMI include:

• Patterning spatially varying DMI: By lithographically structuring adjacent layers, spatial DMI modulation allows for confinement and shaping of domain walls, magnonic waveguides, and skyrmion racetracks (Mulkers et al., 2017).
• Current-induced control: External currents can modify the equilibrium spin current at interfaces (spin Doppler effect), tuning the DMI and enabling dynamic control for device applications (Kato et al., 2018).
• Composition gradients (g-DMI): Introducing a controlled compositional gradient yields a robust, thickness-enhanced DMI with programmable chirality, stabilizing spin spirals and skyrmions and enabling field-free spin–orbit torque switching (Liang et al., 2022).
• Chirality-inverted DMI: The sign of DMI can invert for interactions at different atomic distances, dictating complex spin textures at the nanoscale (Zakeri et al., 2023).

System Type	Main DMI Mechanism	Typical Features/Control
3$d$–5$d$ zigzag chains	Hybridization + SOC	DMI magnitude/sign tunable by 5$d$ choice
Magnetic thin films with HM contacts	Interfacial SOC + ISB	Patterned DMI, current control
Centrosymmetric/gradient bulk layers	Hidden/“g-” symmetry-breaking	Linear thickness dependence, field-free SOT, chirality control
Nanocrystalline, polycrystalline systems	Grain boundary DMI	Chiral misalignment, magnetic softness

8. Outlook and Implications

DMI is central to the stabilization and control of chiral magnetic phases across a vast range of materials, including low-dimensional conductors, complex oxides, multilayered heterostructures, and nanocrystalline aggregates. Its magnitude, sign, and spatial profile can be engineered via electronic structure design, external stimuli, and symmetry manipulation. Extensions to multipolar moments, control by superconducting phase bias (DMI-like terms emergent from mixed-parity superconductivity) (Ouassou et al., 9 Jul 2024), and appearance in the absence of explicit SOC all illustrate the fundamental and practical richness of antisymmetric exchange. Mastery of DMI mechanisms underpins the future of topological spintronics, robust memory/logic architectures, and quantum information schemes predicated on noncollinear spin order.