Antisymmetric Dzyaloshinskii–Moriya Interaction

• Antisymmetric Dzyaloshinskii–Moriya interaction is a relativistic exchange mechanism produced by strong spin–orbit coupling and broken inversion symmetry.
• It competes with symmetric exchange to stabilize noncollinear, chiral magnetic textures such as skyrmions, spin spirals, and domain walls.
• Experimental methods like SPEELS, BLS, and neutron scattering provide quantifiable insights into DMI magnitude and chirality.

The antisymmetric Dzyaloshinskii–Moriya interaction (DMI) is a relativistic exchange mechanism that arises in magnetic systems with strong spin–orbit coupling and broken inversion symmetry, leading to noncollinear spin alignments and stabilizing topologically nontrivial magnetic textures. The DMI is described by a bilinear, antisymmetric exchange term in the magnetic Hamiltonian, contrasting sharply with the symmetric Heisenberg exchange. It is essential for understanding and engineering a range of chiral magnetic phenomena in ultrathin films, multilayers, bulk compounds, and defect-rich environments, and is directly quantifiable via spectroscopic, scattering, and transport techniques. DMI is now a central tool in the synthesis, analysis, and control of spin textures for next-generation spintronic and magnonic devices (Zakeri, 2017).

1. Mathematical Formulation and Emergent Symmetry

The general spin Hamiltonian for a system of discrete localized moments $\mathbf{S}_i$ comprises a symmetric Heisenberg exchange and an antisymmetric Dzyaloshinskii–Moriya term,

$$\mathcal{H} = -\sum_{i\neq j} J_{ij}\,\mathbf S_i\cdot\mathbf S_j -\sum_{i\neq j} \mathbf D_{ij}\cdot (\mathbf S_i\times\mathbf S_j)$$

where $J_{ij}$ is symmetric under $i\leftrightarrow j$ and $\mathbf{D}_{ij} = -\mathbf{D}_{ji}$ is an axial vector determined by local symmetry and spin–orbit effects (Zakeri, 2017, Zakeri et al., 2023). DMI explicitly breaks inversion symmetry: under spatial inversion, the cross product changes sign, enforcing the requirement that $\mathbf{D}_{ij}$ must vanish in centrosymmetric environments (Mazurenko et al., 2021).

For continuous descriptions, especially in thin films and noncentrosymmetric bulk crystals, the DMI contribution is written as a density,

$$w_\mathrm{DMI}^{\rm bulk} = D\,\mathbf{m}\cdot(\nabla\times\mathbf{m}),\qquad w_\mathrm{DMI}^{\rm int} = D[m_z(\nabla\cdot\mathbf{m})-(\mathbf{m}\cdot\nabla)m_z]$$

where $\mathbf{m}(\mathbf{r})$ is the normalized magnetization (Gallardo et al., 2018). The sign and orientation of $D$ and $\mathbf{D}_{ij}$ are dictated by crystallographic details.

2. Microscopic Origin and Electronic Pathways

The DMI arises when two neighboring magnetic moments exchange virtually through a nonmagnetic atom with strong spin–orbit coupling in a non-inversion symmetric environment. The prototypical mechanism (Moriya’s theory) involves three-center hopping and SOC, yielding,

$$\mathbf{D}_{ij} \propto \lambda_\mathrm{SOC} [\mathbf{r}_{i\ell} \times \mathbf{r}_{\ell j}]$$

where intermediate states at site $l$ mediate the interaction between $i$ and $j$ (Zakeri, 2017, Kashid et al., 2014, Mazurenko et al., 2021). The direction and sign of $\mathbf{D}_{ij}$ depend on the geometry defined by the spin sites and the intermediate atom(s), as formalized by symmetry rules (e.g., Moriya’s rules).

DMI strength scales with atomic SOC and the degree of hybridization between magnetic and heavy-metal orbitals. For interfaces, materials with large $5d$ elements (Pt, W, Ir) and sharp inversion breaking (oxide capping, abrupt interfaces) exhibit strong DMI (Zakeri, 2017, Zakeri et al., 2023).

3. Manifestations in Magnetic Excitations and Order

DMI competes with symmetric exchange, resulting in energetically preferred chiral arrangements:

• Magnon Dispersion: The magnon frequency becomes nonreciprocal: $$\omega(\mathbf{k}) = \omega_0 + \frac{2JS}{\hbar}|\mathbf{k}|^2 + \frac{2DS}{\hbar}(\hat{e}\cdot\hat{z})|\mathbf{k}|$$ so that $\omega(k)\neq\omega(-k)$. This leads directly to frequency/energy asymmetries in the magnon spectrum, observable by spectroscopic means (Zakeri, 2017, Kim et al., 2018, Böttcher et al., 2020).
• Chiral Spin Textures: Finite DMI stabilizes Néel-type domain walls, spin spirals, and magnetic skyrmions. The characteristic spiral period is $\lambda=4\pi A/D$ (with $A$ the exchange stiffness) (Turgut et al., 2018, Gallardo et al., 2018).
• Low-dimensional Ordering: DMI can promote spontaneous long-range magnetic order even in 1D and 2D systems by lifting the Mermin–Wagner restriction, as it introduces a linear-in-$k$ term in the spin-wave stiffness that regularizes thermal fluctuations (Torres et al., 2014).
• Anisotropy in Antiferromagnets: At nonmagnetic/AFM interfaces, specific DMI components reduce (in some crystal orientations) to an effective uniaxial anisotropy for the AFM order parameter (Yamane et al., 16 Feb 2025).

4. Experimental Determination and Quantification

Spectroscopic Approaches

Method	Principle	Observable
SPEELS	Spin-polarized EELS at defined $Q$	$\Delta\varepsilon(Q) = 2D_\mathrm{eff} Q$ (Zakeri, 2017, Zakeri et al., 2023)
BLS	Inelastic photon scattering off magnons	$\Delta f = (2\gamma/\pi M_s) D_\mathrm{eff} k$ (Zakeri, 2017, Böttcher et al., 2020)
Domain-wall velocity	Field/current-driven DW motion	$H_\mathrm{DMI} = D/(\mu_0 M_s \Delta)$ (Kato et al., 2018, Kim et al., 2018)
Neutron Scattering	Polarized reflectometry/SANS	Chiral asymmetry in scattering cross section, $\delta R\propto D$ (Tatarskiy, 2019, Sinaga et al., 2024, Michels et al., 2018)
Conventional Magnetometry + ML	FORC/minor-loop fingerprints	CNN-extracted $D$ from loop families (Fugetta et al., 2023)

Table: Principle methods to quantify interfacial and bulk DMI.

Data-Driven DMI Constants (Illustrative)

System	$D_\mathrm{eff}$ (THz·Å)	Method	Reference
Fe(2ML)/W(110)	$2.9\pm0.5$	SPEELS	(Zakeri, 2017)
Ni$_{80}$Fe$_{20}$/Pt	$\sim$0.15	BLS	(Zakeri, 2017)
Pt/Co(1nm)/AlO$_x$	$\sim$1.25	BLS	(Zakeri, 2017)

Reported DMI magnitudes range from $\sim$0.1 THz·Å in weak-SOC systems to $3$ THz·Å with strong SOC; chirality and sign depend on interface composition and stacking (Zakeri, 2017).

Novel and Advanced Methodologies

• ML-Based Metrology: CNNs trained on minor loops can extract $D$ with $\sim$15% uncertainty, leveraging commonly available magnetometry data (Fugetta et al., 2023).
• Chirality Inversion Detection: Atomistic DMI constants can invert sign with neighbor shell—analogous to RKKY oscillations—implying that effective mesoscale $D_\mu$ can be zero even with large atomistic DMI (Zakeri et al., 2023).

5. Tuning, Engineering, and Physical Control

• Materials Chemistry: The DMI is strongly sensitive to chemical composition, interface sharpness, and heavy-metal capping. In B20 thin films, $D$ is tuned by stoichiometry, enabling control over the skyrmion size across two decades in length (Turgut et al., 2018).
• Interlayer DMI: IL-DMI can be engineered in synthetic antiferromagnets with asymmetric stacking (e.g., Ta/Pt/Co/Ir/Co), enabling chiral bias fields (~1–2 mT) and deterministic, field-free SOT switching (Li et al., 2024).
• Current Modulation: Interfacial $D$ can be modulated dynamically by in-plane currents, consistent with the spin Doppler effect: $D \propto J_s$, where $J_s$ is the equilibrium or injected spin current (Kato et al., 2018).
• Defect Engineering: High densities of inversion-breaking defects induce local DMI, allowing chiral texture stabilization even in centrosymmetric or polycrystalline magnets (Michels et al., 2018).

6. Impact on Magnetic Textures and Device Phenomena

• Stabilization of Topological Structures: Sizable DMI is a prerequisite to stabilize skyrmions, chiral domain walls, spin spirals, hopfions, and Néel walls at room temperature, which are robust against defects and thermal fluctuations (Zakeri, 2017, Sinaga et al., 2024, Turgut et al., 2018).
• Wave Propagation Engineering: Periodic modulation of DMI (e.g., by lithographic patterning of heavy-metal wires) yields magnonic crystals with indirect bandgaps, flat bands, and spatially hybridized wavefunctions, offering new approaches to magnonic logic and signal processing (Gallardo et al., 2018).
• Spin Seebeck and Transport Effects: The magnitude and symmetry of the DMI at FM|NM or AFM|NM interfaces control the efficiency and directionality of spin transfer in thermally driven spin-Seebeck devices, depending on detailed orientation of $\mathbf{D}_{ij}$ vis-à-vis interface and magnetic order (Ma et al., 2019, Akanda et al., 2020).
• Universal Applicability: DMI extends far beyond traditional weak ferromagnets. Its effects are generalized to higher-rank multipoles, multipolar DM interactions in $5d^1$ perovskites, and appears as a uniaxial anisotropy in noncollinear antiferromagnets with suitable interface engineering (Hosoi et al., 2018, Yamane et al., 16 Feb 2025).

7. Trends, Challenges, and Future Perspectives

Key trends emerging from systematic experimental and theoretical analyses include:

• DMI strength and sign are tunable via interface engineering, heavy-metal composition, and atomic-level stacking order (Zakeri, 2017, Turgut et al., 2018, Li et al., 2024).
• The sign of DMI (chirality) is not always fixed—chirality inversion with neighbor shell complicates the design rules for mesoscale spin textures (Zakeri et al., 2023).
• DMI is subject to dynamic control through spin currents, enabling "on-the-fly" manipulation in operating devices (Kato et al., 2018).
• First-principles quantification is challenging due to strong sensitivity to SOC and charge density, but methodological advances (DFT+U, Green’s function techniques, ML extraction) are increasingly closing the gap with experiment (Mazurenko et al., 2021, Fugetta et al., 2023).
• Predictive understanding of DMI in antiferromagnetic systems and synthetic heterostructures holds promise for further broadening the palette of chiral spin textures and corresponding device functionalities (Akanda et al., 2020, Yamane et al., 16 Feb 2025).

The DMI is thus not only a fundamental quantum interaction but also a versatile and highly engineerable resource for the stabilization, manipulation, and detection of chiral magnetic order in condensed matter and applied nanomagnetism (Zakeri, 2017, Mazurenko et al., 2021, Zakeri et al., 2023).