Dzyaloshinskii–Moriya Interaction in Chiral Materials

• Dzyaloshinskii–Moriya interaction is an antisymmetric exchange coupling arising from broken inversion symmetry and spin–orbit coupling that stabilizes chiral spin textures like skyrmions and spin spirals.
• It is characterized both atomistically and in continuum form, with experimental approaches such as resonant x-ray scattering and Brillouin light scattering used for its quantification.
• Recent theoretical and experimental advances extend DM-type interactions to electric and superconducting systems, opening new paths for chiral spintronics and topological device applications.

The Dzyaloshinskii-Moriya (DM) type interaction is an antisymmetric exchange coupling between localized moments that arises fundamentally from broken inversion symmetry and, in the canonical case, relativistic spin-orbit coupling (SOC). At both the atomistic and continuum scales, DM-type interactions play a key role in stabilizing non-collinear, chiral spin textures such as spin spirals, domain walls of fixed handedness, and magnetic skyrmions. They have recently been detected and characterized in a wide array of materials systems, including ultrathin magnetic films, bulk noncentrosymmetric magnets, magnetic heterostructures, and even beyond-magnetic contexts in the form of electric Dzyaloshinskii-Moriya interactions and superconductivity-mediated analogs.

1. Fundamental Definition and Symmetry

At the microscopic level, the DM-type exchange interaction between magnetic moments $\mathbf{S}_i$, $\mathbf{S}_j$ takes the form

$$H_\mathrm{DM} = \sum_{\langle i,j \rangle} \mathbf{D}_{ij} \cdot (\mathbf{S}_i \times \mathbf{S}_j)$$

where $\mathbf{D}_{ij}$ is the DM vector, an axial vector dictated by the local symmetry of the underlying crystal or interface (Mazurenko et al., 2021). The presence of this antisymmetric exchange requires the absence of inversion symmetry at the bond center and is allowed whenever

• the exchange path allows spin-orbit mediated virtual hopping processes,
• the lattice or interface lacks a center of symmetry at the midpoint between $\mathbf{S}_i$ and $\mathbf{S}_j$.

The orientation and allowed components of $\mathbf{D}_{ij}$ are directly fixed by the crystal symmetry according to Moriya’s rules (Mazurenko et al., 2021). For instance, if there is an inversion center midway between $i$ and $j$, then $\mathbf{D}_{ij}=0$. The sign of $\mathbf{D}_{ij}$ determines the preferred handedness (chirality) of the canting.

In the continuum, the DM-type term can be written as an energy density

$$E_\mathrm{DMI} = D\, \mathbf{m} \cdot (\nabla \times \mathbf{m})$$

for a vector magnetization field $\mathbf{m}(\mathbf{r})$ (Gross et al., 2016, Mazurenko et al., 2021), or, equivalently,

$$E_\mathrm{DMI} = D [ m_z \partial_x m_x - m_x \partial_x m_z + m_z \partial_y m_y - m_y \partial_y m_z ]$$

for thin films with perpendicular anisotropy (Gross et al., 2016).

2. Physical Origin: Microscopic and Effective Theories

The microscopic origin of the DM-type interaction is rooted in the interplay of spin-orbit coupling and broken inversion symmetry. In Moriya’s superexchange framework, a virtual hopping process whereby an electron moves from $i$ to $j$ via a ligand with strong SOC can pick up an orbital angular momentum “kick” that is antisymmetric under inversion (Mazurenko et al., 2021). The most general microscopic Hamiltonian may include

$$H = \sum_{ij, \sigma\sigma'} t_{ij}^{\sigma\sigma'} a_{i\sigma}^\dagger a_{j\sigma'} + \frac{U}{2} \sum_{i,\sigma \neq \sigma'} n_{i\sigma} n_{i\sigma'}$$

with SOC encoded in $t_{ij}^{\sigma\sigma'}$. To second order in hopping and first order in $\lambda_\mathrm{SOC}$, one obtains the antisymmetric exchange term (Mazurenko et al., 2021, Kim et al., 2017).

At interfaces or heterostructures, inversion symmetry is often broken locally (e.g. at the FM/HM interface in layered systems). There, the DM vector is set by the atomic registry and local orbital occupancy, with its sign and magnitude controlled by the strength of SOC, orbital hybridization, and interface-specific details (Kim et al., 2017).

A distinctive microscopic feature of DM-type interactions is their correlation with orbital anisotropy and intra-atomic asphericity, as quantified by quantities like the magnetic dipole moment $m_T$ and the difference in orbital moments $\Delta m_o = m_o^{\perp} - m_o^{\parallel}$ (Kim et al., 2017).

Beyond conventional SOC-mediated origins, non-collinear magnetization textures themselves can induce substantial DM-type interactions even in the absence of SOC, via spontaneously generated spin and charge currents; these non-relativistic spin-current contributions can be the dominant mechanism in certain systems (Cardias et al., 2020, Cardias et al., 2020). This extends the concept of DM-type exchange far beyond its original relativistic setting.

3. Extensions: Multipolar, Electronic, and Superconducting DM-Type Interactions

The DM-type mechanism is not restricted to spin-dipole moments. In $5d^1$ systems with strong SOC and electron correlations, antisymmetric couplings can appear between rank-2 (quadrupolar) and rank-3 (octupolar) moments (Hosoi et al., 2018): $$H_\mathrm{DM}^Q \sim D^Q\, \mathbf{Q}_i \times \mathbf{Q}_j, \qquad H_\mathrm{DM}^O \sim D^O\, \mathbf{O}_i' \times \mathbf{O}_j'$$ These multipolar DM terms obey the same symmetry rules but arise from orbital–only channels, and can be present with or without spin canting. Their magnitudes and presence can depend delicately on the lattice geometry and strength of SOC.

A parallel class of interactions—electric Dzyaloshinskii-Moriya interactions (eDMI)—can be defined in polar materials, coupling local atomic displacements via an antisymmetric energy term (Chen et al., 2022): $$E_\mathrm{eDMI} = \frac{1}{2}\sum_{i<j} \mathbf{D}_{ij}\cdot(\mathbf{u}_i\times\mathbf{u}_j)$$ with $\mathbf{u}_i$ atomic displacements. eDMI is trilinear in displacements (third-order) and does not require SOC, arising from electronic mediation under local inversion breaking (Chen et al., 2022). eDMI can stabilize chiral electric textures (“bobbers”), topologically distinct from standard domain walls, and exhibits unique group-theoretical symmetries.

In superconducting contexts, DM-type couplings can emerge purely from mixed-parity ($s+ip$) superconducting condensates, without any particle–level SOC. The effective impurity spin Hamiltonian contains a DM vector proportional to the imaginary part of the product of singlet- and triplet-pairing Green’s functions and can be toggled by controlling the superconducting phase or order parameter admixture (Ouassou et al., 9 Jul 2024, Ouassou et al., 2023).

4. DM-Type Interactions in Materials: Classes and Tuning

Different classes of DM-type interactions appear depending on the spatial origin of inversion-symmetry breaking and the material stack:

• Bulk-type DMI: Occurs in noncentrosymmetric crystals (e.g., B20 system, MnSi, FeGe). The continuum energy density is $$E_d^b = D_b \mathbf{m}\cdot(\nabla\times\mathbf{m})$$. In such materials, the bulk DM term stabilizes long-period helical spin arrangements and complex skyrmion lattices (Dmitrienko et al., 2011, Mazurenko et al., 2021, Yershov et al., 12 Feb 2025).
• Interfacial DMI: Arises at heavy metal/ferromagnet (HM/FM) or HM/antiferromagnet (HM/AFM) interfaces (e.g., Pt/Co, W/MnPt). Here, inversion symmetry is broken at the boundary, allowing for large $D$ values. The energy density (per area) takes the form $$w_\mathrm{iDMI} = D [ n_z \nabla \cdot n - (n\cdot \nabla) n_z ]$$, and the strength can be engineered via underlayer material, thickness, or doping (Gross et al., 2016, Akanda et al., 2020). Interfacial DMI can be comparable in magnitude in AFM/HM and FM/HM systems (Akanda et al., 2020).
• Interlayer DMI: Heterostructures can support DMI terms that couple moments across a non-magnetic spacer, mediated by conduction electrons via the Levy-Fert three-site mechanism (Vedmedenko et al., 2018). The resulting interlayer DMI can lead to genuinely three-dimensional chiral textures and is sensitive to stacking geometry, spacer thickness, and the electronic structure of the NM spacer.
• Gradient-induced and bulk-like DMI in centrosymmetric systems: Uniform magnetic layers lacking both global inversion-breaking and composition gradient can still exhibit robust, thickness-dependent DMI that exceeds previously reported interfacial DMI values (Zhu et al., 2022). A composition gradient (g-DMI), even in a disordered system, can yield additive, bulk DMI scaling with both gradient and film thickness (Liang et al., 2022).
• Curvature-induced DMI: In curved geometries such as magnetic nanotubes, both intrinsic (bulk/interfacial) and “mesoscale” DMI terms emerge from interplay between intrinsic interactions and geometric curvature (Yershov et al., 12 Feb 2025). Such curvature-induced DM terms break wall-helicity symmetry and shift dynamical thresholds (e.g., Walker breakdown fields).

5. Measurement and Computation: Methodologies and Challenges

Measuring both the magnitude and sign of DM-type interactions requires methods sensitive to the relative phase and the chirality of spin canting:

• Neutron diffraction and Mössbauer spectroscopy: By exploiting interference effects between magnetic and nuclear structure factors under appropriate field and polarization, both the sign and magnitude of DMI can be determined in weak ferromagnets (e.g., MnCO$_3$, Fe$_2$O$_3$) (Dmitrienko et al., 2010). Forbidden (e.g., $00l$, $l=2n+1$) reflections provide direct evidence for small antiferromagnetic tilts induced by the DM interaction (Dmitrienko et al., 2011).
• Resonant x-ray scattering: Enables element- and orbital-resolved detection of DMI-induced chirality and phase, especially in thin films and multilayers (Dmitrienko et al., 2010, Zakeri et al., 2023).
• Brillouin light scattering (BLS) and spin polarized EELS: Offers direct access to DMI by measuring nonreciprocal magnon dispersion, with quantitative extraction via $\Delta f_\mathrm{DMI} \propto D k$ (Zhu et al., 2022, Zakeri et al., 2023). High-resolution experiments in Co/Ir(001) can resolve not only the magnitude but also the “chirality-inversion” of DMI with distance (Zakeri et al., 2023).
• Scanning-NV magnetometry: Atomic-scale mapping of stray fields above domain walls enables spatially resolved measurements of local DMI with ∼50–150 nm resolution (Gross et al., 2016).
• Electronic structure theory (DFT+U, tight binding, Green’s function): DMI can be computed ab initio via the mapping of total energies of spin spirals, or by evaluating variations of the electronic structure under infinitesimal rotations of non-collinear magnetic configurations (Kim et al., 2017, Cardias et al., 2020). Beyond SOC-driven mechanisms, Green’s function and density-matrix approaches reveal nonrelativistic, spin-current-induced DMI that remains finite without SOC (Cardias et al., 2020).

A technical challenge is the configuration dependence of both magnitude and direction of $\mathbf{D}_{ij}$ for low-symmetry and noncollinear cases, which demands calculations and measurements in the relevant magnetic ground states.

6. Applications and Physical Manifestations

DM-type interactions are central to the stabilization and dynamics of a wealth of chiral magnetic structures:

• Domain wall chirality and current-driven motion: The sign of D determines whether right- or left-handed Néel walls are stable, leading to efficient spin-orbit torque-driven unidirectional DW motion. Sufficiently large D stabilizes homochiral skyrmions, which are robust against fluctuations and suited for high-density devices (Gross et al., 2016).
• Noncollinear antiferromagnets: Interfacial DMI can act as emergent uniaxial magnetic anisotropy in noncollinear AFMs (e.g., stacked-Kagome AFM/NM bilayers), providing a new handle for engineering switching thresholds and auto-oscillation dynamics (Yamane et al., 16 Feb 2025).
• Chirality inversion and oscillatory DMI: The sign and magnitude of DMI couplings can oscillate with distance, reminiscent of RKKY exchange, leading to complex spin textures, stable skyrmions, bimerons, and engineered multi-Q states in nanostructures (Zakeri et al., 2023).
• Curvilinear and 3D textures: In magnetic nanotubes, the interplay of intrinsic, curvature-induced, and mesoscale DMI terms determines wall type, chirality and dynamical breakdown thresholds, enabling new modalities in curvilinear magnonics and data storage (Yershov et al., 12 Feb 2025).
• Electric and superconductivity-mediated DMI: eDMI enables stabilization of topological polar textures (bobbers, skyrmions) without SOC (Chen et al., 2022). In superconductors, DM-type interactions induced by mixed-parity order open prospects for phase-controllable spin manipulation and quantum computation (Ouassou et al., 9 Jul 2024).
• Chiral order in multipolar and hybrid systems: High-rank multipolar DM-type couplings can establish chiral multipolar order and lattice distortions, with detection possible via higher-order resonant x-ray scattering and second harmonic generation (Hosoi et al., 2018).

7. Outlook and Broader Implications

The DM-type interaction landscape has expanded well beyond its original SOC-based context:

• The recognition of non-relativistic, spin-current-driven DMI mechanisms generalizes the concept to many noncollinear magnetic materials (Cardias et al., 2020, Cardias et al., 2020).
• Observations of strong DMI in centrosymmetric, compositionally uniform films suggest that local inversion breaking, strain, or chemical order suffices to realize large bulk DMI, broadening the class of functional materials (Zhu et al., 2022).
• The ability to tune DMI via interface engineering, composition gradients, curvature, or even electronic phase control (in superconductors) provides a diverse toolkit for chiral spintronic and polar ferroic technologies.

Theoretical, computational, and experimental advances now permit systematic identification, measurement, and exploitation of these interactions across materials classes and device geometries, paving the way for controlled realization of exotic topological textures, high-speed low-power data storage, and new quantum functionalities.