--- title: Dzyaloshinskii–Moriya Interaction in Chiral Materials url: https://www.emergentmind.com/topics/dzyaloshinskii-moriya-type-interaction type: topic --- # Dzyaloshinskii–Moriya Interaction in Chiral Materials 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 [2106.00398]. 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 [2106.00398]. 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 $\mathbf{m}(\mathbf{r})$ [1605.07044, 2106.00398], 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 [1605.07044]. ## 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 [2106.00398]. 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 [2106.00398, 1704.02900]. 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 [1704.02900]. 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 $\Delta m_o = m_o^{\perp} - m_o^{\parallel}$ [1704.02900]. 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 [2003.04680, 2007.00358]. 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 [1804.04874]: \[ H_\mathrm{DM}^Q \sim D^Q\, \mathbf{Q}_i \times \mathbf{Q}_j, \qquad H_\mathrm{DM}^O \sim D^O\, \mathbf{O}_i' \times \mathbf{O}_j' \] 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 [2211.13099]: \[ E_\mathrm{eDMI} = \frac{1}{2}\sum_{i