Electric Dzyaloshinskii–Moriya Interaction

• Electric Dzyaloshinskii–Moriya interaction is a chiral exchange mechanism where local polar displacements and electric fields replace spins to generate antisymmetric couplings.
• It bridges lattice chirality, magnetoelectric coupling, and spin–orbit physics, enabling dynamic voltage tuning and electric control of magnetic textures.
• Recent studies reveal its role in reversing DMI chirality, tuning skyrmion structures, and stabilizing novel topological defects via polarization switching and interfacial asymmetry.

Searching arXiv for recent and foundational papers on electric Dzyaloshinskii-Moriya interaction. Electric Dzyaloshinskii-Moriya interaction denotes a class of antisymmetric chiral couplings in which electric variables—local polar displacements, electric polarization, or electric-field-controlled interfacial asymmetry—play the role ordinarily associated with magnetic spin chirality in the conventional Dzyaloshinskii-Moriya interaction (DMI). In current usage, the term covers two closely related but distinct regimes. In the narrow microscopic sense, it refers to a purely ionic-displacement interaction of the form $$E^{A}=\tfrac12\sum_{i,j}\bm{\mathcal D}(i,j)\!\cdot(\bm u_i\times \bm u_j)$$, whose leading invariant is cubic in polar displacements and does not require spin-orbit coupling (Chen et al., 2022). In a broader multiferroic and spintronic sense, it also encompasses electrically generated, electrically tuned, or polarization-reversed magnetic DMI, where an external electric field, a ferroelectric order parameter, or an interfacial dipole controls the magnitude, sign, or chirality of $\bm D_{ij}$ in $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$ (Yang et al., 2016). The topic therefore sits at the intersection of lattice chirality, magnetoelectric coupling, interfacial spin-orbit physics, and topological textures in ferroelectrics and magnets.

1. Conceptual scope and definitions

The conventional magnetic Dzyaloshinskii-Moriya interaction is an antisymmetric exchange term,

$$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$

which requires broken inversion symmetry and, in its standard form, spin-orbit coupling. In multiferroics and oxide interfaces, electric polarization and electric fields enter this framework by determining either the allowed direction of $\bm D_{ij}$ or its magnitude and sign (Meyer et al., 2022). In BiFeO$_3$, for example, the spin-current form

$D_{ij}=C_{ij}\,(u\times e_{ij})$

directly ties the DM vector to the ferroelectric polarization direction $u=P/|P|$, so reversing $P$ reverses every DM vector (Meyer et al., 2022).

A distinct meaning was formalized in the microscopic theory of the electric Dzyaloshinskii-Moriya interaction, abbreviated eDMI. There the antisymmetric object is not built from spins but from local polar displacements $\bm u_i$. The chiral part of the harmonic energy can be written as

$\bm D_{ij}$0

with $\bm D_{ij}$1 (Chen et al., 2022). In perovskites the lowest-order invariant becomes

$\bm D_{ij}$2

which is third order in displacements (Chen et al., 2022). This sharp distinction—bilinear in spins for mDMI, cubic in displacements for eDMI—is central to the present literature.

The broader literature also identifies electric dipoles as descriptors of interfacial DMI. In metallic $\bm D_{ij}$3/Co/Pt multilayers, the out-of-plane electric dipole moment $\bm D_{ij}$4 of Pt correlates nearly linearly with the Pt-layer DMI contribution $\bm D_{ij}$5, establishing an electrostatic quantity as a practical proxy for chiral magnetic exchange (Jia et al., 2019). This suggests that “electric DMI” is now used across three layers of description: a purely electric lattice interaction, polarization-controlled magnetic DMI in multiferroics, and interfacial DMI whose strength is encoded by an electric dipole or tuned by an electric field.

2. Microscopic origin of the purely electric interaction

The microscopic origin of eDMI is formulated in terms of the Born-Oppenheimer Hessian. For ionic displacements $\bm D_{ij}$6, the force constants are

$\bm D_{ij}$7

and the antisymmetric part is

$\bm D_{ij}$8

This antisymmetric component survives because of the ion-electron contribution to the Hessian; it is therefore an electron-mediated effect rather than a purely ionic geometric one (Chen et al., 2022).

At the microscopic level, local inversion-symmetry breaking activates virtual hopping loops that would otherwise cancel. In the tight-binding plus Green’s-function description, each contribution to $\bm D_{ij}$9 arises from a loop of virtual hops $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$0. Certain mirror operations reverse the sign of these loops, and this sign structure enforces an antisymmetric force constant (Chen et al., 2022). The mechanism is analogous to the structural logic behind mDMI—off-diagonal hopping and local inversion breaking are essential in both cases—but eDMI does not require spin-orbit coupling (Chen et al., 2022).

The derived energy form is not merely a formal rewriting of a harmonic force matrix. Symmetry analysis in perovskites shows that the chiral invariant first appears at cubic order in the local polar displacements. The coefficient $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$1 is material-specific and was fitted to first-principles data; for PbTiO$$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$2 it is reported as $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$3 Hartree/Bohr$$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$4 (Chen et al., 2022). The tight-binding model reproduces antisymmetric force constants extracted from density-functional perturbation theory to $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$5 using ONCV pseudopotentials, 50/400 Ry cutoffs, a $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$6 $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$7-mesh, and a $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$8 $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j)$$9-mesh (Chen et al., 2022).

A common misconception is to treat eDMI as a straightforward electrical analogue of spin DMI obtained by replacing $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$0 with $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$1. The microscopic analysis shows that this is not correct. The magnetic interaction is bilinear in $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$2 and $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$3, whereas the electric interaction is at least third order in displacements because local inversion breaking must itself be encoded by a displacement field (Chen et al., 2022). This difference alters both the allowed invariants and the topology of the resulting textures.

3. Relation to magnetic DMI, spin-current mechanisms, and magnetoelectric polarization

In multiferroics, electric and magnetic DMI are often intertwined rather than separate. The spin-current formalism of Katsura-Nagaosa-Balatsky appears explicitly in both molecular and bulk settings. In the tetrahedral single-molecule magnet $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$4, Berry-phase density-functional calculations give a spin-dependent dipole

$$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$5

with $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$6–$$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$7 $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$8Å (Yu et al., 2021). Here the same noncollinear spin texture that enters the DM term also generates an electric polarization. Collinear spin states give $$E_{\rm mDMI}=\sum_{\langle i,j\rangle}\bm D_{ij}\cdot(\bm S_i\times\bm S_j),$$9 and hence $\bm D_{ij}$0 (Yu et al., 2021).

For the Mn$\bm D_{ij}$1 tetrahedron, the spin Hamiltonian contains Heisenberg exchange, DMI, and single-ion anisotropy,

$\bm D_{ij}$2

with each $\bm D_{ij}$3 perpendicular to the Mn-Mn bond and magnitude $\bm D_{ij}$4 meV extracted from total energies of 12 non-collinear configurations (Yu et al., 2021). The resulting magnetoelectric response is quadratic in external fields: $\bm D_{ij}$5 and $\bm D_{ij}$6, with a predicted fractional change $\bm D_{ij}$7 at $\bm D_{ij}$8 K and $\bm D_{ij}$9 V/cm (Yu et al., 2021). This is not eDMI in the narrow ionic sense, but it demonstrates how DM-mediated spin chirality produces an electric response.

BiFeO$_3$0 provides the bulk multiferroic counterpart. The first-principles study finds only one relevant DM contribution from the spin-current model,

$_3$1

with $_3$2 and dominant nearest-neighbor coefficient $_3$3 meV/Fe (Meyer et al., 2022). Exchange is isotropic, but the DM interaction and anisotropy confine the preferred propagation and magnetization directions to the full $_3$4 plane (Meyer et al., 2022). Reversing the ferroelectric polarization flips the sign of the DM vectors and reverses the nonreciprocal magnon dispersion shift $_3$5 (Meyer et al., 2022). A plausible implication is that bulk ferroelectrics realize an experimentally accessible bridge between the strict eDMI concept and electrically reversible magnetic DMI.

4. Electric fields, interfacial asymmetry, and voltage control of magnetic DMI

Electric control of interfacial DMI is a major experimental manifestation of the broader electric-DMI program. In MgO/Co/Pt trilayers, first-principles calculations show that an electric field normal to the interface modifies the DMI approximately linearly,

$_3$6

or, in the micromagnetic description,

$_3$7

with $_3$8 fJ/(V m) (Yang et al., 2016). The zero-field values are $_3$9 meV/atom and $D_{ij}=C_{ij}\,(u\times e_{ij})$0 mJ m$D_{ij}=C_{ij}\,(u\times e_{ij})$1, and under $D_{ij}=C_{ij}\,(u\times e_{ij})$2 V nm$D_{ij}=C_{ij}\,(u\times e_{ij})$3 the DMI increases to about $D_{ij}=C_{ij}\,(u\times e_{ij})$4 mJ m$D_{ij}=C_{ij}\,(u\times e_{ij})$5 (Yang et al., 2016). The mechanism at the oxide/Co interface is described as Rashba-type and differs from the Fert-Levy mechanism at heavy-metal/ferromagnet interfaces (Yang et al., 2016).

The same theme appears experimentally in Pt/Co/AlO$D_{ij}=C_{ij}\,(u\times e_{ij})$6, where electric fields alter labyrinthine stripe domains observed by polar MOKE. Using the equilibrium stripe width $D_{ij}=C_{ij}\,(u\times e_{ij})$7, saturation magnetization $D_{ij}=C_{ij}\,(u\times e_{ij})$8, effective anisotropy $D_{ij}=C_{ij}\,(u\times e_{ij})$9, and assumed exchange stiffness $u=P/|P|$0, the interfacial DMI constant $u=P/|P|$1 is inferred from the domain-wall energy (Schott et al., 2020). For $u=P/|P|$2 pJ/m, $u=P/|P|$3 changes from $u=P/|P|$4 to $u=P/|P|$5 mJ m$u=P/|P|$6 between 6 and 14 V, yielding $u=P/|P|$7 fJ/(V·m); for $u=P/|P|$8 pJ/m, the corresponding values are $u=P/|P|$9 to $P$0 mJ m$P$1 and $P$2 fJ/(V·m) (Schott et al., 2020). The interpretation is that charge accumulation or depletion in the top Co monolayer modifies orbital hybridization and Rashba-type interfacial spin-orbit fields (Schott et al., 2020).

Hybrid multiferroic structures extend this to strain-mediated electric control. In Pt/Co/Pt deposited on PMN-PT, Brillouin light scattering reveals field-tunable interfacial DMI from $P$3 to $P$4 mJ m$P$5 (Udalov et al., 2024). On the [001] cut, the change is isotropic, following approximately $P$6 with $P$7 mJ m$P$8 and $P$9 (mJ m$\bm u_i$0)/(MV m$\bm u_i$1) (Udalov et al., 2024). On the [011] cut, anisotropic strain yields $\bm u_i$2 and $\bm u_i$3 with $\bm u_i$4 and $\bm u_i$5 (mJ m$\bm u_i$6)/(MV m$\bm u_i$7) (Udalov et al., 2024). The appearance of unusual domain structures and skyrmion lattices underlines that electric-field-induced DMI variation can directly reorganize topological states (Udalov et al., 2024).

A related theoretical route uses a Rashba-coupled magnetic bilayer under gate voltage. There the Rashba coefficient $\bm u_i$8 is voltage-dependent, and the induced torque can be recast as a static or dynamical interfacial DMI,

$\bm u_i$9

with $\bm D_{ij}$00 and $\bm D_{ij}$01 (Takeuchi et al., 2019). For typical parameters, the estimated static gate modification is $\bm D_{ij}$02–$\bm D_{ij}$03 meV per bond (Takeuchi et al., 2019). This suggests that voltage control can operate not only through equilibrium structural asymmetry but also through explicitly time-dependent interfacial spin-orbit coupling.

5. Polarization reversal, chirality switching, and topological textures

One of the clearest signatures of electric control over DMI is chirality reversal under polarization switching. In Co(MoS$\bm D_{ij}$04)$\bm D_{ij}$05, first-principles calculations identify two degenerate ferroelectric states, FE1 and FE2, with opposite out-of-plane polarization and opposite DMI: $\bm D_{ij}$06 for FE1 and

$\bm D_{ij}$07

for FE2 (Shao et al., 2022). Along the switching pathway, the DMI follows an approximately linear relation $\bm D_{ij}$08 (Shao et al., 2022). Reversing polarization therefore flips the sign of every bond DM vector.

Micromagnetic simulations with these parameters show that at $\bm D_{ij}$09 T both FE states host worm-like chiral domains, while at $\bm D_{ij}$10–$\bm D_{ij}$11 T isolated skyrmions appear; FE1 supports anti-clockwise-twisted skyrmions and FE2 clockwise-twisted skyrmions (Shao et al., 2022). Above about $\bm D_{ij}$12 T skyrmions collapse (Shao et al., 2022). The stability criterion is given as $\bm D_{ij}$13 with exchange stiffness $\bm D_{ij}$14 meV·Å$\bm D_{ij}$15 (Shao et al., 2022). This is an especially direct realization of electric chirality control because the two ferroelectric minima map one-to-one onto opposite DMI chiralities.

The purely electric eDMI theory predicts an electrical topological defect: the chiral electric bobber. In an effective Hamiltonian for PbTiO$\bm D_{ij}$16 containing the eDMI term, Monte Carlo or zero-temperature relaxations show that sufficiently large $\bm D_{ij}$17 stabilizes a surface-localized half-skyrmion-half-singularity object with chirality fixed by the sign of $\bm D_{ij}$18 (Chen et al., 2022). This defect exists at the top or bottom surface of a uniformly polarized film and does not require external fields (Chen et al., 2022). The analogy to magnetic bobbers is deliberate, but the order parameter is the local polar displacement rather than the magnetization.

Experiments on strain-tunable Pt/Co/Pt/PMN-PT likewise connect electric-field-driven DMI variation to domain morphology. On the [011] cut, when $\bm D_{ij}$19 MV/m gives $\bm D_{ij}$20 and $\bm D_{ij}$21 mJ m$\bm D_{ij}$22, domains collapse into $\bm D_{ij}$23 nm circular bubbles described as skyrmion-like; when $\bm D_{ij}$24 MV/m gives $\bm D_{ij}$25 and $\bm D_{ij}$26 mJ m$\bm D_{ij}$27, a zig-zag stripe pattern emerges (Udalov et al., 2024). These observations indicate that electric control of DMI can tune not only chirality but also anisotropy in the chiral interaction itself.

6. Descriptors, alternative formulations, and open directions

A notable theoretical development is the use of electric dipoles as predictors of magnetic DMI. For $\bm D_{ij}$28/Co/Pt trilayers, the interfacial DMI functional

$\bm D_{ij}$29

is related perturbatively to matrix elements involving the spin-orbit operator and the position operator $\bm D_{ij}$30 (Jia et al., 2019). Because the same hybridizations govern the local electric dipole

$\bm D_{ij}$31

the Pt dipole moment $\bm D_{ij}$32 becomes a descriptor for $\bm D_{ij}$33 (Jia et al., 2019). The fitted relation

$\bm D_{ij}$34

and the electronegativity relation

$\bm D_{ij}$35

provide design heuristics for multilayers (Jia et al., 2019). The reported Pearson correlations are $\bm D_{ij}$36 for Pt and $\bm D_{ij}$37 for Co (Jia et al., 2019). This does not mean that the electric dipole causes the DMI in a simple electrostatic sense; rather, both quantities scale with the same interface hybridization amplitudes.

Beyond static gate tuning, dynamic electric fields can modulate DMI and thereby mediate hybrid excitations. In a two-dimensional ferromagnet, an electric field associated with plasmons modifies the DMI as

$\bm D_{ij}$38

leading to magnon-plasmon coupling $\bm D_{ij}$39 (Rudziński et al., 13 Jun 2025). In a VSe$\bm D_{ij}$40 monolayer on NbSe$\bm D_{ij}$41 under 2% strain, the reported DMI parameters include $\bm D_{ij}$42 meV, $\bm D_{ij}$43 meV, and $\bm D_{ij}$44, with gate-controlled $\bm D_{ij}$45 DMI shifts and anticrossing gaps of order $\bm D_{ij}$46–$\bm D_{ij}$47 meV (Rudziński et al., 13 Jun 2025). This suggests that electric manipulation of DMI can enter the regime of bosonic hybridization rather than merely static texture control.

Another alternative formulation is the spin-Doppler picture, in which each DMI tensor component is equivalent to an equilibrium spin-current density, $\bm D_{ij}$48 (Kato et al., 2018). In W/CoFeB heterostructures, the interfacial DM constant increases linearly with low in-plane current density, with $\bm D_{ij}$49 mJ m$\bm D_{ij}$50/(10$\bm D_{ij}$51 A m$\bm D_{ij}$52) (Kato et al., 2018). Since the effect is current-driven rather than field-driven, it is not an eDMI in the strict sense, but it reinforces a broader theme: antisymmetric exchange can be reinterpreted through electric observables such as dipoles, voltages, or spin currents.

A final conceptual extension is the existence of DM-like antisymmetric interactions between higher multipoles. In inversion-broken $\bm D_{ij}$53 systems, exact diagonalization and perturbation theory reveal cross-product terms not only for dipoles $\bm D_{ij}$54 but also quadrupoles $\bm D_{ij}$55 and octupoles $\bm D_{ij}$56,

$\bm D_{ij}$57

(Hosoi et al., 2018). These are not electric-displacement couplings, but they broaden the DM paradigm beyond ordinary spin dipoles. A plausible implication is that the electric Dzyaloshinskii-Moriya interaction should be understood as part of a larger family of antisymmetric exchange phenomena acting on different order parameters whenever inversion breaking activates chiral virtual hopping paths.

Taken together, the literature defines electric Dzyaloshinskii-Moriya interaction as both a precise microscopic interaction among polar displacements and a broader research program on electric control of chiral exchange. The narrow eDMI framework establishes a genuinely electric, electron-mediated, third-order chiral invariant capable of stabilizing polar topological defects (Chen et al., 2022). The broader multiferroic and interfacial literature shows that electric polarization, interfacial dipoles, strain, and gate fields can tune or reverse magnetic DMI, enabling electrically controlled skyrmions, nonreciprocal magnons, and hybrid magnon-plasmon states (Shao et al., 2022). The unifying principle is that local inversion breaking, encoded electrically or structurally, governs the handedness of the underlying interaction.