Dzyaloshinskii–Moriya & exchange anisotropy

The antisymmetric part of the bilinear spin interaction: the term in the energy that, instead of favouring spins parallel or antiparallel, favours them perpendicular with a fixed handedness. Writing the most general bilinear coupling on a bond as ${\mathbf{S}}_i^{\top }{\mathbf{J}}_{ij}{\mathbf{S}}_j$ with a $3\times 3$ tensor ${\mathbf{J}}_{ij}$, the antisymmetric part is parameterised by a single axial vector ${\mathbf{D}}_{ij}$,

$$H_{\mathrm{DM}}=\sum _{⟨ij⟩}{\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j),D_{ij}^{\gamma }=\frac{1}{2}{ϵ}^{\gamma \alpha \beta }{J}_{ij}^{\alpha \beta }=\frac{1}{4}{ϵ}^{\gamma \alpha \beta }(J_{ij}^{\alpha \beta }-J_{ij}^{\beta \alpha }),$$

so ${\mathbf{D}}_{ij}$ is the dual of the antisymmetric matrix $\frac{1}{2}({\mathbf{J}}_{ij}-{\mathbf{J}}_{ij}^{\top })$. This is the Dzyaloshinskii–Moriya interaction (DMI). It is the same object that, sitting in the full ${\mathbf{J}}_{ij}$, gaps the magnon spectrum in LSWT and is the order-restoring ingredient that lets 2D magnets evade Mermin–Wagner; this note is about where it comes from and what its symmetry allows. I fix the sign/index conventions here once (antisymmetric part of ${\mathbf{J}}_{ij}$, $D^{\gamma }=\frac{1}{2}{ϵ}^{\gamma \alpha \beta }{J}^{\alpha \beta }$) and reuse them in the companion notes — the full tensor decomposition lives in exchange tensors from DFT.

Prerequisites

The symmetric Heisenberg / super-exchange Hamiltonian $H={\sum }_{ij}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ · spin–orbit coupling $\xi \mathbf{L}\cdot \mathbf{S}$ as a relativistic perturbation · elementary point-group / bond symmetry

Dzyaloshinskii’s argument: a Landau-allowed invariant

Dzyaloshinskii’s 1958 paper is phenomenological and exact within its assumptions: it asks which terms the magnetic free energy is allowed to contain, given the crystal symmetry, with no reference to a microscopic mechanism. The order parameters are the sublattice magnetisations — for a two-sublattice antiferromagnet, the Néel vector $\mathbf{L}={\mathbf{M}}_1-{\mathbf{M}}_2$ and the net moment $\mathbf{M}={\mathbf{M}}_1+{\mathbf{M}}_2$. A free-energy density built to be invariant under the crystal’s point group and under time reversal may, in low-symmetry magnets, contain a term linear and antisymmetric in the two order parameters — schematically $\mathbf{d}\cdot (\mathbf{M}\times \mathbf{L})$, a so-called Lifshitz invariant. Such a term is a genuine invariant only when the site symmetry permits it; in a centrosymmetric arrangement of the magnetic ions it is forbidden.

When it is allowed, its consequence is immediate and was Dzyaloshinskii’s point: a nominally collinear antiferromagnet cannot stay collinear. Minimising $F$ with the cross-product term present tilts the two sublattices slightly away from antiparallel, producing a small net moment perpendicular to $\mathbf{L}$ — weak ferromagnetism of the kind seen in $\alpha {\text{-Fe}}_2{\text{O}}_3$ (hematite) and the orthoferrites. Dzyaloshinskii identified the effect as symmetry-dictated canting and predicted which crystal classes show it, but the size of $\mathbf{d}$ was a free parameter of the theory. Supplying its magnitude and microscopic origin was Moriya’s contribution two years later.

Moriya’s mechanism: superexchange to first order in spin–orbit coupling

Moriya derived the interaction microscopically by redoing Anderson’s superexchange theory with spin–orbit coupling retained. The skeleton: in pure (spin-independent) superexchange, virtual hopping of electrons between magnetic ions through a ligand, combined with on-site Coulomb repulsion $U$ and Hund’s rule, generates the isotropic $J{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Now switch on the on-site SOC $\lambda \mathbf{L}\cdot \mathbf{S}$. Treating it perturbatively, the SOC mixes a small amount of excited orbital character into the magnetic ground orbitals, so the hopping integrals acquire spin-dependent, non-collinear corrections. Carrying the perturbation theory to the order that produces a term antisymmetric under $i\leftrightarrow j$ yields precisely ${\mathbf{D}}_{ij}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)$.

Two structural facts fall straight out of the derivation and are worth keeping:

• DMI is first order in SOC. One power of $\lambda$ enters (the orbital admixture on one of the two sites), so the natural scale is $$|{\mathbf{D}}_{ij}|\sim \frac{\Delta g}{g}{J}_{ij}\sim \frac{\lambda }{\Delta }{J}_{ij},$$ with $\Delta g$ the deviation of the ion’s $g$-factor from the free-electron value (itself $\sim \lambda /\Delta$, $\Delta$ a crystal-field splitting). For $3d$ ions $\Delta g/g$ is typically a few percent, so $|\mathbf{D}|$ is usually one to two orders of magnitude below $J$ — small, but its competition with $J$ is what produces all the interesting textures. By contrast the symmetric-anisotropic exchange $\mathbf{\Gamma }$ (the symmetric-traceless part of ${\mathbf{J}}_{ij}$) is second order in SOC, $\sim (\Delta g/g{)}^{2}J$, hence usually smaller still.
• DMI requires a relativistic, multi-orbital ion. With $\lambda \to 0$ the term vanishes identically: a pure single-band Hubbard model has no DMI. This is the same statement as “you need SOC in the DFT” that drives the upstream extraction in exchange tensors from DFT.

Moriya’s symmetry rules

The microscopic theory also fixes when ${\mathbf{D}}_{ij}=0$ and, when nonzero, which direction ${\mathbf{D}}_{ij}$ may point, purely from the local symmetry of the bond connecting sites $i$ and $j$ (let $A$ be the midpoint of the bond, ${\hat{\mathbf{u}}}_{ij}$ the unit vector along it). Moriya’s rules, in his own enumeration:

1. Inversion centre at the bond midpoint $A$ $\Rightarrow$ ${\mathbf{D}}_{ij}=0$. This is the headline rule. ${\mathbf{D}}_{ij}$ is an axial vector tied antisymmetrically to the bond; inversion through $A$ exchanges $i\leftrightarrow j$, under which $H_{\mathrm{DM}}$ is odd (${\mathbf{S}}_i\times {\mathbf{S}}_j\to {\mathbf{S}}_j\times {\mathbf{S}}_i=-{\mathbf{S}}_i\times {\mathbf{S}}_j$), so an inversion symmetry forces ${\mathbf{D}}_{ij}=-{\mathbf{D}}_{ij}=0$. DMI is a fingerprint of broken inversion symmetry on the bond.
2. Mirror plane containing $i$ and $j$ (the bond lies in the plane) $\Rightarrow$ ${\mathbf{D}}_{ij}$ is perpendicular to that mirror plane. (An axial vector lying in a mirror plane would be reversed by the reflection; only the component normal to the plane survives.)
3. Mirror plane perpendicular to the bond, through $A$ $\Rightarrow$ ${\mathbf{D}}_{ij}$ lies in that plane (i.e. is perpendicular to ${\hat{\mathbf{u}}}_{ij}$).
4. Two-fold rotation axis $\perp$ to the bond through $A$ $\Rightarrow$ ${\mathbf{D}}_{ij}\perp$ to that $C_2$ axis.
5. $n$-fold axis ($n\ge 2$) along the bond ${\hat{\mathbf{u}}}_{ij}$ $\Rightarrow$ ${\mathbf{D}}_{ij}\parallel {\hat{\mathbf{u}}}_{ij}$.

Practically: identify the surviving point-group elements of the bond (the intersection of the site symmetries with the bond geometry), and each one projects out forbidden components of ${\mathbf{D}}_{ij}$, leaving the allowed direction. A useful geometric mnemonic for a superexchange path $i-\ell -j$ through a ligand $\ell$ is ${\mathbf{D}}_{ij}\propto {\mathbf{x}}_i\times {\mathbf{x}}_j$ (Keffer/Moriya), where ${\mathbf{x}}_i,{\mathbf{x}}_j$ point from the ligand to the two magnetic ions: a perfectly straight ($180^{\circ }$) bond has the ligand on the inversion centre and gives $\mathbf{D}=0$, while a buckled bond breaks that inversion and lets $\mathbf{D}$ grow with the buckling angle — which is exactly why displacing the bridging ligand (or the substrate, at an interface) tunes the DMI.

Where inversion is broken: bulk vs. interface

Rule 1 makes broken inversion symmetry the necessary condition for a net DMI, and it can be broken two ways:

• Bulk noncentrosymmetric crystals — the magnetic lattice itself lacks an inversion centre. The B20 compounds (MnSi, FeGe, ${\text{Fe}}_{1-x}{\text{Co}}_x\text{Si}$, chiral space group $P{2}_{1}3$) are the paradigm: a uniform bulk DMI of fixed handedness twists the ferromagnet into a long-period helix and, in a field, into a Bloch-type skyrmion lattice. Multiferroics and the orthoferrites are the older, weak-FM examples of Dzyaloshinskii’s canting.
• Interfacial / Rashba systems — inversion is broken by the surface even if both constituent crystals are centrosymmetric. At a heavy-metal / ferromagnet interface (Pt, W, Ir under Co or Fe), the broken out-of-plane symmetry plus the strong SOC of the heavy layer give an interfacial DMI; here rule 2 applies with the mirror plane normal to the interface, so ${\mathbf{D}}_{ij}$ lies in the interface plane, perpendicular to the bond — the configuration that stabilises Néel-type (hedgehog) skyrmions and fixes the chirality of domain walls. The underlying microscopic mechanism — a three-site DMI from spin–orbit scattering of conduction electrons off the heavy metal — is the Fert–Lévy mechanism; its development into interfacial-DMI skyrmionics and spin-orbitronics is the later thin-film lineage (Fert, Cros & Sampaio).

Competition with symmetric exchange: canting, spirals, skyrmions

The physics is set by the dimensionless ratio $|\mathbf{D}|/J$ and the competition between three energy scales: symmetric exchange $J$ (wants collinear), DMI $|\mathbf{D}|$ (wants a fixed-handed $90^{\circ }$ twist), and anisotropy / Zeeman. Minimising $J{\mathbf{S}}_i\cdot {\mathbf{S}}_j+\mathbf{D}\cdot ({\mathbf{S}}_i\times {\mathbf{S}}_j)$ on a bond rotates the two spins to a relative angle

$$\tan \theta =\frac{|\mathbf{D}|}{J},$$

small when $|\mathbf{D}|\ll J$. The three regimes:

• Canting / weak ferromagnetism. In an antiferromagnet a small uniform $\mathbf{D}$ tilts the sublattices by $\theta \approx |\mathbf{D}|/J$, producing the net moment of hematite and the orthoferrites — exactly Dzyaloshinskii’s effect, now with a number attached.
• Spin spirals. In the continuum, a uniform bulk DMI adds a Lifshitz invariant $w_{\mathrm{DM}}=D\mathbf{m}\cdot (\nabla \times \mathbf{m})$ (Bloch/bulk form) or the interfacial $D[m_z\nabla \cdot \mathbf{m}-(\mathbf{m}\cdot \nabla )m_z]$ to the exchange stiffness term $A(\nabla \mathbf{m}{)}^2$. Minimising favours a helical modulation with pitch $$q\sim \frac{D}{A},\text{wavelength }{\lambda }_{\mathrm{h}}=\frac{2\pi }{q}\sim \frac{2\pi A}{D},$$ i.e. the spiral period is long when DMI is weak relative to exchange. The same ratio fixes domain-wall chirality and width: DMI selects Néel over Bloch walls (interfacial) or Bloch over Néel (bulk) and a fixed rotation sense, which is what makes current-driven wall motion efficient.
• Skyrmions. Adding anisotropy and a field to the DMI–exchange competition stabilises a lattice of topologically nontrivial swirls — Bloch-type from bulk DMI, Néel-type from interfacial DMI. Their size again scales as $\sim A/D$, so larger $|\mathbf{D}|/J$ means smaller skyrmions. This is the technological payoff line of the whole subject.

The throughline for the pipeline I build elsewhere: the same ${\mathbf{D}}_{ij}$ that Dzyaloshinskii’s symmetry analysis permits and Moriya’s mechanism sizes is what a broken-symmetry SOC DFT calculation must reproduce, what gaps the Goldstone magnon in LSWT, and what supplies the order-stabilising anisotropy that Mermin–Wagner says a 2D magnet needs.

Builds toward: exchange tensors from DFT · LSWT · Mermin–Wagner

Key references

• I. Dzyaloshinsky, “A thermodynamic theory of weak ferromagnetism of antiferromagnetics,” J. Phys. Chem. Solids 4, 241 (1958) — the symmetry/Landau origin and weak ferromagnetism.
• T. Moriya, “Anisotropic superexchange interaction and weak ferromagnetism,” Phys. Rev. 120, 91 (1960) — the SOC + superexchange mechanism, the $|\mathbf{D}|\sim (\Delta g/g)J$ estimate, and the symmetry rules for ${\mathbf{D}}_{ij}$.
• A. Fert & P. M. Lévy, “Role of anisotropic exchange interactions in determining the properties of spin-glasses,” Phys. Rev. Lett. 44, 1538 (1980) — the three-site, spin–orbit-scattering (RKKY-type) DMI mechanism.
• A. Fert, V. Cros & J. Sampaio, “Skyrmions on the track,” Nat. Nanotechnol. 8, 152 (2013) — the interfacial-DMI / skyrmion-racetrack lineage that applies the mechanism to thin films.