Interfacial Dzyaloshinskii–Moriya interaction¶

Physical problem¶

Let \(\Omega_m\) be the magnetic domain, \(\mathbf m=\mathbf M/M_s\) the reduced magnetization, and \(\hat{\mathbf n}\) the unit interface-symmetry normal. FullMag uses

For the current FDM orientation \(\hat{\mathbf n}=\hat{\mathbf z}\) this is

The \(z\)-derivative terms cancel for this fixed axis. FEM retains the complete general-normal contraction and therefore supports a tilted, non-zero normalized normal.

Governing equations¶

Writing \(G_{ab}=\partial_bm_a\) and \(m_n=\mathbf m\cdot\hat{\mathbf n}\), the first variation implemented by the FEM residual is

After integration by parts, the volume effective field is

For \(+\hat{\mathbf z}\):

The integrated variation also contains the natural boundary contribution

with \(\boldsymbol\nu\) the outward magnetic-boundary normal. When exchange is present, the free boundary stationarity law is

The native FEM weak residual contains the volume first variation and therefore carries the natural boundary physics. Adding a separate identical DMI boundary term would double-count it.

Symbols and SI units¶

Assumptions and validity¶

• D is a surface-energy coefficient in \(\mathrm{J\,m^{-2}}\), not exchange stiffness or volume anisotropy.

• M_s is finite and positive; the field scales as \(1/M_s\) and the energy has no hidden \(\mu_0\).

• The public normal is length three and finite. FEM rejects zero and normalizes any other non-zero vector. FDM rejects every orientation except one equivalent to \(+\hat{\mathbf z}\).

• A missing normal resolves to \((0,0,1)\) in FEM and is the only executable FDM orientation.

• FDM uses centered differences and substitutes the center value for inactive/non-periodic missing neighbors. This is an implemented stencil closure, not a general continuum boundary proof.

• FEM evaluates the weak residual at quadrature points and projects the field with lumped mass.

• FEM Dind_field length and finite values are validated against the resolved mesh; it is not an arbitrary-length Python interpolation array.

Python API¶

The constructor and IR contract are in Interfacial DMI. In the stage-first body API, assigning Dind activates the interfacial-DMI term. This complete thin-film scenario starts from a Neel skyrmion texture, declares the canonical FDM interface orientation implicitly as \(+\hat{\mathbf z}\), and records the realized DMI field and energy during relaxation.

# %% Imports and units
import fullmag as fm

nm = 1.0e-9

# %% Thin-film FDM study
study = fm.study("interfacial_dmi_neel_skyrmion")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
study.objects.mesh.defaults(cell_size=(2 * nm, 2 * nm, 1 * nm))

# %% Geometry, material, initial texture, and interactions
film = study.geometry(
    fm.Box(size=(128 * nm, 128 * nm, 1 * nm), name="film"),
    name="film",
)
film.Ms = 5.8e5
film.Aex = 15.0e-12
film.Dind = 3.0e-3
film.alpha = 0.3
film.m = fm.texture.neel_skyrmion(
    radius=24 * nm,
    wall_width=8 * nm,
    chirality=1,
    core_polarity=-1,
)
study.exchange()
study.demag()

# %% Ordered relaxation stage and DMI observables
study.stages.add_relax(
    stage_id="relax_skyrmion",
    algorithm="projected_gradient_bb",
    max_steps=2_000,
    tolT=1.0e-6,
).autosave(
    fm.StageAutosave(
        table=fm.TableAutosave(
            every_steps=20,
            quantities=[
                "step",
                "mx",
                "my",
                "mz",
                "e_ex",
                "e_demag",
                "e_dmi",
                "e_total",
                "max_torque_T",
            ],
        ),
        fields=[fm.FieldAutosave("H_dmi", every_steps=50)],
    )
)

The public interaction matrix is:

ProblemIR¶

An explicit term lowers to:

{"kind": "interfacial_dmi", "D": 0.003, "interface_normal": [0.0, 0.0, 1.0]}

If the normal is omitted, the key is absent in Python IR and the FEM planner resolves the normalized default. A material-owned route is separate:

{"interfacial_dmi": 0.003, "dind_field": null}

The explicit term and material fallback are alternative resolution inputs, not summed energy terms. Requested intent, normalized normal, coefficient location, solver, device, precision, and output decisions belong in resolved provenance.

Round-trip and failure semantics¶

Canonical script export preserves whether the user authored an explicit InterfacialDMI term or material-owned Dind/Dind_field. Resolved execution records the backend-selected coefficient route and normalized FEM normal; it must not invent a rotated FDM stencil.

Validation errors include non-finite coefficients, malformed/zero normals, non-+z FDM normals, duplicate iDMI terms, invalid Dind_field cardinality or values, missing FEM mesh/context, and missing GPU-resident buffers. Unsupported combinations are planner/runtime errors, not silent fallbacks.

Discrete realization¶

Implementation mapping¶

The adjacent source map binds each claim to stable path-plus-symbol identities. InterfacialDMI and Material own authoring; plan_fdm owns strict +z legality; plan_fem owns normal and material-field resolution; FDM CPU/CUDA own stencil/reduction; FEM CPU/CUDA own weak residual, projection, device field, and energy reduction.

Validation¶

Verify uniform magnetization gives zero volume field and density; \(D\to-D\) reverses field and energy; linear \(m_z(x,y)\) reproduces field signs; chiral wall reflection changes energy sign; FEM satisfies \(-\mu_0\sum_iM_{s,i}w_i^{\mathrm{lump}}\mathbf H_i\cdot\mathbf v_i=R_{\mathrm i,h}\); finite differences of FEM energy match the residual; tilted FEM normals are normalized; and non-+z FDM normals are rejected. GPU qualification requires device identity, executed kernels, matched state/precision, and a documented tolerance.

Limitations¶

• FDM supports only the canonical +z interface normal.

• Spatial normal fields, tensor DMI, curved-surface corrections, and region-interface DMI are not public semantics.

• Dind_field is currently a FEM material realization, not a native FDM scalar-field route.

• FEM uses element-averaged nodal Dind_field; higher-order coefficient quadrature and consistent mass projection remain deferred.

• Explicit user-selectable non-natural DMI boundary operators are not implemented here.

Scientific bibliography¶

• Rohart, S. and Thiaville, A., “Skyrmion stability, metastability and dynamics in ultrathin magnetic films,” Physical Review B 88, 184422 (2013), DOI: 10.1103/PhysRevB.88.184422.

• Bogdanov, A. N. and Rößler, U. K., “Chiral symmetry breaking in magnetic thin films and multilayers,” Physical Review Letters 87, 037203 (2001), DOI: 10.1103/PhysRevLett.87.037203.

• FullMag canonical note: docs/physics/0404-interfacial-dmi.md.

Control Room crosswalk¶

Use Model Explorer -> Objects -> <object> -> Physics when PhysicsInteractionPanel exposes the interaction. Status: partial. frontend support is not implemented applies to physical parameters without a matching control. See {doc}/frontend/capability-register; do not infer UI support from backend or Python availability.

Python/API crosswalk¶

The linked Python API page is authoritative for exact functions, arguments, units, and failure semantics. If this page is a foundation or category overview, runnable Python is ot applicable here and must be taken from the terminal API page.

Bibliography and source scope¶

Use the scientific bibliography and source-code index on the linked terminal page. This block adds no new equation or unverified implementation claim.

Source-code index¶