DMI boundary conditions¶

Governing equations¶

Physical domain and scope¶

The boundary operator belongs to the combined exchange-DMI variational problem. It is not a free-standing boundary value that can be added after computing a DMI field. The surface normal, interface-symmetry normal, coefficient convention, and solver realization must be resolved together. The natural condition is homogeneous only when the DMI surface contribution vanishes.

Bulk DMI boundary condition¶

The combined exchange-plus-bulk-DMI natural boundary term is

This is the isotropic analogue: the DMI surface correction is a tangential rotation proportional to the DMI constant and the surface normal.

Characteristic length¶

The DMI boundary condition introduces a characteristic length

that determines the spatial extent of the boundary twist. When the sample dimension is much larger than \(\ell_{\mathrm{DMI}}\), the boundary modification is confined to a surface layer. When the sample is comparable to \(\ell_{\mathrm{DMI}}\), the DMI boundary condition significantly modifies the bulk magnetization profile.

Impact on skyrmion confinement¶

In confined geometries (nanodiscs, nanowires), the DMI boundary condition determines whether skyrmions are attracted to or repelled from the sample edge. The Rohart–Thiaville analysis shows that the boundary condition creates an effective edge potential for skyrmion centres.

Symbols and SI units¶

Assumptions and validity¶

• \(A\) is the bulk exchange stiffness and \(D\) is the coefficient used by the selected DMI convention; their units and sign are not converted by the Python object.

• The interfacial symmetry normal is normalized by FEM when non-zero. Current FDM planning accepts only the documented canonical orientation; a tilted FDM normal is rejected.

• The boundary formula assumes a smooth enough surface for a trace normal. Corners and non-conforming interfaces require a mesh-specific treatment and must not be described as the smooth formula without qualification.

• The characteristic length is a scale estimate, not a replacement for resolving the boundary layer. Mesh or cell size must be reported relative to \(\ell_{\mathrm{DMI}}\).

• Python serialization, source contracts, and a skipped GPU test are not boundary-condition qualification evidence.

Python API and boundary request¶

The DMI boundary operator is not registered independently. In the stage-first API, assigning Dind to a magnetic body activates the interfacial-DMI term, and the resolved FDM or FEM realization supplies its natural boundary treatment together with exchange. This complete FDM scenario relaxes a Neel wall in an open nanostrip and records the field and energy needed to inspect the edge twist.

# %% Imports and units
import fullmag as fm

nm = 1.0e-9

# %% Open-boundary FDM study
study = fm.study("interfacial_dmi_boundary_twist")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
study.objects.mesh.defaults(cell_size=(2 * nm, 2 * nm, 2 * nm))

# %% Geometry, material, chiral state, and interactions
strip = study.geometry(
    fm.Box(size=(120 * nm, 40 * nm, 2 * nm), name="strip"),
    name="strip",
)
strip.Ms = 5.8e5
strip.Aex = 15.0e-12
strip.Dind = 3.0e-3
strip.alpha = 0.3
strip.m = fm.texture.domain_wall(
    width=12 * nm,
    kind="neel",
    normal_axis="x",
    left=(0.0, 0.0, 1.0),
    right=(0.0, 0.0, -1.0),
)
study.exchange()
study.demag(enabled=False)

# %% Ordered relaxation stage and boundary-sensitive observables
study.stages.add_relax(
    stage_id="relax_boundary_twist",
    algorithm="projected_gradient_bb",
    max_steps=1_000,
    tolT=1.0e-6,
).autosave(
    fm.StageAutosave(
        table=fm.TableAutosave(
            every_steps=10,
            quantities=["step", "e_ex", "e_dmi", "e_total", "max_torque_T"],
        ),
        fields=[fm.FieldAutosave("H_dmi", every_steps=20)],
    )
)

ProblemIR lowering¶

The explicit terms lower independently and preserve their signed coefficients:

{
  "energy_terms": [
    {"kind": "interfacial_dmi", "D": 0.0025, "interface_normal": [0.0, 0.0, 1.0]},
    {"kind": "bulk_dmi", "D": 0.001}
  ]
}

The planner resolves the normal policy and boundary realization after lowering. It must not silently turn bulk DMI into interfacial DMI or rotate an FDM stencil to accommodate a normal that the FDM lane does not support.

Round-trip and failure semantics¶

Requested intent is the term kind, signed coefficient, and authored normal. Resolved execution adds solver, device, normalized normal, boundary operator, precision, and field output policy. Validation errors include non-finite coefficients, malformed vectors, zero FEM normals, and unsupported FDM normal orientation. Unsupported combinations are planner errors; no CPU fallback or boundary omission is allowed. Resolved execution is recorded separately from requested intent. Unsupported combinations are rejected before a solver starts.

Discrete realization¶

FEM obtains the natural boundary term from the weak residual and integrates it on the resolved surface trace. FDM represents the boundary through one-sided/reflected stencil values and active-neighbour rules. These are separate realizations and require separate convergence evidence; the shared continuum equation does not prove discrete parity.

Implementation mapping¶

The Python classes own coefficient and normal serialization. FEM interfacial and bulk DMI residuals are implemented in separate interaction modules; FDM planning owns the canonical normal legality rule. The source map below gives stable path-plus-symbol identities.

Validation¶

Validate uniform-state zero field, sign reversal under \(D\to-D\), analytic linear gradients, boundary-layer convergence against \(\ell_{\mathrm{DMI}}\), FEM energy finite differences, FDM boundary-stencil symmetry, tilted-normal rejection, and matched precision/device evidence.

Limitations¶

The public contract does not yet expose a spatial normal field, curved-surface DMI operator, or an arbitrary FDM interface orientation. The boundary equations describe the current physical target; lane status must be read from the canonical DMI realization pages.

Scientific bibliography¶

1. S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction,” Physical Review B 88, 184422 (2013). doi:10.1103/PhysRevB.88.184422.

2. A. O. Leonov, T. L. Monchesky, N. Romming, A. Kubetzka, A. N. Bogdanov, and R. Wiesendanger, “The properties of isolated chiral skyrmions in thin magnetic films,” New Journal of Physics 18, 065003 (2016). doi:10.1088/1367-2630/18/6/065003.

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¶