Spin-orbit torque¶

Meaning of M1 and M2 in Fullmag¶

These names describe coupling levels of the drift-diffusion model, not two formula variants of PrescribedSpinOrbitTorque.

M1: one-way solved transport. CurrentTransport(coupling="one_way") produces the charge-current field first. SpinDriftDiffusion then consumes that named current source, evaluates direct spin Hall injection, diffusion and active spin reactions, and supplies absorbed interface spin flux to DriftDiffusionSpinTorque. The spin solution does not alter the already solved charge field.

M2: reciprocal solved transport. CurrentTransport(coupling="bidirectional") selects the reciprocal constitutive contract. Charge and spin belong to one coupled steady problem; the charge block includes the authored magnetoresistive conductivities and the spin block can return an inverse-spin-Hall contribution. In the public Python contract, bidirectional current transport requires model="ohmic_poisson", complete anisotropic conductivity data, and a block_gmres charge policy. SpinDriftDiffusion.to_ir() records transport_constitutive.reciprocal.fullmag.v1 instead of the M1 transport_constitutive.one_way.fullmag.v1 contract.

Prescribed SOT: no transport solve. The module directly evaluates a local DL/FL source from a signed current, a polarization axis, two effective efficiencies and the physical FM thickness. It does not create charge potential, spin potential, spin current, interface backflow, or inverse-SHE observables.

Side-by-side physical comparison¶

The table compares physics only. Executable discretization, device, precision, solver, and interface support are lane-specific and fail closed. Consult the drift-diffusion transport page before selecting an M1 or M2 backend.

What prescribed SOT removes¶

The reduction replaces the solved transport chain with effective coefficients:

bulk charge conversion
+ spin diffusion and relaxation
+ interface transmission and backflow
+ unresolved stack losses
                         -> xi_DL and xi_FL

This replacement is useful only when those coefficients are known for the stack, temperature, thickness range and sign convention being simulated. It also changes what can be predicted. Prescribed SOT can answer how a magnetic body responds to an assumed torque efficiency. It cannot predict how changing HM conductivity, HM thickness, spin-diffusion length, interface mixing conductance or electrode geometry changes that efficiency. M1 or M2 is needed for those questions.

There is generally no unique parameter-by-parameter conversion from M1/M2 to prescribed SOT. An effective xi_dl may be extracted by matching an integrated or averaged damping-like torque for one operating point. An effective xi_fl may be fitted in the same way. The resulting pair need not reproduce local hotspots, edge accumulation, thickness dependence, current crowding, backflow, or a different magnetic state.

When the models may agree¶

Prescribed SOT can approximate M1 or M2 when all of the following are justified:

• the FM is thin enough that a volume-averaged torque is meaningful;

• the in-plane current and injected spin polarization are approximately uniform;

• transport relaxes much faster than the magnetic dynamics of interest;

• interface and bulk losses can be represented by fixed effective efficiencies;

• reciprocal modification of the charge path is negligible, or already folded into a calibration performed at the same operating point;

• the observable depends on the integrated torque rather than its local profile.

Agreement should be demonstrated against a declared observable, not assumed from similar-looking trajectories. Useful calibration targets include the volume-integrated DL/FL torque, initial angular acceleration, switching threshold, or a selected harmonic response. A coefficient fitted to one target is not automatically validated for the others.

When M1 or M2 is required¶

Choose M1 when spatial charge and spin transport matters but one-way charge-to-spin coupling is an acceptable approximation. Typical reasons are current crowding, finite spin-diffusion length, multilayer spin absorption, mixing-conductance interfaces, nonuniform torque, or a geometry in which a single authored sigma is insufficient.

Choose M2 when the reciprocal charge response is itself part of the physics or observable. Examples include inverse-SHE voltage/current, simultaneous AMR/PHE/AHE transport, or a device state in which magnetization-dependent charge redistribution materially changes spin injection. M2 is not automatically “more accurate”: it is a larger constitutive model requiring additional parameters, boundary data, solver policy, and validation.

Choose prescribed SOT when the scientific input is already an effective torque efficiency, when reproducing a MuMax-style local SOT experiment, or when large parameter scans would make a solved transport domain unnecessary. Do not use it merely to avoid supplying unknown transport data while still claiming predictions about that missing transport physics.

Python authoring: the decisive differences¶

The following fragments show the model-selection fields. They are deliberately short; complete M1 and M2 examples, including materials, interfaces, electrodes and solver restrictions, are maintained on the drift-diffusion page.

study.spin_torque(fm.PrescribedSpinOrbitTorque(
    name="local_sot",
    target=fm.RegionRef("fm"),
    drive=fm.SignedScalarDrive(
        current_density_Apm2=1.0e11,
        sigma=(0.0, 1.0, 0.0),
    ),
    xi_dl=0.12,
    xi_fl=0.01,
    free_layer_thickness_m=1.0e-9,
))

charge = study.current_transport(
    name="charge",
    model="ohmic_poisson",
    coupling="one_way",
    # domain, materials, boundaries, gauge, solver
)
spin = study.spin_transport(fm.SpinDriftDiffusion(
    id="spin",
    current_source_id=charge.name,
    # domain, materials, interfaces, boundaries, solver
))
study.spin_torque(fm.DriftDiffusionSpinTorque(
    id="transport_torque",
    solve_id=spin.id,
    target=fm.RegionRef("fm"),
))

charge = study.current_transport(
    name="charge",
    model="ohmic_poisson",
    coupling="bidirectional",
    # complete anisotropic materials and block_gmres policy
)
spin = study.spin_transport(fm.SpinDriftDiffusion(
    id="spin",
    current_source_id=charge.name,
    mode="steady",
    # reciprocal operator and lane-specific solver policy
))
study.spin_torque(fm.DriftDiffusionSpinTorque(
    id="transport_torque",
    solve_id=spin.id,
    target=fm.RegionRef("fm"),
))

Do not double count the torque¶

DriftDiffusionSpinTorque consumes the absorbed spin flux from a named M1/M2 solve. PrescribedSpinOrbitTorque creates an independent local source. Adding both to the same target sums both terms in the magnetization equation; Fullmag does not infer that one replaces the other. Such a combination is valid only when the user can identify two distinct physical sources, for example a solved HM torque plus a separately calibrated torque from another unmeshed layer.

For a comparison study, run separate problems or stages with one torque model active at a time, preserve the current and normal sign convention, and compare the same torque or magnetization observable.

Base current-to-frequency conversion¶

This equation answers only: “how large is the torque frequency before DL/FL efficiencies are applied?” Its sign comes from \(J_{\mathrm{signed}}\). Increasing \(M_s\) or \(t_F\) reduces the same injected angular momentum per magnetic volume. The constants \(e\), \(\hbar\), and \(\gamma_e\) are supplied by the backend and are not Python inputs.

Damping-like rate¶

xi_dl is an effective dimensionless damping-like efficiency. In a reduced model it absorbs bulk conversion, interface transmission, spin backflow, and other stack-specific losses. It is usually inferred from harmonic Hall, spin-torque ferromagnetic resonance, switching, or another calibrated torque measurement. It is not automatically equal to the bulk spin Hall angle.

Field-like rate¶

xi_fl is the signed effective field-like efficiency. Its sign and magnitude can depend strongly on interfaces, stack order, annealing, and the convention used for current and interface normal. Do not infer it from xi_dl unless the chosen physical model or measurement explicitly supplies that relation.

Damping-like Gilbert torque¶

The vector \(\mathbf m\times(\hat{\boldsymbol\sigma}\times\mathbf m)\) is the component of \(\hat{\boldsymbol\sigma}\) transverse to \(\mathbf m\). The DL term therefore pushes \(\mathbf m\) toward or away from that transverse polarization, depending on the total sign of \(J_{\mathrm{signed}}\xi_{\mathrm{DL}}\).

Field-like Gilbert torque¶

The vector \(\mathbf m\times\hat{\boldsymbol\sigma}\) has the same geometry as precession about a field parallel to \(\hat{\boldsymbol\sigma}\). This explains the name “field-like”; xi_fl remains a torque efficiency, not a magnetic-field input in tesla.

Total Gilbert source¶

The source has units \(\mathrm{s^{-1}}\) and is added to the magnetization time derivative. Reversing the signed current, reversing sigma, or reversing one efficiency reverses the corresponding term. Reversing both current and sigma leaves the torque unchanged.

Gilbert-to-explicit conversion¶

Backends add an explicit right-hand-side contribution after solving the Gilbert form once. Define the explicit damping-like coefficient separately:

Define the explicit field-like coefficient independently:

The actual explicit source is then

The cross-coupling by \(\alpha\) is an algebraic consequence of converting the Gilbert equation to an explicit time derivative. It is why MuMax-style code may contain apparent DL/FL compensation. Author the physical xi_dl and xi_fl; do not pre-apply these formulas in Python.

Signed scalar drive¶

SignedScalarDrive is the direct reduced-model input:

J_0 is the signed current-density amplitude controlled by the script or experiment. \(f(t)\) is the dimensionless envelope evaluated at stage time.

The authored vector \(\boldsymbol\sigma\) states the spin-polarization direction; only its direction is retained. J_0 may be positive, negative, or zero. sigma is normalized during authoring. The optional envelope must be one of the canonical Fullmag envelope objects: ConstantEnvelope, SinusoidalEnvelope, PulseEnvelope, PiecewiseLinearEnvelope, SincEnvelope, or TabulatedEnvelope.

Vector-current binding¶

VectorCurrentDrive binds the torque to a named vector-current source while making the geometric convention explicit:

Only the component of the named current source along drive_direction contributes to this reduced torque.

drive_direction defines \(\hat{\mathbf t}\) and interface_normal defines the oriented normal \(\hat{\mathbf n}_{NF}\) from the nonmagnetic layer toward the ferromagnet. Both authored axes are normalized. They must be nonzero and must not be parallel within the canonical axis tolerance \(10^{-12}\). The projection preserves current reversal; the polarization axis does not flip when only the source current reverses.

Which values come from where?¶

The following classification is the practical minimum for preparing a script. “Literature” means a starting point for a comparable stack, not a universal material constant.

Fullmag supplies \(e\), \(\hbar\), and \(\gamma_e\). Fullmag derives \(\Omega_0\), \(\Omega_{\mathrm{DL}}\), \(\Omega_{\mathrm{FL}}\), \(C_{\mathrm{DL}}\), and \(C_{\mathrm{FL}}\). None of those constants or derived rates belongs in the Python constructor.

How to use literature values safely¶

Nguyen, Ralph, and Buhrman reported a peak damping-like efficiency per current density of \(\xi_{\mathrm{DL}}^j=0.12\) for their Pt/Co samples at Pt thickness \(2.8\)-\(3.9\,\mathrm{nm}\). This supports xi_dl=0.12 as a realistic literature seed for a related Pt/FM stack, but it does not determine the Co thickness, \(M_s\), \(\alpha\), xi_fl, or the sign convention of a different sample. Their measured efficiency also changes with Pt thickness and resistivity.

Garello and co-workers measured both DL-like and FL-like components and found strong dependence on stack composition, magnetization angle, and annealing. Consequently, the example xi_fl=-0.03 is explicitly illustrative; it is not presented as a transferable Pt/Co constant.

Use this order when building a real case:

1. Enter measured geometry, \(M_s\), \(A_{\mathrm{ex}}\), and \(\alpha\) for the same sample.

2. Fix and test the current, interface-normal, and spin-polarization sign convention.

3. Use literature xi_dl and xi_fl only as initial estimates from a comparable stack.

4. Fit or measure both effective efficiencies for the actual stack.

5. Sweep their uncertainty separately from mesh, timestep, and current uncertainty.

Complete stage-first example¶

# %% Study and execution policy
import fullmag as fm

nm = 1.0e-9
study = fm.study("prescribed_sot_reference")
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, and initial state
body = study.geometry(fm.Box(40 * nm, 20 * nm, 4 * nm), name="film")
body.Ms = 8.0e5
body.Aex = 13.0e-12
body.alpha = 0.02
body.m = fm.texture.uniform(1.0, 0.0, 0.0)

# %% Canonical prescribed SOT
sot = fm.PrescribedSpinOrbitTorque(
    name="hm_sot",
    target=fm.RegionRef("film"),
    drive=fm.SignedScalarDrive(
        current_density_Apm2=-4.0e11,
        sigma=(0.0, 1.0, 0.0),
    ),
    xi_dl=0.12,
    xi_fl=-0.03,
    free_layer_thickness_m=1.5 * nm,
)
study.spin_torque(sot)

# %% Ordered run stage
study.stages.add_run(stage_id="drive", until=1.0e-12)

The sign in current_density_Apm2 is retained in ProblemIR. The authored sigma=(0, 1, 0) is normalized and serialized as sigma_hat.

What the example numbers produce¶

Using the constants embedded by Fullmag, the example inputs produce the following signed rates. These values check units and sign propagation; they do not predict a switching time by themselves.

The \(1\,\mathrm{ps}\) stage in the example is deliberately short. A switching, oscillation, or relaxation study must choose the stage duration from the actual field, anisotropy, damping, geometry, and observable, then demonstrate temporal and spatial convergence.

Binding to an existing vector-current source¶

If the study already contains a current module named charge, replace the drive object only:

# %% Alternative object-level vector-current binding
drive = fm.VectorCurrentDrive(
    current_source="charge",
    drive_direction=(1.0, 0.0, 0.0),
    interface_normal=(0.0, 0.0, 1.0),
)

sot = fm.PrescribedSpinOrbitTorque(
    name="hm_sot_from_charge",
    target=fm.RegionRef("film"),
    drive=drive,
    xi_dl=0.12,
    xi_fl=-0.03,
    free_layer_thickness_m=1.5e-9,
)
sot.to_ir_module()

For these axes, \(\hat{\boldsymbol\sigma}=\hat{\mathbf z}\times\hat{\mathbf x} =\hat{\mathbf y}\) and only the \(x\) projection of the named current contributes. VectorCurrentDrive has no independent envelope argument; timing belongs to the named current source.

Constructor reference¶

SpinOrbitTorque is a deprecated compatibility constructor. New scripts and canonical exports must use PrescribedSpinOrbitTorque.

FDM CPU¶

The reference evaluator consumes a per-cell active mask, local \(M_s\) and \(\alpha\), the signed scalar amplitude, normalized polarization, and the stage-time envelope.

FDM GPU¶

The native CUDA path selects prescribed_sot.fullmag.v1 explicitly and is bounded by fixed-trajectory CPU parity and current-scaling tests when a CUDA device is present.

FEM CPU¶

The MFEM local-interaction path evaluates the same SI prefactor at magnetic nodes, applies the active mask, and performs the Gilbert-to-explicit conversion once.

FEM GPU¶

The CUDA RK source adds the canonical local RHS on device. Its bounded contract checks the independent SI oracle, active-mask exclusion, and CPU parity; this does not by itself qualify every runtime or mesh configuration.