Spin Hall drift-diffusion transport¶

Physical problem¶

A heavy metal (HM) carries conventional charge current. Spin-orbit scattering generates transverse spin current and spin accumulation \(\boldsymbol\mu_s\). At an oriented HM/FM interface, longitudinal spin is transmitted or reflected while absorbed transverse spin torques the ferromagnet (FM).

Fullmag uses \(e>0\), conventional current, and \(\mathbf E=-\nabla V\). Here \(\sigma\) is charge conductivity, distinct from spin conductivity \(\sigma_s\). The direct-SHE contribution is proportional to \(\theta_{\mathrm{SH}}\sigma(\mathbf n\times\mathbf E)\). Reversing \(\theta_{\mathrm{SH}}\), current, \(\mathbf n\), or HM/FM orientation reverses the corresponding polarization or flux. A material sign is meaningful only with these definitions.

Governing equations¶

\(\boldsymbol\mu_s\) is the full spin-channel voltage splitting: the channels are \(V+\boldsymbol\mu_s/2\) and \(V-\boldsymbol\mu_s/2\). \(Q_{ia}\) is charge-equivalent spin current; \((\hbar/2e)Q_{ia}\) is angular-momentum current.

Symbols and SI units¶

Boundaries and interfaces¶

Authoring includes SpinInsulating, SpinSink, SpecifiedSpinPotential, outward SpecifiedSpinFlux, and PeriodicSpin. A negative-axis face reverses outward flux when stored in positive-axis form. Transparent interfaces enforce continuity. With \(\Delta V=V_N-V_F\) and \(\Delta\boldsymbol\mu_s=\boldsymbol\mu_{s,N}-\boldsymbol\mu_{s,F}\), M1 longitudinal injection/backflow is

The \(G_r,G_i\) transverse absorption law is

For magnetic cell \(K\) adjacent to face \(f\), the interface torque contribution is

Optional spin-memory-loss reservoir data exists in authoring/reference FDM but is not generally executable. The spin domain must be a subset of the conducting charge domain; material coverage must be complete and non-overlapping.

Assumptions and validity¶

The model is diffusive and continuum-scale. Active conductivities and lengths are positive; absent reactions use None. A realistic HM/FM model needs separate regions, oriented interface, charge contacts/gauge, HM theta_sh, FM \(M_s\) and \(\mathbf m\), and complete materials. FDM cells must resolve each layer and shortest active length. FEM needs a conforming interface and local refinement. Mesh convergence must be demonstrated.

Python API¶

# %% Stage-first HM/FM model
import fullmag as fm

nm = 1.0e-9
study = fm.study("hm_fm_m1")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
hm_body = study.geometry(
    fm.Box(size=(80 * nm, 40 * nm, 3 * nm), name="hm").translate((0, 0, 1.5 * nm)),
    name="hm",
)
fm_body = study.geometry(
    fm.Box(size=(80 * nm, 40 * nm, 1 * nm), name="fm").translate((0, 0, 3.5 * nm)),
    name="fm",
)
fm_body.Ms, fm_body.Aex, fm_body.alpha = 5.8e5, 1.5e-11, 0.03
fm_body.m = fm.texture.uniform(1.0, 0.0, 0.0)
hm, ferromagnet = fm.RegionRef("hm"), fm.RegionRef("fm")

# %% Solved charge, spin, and named-solve torque
charge = study.current_transport(
    name="charge", model="ohmic_poisson", coupling="one_way",
    domain=[hm, ferromagnet],
    materials=[
        fm.ChargeTransportMaterialAssignment(hm, fm.ChargeTransportMaterial(5.0e6)),
        fm.ChargeTransportMaterialAssignment(ferromagnet, fm.ChargeTransportMaterial(1.0e6)),
    ],
    boundaries=[], gauge=fm.ChargePotentialGauge("zero_mean"),
    solver=fm.ChargeSolverPolicy(engine="cg"),
)
spin = study.spin_transport(fm.SpinDriftDiffusion(
    id="spin", current_source_id=charge.name, domain=[hm, ferromagnet],
    materials=[
        fm.SpinTransportMaterialAssignment(
            hm, fm.SpinTransportMaterial(5.0e6, 0.0, 0.20, 1.5 * nm)
        ),
        fm.SpinTransportMaterialAssignment(
            ferromagnet, fm.SpinTransportMaterial(1.0e6, 0.4, 0.0, 5 * nm, 1 * nm, 1 * nm)
        ),
    ],
    interfaces=[fm.MixingConductanceSpinInterface(
        id="hm_fm", normal_to_ferromagnet=(0, 0, 1),
        normal_side=hm, ferromagnet_side=ferromagnet,
        g_up_Spm2=2.5e14, g_down_Spm2=2.5e14,
        g_r_Spm2=5.0e14, g_i_Spm2=5.0e13,
    )],
    requested_execution=fm.TransportExecution("fdm", "cpu", "double", "strict"),
))
study.spin_torque(fm.DriftDiffusionSpinTorque("transport_torque", spin.id, ferromagnet))
study.stages.add_run(1.0e-15, stage_id="m1_run")

# %% FEM CPU reciprocal reference
import fullmag as fm

study = fm.study("fem_m2_reference")
study.engine("fem")
study.device("cpu", precision="double")
study.mode("strict")
body = study.geometry(fm.Box(size=(30e-9, 20e-9, 4e-9), name="film"), name="film")
body.Ms, body.Aex, body.alpha = 8.0e5, 13.0e-12, 0.02
body.m = fm.texture.uniform(1.0, 0.0, 0.0)
region = fm.RegionRef("film")
operator = "fem_charge_spin_conforming_h1_p1.reciprocal_m2.v1"
charge = study.current_transport(
    name="m2_charge", model="ohmic_poisson", coupling="bidirectional",
    domain=[region],
    materials=[fm.ChargeTransportMaterialAssignment(
        region, fm.ChargeTransportMaterial(4.0e6, 4.0e6, 4.0e6, 0.0)
    )],
    boundaries=[], gauge=fm.ChargePotentialGauge("dirichlet_reference"),
    solver=fm.ChargeSolverPolicy(engine="block_gmres", operator_version=operator),
)
spin = study.spin_transport(fm.SpinDriftDiffusion(
    id="m2_spin", current_source_id=charge.name, domain=[region],
    materials=[fm.SpinTransportMaterialAssignment(
        region, fm.SpinTransportMaterial(3.0e6, 0.2, 0.1, 4e-9, 1e-9, 1e-9)
    )],
    solver=fm.SpinSolverPolicy(engine="block_gmres", operator_version=operator),
    requested_execution=fm.TransportExecution("fem", "cpu", "double", "strict"),
))
study.spin_torque(fm.DriftDiffusionSpinTorque("m2_torque", spin.id, region))
study.stages.add_run(1.0e-15, stage_id="m2_run")

# %% Local prescribed SOT, without drift diffusion
import fullmag as fm

study = fm.study("prescribed_sot_comparison")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
film = study.geometry(fm.Box(40e-9, 20e-9, 1e-9), name="film")
film.Ms, film.Aex, film.alpha = 8.0e5, 13.0e-12, 0.02
film.m = fm.texture.uniform(1.0, 0.0, 0.0)
target = fm.RegionRef("film")
drive = fm.SignedScalarDrive(1.0e11, sigma=(0.0, 1.0, 0.0))
study.spin_torque(fm.PrescribedSpinOrbitTorque(
    "local_sot", target, drive, xi_dl=0.12, xi_fl=0.01,
    free_layer_thickness_m=1.0e-9,
))
study.stages.add_run(1.0e-15, stage_id="sot_run")

ProblemIR¶

SpinDriftDiffusion lowers under spin_transport_modules[] with schema spin_transport.v1, current source, mode, domains, materials, interfaces, boundaries, solver, requested execution, and the one-way or reciprocal constitutive version. Canonical torque lowers with kind drift_diffusion_spin_torque, schema drift_diffusion_spin_torque.v1, solve_id, target, and formula transport_torque_angular_momentum.fullmag.v1.

{
  "spin_transport_modules": [{
    "schema_version": "spin_transport.v1",
    "id": "spin",
    "current_source_id": "charge",
    "constitutive_version": "transport_constitutive.one_way.fullmag.v1"
  }],
  "spin_torque_modules": [{
    "kind": "drift_diffusion_spin_torque",
    "id": "transport_torque",
    "solve_id": "spin",
    "formula_version": "transport_torque_angular_momentum.fullmag.v1"
  }]
}

Round trip and failure semantics¶

Requested intent preserves signs, orientations, materials, coupling, target, and execution request. Resolved execution records backend, device, precision, operator, realization, and capability. Validation errors reject malformed coverage, gauges, interfaces, and coefficients. Unsupported combinations fail closed; they are not dropped, converted to prescribed SOT, or rebound to CPU. Opaque round trip is not execution support.

Discrete realization¶

FDM M2 requires equal charge/spin domains, Dirichlet voltage gauge, no periodic spin, no transparent cross-material interface, explicit mixing laws, reciprocal policy, and complete anisotropic charge coefficients. FEM M2 requires one conforming domain, no internal interfaces, uniform anisotropic coefficients, and a monolithic linear policy. Reciprocal transient is unavailable.

Implementation mapping¶

Python owns typed authoring and serialization. ProblemIR owns canonical intent and graph edges. The planner validates domains, policies, and capability. Backend operators own equations and signs. API/UI preservation does not prove runtime realization.

Validation¶

Python tests verify serialization, named-solve references, material validation, and round trip; round-trip tests do not run a solver. FDM CPU tests cover one-dimensional SHE profile/sign, exact removal at \(\theta_{\mathrm{SH}}=0\), balance, and torque sign. Managed recipes exist, but this page does not claim they ran for this revision.

Promotion requires analytic diffusion and mesh convergence; zero-SHE and sign reversal; charge conservation/gauge invariance; spin residuals; interface and angular-momentum balance; M1 recovery from M2; FDM/FEM common-limit comparison; and actual-device GPU parity with no-fallback provenance.

Limitations¶

No drift-diffusion lane is documented as scientifically validated or production-qualified. Source, schema, planner, unit-test, runtime, and production evidence are separate. Live field publication, full FEM/CUDA parity, and browser rendering were not established. Experimental parameters must be converted into these units and signs.

Scientific bibliography¶

1. S. Zhang, P. M. Levy, and A. Fert, Physical Review Letters 88, 236601 (2002), doi:10.1103/PhysRevLett.88.236601.

2. C. Abert et al., Scientific Reports 5, 14855 (2015), doi:10.1038/srep14855.

3. C. Abert et al., Scientific Reports 6, 16 (2016), doi:10.1038/s41598-016-0019-y.

4. J. Sinova et al., Reviews of Modern Physics 87, 1213 (2015), doi:10.1103/RevModPhys.87.1213.

5. S. Zhang, Physical Review Letters 85, 393 (2000), doi:10.1103/PhysRevLett.85.393.

Source-code index¶