Micromagnetics and spintronics: Models and numerical methods

• Micromagnetics combines a continuous magnetization field with quantum effects such as exchange, assuming magnetic forces align magnetization over a length scale above the lattice spacing.At zero temperature, the magnetization magnitude is the material's saturation magnetization.Source: §1 · p. 4
• Resolving magnetic domains and domain walls supports hysteresis computations and micron-scale studies of switching and domain-wall motion in storage and sensing technologies.The introduction names MRAM, read-head sensors, and angle sensors as applications.Source: §1 · p. 4
• The article surveys analytical and numerical micromagnetics, including finite-difference and finite-element discretizations and their advantages, disadvantages, and implementation pitfalls.Source: §1 · p. 4
• Its main focus is existing spin-transport models integrated with micromagnetics, including their applicability, limits, and discretization.Source: §1 · p. 5

2 Energetics of a ferromagnet

• The total energy combines classical magnetostatic terms with quantum-mechanical exchange and magnetocrystalline anisotropy contributions.The section surveys typical material-dependent energy contributions in the micromagnetic model.Source: §2 · p. 5

2.1 Zeeman energy

• Zeeman energy represents the dependence of a magnetic body’s energy on the external field.It is formulated over the magnetic region using the magnetization and applied field.Source: §2.1 · p. 5

2.2 Exchange energy

• Exchange energy favors parallel alignment and, in the isotropic continuum model, penalizes spatially inhomogeneous magnetization.The continuum expression follows from the discrete Heisenberg interaction; its exchange coefficients depend on the lattice structure.Source: §2.2 · pp. 5–6

2.3 Demagnetization energy

• Demagnetization energy describes dipole interactions, and its competition with exchange produces domains such as vortices with an out-of-plane core.For localized magnetization, the Poisson formulation may have a singular boundary divergence and is well defined only distributionally.Source: §2.3 · pp. 6–8

2.4 Crystalline anisotropy energy

• Anisotropy energy favors alignment with material-dependent easy axes, while interface anisotropy is integrated over the interface rather than the magnetic volume.The section gives uniaxial and cubic forms and notes that higher-order constants are often set to zero in applications.Source: §2.4 · p. 9

2.5 Antisymmetric exchange energy

• Antisymmetric exchange counteracts regular exchange and gives rise to skyrmion configurations with nonzero topological charge.The section describes interface and bulk DMI continuum forms, with interface coupling tied to the interface normal.Source: §2.5 · pp. 10–11

2.6 Interlayer-exchange energy

• RKKY interaction couples magnetic layers across a nonmagnetic spacer, with coupling sign and strength dependent on spacer thickness.Choosing a thickness that makes the coupling constant negative yields antiferromagnetic coupling used in synthetic antiferromagnets.Source: §2.6 · p. 11

2.7 Other energy contributions and effects

• The overview excludes other potentially important effects, including magnetostriction, current-related eddy currents and Oersted fields, and finite-temperature micromagnetics.The article identifies Langevin dynamics and the Landau-Lifshitz-Bloch equation among finite-temperature approaches, but leaves that topic out of scope.Source: §2.7 · p. 12

3 Static micromagnetics

• Static micromagnetics finds stable configurations by minimizing total free energy subject to the magnetization unit-sphere constraint.The formulation uses variational calculus and Lagrange multipliers to enforce volume and boundary constraints.Source: §3 · pp. 12–13

3.1 Zeeman energy

• The Zeeman-energy variation contributes a functional derivative without an additional boundary term.The variation is proportional to the applied field.Source: §3.1 · p. 14

3.2 Exchange energy

• Exchange-energy variation requires integration by parts, producing both a functional derivative and a boundary term.For spatially constant exchange stiffness, the derivative simplifies to −2AΔm.Source: §3.2 · p. 14

3.3 Demagnetization energy

• The demagnetization-field variation yields its energy derivative, with the field’s linear dependence on magnetization producing a factor of two in the differential.The section then identifies the corresponding derivative and boundary term.Source: §3.3 · p. 15

3.4 Anisotropy energy

• The section gives functional derivatives and boundary terms for uniaxial and cubic anisotropy, while interface anisotropy contributes through a boundary term.For interface anisotropy, the volume variational derivative vanishes.Source: §3.4 · p. 15

3.5 Antisymmetric exchange energy

• Antisymmetric-exchange variations are integrated by parts to identify the interface and bulk DMI derivatives and their boundary terms.The interface and bulk cases have distinct boundary contributions.Source: §3.5 · pp. 15–16

3.6 Energy minimization with multiple contributions

• For multiple energy contributions, Brown’s condition and a single composite boundary condition must be satisfied; adding terms modifies that boundary condition rather than adding new ones.Exchange, demagnetization, and antisymmetric exchange contributions are combined in the stated conditions.Source: §3.6 · p. 16

4 Dynamic micromagnetics

• The derivation uses a rigid-body representation of magnetization, with spherical coordinates and a fixed-body frame whose third axis follows m.It sets the transverse moments of inertia to zero and identifies the remaining angular momentum with saturation magnetization.Source: Figure 5 · p. 17, §4 · pp. 17–18
• The Gilbert-form LLG follows from a variational derivation with a Rayleigh damping term and an effective field defined from the energy.The derivation imposes vanishing transverse moments of inertia and relates the remaining axial inertia to saturation magnetization.Source: §4 · pp. 17, 20
• The LLG combines precession around the effective field with dissipative motion toward it.The section presents the Gilbert and Landau–Lifshitz formulations as equivalent under a proper parameter transformation.Source: §4 · p. 17, Figure 6 · p. 21

4.1 Properties of the Landau-Lifshitz-Gilbert Equation

• The LLG conserves the unit magnetization magnitude, |m| = 1, during its evolution.The passage derives this by substituting the explicit LLG into the time derivative of |m|².Source: §4.1 · p. 21
• For α > 0 and a time-independent effective field, the LLG makes system energy non-increasing; at α = 0, it preserves energy.The damped case has Lyapunov structure, while the undamped case has Hamiltonian structure.Source: §4.1 · pp. 21–22

5 Spintronics in micromagnetics

• Spin torque splits into fieldlike and dampinglike contributions: the former is dominated by precession, while the latter directly rotates magnetization toward polarization.The torques arise when polarized currents pass through magnetic regions; their naming depends on the current’s origin.Source: §5 · p. 22

5.1 Spin-transfer torque in multilayers

• In multilayers, spin-dependent scattering polarizes electrons: antiparallel layers can torque FM2, while in parallel configurations scattered electrons can torque FM1.The described device has two magnetic layers separated by a nonmagnetic spacer; switching requires torque above a critical strength.Source: §5.1 · pp. 22–23
• Slonczewski’s model extends the LLG with dampinglike and fieldlike torques on a free-layer magnetization polarized by a second magnetic layer.The macrospin model uses a single vector for the free layer and a polarization vector for the polarizing layer.Source: §5.1 · pp. 23–24
• Slonczewski’s macrospin approach suits small single-domain systems; for thin films below the exchange length, it generalizes by evaluating the local angle between polarization and magnetization.The volume-torque treatment is justified for free layers below the exchange length, with torque strength scaled by reciprocal layer thickness.Source: §5.1 · p. 25, Figure 9 · p. 25
• Slonczewski-like models neglect lateral spin diffusion, can be inaccurate for strongly inhomogeneous magnetization, and use free parameters tied nontrivially to device geometry and materials.The volume-term treatment is only justified for free-layer thicknesses below the exchange length.Source: §5.1 · p. 25

5.2 Spin-transfer torque in continuous media

• For continuous magnets, Zhang–Li torque depends on local magnetization gradients and describes current-driven domain-wall motion, but neglects spin diffusion.The model is not suited to multilayers requiring transport across a nonmagnetic spacer and can be inaccurate for highly inhomogeneous configurations.Source: §5.2 · p. 26

5.3 Spin-diffusion

• Spin accumulation provides a more general spin-torque description than Slonczewski and Zhang–Li models, which neglect spin-polarization diffusion to some extent.The spin-accumulation vector describes the conducting electrons’ polarization deviation from equilibrium and is zero when no current is applied.Source: §5.3 · p. 27
• The spin-diffusion model couples spin accumulation to magnetization and spin current across the complete sample, including nonmagnetic leads and spacers.Material parameters may vary spatially in composite media; the spin-current definition also depends on electric potential and material parameters.Source: §5.3 · p. 27
• Because spin accumulation relaxes two orders of magnitude faster than magnetization, it can be treated as instantaneously equilibrated for magnetization-focused calculations.Setting its time derivative to zero makes spin accumulation depend on magnetization rather than requiring coupled time integration.Source: §5.3 · pp. 27, 29
• A self-consistent model solves for spin accumulation and electric potential together, using boundary conditions to prescribe potential or current inflow.This avoids prescribing current or potential throughout the sample when magnetization-dependent resistance effects are to be accounted for.Source: §5.3 · p. 29

5.4 Boundary conditions for the spin-diffusion model

• Dirichlet or Neumann conditions on electric potential specify contact voltage or charge-current inflow, respectively; non-contact boundaries use homogeneous Neumann conditions.The same spin-accumulation boundary conditions apply to simplified diffusion models with prescribed potential or current.Source: §5.4 · pp. 29–30
• Homogeneous Neumann conditions for spin accumulation correspond to zero spin-current flux, but electric contacts may need Robin conditions when their nonmagnetic regions are thinner than the spin-flip relaxation length.The zero-flux assumption is reasonable at contacts when the nonmagnetic region exceeds that length.Source: §5.4 · p. 30

5.5 Spin-orbit torque

• Spin-Hall and inverse spin-Hall effects extend spin diffusion to charge-to-spin and spin-to-charge conversion, respectively, including in heavy-metal/magnetic multilayers.The extended current definitions are inserted into the source equations to form a self-consistent model across magnetic and nonmagnetic regions.Source: §5.5 · p. 31

5.6 Material parameters in the spin-diffusion model

• Spin-flip relaxation and exchange coupling are parameterized by characteristic lengths or times, and material parameters can vary spatially across composite systems.The material-parameter fields may jump across interfaces because they are not spatially differentiated in the governing equations.Source: §5.6 · p. 32

5.7 Spin dephasing

• Spin dephasing adds a dephasing-time term to the spin-diffusion model, which can be solved in equilibrium or nonequilibrium and with prescribed or self-consistent charge current.Source: §5.7 · p. 32

5.8 Valet-Fert model

• Valet–Fert spin potential is collinear with magnetization inside magnetic regions, so torque arises only at magnetic–nonmagnetic interfaces.Its complex spin-mixing conductance sets dampinglike and fieldlike torque strengths, but their ratio is an input rather than a model prediction; the model also allows potential jumps at interfaces.Source: §5.8 · pp. 33–34

5.9.1 Slonczewski model

• The original Slonczewski angular dependence fits symmetric multilayers, while asymmetric layers require its generalized form.The comparison with spin-diffusion results uses equal 3 nm magnetic layers for the symmetric case and 3 nm and 5 nm layers for the asymmetric case.Source: §5.9.1 · p. 34, Figure 15 · p. 35

5.9.2 Zhang-Li model

• With diffusion D0 set to zero, the Zhang–Li model exactly reproduces spin diffusion and is restricted to local torque phenomena such as domain walls.Source: §5.9.2 · pp. 34, 36

5.9.3 Valet-Fert model

• Valet–Fert and Zhang–Levy–Fert share bulk equations and can be mapped across magnetic and nonmagnetic regions, but differ in interface potential continuity.Effective thin layers can mimic Valet–Fert interface jumps in Zhang–Levy–Fert; the effective-layer spin-flip and exchange parameters can approximately reproduce behavior, but their dependence on spin-mixing conductance is nontrivial and best resolved by fitting.Source: §5.9.3 · pp. 36–38
• For collinear magnetization, the parameter mapping is exact and the spin-mixing term vanishes, consistent with zero spin torque.In the small-layer-thickness limit, the mapped models also yield the same spin accumulation and electric potential.Source: §5.9.3 · p. 38
• The Zhang–Levy–Fert model describes spin transport and torque more generally through bidirectional torque, while additional thin layers can model Valet–Fert interface properties.The passage notes that Valet–Fert interface properties are discussed and determined for various material systems.Source: §5.9.3 · p. 38

• The spin-diffusion model is restricted to diffusive transport, although additional terms could represent some compatible effects such as in-plane GMR, spin pumping, and anomalous Hall effect.Source: §5.10 · p. 38
• Tunnel-junction transport is non-diffusive, and ab initio models are computationally challenging and do not integrate well with semiclassical micromagnetics.Source: §5.10 · p. 38
• A model that integrates tunnel-junction transport with spin diffusion and micromagnetics remains an ongoing research topic.Source: §5.10 · p. 38

• Naive demagnetization-field computation scales as O(n^2), while proposed methods reduce complexity to O(n log n) or O(n).Source: §6 · p. 39
• Second-order exchange coupling creates high stiffness, demanding careful numerical time integration despite its role in domain generation alongside demagnetization.Source: §6 · p. 39
• Nonlinear energy landscapes complicate energy minimization and can cause quasistatic hysteresis computations to miss nearby local minima.Source: §6 · p. 39
• Finite differences typically use regular cuboid meshes and finite elements irregular tetrahedral meshes; either requires cells small enough to resolve domain walls.The exchange length characterizes domain-wall width, and effective-field computation is needed only in the magnetic domain.Source: §6 · pp. 39–40

6.1 The finite-difference method

• Regular meshes preserve the demagnetization integral’s convolution structure, while a direct dense operator would require O(n^2) storage and computation.The mesh restriction can make finite differences unsuitable for some complex geometries, but enables fast algorithms.Source: §6.1 · p. 40, §6.1.1 · p. 41

6.1.1 Demagnetization field

• FFT evaluation turns the discrete demagnetization convolution into Fourier-space cellwise multiplication, reducing computation to O(n log n).Zero-padding avoids unphysical negative-distance contributions; physical field values occupy only the first n_i entries.Source: §6.1.1 · pp. 42–43, Figure 18 · p. 42
• Analytical demagnetization-tensor formulas for cuboid cells are exact, but floating-point errors can affect large cell distances and degenerated cells.Numerical integration is described as a way to avoid these errors; computing the field from a scalar potential reduces Fourier transforms but compromises accuracy.Source: §6.1.1 · pp. 43–44

6.1.2 Local field contributions

• Finite differences approximate local derivatives for short-range fields, and second-order approximations are commonly used despite possible convergence gains from higher orders.Virtual cells implement boundary conditions so derivatives can be evaluated without special treatment at boundary cells; their values depend on the field contribution.Source: §6.1.2 · pp. 44–45, Figure 19 · p. 45

6.1.3 Spintronics

• Finite-difference spin-torque terms depend on local magnetization and derivatives, while spin-diffusion discretization must account for discontinuous material parameters at interfaces.The spin-current may jump across interfaces, so the discretized divergence must account for those jumps in a distributional sense.Source: §6.1.3 · pp. 45–46

6.1.4 Existing software packages

• Finite-difference software includes CPU packages such as OOMMF, Fidimag, and MicroMagus, and GPU codes such as MuMax3 and magnum.fd.OOMMF also has a GPGPU extension.Source: §6.1.4 · p. 46

6.2 The finite-element method

• The finite-element stiffness matrix is sparse because basis functions couple only neighboring nodes, reducing storage from O(n^2) to O(n) and enabling efficient iterative solvers.Iterative methods avoid direct matrix inversion and use matrix–vector multiplications, which scale better than direct methods.Source: §6.2 · p. 48

6.2.1 Demagnetization field

• The demagnetization potential requires open-boundary treatment: truncation approximates the exterior with a finite region, while shell transformation maps a finite shell to the infinite exterior.A truncation region about five times the magnetic region is a reported accuracy–cost trade-off [93]; shell transformation represents infinity finitely but its singular outer-boundary metric tensor worsens matrix conditioning.Source: §6.2.1 · pp. 48–50
• The hybrid finite-element–boundary-element method discretizes only the magnetic region, but its dense boundary operator loses sparsity; projecting the potential gradient yields a continuous field on the magnetization mesh.The hybrid calculation evaluates a double-layer potential on the boundary and solves a Laplace problem; projection uses mass lumping to avoid an additional linear solve.Source: §6.2.1 · pp. 50–52

6.2.2 Local field contributions

• For local fields, integration by parts avoids second spatial derivatives, and multiplying by µ0Ms before integration by parts addresses divergence at interfaces where saturation magnetization is discontinuous.The energy-differential formulation embeds exchange and antisymmetric-exchange boundary conditions without requiring them to be applied explicitly; mass lumping can reduce quadratic energy field calculations to a sparse matrix–vector multiplication.Source: §6.2.2 · pp. 52–53

6.2.3 Interface contributions

• Interface energy contributions can be represented as effective fields by requiring the field-generated energy to equal the interface energy, then applying basis-function discretization and mass lumping.This gives a system with the same form as a local bulk-field contribution.Source: §6.2.3 · pp. 53–54

6.2.4 Spintronics

• Simplified Slonczewski and Zhang–Li torques can be incorporated as local fields, while Slonczewski torque as an interface effect uses the interface-field formulation.Source: §6.2.4 · p. 54
• For spin diffusion, the equilibrium spin accumulation is computed from a Galerkin weak form, with integration by parts avoiding derivatives of discontinuous magnetization and material parameters.Material parameters are represented by piecewise-constant functions; magnetization is piecewise affine and globally continuous within the magnetic region, and integrals involving it are restricted to that region.Source: §6.2.4 · p. 54
• Self-consistent spin diffusion uses continuous piecewise-affine fields for spin accumulation and electric potential, coupled weak formulations, and a 4n × 4n sparse system with contact boundary conditions.Spin accumulation is assumed continuous across material interfaces; current inflow uses a natural Neumann condition, prescribed potential a Dirichlet condition, and homogeneous Neumann conditions apply to spin accumulation.Source: §6.2.4 · pp. 54–55

6.2.5 Existing software packages

• General-purpose finite-element tools include mesh generators Gmsh and NetGen, full-stack ONELAB and NGSolve, Escript, MFEM, and FEniCS with PETSc.BEM++ [110] and H2Lib provide boundary-element matrix assembly and hierarchical-matrix compression for hybrid demagnetization methods.Source: §6.2.5 · p. 55
• Micromagnetic finite-element packages include FinMag, magnum.fe and FEELLGOOD, plus Magpar, NMag, FEMME, Tetramag and Fastmag.FinMag focuses on classical micromagnetics; magnum.fe and FEELLGOOD also solve spin-diffusion models, while GPU-based Tetramag and Fastmag accelerate computations.Source: §6.2.5 · pp. 55–56

6.3 Other spatial discretization methods

• The non-uniform FFT aims to combine FFT-based convolution speed with the flexibility of irregular meshes.Both FFT-accelerated convolution and finite-element demagnetization-field computation have at least O(n log n) complexity.Source: §6.3 · p. 56
• The fast multipole method, used for particle-cloud interactions, has also been shown applicable to the continuous demagnetization-field problem and scales with n.Source: §6.3 · p. 56
• Low-rank tensor approximations are another class of methods with complexity below O(n log n).Source: §6.3 · p. 56

6.4.1 Explicit Runge-Kutta methods

• Explicit Runge–Kutta methods are computationally cheap and standard on finite-difference grids; adaptive RKF and Dormand–Prince use fourth/fifth-order estimates, but stiffness can threaten stability.Higher-order methods require more effective-field evaluations but can permit larger time steps without compromising accuracy.Source: §6.4.1 · pp. 57–58

6.4.2 Implicit midpoint scheme

• The implicit midpoint rule exactly preserves |m| for any time-step size and preserves quadratic magnetic energy when damping α=0, but requires costly Newton iterations.Each Newton iteration requires an effective-field evaluation.Source: §6.4.2 · p. 58

6.4.3 Tangent-plane integration

• Tangent-plane integration restricts the weak-form solution and test spaces to vectors orthogonal to magnetization, supporting finite-element time stepping.Separate θ choices allow stiff exchange to be implicit and demagnetization explicit; coupling with spin-accumulation time marching yields a method for dynamic spin diffusion. Implementations include Lagrange multipliers or local tangent-plane mappings, while higher-order variants lose linearity in the solution variable.Source: §6.4.3 · pp. 58–59

6.4.4 Backward differentiation formula

• BDF is a multistep method using preceding magnetization snapshots; its nonlinear solve can use Newton iterations and preconditioning, making each step expensive but effective across many finite-element problems.The preconditioner approximates the system with an easily invertible operator, often retaining local linear field contributions such as exchange.Source: §6.4.4 · pp. 59–60

6.4.5 Existing software packages

• Explicit schemes are straightforward to implement, whereas implicit methods need nonlinear-system solvers; PETSc supports Newton methods and SUNDIALS provides adaptive-step BDF.Source: §6.4.5 · p. 60

6.5 Energy minimization and barrier computation

• Quasistatic hysteresis is computed by stepping the external field and minimizing energy, since slow field sweeps make dynamics infeasible; local rather than global minima preserve hysteresis.Source: §6.5 · pp. 60–61
• The string method discretizes a transition path into magnetization images, relaxes them, then reparametrizes the path to prevent images collapsing into endpoint minima and losing barrier information.Source: Figure 24 · p. 61, §6.5 · p. 62
• Gradient methods start from a given magnetization and converge to local minima; projected descent improves unit-sphere handling, while tailored methods target performance and reduce risk of missing local minima.Source: §6.5 · p. 61
• For magnetic storage states, the barrier between minima defines information lifetime through the Néel–Arrhenius equation; the string method estimates the barrier using a minimum-energy path.Source: §6.5 · pp. 61–62
• Energy-weighted reparametrization concentrates images near the barrier, whereas nudged elastic band uses spring forces to maintain homogeneous path spacing.Source: §6.5 · p. 62

7.1 Standard problem #4

• FDM and FEM agree on the standard problem #4 magnetization dynamics, but FDM has lower total simulation time despite requiring more RHS evaluations as the mesh is refined.For this thin domain, FDM reduces demagnetization computation to a 2D convolution, while FEM is dominated by the boundary-element method’s inferior scaling.Source: §7.1 · pp. 63, 65, Figure 25 · p. 63, Figure 27 · p. 64

7.2 Standard problem #5

• For standard problem #5, larger diffusion lengths produce deviations from Zhang–Li vortex equilibria; the authors recommend the full diffusion model for accurate descriptions.Deviations are most significant in the x-shift for ξ = 0.05 and in the y-shift for ξ = 0.5.Source: §7.2 · p. 66, Figure 29 · p. 66

7.3 Spin-torque oscillator

• In the spin-torque oscillator simulations, the spin-diffusion model captures changes with polarizing-layer thickness and material parameters, while Slonczewski’s model is less computationally complex.The resulting oscillation frequencies and tilting angles vary with polarizing-layer thickness.Source: §7.3 · p. 67, Figure 31 · p. 67

7.4 Spin-orbit torque MRAM

• The SOT MRAM field simulation shows opposite curl-like structures for spin accumulation and the Oersted field, with spin accumulation much smaller in the stack.The text attributes the smaller spin accumulation in the stack to its interplay with magnetization in the magnetic layers.Source: §7.4 · p. 68, Figure 33 · p. 69
• The SOT MRAM write simulation shows the free layer switching during the current pulse while the pinned-layer magnetization remains fixed.The pulse is applied for 0.5 ns, then decays linearly to zero over a further 0.5 ns.Source: Figure 34 · p. 69, §7.4 · p. 70
• The simulated SOT MRAM readout shows a 23% resistance difference between antiparallel and parallel configurations.The potential, and therefore resistance, is higher for antiparallel alignment.Source: §7.4 · p. 70, Figure 35 · p. 70

8 Conclusion

• The spin-diffusion model is among the most complete approaches discussed, but its applicability and limitations require careful consideration.Source: §8 · p. 70
• The spin-diffusion model is limited to diffusive transport and is not suitable for accurate descriptions of tunnel barriers.The conclusion identifies efficient integration of ab initio methods with micromagnetics as an important future task for tunnel-barrier transport.Source: §8 · pp. 70–71
• Model complexity and discretization should be chosen for the problem: complex models may provide detail, while simpler ones can support many simulations quickly.Discretization affects accuracy and computation time.Source: §8 · p. 71