Refined Thiele Approach

• The refined Thiele approach is an advanced modeling strategy that reduces high-dimensional micromagnetic equations to collective coordinates for predicting vortex dynamics.
• It employs a deformation-sensitive ansatz to capture nonlinear effects such as damping anisotropy and amplitude-dependent gyrotropic evolution in spin-torque vortex oscillators.
• This semi-analytical method, validated by micromagnetic simulations, provides a scalable framework for designing robust neuromorphic circuits.

The refined Thiele approach is an advanced modeling strategy for predicting the nonlinear dynamics of magnetic vortices in spin-torque vortex oscillators (STVOs), with special emphasis on the amplitude-dependent evolution of gyrotropic and damping contributions under core deformation. This formalism enables semi-analytical quantification of vortex dynamics that reproduces essential nonlinear effects, such as damping anisotropy, typically seen in micromagnetic simulations but at significantly reduced computational cost, making it pivotal for scalable neuromorphic circuit design (Ducarme et al., 20 Aug 2025).

1. Collective-Coordinate Reduction and Formal Equation

At the core of the refined Thiele method is the reduction of a high-dimensional micromagnetic PDE system to a set of collective coordinates representing the vortex core position $X$ (Cartesian or polar $s,\,\phi$). The general form of the refined equation of motion is: $$(\dot{X} \times G) + D \cdot \dot{X} = -\nabla W + F_{\mathrm{ST}}$$ where

• $G$ is the gyrotropic tensor, capturing topological winding,
• $D$ is the damping tensor, reflecting energy dissipation,
• $W$ is the total magnetic potential energy,
• $F_{\mathrm{ST}}$ is the external spin-transfer torque force.

Under cylindrical symmetry, $G(s) = G_{yx}(s)$ is often used while $D$ may bifurcate into azimuthal ($D_\phi$) and radial ($D_\rho$) contributions for a deformed vortex core.

2. Vortex Magnetization Profile Ansatz

The innovation in the refined approach lies in constructing a deformation-sensitive ansatz for the out-of-plane magnetization $m_z(n_r)$: $$m_z = P A_c\, \exp\left[-\frac{(n_r - s)^2}{P_{c,r}^2}\right] - A_a \exp\left[-\frac{(n_r - (s - d))^2}{P_{a,r}^2}\right]$$ with:

• $P$ the vortex polarity,
• $A_c$ normalization compensating for the dip amplitude and separation,
• $n_r$ radial coordinate,
• $s$ normalized orbit radius,
• $d$ core-dip separation,
• $P_{c,r}, P_{a,r}$ are width parameters for the core and dip. This construction effectively represents both the primary core and the "dip" region of opposite polarity, which becomes substantial at large oscillation amplitudes.

3. Semi-Analytical Evaluation of Gyrotropic and Damping Tensors

Using the deformed profile, effective dynamical tensors are extracted as:

• Gyrotropic Tensor:

$$G_{ab}(X) = \int dV\, m \cdot \left( \partial_a m \times \partial_b m \right)$$

For increasing $s$, in-plane magnetization tilts radially and winding number density rises, resulting in $G(s) > G_0$ (rigid core value).

• Damping Tensor:

$$D_{as}(X) = \frac{\alpha M_S}{\gamma} \int dV\, (\partial_a m \cdot \partial_s m)$$

Under core deformation, $D_xx \neq D_yy$ (anisotropy), with $D(s)$ growing as tail distortions and dip regions amplify at larger displacements. Typically, $D_\phi(s)$ (azimuthal) is most pertinent for circular vortex motion.

4. Direct Parameter Extraction from Micromagnetic Simulations

The refined scheme is complemented by direct numerical extraction:

• Detailed micromagnetic simulations (e.g., via mumax+) are performed for different $s$ (0.1–0.8) capturing steady-state $m_z$ fields.
• Radial basis function (RBF) interpolation maps scattered core and magnetization data to a uniform grid, allowing precise spatial derivatives.
• Calculated gradients via Eqs. above yield $G(s)$ and $D(s)$, which validate or fine-tune the ansatz.
• For larger $s$, simulations reveal pronounced rises in $D(s)$ beyond semi-analytic predictions, highlighting long-range deformation effects.

5. Damping Anisotropy and Nonlinear Dynamical Consequences

A principal feature uncovered is the clear anisotropy in damping under vortex deformation:

• In linear (rigid) regime: $D_0 = \alpha n |G_0|$, where $n$ is geometric ($n = 1/2\ln(R/2l_{ex}) + 3/8$).
• For deformed vortices: $D$ splits into $D_\phi(s)$ and $D_\rho(s)$, with $D_\phi(s)$ escalating with amplitude.
• This anisotropic damping modifies oscillation thresholds, phase noise, and tuning response, impacting coupled STVO arrays and their consistency in neuromorphic architectures.

6. Implications for Neuromorphic Computing and Predictive Design

The refined Thiele approach provides actionable advances:

• Semi-analytical $G(s)$ and $D(s)$ capture amplitude- and anisotropy-induced nonlinearities critical in high-density oscillator arrays.
• Direct simulation-driven adjustment allows rapid benchmarking against MMS, enabling scalable simulation of complex ensembles.
• Predictive modeling tools derived from this framework furnish robust designs for pattern recognition, signal processing, and time-domain neuromorphic tasks where oscillator nonlinear responses define system behavior.

7. Synthesis and Future Outlook

The approach achieves synthesis of detailed micromagnetic simulation insight and analytical tractability:

• Incorporation of vortex core deformation via ansatz (Eq. [2]) ensures the dynamical model remains valid in both small and large amplitude regimes.
• Efficient extraction and validation with MMS close critical accuracy gaps while maintaining computational efficiency.
• The refined Thiele formalism, particularly its treatment of damping anisotropy and nonlinear $G(s), D(s)$ evolution, supports the engineering of next-generation neuromorphic circuits where physical fidelity and simulation throughput are paramount.

A plausible implication is that additional generalizations could explore stochasticity or temperature-related effects utilizing this ansatz/model-coupling strategy, further expanding predictive utility in practical device design.