First-Principles Atomistic Model

• The first-principles-based atomistic model is a computational method that derives atomic-scale interactions directly from quantum mechanical principles, ensuring parameter-free predictions.
• It incorporates damping and inertial effects by linking microscopic spin dynamics with electronic structure calculations, such as DFT and Green’s functions.
• This approach enables quantitative simulations of ultrafast magnetization and emergent phenomena, advancing the design of high-speed magnetic and optospintronic devices.

A first-principles-based atomistic model is a computational framework where the physical behavior of a material or molecular system is described at the atomic scale with all essential interactions, parameters, and dynamical rules derived strictly from quantum-mechanical or statistical-mechanical first principles. In such models, key ingredients such as interatomic forces, magnetic exchange, damping, and other couplings are systematically computed from fundamental electronic structure calculations—usually density functional theory (DFT) or many-body Green’s function approaches—without recourse to empirical fitting. The result is an atomistic simulation method of predictive fidelity, capable of treating ultrafast magnetization dynamics, lattice vibrations, phase transitions, and other emergent phenomena rooted in the underlying electronic structure.

1. Generalized Atomistic Spin Dynamics: Modified LLG Equation

The prototypical application discussed in (Bhattacharjee et al., 2011) is the treatment of ultrafast spin dynamics, where the familiar Landau–Lifshitz–Gilbert (LLG) equation is extended to account for femtosecond processes by incorporating both damping and moment-of-inertia effects. The generalized equation of motion for the magnetization $\mathbf{M}$ in its uniform limit reads:

$$\dot{\mathbf{M}} = \mathbf{M} \times \left( -\gamma\,\mathbf{B} + \hat{\mathbf{G}}\,\dot{\mathbf{M}} + \hat{\mathbf{I}}\,\ddot{\mathbf{M}} \right)$$

Here, $\gamma$ is the gyromagnetic ratio, $\mathbf{B}$ is the effective field (encompassing external, exchange, and anisotropy contributions), $\hat{\mathbf{G}}$ is the Gilbert damping tensor, and $\hat{\mathbf{I}}$ is the moment of inertia tensor. The $\ddot{\mathbf{M}}$ inertial term accounts for nutation—a rapid "wobble" of the magnetization vector superimposed on its usual precessional motion—and significantly influences the magnetization dynamics at ultrashort timescales.

2. Origin of Damping and Inertial Terms from First Principles

The model for $\hat{\mathbf{G}}$ and $\hat{\mathbf{I}}$ goes beyond phenomenological approaches. Damping and inertia emerge naturally from a frequency-dependent, retarded spin-spin correlation function; in essence, they are tied to the frequency response of the system’s generalized exchange tensor $\chi_{ij}(\omega)$. These tensors are obtained via a low-frequency expansion:

• Effective internal field:

$$\mathbf{B}_{\text{int}} = \mu\,\lim_{\omega \to 0}\chi(\omega)$$

• Gilbert damping tensor:

$$\hat{\mathbf{G}} = \gamma\,\lim_{\omega\to 0} i\,\partial_\omega\chi(\omega)$$

• Moment of inertia tensor:

$$\hat{\mathbf{I}} = -\frac{\gamma}{2}\,\lim_{\omega\to 0}\partial_\omega^2\chi(\omega)$$

For a ferromagnetic state with slowly varying spin-resolved density of states $\rho_\sigma(\varepsilon)$ near the Fermi energy, these quantities can be approximated as (with $D$ an electronic bandwidth parameter and Sp denoting a trace in spin space):

$$\hat{\mathbf{G}} \sim 2\gamma\pi\,\mathrm{Sp}\Bigl[ \langle \partial_i H_0\rangle\,\rho\,\langle \partial_j H_0\rangle\,\rho \Bigr]_{\varepsilon=\varepsilon_F}$$

$$\hat{\mathbf{I}} \sim -\frac{\gamma}{D}\,\mathrm{Sp}\Bigl[ \langle \partial_i H_0\rangle\,\rho\,\langle \partial_j H_0\rangle\,\rho \Bigr]_{\varepsilon=\varepsilon_F}$$

This approach situates both damping and inertial effects within a rigorous theoretical setting, directly linked to electron dynamics and not requiring ad hoc parameter choices.

3. s–d-Like Exchange Interaction and Nonlocal Effects

The coupling between localized magnetic moments and itinerant electrons is described via an "s–d-like" interaction Hamiltonian:

$$\mathcal{H}_{\text{int}} = -\int J(\mathbf{r},\mathbf{r}')\,\mathbf{M}(\mathbf{r},t)\cdot\mathbf{s}(\mathbf{r}',t) \,d\mathbf{r}\,d\mathbf{r}'$$

Here, $J(\mathbf{r},\mathbf{r}')$ is the effective electron-mediated exchange (closely related to the spin-dependent exchange–correlation potential in DFT), and $\mathbf{s}(\mathbf{r}',t)$ is the electronic spin density. Integrating out electronic degrees of freedom using, for example, a Keldysh contour approach, one obtains an effective action for $\mathbf{M}$ in which both spatial and temporal nonlocalities are present. A slow-time expansion yields instantaneous (exchange field), dissipative (damping), and inertial terms.

The generalized evolution equation hence takes the form

$$\dot{\mathbf{M}}(\mathbf{r},t) = \mathbf{M}(\mathbf{r},t) \times \left[-\gamma \mathbf{B}_\text{ext}(\mathbf{r},t) + \int {\cal D}^r(\mathbf{r},\mathbf{r}';t,t') \cdot \mathbf{M}(\mathbf{r}',t') \,dt'\,d\mathbf{r}'\right]$$

where ${\cal D}^r$ is a retarded dyadic propagator (electron-mediated) that encapsulates both nonlocal exchange and memory effects.

4. First-Principles Calculation of Model Parameters

All prominent interaction terms—static exchange, Gilbert damping, and moment of inertia—are reduced to integrals over the electronic Green’s functions and $J(\mathbf{r},\mathbf{r}')$:

${\cal D}^r(\mathbf{r},\mathbf{r}';\varepsilon) = 4\,\mathrm{Sp} \int J_{\mathbf{r}\mathbf{r}_1} J_{\mathbf{r}_2\mathbf{r}'} \frac{f(\omega)-f(\omega')}{\varepsilon-\omega+\omega'+i\delta} \, \bfsigma\,\mathrm{Im}\,\mathbf{G}^r_{\mathbf{r}_2\mathbf{r}_1}(\omega)\, \bfsigma\,\mathrm{Im}\,\mathbf{G}^r_{\mathbf{r}_1\mathbf{r}_2} (\omega') \,\frac{d\omega}{2\pi}\frac{d\omega'}{2\pi}\,d\mathbf{r}_1\,d\mathbf{r}_2$

By applying Kramers–Kronig relations, one extracts the first ($\hat{\mathbf{G}}$) and second ($\hat{\mathbf{I}}$) frequency derivatives of ${\cal D}^r$ at zero frequency, with explicit formulas provided for use in practical electronic structure calculations.

5. Dynamical Regimes and Physical Interpretation

The central innovation of this first-principles atomistic model is that it enables simulation of magnetization dynamics at femtosecond timescales, where conventional LLG dynamics is insufficient. The moment-of-inertia term, negligible at long times, becomes essential for capturing nutation and transient dynamics at sub-picosecond intervals. The parameter-free nature of the model (with all terms accessible from DFT or Green’s function calculations) permits quantitative treatment of dissipation and inertial phenomena in realistic material contexts.

Such models are directly suited for studying ultrafast magnetization switching, spin–orbit torque phenomena, or laser-induced demagnetization, where accurate representation of both energy relaxation (damping) and angular momentum dynamics (inertia) is critical. The generalized framework can naturally be extended to non-collinear, non-uniform, or even strongly disordered magnets via appropriate spatial generalization of the tensors and numerical evaluation of the requisite response functions.

6. Comparative Advantages and Limitations

Relative to purely phenomenological atomistic spin dynamics, the first-principles strategy:

• Supplies all key model parameters from underlying electronic structure, eliminating empiricism.
• Reveals clear physical origin of damping and inertia as consequences of retarded electron-mediated exchange.
• Fully incorporates spatial and temporal nonlocality when needed for detailed modeling.
• Enables parameter-free, material-specific predictions for ultrafast magnetization dynamics, justifying inclusion of inertial effects when comparing to experiment.

The principal limitation remains computational: accurate evaluation of retarded propagators and response tensors may be demanding, especially for large, complex, or low-symmetry materials, but ongoing advances in high-performance electronic structure methods and interpolation strategies continue to alleviate this barrier.

7. Impact and Applications

First-principles-based atomistic spin dynamic models are now an essential component in the predictive simulation of ultrafast spin phenomena, with direct impacts on the microscopic understanding and engineering of high-speed magnetic devices, optospintronic systems, and thermally driven magnetization processes. Their transferability, accuracy, and transparency distinguish them as a benchmark for ultrafast spin–lattice simulations in metallic, insulating, and low-dimensional magnetic materials.