Multiscale Numerical Modelling of Ultrafast Laser-Matter Interactions: Maxwell Two Temperature Model Molecular Dynamics (M-TTM-MD)

In this work, we present a comprehensive numerical framework that couples numerical solutions of Maxwell’s equations using the Finite-Difference Time-Domain (FDTD) approach, Molecular Dynamics (MD), and the Two-Temperature Model (TTM) to describe ultrafast laser-matter interactions in metallic systems at the atomic scale. The proposed Maxwell-Two-Temperature Model-Molecular Dynamics (M-TTM-MD) bridges the gap between electromagnetic field propagation, electron-phonon energy exchange, and atomic motion, allowing for a self-consistent treatment of energy absorption, transport, and structural response within a unified simulation environment. The calculated electromagnetic fields incorporate dispersive dielectric properties derived using the Auxiliary Differential Equation (ADE) technique, while the electronic and lattice subsystems are dynamically coupled through spatially and temporally resolved energy exchange terms. The changes in the material topography are then reflected in the updated grid for the FDTD scheme. The developed M-TTM-MD model provides a self-consistent numerical framework that offers insights into laser-induced phenomena in metals, including energy transport and surface dynamics under extreme nonequilibrium conditions.

💡 Research Summary

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The paper introduces a comprehensive multiscale simulation framework—named M‑TTM‑MD—that self‑consistently couples three numerical methods to describe ultrafast laser–matter interactions in metals at the atomic level. The electromagnetic field propagation is solved with a Finite‑Difference Time‑Domain (FDTD) scheme on a Yee grid, incorporating dispersive material response via an Auxiliary Differential Equation (ADE) formulation. The dielectric function is made temperature‑ and frequency‑dependent, allowing real‑time updates of optical constants as the electron temperature rises during the laser pulse.

Energy absorbed from the electromagnetic wave is evaluated as the divergence of the Poynting vector (Qₗₐₛ = –∇·S) on the fine FDTD mesh. Because the FDTD cell size is typically smaller than the Two‑Temperature Model (TTM) cell, the local energy flux is volume‑averaged over groups of FDTD cells before being transferred to the corresponding TTM cell. This guarantees energy conservation across different spatial resolutions.

The TTM solves the electron heat diffusion equation Cₑ(Tₑ)∂ₜTₑ = ∇·(Kₑ∇Tₑ) – G(Tₑ – Tₗ) + Qₗₐₛ, where Cₑ and Kₑ are temperature‑dependent electron heat capacity and conductivity, and G is the electron‑phonon coupling factor. The lattice temperature Tₗ evolves through coupling to the electron subsystem and through direct energy exchange with the atomistic Molecular Dynamics (MD) module.

MD is performed with Embedded‑Atom Method (EAM) potentials, integrating Newton’s equations of motion. A coupling term ξ, proportional to the local electron‑lattice temperature difference, injects the electron‑phonon energy into the atomic degrees of freedom without requiring an explicit lattice heat capacity. Atomic kinetic temperature and pressure are extracted via the equipartition theorem and the virial theorem, respectively, providing thermodynamic feedback to the TTM.

A crucial feature of the framework is the feedback loop: as the MD simulation modifies the surface topography and local density, the material’s relative permittivity εᵣ in the FDTD domain is updated (εᵣ = ρ_mat/ρ₀ · ε_mat). Consequently, the scattering and absorption of subsequent portions of the laser pulse change, which in turn alters the source term Qₗₐₛ for the next TTM update. This loop continues throughout the pulse and across multiple pulses, enabling the study of cumulative effects such as Laser‑Induced Periodic Surface Structures (LIPSS), spallation, and evaporation.

Temporal scales are carefully synchronized: the FDTD timestep is constrained by the Courant condition to ~10⁻³ fs, the TTM timestep to ~10⁻² fs, and the MD timestep to tens of femtoseconds. A counter variable aligns the three solvers, ensuring that the energy deposited by the electromagnetic field is correctly mapped onto the electron temperature field before the MD step proceeds.

Parallelization is achieved with MPI domain decomposition. Each processor holds a sub‑domain of the FDTD, TTM, and MD grids, with “skin layers” that store one‑cell‑wide ghost regions for fields and atoms. Data exchange occurs every timestep for the FDTD curl updates and every few timesteps for MD forces, allowing the code to scale to simulations containing millions of atoms and large electromagnetic domains.

The authors validate the method by simulating femtosecond laser irradiation of a gold surface. They demonstrate that the model captures the rapid rise of electron temperature, subsequent lattice heating, melting, and the formation of periodic surface ripples whose spacing matches the predicted plasmon‑interference wavelength. Energy balance checks confirm that total absorbed laser energy equals the sum of electronic, lattice, and kinetic contributions within numerical tolerance.

In summary, the M‑TTM‑MD framework bridges the gap between macroscopic electromagnetic wave propagation, mesoscopic electron‑phonon heat transfer, and microscopic atomic dynamics. By dynamically updating material optical properties and incorporating surface‑topography feedback, it provides a powerful tool for predictive modeling of ultrafast laser processing, enabling quantitative studies of energy transport, phase transitions, and nanostructure formation under extreme nonequilibrium conditions.