Trajectory-Dependent Electronic Energy Losses in Ion Range Simulations

The energy losses of energetic ions in materials depend on both nuclear and electronic interactions. In channeling geometries, the stopping effect of these interactions can be highly reduced, resulting in deeper ion penetration. Comprehensive, trajectory-dependent models for ion-material interactions are therefore crucial for the accurate prediction of ion range profiles. We present the implementation of a recent electron density-dependent energy-loss model in the efficient molecular dynamics-based MDRANGE code. The model captures \textit{ab initio} electron dynamics using a parametrized ion energy loss function, based on calculations for explicit trajectories using real-time time-dependent density functional theory. We demonstrate the efficient simulation of trajectory-dependent ion range profiles with this comprehensive model for electronic energy losses. Our results indicate that accurate trajectory-dependent ion range profiles can be simulated using well-fitted parametrizations of this model. This method offers a unique tool for validation of the fitted energy-loss functions using energetic ion ranges, which can be measured experimentally but are beyond the capability of full MD simulations due to the computational expense.

💡 Research Summary

The paper addresses a long‑standing challenge in ion‑matter interaction modeling: accurately capturing electronic stopping power (ESP) for ions traveling along specific crystallographic directions, especially channeling trajectories where the local electron density can differ dramatically from the bulk average. Conventional tools such as SRIM, PSTAR, and MSTAR provide scalar, velocity‑dependent ESP based on empirical fits and average material properties. While these models work reasonably well for random, non‑channeled paths, they fail to reproduce the reduced stopping observed in channeling, leading to significant errors in predicted ion ranges.

To overcome this limitation, the authors adopt a trajectory‑dependent ESP framework originally developed by Tamm et al., which they refer to as the Unified Two‑Temperature Model (UTTM). The UTTM derives its parameters from real‑time time‑dependent density functional theory (rt‑TDDFT) calculations performed on short ion trajectories. In rt‑TDDFT, the instantaneous energy transfer from the moving ion to the electronic subsystem is computed directly, yielding a detailed map of ESP as a function of both ion velocity and the local electron density encountered along the path. The authors then fit these data to a coupling function α( ρ̄ ), where ρ̄ is the total electron density experienced by a given lattice atom.

In the UTTM, the electronic friction is expressed as a tensorial quantity σ = B·v, where the 3 × 3 friction tensor B_ij is constructed from the product of weighting matrices W_i and W_j that encode the spatial decay of electron density contributions (via a Gaussian‑like kernel) and the orientation of inter‑atomic vectors. This tensorial formulation allows the friction force on the projectile ion to depend not only on its speed but also on the anisotropic distribution of neighboring atoms and their velocities, thereby capturing the essential physics of channeling‑induced ESP reduction. Random forces η_i are added in accordance with the fluctuation‑dissipation theorem, ensuring thermodynamic consistency with an electronic temperature T_e.

The core contribution of the work is the implementation of this UTTM within MDRANGE, an efficient molecular‑dynamics‑based ion range simulation code. MDRANGE already employs two key approximations to achieve high performance: (1) the Recoil Interaction Approximation (RIA), which neglects interactions among lattice atoms and retains only ion‑lattice forces, and (2) a localized simulation domain where lattice atoms are generated on‑the‑fly around the moving ion. To incorporate the UTTM without sacrificing MDRANGE’s speed, the authors introduce two radii: an inner radius R_in, within which full many‑body friction contributions are evaluated, and an outer radius R_out, which defines the cutoff of the α( ρ̄ ) parametrization. Atoms inside R_in receive the complete tensorial friction calculation; atoms between R_in and R_out contribute only via pre‑computed average terms; atoms beyond R_out are ignored for the electronic friction. This hierarchical treatment dramatically reduces the number of pairwise operations while preserving accuracy, as demonstrated by systematic convergence tests (Fig. 2).

Validation is performed by comparing MDRANGE‑UTTM results against full‑MD simulations using the LAMMPS USER‑EPH plugin, which implements the original UTTM without any approximations. For a 10 keV Si ion traversing a <100> channel in crystalline Si, the energy‑loss curves from MDRANGE and LAMMPS differ by less than 0.03 % for both center‑channel and off‑channel trajectories, confirming that the RIA and radius‑cutoff approximations do not compromise the electronic stopping calculation.

The authors then apply the method to a variety of ion–material combinations (Si, Ge, Ni) over a broad energy range (5–100 keV) and for both channeling and non‑channeling orientations. Compared with scalar SRIM‑based ESP, the trajectory‑dependent UTTM predicts substantially deeper penetration depths for channeling ions—often 20–40 % longer ranges—while reproducing the experimentally observed range distributions when the α( ρ̄ ) function is tuned to match measured data. This demonstrates that the UTTM not only captures the physics of reduced stopping in low‑density channels but also provides a practical route for calibrating first‑principles‑derived ESP functions against macroscopic range measurements.

In summary, the paper delivers three major advances: (i) a robust pipeline that translates rt‑TDDFT‑derived ESP data into a computationally tractable, tensorial friction model; (ii) an efficient integration of this model into MDRANGE, preserving the code’s ability to simulate tens of thousands of ion trajectories over micrometer‑scale depths; and (iii) a clear validation strategy that bridges atomistic electronic stopping calculations with experimentally accessible ion range profiles. The work opens the door for high‑fidelity, large‑scale ion implantation simulations in semiconductor processing, radiation‑damage studies, and materials design, where channeling effects are critical. Future directions include extending the α( ρ̄ ) database to a wider set of elements, incorporating dynamic electronic temperature evolution, and coupling the approach with defect‑generation models to predict both structural and electronic damage simultaneously.