Trajectory-Dependent Electronic Energy Losses in Ion Range Simulations

TL;DR

This work integrates a trajectory-dependent electronic stopping power model, the unified two-temperature model (UTTM), into the MDRANGE molecular-dynamics framework to predict ion range profiles with ab initio-informed precision. By combining the Langevin-type, tensorial friction of UTTM with MDRANGE's recoil-approximation and localized domain, the authors enable micrometer-scale simulations of thousands of ion trajectories across channeling and random geometries. Comparisons against scalar ESP approaches, rt-TDDFT data, and SRIM benchmarks demonstrate clear trajectory dependence of electronic energy losses and highlight the need for robust $oldsymbol{eta}(ar{ ho})$ parametrizations. The approach provides a computationally efficient platform for validating and refining energy-loss fittings against experimental ion-range measurements and for benchmarking trajectory-specific electronic stopping in complex materials.

Abstract

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.

Figures (6)

• Figure 1: Comparison of the rate of energy loss of 10 keV Si ions traveling along center-channel and off-channel $\langle 100 \rangle$ trajectories in an Si lattice, simulated using LAMMPS and MDRANGE. The ESP was modeled using the UTTM model with a quadratic $\alpha (\bar{\rho})$ function developed by Jarrin et al. jarrin_2021. The target was initialized at 0 K without thermal displacements. In the off-channel trajectory, the projectile ion was shifted toward a neighboring lattice atom by $50\%$ of the distance between them in the center-channel trajectory.
• Figure 2: (a) Illustration of a kinetic ion in a lattice, annotated with the radii used for the calculation of the ion's neighbors. The outer radius $R_{out}$ portrays the cutoff of the UTTM parametrization $\alpha(\bar{\rho})$. The inner radius $R_{in}$ portrays the user-defined cutoff radius implemented in our approximation for the calculation of full friction components of the UTTM model in MDRANGE. (b) The energy loss, as a function of distance traveled, of a single 25 keV Ni ion in the $\langle 100 \rangle$ center channel in Ni without thermal displacements. (c) Calculated integral range profile of 25 keV Ni ions along the $\langle 100 \rangle$ channeling direction in Ni, thermalized at 300 K, for varying $R_{in}$. Ranges were calculated using the method described in Sec. \ref{['sec:methods']} for 50,000 ion implants. We employed the Ni $\alpha(\bar{\rho})$ parameterization developed by Caro et al. tamm_2019_eph1, which has a cutoff of $5.0$ Å. Curves for $R_{in} = 3.0$ and $5.0$ Å coincide for the energy loss and range profile.
• Figure 3: Integral range profiles of 10 keV As ions in GaAs along the $\langle 100 \rangle$ principal channeling direction using different constant values for the temperature of the electronic system. The atomic system was thermalized to 300 K in all cases. The range profiles were calculated using the method described in Sec. \ref{['sec:methods']} for 100,000 ion implants, with the UTTM model for the ESP. The GaAs coupling function used here for the UTTM model was developed by Teunissen et al. teunissen2023.
• Figure 4: Calculated integral ion range profiles of 10 keV Si ions in diamond Si along the (a) $\langle 100 \rangle$, (b) $\langle 111 \rangle$, and (c) $\langle 110 \rangle$ channeling directions, and (d) along a random direction, using different models for the ESP. Each channeling plot contains the range profiles predicted using the UTTM model with two parametrizations, both produced by Jarrin et al. jarrin_2021, which differ in the functional form of the $\alpha (\bar{\rho})$ coupling function. Range profile predictions using the scalar ESP calculated using SRIM are plotted for all directions, and using scalar TDDFT calculated by Nuñez-Palacio et al. Nunez2025 for the channeling directions. In the $\langle 100 \rangle$ and $\langle 110 \rangle$ directions, the stopping corresponds to the center-channel, and in the $\langle 111 \rangle$ direction it corresponds to the “ half-center” channel. Note the different axes scales, particularly the logarithmic x-axis scale in (c).
• Figure 5: Calculated integral ion range profiles of 25 keV Ni ions in FCC Ni along the $\langle 111 \rangle$ channel and a random direction, calculated using different models for the ESP. The $\langle 111 \rangle$ channeling plot contains the predicted ion range profile using the UTTM model with the parametrization of ab initio TDDFT data produced by Caro et al. tamm_2019_eph1, and the scalar TDDFT data calculated by Ullah et al. ullah2018. Range profile predictions using the scalar ESP calculated using SRIM are plotted for both the channel and random directions.