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Trajectory-Dependent Electronic Stopping in Simulations of Self-Ion Ranges in Elemental Semiconductors

This paper presents a parametrized unified two-temperature molecular dynamics model for self-ions in elemental semiconductors that accurately reproduces trajectory-dependent electronic stopping values from rt-TDDFT simulations, thereby generating ion range profiles that closely match experimental data.

Original authors: Rafael Nuñez-Palacio, Glen Kiely, Artur Tamm, Andrea Sand

Published 2026-08-13

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Rafael Nuñez-Palacio, Glen Kiely, Artur Tamm, Andrea Sand

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to sneak a marble through a crowded, twisting hallway. If the marble rolls down the exact center of the corridor, it might glide smoothly, barely bumping into anyone. But if it veers even slightly toward the walls, it will crash into people, lose its speed, and stop much sooner. This is the basic idea behind how high-speed particles move through solid materials like the silicon chips inside your phone. Scientists call this "ion implantation," a process used to build the tiny circuits that power our modern world. To do this right, engineers need to know exactly how deep a particle will travel before it stops. However, the rules of this game are tricky. The particles don't just bounce off atoms like billiard balls; they also interact with a "sea" of electrons that surround the atoms. This interaction acts like invisible friction, slowing the particle down. The problem is that this friction isn't the same everywhere. It changes depending on whether the particle is gliding through an open channel or crashing into a dense crowd of atoms. Getting this wrong means your computer chip might not work, or your medical device could fail.

This paper tackles that tricky "invisible friction" by creating a new, smarter way to simulate it. The researchers, working with materials like silicon, germanium, and diamond, realized that old computer models were too simple. They treated the friction as if it were the same in every direction, like a blanket of fog that slows you down equally no matter which way you turn. But in reality, the "fog" is patchy and uneven. The team developed a new simulation tool called the Unified Two-Temperature Model (UTTM) that acts more like a high-definition map. Instead of a blanket, this map knows exactly how dense the electron "crowd" is at every single point the particle travels. They tested this new map against super-accurate, but incredibly slow, quantum physics calculations (called rt-TDDFT) and found that their new model could predict how far particles would travel with much higher accuracy than before. When they compared their results to real-world experiments with silicon, their new model matched the data almost perfectly, while the old models were often too optimistic about how deep the particles could go.

The Invisible Traffic Jam

To understand what these scientists did, let's picture a high-speed train (the ion) zooming through a tunnel (the crystal). In the old way of thinking, scientists assumed the train faced the same amount of air resistance no matter where it was in the tunnel. If the train was in the middle of the track, or if it was swerving near the wall, the air resistance was just a fixed number. But in the real world of atoms, the "air" is actually a cloud of electrons. When the train is in the middle of the tunnel (a "channeling" path), it flies through a low-density area where the electron cloud is thin. It glides fast and goes deep. But if the train hits a patch where the electron cloud is thick (near the atoms themselves), it slams into a traffic jam and stops abruptly.

The problem is that the old computer models used a "one-size-fits-all" friction rule. They didn't know the difference between the smooth, open highway and the crowded, bumpy backroads. This paper introduces a new set of rules that changes the friction based on exactly where the particle is. It's like upgrading from a simple speedometer to a GPS that knows the traffic conditions in real-time.

The New Map: Reading the Electron Cloud

The researchers focused on three materials: Carbon (in the form of diamond), Silicon, and Germanium. These are the building blocks of many electronic devices. They wanted to know: "If we shoot a particle at 100,000 electron volts (a specific high energy) into these materials, how deep will it go?"

To answer this, they didn't just guess. They built a detailed 3D map of the electron density for each material. Think of this map as a topographical chart, but instead of mountains and valleys, it shows "hills" of high electron density (near the atomic nuclei) and "valleys" of low density (the open channels between atoms).

They discovered that the friction the particle feels depends heavily on which "terrain" it is crossing:

The Out-of-Bond Region: This is the open valley between atoms. Here, the electron density is low, and the friction is relatively gentle.
The Bond Region: This is the space where atoms are holding hands. The density is higher, and the friction increases.
The Core Region: This is right next to an atom's nucleus. The density is massive, and the friction is intense.
The Deep-Core Region: This is a collision course. If the particle gets too close to the nucleus, the friction spikes dramatically.

The team created a special "coupling function" (a fancy math term for a rule that links density to friction) that changes its behavior depending on which of these four regions the particle is in. They tested this rule against incredibly precise quantum simulations (rt-TDDFT) that track every single electron's movement. The result? Their new rule matched the quantum simulations almost perfectly, capturing the subtle differences between a particle gliding down a channel and one crashing into a wall.

Why the Old Maps Failed

The paper explicitly points out where previous models went wrong. One popular model, called SRIM, is like a map that only knows about random, messy paths. It was built using data from particles that were bouncing around randomly, avoiding the clean channels. When scientists tried to use SRIM to predict how particles move down a clean channel, it failed. It assumed the friction was too high, predicting that particles would stop much sooner than they actually do.

Another issue was with older versions of the new model (the UTTM). These older versions used a "vacuum" density, which is like looking at an atom floating in empty space. But inside a solid crystal, atoms are squished together, and their electron clouds merge. The old models didn't account for this merging, leading to errors. The new paper fixes this by using a "bulk-fitted" density, which accurately reflects how the electron clouds look when the atoms are packed together in a real crystal.

The Results: A Perfect Match

When the team ran their new simulations, the results were impressive. They simulated millions of particle paths and compared the depth at which the particles stopped to real experimental data.

For Silicon: They shot 100 keV silicon ions into a silicon crystal. The old models (SRIM and previous UTTM versions) predicted the ions would stop at the wrong depth or spread out too much. The new model, however, produced a profile that matched the experimental data almost perfectly. It correctly predicted that the ions would travel deep into the crystal if they stayed in the channel, but would stop sooner if they hit the dense regions.
For Carbon and Germanium: They also mapped out the stopping power for diamond and germanium. They found that for germanium, the inner electrons (the "core" electrons) start to play a big role in slowing down the particle once it gets very close to the nucleus (within 1.3 Ångströms). This is a detail that older models missed.

The paper also noted that the friction isn't just about speed; it's about direction. A particle moving down the <110> channel in silicon behaves differently than one moving down the <001> channel. The new model captures these directional differences, whereas the old models treated all directions as if they were the same.

The Bottom Line

This paper doesn't claim to have solved every mystery in particle physics. It doesn't say that we now know everything about how electrons behave. Instead, it offers a much better tool for engineers and scientists. By creating a simulation that understands the "terrain" of the electron cloud, they have made it possible to predict exactly how deep ions will penetrate into materials like silicon and germanium.

This is crucial for making better computer chips. If you can predict exactly where the ions will stop, you can build transistors that are smaller, faster, and more reliable. The authors suggest that this method could be extended to other materials, especially those where the electron density changes drastically depending on the direction you look. In short, they've replaced a blurry, guesswork map with a high-definition GPS, ensuring that the next generation of technology is built on a foundation of precise knowledge.

Technical Summary: Trajectory-Dependent Electronic Stopping in Simulations of Self-Ion Ranges in Elemental Semiconductors

Problem Statement
Accurate prediction of ion range profiles is critical for semiconductor device fabrication, where implantation depth dictates material properties such as conductivity and mechanical strength. While ion stopping has been extensively studied, fundamental challenges remain regarding the transition from linear response regimes to stopping power peaks, the dynamic evolution of projectile charge states, and the highly anisotropic nature of electronic stopping. Traditional Molecular Dynamics (MD) simulations often rely on isotropic, empirical friction models (e.g., SRIM) or simple damping forces that fail to capture path-dependent effects and ignore electron–phonon interactions. Although Real-Time Time-Dependent Density Functional Theory (rt-TDDFT) offers high accuracy in calculating electronic stopping power, its computational cost restricts system sizes, making it impractical for direct large-scale ion range simulations. The Unified Two-Temperature Model (UTTM) attempts to bridge this gap by coupling classical ions to an electron bath, but previous parameterizations for semiconductors (specifically Silicon and GaAs) have struggled to accurately reproduce stopping in both low-density channeling regions and high-density collision-like regions simultaneously.

Methodology
The authors develop a refined parameterization of the UTTM for self-ions in elemental semiconductor crystals (Carbon, Silicon, and Germanium) to reproduce rt-TDDFT electronic stopping values across various trajectories.

rt-TDDFT Calculations: The study first establishes accurate electronic stopping power data using rt-TDDFT. The authors systematically determined the necessary pseudopotentials for each element:
- Carbon: An all-electron ONCV pseudopotential (C6).
- Silicon: Si4 (4 explicit electrons) for standard conditions and Si12 (12 explicit electrons) for close approaches (<0.8 Å) or high energies (>200 keV).
- Germanium: Ge4 (4 electrons), Ge14 (14 electrons), and Ge22 (22 electrons). The study found that Ge22 is required to capture core-electron contributions (3s, 3p, 3d) during close collisions and at energies above 200 keV.
- Functionals: The Perdew–Burke–Ernzerhof (PBE) functional was selected over LDA, as LDA was found to overestimate stopping in Germanium.
- Trajectories: Calculations were performed for four major centered channeling directions ( $\langle 110 \rangle, \langle 001 \rangle, \langle 112 \rangle, \langle 111 \rangle$ ), half-centered and off-centered channels, and two incommensurate paths (one involving a close collision).
UTTM Parameterization:
- Electron Density ( $\rho$ ): Instead of using vacuum atomic densities, the authors iteratively fitted spherical radial density functions to match the 3D ground-state DFT electron density of the bulk crystal. This ensures the MD density map aligns with DFT results.
- Coupling Function ( $\alpha$ ): A piecewise coupling function $\alpha(\rho)$ $α (ρ)$ was developed, divided into four distinct density regions based on local geometry:
  - Out-of-bond: Low density, contributions from $\ge 3$ equidistant neighbors (e.g., $\langle 110 \rangle$ channel).
  - Bond: Intermediate density, contributions from 2 neighbors (e.g., $\langle 001 \rangle$ channel).
  - Core: High density, single dominant neighbor (e.g., off-centered $\langle 110 \rangle$ ).
  - Deep-core: Very high density, collision-like events (e.g., incommensurate path A).
- The coupling values for each region were optimized to minimize the error between UTTM and rt-TDDFT stopping powers.
Range Simulations: Ion range profiles were generated using the MDRANGE code, implementing the new UTTM parameterizations. Simulations were conducted for self-ions in C, Si, and Ge along principal channels and random (polycrystalline) directions at 4 K to maximize channeling effects.

Key Contributions and Results

Accurate Stopping Reproduction: The new parameterization successfully reproduces rt-TDDFT electronic stopping values for all tested trajectories, including low-density channels and high-density collision events. The deviation between UTTM and rt-TDDFT is generally below 10%, a significant improvement over previous models which showed errors up to 100% in certain regimes.
Germanium Characterization: The study characterizes electronic stopping in Germanium, revealing that while valence electrons govern channeling, the entire $n=3$ shell becomes active when the ion-target separation drops below 1.3 Å. This contrasts with Silicon, where core participation is only significant below 0.8 Å.
Ion Range Validation: For Silicon self-ions at 100 keV, the UTTM predictions closely match experimental concentration profiles. The model correctly captures the deep penetration of channeled ions and the concentration peak near the surface, outperforming SRIM and previous UTTM parameterizations which either underestimated or overestimated ranges depending on the trajectory.
Trajectory Dependence: The results highlight that stopping power is not uniform; it varies significantly based on the local electron density sampled by the trajectory. The study demonstrates that using a density-dependent coupling function is essential to capture these variations, whereas constant or scalar stopping models fail to reproduce the correct range profiles, particularly for channeled ions.

Significance
The paper claims that this work demonstrates the feasibility of transferring the full trajectory-dependent variability of rt-TDDFT electronic stopping calculations into large-scale molecular dynamics simulations. By resolving the density-dependent behavior of stopping from low-density channels to high-density nuclear regions, the proposed method improves the accuracy of general MD simulations where electronic energy dissipation is a key factor. The authors suggest that this geometry-based coupling approach can be extended to other semiconductors and insulators where dissipation varies strongly among different local environments (bonds, interstitials, and atomic cores). The study provides a validated framework for predicting ion ranges in elemental semiconductors that aligns with experimental data, addressing the limitations of isotropic friction models.