Heuristic Used in Simulations of Ions Passing Through Solids

In the labyrinthine world of condensed-matter physics, where particles dance and materials transform under duress, the binary collision approximation (BCA) emerges as a rather blunt instrument—a heuristic designed for the specific, and often tedious, task of efficiently simulating the penetration depth and the subsequent defect production caused by energetic ions as they hurtle through solid materials. These aren't just any ions; we're typically discussing those possessing substantial kinetic energies, often measured in the kilo-electronvolt (keV) range or even higher, for which a more granular, atom-by-atom approach would be computationally prohibitive. The core premise of this method is deceptively simple: the incoming ion's journey through the dense atomic lattice of a material is approximated as a discrete sequence of independent, pairwise collisions. Each interaction is treated as a distinct binary event between the swift ion and a stationary sample atom—or more precisely, its atomic nucleus. Between these well-defined collisional encounters, the ion is rather optimistically assumed to maintain a perfectly straight trajectory, its energy loss solely attributed to electronic stopping power. Crucially, within these inter-collisional segments, no energy is considered to be lost in collisions with nuclei, a simplification that allows for the remarkable computational efficiency that underpins the BCA's utility. This model, while a pragmatic compromise, has proven indispensable for understanding fundamental processes in fields ranging from materials science to nuclear engineering. [1] [2] [3]

[[File:BCA collision.svg|thumb|Schematic illustration of independent binary collisions between atoms. The incoming ion (purple) interacts sequentially with target atoms (red), altering its path and losing energy.]]

Simulation Approaches

Within the BCA framework, the very essence of a single collision between an incoming ion and a target atom (its nucleus, to be precise) is distilled down to the elegant, albeit somewhat laborious, solution of a classical scattering integral. This integral, a mathematical expression of the forces at play, is solved for the specific impact parameter of the incoming ion, which is effectively the perpendicular distance between the ion's trajectory and the center of the target atom if the ion were to continue unimpeded. The solution of this integral yields two critical pieces of information: the scattering angle, which dictates the new direction of the ion's travel, and the precise amount of energy lost by the ion to the sample atom during that interaction. Consequently, one can precisely determine the ion's energy after the collision relative to its energy before it. [1]

This classical scattering integral finds its definition most conveniently within the centre-of-mass coordinate system. Here, the complex interaction of two distinct particles can be mathematically reduced to the motion of a single effective particle under the influence of a shared interatomic potential. This potential, a function describing the repulsive and attractive forces between the two atoms as a function of their separation, is the cornerstone upon which the scattering dynamics are built.

The scattering angle, denoted by $\Theta$, is thus determined from this repulsive pair interatomic potential $V(r)$—where $r$ is the distance between the particles—as a function of the impact parameter $b$ by the following integral:

$\Theta = \pi - \int_{-\infty}^{\infty} {\frac {b\,dr}{r^{2}{\sqrt{1-{\frac {b^{2}}{r^{2}}}-{\frac {V(r)}{E_{CM}}}}}}}$

Here, $E_{CM}$ represents the total system energy within the center-of-mass coordinates. [4] From this calculated scattering angle, the energy transferred ($T$) from the incoming ion to the target atom can then be elegantly obtained using the conservation laws of momentum and energy, expressed as:

$T=4{\frac {M_{1}M_{2}}{(M_{1}+M_{2})^{2}}}E\cos ^{2}\theta _{2}$

where $\theta_{2}$ is defined as $\theta_{2}={\frac {\pi -\Theta }{2}}$. In this equation, $E$ denotes the initial energy of the incoming ion, which possesses a mass of $M_{1}$, and $M_{2}$ is the mass of the stationary material atom with which this particular collision event is occurring. This formula explicitly quantifies the kinetic energy imparted to the target atom, which subsequently recoils.

Beyond just the energy and angle, it is also entirely possible—and often necessary—to solve the time integral of the collision. This calculation reveals the precise duration that has elapsed during the interaction, a piece of information that becomes particularly crucial when the BCA is deployed in its "full cascade" mode, a more comprehensive simulation approach discussed further below. [3]

The energy loss attributable to electrons, known as electronic stopping power, is handled with a degree of flexibility within BCA simulations. This can be addressed through sophisticated models where the electronic stopping is explicitly dependent on the impact parameter of the ion's trajectory relative to the target electron cloud. [5] Alternatively, a simpler, more generalized approach subtracts a stopping power value that depends solely on the ion's velocity, applied continuously between the discrete nuclear collisions. [4] A hybrid approach, combining elements of both, is also commonly employed, allowing for a balance between accuracy and computational efficiency depending on the specific problem at hand.

The methodology for selecting the next colliding atom's position and its associated impact parameter fundamentally divides BCA codes into two primary categories: the "Monte Carlo" BCA codes and the more structurally aware crystal-BCA codes.

In what is somewhat misleadingly termed the "Monte Carlo BCA" approach—misleading because "Monte Carlo" can refer to a vast array of Monte Carlo simulation techniques—the distance to, and the impact parameter of, the next atom destined for a collision are chosen randomly. This selection is governed by a probability distribution that relies solely on the average atomic density of the material. This method, by its very nature, effectively simulates the ion's passage through a material that is considered to be fully amorphous, lacking any long-range crystalline order. Prominent examples of codes employing this approach include SRIM and SDTrimSP, which have become workhorses for many applications requiring quick, statistically robust results for disordered systems.

It is, however, also possible—though undeniably more complex and computationally demanding—to implement BCA methods for materials possessing a defined crystalline structure. In these "crystal-BCA" codes, the moving ion's precise position within the crystal lattice is tracked, and consequently, the distance and impact parameter to the next colliding atom are meticulously determined to correspond to an actual atom located at a specific, predictable site within that crystal. This sophisticated approach allows BCA to accurately simulate phenomena such as channelling, where ions can travel considerable distances along open crystallographic directions with minimal energy loss. Codes like MARLOWE are exemplary in their ability to operate with this level of structural detail.

Furthermore, the inherent flexibility of the binary collision approximation can be extended to model the dynamic evolution of a material's composition. This is particularly relevant under prolonged ion irradiation, where the continuous bombardment can lead to significant alterations in the local atomic ratios, driving processes such as ion implantation (the introduction of foreign ions into the host material) and sputtering (the ejection of host atoms from the surface). [6]

However, like any approximation, the BCA has its limits. At lower ion energies, the fundamental assumption of independent, isolated collisions between atoms begins to fray at the edges. The interactions become less distinct, and the probability of multiple, simultaneous, or closely overlapping collisions increases significantly. To some extent, this issue can be mitigated by modifying the collision integral to account for these more complex, multi-body interactions. [3] [7] Nevertheless, there is a hard boundary: at very low energies (typically below approximately 1 keV, though a more precise estimate can be found in reference [8]), the BCA approximation fundamentally breaks down. In these regimes, where the kinetic energy of the ions is comparable to or less than the binding energies of the atoms in the lattice, and interactions are truly collective, one must abandon the BCA in favor of more robust simulation approaches, specifically molecular dynamics (MD). MD simulations, by their very design, explicitly handle the many-body interactions of arbitrarily numerous atoms, providing a far more accurate representation of the chaotic atomic dance at low energies. These MD simulations can either focus solely on the incoming ion and its immediate primary recoils (a technique known as the recoil interaction approximation or RIA [9]) or, for a truly comprehensive picture, simulate all atoms participating in a collision cascade, capturing the full, intricate ripple effect through the material. [10]

BCA Collision Cascade Simulations

The realm of BCA simulations can be further delineated based on their scope: whether they merely track the journey of the initial incoming ion, or if they take the more ambitious route of also following the trajectories of the secondary, tertiary, and subsequent recoils produced by that ion. This latter, more comprehensive approach is known as the "full cascade mode," a feature famously implemented in popular BCA codes such as SRIM. When a code does not account for these secondary collisions, limiting its scope to only the primary ion's path, the number of resulting defects is then typically estimated using the Robinson extension of the well-established Kinchin-Pease model, a theoretical framework for calculating displacement damage.

The suitability of BCA for simulating these complex atomic disturbances, or collision cascades, is contingent upon specific conditions. If the initial recoil or ion mass is relatively low, and the material through which the cascade propagates possesses a low atomic density (meaning the combination of recoil and material results in a low stopping power), then the collisions between the initial recoil and the sample atoms will occur infrequently enough to be reasonably understood as a sequence of independent binary collisions. Under these ideal circumstances, a collision cascade can be theoretically well-described and accurately modeled using the BCA.

[[File:Linear collision cascade.svg|thumb|Schematic illustration of a linear collision cascade. The thick line indicates the material surface, while thinner lines depict the ballistic trajectories of atoms from their initiation until they come to rest. The purple circle represents the incoming ion. Red, blue, green, and yellow circles denote primary, secondary, tertiary, and quaternary recoils, respectively, showcasing the branching nature of the cascade. Between these ballistic collision events, the ions are assumed to travel in straight paths. BCA, particularly in its "full cascade mode," is well-suited to describe these linear collision cascades.]]

Damage Production Estimates

BCA simulations are inherently designed to provide a wealth of spatially resolved information. They naturally yield the ion's penetration depth into the material, its lateral spread from the point of incidence, and critically, the spatial distributions of both nuclear and electronic energy deposition. Beyond these fundamental kinematic and energetic outputs, BCA codes are also frequently leveraged to estimate the amount of damage induced within materials. This estimation typically relies on a straightforward, if somewhat simplistic, assumption: any recoil atom that receives an energy transfer exceeding the material's characteristic threshold displacement energy is presumed to produce a stable, permanent defect in the lattice.

However, this approach, while convenient, must be wielded with considerable caution, for several compelling reasons. Firstly, it entirely neglects the dynamic, thermally activated recombination of damage. In real materials, particularly at elevated temperatures or over extended periods, many initially created defects (such as closely spaced Frenkel pairs) can spontaneously annihilate, undoing the damage counted by the simulation. Secondly, and perhaps more significantly, the model often overestimates damage production, especially in metals at high energies. It is a well-established experimental fact that the actual damage produced in metals under energetic irradiation is often only around 20% of what the idealized Kinchin-Pease model predicts. [11] Moreover, this simplistic method operates under the assumption that all defects are isolated Frenkel pairs—a vacancy and an interstitial atom. In reality, especially in dense collision cascades, the initial damage state frequently manifests not as isolated point defects, but as complex defect clusters, extended loops, or even embryonic dislocations. [12] [13] To address these limitations, modern BCA codes can, and often do, incorporate more sophisticated damage clustering and recombination models, which significantly improve their reliability and predictive power in this regard. [14] [15] Finally, and perhaps most frustratingly for those seeking precision, the average threshold displacement energy itself is notoriously difficult to determine with high accuracy for most materials, introducing an inherent uncertainty into any damage calculation built upon it.

BCA Codes

Several specialized software packages implement the binary collision approximation, each with its own strengths, weaknesses, and intended applications.

• SRIM: Standing for "Stopping and Range of Ions in Matter," SRIM is arguably the most widely utilized BCA code currently available, largely due to its user-friendly graphical user interface (GUI) and broad applicability. It is capable of simulating linear collision cascades in amorphous materials for virtually any ion and any material, handling ion energies extending up to an impressive 1 GeV. However, despite its widespread popularity, it is crucial to recognize SRIM's inherent limitations. It does not account for critical effects such as channelling in crystalline structures, nor does it typically model damage directly resulting from electronic energy deposition—a phenomenon that is absolutely necessary for accurately describing swift heavy ion damage in many materials, particularly insulators. Furthermore, damage produced by excited electrons, a non-negligible factor in some scenarios, is also outside its purview. Users should also be aware that the sputter yields calculated by SRIM may, in certain cases, be less accurate compared to those obtained from other, more specialized codes. [17]

• MARLOWE: In stark contrast to SRIM's amorphous focus, MARLOWE is a significantly more extensive and complex code renowned for its ability to meticulously handle crystalline materials. Its sophisticated architecture supports a vast array of different physics models, allowing for highly detailed simulations that can capture subtle effects like channelling and crystallographic dependencies in ion-solid interactions. This makes MARLOWE an invaluable tool for researchers delving into the fundamental physics of ion beam modification of single-crystal materials. [2[ [3]

• TRIDYN / SDTrimSP: Initially developed as TRIDYN, and with newer, more advanced versions known as SDTrimSP, this BCA code distinguishes itself through its unique capability to accurately simulate dynamic composition changes within a material. This feature is particularly vital for understanding processes where the material's stoichiometry evolves significantly under ion bombardment, such as during prolonged ion implantation or aggressive sputtering processes, where surface atoms are continuously ejected and implanted ions accumulate. [6] Its ability to track these changes makes it indispensable for applications in thin-film deposition, surface modification, and semiconductor processing.

• DART: A French code developed by the CEA (Commissariat à l'Energie Atomique) in Saclay, DART offers an alternative perspective on BCA simulations. It diverges from codes like SRIM in its specific treatment of electronic stopping power and its analytical resolution of the scattering integral. While the nuclear stopping power is derived from the widely accepted universal interatomic potential (the ZBL potential), its electronic stopping power relies on Bethe's equation for protons and the Lindhard-Scharff model for heavier ions. The amount of defects produced by DART is determined through calculations based on elastic cross sections and the atomic concentrations of the constituent atoms, providing another robust approach to assessing radiation damage.

See also

• Collision cascade
• Molecular dynamics
• Stopping power (particle radiation)