Early Stages of Low-Energy Pattern Formation in Ar Cluster Bombardment

José Carlos Jiménez-Sáez ^1,* , Sagrario Muñoz ² , Pablo Palacios ¹ 

1. Dept. of Applied Physics in Aeronautical and Naval Engineering, ETSIAE, Universidad Politécnica de Madrid (UPM), 28040 Madrid, Spain

2. Dept. of Structure of Matter, Thermal Physics and Electronics, Faculty of Physical Sciences, Universidad Complutense de Madrid (UCM), 28040 Madrid, Spain

* Correspondence: José Carlos Jiménez-Sáez 

Academic Editor: Desmond K. Loke

Special Issue: Molecular Dynamics Simulations: Bridging Theory and Real-World Applications

Received: March 04, 2026 | Accepted: April 20, 2026 | Published: April 24, 2026

Recent Prog Sci Eng 2026, Volume 2, Issue 2, doi:10.21926/rpse.2602006

Recommended citation: Jiménez-Sáez JC, Muñoz S, Palacios P. Early Stages of Low-Energy Pattern Formation in Ar Cluster Bombardment. Recent Prog Sci Eng 2026; 2(2): 006; doi:10.21926/rpse.2602006.

© 2026 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

When the flat surface of a solid is subjected to ion bombardment at oblique incidence, periodic modulations or surface ripples develop. The materials produced in this manner are nanostructured and have promising applications across diverse fields such as photonics, electronics, and new energy, due to their unique properties. This ion beam-induced patterning effect was first observed by Cunningham [1] on Au using 8 keV-Ar⁺ at an off-normal incidence of 70°. The first model proposed to explain this phenomenon was the Bradley-Harper model [2], based on Sigmund’s sputtering theory, which posited that surface instability arises from higher sputtering in the troughs than on the crest, the latter being mitigated by diffusion-based surface relaxation. Carter and Vishnyakov introduced the impact-induced mass redistribution of recoiling atoms in the first phase of bombardment to address the shortcomings of the BH theory in predicting pattern formation at low angles of incidence. Norris et al. [3] developed the formalism of the crater function, which recognized the roles of sputtering and ion-induced mass redistribution, thereby allowing the prediction of the surface morphology of patterns and the corresponding wavelength of these periodic structures. This algorithm has recently been placed on a rigorous mathematical foundation [4], enabling its use in the study of ripple formation for ion energies below the sputtering yield threshold by deriving the equation of motion for the surface [5].

Recently, low-energy Si bombardment simulations (30 eV) have been performed using molecular dynamics (MD) to understand ripple formation when sputtering is negligible, and only redistribution processes are involved [6,7]. The latter work specifically concludes that surface mass migration plays a fundamental role in ripple formation and that surface erosion can be neglected.

In this work, we study the phenomenon of very low-energy bombardment of a metallic substrate. We use clusters of very few atoms to accelerate the formation of surface ripples. The reader may find a review of this bombardment technique in reference [8]. As shown in the aforementioned article, the typical experimental cluster size and energy are usually several orders of magnitude higher than those used in this work. In previous works, we did not account for longitudinal effects in pattern formation, having used transverse bombardment [9,10]. In this work, we have included these effects by bombarding a Cobalt substrate with Ar clusters at 250 eV along the bombardment direction. We then focus on the effects that occur when the cluster energy is reduced to the point where sputtering becomes negligible at grazing angles.

2. Model

In this study, a face-centered cubic (FCC) Co surface oriented in the <110> direction was bombarded. The cobalt substrate was selected due to its wide range of magnetic applications, among others [11]. The crystallographic axis [001] corresponded to the y-axis (see Figure 1), and the two other directions <110> corresponded to the remaining two Cartesian axes. The dimensions of the Co substrate in unit cells along the x, y, and z axes were 56 × 28 × 14 (14.0 × 9.9 × 3.4 nm). Periodic boundary conditions were applied in the x and y directions. The three bottommost layers of the solid along the z-axis remained fixed, and above them, several thermalization layers maintained a temperature of 300 K using a generalized Langevin equation [12].

Figure 1 Ar cluster and Co(110) surface before the first cluster was launched. The graph shows the impact area (pink), the region of fixed atoms (green), and the region of thermalizing atoms (blue).

The potential describing the interaction of the Co atoms was of the second-moment tight-binding type [13], except for short distances where it was coupled to the ZBL potential. The latter potential also models the interactions between Ar and Co atoms. The Ar clusters consisted of six atoms in an octahedral geometry without an atom in the center. The results are independent of the cluster shape, but not of its size, as described in a previous work [9]. The Lennard-Jones potential described in [14] allowed for the calculation of the forces between these atoms.

Initially, the surface was free of stress and prepared in accordance with other works [9]. A bombardment event with a cluster lasted 25 ps for bombardments at 250 eV and 20 ps for bombardments at lower energies. For the first 15 ps (10 ps for lower energies), the time step was 0.5 fs. After that time, this step was doubled because the collisional processes involved lower energies. At 20 ps (15 ps for lower energies), the system was scaled to 300 K and then relaxed for 5 ps to ensure the same temperature conditions when launching the next cluster. The system evolved in the microcanonical ensemble coupled to a thermal bath at the bottom of the substrate to maintain a constant temperature.

In this work, unlike previous studies [9,10], we analyzed the longitudinal effects of bombardment, namely the effects along the launch direction of the clusters. The chosen model attempts to minimize the temporal response of the surface in two ways: first, by using cluster bombardment, and second, by significantly shortening response times by avoiding bombardment of the entire surface and reducing the system to its fundamental characteristics. Bombarding a surface strip, rather than the entire surface, reduces simulation time by a considerable amount to achieve the same fluence. The impact of each cluster always occurred at the center of the substrate along the x coordinate. To achieve this, the reference frame was moved in the x direction. The y coordinate where the cluster’s geometric center was randomly chosen between -5 and 5 Å. In this way, the cluster could affect a longitudinal strip with 10 Å width. The implanted fluence was approximately 10¹⁶ atoms/cm² in all cases. This value was set by analyzing surface effects and the work of other authors [6]. At this point, it should be noted that patterns have been observed in Co for higher fluences and energies [15]. However, by concentrating the excitation of the surface, we obtain incipient surface ripples in that area.

The conditions of this work (low surface temperature) are such that the erosive (or collisional) regime predominates over the diffusive regime [16]. From a collisional perspective, the pattern-formation process involves two physical phenomena: sputtering, which removes atoms, and surface mass redistribution. To assess the magnitude of both physical processes, we evaluated two physical quantities: on the one hand, the sputter yield Y (the number of atoms that leave the solid per incident Ar atom), and on the other hand, the displacement of the substrate atoms in the horizontal direction [17], d_x. This quantity is calculated as follows:

\[ d_x=\sum_{i=1}^{N_d}(x_i-x_{i0}) \tag{1} \]

where N_d is the number of displaced atoms, and x_i0 and x_i are the initial and generic horizontal coordinates of atom i of the solid, respectively. In some cases, the vertical displacement was also calculated by replacing the x coordinate with the z coordinate in the previous formula.

3. Results and Discussion

3.1 Dependence on the Angle of Incidence

After bombardment, the final surface state should reflect a set of approximately periodic surface ripples. The model would be complete if these ripples disappeared at small angles of incidence. However, this effect has been observed only in the absence of sputtering effects in the process [7]. Figure 2 shows the final the surface state for the different angles of incidence. Note that the initial state would be the flat surface shown in Figure 1. The craters created in the lattice are clearly visible in Figure 2c. These craters are interspersed with ridges. At grazing angles, the craters become shallower and fewer in number. Thus, at 70°, the ripples are so faint that they appear not to have been nucleated. However, we observed that doubling the fluence results in these ripples emerging, even though surface erosion is already considerable. A more interesting case is shown in Figure 2b, corresponding to 40°. In this graph, one can observe how the number of craters increases and how they coalesce. However, this increase can lead to a poorly defined surface structure, i.e., crater collapse. This collapse may lead to the disappearance of the surface pattern structure through other diffusion processes, in addition to those simulated by molecular dynamics. This collapse is not evident in Figure 2a, which corresponds to 30°. However, in this graph, a clear aperiodicity of the surface ripples is observed.

Figure 2 xz projections of the bombarded longitudinal strip for different angles of incidence: 30° (a), 40° (b), 50° (c), 60° (d), and 70° (e). The crystallographic viewing direction is <100>. A surface line has been drawn with atoms distinguished by color to guide the eye.

The sputtering yield primarily determines the surface morphology. Figure 3 shows the dependence of this quantity on the number of clusters for different angles of incidence. The x-axis shows the number of clusters launched, which corresponds to the simulation time (1 cluster lasts 25 ps). Consequently, it is also a measure of fluence. The dependence of this quantity on the angle of incidence was discussed in a previous work [9] and shows a clear maximum at 60°. Sigmund’s theory of sputtering of amorphous targets states that the sputter yield decreases from a maximum at grazing angles. However, the exact values at which this occurs depend on the material's crystallinity [18]. In this case, its value over time is highest at 50°, and therefore, this surface exhibits the largest surface craters. The cumulative sputtering would also explain the result for the 70° curve, where erosion begins very slowly and increases to values close to those of the other curves. This suggests that, for the fluence used, the sputtered fraction was lower, but from this point onward, extraction returns to normal values. This would explain why patterns are not observed until higher fluences are reached for this angle of incidence.

Figure 3 Sputter yields as a function of the number of clusters for different angles of incidence.

The redistribution of atoms is another factor that determines the surface configuration. Figure 4 shows the horizontal displacement for different angles of incidence. However, longitudinal effects differ from transverse effects. When analyzing the latter, we found a maximum displacement at angles of 50° [9]. Now, when describing longitudinal effects, this maximum disappears, and this quantity exhibits a monotonically decreasing dependence on the angle of incidence. When the damage function ceases to be local and extends beneath a surface, it displaces more atoms over a greater distance. Notably, the horizontal displacement of the 70° curve is the smallest of all. This curve failed to produce surface ripples. Therefore, we attribute the absence of these ripples to both the low sputtering yield and the low redistribution. The behavior of these curves is similar to that found in other work on Si and is not correlated with the existence of a critical angle [6]. The decreasing slope of the curves shows reduced momentum transfer in the cluster beam direction. The experimental work on the subject focuses on investigating the compositional morphology of ripples and is fundamentally centered on Si [19].

Figure 4 Horizontal displacements in the strip as a function of time for different angles of incidence.

The analysis of volume conservation also sheds light on the relative importance of the phenomena involved during the bombardment [10,20]. We disregard the volume of the implanted Ar atoms, since the fluence is small and, therefore, the volume they occupy is negligible in relation to the substrate atoms. Any atom that formerly occupied a volume in an eroded area must have been extracted by sputtering or redistributed within the solid. That is: V_em = V_sp + V_rd, where V_sp is the volume occupied by the sputtered atoms, and V_rd is the redistributed volume. As in previous works [10], we define the redistribution factor as follows:

\[ F_{rd}=\frac{V_{rd}}{V_{em}} \tag{2} \]

This factor measures the specific weight of redistribution in the process relative to sputtering. In Figure 5a, we show the eroded volume of the solid for different angles of incidence. This extracted volume would include, on the one hand, the volume of the sputtered layers and, on the other hand, the volume of the craters created in the strip. In this graph, the substrate with the greatest volume loss is the one corresponding to 50°. In fact, the extracted volumes are ordered by angle of incidence according to the cumulative sputtering yield obtained in Figure 3. Thus, the lowest values occur for the 60° and 70° cases. Both the sputtered volume and the redistributed volume of the strip zone exhibit the same functional dependence on time as the total vacated volume; they differ only in their relative ordering for the different angles of incidence. The sputtered volume curves are ordered from highest to lowest with the angle of incidence in the same way as the sputtering yield (Figure 3) and the redistributed volume curves in the same way as the horizontal displacement (Figure 4).

Figure 5 (a) Emptied volume normalized by the volume of the Co unit cell (a³) and (b) redistribution factor in the strip as a function of time for different angles of incidence.

In Figure 5b, we presented the system redistribution factor for the different angles of incidence. When longitudinal effects were not taken into account, this factor indicated a higher weight for sputtering for values below 0.35 [10]. Now, taking these effects into account, we see that this factor is on the order of 0.5. In any case, for the 60° and 70° cases, the predominant factor appears to be sputtering, while in the remaining cases, it is redistribution.

3.2 Pattern Formation in the Absence of Sputtering

In this work, we analyze whether the effects of mass redistribution are sufficient to provoke a pattern formation phenomenon when erosion is practically negligible, specifically when the energy of the Ar atoms is close to the sputtering threshold. Recent work [7] has succeeded in forming surface patterns by bombarding Si with energies (30 eV/atom) such that the sputtering was negligible; thus, only the mass redistribution effect was responsible for the formation of these structures. Although the Si lattice is different from that of a metal, since ion bombardment is a process far from equilibrium, it is not clear whether the results in metals will differ significantly. This work aims to determine whether it is possible to form such structures in metals in the absence of sputtering, considering only redistribution effects. To examine this circumstance, we have bombarded our system at an angle of incidence of 60° with projectiles at different energies. The results are shown in Figure 6. This graph shows that, for the fluence used, surface ripples are not observed at energies below 100 eV/atom. The simulation at 50 eV/atom was continued up to twice the fluence used elsewhere in this work, without finding any appreciable changes.

Figure 6 xz projections of the bombarded longitudinal strip for different bombardment energies: 50 eV/atom (a), 100 eV/atom (b), 150 eV/atom (c), 200 eV/atom (d), and 250 eV/atom (e). The crystallographic viewing direction is <100>. A surface line has been drawn with atoms distinguished by color to guide the eye.

We must evaluate how the two fundamental parameters that measure the phenomena associated with pattern formation decrease to determine their influence on the process.

Regarding the sputter yield, Table 1 shows that it decreases almost proportionally with energy. According to Sigmund’s sputter yield theory, this quantity is directly proportional to the depth distribution of deposited energy. Therefore, as long as there is energy deposited near the surface, extraction of atoms will occur [21]. Table 1 shows the sputter yield as a function of the cluster atom's energy. Furthermore, at a sputter yield of approximately 0.25 atoms/ion, patterns are no longer formed for the fluence considered here. Even more notable are the horizontal displacement curves shown in Figure 7a. The curves corresponding to 150 eV/atom and 100 eV/atom have similar displacements, and yet only the former shows ripples. This fact would rule out redistribution as the sole cause of the phenomenon. The vertical displacements are more revealing, as represented in Figure 7b. Specifically, the curves that show patterns exhibit an increasing trend or are about to. Small displacements outward from the surface appear to explain the formation of ridges [6].

Table 1 Sputter yields as a function of the energy of the atoms in the cluster.

Figure 7 (a) Horizontal displacement and (b) vertical displacement in the strip as a function of time for different projectile energies.

We have examined the role of sputtering in the process. To this end, Figure 8 shows the atoms that leave a given layer in the z-direction either by sputtering or by redistribution, as well as those that reach that layer by redistribution for two cases: one exhibiting pattern formation (200 eV/atom, Figure 8a) and another without (100 eV/atom, Figure 8b) [22]. Due to lattice damage, the relocated atom functions become wider in the former case, which influences the generation of internal voids, especially regarding sputtering. The graph shows that redistribution fills the innermost layers of the material, while sputtering and redistribution deplete the upper layers. However, it is in these upper layers where the fundamental differences appear: sputtering vacates more volume as energy increases. Furthermore, the curves for atoms relocated inward and outward have different slopes in the two cases. Consequently, at higher energies, more atoms are extracted at greater depths and fewer at shallower depths, while the inverse is true at lower energies. This tendency results in the accumulation of surface atoms, creating crests at high energies and the opposite at low energies; that is, it contributes to structure degradation similar to the explanation of ripple dissipation in [7]. Finally, sputtering predominates in the higher layers above the redistributed atoms at higher energies. However, given the net atomic balance, we must highlight sputtering as the primary mechanism that creates surface vacancies necessary for pattern formation. Thus, when this effect diminishes, the appearance of this surface effect through redistribution alone is no longer possible. Given the similarity between all the relocation and sputtering functions across different angles of incidence, we can assume that this holds for any angular range, regardless of whether the predominant effect is redistribution or sputtering.

Figure 8 Sputtered atoms, and outward and inward relocated atoms in the strip as a function of the z-position for the 200 eV/atom system (a) and the 100 eV/atom system (b). The graph also shows the net atomic balance as a function of position, which is obtained by subtracting the incoming atoms from the sputtered and outgoing atoms.

4. Conclusions

In this study, a Co(110) metal surface was bombarded with Ar clusters at different angles of incidence with an energy of 0.25 keV/atom. The system was also bombarded at lower energies until negligible sputtering was achieved. To accelerate the simulation, only a 10 Å-wide longitudinal strip of the target was bombarded. In all cases, a fluence of 10¹⁶ atoms/cm² was employed.

Low-temperature bombardment conditions rule out the influence of diffusive effects on the process. For the given fluence, surface ripples are excited throughout the angular range except at 70°. At this angle, cumulative sputtering and horizontal displacement are significantly lower than for the other angles of incidence. The vacated volume that contributes to ripple formation shows maxima for intermediate angles near 50°, as does cumulative sputtering. The redistributed volume exhibits the same angular dependence as the horizontal displacement and decreases with the angle of incidence. The redistribution factor indicates a preponderance of sputtering at grazing angles.

When the system is bombarded at lower energies, the sputter yield decreases proportionally with energy. Horizontal displacement also decreases, and this is not a key factor in the disappearance of ripples. In contrast, vertical displacement shows an increasing trend when ripples form. Finally, the study of the net balance of moving atoms allows us to conclude that sputtered atoms are primarily responsible for the vacating of the surface craters and, therefore, for the formation of patterns. For this reason, surface patterns are not observed at low energies in the material studied. In this work, a series of simplifications have been used, such as studying a single angle of incidence at low energy or disregarding cluster fragmentation, which should be addressed in future work.

Acknowledgments

This work was supported by the Universidad Complutense of Madrid under the Project for Research Groups (Bioelectromagnetism Research Group 910305).

Author Contributions

Methodology, J.C.J.-S.; Software, J.C.J.-S.; Investigation, J.C.J.-S, S.M. and P.P.; Writing – review & editing, J.C.J.-S, S.M. and P.P.; Funding acquisition, S.M.

Competing Interests

The authors have declared that no competing interests exist.