# Pattern Formation on Alloys by Cluster Ion Beam Irradiation
## 2 Model  
In this study, an Au-Cu alloy surface has been bombarded in the form of an fcc lattice oriented in the <110> direction. The crystallographic axis <001> aligns with the \(y\)-axis, while the two <110> directions coincide with the remaining coordinate axes. The number of unit cells of the Au-Cu substrate along the \(x\), \(y\), and \(z\)-axes was 56 \(\times\) 28 \(\times\) 14. The cell length depended on the composition and was taken from the reference [23]. The \(x\) and \(y\)-directions extended the crystal using periodic conditions. The initial system before the bombardment can be visualized in other works [20; 21]. For the most part, the details of the model are the same as those found in these works. The substrate was fixed, keeping the three lower layers fixed perpendicular to the \(z\)-direction. On top of these, there were five layers that thermally controlled the system using a generalized Langevin equation [24]. These layers absorbed heat to maintain the temperature at 300 K as in other works [20]. The second-moment tight-binding potential in the framework of the Finnis-Sinclair model described the interactions between Au-Cu atoms [25]. At short distances, approximately between 1.7 and 2.0 A depending on the material, this potential is smoothly coupled to the ZBL potential via a bond function. Ar-Cu and Ar-Au interactions are described by the ZBL potential for any distance. The bombarded surface is free from all stress. First, a periodic fcc network at 5 K with the desired composition is randomly generated. Subsequently, the system is heated to 300 K by scaling the temperature in six steps and stabilizing the system for 50 ps. In this process, the canonical ensemble is used with an inertia factor associated with temperature (adjustable parameter) equal to \(Q=10^{-36}\) g cm\({}^{2}\). The system is then relaxed at zero pressure for 300 ps in the canonical isobaric ensemble with the associated inertia factor (also adjustable) \(W=10^{19}\) g [26]. Finally, we remove periodic conditions along the \(z\)-axis, fixing the bottom and relaxing the system in the microcanonical ensemble coupled to a thermal bath. Each substrate-cluster collision lasted 25 ps. This time interval is similar to that used in other works [20; 21]. The time step was 0.5 fs for the first 15 ps. The step was then doubled since the collisional processes were less energetic. The system was scaled to 300 K at 20 ps and subsequently relaxed for 5 ps before the next cluster was launched. This ensured similar temperature conditions for each launch. In any case, the system evolves in the microcanonical ensemble coupled to a thermal bath. In each launch, the impact always occurred in the center of the solid in the \(y\)-direction, for which the frame of reference was moved. Statistically, the results should be independent of this coordinate. The launched clusters of six atoms had an octahedral shape (without an atom in the center). This is a fairly common shape, although the results do not depend on this fact, since the impacts always occurred randomly. Clusters were selected instead of atoms to accelerate the surface response and shorten the simulation time. Obviously, it is always possible to choose larger clusters; however, this could interfere with the boundary conditions. However, their influence has been studied in a previous article [20]. The potential that models the interaction in the clusters is the Lennard-Jones [27], taking the interatomic separation that minimizes its energy. The angle of incidence of the clusters varied between 30\({}^{\circ}\) and 70\({}^{\circ}\) to study their influence on specific parameters. A total of 336 clusters were launched on the target; that is, the surface was bombarded with approximately 2000 Ar atoms in total, each with 250 eV. Therefore, the influence of the number of implanted Ar atoms is very small. The bombardment simulated a linearly focused ion beam by randomly impacting the \(x\)-coordinate of the geometric center of the cluster within a range between \(-10\) and 0 A.