## 3 Results
### Impurity-Cluster Binding Energies
#### 3.2.2 2D-Cu-Clusters in 128- and 192-atom Supercells  
Lattice relaxations within the habit plane of the Cu-platelet cause an interaction between neighboring supercells. Hence, a 108-atom supercell (\(3\times 3\times 3\) unit cells) is already quite small to accommodate a Cu-platelet consisting of more than 5 atoms. Thus, we have repeated some of the calculations in enlarged supercells to keep the growing platelets farther apart from each other between the periodic images of the supercells.  
On the one hand, we employed a supercell of 128 atoms (\(4\times 4\times 2\) unit cells) increasing the lateral distance of the periodic images, while reducing the distance perpendicular to the platelets in the \(<001>\)-direction. On the other hand, a supercell of 192 atoms (\(4\times 4\times 3\) unit cells) increases the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Size (atoms)** & **Size (Unit Cell)** & **MP-mesh** \\ \hline standard & 108 & \(3\times 3\times 3\) & \(4\times 4\times 4\) \\ flattened & 128 & \(4\times 4\times 2\) & \(4\times 4\times 6\) \\ elevated & 144 & \(3\times 3\times 4\) & \(3\times 3\times 2\) \\ widened & 192 & \(4\times 4\times 3\) & \(2\times 2\times 3\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Used supercells with corresponding MP-meshes.  
Figure 2: Energy gain during Cu-cluster growth for a supercell of 108 atoms in sequences 1 and 2. (**left**) The energy gain due to the last attached Cu atom, (**right**) the total energy gain per Cu atom in the growing cluster. Sequences 1 and 2 are explained in Figure 1.  
lateral distance while keeping the distance in between Cu-platelets in the \(<001>\)-direction the same as for the 108-atoms cell (cf. Table 5). The results for these supercells are shown in Figures 3 and 4.  
For the flattened supercell of 128 atoms (see Table 5) the separation between the copper platelets in the habit plane increases compared to the supercell of 108 atoms, while the distance between the platelets and their periodic images becomes obviously too small. Figures 3 and 4 show that the energy gain by relaxation is significantly smaller for the supercell of 128 atoms, which artificially suppresses the energy-lowering relaxation perpendicular to the platelets (see Figure 5).  
The use of the supercell of 144 atoms, on the other hand, just increases the separation perpendicular to the copper platelets (see Table 5). Thus, the size of the platelets has to be limited to five Cu atoms and therefore this configuration is not considered further. Finally, the use of the supercell of 192-atoms (\(4\times 4\) lattice constants wide, but three lattice constants in height as well) reduces the interactions between copper platelets in the habit plane, while the separation of the platelets in the \(<001>\)-direction is the same as for the supercell of 108 atoms. Thus, this supercell gives the largest energy gain as seen from Figures 3 and 4.  
Since the energy differences are nevertheless small, i.e., less than 10% between the supercells of 108 and 192 atoms, we have chosen the computationally more feasible supercell of 108 atoms for the following discussion.  
Figure 4: Energy gain during Cu-cluster growth. The results corresponding to the supercells of 108, 128, and 192 atoms are compared. (**left**) Energy gain of the last Cu atom attached, (**right**) total energy gain per Cu-atom of a growing cluster. Sequence 2 is explained in Figure 1.  
Figure 3: Energy gain during Cu-cluster growth. Results corresponding to supercells of 108, 128, and 192 atoms are compared. (**left**) Energy gain due to the last Cu atom attached, and (**right**) total energy gain per Cu-atom of a growing cluster. Sequence 1 is explained in Figure 1.