## 3 Results
### Impurity-Cluster Binding Energies  
The reliability of our calculations for Cu clusters is based on the tests described above. While for VASP the transferability of the pseudo potentials is well established, this is not the case for other methods. Especially, for SIESTA [33] the employed pseudo potentials have to be tested in well-known Al-Cu binding configurations of Al\({}_{2}\)Cu as in ref. [34]. To begin with, we give in Table 3 the binding energies for a pair of Cu atoms with respect to two separate Cu atoms. These energies are obtained by optimizing the lattice constant for Al for each supercell size and \(\mathbf{k}\)-point mesh. All the supercell sizes lead to a binding energy of around \(50\,\mathrm{meV}\). For the 128 atom supercell the binding energy is the smallest one reflecting the small spacing between the adjacent Cu habit planes of the periodic images and the ensuing artificial interaction. We will give the total binding energies of two- (2D) and three-dimensional (3D) copper clusters as the energy gain relative to separated Cu atoms in aluminum. From this we calculate the binding energy also per Cu-atom, i.e., the average over the cluster. We calculate also the binding energy of the last Cu-atom attached to a cluster, which indicates, if it is energetically favorable for an already existing cluster to grow further by attaching another Cu atom. This energy has to be compared to the thermal energy at room temperature of \(3/2\,\mathrm{k}T=40\,\mathrm{meV}\). The construction scheme of the 2D Cu-platelets on the \(\{100\}\)-plane of the fcc Al-lattice is shown in Figure 1. It is based on well-established experimental facts on Cu-platelet formation [3; 5; 6; 39]. Firstly, we have performed calculations on small agglomerates up to 4 Cu atoms. The results are presented in Table 4. We observe that two Cu-atoms on the 1 NN positions are bound together with a binding energy of about \(50\,\mathrm{meV}\), while there is a weak binding of 2 NN Cu-atoms of about \(10\,\mathrm{meV}\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Supercell Size (atom)** & \(\Delta H_{\text{mix}}\) **(meV)** \\ \hline
108 & \(-50.5\) \\
128 & \(-54.2\) \\
144 & \(-53.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Solubility enthalpy \(\Delta H_{\text{mix}}\) of Cu in Al calculated by using Equation (2) and different supercell sizes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Scheme** & **Number Atoms** & **Size Unit Cells** & **k-points** & \(E_{\text{bind}}\) **(meV)** \\ \hline LDA & 108 & \(3\times 3\times 3\) & \(4\times 4\times 4\) & \(-50.3\) \\ LDA & 128 & \(4\times 4\times 2\) & \(4\times 4\times 8\) & \(-46.3\) \\ LDA & 144 & \(3\times 3\times 4\) & \(4\times 4\times 4\) & \(-56.2\) \\ LDA & 192 & \(4\times 4\times 3\) & \(3\times 3\times 6\) & \(-54.7\) \\ GGA & 108 & \(3\times 3\times 3\) & \(6\times 6\times 6\) & \(-51.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Binding energy of two Cu solute atoms in Al on nearest neigbor positions in fcc Al. The c-direction is perpendicular to the habit plane of the Cu atoms. Negative signs indicate binding. as well. However, the most important result is that 2D agglomerates of 4 Cu on the \(\{100\}\)-plane of the fcc Al are preferred instead of the 3D tetrahedron structure (h) with an energy difference of 259 meV for the triangle configuration (g) and with 398 meV for the rectangle configuration (f) in Table 4 (cf. also Figure 1). Experimentally, there is, since the early investigations by [PERSON] and [PERSON] [5; 6] a long-standing agreement on the fact that Cu prefers to precipitate as 2D platelets on the \(\{100\}\)-planes of the fcc Al [40; 41]. Recently, this has been even confirmed by HR-TEM [39] showing mono-atomic platelets on the three equivalent \(\{100\}\)-planes in the fcc lattice. Our results presented in Table 4 confirm this perception. Of the triangular structures (d) and (e) the platelet structure (d) in the \((100)\)-plane is favored by nearly 40 meV compared to the platelet structure (e) on a \((111)\)-plane.