# The Correlation Factors and Mechanisms of Diffusion for P and S in the Cu Single Crystal
## 3 Results  
### Vacancy Formation Energies and Vacancy-Impurity Interactions  
The vacancy formation energies (\(\Delta E_{\text{vac}}\)) and the binding--or absorption--energies (\(\Delta E_{\text{FS}}\)) of P and S to the Cu lattice, and the interaction energies between P and S with a vacancy (\(\Delta E_{\text{PS}}\)) at the nearest neighbor site are shown in Table 1.  
The vacancy formation energies of Cu, the absorption energies of P and S and the interaction energies between P, S and the vacancies agreed well with our and other authors' previously published data [33; 34; 35; 36]. The absorption energies \(\Delta E_{\text{P}}\) and \(\Delta E_{\text{S}}\) were \begin{table}
\begin{tabular}{c c c c c} \hline \(\Delta E_{\text{P}}\) & \(\Delta E_{\text{S}}\) & \(\Delta E_{\text{vac}}\) & \(\Delta E_{\text{vac}\text{-P}}\) & \(\Delta E_{\text{vac}\text{-S}}\) \\ \hline \(-\)0.653 & \(-\)0.511 & 1.176 & \(-\)0.229 & \(-\)0.432 \\ \hline \end{tabular}
\end{table} 
Table 1: Vacancy formation energies (\(\Delta E_{\text{vac}}\)) for Cu and interaction energies between P and S with a vacancy at the nearest neighbor site (\(\Delta E_{\text{vac}\text{-PS}}\)). Values in eV. presented here for contextualization but we alert the reader to the large uncertainty of these values because of the computation of single S and P atoms in a vacuum, especially because these two elements have different spin states and DFT cannot account fully for these configurations with accuracy [37; 38]. Considering the underlying uncertainties, the \(\Delta E_{\text{P}}\) and \(\Delta E_{\text{S}}\) of fcc Cu were approximately the same when no vacancies are near. However, when the vacancies were the nearest neighbors to the impurities, this picture changes. Both P and S could stabilize a vacancy as the nearest neighbor, which is the result of an attractive interaction between these elements and the vacancy. This has also reflected in the jumps that involve the impurities whose activation energies follow the Bronsted-Evans-Polanyi (BEP) principle [39], as will be shown later. As expected, \(\Delta E_{\text{vac\_S}}\) was more negative than \(\Delta E_{\text{vac\_P}}\)[40]. This was also observed for the segregation of these two elements at the grain boundaries of Cu where S benefited more from bonding with substitutional sites with a larger volume as compared to P [26; 27; 29]. When bonding at the GB sites, S also tends to form more asymmetrical geometries when compared to P [26]. This happened both for the single impurity and for impurity pairs. This result was in line with the trends on coordination preferences in complexes that contain transition metal centers and these two p-block elements. The fact that the strength of the bonds between the Cu matrix and S benefited from asymmetry, when compared to P, is the factor responsible for the stronger interactions between S and a vacancy when compared to P and a vacancy, as shown in the data presented in Table 1. This also had effects on the diffusion coefficients for these elements, as will be shown below. Correlation Effects: The Effective Diffusion Model or 5-Frequency Model for the Diffusion of P and S in Cu  
The 5-frequency (5-jump or 5-hop) model for diffusion, described in the Methods section, is a good approximation to describe the complex correlation effects that occur when substitutional impurities diffuse in the fcc matrix. This model has been employed both at high and low temperatures with success. Because the results of Equations (3) to (8) depend on the accuracy of the energetic and vibrational data employed, we employed methods that have been thoroughly benchmarked and tested and will not present the details of such benchmarking here. Such details are given in the Methods section and in the cited references of our previous works [26; 27; 36]. The potential energy profiles from which we obtained the \(\Delta E_{a}\) for each of the 5-jumps necessary to describe the correlation effects in the diffusion of S and P in Cu are shown in Figures 2 and 3.