and $L_z$.

The Au-Au interaction is modeled via the many-body Gupta potential[@Gupt81] (GP): $$\begin{aligned}
\label{Gupta}
V(\{r_{ij}\}) &=& \sum_{i}^N
  \sum^N_{j\neq i}A\ e^{-p(r_{ij}/r_0 - 1)}  
\nonumber\\&&\hspace{6ex}-\ 
  \sum_{i}^N
  \sqrt{\sum_{j\neq i} \xi^2\ e^{-2q(r_{ij}/r_0 - 1)}}\ .\end{aligned}$$ The distances $r_{ij} = |{{\boldsymbol r}}_i - {{\boldsymbol r}}_j|$ are measured in units of the bulk first-neighbor distance $r_0 = 2.885$ Å. The indices $i,j \in
\{1,\ldots,N\}$ label the Au atoms at positions ${{\boldsymbol r}}_i$ and ${{\boldsymbol r}}_j$, respectively. The parameterizations for Au as found in the literature,[@CR93] i.e., $A = 0.2061$ eV, $\xi = 1.790$ eV, $p =
10.229$, and $q = 4.036$, has been determined to match the bulk elastic constants and the surface contraction.

The Au-substrate interaction is modeled by two-particle interactions in the form of generalized Lennard-Jones 6-12 and 3-6 potentials. Their derivation is discussed in detail in Sec.\[SubstrateDFT\].

Runs on the atomic level have been performed for up to 10$^{8}$ updates per atom for up to 600 atoms. Runs have been stopped when a metastable configuration is obtained that does not evolve anymore on a reasonably accessible time scale. Depending on the size of the system metastable configurations are attained after a few minutes ($N = 55$) or a couple of days (Au 