Friedel oscillations responsible for stacking fault of adatoms: The case of $\mathrm{Mg}(0001)$ and $\mathrm{Be}\left(0001\right)$

We perform a first-principles study of Mg adatom/adislands on the $\mathrm{Mg}(0001)$ surface, and Be adatom on $\mathrm{Be}\left(0001\right)$, to obtain further insight into the previously reported energetic preference of the fcc faulty stacking of Mg monomers on $\mathrm{Mg}(0001)$. We first provide a viewpoint on how Friedel oscillations influence ionic relaxation on these surfaces. Our three-dimensional charge-density analysis demonstrates that Friedel oscillations have maxima which are more spatially localized than what one-dimensional average density or two-dimensional cross-sectional plots could possibly inform: The well-known charge-density enhancement around the topmost surface layer of $\mathrm{Mg}(0001)$ is strongly localized at its fcc hollow sites. The charge accumulation at this site explains the energetically preferred stacking fault of the Mg monomer, dimer, and trimer. Yet, larger islands prefer the normal hcp stacking. Surprisingly, the mechanism by which the fcc site becomes energetically more favorable is not that of enhancing the surface-adatom bonds but rather those between surface atoms. To confirm our conclusions, we analyze the stacking of a Be adatom on $\mathrm{Be}\left(0001\right)$—a surface also largely influenced by Friedel oscillations. We find, in fact, a much stronger effect: The charge enhancement at the fcc site is even larger and, consequently, the stacking-fault energy favoring the fcc site is quite large—$44\mathrm{meV}$.

Figure 1

Upper insets: Difference between the charge density of bulk Mg and that of a nonrelaxed bulk-terminated $\mathrm{Mg}(0001)$ surface. (a) Isosurfaces. The $z$ axis is perpendicular to the surface. The (blue) balls represent the first four atomic layers of the slab. The pocket (red) indicates the region of $\mathrm{Mg}(0001)$ that displays more charge density than the corresponding one in bulk Mg. (b) [0001], cross section of the isosurface in (a); the fcc region (red) displays charge accumulation and the hcp region (blue) displays charge depletion. Lower insets: [0001], cross-sectional planes of the total charge density around (c) the fully relaxed $\mathrm{Mg}(0001)$ and (d) bulk Mg. Darker (brown) regions in (c) and (d) indicate less charge. In (b)–(d), the plane is located at the height of the surface or bulk atoms under consideration in order to highlight the charge accumulation around the fcc hollow site of $\mathrm{Mg}(0001)$.

Figure 2

[0001], cross section of the total charge density of (a) nonrelaxed bulk-terminated $\mathrm{Mg}(0001)$ and (b) fully relaxed $\mathrm{Mg}(0001)$. Darker (brown) regions indicate less charge. The plane is located at $\sim 1.2\AA$ above the position of the surface atoms.

Figure 3

Three-dimensional charge-density difference isosurfaces showing the Friedel oscillations in (a) $\mathrm{Mg}(0001)$ and (b) $\mathrm{Be}\left(0001\right)$. The charge-density isovalue is the same for both surfaces. The difference is taken between the charge density of bulk Mg and that of a nonrelaxed bulk-terminated $\mathrm{Mg}(0001)$ surface. The $z$ axis is perpendicular to the surface. The balls (light-blue and green) represent the first three layers of the slab. The pockets (red surfaces) indicate the regions in the surface displaying more charge density than the corresponding one in bulk.