Achieving Epitaxy between Incommensurate Materials by Quasicrystalline Interlayers

Epitaxial interfaces of commensurate periodic materials can be characterized by a locking into registry of their atomic structure. This characteristic is identified as a natural framework to capture the essence of epitaxy also for systems including quasicrystalline materials. The resulting general definition for epitaxy requires a matching of reciprocal lattice points. The consequences for the real space structure of an epitaxial interface between quasiperiodic and periodic materials are explored and an experimental realization of such an interface is presented. It is demonstrated that due to their higher number of reciprocal lattice basis vectors (exceeding three), quasicrystals can provide interlayers epitaxially linking incommensurate materials.

Figure 1

(a) SPALEED pattern from Al-Ni-Co(00001) clean surface for electron energy $E_{\mathrm{kin}}=75\mathrm{eV}$, (b),(c) SPALEED pattern from epitaxial AlAs islands on Al-Ni-Co(00001) for $E_{\mathrm{kin}}=130$ and 140 eV, respectively. One of the ten symmetry related sets of facet spots is marked by circles, squares indicate clean surface peaks, dotted circles in (c) the locations of the corresponding facet spots in (b). (d) SPALEED intensity in the reciprocal space plane spanned by the in-surface [10000] direction and the surface normal; gray lines are guides to the eye. (e) Sketch of the derived geometry with faceted substrate [light gray, $\alpha$ is $(35.27\pm 0.15)°$] and the in-plane orientation of the strained AlAs(111) film. (f) Surface reciprocal lattice of the strained AlAs(111) film (vertices of the grid: reciprocal lattice points).

Figure 2

(a) Reciprocal lattice match with circles depicting the reciprocal lattice of Al-Ni-Co projected onto the ($10\overline{2}\overline{2}4$) interface plane and the vertices of the mesh of the AlAs(111) film. The radii of the circles are proportional to the Fourier amplitudes of the atomic density of Al-Ni-Co calculated from the structural model by Yamamoto and Weber [18]. (b) Projected Al-Ni-Co reciprocal lattice vectors.

Figure 3

Epitaxial interfaces between a periodic and a Fibonacci chain. (a) Real space structure for a repulsive interaction (disfavoring on-top sites). (b) Corresponding reciprocal lattice structure with the radii of the black circles proportional to the magnitude of the Fourier components. (c) Average distribution of the Fibonacci chain atoms within the unit cell of the periodic chain. The common reciprocal lattice vector is $\mathrm{G}={\mathrm{G}}_{11}^{\mathrm{Fib}}={\mathrm{G}}_2=4\pi /a$.

Figure 4

$\mathrm{Al}\mathrm{\text{−}}\mathrm{Ni}\mathrm{\text{−}}\mathrm{Co}(10\overline{2}\overline{2}4)$ surface layer atomic structure and structural match with AlAs(111). (a) Fourier transform of the top layer of bulk-truncated $\mathrm{Al}\mathrm{\text{−}}\mathrm{Ni}\mathrm{\text{−}}\mathrm{Co}(10\overline{2}\overline{2}4)$, marked as L(0) in (c). The radii of the circles are proportional to the magnitude of the Fourier amplitudes. Circles at the vertices of the grid belong to the common sublattice. A strong set for potential alternate epitaxy is outlined by squares. (b) Average distribution of the atoms in the Al-Ni-Co interface layer L(0) in the real space interface unit cells of the AlAs(111) film. Those of the first and second Al-Ni-Co subsurface layers $\mathrm{L}(-1)$ and $\mathrm{L}(-2)$ are marked by dashed outlines. (c) $\mathrm{Al}\mathrm{\text{−}}\mathrm{Ni}\mathrm{\text{−}}\mathrm{Co}(10\overline{2}\overline{2}4)$ areal atomic density $\rho (z)$ in the planes perpendicular to the [$10\overline{2}\overline{2}4$] direction. Calculations are based on the Al-Ni-Co structural model from Ref. 18.