Supporting information: Evidence for a C14 Frank-Kasper phase in. One-Size Gold Nanoparticle Superlattices.

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1 Supporting information: Evidence for a C14 Frank-Kasper phase in One-Size Gold Nanoparticle Superlattices. Stéphanie Hajiw, Brigitte Pansu, and Jean-François Sadoc Laboratoire de Physique des Solides, Bât 510, UMR-CNRS 8502, Université Paris-Sud, ORSAY Cedex, FRANCE brigitte.pansu@u-psud.fr The C14 structure In the FK MgZn 2 =C14 structure, the Zn atoms are arranged in parallel hexagonal layers h 1 at z=0 and h 2 at z=1/2 or in Kagome layers K 1 at z=1/4 in B position with respect to h 1 and K 2 at z=3/4 in C position with respect to h 1. Each Zn Kagome layer is sandwitched between two hexagonal layers made by Mg atoms as for instance H 1 -K 1 -H 1. H 1 and H 1 are situated at z=±3/16 on each side of the Kagome layer K 1, exactly above the holes of the Kagome layer, that is in the same position B with respect to h 1. The hexagonal cell thus contains 8 Zn atoms and 4 Mg atoms. Their positions is detailed in Table 2. There are three equivalent positions with different environments. Each Mg atom has 16 neighbors: 4 Mg, 9 Zn (in the Kagome layers), 3 Zn (in a hexagonal layer). Each Zn atom in a Kagome layer has 12 neighbors: 6 Mg, 4 Zn (K-type), 2 Zn (h-type). Each Zn atom in a hexagonal layer (h type) has also 12 neighbors: 6 Mg and 6 Zn (K-type). For the Zn sites, the coordination To whom correspondence should be addressed 1

2 polyhedra are thus deformed icosahedra. Let a be the lattice parameter in the hexagonal plane, therefore c = 8 3 a. The Voronoi cell volume around each atom is a3 for the 4 Mg sites, a 3 for the 2 Zn sites (h-type), a 3 for the 6 Zn sites (K-type) for a cell volume equal to 2a 3. Table 1: Position of the 12 atoms in the hexagonal cell (a, a, 8 3 a). atom (x,y,z) label 2Zn (0,0,0) h1 (0,0,0.5) h2 6Zn (0.833,1.666,0.25) K1 (-1.666,-0.833,0.25) K1 (0.833,-0.833,0.25) K1 (-0.833,-1.666,0.75) K2 (1.666,0.833,0.75) K2 (-0.833,0.833,0.75) K2 4 Mg (0.333,0.667,0.0625) H1 (0.667,0.333,0.5625) H2 (0.667,0.333, ) H 2 (0.333,0.667,0.4375) H 1 Coordination in the C14 Laves phase The prototype of this Frank and Kasper phase, first proposed by Friauf and Laves, is the Zn 2 Mg structure. It is related to the Cu 2 Mg structure (C15) in the same way as HCP is related to FCC. In both structure, the Mg atoms are on the positions of atoms in tetravalent structures : cubic diamond structure for C15 or wurtzite (the hexagonal version of diamond) for C14. The other atoms (Zn or Cu) are filling voids in the loosely packed diamond or wurtzite structures. Figure 1 shows the coordination shells for the different types of atoms in the C14 phase shown on Fig 3 of the paper. Like in FCC and HCP close packing which could be described as a stacking of hexagonal planes: ABC for fcc and AB for hcp the two structures C15 and C14 could be 2

3 described by stacking of (more decorated) planes following the same rules. As in close packed structures there are the possibility of more complex sequences of A, B, C planes leading to what is called polytype. This is not excluded in NP structures. Figure 1: Coordination shells (top) for the different types of atoms in the C14 structure (prototype Zn 2 Mg) and there corresponding Voronoi cells (bottom). The Zn atoms coordination is Z12 (icosahedral) and the M g atom coordination is Z16. On the coordination shell Z12 are shown in yellow and Z16 in red. The coordination shell Z12 for Zn h1 contains a ring of six Z16 in chair configuration sandwiched between two triangles of Z12. The Zn k1 shell has a ring of six Z12 in boat configuration sandwiched between two and four Z12. The Mg (Z16) atoms neighbors are four Z16 and twelve Z12. The Z12 sites are connected together in a wurtzite network by a so called Frank-Kasper network. The Z phase In the FK Zr 4 Al 3 =Z structure (Figure 2, the Al are stacked in Kagome-mesh layers (K,blue) at z=0, half Zr are situated in a hexagonal mesh (Hex,red) at z=1/4 and z=3/4. The other half Zr are stacked in layers (green) at z=0.5 and are situated just above Al triangles of the Kagome layers. 3

4 Figure 2: Position of the atoms in the Z phase cell (Zr 4 Al 3 ): 3 Al (in blue) at z = 0, 1 Zr (in red) at z = 1/4 and 1 Zr at z = 3/4, 2 Zr (in green) at z = 0.5. Table 2: Position of the 7 atoms in the hexagonal cell (a, a, 0.993a). atom (x,y,z) label 3Al (1/3,1/6,0) K (5/6,1/6,0) K (5/6,2/3,0) K 2Zr (1/3,2/3,0.25) Hex (1/3,2/3,0.75) Hex 2 Zr (0,0,0.5) (2/3,1/3,0.5) Bragg peaks tables in C14, C15 and Z phases The position of the Bragg peaks are given in Table 3 for the C14 phase, in Table 4 for the C15 phase and in Table 5 for the Z phase. 4

5 Table 3: Bragg peaks for the C14 structure with a = 6.45nm hkl q in Å 1 distance/a Intensity multiplicity Powder peak intensity Figure 3: Indexation of Bragg peaks of Fig. 2b using classical hexagonal notations (100=10 10, 110=11 20). The Bragg peak intensities on a single domain pattern cannot be easily related to Table 3 due to the scattering conditions (intersection of the reciprocal plane by the Bragg sphere). 5

6 Table 4: Bragg peaks for the C15 structure with a = 9.12nm, at the same density as the C14 structure with a = 6.45nm. hkl q in Å 1 Intensity multiplicity Powder peak intensity Table 5: Scattering pattern for the Z structure with a = 6.45nm. hkl q in Å 1 Intensity multiplicity Powder peak intensity

7 Projection of the C14 structure along various directions a) b) c) Figure 4: Projection of the C14 structure a) in the ( a, b) plane b) in the ( b, c) plane, c) in the ( a, c) plane. a) b) Figure 5: a) Cryo-TEM image with a hexagonal-type pattern (scale bar=10nm) b) Projection of a slice (c/2) of the C14 structure in the ( a, b) plane 7

8 a) b) Figure 6: a) Cryo-TEM image with periodic lines, period close to a/2 (scale bar=10nm) b) Projection of the C14 structure in a plane perpendicular to [sin5, cos5 cos32.15, -cos5 sin32.15 ] containing the (013) direction (=[0, sin32.15, cos32.15 ]) 8