AUXILIARY MATERIALS
Double-diamond NaAl via pressure:
understanding structure through Jones-zone activation
Ji Feng1, Roald Hoffmann2 and N. W. Ashcroft3
June 15, 2009
1
Department of Materials Science and Engineering, School of Engineering and Applied Science,
University of Pennsylvania, Philadelphia, PA 19104, USA
2
Department of Chemistry and Chemical Biology, Baker Laboratory. Cornell University. Ithaca,
NY 14853-1301, USA
3
Laboratory of Atomic and Solid State Physics and Cornell Center for Materials Research, Clark
Hall. Cornell University, Ithaca, NY 14853-2501, USA
A1. Wigner-Seitz spheres for projected DOS and projected charge density
To calculate the projected DOS and projected charge density associated with each atom, we
use the Wigner-Seitz method. A Wigner-Seitz sphere is defined for each atom, whose radius is
A1
proportional to the atomic radius of the corresponding element, scaled such that the sum of
volumes the Wigner-Seitz spheres of all atoms cover equals the total volume. The atomic radii of
and Al are 1.86 and 1.43 Å, respectively. Define a scaling factor for the projection sphere radii:
c(Na) = 2×1.86/(1.86+1.43) = 1.1307, and c(Al) =2×1.43/(1.86+1.43) = 0.8693. The interatomic
separation in the double-diamond structure between nearest neighbors (Na-Al), dNN = 31/2a/4,
where a is the lattice constant of the cubic unit cell. The total volume covered by Wigner-Seitz
spheres is ΩWS,tot = 8 × 4π[RWS(Na)3 + RWS(Al)3]/3, for a total of 8 formula units per cubic unit cell.
If we require ΩWS,tot = a3 (the volume of a cubic unit cell), we have a3 = 32πR3[ c(Na)3 + c(Al)3]/3,
where R is the unknown, and is related to the Wigner-Seitz radii by RWS(Na) = c(Na)R and
RWS(Na) = c(Al)R. Solving for R, we have R = 0.2421a. Now we have:
RWS(Na) = 0.316a = 0.632dNN;
(A1)
RWS(Al) = 0.210a = 0.486dNN.
(A2)
and
A2. Band structure of NaAl in the double-diamond structure
The band structure of NaAl in the double-diamond structure at ordinary pressure is shown in
Fig. A2. The subvalence band gap at high-symmetry point L(0.5,0.5,0.5) (at ~ -5 eV) is a result of
the level splitting due to (111) component of the lattice potential.
A2
Figure A2. DFT-GGA band structure of NaAl in the double-diamond structure at close-to-ambient
pressure. High-symmetry k-points are, expressed in terms primitive reciprocal lattice vectors:
G(0,0,0), X(0.5,0,0.5), W(0.5,0.25,0.75) and L(0.5,0.5,0.5).
A3. More extensive structural search
Our structural search for potential Na-Al phases takes structures from two sources. First, we
take our structures from structure maps, near the location of the Na-Al on these maps. [A1]
Second, we start with randomly generated structures. [A2] In both cases, we optimize the
structures with the DFT method using the valence-only pseudopotential for sodium. We compare
the results for selected structures with those obtained from Na pseudopotential with 2s and 2p as
valence electrons. Pettifor's structure map is based on a chemical scale, termed χ, that resembles
the Mendeleev atomic numbers of the periodic table. By arranging elements according to the
phenomenological chemical scales along both x- and y-axis, it is noticed that structure types of
correspond binary alloys are approximately separated. Though phenomenological and based
entirely on the structures at ambient pressure, Pettifor's structure maps of binary alloys have been
A3
shown to give useful chemical advice in terms of predicting the structures of compounds, at
ambient pressure and recently, at high pressure.
A section of AB structure map near the Na-Al combination is shown in Fig. A4 a. We choose
a large search radius, as indicated by In Fig. A3 a, we show a section of Pettifor's binary structure
map for AB intermetallic compounds, in the neighborhood of where Na-Al is located. We see that
this structure map suggests a few likely structure types: NaTl, KGe, NaPb, CrB, AuCu and CsCl
(AuCu is the tetragonally distorted version of CsCl structure). These structures span a range of
complexity, from small simple unit cells, to large complex unit cells.
We describe here each of these structures here. CsCl structure (Figure A3 (b)) has a cubic unit
cell with Cs atom occupying the corners and Cl atom the center of the cell. AuCu (Fig. A3 c)
structure is a tetragonally elongated form of CsCl. In CrB structure (Figure A3 (c)), Cr atom forms
a zigzag chain, propagating in hexagonal channels of the B matrix.
KGe (Figure A3 (e-f)) and NaPb (Figure A3 (g-h)) structures are more complex, and are
related. They both have tetrahedral cages formed by the more electronegative elements (Ge in KGe
and Pb inNaPb). In both structures each tetrahedral cage is face-capped by the more electropositive
elements (Figure A3 (f) and (h))). These eight-atom clusters, approximately stella octangula in
shape, are the basic motifs of the extended structures.
A4
Figure A3. (a) Pettifor structure
map for AB alloy (adapted from
Villar's book). (b) CsCl structure.
(c) AuCu structure. d. CrB
structure. (e)-(f) KGe structure.
g-h. NaPb structure. Under each
structure we list, in that order, the
archetypical compound, the
symbol used in Villar's book,
Pearson symbol and the space
group of that structure. We use
blue color for the more
electropositive element, and green
for the more electronegative
element.
A5
In KGe, the stella octangula K4Ge4 clusters can be viewed as separated from each other,
without common atoms. They occupy two symmetry-distinct positions in a cubic unit cell. One
type of K4Ge4 clusters occupy the corner and center of the unit cell. The other kind of K4Ge4
clusters occur as pairs, forming a large icosahedron with these clusters as vertices (Fig. A3 e). This
resembles the structure of CrSi3, if we think of type 1 cluster as Cr, and type 2 cluster as Si.
In NaPb, however, each pair of neighboring Na4Pb4 clusters share Na, thus forming a
tetrahedrally connected network network (Figure A3 (h)). Half of the Na atoms are not occupying
the capping position of the Pb tetrahedra, and they are located in the void space of the diamondoid
network of the connected Na4Pb4 clusters. For the NaAl calculations in various structures here, we
let Na occupy the more electropositive positions and Al the more electronegative positions.
We also performed a random structure search (RSS) [2] for potential Na-Al intermetallic
compounds at 20, 40, 60 and 80 GPa. In the RSS calculations, we use a pseudopotential for Na that
only includes the 3s electron. We calculated the enthalpy of formation of the NaAl phase in
double-diamond structure with the valence only pseudopotentials. As shown in Figure 7 of the
text, the enthalpies of formation from valence-only calculations and those that include 2s and 2p
electrons of sodium compare very closely. Therefore, it is a valid approach for this system to use
the valence-only pseudopotentials for faster screening in the RSS. Subsequently, we confirm the
results from the valence-only structural search by calculations including the sodium core states.
We briefly describe the procedure for generating a random structure here. First, we generate
the lengths of the three lattice vectors, each taking a uniformly random value between 0.5 and 1.5
A6
(arbitrary units). Then we generate the three angles (α, β and γ ) made by the lattice vectors, each
taking a uniformly random value between π/6 and 5π/6, subject to the constraint that the sum of the
three angles be no greater than 2π. Then we proportionally scale the lengths of the lattice vectors to
a specified unit cell volume, thereby completely specifying the lattice parameters. The position of
an ion in the unit cell is defined by three fractional coordinates, (x, y, z), each corresponding to a
random number (uniformly random between 0 and 1).
Reference
A1 (a) D.G. Pettifor, in “Physical Metallurgy”, ch. 3 (eds. R. W. Cahn and P. Haasen),
North-Holland (Amsterdam), 1983. (b) D. G. Pettifor. Solid State Comm. 51, 31 (1984).
A2 C. J. Pickard and R. J. Needs, Phys. Rev. Lett. 97, 45504 (2006).
A7