Algorithm for generating derivative structures

We present an algorithm for generating all derivative superstructures—for arbitrary parent structures and for any number of atom types. This algorithm enumerates superlattices and atomic configurations in a geometry-independent way. The key concept is to use the quotient group associated with each superlattice to determine all unique atomic configurations. The run time of the algorithm scales linearly with the number of unique structures found.

Figure 1

An example parent lattice (left) and a superstructure (right). The parent lattice is fcc and the superlattice is defined by the (doubled) unit cell outlined in gray. The two interior points of the superlattice are occupied by one black atom and one gray atom. Together the superlattice and atoms constitute a derivative structure. The superstructure of this example is that of CuAu.

Figure 2

(Color) The first 17 binary structures derived from the fcc lattice. All have 4 atoms/cell or less. Structures shown with a green plane can be characterized as a stacking of pure A and B atomic layers. For example, the $L{1}_0$ structure (upper left) is an alternating $\left(A_1{B}_1\right)$ sequence of layers stacked in the [001] direction. All of the 2 and 3 atoms/cell structures have physical manifestations. Of the 4 atoms/cell structures only four have physical manifestations. Three of the others (yellow backgrounds) have been predicted to exist (Ref. 14) but not yet observed. The other five (blue backgrounds) have never been observed or predicted to exist in any system.

Figure 3

Required CPU time as a function of the number of unique derivative superstructures found. The scaling is linear, the best possible scaling for this type of problem.

Figure 4

(Color online) (I) One-dimensional example of a parent lattice, (II) a derivative superlattice (index $n=4$), and (III) one possible superlattice labeling.

Figure 5

(Color online) Three labelings of the superlattice of Fig. 4. Lines II and III are identical to line I except the colors (labels) are shifted. In line II the colors have been shifted two units to the right. In line III, the shift has been effected by translating the lattice two units to the left.

Figure 6

A superlattice whose SNF is nontrivial (noncyclic). Because the quotient group of the superlattice is noncyclic, translations of the lattice are not cyclic permutations. Instead the sites are permuted in groups of two or four with each translation (rather than permuting as a single group of 8). In the figure, three different pictures of the same superlattice are shown. In each case a different origin is chosen. In the second case, the translation permutes two groups of four sites and in the third, four pairs are permuted.

Figure 7

(Color online) (I) One-dimensional example of a parent lattice, (II) a derivative superlattice (index $n=4$), and (III) a superperiodic (or nonprimitive) labeling. Although the index of the superlattice is $n=4$, the structure can be represented by a superlattice labeling of period 2 instead of 4. The superstructure of line III would have been found as an index $n=2$ derivative structure and is therefore a duplicate.

Figure 8

(Color online) Two identical $4\times 4$ superlattices (dotted lines) with rotationally equivalent labelings (red and blue circles). Although the superlattices themselves are unchanged, a $90°$ rotation applied to the left labeling yields that shown on the right. Thus the second is a duplicate of the first and should be removed from the list of labelings.

Figure 9

Symmetrically unique HNF matrices as a function of the index $n$ (determinant size) for an fcc parent lattice. Asymptotically, the fraction of unique superlattices approaches $2/N=1/24\approx 5\%$, where $N=48$ is the number of symmetry operations of the fcc parent lattice.

Figure 10

(Color online) Left axis (red): number of HNF matrices as a function of determinant size. Right axis (blue): number of inequivalent fcc superlattices as a function of volume.

Figure 11

The first 21 binary structures derived from the simple-cubic lattice. All have 4 atoms/cell or less. Structures marked with a crystallographic direction $\left(hkl\right)$ can be characterized as a stacking of pure A and B atomic layers (black and white spheres). For $n=4$ all of the stacked structures occur in pairs, $A_3{B}_1$ and $A_2{B}_2$. The last three structures, labeled 19, 20, and 21, cannot be characterized as pure stackings. These structures have basis vectors whose corresponding quotient group is $G={\mathbf{Z}}_2\oplus {\mathbf{Z}}_2$ (rather than ${\mathbf{Z}}_4$).

Figure 12

Two-dimensional superlattices of index $n=8$. The first and third share the same SNF, while the second has the trivial SNF, implying a purely cyclic quotient group. In contrast to the first and third superlattices, the lattice points in layers parallel to the superlattice edge (dotted line) must all have the same label.