Superlattice pseudouniform orderings as modulated structures: Stripe and checkerboard arrangements

Compositional uniformity, domain patterning and the mechanism underlying nano-chessboard arrays

“…17. The rules are based on the decomposition of the overall B fraction x in irreducible terms according to the Farey tree construction 17,18 : for example, x=5/12 can be decomposed as 1/2 ⊕ 1/2 ⊕ 1/3 ⊕ 1/2 ⊕ 1/3, where the ⊕ notation indicates a simultaneous addition of numerator and denominator in the fractions (e.g. n 1 /m 1 ⊕ n 2 /m 2 = (n 1 +n 2 )/(m 1 +m 2 ), which physically corresponds to the concatenation of two subsets of motifs of concentrations x 1 =n 1 /m 1 and x 2 =n 2 /m 2 ) This leads to the so-called uniform sequence [22323], which compactly indicates the composition and arrangement of the basic tiles in a pseudouniform stripe configuration.…”

“…In most cases the arrangement involves ir-reducible tiles of compositions 1/n and 1/(n+1) bracketing the overall B fraction x. 17 . These tiles can be seen as encoding at a local level the drive for dilution, and their overall arrangement according to the rules results in a pseudouniform intergrowth.…”

“…17 is that the arrangements of the type shown in Figure 3 can be considered as compositionally modulated structures with a single composition-dependent primary modulation wave vector (a 1D modulation). 17 The Fourier spectra (square modulus of the geometric structure factor) of these two examples are shown in Figure 3(c) and 3(d). One can see that the wave vectors (5/12)(1,1) and (3/14)(2,1), respectively, which are approximately orthogonal to the stripe direction, determine the location of the most intense satellite reflections stemming from the ordering.…”

“…Within the superspace formalism originally developed to describe incommensurately modulated structures, [19][20][21] these pseudouniform orderings can be described as longperiod but commensurate structures with a simple steplike occupational modulation, with values either "atom A" or "atom B" in the relevant proportion, and using those vectors as primary modulation wave vectors. More specifically, a pseudouniform striped ordering is fully defined by an occupational modulation function f (x 4 ) of period 1 with value "atom B" in the interval − x/2 < x 4 < x/2 and value "atom A" in the interval x/2 < x 4 < 1 − x/2 and a modulation vector q that depends on x according to definite rules, 17 such that the kind of atom occupying a site m of the lattice is determined by the value of this function at x 4 =q•m. The continuous variable x 4 of the modulation function can be identified with the coordinate along the internal space used in the superspace formalism.…”

“…In the last decade it has been shown 9,12 that this type of one-dimensional uniformity-driven arrangement can be most efficiently described by means of step-like atomic occupation modulations with wave vectors and forms for the modulation functions that can be systematized according to simple rules using the superspace formalism. [14][15][16] Very recently, the main ideas have been extended to the analysis of the distribution of minority motifs in two-dimensional systems, 17 showing that pseudouniform arrangements can originate as the result of a simple quest for maximal dilution of a minority motif within lattice- restricted sites. In wide ranges of composition, the dilution drive was shown to stabilize 1D stripe arrangements oblique to the underlying lattice, following predictable rules based on the alternation of sub-units of different compositions on the nanoscale, but still keeping an overall identity as one single phase.…”