Design of crystal-like aperiodic solids with selective disorder–phonon coupling

Functional materials design normally focuses on structurally ordered systems because disorder is considered detrimental to many functional properties. Here we challenge this paradigm by showing that particular types of strongly correlated disorder can give rise to useful characteristics that are inaccessible to ordered states. A judicious combination of low-symmetry building unit and high-symmetry topological template leads to aperiodic ‘procrystalline' solids that harbour this type of disorder. We identify key classes of procrystalline states together with their characteristic diffraction behaviour, and establish mappings onto known and target materials. The strongly correlated disorder found in these systems is associated with specific sets of modulation periodicities distributed throughout the Brillouin zone. Lattice dynamical calculations reveal selective disorder-driven phonon broadening that resembles the poorly understood ‘waterfall' effect observed in relaxor ferroelectrics. This property of procrystalline solids suggests a mechanism by which strongly correlated topological disorder might allow independently optimized thermal and electronic transport behaviour, such as required for high-performance thermoelectrics.

Pauling number (p) 0.75 Two dimensional point group of parent node D 6 Two dimensional point group of node C 1 Supplementary Figure 16: (A) Configuration and (B) corresponding (hk)* diffraction pattern for the T 3 U X procrystalline system. Though not long-range ordered, this lattice has rectangular symmetry at the macroscopic scale.
Procrystalline symmetry: Pauling number (p) 0.25 Point group of parent node D 6h Point group of node D 3h Supplementary Figure 17: A description of the T 3 S state, which can not form a configuration on the triangular lattice.   Though not long-range ordered, this lattice has tetragonal symmetry at the macroscopic scale.
Known example for C 2 T and C 4 T:         Table 3: A summary of procrystalline phases including node symmetry and Pauling number. * For these chiral phases the absence of a point-symmetry element to interconvert enantiomeric nodes reduces the Pauling number by a factor of two.
Phase Parent Symmetry Node Symmetry Pauling number (p) occ.

Supplementary Notes 1 Neighbourhood geometries
For high-symmetry lattices, one method of constructing the neighbourhood is as follows. The Dirichlet-Voronoi 15-17 cell is identified and augmented to include those neighbouring lattice points to which the node at the centre of the cell is connected. Note that the Dirichlet-Voronoi cells provide a dense covering of space, and so the neighbourhoods also provide a dense covering. The difference is that the neighbourhood tiling is an overlapping tiling, where the overlap corresponds to the regions of augmentation. This is essentially a geometric means of enforcing "matching rules" as implemented in other aperiodic tilings of 2-and 3-space. 18 Note that, by design, the point symmetries of neighbourhood and lattice node coincide. Dirichlet-Voronoi constructions of neighbourhoods for the various high-symmetry lattices considered in our study are given in Supplementary Figures 38 and 39. For more complex lattice geometries, it is possible that the Dirichlet-Voronoi cell cannot be used to identify the correct neighbourhood. This is the case, for example, whenever connected lattice points do not share Voronoi faces. In such cases, we anticipate that other fundamental tilings of the underlying lattice-such as affine transformations of the Voronoi decompositionwill likely provide the relevant starting point for neighbourhood identification.

Procrystalline system notation
In order to facilitate discussion of specific procrystalline states, we have devised a working notation which aims to identify succinctly the relevant distinguishing features of each state. Each descriptor is of the form A n (B C ), where A denotes the underlying neighbourhood lattice, n gives the number of linkers distinguished for each node, B clarifies the geometric arrangement of those linkers, where necessary, and C identifies the use of chiral ('X') or racemic ('R') arrangements, where necessary.
A list of corresponding terms that we have used in this document is given in Supplementary Table 2. We make no claim that this notation is anything but a working system, devised to facilitate our own book-keeping. There are obvious limitations: the nomenclature is not unique since e.g. S 1 and S 3 identify the same procrystalline state; moreover, the existence of vernacular terms (and hence abbreviations) for specific geometries is only guaranteed for high-symmetry lattices.

Pd(CN) 2 and Pt(CN) 2 -Background
The structures of platinum(II) cyanide and platinum(II) cyanide have long been predicted to be related to the layered structure of Ni(CN) 2 . 19 This structure arises from the connection of square planar [M(CN) 4 ] 2units into grid-like layers; for Ni(CN) 2 these layers stack in the c crystallographic direction with no true periodicity, but such that cyanide ions of one layer are positioned directly above Ni 2+ cations of the layer below. 20,21 Periodicity is lost due the the fact that there are twice as many CN groups as Ni atoms, such that there are two possible arrangements for each pair of layers. 20 Intriguingly, a similar structural model is not found to account for the diffraction patterns of Pd(CN) 2 and Pt(CN) 2 due to the absence of sharp (00l) peaks. 14 Rather, a very broad diffraction pattern is observed for both compounds, and for the mixed-metal cyanide Pd 1/2 Pt 1/2 (CN) 2 . As noted in Ref. 14, these broadened peak profiles make structure determination from powder diffraction data alone difficult. Using a combination of powder diffraction, total scattering and spectroscopic data the authors of Ref. 14 proposed a nanocrystalline model in which the crystallite dimensions in the c direction are very small-therefore broadening the (00l) reflections such that they are not seen in the diffraction pattern-but the tetragonal average symmetry of Ni(CN) 2 is retained. 14 The broadening term used was so severe that the corresponding structural interpretation is that all Pt/Pd cyanides should be considered completely delaminated. Analysis of the experimentally determined pair distribution functions (T (r)) confirms the square-planar nature of the local bonding arrangement with significant broadening in the third dimension. An off-set stacking model (retaining p4mm sheet symmetry) was used to model the PDFs, though now with finite correlations in the stacking direction. 14 Considering the alternative model where neighbouring layers are in register, i.e. metals above each other, creates a series of metal nodes that form a nearly cubic lattice. Decorating with the square planar units in such a way that each unit is fully connected results in a network directly related to the C 4 T procrystalline model. We find that this model better describes the experimental powder diffraction data, as described below. Such a network might be considered a defect Prussian blue structure where known guests (NH 3 or H 2 O, depending on synthesis method) can be incorporated within the structure. Peak broadening therefore arises from a combination of procrystalline correlated disorder, isotropic strain and crystallite size effects.

Supplementary Methods 1 Phonon calculations
The GULP 22 input for the conventional unit cell implementation was as follows: In the supercell implementation of the mean-field case the projection gave phonon dispersion curves identical to those obtained using the conventional unit cell calculation discussed above as shown in Supplementary Figure 40. This validates the supercell implementation as a method for explicitly including disorder in phonon dispersion curve calculations.

Pd(CN) and Pt(CN) -Extraction and refinement of data
Powder diffraction data as a function of 2θ (Cu-K α1 radiation, λ = 1.540562Å) were extracted 23 from Figure 2 of Ref. 14. Data were interpolated, rebinned (∆2θ = 0.1 • ) and fitted using TOPAS v4.1. 24 The average structure of the C 4 T procrystalline M(CN) 2 model is unstable with respect to ferroelastic distortion; 25 the ferroelastic state is best described by space group Cmmm, which is a maximal subgroup of the parent P 4/mmm procrystal approximant. Full structural details determined using Rietveld refinement in this space group are given in Supplementary Tables 4  and 5; data and fits are presented in Supplementary Figure 42. The powder X-ray diffraction data were of sufficient quality only to refine lattice parameters and peak shape profile variables (including a Stephens (hkl)-dependent broadening term 26 and preferred orientation parameters), while making use of (fixed) atomic coordinates. Consequently the fits obtained are not of the same quality as Pawley refinements, but are more meaningful in the sense that a physicallysensible structural model is used to determine intensities. We note that large errors on refined lattice parameters are expected from extracted data.