The diversity of three-dimensional photonic crystals

Many butterflies, birds, beetles, and chameleons owe their spectacular colors to the microscopic patterns within their wings, feathers, or skin. When these patterns, or photonic crystals, result in the omnidirectional reflection of commensurate wavelengths of light, it is due to a complete photonic band gap (PBG). The number of natural crystal structures known to have a PBG is relatively small, and those within the even smaller subset of notoriety, including diamond and inverse opal, have proven difficult to synthesize. Here, we report more than 150,000 photonic band calculations for thousands of natural crystal templates from which we predict 351 photonic crystal templates – including nearly 300 previously-unreported structures – that can potentially be realized for a multitude of applications and length scales, including several in the visible range via colloidal self-assembly. With this large variety of 3D photonic crystals, we also revisit and discuss oft-used primary design heuristics for PBG materials.

Color indicates bands between which the PBG occurs, with plotting marker corresponding to data source.
(a) PBG between bands 2-3 (red), 8-9 (teal) and 14-15 (lavender) for Diamond at ε = 12.96, as reported by 1 and generated by the example code given by 4 . PBG sizes reported in Ref. 1 were found to be overestimated in 5 , therefore a better benchmark for diamond is provided by 4 . (b) PBG between bands 2 and 3 in Inverse Diamond at ε = 12.96, as reported by 1 . (c) PBG between bands 5 and 6 in Inverse Simple Cubic at ε = 13, as reported by 2 . (d) PBG between bands 16 and 17 in Hexagonal Diamond at ε = 12, as reported by 3 . PBG between bands 4 and 5 (orange) were unreported in 3 , with no indication if calculations for corresponding φ were run. (e) PBG between bands 2-3 (red) and 8-9 (teal) for Diamond at φ ≈ 0.34, as reported by 1 and generated by the example code given by 4 . PBG sizes reported in Ref. 1 were found to be overestimated in 5 , therefore a better benchmark for diamond is provided by 4 . (f) PBG between bands 2 and 3 in Inverse Diamond at φ ≈ 0.19, as reported by 1 . (g) PBG between bands 8 and 9 in Inverse Opal at φ ≈ 0.26, as reported by 5,6 .
In Fig. 2, we compare the PBG sizes computed using our methodology and previously published in literature within references 1-6 . Any discrepancies between the computations reported and previous literature can be due to differing smoothing functions of ε space, using a more exhaustive IBZ, and small errors in transcribing previous literature, which were available in figure, and not table, form.

Data Management with Signac
The data for this project was managed using signac and the workflow managed by signacflow in a multi-level project. 7 The top level of the project consisted of statepoints consisting of structural data. Inside each structure statepoint was an additional project managing the statepoints containing radii and dielectric constant. Supplementary Figure 3. Organization of Project Using signac and signac-flow. Operations and labels were stored individually for the structure at large and independent radii and dielectric constants, as was appropriate. Typical structure-level operations included symmetry calculations, PBG atlas generation, and summarizing lower level data. Typical lower level operations include running and analyzing MPB, computing fill fraction, and field analysis. For each mode motif in the main text, we show the reduction in visual complexity from the vector fields to a flow diagram to a mode motif. For each motif we attempted to provide a visual that best represents the motif, with many structures exhibiting variations on these motifs or mixed motif behavior.

C. Extended Analysis
For this section, we will report percentages of structures that have a particular structural fingerprint across all structures studied and the 351 that have PBGs. We will also report relative probabilities, i.e. the probability a given fingerprint is found in structures with PBGs divided by its probability of being found in all structures. We compared the sample of structures used in this project with a rough estimate of the data space available through ICSD and COD to show that our sample set can represent a general distribution.

Relative Probabilities Across Space Groups
Structures with a PBG All Structures Studied  C. cF136-Si (Inverse) Inverse Clathrate-II Image of cF136-Si, generated by Vesta

Inverse Opal (FCC)
Image of cF4-Cu, generated by Vesta

F. cF8-C (Direct)
Diamond Image of cF8-C, generated by Vesta

Inverse Silicon (II)
Image of cI16-Si, generated by Vesta