Self-assembled (opal-templated) silicon inverse opals were predicted by Busch and John (1998, 1999; reviewed in John and Busch, J. Lightwave Technol. 1999) to sustain a complete three-dimensional photonic band gap of roughly 10% of the gap-center frequency when the opal template is incompletely infiltrated with silicon, and — when the void network is infiltrated with an optically birefringent nematic liquid crystal — to exhibit a completely field-tunable photonic band gap, a 'tunable electromagnetic vacuum' that would also enable engineered spontaneous-emission dynamics of embedded emitters. More than two decades later this platform has only been partially realized: experiments have produced silicon inverse opals with computed complete gaps, have tuned stop bands (frequency intervals in which light is blocked only for a limited range of propagation directions, unlike a complete gap) by liquid-crystal infiltration, and have shown that structural disorder can destroy the gap, but the quantitative disorder tolerances, the electro-optic opening and closing of a complete gap via nematic-director reorientation, and the predicted complete-gap local-density-of-states (LDOS) effects in large-area samples remain undemonstrated. The problem is to determine and experimentally validate the conditions — structural tolerances, infiltration and liquid-crystal alignment control, emitter placement — under which the predicted complete, tunable band gap and its LDOS phenomenology survive in realistic self-assembled samples.

Background

A photonic crystal is a dielectric material with a periodic refractive-index modulation on the scale of the optical wavelength; a complete photonic band gap (PBG) is a frequency interval in which light of any propagation direction and polarization is forbidden, enabling full three-dimensional control of spontaneous emission and light guiding. Self-organizing routes to such crystals avoid lithography: colloidal microspheres (silica or polymer) sediment or evaporate into close-packed face-centered-cubic (fcc) opals, whose void network can then be infiltrated with a high-index semiconductor such as silicon and the template removed, leaving an inverse opal — an fcc lattice of air spheres in silicon. If the spheres are sintered (partially fused) before infiltration, small 'necks' connect neighboring voids and strongly affect the resulting band structure.

The foundational calculations for this platform are those of Busch and John: 'Photonic band gap formation in certain self-organizing systems' (Phys. Rev. E 58, 3896 (1998)) used a plane-wave expansion of the electromagnetic eigenproblem (the standard method that expands the periodic dielectric function and fields in Fourier components and diagonalizes the resulting matrix eigenproblem, here with about 531 plane waves) to show that a complete three-dimensional PBG spanning roughly 10% (silicon) and 15% (germanium) of the gap-center frequency is attainable only for a specific incomplete infiltration of the opal — the silicon filling fraction must be optimized rather than maximal — and to map how sintering, strain, and fcc versus hexagonal close packing modify the gap. They also computed the local density of optical states (LDOS, the quantity that, by Fermi's golden rule, sets the spontaneous-emission rate of an emitter at a given position), showing substantial LDOS pseudogap structure — a 'colored vacuum' — even where no complete gap exists. In 'Liquid-crystal photonic band gap materials: the tunable electromagnetic vacuum' (Phys. Rev. Lett. 83, 967 (1999)) the same authors showed that infiltrating the void network of a silicon inverse opal with a nematic liquid crystal (an anisotropic fluid whose rod-like molecules share a mean orientation, the 'director', and whose refractive indices depend on polarization relative to that director) makes the complete PBG fully tunable: an applied electric field that rotates the director relative to the silicon backbone shifts the gap, and it can be opened or closed outright, while orientational disorder of the director can localize light. John and Busch reviewed this program, and its open experimental requirements, in J. Lightwave Technol. 17, 1931 (1999) ('Photonic bandgap formation and tunability in certain self-organizing systems').

Experiments followed quickly: Blanco et al. (Nature 405, 437 (2000)) synthesized centimeter-scale single-crystal silicon inverse opals by chemical infiltration of sintered silica opals, with a computed complete gap centered near 1.46 µm; Vlasov et al. (Nature 414, 289 (2001)) demonstrated planar on-chip natural assembly with defect densities low enough that the band gap survives, while warning that colloidal assembly generally yields polycrystalline, defective crystals in which structural disorder 'can destroy the photonic bandgap'. Tunability experiments — liquid-crystal infiltration of macroporous silicon (silicon perforated by deep, regular cylindrical pores, an etched rather than self-assembled structure) and of inverse opals, plus ferroelectric and dual-frequency liquid-crystal variants — achieved reversible shifts of stop bands (frequency intervals blocked only for a limited range of propagation directions, as opposed to a complete band gap). Emitters embedded in complete-gap silicon crystals have shown strongly inhibited spontaneous emission, but only in top-down fabricated inverse-woodpile structures (two interpenetrating perpendicular pore arrays etched into silicon). The idealized assumptions of the 1998–1999 theory — perfect fcc periodicity, exact filling fraction, homogeneous conformal liquid-crystal coatings, coherent field-driven director reorientation, and emitters placed at favorable LDOS positions — have not been validated together in any experimentally produced complete-gap self-assembled material.

Problem Statement

The objective is to establish, with coupled computation and experiment, whether the complete, field-tunable photonic band gap predicted for self-assembled silicon inverse opals can be realized and preserved in realistic material samples, and what conditions that requires. Concretely, within the fcc inverse-opal geometry studied by Busch and John (opal template of close-packed spheres connected by sintering necks, incompletely infiltrated with silicon to the gap-optimal filling fraction, optionally with a nematic liquid crystal in the remaining void network), determine and demonstrate: (1) the quantitative tolerance of the complete photonic band gap to realistic fabrication imperfections — sphere polydispersity (sphere-to-sphere size variation), lattice disorder and stacking faults (local departures from the fcc stacking sequence), domain boundaries, nonuniform or incomplete infiltration, and neck-radius variations — together with the tolerances and reproducibility actually achievable in large-area samples; (2) electro-optic tunability of the complete gap: reversible opening and closing of the complete three-dimensional gap by electric-field-driven reorientation of an infiltrated nematic liquid crystal, treated with realistic director configurations (anchoring at silicon interfaces — i.e. the preferred director orientation imposed by the solid surfaces — elastic distortions, director defects) and with device-relevant electrode geometries and response times, rather than the idealized homogeneous alignment assumed in the original theory; and (3) the predicted local-density-of-states phenomenology in the completed material — suppression and restructuring of the spontaneous-emission rate of emitters placed at controlled positions within the gap or near gap edges, with the quantitative conditions (emitter density, coherence and dephasing, material absorption and thermal robustness) under which this suppression and restructuring remain observable. These three results are jointly the validation of the self-assembled tunable-photonic-band-gap platform, and all three are mandatory parts of a solution for the same class of self-assembled material: the quantitative disorder-tolerance analysis, the field-driven opening and closing of the complete gap, and the emitter-level LDOS response. Each part is to be established by coupled computation and experiment.

References

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