Soft self-assembly of Weyl materials for light and sound

Abstract

Soft materials can self-assemble into highly structured phases that replicate at the mesoscopic scale the symmetry of atomic crystals. As such, they offer an unparalleled platform to design mesostructured materials for light and sound. Here, we present a bottom-up approach based on self-assembly to engineer 3D photonic and phononic crystals with topologically protected Weyl points. In addition to angular and frequency selectivity of their bulk optical response, Weyl materials are endowed with topological surface states, which allow for the existence of one-way channels, even in the presence of time-reversal invariance. Using a combination of group-theoretical methods and numerical simulations, we identify the general symmetry constraints that a self-assembled structure has to satisfy to host Weyl points and describe how to achieve such constraints using a symmetry-driven pipeline for self-assembled material design and discovery. We illustrate our general approach using block copolymer self-assembly as a model system.

Keywords: colloids; metamaterials; polymers; semimetal; topological matter.

Fig. 1.

Bulk Weyl points and arc surface states. (A) Sketch of an interface between a band gap material and a Weyl material. Arc surface states (purple) appear at this interface. (B) In a time-reversal invariant system, inversion symmetry has to be broken for Weyl points to exist. The simplest situation consists of four Weyl points with charges $\pm 1$ (red and blue, respectively) in the bulk Brillouin zone (bulk BZ). A plane interface preserves space periodicity in two directions and is hence described by a 2D surface Brillouin zone (surface BZ). Crucially, topological arc surface states (represented in purple) appear between Weyl points of opposite charge on the surface BZs. (C) The surface dispersion relation at the interface between a Weyl material and a gapped system features conical dispersions relations, which are the projections of the Weyl points. In addition, a manifold of topological arc surface states (light purple) appears. The intersection of this manifold with a plane of constant frequency (or energy) is sometimes called a Fermi arc in reference to the situation in electronic solid-state physics, where this plane is set at the Fermi energy. (D) The arc surface states may be observed by creating defects at the interface to couple them with incident waves.

Fig. 2.

Self-assembly process and effect of strain. (A) The self-assembly of triblock terpolymers leads to “colored” double gyroids, where the two minority networks (red and blue) are chemically distinct (68, 69). Starting from the self-assembled structure, a series of selective etching, partial dissolution, and backfilling steps leads to an asymmetric double gyroid made of high dielectric constant materials, which constitute a 3D photonic metacrystal. Crucially, the photonic band structure of such a system has a threefold degeneracy at the center of the Brillouin zone (the $\mathrm{\Gamma }$ point), which is represented in purple in B, Upper Right. This threefold degeneracy can be split into a set of Weyl points by an appropriate strain (in this case, pure shear) represented in red and blue (for Weyl points of charge 1 and $-1$, respectively) in B, Lower Right.

Fig. 3.

Reducing the symmetry. In A–C, we show the various structures of interest: (A) the (symmetric) double gyroid, (B) the asymmetric double gyroid, and (C) the strained asymmetric double gyroid (with shear strain). For the double gyroid, the group of the wave vector $\mathrm{\Gamma }$ is the octahedral group O_h (m$\overline{3}$m in Hermann–Mauguin notation), while it is the octahedral rotation group O (432) for the asymmetric double gyroid. When strain is applied to the asymmetric double gyroid, its symmetry is reduced, which corresponds at the $\mathrm{\Gamma }$ point to a subgroup of the octahedral rotation group O. Such subgroups are organized in a Hasse diagram. (The octahedral rotation group is indeed a subgroup of the full octahedral group O_h, which has more subgroups that are not relevant here and were not represented. As strain preserves inversion symmetry, any strained symmetric double gyroid still has an inversion center. The point group at $\mathrm{\Gamma }$ is then the product of the inversion group S₂ with a subgroup of O. Such situations can be achieved by starting from the symmetric double gyroid. In addition, there are other mixed subgroups of O_h, which cannot be directly realized through our method.) In D–F, the method of invariants predicts the qualitative features of the band structure of the modified double gyroids (B and C).

Fig. 4.

Photonic band structures. Photonic band structures of (A) the symmetric double gyroid and (B) the shear-strained asymmetric double gyroid. The threefold quadratic band crossing at the $\mathrm{\Gamma }$ point of the band structure of the unperturbed double gyroid is split into Weyl points on the $\mathrm{\Gamma }-N\prime$ and $\mathrm{\Gamma }-H\prime$ lines (in contrast, there is no crossing on the $\mathrm{\Gamma }-P$ line, which distinguishes the pair of Weyl points from a nodal line). The first eight bands of the band structures were computed with the MPB package (102) on a ${(64\times 3)}^3$ grid, with (A) ${\epsilon }_A={\epsilon }_B=16$, $t_A=t_B=1.1$, and $\theta =0$ and (B) ${\epsilon }_A=20.5$, ${\epsilon }_B=11.5$, $t_A=t_B=1.1$, and $\theta =0.3$. Here, ${\omega }_0=2\pi c/a$, where $c$ is the speed of light in vacuum.

Fig. 5.

Evolution of the main features of the photonic band structures. Evolution of characteristic features of the Weyl points, represented in C, with (A) the strain angle $\theta$ and (B) the dielectric asymmetry $\delta \epsilon \equiv {\epsilon }_B-{\epsilon }_A$. Here, the band structures are computed with the set of parameters in Fig. 4B on a ${(32\times 3)}^3$ grid. We plot both the minimum of the local gap between the fourth and the fifth bands, $\mathrm{\Delta }{f}_{45}^{\mathrm{\Gamma }P}=\min \{|f_5(k)-f_4(k)|\mid k\in \mathrm{\Gamma }-P\}$, and the normalized positions of the two inequivalent Weyl points, ${\widehat{w}}_{+}=w_{+}/\parallel \mathrm{\Gamma }N\prime \parallel$ and ${\widehat{w}}_{-}=w_{-}/\parallel \mathrm{\Gamma }H\prime \parallel$. The gray dashed lines correspond to the value at which each parameter is kept constant in other figures. The abrupt jump in the position of one of the Weyl points near $\theta =0.4$ is an artifact: another set of band crossings appears on the $\mathrm{\Gamma }-N\prime$ and $\mathrm{\Gamma }-H\prime$ lines near this value (SI Appendix). For the local gap, light green regions delimited by dashed lines correspond to the order of magnitude of symmetry-breaking numerical errors (SI Appendix). The data are not meaningless below this threshold, but the effects of the strain and structural asymmetry are not distinguishable from the spurious numerical reduction of symmetry. Similarly, ${\widehat{w}}_{\pm }$ should both vanish at $\theta =0$ (which is clearly not the case). This provides an order of magnitude of the uncertainty on both observables.

Fig. 6.

Band structures of the unperturbed double gyroid for different waves. (A) Dispersive photonic band structure of a metallic double-gyroid structure made of a Drude metal with the plasma frequency of gold standing in vacuum. (B) Phononic band structure for an elastic double gyroid in steel embedded in an epoxy elastic matrix. (C) Acoustic band structure for sound in air confined outside of a double gyroid with hard wall boundary conditions. In the dispersive photonic and phononic band structures (A and B), a threefold degeneracy (highlighted by gray circles) is found. As such, we expect such systems to exhibit Weyl points when strained. In A, ${\omega }_0=2\pi c/a$, where $c$ is the speed of light in vacuum. We use the plasma frequency of gold, ${\omega }_{\text{p}}/2\pi \simeq 2.19\times {10}^{15}\text{Hz}$ (108), and $a\simeq 500\text{nm}$. The loss term $\mathrm{\Gamma }$ is initially set to 0, and the results show no significant deviations from the case computed with the tabulated value $\mathrm{\Gamma }/2\pi =5.79\times {10}^{12}\text{Hz}$ (108). In B, ${\omega }_0=2\pi c_{\text{t}}/a$, where $c_{\text{t}}$ is the speed of transverse waves in epoxy. The values assumed for the longitudinal and transverse speeds of sound in steel and epoxy are obtained from the components of elastic tensor $C_{IJ}$ as $c_{\text{t}}^2=C_{44}/\rho$ and $c_{\mathrm{\ell }}^2=C_{11}/\rho$ from the values in refs. –, namely ${\rho }^{\text{epoxy}}=1180{\text{kg m}}^{-3}$, $C_{11}^{\text{epoxy}}=7.61\text{GPa}$, and $C_{44}^{\text{epoxy}}=1.59\text{GPa}$ and ${\rho }^{\text{steel}}=7780{\text{kg m}}^{-3}$, $C_{11}^{\text{steel}}=264\text{GPa}$, and $C_{44}^{\text{steel}}=81\text{GPa}$. In C, ${\omega }_0=2\pi c_{\text{air}}/a$, where $c_{\text{air}}$ is the speed of sound in air. All computations are performed with a $48\times 48\times 48$ grid. More details on the model and computation are in SI Appendix.

Fig. 7.

Symmetry-driven mesostructured material discovery pipeline. To obtain mesostructured materials with a set of desired properties, we suggest the following automated discovery pipeline. We start from a library of self-assembled structures, which is scanned for candidates matching symmetry requirements for a set of target properties. This requires us to automatically determine the space group of each structure: a script space_group.py does this job for structures represented as a skeletal graph (SI Appendix). A best candidate for the initial structure is then selected, and its properties are numerically computed. For example, we compute the band structure, from which the topological charges of the Weyl points (if any) are determined by a script weyl_charge.py. An effective description is then extracted from the numerical data: here, we need to determine the irreducible representations of the numerical eigenvectors, a job performed by the script irreps.py (SI Appendix). The effective description then allows us to determine which modifications should lead to the desired properties (for example, through a symmetry reduction). Here, this step could also be automated using https://github.com/greschd/kdotp-symmetry. Finally, the properties of the modified structure are numerically determined and compared with the desired properties. In case of failure, a new initial structure is selected from the library, and the process is iterated.