Cubic metamaterial crystal supporting broadband isotropic chiral phonons

Starting from the achiral body-centered-cubic Wigner-Seitz cell, a truncated octahedron, we design a chiral simple-cubic metamaterial crystal which exhibits chiral phonons for all phonon propagation directions in three-dimensional (3D) space over a broad frequency range as well as nearly isotropic acoustical activity. The resulting blueprint has simple-cubic crystal symmetry and is radically simpler than previous proposals based on 3D quasicrystalline or 3D triclinic truss lattices, making it amenable to state-of-the-art 3D additive manufacturing. To emphasize this aspect, we present electron micrographs of microstructured metamaterial samples fabricated by using 3D laser nanoprinting.

Figure 1

Illustration of our blueprint for a 3D chiral mechanical metamaterial supporting isotropic chiral phonons. (a) Wigner-Seitz cell of a bcc lattice, a truncated octahedron. (b) The red and blue lines in panel (a) are replaced by red and blue cylindrical elastic rods with radii $r_1$ and $r_2$, respectively. The colors are for illustration only. Both red and blue rods are made from the same ordinary elastic material, for example, by a polymer. (c) The same as (b), but the red rods are twisted to obtain a chiral lattice. The length of the diagonal $d$ on the square facets of the truncated octahedron is indicated. All 12 red rods are completely within the unit cell, whereas each of the 24 blue rods is only half within the unit cell, leading to a total of 24 rods per unit cell. This crystal is composed of a single constituent material for which we use the parameters $E=4.18\mathrm{GPa},\rho =1.15\mathrm{g}/{\mathrm{cm}}^3$, and $\nu =0.4$, corresponding to a typical polymer, throughout this paper. Isotropic acoustical activity and isotropic chiral phonons are achieved with the optimized geometry parameters $r_1/a=0.0436,r_2/a=0.0533$, and $d/a=0.543$. For all of the numerical calculations below, we have chosen the lattice constant $a=100\mu \mathrm{m}$.

Figure 2

(a) Rendered 3D metamaterial crystal composed of $6\times 6\times 6=216$ simple-cubic unit cells as illustrated in Fig. 1. (b) Enlarged view of the rectangle region enclosed by yellow lines in (a). (c) and (d) Scanning electron micrographs of a nanoprinted polymer-based metamaterial crystal sample. These images can directly be compared with their blueprints shown in panels (a) and (b). The good agreement demonstrates that the design approach outlined in this paper is amenable to state-of-the-art 3D additive manufacturing on the micrometer scale.

Figure 3

Numerically calculated phonon band structures for elastic waves propagating in the cubic chiral metamaterial along: (a) the principal cubic direction [001], (b) the face-diagonal direction [110], and (c) the space-diagonal direction [111]. The two transverse bands are marked in red, and the longitudinal band is marked in black. Other higher-order bands are marked in gray. In the long-wavelength or low-frequency limit, the frequency splitting between the two transverse bands along the three directions is nearly the same. This behavior corresponds to an effectively isotropic chiral metamaterial.

Figure 4

(a) Relative frequency splitting, $2(f_2-f_1)/(f_2+f_1)$, between the two transverse bands, as shown in Fig. 3, along the principal cubic direction [001], the face-diagonal direction [110], and the space-diagonal direction [111]. We find nearly the same relative frequency splitting over the shadowed range of wave-numbers $\left|\mathbf{k}\right|<0.2\pi /a$ (compare different colors). (b) Calculated rotatory power (absolute value of the rotation angle per lattice constant $a$) for incident linear polarization for phonon propagation along the three directions as in (a). Below a frequency of 0.15 MHz, the three curves nearly overlap, indicating that we have achieved the aimed-at isotropic acoustical activity.

Figure 5

Direction dependence of the eigenfrequencies $f_i={\omega }_i/\left(2\pi \right)$ for the first three acoustic bands with $i=1,2,3$, shown for three selected circular wave-vector trajectories in reciprocal space for fixed length of the wave-vector $\left|\mathbf{k}\right|=0.2\pi /a$. (a) Illustration of the three chosen trajectories. Blue curve: Plane perpendicular to the principal cubic direction [001]. Red curve: Plane perpendicular to the face-diagonal direction [110]. Yellow curve: Plane perpendicular to the space-diagonal direction [111]. (b) Polar plot of the first two transverse eigenfrequencies for wave vectors for these three trajectories (see the colors). The first transverse eigenfrequency is indicated by the thin inner curves, and the second one is indicated by the thick outer curves. Here, zero azimuthal angle corresponds to the wave-vector direction indicated by the black line in (a). (c) Third eigenfrequency, corresponding to the longitudinal eigenmodes, plotted in the same manner. Note that we have intentionally largely stretched the depicted frequency range in panels (b) and (c) to make small deviations from isotropic behavior visible. For example, for the thick red curve in panel (b), the peak-to-peak frequency variation is less than 1.2 kHz around a mean value of about 136.4 kHz, corresponding to a relative peak-to-peak variation of the eigenfrequency of less than 0.9%. The same results are represented in a different graphical manner in Fig. 6.

Figure 6

(a) Direction dependence of the eigenfrequencies $f_i={\omega }_i/\left(2\pi \right)$ for the first three acoustic bands with $i=1\text{–}3$ (from left to right) for a fixed length of the wave vector, i.e., $\left|\mathbf{k}\right|=0.2\pi /a$, shown in three-dimensional representation. For any point on the surface, the distance to the origin is proportional to the value of the eigenfrequency for that propagating direction. A sphere would represent the perfect isotropic behavior. Obviously, our designed cubic chiral metamaterial crystal comes close to this ideal. (b) On the left-hand side, the relative frequency splitting $2(f_2-f_1)/(f_2+f_1)$ is plotted in the same manner. In the middle, we show the ratio $\zeta$ of the minor and major axes of the ellipse formed by the tip of the displacement vector of the first transverse eigenmode, averaged over one unit cell versus time. The ratio $\zeta$ is false-color coded onto a sphere. $\zeta =1$ corresponds to circular polarization, and $\zeta =0$ corresponds to linear polarization. Elliptical polarization leads to a value of $\zeta$ in between 0 and 1. On the right-hand side, we show the displacement vector trajectory on a plane perpendicular to the wave vector, again averaged over one unit cell, on a sphere in reciprocal space. Black and red trajectories are for $i=1$ and $i=2$, respectively. For all propagating directions in three dimensions, we observe very nearly circularly polarized phonon modes. A close inspection does, however, reveal small deviations from circular polarization. For example, these deviations are especially pronounced for phonon propagation along the face-diagonal directions.

Figure 7

To investigate the robustness of the behavior found in Fig. 6 against possible fabrication tolerances, we study parameters different from the ideal. As an example, we depict results for varying the normalized rod radius $r_2/a$. (a) Chiral polarization degree $\zeta$, plotted as in Fig. 6 but for variation of $r_2/a$ by $\left|\mathbf{k}\right|=0.2\pi /a$ and $+5$% (from left to right) relative to the optimum value of $r_2/a=0.0533$. (b) Plot of the first and second transverse eigenmodes for the same parameters and represented as in the right-hand side of Fig. 6. For relative variations not larger than $\pm 2\%$, the isotropic chiral behavior is rather robust.