Snowflake phononic topological insulator at the nanoscale

We show how the snowflake phononic crystal structure, which recently has been realized experimentally, can be turned into a topological insulator for mechanical waves. This idea, based purely on simple geometrical modifications, could be readily implemented on the nanoscale.

Figure 1

Setup and bulk band structures (FEM simulations). (a) Snowflake crystal design defined by the parameters $(a,r,w,d,\mathrm{\Delta }r)$, where $d$ is the slab thickness. For $\mathrm{\Delta }r=0$ ($\mathrm{\Delta }r\ne 0$), the border of the Wigner-Seitz cell is drawn in blue (red). (b) Reciprocal space. The edges of the corresponding Brillouin zones are drawn in the same colors. Also plotted are the contours along which the band structures in (c) (blue contours) and (d) and (e) (red contours) are calculated. (c) Band structure for $\mathrm{\Delta }r=0$. (d) Same band structure, but folded into the smaller BZ. The Dirac cones at the $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points are now mapped onto the $\mathbf{\Gamma }$ point. (e) Band structure for $\mathrm{\Delta }r\ne 0$. The Dirac cones are now gapped. Note that the same behavior is observed for several Dirac cones (highlighted by dotted boxes). The modes symmetric to the $xy$ plane are displayed in darker colors. [Here, we have considered a silicon crystal slab with a Young' s modulus of $170\text{GPa}$, mass density $2329{\mathrm{kg}/\mathrm{m}}^3$, Poisson's ratio 0.28, and geometrical parameters $(a,r,w,d)=(5000,1800,750,220)\text{nm}$.]

Figure 2

Band inversion (FEM simulations): Frequencies of the $|p^{\pm }\rangle ,|d^{\pm }\rangle$ modes at the $\mathrm{\Gamma }$ point evolving for a sweep of $\mathrm{\Delta }r$. Snapshots of the corresponding displacement fields for $\mathrm{\Delta }r=\mp 200\text{nm}$ are also shown. The in-plane displacement field is directly visualized by the deformation, whereas the out-of-plane displacement is encoded in the color scale. $d\left(p\right)$ orbitals are (anti)symmetric under rotation by ${180}^{\circ }$.

Figure 3

FEM simulations for an infinite strip. (a) Snowflake strip configuration comprising two different domains, with the lower (upper) domain containing smaller (larger) snowflakes with $\mathrm{\Delta }r=-200\text{nm}$ ($\mathrm{\Delta }r=+200\text{nm}$). The depicted strip hosts $n=11$ resized snowflakes ($n_l=6$ blue and $n_u=5$ red). (b) Acoustic band structure of the blue domain only, with a system size $n=17$. The blue shaded area indicates the strip's bulk bands, whereas the sole remaining bands are degenerate pairs of edge states localized at both boundaries of the strip. (c) Counterpart to (b) for the upper domain, essentially the strip version of Fig. 1. A strip comprising two domains ($\mathrm{\Delta }{r}_{l/u}=\mp 200\text{nm}$, $n_l=18$, $n_u=17$) has the band structure depicted in (d) and (e). In addition to the bulk modes and the edge modes at the physical boundaries (g), it reveals the two topologically protected counterpropagating modes localized at the domain wall [wave function in (f)]. For clarity, in all these band structures we just depicted the modes symmetric to the sample plane. For all FEM calculations fixed boundaries ($\mathbf{u}=\mathbf{0}$) were used at the upper and lower end of the silicon slab.

Figure 4

Finite size sample with an arbitrarily shaped sharp domain wall, simulated using a tight-binding model. One polarization ($p^{-}$) is injected and propagates to the right, which can be identified with the wave function of Fig. 3. (a) Detailed representation, showing all six triangles inside each hexagonal unit cell. Color indicates the square of the mechanical wave amplitude, i.e., the energy. (b) Extracting the component of the $p^{-}$ mode inside each unit cell. (c) Scattering into the other helicity ($p^{+}$) is present, but still strongly suppressed even at sharp corners. Note that in (c) we enhanced the depicted energy by a factor of 10 to make the weak scattered component visible. (d) Weak component of opposite polarization ($p^{+}$) appearing at a corner (color scale different from before). (e) Domain wall geometry (transverse to corner). (f) The total fraction of $p^{+}$ polarization decreases for a smoother domain wall.