Novel Pyrochlorelike Crystal with a Photonic Band Gap Self-Assembled Using Colloids with a Simple Interaction Potential

Using computer simulations, we investigate the phase behavior of a system of particles interacting with a remarkably simple repulsive square-shoulder pair potential and report the formation of a novel (and stable) pyrochlorelike crystal phase. The lattice structure of the pyrochlorelike phase formed in our simulations possesses two inherent length scales corresponding to the inter- and intratetrahedral neighbors. We show that it can be used to fabricate a photonic crystal which displays complete photonic band gaps in both the direct and inverted dielectric structures.

Figure 1

Phase diagram in the temperature-density representation obtained for the HCSS system with shoulder width $\delta =2.10{\sigma }_{HS}$. The reduced quantities are defined as $T^{*}=k_{B}T/\epsilon$ and ${\rho }^{*}=N{\sigma }_{HS}^3/V$. The phases shown are FL, bcc, hfcc, mfcc, lfcc, PYR, and col. The gray regions denote the two-phase coexistence regions.

Figure 2

(a) Radial distribution function $g(r)$ for the iPYR and the PYR obtained in our simulations at densities where the positions of the peaks at $r=2.10{\sigma }_{HS}$ match. The intra- (purple) and intertetrahedral (yellow) neighbors of a particle (green) corresponding to a certain peak are highlighted in the sample configurations given in the insets. (b) A sample configuration of PYR showing the intra- (purple) and intertetrahedral (yellow) neighbors of a particle (green) along with their coronas.

Figure 3

Comparison of the photonic properties of pyrochlore structures, namely, the iPYR and the PYR obtained in our simulations: (a),(c) relative gap width $\mathrm{\Delta }\omega /{\omega }_m$ as a function of (reduced) density ${\rho }^{*}=N{\sigma }_{HS}^3/V$ and (b),(d) band gap map calculated for (a),(b) a direct structure of silicon spheres in air and (c),(d) an inverse structure of air spheres in silicon. In (a),(b), the densities where the particles start to overlap in iPYR $({\rho }_{\text{iPYR}}^{*})$ and where the particles in a tetrahedron $({\rho }_{\mathrm{PYR}\text{−}1}^{*})$ and between neighboring tetrahedra $({\rho }_{\mathrm{PYR}\text{−}2}^{*})$ start to overlap in a PYR lattice are individually marked. The reduced frequency is written as ${\omega }^{*}=\omega a/2\pi c$. A representation of the configuration of PYR pertaining to the density at which the maximum band is obtained is given in (b) for the direct (${\rho }^{*}=1.18$) structure and in (d) for the inverse (${\rho }^{*}=2.52$) structure. Here, the green and red colors, respectively, denote silicon and air.