Lattice dynamics of two-dimensional colloidal crystals subject to external light potentials

By exposing two-dimensional crystals to tunable substrate potentials one can selectively manipulate the crystal’s phonon band structure. We explore this idea and study the overdamped lattice dynamics of colloidal crystals subject to commensurate substrate potentials with sinusoidal modulations in up to two spatial directions. We furthermore show that the mean-square displacement of colloids in the crystal can be understood as the Laplace transform of the phonon spectrum and discuss how our results can best be verified experimentally.

Figure 1

Unit vectors ${\vec{e}}_x$ and ${\vec{e}}_y$, lattice constant $a$, and the first-neighbor shell lattice points of a hexagonal lattice.

Figure 2

The first Brillouin zone of the reciprocal lattice of a 2D hexagonal crystal (gray solid line), for a free crystal and for crystals subject to commensurate periodic potentials with modulations in different directions (dashed line). The path enclosing the irreducible section of the first Brillouin zone is plotted as a thick solid line. The numbers label the sections of this path along which the band structure is determined.

Figure 3

Phonon band structures $\lambda \left(\vec{q}j\right)$ of a 2D hexagonal crystal (dashed line) and of a crystal subject to commensurate external potentials (solid line) for the six cases specified in Fig. 2. The gray numbers correspond to the numbers in Fig. 2, identifying the sections of the path along which $\lambda \left(\vec{q}j\right)$ is determined. $(k_1∕k_0,k_2∕k_0)$ is (4,0), (0,4), (4,4), (4,4), (4,4), (4,4) in (a)–(f).

Figure 4

Band structures for case (e) in Fig. 2 and corresponding phonon spectra, for various substrate potential strengths. $(k_1∕k_0,k_2∕k_0)$ is [from (a) to (f)] (0,0), (2,0), (4,0), (4,2), (4,4), (4,6). The gray numbers correspond to the numbers in Fig. 2, case (e), identifying the sections of the path along which $\lambda \left(\vec{q}j\right)$ is determined.

Figure 5

2D contour plots of the band structure in the first Brillouin zone for case (e) in Fig. 2 and for the following combinations of $(k_1∕k_0,k_2∕k_0)$: (0,0), (4,0), (4,4), (4,6) from (a) to (d). (a) is the free crystal; (b) represents also case (a) of Fig. 2. The left (right) column of graphs displays the higher (lower) bands.

Figure 6

(a) Time dependence of the mean-square displacement ${\delta }_{\mathrm{MSD}}\left(\tau \right)$ for the six cases considered in Fig. 4. (b) Mean-square displacement ${\delta }_{\mathrm{MSD}}\left(\infty \right)$ as a function of the strength $k_1$ of the substrate potential for the six cases specified in Fig. 2. If required, $k_2$ is set to $4k_0$. (c) shows for cases (a) and (e) in Fig. 2 the dependence of ${\delta }_{\mathrm{MSD}}\left(\infty \right)$ on the bin size $d\lambda$ of the phonon spectrum $G\left(\lambda \right)$ for four different bin sizes (see plot legend for details).