Figure 1

Illustration of the colloidal system subjected to commensurate optical interference patterns. Gray dots represent colloidal particles and gray lines indicate the minima of the light-induced 1D periodic potentials. The lattice constants $d_1$ and $d_2$ of the 1D periodic potentials are chosen to be commensurate with the mean particle distance $a$.Reuse & Permissions

Figure 2

Experimental realization of a 2D colloidal crystal on a 1D periodic substrate [see Fig. 1a]. (a) Micrograph of the 2D colloidal crystal in the central region of the field of view ($300\mu \mathrm{m}\times 200\mu \mathrm{m}$). (b) Zero and first order Bragg peaks (black dots) and first Brillouin zone (gray area). The black solid line represents the irreducible path along which the phonon band structure is plotted and the path sections are marked by numbers. (c)–(e) 2D correlation function $g(x,y)$ and (f)–(h) phonon band structure ${\lambda }_s(q)$ along the irreducible path for (c),(f) the spontaneous crystal, i.e., $k_1/k_0=0$, (d),(g) $k_1/k_0=1$, and (e),(h) $k_1/k_0=2.5$. Closed and open symbols refer to the upper and lower band, respectively. Solid lines represent band structures calculated in the framework of harmonic lattice dynamics [14].Reuse & Permissions

Figure 3

Experimental realization of a 2D colloidal crystal in the presence of two 1D periodic substrates rotated by 90° (see Fig. 1). The graphs are 2D contour plots of the phonon band structure in the first Brillouin zone derived from experimental (left column) and analytical data (right column). (a) $\{k_1/k_0,k_2/k_0\}=\{0,0\}$. (b) $\{k_1/k_0,k_2/k_0\}=\{1.4,0.2\}$. (c) $\{k_1/k_0,k_2/k_0\}=\{1.5,1.5\}$. (d) $\{k_1/k_0,k_2/k_0\}=\{0.5,1.5\}$. Experimental contour plots were determined for approximately 800 $\overrightarrow{q}$-values. Calculated plots were obtained in the framwork of harmonic lattice dynamics using approximately 3000 $\overrightarrow{q}$-values [14].Reuse & Permissions

Figure 4

Phonon band structure (solid symbols) of a paramagnetic 2D binary crystal (see inset) calculated in the harmonic approximation. The black particles in the inset have a magnetic susceptibility 20 times larger than the susceptibility of the gray particles. The magnetic field $\overrightarrow{B}$ is perpendicular to the system plane. The band gap can be closed for $\overrightarrow{B}=\overrightarrow{0}$ and $\lambda >0$ if all the particles additionally interact via a Lennard-Jones potential (open symbols).Reuse & Permissions