Localized and Delocalized Motion of Colloidal Particles on a Magnetic Bubble Lattice

We study the motion of paramagnetic colloidal particles placed above magnetic bubble domains of a uniaxial garnet film and driven through the lattice by external magnetic field modulation. An external tunable precessing field propels the particles either in localized orbits around the bubbles or in superdiffusive or ballistic motion through the bubble array. This motion results from the interplay between the driving rotating signal, the viscous drag force and the periodic magnetic energy landscape. We explain the transition in terms of the incommensurability between the transit frequency of the particle through a unit cell and the modulation frequency. Ballistic motion dynamically breaks the symmetry of the array and the phase locked particles follow one of the six crystal directions.

Figure 1

(a) Scheme showing a magnetic bubble lattice [radius $R$, lattice constant $a$, one unit cell is shaded in blue (gray)] with paramagnetic particles deposited on top of it. The magnetic field $\mathbf{H}$ precesses with frequency $\Omega$ under an angle $\vartheta$ around the $z$ axis. (b) Continuous domain area fraction $1-\Phi$ (filled squares) and lattice constant (open circles) for a bubble array at different values of the $z$ component of the field, $H_z$. The insets show polarization microscopy images of the film at different fields $H_z$.

Figure 2

Correlation function versus time for different values of the normal field $H_z$ ($\Omega =21.9{\mathrm{s}}^{-1}$, $H_{xy}=1432\mathrm{A}/\mathrm{m}$). The continuous line shows a power law fit for the ballistic (open squares) and superdiffusive (open circles) behavior. The green (gray) lines correspond to $C(\tau )$ for the localized state obtained at low (open triangle) and high (closed triangles) field strength.

Figure 3

Phase diagram of the exponent $\alpha$ in the ($H_z/M_s$, $\Omega$) plane. White region denote localize colloidal state, cyan (gray) superdiffusive motion and blue (black) ballistic motion. The inset shows the behavior of $\alpha$ with $H_z$ along the red (gray) cut in the phase diagram. Particle trajectory of the localized [(a),(d) in green-white] and delocalized states [(b),(c) in gray] are superimposed on the images on the bottom.

Figure 4

Orbit and transit frequency through the unit cell as a function of the normal field $H_z$. The squares and triangles are for orbital frequency, ${\omega }_o$, the circles for frequency of transit ${\omega }_t$. On the bottom line are schematics (a)–(d) that show the corresponding particle trajectory.