Localized and Delocalized Motion of Colloidal Particles on a Magnetic Bubble Lattice Pietro Tierno, 1 Tom H. Johansen, 2 and Thomas M. Fischer 1, * 1 Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida, USA 2 Department of Physics, University of Oslo, Blindern, Norway (Received 28 March 2007; published 20 July 2007) 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 super- diffusive 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. DOI: 10.1103/PhysRevLett.99.038303 PACS numbers: 82.70.Dd, 66.20.+d, 67.40.Hf A particle driven through a periodic potential with pin- ning sites reveals fascinating dynamics which encompass localization at the pinning sites, ballistic motion through the lattice, or even superdiffusive (chaotic) motion. Theoretical models cover a broad range of phenomena, e.g., atomic diffusion on metallic surfaces [1], flux flow in Josephson-junction arrays [2], and the motion of phonons in the Frenkel-Kontorova model [3]. Recent experiments with holographic optical tweezer [4,5], have shown that colloidal particles represent an experimentally accessible model system to investigate how the particle dynamics is affected by a corrugated potential. There, a periodic po- tential was generated with a laser using diffractive ele- ments. However there is a rich variety of means to achieve miniaturized periodic patterns resulting in, e.g., high di- electric or magnetic susceptibility contrast modulations. In particular, cylindrical magnetic ‘‘bubble’’ domains in uni- axial garnet films provide strong and externally tunable magnetic pinning sites at the location of the domain walls of the magnetic bubbles. External modulations alter the bubble domain pattern and drive paramagnetic particles placed above the film through the bubble array. The ability to tune the particle interaction with the pinning lattice and to vary the strength or frequency of the external driving modulation leads to novel implications for both device fabrication [6] and fundamental studies [7]. Here we report on the externally driven dynamics of paramagnetic colloidal particles in an aqueous solution above a regular lattice of magnetic bubbles. A precessing external magnetic field simultaneously controls the bubble array pattern and the colloidal particle motion. The field precession lets the colloidal particles either orbit around one localized magnetic bubble, stochastically or ballisti- cally move through the bubble lattice, or confines the motion to the interstitial region between the neighboring bubbles. We show that the different regimes of motion are controlled by the precession frequency , and the normal component of the magnetic field H z . Using video micros- copy with a particle tracking routine, we identify the transitions between the various regimes and determine the full dynamical phase diagram of the driven system. To create a magnetic bubble lattice we use a ferrite garnet film with uniaxial anisotropy and composition Y 2:5 Bi 0:5 Fe 5q Ga q O 12 (q  0:5–1), thickness 5 m and saturation magnetization M s  1:7  10 4 A=m. The equilibrium domain pattern of such a film is a stripe labyrinth pattern. High frequency ( 12  10 3 s 1 ) mag- netic field pulses of amplitude 10 5 A=m normal to the film enforce the formation of the metastable magnetic bubbles inside the garnet film. The bubbles are cylindrical domains, with diameter 2R  8:2  0:1 m, of reverse magnetization separated by a continuous magnetized film (see Fig. 1)[8]. Application of an external magnetic field H z parallel (antiparallel) to the bubble magnetization di- rection increases (decreases) the size of the cylindrical domains. For the magnetic field strengths used here, H z < 7000 A=m, the bubble shape remains circular and the lattice keeps its hexagonal form, as seen from the micro- scope images in Fig. 1. The bubble array pattern is then fully described by the lattice constant a and the bubble area fraction  [9]. In part (b) of Fig. 1 we show the con- tinuous domain area fraction 1   (squares) and the lat- tice constant a (circles) of the lattice measured as a func- tion of the external magnetic field. The bubble area fraction linearly decreases with increasing magnetic field H z nor- mal to the film while the lattice constant only changes marginally [10]. The colloidal suspension consist of polystyrene para- magnetic particles with a diameter D  2:8 m and ef- fective magnetic susceptibility   0:17 (Dynabeads M-270). In aqueous solution the particles are electrostatic stabilized by the dissociation of surface carboxylic groups (COO  ). We coat the film with a thin layer of polysodium- 4-styrene sulfonate to prevent adhesion of the particles to the film [11]. A water drop containing 7  10 6 beads=ml was placed on top of the garnet film and the particle PRL 99, 038303 (2007) PHYSICAL REVIEW LETTERS week ending 20 JULY 2007 0031-9007= 07=99(3)=038303(4) 038303-1  2007 The American Physical Society