Defect Dynamics in Artificial Colloidal Ice: Real-Time Observation, Manipulation, and Logic Gate

Abstract

We study the defect dynamics in a colloidal spin ice system realized by filling a square lattice of topographic double well islands with repulsively interacting magnetic colloids. We focus on the contraction of defects in the ground state, and contraction or expansion in a metastable biased state. Combining realtime experiments with simulations, we prove that these defects behave like emergent topological monopoles obeying a Coulomb law with an additional line tension. We further show how to realize a completely resettable "NOR" gate, which provides guidelines for fabrication of nanoscale logic devices based on the motion of topological magnetic monopoles. DOI: 10.1103/PhysRevLett.117.168001 Geometric frustration is a complex phenomenon which encompasses a broad range of systems, from magnetic materials [1], to ferroelectrics [2], trapped ions [3], confined microgel particles [4], and folding proteins [5]. It emerges when the spatial arrangement of the system elements cannot simultaneously minimize all interaction energies, and leads to exotic phases of matter with a low-temperature degenerate ground state, such as spin ice [6][7][8]. Artificial spin ice systems (ASI) are lattices of interacting nanoscale ferromagnetic islands, recently introduced as a versatile model to investigate geometrically frustrated states [9,10], including the role of disorder [11,12], thermalization [13][14][15], and the excitation dynamics [16][17][18][19][20]. In opposition to bulk spin ice such as pyrochlore compounds, ASI allow us to directly visualize the spin textures and to tailor the spatial arrangement of the system elements.An intriguing aspect in ASI, which is attracting much theoretical interest, is the dynamics of defects [21][22][23][24][25][26][27][28]. The interactions between pairs of defects is one of the distinctive features between three dimensional (3D) and two dimensional (2D) spin ice. In a 3D pyrochlore compound, the spins are located on a lattice of corner-sharing tetrahedra, and can point either towards the tetrahedra center (spin in), or away from it (spin out). Thus the ground state (GS) follows the "ice rules," with two spins coming in and two going out of each vertex in order to decrease the vertex energy. At finite temperature, defects that behave like "magnetic monopoles" [29,30] can emerge when a spin flips, producing a local increase of the magnetic energy. A way to overcome the system complexity is to use the "dumbbell" model [31], which only considers the magnetic charge distribution at the vertices of the lattice. Within this formalism, it was shown that in 3D spin ice, a pair of defects connected by strings of flipped spins only interact through a magnetic Coulomb law at low temperature. In contrast, numerical simulations show that for a 2D square ASI, i.e., a projection of the 3D ice system on a plane, such a string requires an additional energetic term in the form of a line tension [21]. The reason is that, while in a 3D system all spin configurations that satisfy the ice rules have equal ener...

Colloquium : Ice rule and emergent frustration in particle ice and beyond

“…(13) is zero. That is certainly true if a lattice has point reflection symmetry in the middle points {x} of each edge, and explains why the kagomé and square particle-based ices follow the ice rule, as found previously numerically and experimentally (Libál et al, 2006;Loehr et al, 2016b;Ortiz-Ambriz and Tierno, 2016).…”

“…(a) Colormap showing the vertex charges for a line of flipped particles connecting two q = ±2 defects. These defects annihilate after t = 4.5 s due to the external field, applied at t = 0 s. Image edited with permission fromLoehr et al (2016b). (b) Microscope images showing a square ice state prepared in its ground state of type IV (q = 0) vertices (left), or in a metastable state with a grain boundary (right).…”

“…(e,f) Images showing a 0 output obtained from (1, 0) input. In (e) the 1 input of the first row containing particles with high susceptibility is induced by an extra charge coming from outside the region, fromLoehr et al (2016b).…”

Information transmission and signal permutation in active flow networks

“…The planar and non-planar network designs proposed and investigated here could be implemented and tested in microfluidic chips, exploiting recent progress in 3D printing [54] and in the geometric control of collective transport in dense suspensions of microorganisms [18,19] and ATP-powered microtubule bundles [23]. Furthermore, recent progress in experimental realisation of artificial magnetic and colloidal spin-ice systems [40,55,56] suggests that the input-output spin-ice model studied here could itself be directly realised. More broadly, however, the above results establish a direct link between active matter and ostensibly unrelated mathematical concepts in information and group theory, thus promising novel symmetry-based approaches to autonomous network design.…”

Triple Junction at the Triple Point Resolved on the Individual Particle Level

“…In these systems, a face-centered cubic (fcc) crystal, a bodycentered cubic (bcc) crystal and a fluid can coexist. Colloids have long been used as model systems to explore such situations, including nucleation [26][27][28][29], crystallization [30][31][32][33][34], melting [16,35,36], defects [37], glass transition [38][39][40][41], solid-liquid interfaces [42][43][44][45][46], solid-solid phase transformations [47][48][49][50] as well as the microscopic response to external forces [51][52][53][54].…”