Artificial spin ice allows us to probe and manipulate single spins and to observe the statistics of its ground state. It is for example possible to visualize the location of defects in response to different external magnetic fields [127]. A central aspect however remains elusive. Due to the extreme fast spin flipping processes all the dynamics of artificial spin ices are hardly experimentally accessible.
Artificial colloidal ice overcomes this limitation. In this system the nanomagnetic islands are replaced by interacting colloids that are confined in double well traps. As I will show in the following, such a system can exhibit a collective behavior similar to artificial spin ice. The use of micron sized colloidal particles has the major advantage that they can be directly observed with video microscopy and that their timescales are easily experi-mentally accessible. Also the particles as well as their interactions can be manipulated in real time.
The first proposal to realize such a colloidal system came from Libal et.al. [128]. It is based on electrostatically interacting colloids that are confined in a lattice of bistable optical traps. However, the experimental realization of optically based colloidal ice turned out to be challenging. On the one hand, it is difficult to stabilize and control the electric charges of the colloids. On the other hand, the huge number of optical traps would require a very high laser power.
Antonio Ortiz and Pietro Tierno succeeded to experimentally realize an alternative sys-tem [129]. Instead of optical traps they used soft lithography to create gravitational double well structures (Figure 4.8). These are elliptic cavities (21µm by 11µm) with a small hill in the center (figure4.8a-c). A colloidal particle placed in these traps has two possible stable positions, one in each side of the hill. The depth of the trap is chosen such that the interaction with the other colloids can push the particle over the central hill but the particles can never escape the trap. Thermal fluctuation cannot induce a jump over the hill which makes the system athermal like artificial spin ice.
4.5 Artificial colloidal ice
Figure 4.8: a) Scheme of colloidal spin ice in honeycomb geometry. The external magnetic fieldB(purple arrow) induces magnetic moments in the paramagnetic colloids.
b)Profilometer measurement of the soft lithographic structure. c)Cross section of the profile along a double well trap (along the blue line in (a) and (b)). The two stable positions and the central hill are clearly visible. d,e)Spin configurations of honeycomb ice (lattice constant a = 44µm) and square ice (a = 33µm). Blue arrows denote the assigned spin direction. A ground state vertex is highlighted in both cases. f,g)Vertex configuration for honeycomb and square ice sorted according to the corresponding energy.
Scale bar for all images is 20µm. Picture adapted from [129].
The double well traps are then arranged either in a square (figure4.8e) or a honeycomb lattice (figure 4.8d). The colloidal ice can be compared to conventional artificial spin ice by simply assigning arrows to each trap pointing from the vacant to the filled site (figure 4.8d,e). The system can also be described by the dumbbell model introduced in section 4. In this picture a particle close to the vertex would be a positive elementary
charge and a particle on the remote position a negative charge1.
The interaction between the colloids is introduced by using paramagnetic colloids (di-ameter 2r = 10.3µm). Under application of an external magnetic field B the particles acquire magnetic moments m=V χB/µ0, where V is the particle volume and χ ∼0.1 the volume susceptibility. The magnetic field is applied perpendicular to the crystal plane (figure4.8a). The particles will therefore interact via a repulsive isotropic dipolar potential
U = µ0m2
4πr3ij ∝B2. (4.3)
The advantage in comparison to artificial spin ice is obvious, changing the interaction strength no longer requires changing the lithographic pattern but can be easily done by varying the applied magnetic field.
The pairwise repulsion gives rise to four energetically different vertex configuration in the honeycomb lattice (fig. 4.8f) and 6 different configuration in the square lattice (fig. 4.8g). On a single vertex level the repulsive interaction tries to maximize the distance between the particles and therefore negatively charged vertices are energetically favorable. Considering the whole lattice this is no longer true. In contrast to artificial spin ice the overall ground state cannot be derived from a single vertex consideration but arises from a collective effect of all interacting particles. The reason is that the colloidal particles move in contrast to the spins, which results in a higher energy for positive charges than for negative ones. Both are though topologically connected. To maintain overall charge neutrality a negatively charged vertex always has to be balanced by a positive one. In consequence the ground state is not made of the lowest energy vertices but of unchargedSIII and SIV vertices (KII and KIII for the honeycomb lattice), that obey the (pseudo) ice rule.
In [129] this was experimentally verified. The results are shown in figure 4.9. In the square ice the SIII type vertices dominate as the interaction between the particles is increased. The reason for the suppression of SIV vertices is the artifact of the two dimensional projection, which was discussed in the previous section. In the honeycomb colloidal iceKII and KIII vertices dominate in agreement with the pseudo ice rule.
In summary colloidal ice also follows the (pseudo) ice rule. Like conventional artificial spin ice it is a suitable two dimensional model system for spin ice, which additionally has access to the dynamics of the system.
1Since the underlaying degrees of freedom in this case are not magnetic moments or spins, the elementary charges introduced here are not magnetic monopoles. Hence also excited vertices will not carry an overall magnetic charge and are therefore not magnetic monopoles. They can be considered as topological charges or monopoles. Nonetheless, publication [P6] shows that these charges closely resemble the behavior of magnetic monopoles.