As shown in the previous sections spin ice has a lot of interesting properties. However it is not possible to directly observe single spins and their frustrated arrangement in the bulk of the material. Therefore all measurements are restricted to indirect observation, e.g. by neutron scattering [118] or the measurement of averaged quantities such as the heat capacity [119].
4.4 Artificial spin ice
One way to overcome this problem is to design two dimensional artificial model systems that do not show these limitations. The idea that came up in 2006 was to model the classical behavior of the spins with nanoscale magnets. To enable the observation of single spins the three dimensional tetrahedral arrangement is projected into two dimen-sions. There are two different possibilities of doing so, the spins can be arranged either into a square lattice [120] or a honeycomb lattice [121].
Figure 4.5: a)Arrangement of the nanoislands into a two dimensional square lattice.
Each vertex has four adjacent spins. b)Energetically favorable and unfavorable arrange-ment of a pair of either nearest neighbor spins or opposite spins at the same vertex. c) The 16 possible spin configuration are divided into 4 topologically distinct types sorted according to their energy. The percentage given corresponds to the fraction of vertices in a system of non interacting spins. Picture adapted from [120]
To realize the artificial square ice, Wang and co workers used elongated permalloy nanois-lands (80 nm by 220 nm laterally and 25 nm thick) that were built using lithographic methods [120]. The size is chosen such that each island has a single-domain ferromagnetic moment which is stable at room temperature. This stability of the magnetic moments renders artificial spin ice athermal. The shape anisotropy restricts the magnetic moment to point along the long axis of the island. Such islands therefore behave like nanoscale analogues of the Ising spins in spin ice. The direction of the spin could be visualized by magnetic force microscopy (MFM).
The islands are then arranged in a two dimensional square lattice as shown in Figure4.5a.
Each vertex of the lattice has, like in 3D spin ice, four incoming spins which results in 16 possible configurations. However, in the two dimensional projection the interaction between the spins is no longer equal for all pairs. The interaction between two per-pendicular nearest neighbors is stronger than the between two opposite parallel spins,
because the latter have a longer distance. Favorable and unfavorable pair alignments for both situations are shown in figure 4.5b. All 6 pairwise interactions add up to the total vertex energy. The 16 possible spin configurations are divided into 4 topologically distinct classes, which are sorted according to their energy in figure 4.5c. Type I and type II fulfill the two in two out ice rule.
Figure 4.6: Excess percentage of the different vertex types, which is defined as the difference between the observed vertex fraction and the one expected for a random dis-tribution. With an increasing interaction between the spins (decreasing lattice spacing) the percentage of vertices that obey the ice rule (type I and II) increases while the other vertex types are suppressed. Picture adapted from [120]
Wang et. al. could prove that their system in fact mimics the behavior of real 3D spin ice. When the interaction strength between the magnetic moments is increased (which is realized by reducing the lattice spacing) the number of type I and II vertices increases, while the energetically less favorable type III and IV vertices are suppressed. Thus the system follows the ice rule. Its ground state is made of vertices with a two in and two out spin configuration.
A second possibility for artificial spin ice is the honeycomb structure shown in figure 4.7e. Tanaka and co-workers implemented it with a continuous network of ferromagnetic wires [121]. The idea remains the same as in the square artificial spin ice, nanoscale magnetic moments emulate the role of the spins. In contrast to the previous case only three spins meet at each vertex (Figure4.7). Thus the ground state can no longer follow the ice rule and obeys a pseudo ice rule instead. Three in (Figure 4.7a) and three out vertices (Figure4.7d) are suppressed. The ground state is made of one in (Figure4.7c) and two in (Figure 4.7b) sites. However they did not find long range ordering of the two different types of ground state vertices. Note that within the dumbbell model with magnetic charges only the square ice ground state vertices are uncharged. In the honeycomb lattice vertices always carry a charge.
4.4 Artificial spin ice
Figure 4.7: a)-d) Possible configurations of the three spins meeting at each vertex.
The pseudo ice rule favors the ’one in, two out’ and the ’two in, in out’ configurations.
e) Arrangement of the spins on the honeycomb geometry. While the vertices sit on the corners of a honeycomb lattice, the center of the spins are located on the sites of a kagome lattice. Picture adapted from [121]
Lack of residual entropy in artificial spin ice The biggest drawback in all artificial spin ices is the loss of the ground state degeneracy. In square ice the reason for this is quite obvious. In the three dimensional configuration a spin has the same distance to the other three spins on the same tetrahedron. This is no longer true in the square lattice. The opposite spins have a larger distance than the two neighboring, perpendicular spins.
In consequence type I vertices have lower energy than type II vertices (Figure 4.5c).
The system’s unique ground state is therefore made of a alternating pattern of the two different type I vertices.
In the honeycomb lattice the situation is less obvious. On the vertex level all three spins are equal. The degeneracy is nonetheless lifted because every ground state vertex carries a net charge and also a net dipole moment. Upon cooling honeycomb ice it will first reach ice I phase where the system follows the pseudo ice rule. Further cooling will result in ordering due to the Coulombic interaction of the charged vertices. The system reaches the ice II phase where positive charges reside on one sublattice of the honeycomb and negative on the other one. At even lower temperatures the dipole moments of the vertices start to order and form a long range ordered spin ’solid’ without any remaining entropy [43]. First indications for the ice II were found in experiments [122] while the spin ’solid’ state was so far only observed in simulations [43].
There have been different approaches to regain the residual entropy in artificial spin ices.
One approach is to change the geometry of the two dimensional lattice towards a shakti lattice which in fact shows residual entropy [123]. However it is clear that a full analogue to spin ice with four equal spins per vertex requires a 3D arrangement of the spins.
One possibility of realizing a three dimensional version is through self assembly [124].
Unfortunately this allows for little control over the system and its symmetry. Another approach is to start from the two dimensional lithographic patterns and add the third
dimension by stacking multiple layers [125]. This however involves technical problems in the lithographic nano-fabrication process which makes the experimental realization challenging [43]. Recently there were nonetheless some first samples realized with this difficult technique [126].
In section4.7I present a new approach to restore the degeneracy in our two dimensional colloidal model system. It preserves the square geometry and the four coordinated vertices and nonetheless allows to stick to two dimension. The latter should pave the way far a fast experimental realization.