Artificial spin ice and especially colloidal spin ice offer a lot of new opportunities com-pared to real three dimensional spin ice. They allow for direct observation and manipu-lation of the single spin degrees of freedom and even open up the possibility to directly observe the dynamics of monopole defects. However, as discussed two dimensional model
3It is clear that due to their low Reynolds numbers (Re∼10−6) the colloidal motion is overdamped, that is inertia is negligible. This suggests that defects in colloidal ice follow the same dynamics. In the supplementary material of [P6] we validated this assumptions with the help of numerical simulations
4.7 Restoring the residual entropy in artificial colloidal ice
systems have one major drawback. As discussed in the chapter on artificial spin ice (sec-tion 4.4) they do not reproduce one of the most intriguing features of 3D spin ice, its residual entropy.
In this chapter I will present a new approach to restore residual entropy in colloidal ice.
In contrast to previous attempts based on artificial spin ice this truly two dimensional approach should be easier to realize in experiments. The idea is to combine two particles of different magnetic moments with two types of traps with different lengths. In the following I will theoretically show that there are combinations of lengths and moments that result in equal energies for all ice rule vertex configurations and hence gives rise to residual entropy. I validate the theoretical result with numerical simulations. This gives strong indications that our approach in fact shows the ground state degeneracy of 3D spin ice.
Figure 4.10: Vertex geometry of the modified spin ice. The horizontal traps are shorter than the vertical traps. The two different types of traps contain particles of different magnetic moments (red and blue). a) All four SIV type vertices have equal energy E1. b,c) The two former ground state configuration (SIII) now have different energies E2 6=E3
A vertex of the modified system is shown in figure 4.10. It consists of traps of different lengths lx along thex-direction andly along they-direction. The length of the trap is defined as the distance between the two stable positions on both sides of the central hill.
The vertices thereby remain in their original square configuration with a lattice constant a.
The different types of traps are occupied by two different types of particles with magnetic moments mx and my. Here again a perpendicular magnetic field B is used to induce dipolar interactions between the paramagnetic colloids. The two magnetic moments mi = χiViB can be realized by either using particles of different volume susceptibility χi or with colloids of different diameter and therefore different volumeVi.
The sum of the 6 pairwise dipolar interaction energies
Uij = µ0mimj
4πr3ij (4.4)
of the 4 particles per vertex gives the total vertex energy E= X4
i,j=1 j>i
Uij (4.5)
whererij is the distance between the particlesiand j. All 4 possible SIV vertices have the same energyE1 (figure 4.10a). But the two former ground stateSIII can now have different energiesE2 6=E3 (figure4.10b,c). The requirement for residual entropy is, that all 6 configurations that fulfill the ice rule must have the same energy. Thus the set of equations
E1 =E2 =E3 (4.6)
has to be solved, where each of the energiesE =E(lx, ly,mmy
x) is a nonlinear function of the system parameters. Note that the two magnetic moment degrees of freedom can be reduced to the ratio since the total moment is anyways scaled by the magnitude of the external field. I used a self made maple program to numerically determine parameters that solve equation (4.6). I obtained several sets of parameters that physically possible solutions to the problem. Although a lot of them contain values that are difficult to be realized in experiments there are still solution that lie in the experimentally accessible range.
I then performed numerical simulations with these solutions. The parameter set that yields the best results in simulations is
lx= 0.291·a;ly= 0.835·a;my
mx = 0.082. (4.7)
The simulations scheme is the same that I also used in [P6]. The particles have the same size 2r= 10.3µm, but different susceptibility with χx = 0.08 for the higher value.
The system of 15 by 15 vertices is bigger than the in experiments accessible size. Only the central 4 by 4 vertices are used to determine the statistics of the ground state. In the beginning the system is in a disordered state. The simulations runs for 30 s (time step ∆t= 0.01 s) with the magnetic field switched. After that the vertex distribution is measured. All results are averaged over 100 simulation runs.
The result is shown in figure4.11. At high magnetic fields the ground state is dominated by SIII and SIV vertices that obey the ice rule. But in contrast to the normal colloidal ice (compare figure4.9) theSIV are no longer suppressed and instead now have approx-imately twice the fraction of theSIII sites. This is expected from the different numbers of possible configurations (4 and 2). Note that the system used for simulation is bigger than the experimental system. But the behavior in a smaller, experimentally accessible system is basically the same except for a slightly higher fraction of non ground stateSII
vertices due to finite size effects. To increase the number of ice rule vertices and to get