New Journal of Physics 14, 025006 (2012)

Multi-Step Ordering in Kagome and Square Artificial Spin Ice

C.J. Olson Reichhardt¹, A. Libál², and C. Reichhardt¹

¹Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
²Faculty of Mathematics and Computer Science, Babes-Bolyai University, RO-400591 Cluj-Napoca, Romania

New Journal of Physics 14 (2012) 025006 (15pp)
Received 30 November 2011
Published 20 February 2012
Online at http://www.njp.org/
doi:10.1088/1367-2630/14/2/025006

Abstract. We show that in colloidal models of artificial kagome and modified square ice systems, a variety of ordering and disordering regimes occur as a function of biasing field, temperature, and colloid-colloid interaction strength, including ordered monopole crystals, biased ice rule states, thermally induced ice rule ground states, biased triple states, and disordered states. We describe the lattice geometries and biasing field protocols that create the different states and explain the formation of the states in terms of sublattice switching thresholds. For a system prepared in a monopole lattice state, we show that a sequence of different orderings occurs for increasing temperature. Our results also explain several features observed in nanomagnetic artificial ice systems under an applied field.

Contents
1. Introduction
2. Orderings for kagome ice
3. Orderings for square ice
4. Summary
Acknowledgments
References

1. Introduction

Spin ices have been extensively studied as ideal systems that exhibit geometric frustration effects since not all the pairwise spin interactions can be satisfied simultaneously [1,2,3,4]. Such systems are termed spin ices due to their similarity with a water ice phase in which proton ordering into corner-sharing tetrahedra is frustrated but obeys the "ice rule" of two protons "in" (close to the O atom) and two protons "out" (far from the O atom) in every tetrahedron [3]. In spin ices, the corresponding spin ice rules determine how many spins point toward or away from each vertex in the system, and these rules vary depending on the geometry of the system. For example, in two dimensional (2D) square ice, ice-rule-obeying states have two spins pointing toward each vertex and two spins pointing away, and it is possible for the spins to organize into an ordered ground state configuration [5,6]. For 2D kagome ice, ice-rule-obeying states have two spins in and one spin out out or two spins out and one spin in; in this case, the lowest energy state is not ordered but contains only vertices that obey these ice rules.

There has recently been growing interest in 2D artificial spin ice systems created using nanomagnetic arrays [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19], 2D colloidal particle assemblies [20,21], or vortices in nanostructured superconductors [22]. In the nanomagnetic arrays, the orientation of the magnetic moment of each nanoisland produces the effective spin ordering. The colloidal and vortex systems more closely resemble the water ice system in that the repulsive interactions of the colloids or vortices at vertices resemble the interactions between charged protons, and the ice rules indicate the number of colloids or vortices that sit close to or away from a vertex. Experimental versions of these systems allow for direct imaging of the microscopic spin configurations as well as for careful control of numerous parameters that are not accessible in real spin systems.

In the work of Wang et al. [5] on 2D square ice, as the interactions between neighboring nanomagnets increased, the system was increasingly filled with ice-rule-obeying vertices; however, the predicted ground state was not observed due to quenched disorder effects in the nanomagnet array. Experiments on kagome artificial ice systems soon followed and showed that the ice rules predicted for that geometry were obeyed [14]. In more recent square ice studies using more advanced techniques, large regions of the sample adopted the square ice ground state while non-ice-rule obeying vertices created grain boundaries [9]. Such grain boundary formation was previously predicted in vortex simulations to occur for weak quenched disorder, while for stronger disorder isolated non-ice-rule defects begin to appear [22].

Lack of thermalization is an issue for the experimental nanomagnet systems since thermal activation of the spin configurations is weak or nonexistent. This can be partially mitigated by applying a changing external field [7,8]. It was recently shown that biased states appear under applied external fields for both square and kagome ice systems, while cycled external fields can be used to generate hysteresis curves [12,11,10,13]. For kagome ice, Zabel et al. [11] observed an ordered monopole state where each vertex has either all spins pointing out or all spins pointing in, with the vertices alternating sign in a crystalline arrangement. Other studies have examined the creation and motion of monopole excitations in the presence of a biasing field for both square and kagome ice systems [12,10]. Recent theoretical studies have found that 2D kagome ice can undergo a two-stage ordering transition from a paramagnetic state to an ice-rule-obeying state followed by an ordered monopole state [23]. The thermal disordering or melting of the ordered states can be readily studied in colloidal artificial ices, and recently materials have been created with a spin ordering temperature that is near room temperature, making it now possible to melt magnetic artificial spin ices [24].

Here we show that a variety of different types of orderings are possible in both kagome and modified square ice systems as a function of external field, temperature, and interaction strength for a colloidal artificial spin ice system composed of arrays of elongated optical traps that each capture one colloid. Experimental studies have demonstrated frustration in buckled 2D colloidal layers [21] while numerical studies have shown that charged colloids in elongated trap arrays exhibit the same spin ice rules observed for nanomagnet arrays [20]. Computational and experimental studies of colloidal ordering on 2D square and triangular substrates have revealed numerous ordered states and multi-step transitions between the states [25,26,27,28,29,30,31,32]. The elongated optical traps we consider here permit each colloid to sit in one of two locations at either end of the trap. Arrays of this type of trap have been created experimentally to study various types of ratchet effects, and preferential ordering of the colloids into a state that minimized the colloid-colloid interactions was observed [33]. An advantage of the colloidal system over the nanomagnetic systems is that thermal activation of the effective spin degrees of freedom is possible. The relative importance of the thermal fluctuations can be varied either by adjusting the temperature or by holding the temperature fixed and modifying the strength of the colloid-colloid interactions. It is also possible to bias the colloidal artificial ice system using an electric field.

Figure 1: (a) Schematic of artificial square ice system consisting of elongated traps with two potential minima. An elementary unit or vertex consists of four traps that each capture one charged colloid. The effective spin direction is defined to be toward the end of the trap where the colloid is sitting. Ice-rule obeying states have two colloids close to the vertex and two colloids away from the vertex; one of the two possible ground state configurations is illustrated. Inset: Arrow indicates orientation of external drive for the square ice. (b) Schematic of an artificial kagome ice system. Ice-rule obeying states have either two colloids close to the vertex and one away from the vertex or two colloids away from the vertex and one close to the vertex, as shown. Inset: Arrow indicates orientation of external drive for the kagome ice.

Simulation- We consider a 2D array of N elongated traps that each contain a single colloid. Each trap has two potential minima where the colloid can sit, as shown in figure 1(a) where we illustrate a single vertex of the square ice system. The effective spin at each site is defined to point toward the end of the trap occupied by the colloid. The colloids interact with each other via a repulsive screened Coulomb or Yukawa potential given by V(r_ij) = r_ij⁻¹exp(−κr_ij)∧r_ij where κ = 4/a₀ is the screening length and a₀ is the average spacing between the vertices. Since the colloids are repulsive it is very energetically costly for all the colloids around a vertex to sit close to the vertex in a positive monopole state; on the other hand, the lowest energy state for a vertex is the negative monopole configuration in which all the colloids sit away from the vertex. Due to geometrical constraints it is not possible for all of the vertices to adopt negative monopole configurations. The most cost-effective way to accommodate the colloid-colloid repulsion in the square ice is for each vertex to have two close colloids and two far colloids in a "two in-two out" state that obeys the square ice rules. These ice rule obeying states can be biased, with the close colloids occupying traps oriented at 90^° to each other, or they can be the slightly lower energy ground state configuration shown in figure 1(a) [20]. For kagome ice, shown schematically in figure 1(b), the ice rule obeying vertices have either two in and one out or one out and two in.

The colloid dynamics evolve according to the following overdamped equation of motion:

dR_i

= F^cc_i + F^ext + F^T_i + F^s_i .

(1)

Here η is the damping constant and the colloid-colloid interaction force is F_i^cc = −F₀q²∑^N_{i ≠ j}∇_i V(r_ij) with F₀ = Z^*2/(4πϵϵ₀a₀²), r_ij=|r_i − r_j|, and ∧r_ij=(r_i−r_j)/r_ij. r_i(j) is the position of particle i(j), Z^* is the unit of charge, ϵ is the dielectric constant of the solvent, and q is the magnitude of the charge on a single colloid. The strength of the repulsion between the colloids can be controlled by varying q. As an example of a physical value for F₀, using the experimental parameters reported in [34] gives F₀=15.6pN. The thermal force F^T arises from Langevin kicks with 〈F^T_i〉 = 0 and 〈F_i(t)F_j(t^′)〉 = 2ηk_BTδ_ijδ(t − t^′). The substrate force F^s_i arises from N traps composed of two parabolic ends capping a cylindrical confining area of length l=1.333a₀ and width d_p=0.4a₀ with a maximum strength of F_p and radius r_p; an additional parabolic barrier of height f_r is placed at the center of the trap to produce two potential minima at each end of the trap [20]. This geometry is similar to that considered in [20]. The external biasing force is given by F^ext=F_ext[cos(θ_ext)∧x + sin(θ_ext)∧y], with θ_ext=0 for kagome ice and θ_ext=45^° for square ice, as illustrated schematically in the insets of figure 1. For the nanomagnetic system, an in-plane applied external field was used as a biasing force that could align the magnetic moments of the nanoislands. For the colloidal system, the sample can be biased by an in-plane electric field, while for a system of vortices in a type-II superconductor the bias would come from an in-plane applied current. In this work we consider external forces that are strong enough to induce hopping over the central barrier of the traps but not strong enough to cause the colloids to escape from the traps.

Figure 2: A kagome ice system with no colloid-colloid interactions (q²=0) and f_r=4.0 for F_ext applied along the x axis. (a) The effective magnetization M for the entire sample (black filled circles) vs time in simulation steps, along with the sublattice magnetizations M₁ (green open squares) and M₂₃ (red x's). (b) Simulation protocol: F_ext vs time. The system is initially driven into an ice rule obeying state which transitions into a monopole state when F_ext becomes negative. (c) N_i/N, the fraction of vertices of type N_i, vs time. Black filled circles: N₀/N; red open squares: N₁/N; green x's: N₂/N; blue +'s: N₃/N. The monopole states are N₀ and N₃.

Figure 3: Black circles: particle locations; open ellipses: trap locations for a 20a₀ ×20a₀ section of a kagome ice sample. Colored circles indicate vertex types: N₀ (blue), N₁ (green), N₂ (yellow), and N₃ (red). (a) The positive biased ice rule obeying state. (b) The monopole state consisting of an ordered lattice of N₀ and N₃ vertices. (c) A finite temperature ice-rule obeying non-biased state. (d) A high temperature paramagnetic state where the vertex types are uncorrelated.

2. Orderings for Kagome Ice

We first consider a kagome ice array in an external field F_ext applied along the x axis, in alignment with the axes of 1/3 of the traps in the sample. In figure 2 we plot the fraction of vertex types N_i/N, the effective magnetization M, and the applied external force F_ext as a function of time for a sample with noninteracting colloids at q²=0. The vertex types are defined as follows: N₀ has no colloids near the vertex; N₁ has one colloid near the vertex and two far from the vertex; N₂ has two colloids near the vertex and one far; and N₃ has three colloids near the vertex. N₁ and N₂ vertices obey the kagome ice rules, while N₀ and N₃ vertices are negative and positive monopoles, respectively.

In figure 2(c), the noninteracting colloids begin in a randomized state with N₀/N = N₃/N = 0.12 and N₁/N = N₂/N = 0.38. As shown in figure 2(b), F_ext is gradually increased from zero. Just above F_ext = 1.0, the system jumps into the biased ice-rule obeying state illustrated in figure 3(a). We define the contribution of each trap to the effective magnetization M according to the component of the effective spin direction that is aligned with F_ext. Each trap is assigned a value s_i=±1 depending on whether it is biased with or against the direction of F_ext, and this value is then weighted by cos(θ_i). The sublattice of trap sites that are aligned with the x axis (marked in grey in figure 4) have θ_i=0 while the remaining trap sites that are oriented at ±60^° to the x axis have θ_i=60^°. In the biased state the net magnetization is M/N_norm = N_norm⁻¹∑_i=1^Ns_icos(θ_i) = 1.0, where N_norm=cos(0^°)N/3 + cos(60^°)2N/3, and in figure 2(a) the sample reaches the biased state at F_ext=3.7.

Figure 4: Schematic of the formation of the ordered monopole state in a kagome ice sample under a biasing drive applied along the x axis. Black circles are colloid positions and ellipses indicate trap locations. The traps can be broken into two sublattices: sublattice A aligned with the x axis (shaded ellipses) and sublattice B oriented at ±60^° from the x axis (open ellipses). For noninteracting colloids (q²=0) the switching or coercive field for sublattice A is lower than for sublattice B. Colored circles indicate vertex types: N₀ (blue), N₁ (green), N₂ (yellow), and N₃ (red). (a) The system begins in the positively biased ice rule obeying state. (b) As F_ext becomes increasingly negative, it crosses the coercive field of sublattice A first. Colloids in sublattice A switch while those in sublattice B do not, creating the ordered monopole state. (c) As F_ext continues to decrease, sublattice B switches and the sample reaches the negatively biased ice rule obeying state.

After the sample is fully biased, we decrease F_ext as shown in figure 2(b). At F_ext=−1.7, the biased state switches into an ordered monopole state where N₀ and N₃ each jump from 0% of the population to 50% of the population while N₁ and N₂ simultaneously drop to zero as shown in figure 2(c). The ordered monopole state, illustrated in figure 3(b), is a lattice of positive and negative monopoles and is the same state observed in experiments on kagome nanomagnetic systems under external drives [11]. The ordered monopole state arises due to the fact that the kagome lattice is effectively composed of two sublattices that have different switching fields under an external x-direction drive. This is illustrated schematically in figure 4 where the darker ellipses indicate the sublattice with a lower switching field, which we term sublattice A. For the case of non-interacting colloids with q²=0, the switching field for sublattice A is simply the maximum force f_r of the barrier in the center of the trap. Since the applied drive is not aligned in the direction of the traps of sublattice B, the switching force for sublattice B has a higher value of f_r/cos(60^°). Note that in the actual simulation, illustrated in figure 3(b), the colloids need not move to the end of the trap but must only cross over the central barrier in order to change the state of the vertex. When we apply a sufficiently large F_ext in the positive x-direction to cause both the A and B sublattices to switch, the system enters the positively biased state illustrated in figure 4(a). After we reduce F_ext back to zero, we begin applying F_ext in the negative x direction. Sublattice A switches in the negative x direction first due to its lower threshold, but since the colloids in sublattice B have not yet switched, the ordered monopole state forms as shown in figure 4(b). For q² ≠ 0 or finite colloid-colloid interactions, creation of monopoles becomes energetically unfavorable, so for sufficiently high q² the switching of sublattice A simultaneously induces a switching of sublattice B, bringing the system directly into the negatively biased state illustrated in figure 4(c) and preventing the formation of the ordered monopole state. For a certain range of nonzero q², however, the ordered monopole state can still be stabilized when F_ext passes through zero. Our results suggest that in the experiments of Zabel et al. [11] the magnetic islands are in the weakly interacting or noninteracting limit.

Figure 5: N_i/N vs temperature T in kagome ice samples with f_r=1 that have been initialized in the ordered monopole state; F_ext is set to zero before the temperature is applied. Black filled circles: N₀; red open squares: N₁; green x's: N₂; blue +'s: N₃. (a) A sample with zero pairwise interactions q²=0. Here the ordered monopole state melts directly into a paramagnetic state. (b) A sample with q²=0.3. As T increases, the monopole density drops to zero while N₁/N and N₂/N pass through peak values of 0.5, indicating that the ordered monopole state melts into an ice-rule obeying state. At higher temperatures, monopoles reappear while N₁/N and N₂/N decrease and approach the value expected for the paramagnetic state.

We next examine the thermal disordering of the ordered monopole state for varied colloid-colloid interaction strengths. In figure 5(a) we plot the vertex populations N_i/N versus temperature T for a noninteracting sample with q²=0, while figure 5(b) shows N_i/N versus T for an interacting sample with q²=0.3. In each case we set F_ext=0 before applying the temperature. In the noninteracting system in figure 5(a), the ordered monopole state persists up to T = 2.5, at which point the N_i/N cross over to the values expected for an uncorrelated random arrangement or paramagnetic state. In figure 5(b) for q² = 0.3, N₃/N and N₀/N drop to zero with increasing T while N₁/N and N₂/N increase to 0.5, indicating a crossover from the ordered monopole state into an ice-rule obeying state. This is not the biased ordered state illustrated in figure 3(a); instead, the ice-rule obeying state lacks true long range order, as shown in figure 3(c). As T is further increased, N₃/N and N₀/N gradually increase while N₁/N and N₂/N decrease to values close to those expected for a random or paramagnetic state of the type illustrated in figure 3(d).

Figure 6: Phase diagram for T vs pairwise interaction strength q² for the kagome ice system with f_r=1 indicating regions where the ordered monopole, ice-rule obeying, and random or paramagnetic states appear. For weak interactions (small q²) the monopole state disorders directly into the random state, while the width of the ice-rule obeying region grows with increasing q². Red symbols: beyond the highest value of q² where the ordered monopole state can be created with a biased drive protocol, the system can be artificially placed into the ordered monopole state and then melted into the ice-rule obeying state.

By conducting a series of simulations, we map out the different orderings as a function of temperature and pairwise interactions as shown in figure 6. The line separating the ice rule obeying state from the random state is defined as the point at which N₁/N and N₂/N reach values that are within 10% of those expected for a random vertex arrangement. For weak interactions or small q² the ordered monopole state melts directly into the random state. As q² increases, the ice-rule obeying region increases in extent while the line marking the crossover between the ordered monopole state and the ice-rule obeying state drops to lower T. These results suggest that the ordered monopole states can only be observed for systems with effective spin-spin interactions that are weak compared to the strength of the switching or coercive field. For q² > 0.4 the ordered monopole state can no longer be prepared by means of the external field protocol since the repulsion between colloids becomes strong enough that switching of the A sublattice immediately induces switching of the B sublattice. Nevertheless, the ordered monopole state is still stable at lower T for q² > 0.4 as can be seen by artificially preparing the lattice in the ordered monopole state and then increasing the temperature. Such artificial preparation could be achieved in colloidal systems by, for example, tuning the colloid-colloid interactions to a lower value in order to create the ordered monopole state and then increasing the interactions to a higher value; however, it is generally not possible to vary the interaction strength in magnetic systems. The line marking the melting into the ice-rule obeying state obtained by using artificially created ordered monopole states is marked with a dashed line in figure 6.

Figure 7: (a) Schematic of a modified square ice sample containing two trap sublattices A (shaded ellipses) with central barrier f_r¹ and B (open ellipses) with central barrier f_r². We take f_r²/f_r¹=2 so that sublattice A has a lower threshold for switching. F_ext is applied along a line oriented at 45^° to the x axis. (b) A positively biased ice rule obeying state is formed by initial driving in the positive direction, θ_ext=45^°. Yellow circles: positively biased N₂^b vertices. (c) When the driving direction is switched to the negative direction, sublattice A flips first producing a checkerboard monopole state. Blue circles: N₀ vertices; red circles: N₄ vertices. The colloid-colloid interactions must be weak in order for the ordered monopole state to remain stable.

Figure 8: A modified square ice system as illustrated in figure 7 with q²=0.4, f_r¹=1.0, and f_r²=2.0 for F_ext applied along a line tilted at 45^° to the x axis. (a) N_i/N, the fraction of vertices of type N_i, vs time in simulation time steps. Black filled circles: N₀; red open squares: N₁; green +'s: biased N₂^b vertices; purple open triangles: ground state N₂^gs vertices; orange open diamonds: N₃; blue x's: N₄. The monopole states are N₀ and N₄. (b) F_ext vs time. The system switches into a positively biased ice rule obeying state with N₂^b/N=1.0 near F_ext=1.3. After the drive has been reversed, the system switches into an ordered monopole state with N₀/N=N₄/N=0.5 at F_ext=−0.65 and then orders into a negatively biased ice rule obeying state at F_ext=−1.3.

3. Orderings For Square Ice

In order to create an easily accessible ordered monopole state for a square ice lattice, we propose a simple extension of the standard square ice geometry where we now have two sublattices as illustrated in figure 7. Each sublattice has a different strength of the central barrier, f_r¹ and f_r², and we take f_r²/f_r¹=2. In figure 7(a), the darker traps have the weaker barriers. Such a geometry could be produced in the nanomagnetic system by using two different sizes or two different materials for the nanoislands, producing two different coercive field values. We apply a biasing force along a line oriented at 45^° to the x axis. In the square ice array, the vertex types N₀, N₁, N₂, N₃, and N₄ have 0, 1, 2, 3, and 4 colloids close to the vertex, respectively. The N₂ vertices are further subdivided into biased vertices N₂^b where the two close colloids are in adjacent traps as in figure 7(b), and ground state vertices N₂^gs where the two close colloids are on opposite sides of the vertex as in figure 1(a). In figure 8 we plot the the external drive F_ext and fraction of vertex types N_j/N versus time. The system begins in a random state with N₄/N + N₀/N = N₂^gs/N = 0.125 and N₁/N = N₂/N = N₃/N = 0.25. As F_ext increases past F_ext=1.3, the sample enters the positively biased ice rule obeying state illustrated in figure 7(b), with N₂^b/N=1. When the external drive reaches F_ext=2.0, we begin decreasing F_ext all the way to F_ext=−2.0. The biased state persists until F_ext = −0.67, at which point a checkerboard monopole state appears as illustrated in figure 7(c). In analogy to the kagome ice system, in the two-sublattice square ice sample the monopole state forms when sublattice A, with its weaker barrier, switches before sublattice B. If the colloid-colloid interaction strength is small enough, the particles in sublattice B remain in their original positions, producing the monopole state shown in figure 7(c). As F_ext is further increased in the negative direction, sublattice B switches at F_ext=−1.3 to form a negatively biased ice rule obeying state aligned with the drive. We note that in our previous work on square ice with only a single trap sublattice, monopole formation was rare due to its high energetic cost.

Figure 9: The vertex populations N_i/N vs temperature T in a modified square ice system with f_r¹=1.0 and f_r²=2.0 that has been prepared in a monopole state with F_ext set to zero. Black filled circles: monopole states N₀/N+N₄/N; red open squares: N₁/N + N₃/N; green x's: N₂^b/N biased vertices; blue +'s: N₂^gs/N ground state vertices. (a) A sample of noninteracting colloids with q²=0 passes directly from a monopole state to a disordered state. (b) A sample with q² = 0.2 shows the formation of the biased triple state followed by the thermally induced biased ice rule obeying state and then a random or paramagnetic state. (c) In a sample with q² = 0.4, the monopole state is followed by the biased triple state, the thermally induced biased ice rule obeying state, the thermally induced ground state, and the paramagnetic state.

Figure 10: Black circles: particle locations; open ellipses: trap locations for a 20a₀ ×20a₀ section of a modified square ice system with f_r¹=1.0, f_r²=2.0, and q²=0.4. The two trap sublattices are indicated by ellipses of different sizes. Colored circles indicate vertex types: N₀ (blue), N₁ (green), biased N₂^b (yellow), ground state N₂^gs (gray), N₃ (orange), and N₄ (red). (a) The ordered monopole state. (b) The biased triple state. (c) The thermally induced biased ice rule obeying state. (e) The thermally induced ground state. (f) The paramagnetic or uncorrelated state.

We next consider thermal effects on the ordered monopole state. To prepare the sample, we sweep F_ext to a value at which the monopole state appears and then switch off F_ext. If the pairwise interactions between colloids are not too strong, the monopole state remains stable even without a biasing drive. Figure 9(a) shows the vertex populations N_i/N versus temperature T for a system of noninteracting colloids with q²=0. Here the monopole state illustrated in figure 10(a) disorders into the paramagnetic state illustrated in figure 10(e), with N₀/N and N₄/N monotonically dropping from 0.5 to around 0.06 but never falling below this value. In samples with interacting colloids, such as the system with q²=0.2 shown in figure 9(b), N₀/N and N₄/N both drop from 0.5 to 0 with increasing T indicating that all monopoles disappear from the system. As T is increased further, N₀/N and N₄/N gradually increase back to the paramagnetic value of 0.0625. For 1.9 ≤ T ≤ 2.3, where N₀/N and N₄/N are dropping to zero, the values of N₁/N and N₃/N pass through a peak. This is followed by a window 2.3 ≤ T ≤ 3.3 where N₁/N and N₃/N drop again while the value of N₂^b/N peaks. A weak peak in N₂^gs/N appears in the vicinity of T ∼ 3.5, while at higher temperatures the vertex populations approach the values expected in a completely random configuration.

The same trends are more clearly evident in figure 9(c) for a sample with larger pairwise interactions, q²=0.4. Here the highest value reached by N₁/N + N₃/N is nearly 0.9, indicating that large portions of the sample are filled with N₁ or N₃ vertices. The corresponding colloid configuration is illustrated in figure 10(b), where local ordering of the N₁ and N₃ vertices into a lattice can be seen. We term this state a biased triple state; it forms due to the preferential switching of colloids in the weaker sublattice A traps. As T increases, N₁/N and N₃/N drop while N₂^b/N peaks at a value of 0.8 corresponding to the thermally induced biased ice rule obeying state, illustrated in figure 10(c). At higher T, N₂^b/N decreases again while N₂^gs/N reaches its peak value of 0.7 corresponding to the thermally induced ground state shown in figure 10(d). For still higher T, the N_i/N approach the values expected for a random configuration and the sample enters a paramagnetic state, illustrated in figure 10(e).

The sequence of ordered states we observe for finite pairwise interactions under finite temperatures can be understood by considering the interaction of the metastable monopole state with the two trap sublattices. Our protocol of sweeping F_ext into the monopole regime and then switching off F_ext at zero temperature causes the sample to be trapped in this metastable state for low temperatures. As T increases from zero, the system first lowers its energy by moving one colloid out of each N₄ vertex into an N₀ vertex, creating N₁ and N₃ vertex pairs and generating the biased triple state. This produces the peak in N₁/N and N₃/N near T=1.6 in figure 9(c), while figure 10(b) indicates that due to the finite temperature, some non-N₁ or N₃ vertices still exist in the sample. In the biased triple state, the lower energetic cost of an N₃ vertex compared to an N₄ vertex means that a higher thermal activation energy is required to destroy N₃ vertices compared to N₄ vertices. The difference in thermal activation energies determines the size of the temperature window where the biased triple state remains metastable. Once the temperature is large enough, colloids jump out of N₃ vertices into neighboring N₁ vertices, resulting in the formation of N₂ vertices. This hopping is more likely to occur in the weaker traps of sublattice A, resulting in the preferential formation of biased N₂^b vertices. As a result, N₂^b/N peaks near T=2.3 in figure 9(c) and the sample enters the biased ice rule obeying configuration (with thermal defects) shown in figure 10(c). Since the N₂^b vertices have a slightly higher energy than the ground state N₂^gs vertices [20], at even higher temperatures additional hopping produces an increased fraction of N₂^gs vertices, placing the system in the thermally induced and thermally defected square ice ground state configuration illustrated in figure 10(d). For sufficiently high T, the correlations between neighboring traps are lost and the sample enters the paramagnetic state shown in figure 10(e).

Figure 11: Phase diagram for T vs q² for the modified square ice sample with f_r¹=1.0 and f_r²=2.0 indicating regions where the ordered monopole, biased triple, biased ice rule obeying, square ice ground state, and random states appear. As q² increases, the width of the ordered monopole state decreases while the transition to the random state shifts to higher T.

In figure 11 we plot a phase diagram of the regions in T and q² space where the different ordered states appear. The lines indicate crossovers and are determined based on when the different N_i/N cross threshold values. As q² increases, the destruction of the ordered monopole state drops to lower values of T. The biased triple state and biased ice rule obeying state maintain roughly constant widths in T as q² increases, although both phases shift to lower ranges of T. In contrast, the square ice ground state, which appears only for larger values of q², increases in width as q² increases. Above q²=0.45 we can no longer stabilize the ordered monopole state using our biasing protocol. Our results indicate that a simple modification of the square ice geometry can produce a wealth of orderings in the artificial spin ice system.

We note that the phase diagrams for the kagome and modified square ice presented here do not exhibit true phases since we have driven the system into a metastable state. In many of the nanomagnetic systems that exhibit hysteresis, most of the observed states are also metastable unless a successful annealing protocol has been applied. All of our samples contained no quenched disorder. We expect that many of the states we observe would be gradually washed out if increasing amounts of quenched disorder are added; however, there should still be observable correlations in the different states that can be detected by analyzing the vertex population densities.

4. Summary

We have described the creation of different types of orderings in artificial kagome and modified square ice systems using external fields, temperature, and particle-particle interaction strength. We show that multiple-step ordering-disordering transitions occur that can be identified by counting the vertex populations. In the kagome ice, an ordered monopole state can be induced by applying a biasing field to samples with pairwise interactions that are not too strong. With increasing temperature, the system crosses into an ice rule obeying state and then into a high temperature paramagnetic state. For stronger pairwise interactions, the monopole state is unstable to the formation of the ice rule obeying state. Our results suggest that recent experimental observations of monopole ordering in kagome nanomagnetic systems were performed in a weakly interacting or noninteracting regime. For square ice, we make a simple modification to the geometry by introducing two sublattices of traps that have different switching fields and show that under the application of an external biasing field, this system can form a checkerboard ordered monopole state. The modified square ice exhibits multi-state ordering as a function of temperature starting from the monopole state, passing through a biased triple state, a thermally induced biased ice rule obeying state, a thermally induced square ice ground state, and a disordered or paramagnetic state. Although our results are obtained specifically for colloids on periodic trap arrays, the behavior should be generic to other artificial ice systems under external drives, including nanomagnetic array systems.

5. Acknowledgements

This work was carried out under the auspices of the NNSA of the U.S. Department of Energy at Los Alamos National Laboratory under contract number DE-AC52-06NA25396. The work of A. Libál was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2011-3-0114.

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