Disordered Kagomé Spin Ice

Artificial spin ice is made from a large array of patterned magnetic nanoislands designed to mimic naturally occurring spin ice materials. The geometrical arrangement of the kagome lattice guarantees a frustrated arrangement of the islands’ magnetic moments at each vertex where the three magnetic nanoislands meet. This frustration leads to a highly degenerate ground state which gives rise to a finite (residual) entropy at zero temperature. In this work we use the Monte Carlo simulation to explore the effects of disorder in kagome spin ice. Disorder is introduced to the system by randomly removing a known percentage of magnetic islands from the lattice. The behavior of the spin ice changes as the disorder increases; evident by changes to the shape and locations of the peaks in heat capacity and the residual entropy. The results are consistent with observations made in diluted physical spin ice materials.


I. INTRODUCTION
2][3][4][5][6] In spin ice pyrochlores, such as Dy 2 Ti 2 O 7 , the magnetic moments of the Dy 3+ ions align along the tetrahedral axes of the crystal and these systems transition from ferromagnetic to antiferromagnetic spin ordering at low temperatures.At low temperatures these systems follow the "spin ice rules", where at each vertex two spins point inward and two spins point outward. 7This 2-in/2-out configuration of spins is six-fold degenerate and results in finite entropy consistent with Pauling's theory for the zero-point entropy of water ice. 8rtificial spin ice is a metamaterial consisting of a patterned array of magnetic islands engineered to mimic the behavior of physical spin ice systems. 9Each elongated island acts as an Ising macrospin and couples with the other islands in the array through dipolar interactions.A key advantage of artificial spin ice is that it is macroscopic and can be visualized more readily than the microscopic systems. 102][13] Square lattices of artificial spin ice have been created and shown to preserve the 2-in/2-out, low temperature configuration, at each vertex. 14By increasing the temperature, or reducing the magnetic coupling by increasing the spacing between islands, magnetic charge excitations (monopoles) can be created which then propagate through the lattice; a subject of intense investigation. 15rtificial kagomé spin ice differs from square ice in that the geometry is hexagonal, and therefore each vertex consists of three islands meeting instead of four. 1,15,16A representation of a segment of the array is shown in FIG. 1a).This cartoon represents the magnetization of the islands (arrows) and the corresponding magnetic charge at the vertices (circles).The primary effect of changing the geometry is that a magnetic charge automatically exists at each of the vertices in the array as each vertex a andrew.kunz@marquette.edu is magnetically frustrated.The frustrated ground state configuration at each vertex is automatically 2-in/1-out (positive magnetic charge), or 1-in/2-out (negative magnetic charge).Higher order energy excitations consist of 3-in, or 3-out magnetic configurations.At low enough temperatures, artificial kagomé spin ice has no high energy excitations (Ice I phase) and as the temperature is further decreased the system becomes antiferromagnetically charge ordered (Ice II) as shown in FIG.1a) and FIG.2a). 11n this work, we model physical systems by investigating the role of disorder in kagomé spin ice by randomly removing a known percentage of magnets from the full array.This compares to real spin ice materials as they have been found to have nonmagnetic defects. 3,4The removal of a single magnet at a vertex leads to local ferromagnetic coupling as the frustration is eliminated at the corresponding vertices; impacting the spin dynamics.A related study has been carried out on doped colloidal artificial spin ice. 17The doped systems effectively pin a charge at each vertex, whereas removing an island removes the charge from the vertices.9][20] In each case the disorder has been shown to lead to changes in the magnetization dynamics, and equilibrium states.

II. SIMULATION
The magnetic islands in artificial spin ice are single domain magnets and as such can be modeled by an Ising macrospin with a magnetic moment dependent on the material and island dimensions.We model rounded rectangular permalloy islands (M s = 8 x 10 5 A/m) with dimensions 63 x 26 x 5 nm 3 .The centers of neighboring islands are separated by a distance of 170 nm.This combination leads to a weakly coupled system that is thermally active above 30 K. 11 Starting from a system containing about 3000 sites, disorder is introduced into the system by randomly selecting sites and setting the magnetic moment of those locations to zero.
We search for equilibrium states via Monte Carlo simulation.Specifically, we follow the Metropolis-Hastings algorithm by calculating the dipole -dipole energy for each pair of interacting magnetic moments, m in the array, separated by a distance r.The total energy in the array is a sum over all the dipole pairs.Following the Monte Carlo algorithm a random macrospin is chosen and the change in the systems energy, ∆E, associated with reversing that spin is calculated. 21If the energy decreases, the flip is accepted.If the energy increases, the Boltzmann factor e −∆E/k B T is calculated and compared to a randomly generated number between 0 and 1.If the random number is less than the Boltzmann factor the flip is accepted, otherwise the system is left unchanged.On average a Monte Carlo step (MCS) corresponds to choosing each one of the magnets in the array one time, and at least 1200 MCS are taken at each temperature.The energy of each site and the total energy are tracked as a function of the system temperature at it decreases from 100 K to 0.2 K.

III. RESULTS
FIG 2a) shows the perfectly ordered Ice II phase with the black and grey circles, corresponding to negative and positive charges respectively, at each of the vertices in the kagomé structure.To visualize the charge ordering, follow the rows of black or grey circles; if the rows are continuous the ordering is complete.
FIG. 3 shows that the phase transition temperatures decrease as the amount of disorder increases.Disorder is recorded as the percentage of magnetic islands removed from the array.The removal of a small number of islands leads to local regions where the frustration is removed.The magnetic charge at a vertex with a missing spin in the Ice II phase is zero as shown in the cartoon of FIG. 1b).A corresponding image of the charges in a region of an array missing 2% of the islands is shown in FIG.2b where a charge of zero at a vertex is shown in red.When one magnet is removed from each vertex it is possible that no dynamic response is necessary; the charge ordering in the region around the missing magnet is unchanged (see FIG. 1b).Inspection of FIG.2b shows that the charge ordering is the same throughout most of the image, but in the lower left quadrant it is different and there are easily identifiable locations where two negative charges or two positive charges are located next to each other.This behavior is explained below when the third magnetic at the vertex is removed.As the number of missing magnets grows, we start to find vertices that are missing two magnets (FIG.1c), and sometimes all three (FIGs.1d).The disordered regions also start to merge as shown in FIG.2c) at 6% and FIG.2d) at 10%.In each case domains, regions of differing charge order start to appear.The reason for this can be seen in the cartoons of FIG.1d).On the removal of the third magnet from the vertex, a +2 charge would appear.This is an energetic excitation that the system wants to remove.We show two potential paths; the first involving a chain of islands reversing (shown with the red arrows and would continue off the image until another missing spin is encountered) maintaining the local charge ordering, and a second where a single reversal takes place but the charge ordering is lost (two positive charges neighbor).
To quantify the effects of the disorder we calculated the entropy, S, in the system.Using the thermodynamic identity we integrate C/T, as shown in the inset of FIG. 3, for each disorder density.We know that at high temperatures each of the magnets in the array acts as an independent spin and that because the spins are Ising like they can only take on one of two orientations.The total multiplicity, Ω, is therefore just 2 N and the entropy at high temperatures (T f ) must be a maximum of where N is the total number of magnetized sites in the array and k B is Boltzmann's constant.FIG. 4 is a plot of the reduced entropy (S/N•k B ) in each case, normalized to the known high temperature value.From (2) we find and a rearrangement leaves residual entropy S 0 as the difference between the maximum entropy and the area under the C/T curve FIG. 4 shows that above about 40 K the entropy in each case is essentially the same and the curve is asymptotically approaching the value of ln(2) as expected.However, at low temperatures, as shown in the inset of FIG. 4, there are clear differences in the entropy and in each case the value does not go to S = 0.This apparent violation of the third law of thermodynamics is consistent with Pauling's discovery of residual entropy in water ice. 8We find that the residual entropy is generally increasing with the amount of disorder in the system; a finding that is consistent with measurements made in physically diluted spin ice systems. 3,4Statistically speaking, the introduction of disorder removes frustration at vertices in the material, which impacts the dynamics in such a way that the path to the perfectly charge ordered state is blocked and the number of arrangements, and hence the residual entropy, increases as the number of missing magnets increases.

IV. CONCLUSION
In summary, artificial spin ice with random disorder can be used to model physical spin ice systems.We show that the removal of random islands mimics the behavior of missing bonds in physical systems.These missing spins lead to differences in the spin dynamics as is evident visually, as well as in heat capacity and entropy calculations.The results of the simulations including disorder are consistent with that measured in physical systems. 3,4 FIG. 1. a) Cartoon of the spin orientation (arrows) and charge distribution for three hexagons of the kagomé array in the ice II phase.b) Orientations with a single spin removed.c) Two spins removed from the central vertex, leaving a dangling spin.d) Two possible reversal processes when the third spin is removed; a chain of reversals to maintain local charge ordering, or a single flip which puts like charges next to one another.

FIG. 2
FIG. 2. a) Perfect charge ordering in the pure kagome lattice.b) At 2% disorder the charge ordering is already affected.By 6% c) and 10% d) differing charge ordered domains are clearly visible in the lattice.

FIG. 3 .
FIG.3.Heat capacity per magnet as a function of temperature.The high temperature peak is the paramagnetic to ice I transition, and the low temperature peak is the ice I to ice II transition.Both transitions shift as the disorder in the system increases.The inset shows the heat capacity over temperature as a function of temperature.

FIG. 4 .
FIG. 4. Entropy of the array (normalized to the maximum entropy) as a function of temperature for different disorder densities.The inset shows the low temperature behavior and the residual entropy in the system.The residual entropy generally increases with disorder.