Abstract:
We study a frustrated dipolar array recently manufactured lithographically by Wang {\em et al.} [Nature {\bf 439}, 303 (2006)] in order to realize the square ice model in an artificial structure. We discuss models for thermodynamics and dynamics of this system. We show that an ice regime can be stabilized by small changes in the array geometry; a different magnetic state, kagome ice, can similarly be constructed. At low temperatures, the square ice regime is terminated by a thermodynamic ordering transition, which can be chosen to be ferro- or antiferromagnetic. We show that the arrays do not fully equilibrate experimentally, and identify a likely dynamical bottleneck.

Abstract:
We investigate the thermodynamics of artificial square spin ice systems assuming only dipolar interactions among the islands that compose the array. The emphasis is given on the effects of the temperature on the elementary excitations (magnetic monopoles and their Dirac strings). By using Monte Carlo techniques we calculate the specific heat, the density of poles and their average separation as functions of temperature. The specific heat and average separation between monopoles and antimonopoles exhibit a sharp peak and a local maximum, respectively, at the same temperature, $T_{p}\approx 7.2D/k_{B}$ (here, $D$ is the strength of the dipolar interaction and $k_{B}$ is the Boltzmann constant). As the lattice size is increased, the amplitude of these features also increases but very slowly. Really, the specific heat and the maximum in the average separation $d_{max}$ between oppositely charged monopoles increase logarithmically with the system size, indicating that completely isolated charges could be found only at the thermodynamic limit. In general, the results obtained here suggest that, for temperatures $T \geq T_{p}$, these systems may exhibit a phase with separated monopoles, although the quantity $d_{max}$ should not be larger than a few lattice spacings for viable artificial materials.

Abstract:
We have studied experimentally the states formed in artificial square ice nanomagnet systems following demagnetization in a rotating in-plane applied magnetic field that reduces to zero in a manner that is linear in time. The final states are found to be controlled via the system's lattice constant, which determines the strength of the magnetostatic interactions between the elements, as well as the field ramping rate. We understand these effects as a requirement that the system undergoes a sufficiently large number of active rotations within the critical field window in which elements may be reversed, such that the interactions are allowed to locally exert their influence if the ground state is to be approached. On the other hand, if quenched disorder is too strong when compared to the interaction strength, any close approach to the ground state is impossible. These results show that it is not necessary for there to be any ac component to the field amplitude that is applied to the system during demagnetization, which is the method almost exclusively employed in field protocols reported to date. Furthermore, by optimizing the parameters of our linear demagnetization protocol, the largest field-generated ground state domains yet reported are found.

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.

Abstract:
The thermally-driven formation and evolution of vertex domains is studied for square artificial spin ice. A self consistent mean field theory is used to show how domains of ground state ordering form spontaneously, and how these evolve in the presence of disorder. The role of fluctuations is studied, using Monte Carlo simulations and analytical modelling. Domain wall dynamics are shown to be driven by a biasing of random fluctuations towards processes that shrink closed domains, and fluctuations within domains are shown to generate isolated small excitations, which may stabilise as the effective temperature is lowered. Domain dynamics and fluctuations are determined by interaction strengths, which are controlled by inter-element spacing. The role of interaction strength is studied via experiments and Monte Carlo simulations. Our mean field model is applicable to ferroelectric `spin' ice, and we show that features similar to that of magnetic spin ice can be expected, but with different characteristic temperatures and rates.

Abstract:
Dynamical effects under geometrical frustration are considered in a model for artificial spin ice on a square lattice in two dimensions. Each island of the spin ice has a three-component Heisenberg-like dipole moment subject to shape anisotropies that influence its direction. The model has real dynamics, including rotation of the magnetic degrees of freedom, going beyond the Ising-type models of spin ice. The dynamics is studied using a Langevin equation solved via a second order Heun algorithm. Thermodynamic properties such as the specific heat are presented for different couplings. A peak in specific heat is related to a type of melting-like phase transition present in the model. Hysteresis in an applied magnetic field is calculated for model parameters where the system is able to reach thermodynamic equilibrium.

Abstract:
Artificial square spin ices are structures composed of magnetic elements arranged on a geometrically frustrated lattice and located on the sites of a two-dimensional square lattice, such that there are four interacting magnetic elements at each vertex. Using a semi-analytical approach, we show that square spin ices exhibit a rich spin wave band structure that is tunable both by external magnetic fields and the configuration of individual elements. Internal degrees of freedom can give rise to equilibrium states with bent magnetization at the edges leading to characteristic excitations; in the presence of magnetostatic interactions these form separate bands analogous to impurity bands in semiconductors. Full-scale micromagnetic simulations corroborate our semi-analytical approach. Our results show that artificial square spin ices can be viewed as programmable and tunable magnonic crystals that can be used as metamaterials for spin waves.

Abstract:
The interactions between an excitation (similar to a pair of Nambu monopoles) and a lattice defect are studied in an artificial two-dimensional square spin ice. This is done by considering a square array of islands containing only one island different from all others. This difference is incorporated in the magnetic moment (spin) of the "imperfect" island and several cases are studied, including the special situation in which this distinct spin is zero (vacancy). We have shown that the two extreme points of a malformed island behave like two opposite magnetic charges. Then, the effective interaction between a pair of Nambu monopoles with the deformed island is a problem involving four magnetic charges (two pairs of opposite poles) and a string. We also sketch the configuration of the field lines of these four charges to confirm this picture. The influence of the string on this interaction decays rapidly with the string distance from the defect.

Abstract:
We report X-ray resonant magnetic scattering studies of a Permalloy artificial square ice nanomagnet array, focussing on the field-driven evolution of the sum Σ and difference Δ signals of left and right handed circularly polarized synchrotron X-rays at different lateral positions in reciprocal space Qx. We used X-rays tuned to the Fe L3 resonance energy, with the scattering plane aligned along a principal symmetry axis of the array. Details of the specular Δ hysteresis curve are discussed, following the system magnetization from an initial demagnetized state. The periodic structure gives rise to distinct peaks at in-plane reciprocal Bragg positions, as shown by fitting Σ(Qx) to a model based on a simple unit cell structure. Diffraction order-dependent hysteresis in Δ is observed, indicative of the reordering of magnetization on the system's two interpenetrating sublattices, which markedly deviates from an ideal Ising picture under strong applied fields.

Abstract:
We examine square and kagome artificial spin ice for colloids confined in arrays of double-well traps. Unlike magnetic artificial spin ices, colloidal and vortex artificial spin ice realizations allow creation of doping sites through double occupation of individual traps. We find that doping square and kagome ice geometries produces opposite effects. For square ice, doping creates local excitations in the ground state configuration that produce a local melting effect as the temperature is raised. In contrast, the kagome ice ground state can absorb the doping charge without generating non-ground-state excitations, while at elevated temperatures the hopping of individual colloids is suppressed near the doping sites. These results indicate that in the square ice, doping adds degeneracy to the ordered ground state and creates local weak spots, while in the kagome ice, which has a highly degenerate ground state, doping locally decreases the degeneracy and creates local hard regions.