Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands

Vertex dynamics in finite two-dimensional square spin ices

Local magnetic ordering in artificial spin ices is discussed from the point of view of how geometrical frustration controls dynamics and the approach to steady state. We discuss the possibility of using a particle picture based on vertex configurations to interpret the time evolution of magnetic configurations. Analysis of possible vertex processes allows us to anticipate different behaviors for open and closed edges and the existence of different field regimes. Numerical simulations confirm these results and also demonstrate the importance of correlations and long-range interactions in understanding particle population evolution. We also show that a mean-field model of vertex dynamics gives important insights into finite size effects.

Tuning the ferromagnetic coupling of Fe nanodots on Cu(111) via dimensionality variation of the mediating electrons

Using in situ magneto-optical Kerr effect measurements and phenomenological modeling, we study the tunability in both the magnetization anisotropy and magnetic coupling of Fe nanodots on a curved Cu(111) substrate with varying vicinity. We observe that, as the terrace width w decreases, the magnetization anisotropy increases monotonically, faster when w is smaller than the nanodot size d. In contrast, the magnetic coupling strength also increases until w approximately d, after which it decreases steeply. These striking observations can be rationalized by invoking the counterintuitive dimensionality variation of the surface electrons mediating the interdot coupling: the electrons are confined to be one dimensional (1D) when w > or = d, but become quasi-2D when w < d due to enhanced electron spillover across the steps bridged by the nanodots.

Non-collinearity and spin frustration in the itinerant kagome ferromagnet Fe(3)Sn(2)

Frustrated itinerant ferromagnets, with non-collinear static spin structures, are an exciting class of material as their spin chirality can introduce a Berry phase in the electronic scattering and lead to exotic electronic phenomena such as the anomalous Hall effect (AHE). This study presents a reexamination of the magnetic properties of Fe(3)Sn(2), a metallic ferromagnet, based on the two-dimensional kagome bilayer structure. Previously thought of as a conventional ferromagnet, we show using a combination of SQUID (superconducting quantum interference device) measurements, symmetry analysis and powder neutron diffraction that Fe(3)Sn(2) is a frustrated ferromagnet with a temperature-dependent non-collinear spin structure. The complexity of the magnetic interactions is further evidenced by a re-entrant spin glass transition ([Formula: see text] K) at temperatures far below the main ferromagnetic transition (T(C) = 640 K). Fe(3)Sn(2) therefore provides a rare example of a frustrated itinerant ferromagnet. Further, as well as being of great fundamental interest our studies highlight the potential of Fe(3)Sn(2) for practical application in spintronics technology, as the AHE arising from the ferromagnetism in this material is expected to be enhanced by the coupling between the conduction electrons and the non-trivial magnetic structure over an exceptionally wide temperature range.

Phase-space networks of geometrically frustrated systems

We illustrate a network approach to the phase-space study by using two geometrical frustration models: antiferromagnet on triangular lattice and square ice. Their highly degenerated ground states are mapped as discrete networks such that the quantitative network analysis can be applied to phase-space studies. The resulting phase spaces share some comon features and establish a class of complex networks with unique Gaussian spectral densities. Although phase-space networks are heterogeneously connected, the systems are still ergodic due to the random Poisson processes. This network approach can be generalized to phase spaces of some other complex systems.

Magnetic ground state of single and coupled permalloy rectangles

We have studied the magnetic domain structure in Permalloy rectangles that reveal flux-closure domain configurations. Arrays with varying spacing between the rectangles are investigated by scanning electron microscopy with polarization analysis as well as by micromagnetic simulation. In contrast to general expectation, rectangles in the flux-closure Landau state show significant coupling and form a magnetic pattern of common chirality. The coupling is due to the stray field that originates from small changes of the magnetization alignment, which is sensitive to the exact shape and the separation of the rectangles.

Creating artificial ice states using vortices in nanostructured superconductors

We demonstrate that it is possible to realize vortex ice states that are analogous to square and kagome ice. With numerical simulations, we show that the system can be brought into a state that obeys either global or local ice rules by applying an external current according to an annealing protocol. We explore the breakdown of the ice rules due to disorder in the nanostructure array and show that in square ice, topological defects appear along grain boundaries, while in kagome ice, individual defects appear. We argue that the vortex system offers significant advantages over other artificial ice systems.

Two-point correlation function for the triangular Ising antiferromagnet

A number of aspects of the two-point correlation behavior of the nearest-neighbor, triangular Ising antiferromagnet are studied using a combination of numerical evaluation of exact expressions and Monte Carlo simulation. Existing asymptotic results for on-axis correlations at finite temperatures are evaluated and shown to be of limited accuracy. The sublattice structure of the off-axis correlation function is clarified, and rotational invariance is studied as a function of temperature. Separations and temperatures for which the correlation function is significant are identified, and a simple functional expression is developed that allows accurate calculation of the correlation function in this region.

Stripes, zigzags, and slow dynamics in buckled hard spheres

We study the analogy between buckled colloidal monolayers and the triangular-lattice Ising antiferromagnet. We calculate free-volume-induced Ising interactions, show how lattice deformations favor zigzag stripes that partially remove the Ising model ground-state degeneracy, and identify the martensitic mechanism prohibiting perfect stripes. Slowly inflating the spheres yields jamming as well as logarithmically slow relaxation reminiscent of the glassy dynamics observed experimentally.

Static magnetic order in Tb2Sn2O7 revealed by muon spin relaxation with exterior muon implantation

Tb2Sn2O7 has been proposed as an ordered spin ice, but the precise nature of the low temperature magnetic state remains uncertain. Recent independent muon spin relaxation (microSR) investigations suggest the possibility of exotic ground states with static order precluded on time scales longer than 10(-6) s. Here the more conventional hypothesis of canted ferromagnetism is tested by means of microSR with the muons stopped outside the sample, as well as ultralow field bulk magnetization measurements. The field cooled state shows conventional static order, while the zero field cooled state may be interpreted in terms of conventional closed domains. These results rule out purely dynamical ground states and illustrate the value of exterior muon implantation as a complement to the conventional technique.

Energy minimization and ac demagnetization in a nanomagnet array

We study ac demagnetization in frustrated arrays of single-domain ferromagnetic islands, exhaustively resolving every (Ising-like) magnetic degree of freedom in the systems. Although the net moment of the arrays is brought near zero by a protocol with sufficiently small step size, the final magnetostatic energy of the demagnetized array continues to decrease for finer-stepped protocols and does not extrapolate to the ground-state energy. The resulting complex disordered magnetic state can be described by a maximum-entropy ensemble constrained to satisfy just nearest-neighbor correlations.

Nonmonotonic zero-point entropy in diluted spin ice

Water ice and spin ice are important model systems in which theory can directly account for "zero-point" entropy associated with quenched configurational disorder. Spin ice differs from water ice in the important respect that its fundamental constituents, the spins of the magnetic ions, can be removed through replacement with nonmagnetic ions while keeping the lattice structure intact. In order to investigate the interplay of frustrated interactions and quenched disorder, we have performed systematic heat capacity measurements on spin ice materials which have been thus diluted up to 90%. Investigations of both Ho and Dy spin ices reveal that the zero-point entropy depends nonmonotonically on dilution and approaches the value of Rln2 in the limit of high dilution. The data are in good agreement with a generalization of Pauling's theory for the entropy of ice.

Classical antiferromagnet on a hyperkagome lattice

Motivated by recent experiments on Na4Ir3O8 [Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, arXiv:0705.2821 (unpublished)], we study the classical antiferromagnet on a frustrated three-dimensional lattice obtained by selectively removing one of four sites in each tetrahedron of the pyrochlore lattice. This "hyperkagome" lattice consists of corner-sharing triangles. We present the results of large-N mean field theory and Monte Carlo computations on O(N) classical spin models. It is found that the classical ground states are highly degenerate. Nonetheless a nematic order emerges at low temperatures in the Heisenberg model (N=3) via "order by disorder," representing the dominance of coplanar spin configurations. Implications for ongoing experiments are discussed.

Ground state lost but degeneracy found: the effective thermodynamics of artificial spin ice

We analyze the rotational demagnetization of artificial spin ice, a recently realized array of nanoscale single-domain ferromagnetic islands. Demagnetization does not anneal this model system into its antiferromagnetic ground state: the moments have a static disordered configuration similar to the frozen state of the spin ice materials. We demonstrate that this athermal system has an effective extensive degeneracy and we introduce a formalism that can predict the populations of local states in this icelike system with no adjustable parameters.

Triangular Ising antiferromagnet: boundary conditions, ground state entropy, and vortices

The ground state entropy density of the triangular Ising antiferromagnet is considered as a function of boundary conditions on domains for which the ground states do not admit a dimer covering. These domains admit a rich set of ground states that cannot be classified in the usual way in terms of nonintersecting strings. Various parametrized boundary conditions and domain shapes are identified that allow the ground state entropy density to be varied between zero and maximal degeneracy. The dependence of degeneracy on boundary spins and/or domain shape is interpreted in terms of strings that are not restricted to be nonintersecting.

Realizing colloidal artificial ice on arrays of optical traps

We demonstrate how a colloidal version of artificial ice can be realized on optical trap lattices. Using numerical simulations, we show that this system obeys the ice rules and that for strong colloid-colloid interactions, an ordered ground state appears. We show that the ice-rule ordering can occur for systems with as few as 24 traps and that the ordering transition can be observed at constant temperature by varying the barrier strength of the traps.

Properties of ferroelectric nanodots embedded in a polarizable medium: atomistic simulations

An atomistic approach is used to investigate finite-temperature properties of ferroelectric nanodots that are embedded in a polarizable medium. Different phases are predicted, depending on the ferroelectric strengths of the material constituting the dot and of the system forming the medium. In particular, novel states, exhibiting a coexistence between two kinds of order parameters or possessing a peculiar order between dipole vortices of adjacent dots, are discovered. We also discuss the origins of these phases, e.g., depolarizing fields and medium-driven interactions between dots.

Exactly solvable interacting spin-ice vertex model

A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.

Artificial square ice and related dipolar nanoarrays

We study a frustrated dipolar array recently manufactured lithographically by Wang 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.