A frustrated system is one whose symmetry precludes the possibility that every pairwise interaction (“bond”) in the system can be satisfied at the same time. Such systems are common in all areas of physical and biological science. In the most extreme cases, they can have a disordered ground state with “macroscopic” degeneracy; that is, one that comprises a huge number of equivalent states of the same energy. Pauling9s description of the low-temperature proton disorder in water ice was perhaps the first recognition of this phenomenon and remains the paradigm. In recent years, a new class of magnetic substance has been characterized, in which the disorder of the magnetic moments at low temperatures is precisely analogous to the proton disorder in water ice. These substances, known as spin ice materials, are perhaps the “cleanest” examples of such highly frustrated systems yet discovered. They offer an unparalleled opportunity for the study of frustration in magnetic systems at both an experimental and a theoretical level. This article describes the essential physics of spin ice, as it is currently understood, and identifies new avenues for future research on related materials and models.

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Spin ice state in frustrated magnetic pyrochlore materials.

Within the past 20 years or so, there has occurred an explosion of interest in the magnetic behavior of pyrochlore oxides of the type $A_{2}^{3+}$$B_{2}^{4+}$O$_{7}$ where $A$ is a rare-earth ion and $B$ is usually a transition metal. Both the $A$ and $B$ sites form a network of corner-sharing tetrahedra which is the quintessential framework for a geometrically frustrated magnet. In these systems the natural tendency to form long range ordered ground states in accord with the Third Law is frustrated, resulting in some novel short range ordered alternatives such as spin glasses, spin ices and spin liquids and much new physics. This article attempts to review the myriad of properties found in pyrochlore oxides, mainly from a materials perspective, but with an appropriate theoretical context.

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Magnetic Pyrochlore Oxides

When a number of interactions compete within a system they can't all prevail, so the resolution of ‘frustrated’ forces is an important determinant of the overall behaviour of a system. In particular, geometrical frustration among spins in magnetic systems can lead to exotic effects such as ‘spin ice’, a state where atomic magnetic moments mimic the frustration of hydrogen ion positions in water ice. Wang et al. have created artificial spin ice using lithographically fabricated arrays of nanoscale magnets. Magnetic moments in the lattice follow the two [pointing]-in/ two-out ‘ice rule’ typical of spin ice. With this model it is possible to study frustration in great detail; this is relevant to magnetic recording, where ferromagnetic elements are being pushed to ever higher densities. On the cover, a magnetic force microscopy representation of the magnetization pattern of artificial spin ice: plateaus and valleys show regions of opposite magnetization. Frustration, defined as a competition between interactions such that not all of them can be satisfied, is important in systems ranging from neural networks to structural glasses. Geometrical frustration, which arises from the topology of a well-ordered structure rather than from disorder, has recently become a topic of considerable interest1. In particular, geometrical frustration among spins in magnetic materials can lead to exotic low-temperature states2, including ‘spin ice’, in which the local moments mimic the frustration of hydrogen ion positions in frozen water3,4,5,6. Here we report an artificial geometrically frustrated magnet based on an array of lithographically fabricated single-domain ferromagnetic islands. The islands are arranged such that the dipole interactions create a two-dimensional analogue to spin ice. Images of the magnetic moments of individual elements in this correlated system allow us to study the local accommodation of frustration. We see both ice-like short-range correlations and an absence of long-range correlations, behaviour which is strikingly similar to the low-temperature state of spin ice. These results demonstrate that artificial frustrated magnets can provide an uncharted arena in which the physics of frustration can be directly visualized.

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

Single-molecule magnets (SMMs) have received much attention owing to their quantum tunneling and slow relaxation. These behaviors can be observed when the molecule has large ground spin state with a uniaxial magnetic anisotropy, namely negative zero-field splitting (ZFS) parameter D. Besides a large ground state, to increase the SMMs energy barrier and blocking temperature, it is of fundamental importance to control the magnetic anisotropy. It is also of great complexity to understand the conditions that determine the anisotropy or zero-field splitting properties for a cluster. As a result, SMMs containing only one spin carrier are of great interest because of the simplification in the analysis of local anisotropy and ZFS. Recently, some molecules with isolated 3d or 4f metal ions were observed to show a direct-current (dc) field-induced slow magnetic relaxation. Studies show that this kind of dc-field-dependent relaxation phenomenon is thermally activated, and the quench of fast quantum tunneling by the dc field could give rise to the slow relaxation. However, as commented by Benelli and Gatteschi, it is an unexpected and puzzling behavior, and the underlying mechanism is still unclear. Ishikawa et al. reported that the phthalocyanine (Pc) double-decker anion complexes [Pc2Ln] with a single Tb or Dy ion show slow relaxation without a dc field. A single Er ion in the polyoxometallate system [ErW10O36] 9 showed similar relaxation behavior in zero static magnetic field. The first single-actinide complex [U(Ph2BPz2)3] (containing the U III ion; Ph2BPz2 = diphenylbis(pyrazolylborate)) reported by Rinehart and Long shows a similar slow relaxation. Owing to the single-ion features, these complexes could be called “single-ion magnets”. In these complexes, ligand fields with a high-order single axis Cn (n> 2; n = 4 for Ref. [8, 9] and n = 3 for Ref. [10]) split the (2J + 1) degenerate ground states into new sublevels, which produces a uniaxial anisotropy, thus giving rise to a higher energy barrier for relaxation. The Dy ion, which possesses a Kramers ground state of H15/2, is an appealing paramagnetic source for the construction of SMMs in a suitable ligand-field symmetry and strength. Moreover, among reported SMMs, some of them are clusters containing Dy : Dy2, [11] Dy3, [12] Dy4, [13] Dy5, [14] Dy-containing chain, and tens of Dycontaining 3d–4f clusters. All three reported types of single-ion magnets are found with a high-order single axis defining the local symmetry. Pursuing this clue, we synthesized a series of mononuclear Ln compounds with a local symmetry close to D4d. Crystal analysis shows that the isomorphous complexes consist of a neutral mononuclear [Ln(acac)3(H2O)2] complex (Ln = Dy, Ho, Er, acac = acetylacetonate) together with an uncoordinated water molecule and an uncoordinated ethanol molecule (Figure 1a). In the Dy complex, Dy is coordinated by eight oxygen atoms with Dy O bonds of 2.311–2.434 , six of which come from the acetylacetonate ligand and two from coordinated water molecules. The eight oxygen atoms form an approximately square-antiprismatic coordination polyhedron, and the local symmetry of Dy is nearly D4d (Figure 1b). Actually, a similar lanthanide acac complex was firstly reported in 1968, but the uncoordinated water and

A Mononuclear Dysprosium Complex Featuring Single-Molecule-Magnet Behavior†

Frustration, defined as a competition between interactions such that not all of them can be satisfied, is important in systems ranging from neural networks to structural glasses. Geometrical frustration, which arises from the topology of a well-ordered structure rather than from disorder, has recently become a topic of considerable interest. In particular, geometrical frustration among spins in magnetic materials can lead to exotic low-temperature states, including ‘spin ice’, in which the local moments mimic the frustration of hydrogen ion positions in frozen water. Here we report an artificial geometrically frustrated magnet based on an array of lithographically fabricated single-domain ferromagnetic islands. The islands are arranged such that the dipole interactions create a two-dimensional analogue to spin ice. Images of the magnetic moments of individual elements in this correlated system allow us to study the local accommodation of frustration. We see both ice-like short-range correlations and an absence of long-range correlations, behaviour which is strikingly similar to the low-temperature state of spin ice. These results demonstrate that artificial frustrated magnets can provide an uncharted arena in which the physics of frustration can be directly visualized.

Artificial spin ice in a geometrically frustrated lattice of nanoscale ferromagnetic islands: addendum

The large degeneracy of states resulting from the geometrical frustration of competing interactions is an essential ingredient of important problems in fields as diverse as magnetism, protein folding and neural networks. As first explained by Pauling, geometrical frustration of proton positions is also responsible for the unusual low-temperature thermodynamics of ice and its measured 'ground state' entropy. Recent work has shown that the geometrical frustration of ice is mimicked by Dy2Ti2O7, a site-ordered magnetic material in which the spins reside on a lattice of corner-sharing tetrahedra where they form an unusual magnetic ground state known as 'spin ice'. Here we identify a cooperative spin-freezing transition leading to the spin-ice ground state in Dy2Ti2O7. This transition is associated with a very narrow range of relaxation times, and represents a new form of spin-freezing. The dynamics are analogous to those associated with the freezing of protons in ice, and they provide a means through which to study glass-like behaviour and the consequences of frustration in the limit of low disorder.

How ‘spin ice’ freezes

We report a study of the low temperature bulk magnetic properties of the spin ice compound ${\mathrm{Dy}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ with particular attention to the $(Tl4\mathrm{K})$ spin freezing transition. While this transition is superficially similar to that in a spin glass, there are important qualitative differences from spin glass behavior: the freezing temperature increases slightly with applied magnetic field, and the distribution of spin relaxation times remains extremely narrow down to the lowest temperatures. Furthermore, the characteristic spin relaxation time increases faster than exponentially down to the lowest temperatures studied. These results indicate that spin-freezing in spin ice materials represents a novel form of magnetic glassiness associated with the unusual nature of geometrical frustration in these materials.

Low-temperature spin freezing in the Dy 2 Ti 2 O 7 spin ice

Single-crystal superconducting tin nanowires with diameters of 40–160 nm have been prepared by electrochemical deposition in porous polycarbonate membranes. Structural characterization through transmission electron microscopy and x-ray diffraction showed that the nanowires are highly oriented along the [100] direction. Although the superconducting transition temperature is close to the bulk value of 3.7 K, the effect of reduced dimensionality is clearly evident in the electrical transport properties of the thinnest wires (40 nm diameter). Magnetization measurements show that the critical field of the nanowires increases significantly with decreasing diameter to ∼0.3 T for the thinnest wires, nearly an order of magnitude larger than the bulk value.

Synthesis and characterization of superconducting single-crystal Sn nanowires

We have studied spin relaxation in the spin ice compound Dy2Ti2O7 through measurements of the ac magnetic susceptibility. While the characteristic spin-relaxation time (tau) is thermally activated at high temperatures, it becomes almost temperature independent below T(cross) approximately 13 K. This behavior, combined with nonmonotonic magnetic field dependence of tau, indicates that quantum tunneling dominates the relaxational process below that temperature. As the low-entropy spin ice state develops below T(ice) approximately 4 K, tau increases sharply with decreasing temperature, suggesting the emergence of a collective degree of freedom for which thermal relaxation processes again become important as the spins become strongly correlated.

https://arxiv.org/pdf/cond-mat/0302453.pdf

Quantum-classical reentrant relaxation crossover in Dy2Ti2O7 spin ice.

We have studied the low temperature a.c. magnetic susceptibility of the diluted spin ice compound Dy(2-x)MxTi2O7, where the magnetic Dy ions on the frustrated pyrochlore lattice have been replaced with non-magnetic ions, M = Y or Lu. We examine a broad range of dilutions, 0 0.4 and persists to x = 1.98. This behavior can be understood as a non-monotonic dependence of the quantum spin relaxation time with dilution. The results suggest that the observed spin freezing is fundamentally a single spin process which is affected by the local environment, rather than the development of spin-spin correlations as earlier data suggested.

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J. Snyder

Papers

How ‘spin ice’ freezes

Low-temperature spin freezing in the Dy 2 Ti 2 O 7 spin ice

Synthesis and characterization of superconducting single-crystal Sn nanowires

Quantum-classical reentrant relaxation crossover in Dy2Ti2O7 spin ice.

Quantum and thermal spin relaxation in the diluted spin ice Dy2−xMxTi2O7 (M=Lu,Y)