Crystal shape-dependent magnetic susceptibility and Curie law crossover in the spin ices Dy2Ti2O7 and Ho2Ti2O7

Low-temperature high-frequency dynamic magnetic susceptibility of classical spin-ice Dy2Ti2O7

Radio-frequency (14.6 MHz) AC magnetic susceptibility,χAC', of Dy₂Ti₂O₇was measured using self-oscillating tunnel-diode resonator. Measurements were made with the excitation AC field parallel to the superimposed DC magnetic field up to 5 T in a wide temperature range from 50 mK to 100 K. At 14.6 MHz, a known broad peak ofχAC'(T)from kHz-range audio-frequency measurements around 15 K for both [111] and [110] directions shifts to 45 K, continuing the Arrhenius activated behavior with the same activation energy barrier ofE_a≈ 230 K. Magnetic field dependence ofχAC'along [111] reproduces previously reported low-temperature two-in-two-out to three-in-one-out spin configuration transition at about 1 T, and an intermediate phase between 1 and 1.5 T. The boundaries of the intermediate phase show reasonable overlap with the literature data and connect at a critical endpoint of the first order transition line, suggesting that these features are frequency independent. An unusual upturn of the magnetic susceptibility atT→ 0 was observed in magnetic fields between 1.5 T and 2 T for both magnetic field directions, before fully polarized configuration sets in above 2 T.

The history of spin ice

This review is a study of how the idea of spin ice has evolved over the years, with a focus on the scientific questions that have come to define the subject. Since our initial discovery of spin ice in 1997, there have been well over five thousand papers that discuss it, and in the face of such detail, it must be difficult for the curious observer to 'see the wood for the trees'. To help in this task, we go in search of the biggest insight to have emerged from the study of spin ice. On the way, we identify highlights and outstanding puzzles, and celebrate the inspirational role that Roger Cowley played in the early years.

Modification of spin-ice physics in Ho2Ti2O7 thin films

We present an extensive study on the effect of substrate orientation, strain, stoichiometry, and defects on spin-ice physics in Ho 2 Ti 2 O 7 thin films grown onto yttria-stabilized-zirconia substrates. We find that growth in different orientations produces different strain states in the films. All films exhibit similar c -axis lattice parameters for their relaxed portions, which are consistently larger than the bulk value of 10.1 Å. Transmission electron microscopy reveals antisite disorder and growth defects to be present in the films, but evidence of stuffing is not observed. The amount of disorder depends on the growth orientation, with the (110) film showing the least. Magnetization measurements at 1.8 K show the expected magnetic anisotropy and saturation magnetization values associated with a spin ice for all orientations; shape anisotropy is apparent when comparing in- and out-of-plane directions. Significantly, only the (110)-oriented films display the hallmark spin-ice plateau state in magnetization, albeit less well defined compared to the plateau observed in a single crystal. Neutron-scattering maps on the more disordered (111)-oriented films show the Q = 0 phase previously observed in bulk materials, but the Q = X phase giving the plateau state remains elusive. We conclude that the spin-ice physics in thin films is modified by defects and strain, leading to a reduction in the temperature at which correlations drive the system into the spin-ice state.

Special temperatures in frustrated ferromagnets

The description and detection of unconventional magnetic states, such as spin liquids, is a recurring topic in condensed matter physics. While much of the efforts have traditionally been directed at geometrically frustrated antiferromagnets, recent studies reveal that systems featuring competing antiferromagnetic and ferromagnetic interactions are also promising candidate materials. We find that this competition leads to the notion of special temperatures, analogous to those of gases, at which the competing interactions balance, and the system is quasi-ideal. Although induced by weak perturbing interactions, these special temperatures are surprisingly high and constitute an accessible experimental diagnostic of eventual order or spin-liquid properties. The well characterised Hamiltonian and extended low-temperature susceptibility measurement of the canonical frustrated ferromagnet Dy₂Ti₂O₇ enables us to formulate both a phenomenological and microscopic theory of special temperatures for magnets. Other members of this class of magnets include kapellasite Cu₃Zn(OH)₆Cl₂ and the spinel GeCo₂O₄.

Competing interactions in frustrated magnets give rise to complex emergent phenomena, which challenge a full microscopic understanding but invite comparison to other systems. Bovo et al. find an analogy to classical gases and identify special temperatures that reveal fine details of the microscopic Hamiltonian.

Harmonic phase in polar liquids and spin ice

Many liquid or liquid-like states remain stable down to temperatures well below the interaction energy scale, where mean-field theory predicts an ordering transition. In magnetism, correlated states such as spin ice and the spin liquid have been described as Coulomb phases, governed by an emergent gauge principle. In the physical chemistry of polar liquids, systems that evade mean field order have, in contrast, been described by Onsager’s theory of the reaction field. Here we observe that in the low-temperature limit, Onsager’s theory may be cast as a prototypical theory of the Coulomb phase. However at finite temperature, it describes a distinct geometrical state, characterised by harmonic functions. This state, labelled here the ‘harmonic phase’, is shown to occur experimentally in spin ice, a dipolar lattice system. It is suggested to be relevant to more general dipolar liquids.

Spin ice materials can be described using idealised models of frustrated magnetism and have motivated a revisiting of the theory of interacting dipolar systems. Bramwell shows that Onsager’s theory of polar liquids describes the Coulomb phase and predicts a distinct ‘harmonic phase’ at finite temperature.

Restoration of the third law in spin ice thin films

A characteristic feature of spin ice is its apparent violation of the third law of thermodynamics. This leads to a number of interesting properties including the emergence of an effective vacuum for magnetic monopoles and their currents – magnetricity. Here we add a new dimension to the experimental study of spin ice by fabricating thin epitaxial films of Dy₂Ti₂O₇, varying between 5 and 60 monolayers on an inert substrate. The films show the distinctive characteristics of spin ice at temperatures >2 K, but at lower temperature we find evidence of a zero entropy state. This restoration of the third law in spin ice thin films is consistent with a predicted strain-induced ordering of a very unusual type, previously discussed for analogous electrical systems. Our results show how the physics of frustrated pyrochlore magnets such as spin ice may be significantly modified in thin-film samples.

In bulk, the spin ice Dy₂Ti₂O₇ has posed an enigma because – due to its slow dynamics – it is unclear whether and how the material will reach a zero entropy state at zero temperature. Here, the authors show that in thin films of Dy₂Ti₂O₇ a zero entropy state is induced at 0.4 K, plausibly by lattice strain.