Pyrochlore structures, of chemical formula A2B2O7 (A and B are typically trivalent and tetravalent ions, respectively), have been the focus of much activity in the condensed matter community due to the ease of substitution of rare earth and transition metal ions upon the two interpenetrating corner-shared tetrahedral lattices. Over the last few decades, superconductivity, spin liquid states, spin ice states, glassy states in the absence of chemical disorder, and metal-insulator transitions have all been discovered in these materials. Geometric frustration plays a role in the relevant physics of all of these phenomena. In the search for new pyrochlore materials, it is the RA/RB cation radius ratio which determines the stability of the lattice over the defect fluorite structure in the lower limit. Under ambient pressure, the pyrochlores are stable for 1.36 ≤ RA/RB ≤ 1.71. However, using high pressure synthesis techniques (1-10 GPa of pressure), metastable pyrochlores exist up to RA/RB = 2.30. Many of these compounds are stable on a timescale of years after synthesis, and provide a means to greatly enhance exchange, and thus test theories of quantum magnetism and search for new phenomena. Within this article, we review new pyrochlore compounds synthesized via high pressure techniques and show how the ground states are extremely sensitive to chemical pressure.

Over the last few decades, the importance of geometrical frustration as a guiding principle for many diverse phenomena has become increasingly recognized within the larger condensed matter community.1–4 There are many current reviews of these systems, with the pyrochlores taking centre stage as one of the most important three dimensional frustrated materials.5 Frustration arises when systems cannot satisfy all of their local constraints. Geometric frustration occurs when the local configuration of magnetic sites is the cause. The canonical example of frustration in two dimensions occurs for Ising-like spins upon the triangular lattice. For the nearest neighbour antiferromagnetic exchange, there is no solution for minimizing the energy defined by the Hamiltonian. As a result, the system tends to find alternate avenues to remove its entropy in the limit of 0 K.6,7 These compromises between local constraints on spins and tendency to follow the 3rd law of thermodynamics is the heart of the exotic magnetism observed.

In three dimensions, the natural analogue to the triangular plaquette of spins is the tetrahedron. There are multiple ways of connecting tetrahedra within extended solids to form geometrically frustrated lattices, but perhaps the most commonly studied are the edge-shared tetrahedral lattice and the corner-shared tetrahedral lattice. Ordered B-site perovskites of the formula A2BB’O6 have become increasingly well-studied over the last decade as face-centred cubic (FCC) structures with networks of edge-shared spins on the B or B’ site.8 Although many of these structures are not cubic (due to differing sizes of A, B, and B’ ions with different tolerance factors), and therefore have non-isotropic exchange, many of these systems have unusual ground states. Sr2CaReO6 and Sr2MgReO6 are spin glasses, for example, with random orientations of S = 1/2 spins freezing out at low temperatures in the absence of chemical disorder.9,10 Other S = 1/2 FCC lattices, such as Ba2Y MoO6, have been identified as valence bond glasses with clear singlet-triplet excitation spectra.11,12 Other ordered FCC systems show significant orbital effects such as Ba2Y OsO6.13 However, while many of these studies of edge-shared tetrahedral lattices are in their infancy, the corner-shared tetrahedral lattice exemplified by the pyrochlore structure A2B2O7 has been an archetypal magnetic system to study geometric frustration for decades.5 New pyrochlore lattices synthesized by high pressure methods form the basis for this review article, with emphasis on comparisons to the well-studied rare earth titanates.

The pyrochlore lattice, with general formula A2B2O7, has been widely studied since antiquity, but it has only been over the last few decades that the electronic and magnetic properties have been elucidated (Table I).14 The cubic crystal structure is preserved for a wide variety of trivalent (A site) and tetravalent (B site) ions; many of which can be easily synthesized under ambient conditions. The rare earth titanates are particularly attractive objects of study due to the relative stability of the Ti4+ ion under ambient conditions, and the ease of crystal growth using, for example, a floating zone image furnace. This has led to detailed studies of the varied effects of exchange interactions, crystal field configurations, and dipolar terms of the Hamiltonian using nearly the entire lanthanide series (with the exception of Nd2Ti2O7 and Pr2Ti2O7, as noted below).16,17 An understanding of the interplay of these terms in determining the ground states of these frustrated compounds has only been possible through considerable modelling and calculation efforts from theoretical condensed matter physicists.

TABLE I.

Summary of the magnetic properties of R2Ge2O7 and R2Pb2O7.

R θN or θC Anisotropy Ground State a (Å)
Pr(Pb)  Easy axis  Quantum spin ice?  10.8721(9) 
Nd(Pb)  0.41 K  Easy axis?  Antiferromagnet?  10.8336(4) 
Gd(Pb)  0.8 K  Isotropic  Ferromagnet?  10.7292(8) 
Tb(Ge)  Easy axis  Spin liquid  9.9617(1) 
Dy(Ge)  Easy axis  Spin ice  9.9290(5) 
Ho(Ge)  Easy axis  Spin ice  9.9026(6) 
Er(Ge)  1.41 K  Easy plane  Antiferromagnet?  9.8767(3) 
Yb(Ge)  0.61 K  Easy plane  Antiferromagnet?  9.8306(6) 
R θN or θC Anisotropy Ground State a (Å)
Pr(Pb)  Easy axis  Quantum spin ice?  10.8721(9) 
Nd(Pb)  0.41 K  Easy axis?  Antiferromagnet?  10.8336(4) 
Gd(Pb)  0.8 K  Isotropic  Ferromagnet?  10.7292(8) 
Tb(Ge)  Easy axis  Spin liquid  9.9617(1) 
Dy(Ge)  Easy axis  Spin ice  9.9290(5) 
Ho(Ge)  Easy axis  Spin ice  9.9026(6) 
Er(Ge)  1.41 K  Easy plane  Antiferromagnet?  9.8767(3) 
Yb(Ge)  0.61 K  Easy plane  Antiferromagnet?  9.8306(6) 

The first thorough study of the model pyrochlore titanate structure was made by Knop on Er2Ti2O7.18 It is fortunate that this compound was chosen as it is a chemically well-ordered example within the rare earth series. The space group Fd-3m was established as the proper cubic symmetry. The positions (0, 0, 0), (1/4, 1/4, 0), (1/4, 0, 1/4), and (0, 1/4, 1/4) are occupied by the A = Er rare earth site, which form the network of corner-shared tetrahedra now synonymous with pyrochlores when translated by a face-centering.19 The local symmetry of the rare earth site is defined by two equidistant oxygen atoms which are oriented along the 〈1, 1, 1〉 axis, and the remaining six oxygen atoms in a plane perpendicular to this axis forming a distorted hexagon. It is these local anions which determine the magnetic symmetry of the rare earth site (with an axial term and a planar term which can both play a role). Holmium and dysprosium titanate, for example, have dominate exchange which leads to a strong axial term and therefore Ising-like moments, while species such as erbium titanate have planar XY-like moments.2,20 These effects can be tuned with different choices of cations upon the A-site. The B-site also forms an interpenetrating network of corner-shared tetrahedra, but this review only considers magnetism associated with the A-site.

Subramanian was among the first to characterize the stability fields of different pyrochlores for varied ratios of RA/RB radii.14,15 This is the dominant factor controlling the ordering of ions within the pyrochlore lattice, which competes with the disordered fluorite lattice for similar cation sizes. As an empirical rule, the stability region under ambient pressure follows the relationship 1.36 ≤ RA/RB ≤ 1.71 (Figures 1 and 2).5,14 However, under pressure, metastable pyrochlore phases can be synthesized with well-ordered sublattices which extend these boundaries.15 The germanate series A2Ge2O7 and the plumbate series A2Pb2O7 are two examples of recent pyrochlores which have been prepared to test the application of chemical pressure to the canonical titanate pyrochlores.21–26 It is these series that will be the focus of the remainder of this review.

FIG. 1.

Schematic of the known pyrochlores of cation valence 3+ or 4+. A figure of the inter-connected A and B site cations is shown in the bottom left hand corner.

FIG. 1.

Schematic of the known pyrochlores of cation valence 3+ or 4+. A figure of the inter-connected A and B site cations is shown in the bottom left hand corner.

Close modal
FIG. 2.

The rare earth stability field of pyrochlores. Species inside the area delimited by a dashed line are prepared under applied pressure. Species outside of the dashed line are stable under ambient pressure conditions.5 

FIG. 2.

The rare earth stability field of pyrochlores. Species inside the area delimited by a dashed line are prepared under applied pressure. Species outside of the dashed line are stable under ambient pressure conditions.5 

Close modal

Throughout this review, the germanates and plumbates are referenced with respect to the pyrochlore structure. It is well known that at the edges of the stability field, chemical disorder is present which can result in tendencies to form the defect fluorite phase or other phases such as A4B3O12.27,28 With these new pyrochlores synthesized under high pressure, there is no evidence for such competing phases, but more rigorous tests using techniques, such as PDF, have not been reported in the literature. There are also possibilities for A/B site mixing, but these scenarios are unlikely for very large differences in RA and RB. The “stuffing” of pyrochlores has also been observed for the titanates, where the A and B sites can mix such as in Y b2Ti2O7.29 This is one of the explanations for the wide range of ratios that can be accommodated in the pyrochlore structure. Throughout this review, such effects are noted when reported in the literature, but, in general, many of the pyrochlores prepared under high pressure techniques have a high degree of A/B site ordering.

Rapid advances in the technology of high pressure synthesis have made such methods more feasible for the modern condensed matter scientist. In general, the term “high pressure” is reserved for appliedpressures of 0.1 GPa or greater, beyond typical hydrostatic methods.14,30 Bridgeman was one of the first pioneers of these methods in the 1940s, combining high pressures and high temperatures to start exploring new phase diagrams of solids.31 The first such methods involved combinations of anvils to achieve pressures for small sample sizes (i.e., 100 μ m) of up to 4 GPa. Walker advanced many of these techniques in the 1990s to increase the size of the samples and pressures to greater than 10 GPa.32 For many of the samples studied in this review article, Walker-type presses were used to obtain single phase materials of sizes typical for characterization and neutron scattering experiments. Diamond anvil cells can also be used, although with smaller product sample sizes. High pressures are essential for the stabilization of these new pyrochlores, which tend to form lower symmetry structures under ambient pressure (such as the tetragonal pyrogermanate Ho2Ge2O7).33 

Spin ices have held the attention of condensed matter physicists and solid state chemists alike for over a decade.2,5 While the original mapping of the proton ordering in water ice was first put forward by Pauling and others, the first connection to spin disorder in pyrochlores was made by Anderson.34 However, it took many years before candidates were discovered which displayed what is known as the “Pauling entropy” at low temperatures.35,36 Both water ice and spin ice have tetrahedral local constraints which lead to 1.68 J/mol K of residual entropy (of proton ordering in the latter case and spin order in the former). Ho2Ti2O7 and Dy2Ti2O7 were later discovered to have precisely the correct Pauling entropy for a “two-in, two-out” state below a temperature scale of ∼1 K, championing the classical statistical mechanics approach for both systems (see Figures 3 and 4).1,37,38 Since these spin ices have a variety of specific constraints to favour such as state—well-isolated Kramers doublets, large dipolar moments, and weak nearest-neighbour ferromagnetic exchange—there have been few other candidates put forward as true spin ices.39 While modern theories, and some experiments, seem to suggest that entropy is recovered at extremely low temperatures in the canonical spin ices, it is widely accepted that within a range of temperatures the local configurations are “ice-like.”40,42

FIG. 3.

Adaptation of the den Hertog/Gingras dipolar spin ice phase diagram, with experimental data added. Smaller lattices with increased antiferromagnetic exchange push the spin ices towards the Q = 0 ordered state phase boundary.40,41

FIG. 3.

Adaptation of the den Hertog/Gingras dipolar spin ice phase diagram, with experimental data added. Smaller lattices with increased antiferromagnetic exchange push the spin ices towards the Q = 0 ordered state phase boundary.40,41

Close modal
FIG. 4.

The two competing phases for spin-ices: (a) the Q = 0 “all-in all-out” phase and (b) the spin-ice “two-in, two-out” phase.

FIG. 4.

The two competing phases for spin-ices: (a) the Q = 0 “all-in all-out” phase and (b) the spin-ice “two-in, two-out” phase.

Close modal

The interest in spin ices piqued recently with the reformulation of the problem into a form in which quasiparticle excitations of the ice-state (namely, a “three-in, one-out” or “one-in, three-out” excitation) can be viewed as a “monopole” excitation.43 This fractionalization of magnetic charge is unique in condensed matter systems and leads to deviations away from the “spin-ice” models which are largely classical in origin.44,45 Recent experiments imply that the monopole model works rather well in describing the heat capacity of Dy2Ti2O7 in particular, which has a higher density of monopoles owing to the smaller lattice (see Figure 5). This is reflected in a lowering of the Weiss temperature as the lattice contracts in size, which effectively lowers the chemical potential for monopole creation. Although the monopole density can be quite low at low temperatures (on the order of parts per thousand to percent spin defects), one attractive route towards increasing the density is by shrinking the lattice (effectively increasing the exchange and therefore lowering the chemical potential for monopole formation).21 

FIG. 5.

(a) Rescaled unitless temperature scale (with respect to the chemical potential) T as a function of calculated monopole density. The germanate pyrochlores have an order of magnitude more monopoles per unit volume compared to their titanate analogues. (b) The magnetic heat capacity of Dy2Ti2O7 (DTO) compared to Dy2Ge2O7 (DGO) (inset). Debye-Huckël theory (DT) can be used alone to calculate the heat capacity of DTO, but Bjerrum corrections (DT + B) are needed for the high density of monopoles in DGO. The dotted line represents the DH + B fit to DGO, with a comparison to an ideal lattice gas with onsite Coulomb repulsion as a dashed line.21 

FIG. 5.

(a) Rescaled unitless temperature scale (with respect to the chemical potential) T as a function of calculated monopole density. The germanate pyrochlores have an order of magnitude more monopoles per unit volume compared to their titanate analogues. (b) The magnetic heat capacity of Dy2Ti2O7 (DTO) compared to Dy2Ge2O7 (DGO) (inset). Debye-Huckël theory (DT) can be used alone to calculate the heat capacity of DTO, but Bjerrum corrections (DT + B) are needed for the high density of monopoles in DGO. The dotted line represents the DH + B fit to DGO, with a comparison to an ideal lattice gas with onsite Coulomb repulsion as a dashed line.21 

Close modal

This was one of the leading arguments for investigating the germanate analogues of the spin ices, where monopole concentrations should increase by an order of magnitude at any given temperatures.21 Ho2Ge2O7 and Dy2Ge2O7 were reported as stable, well-ordered pyrochlore structures prepared under high pressure (8 GPa) and at temperatures of 1000 °C.14 However, there were no reports of the magnetic properties of these compounds. The first detailed report by Zhou et al. appeared in Nature Communications in 2011.21 Both pyrochlores had lattice constants which agreed with earlier reports (a ∼ 10.1 Å) and appeared to have well-ordered sublattices with high quality refinements. The shift in Weiss temperatures to larger, more antiferromagnetic exchange (θCW ∼ 0 K, with the net antiferromagnetic exchange cancelling out the net ferromagnetic nearest neighbour dipole exchange) is consistent with noted trends in pyrochlore structures under chemical pressure. Magnetization results, later detailed by Hallas et al., displayed the characteristic saturation at low temperatures which is also consistent with the “two-in, two-out” spin configuration.24 Finally, in the case of Ho2Ge2O7, detailed polarized elastic neutron scattering experiments have shown that this is the correct local spin structure, and later inelastic time-of-flight experiments indicated that Ho3+ crystal field scheme is indeed similar to the titanate—that of a series of doublets, well-separated from the ground state by an energy scale of ∼200 K.17,24 Detailed neutron scattering experiments of Dy2Ge2O7 are not yet reported in the literature (presumably due to the high thermal neutron absorption cross section of naturally occurring Dy).

For highly concentrated monopole densities, recombination of magnetic charge should occur in pairs in an analogous way that electrolytes display pairing of positive and negative ions in solution. The physical chemistry of such processes was elucidated by Bjerrum in 1926 for describing the activities of concentrated ions in solution as a correction to the standard Debye-Huckël theory.46 The correction for Bjerrum pairs was verified for many electrolytes and is now a well-accepted first-order approximation made for electrolyte solutions. Since the physics of monopoles was based upon Debye-Huckël theory, the parameters linking spin ices and electrolytes have already been established with the moment size, density, and relevant exchange.21 Much of the physics of the spin ices can be recaptured with the definition of two dimensionless parameters, T (temperature) and x (monopole creation). The temperature scale T is defined as

T = 4 π k B T a μ 0 Q 2 ,
(1)

where T is the temperature in Kelvin, a is the lattice parameter, and Q is the elementary monopole charge, given by

Q = 2 μ a .
(2)

The dimensionless monopole density xca3, where c is the concentration. Since the parameters T and x are not independent of each other, each spin ice species follows a trajectory in T-x space as shown in Figure 5(a). At any given temperature, it can be clearly noted that the germanates have markedly higher concentrations of monopoles than their titanate analogues. Therefore, the germanate spin ices, with large concentrations of monopoles at low temperatures where Bjerrum pairing should become significant, provide a direct test of the magnetic charge theory. In the case of Ho2Ge2O7, this correction is not very significant, as the monopole concentration is roughly within the same range as Dy2Ti2O7 where such effects are not important. However, the Bjerrum correction in Dy2Ge2O7 is substantial and dramatically changes the form of the specific heat to a sharper feature. With essentially no fitting parameters (since there were already defined by the spin ice problem), the specific heat of Dy2Ge2O7 can be reproduced to high precision, beautifully demonstrating the validity of the magnetoelectrolyte approach for describing monopole excitations (see Figure 5). The sharpness of the heat capacity peak compared to other pyrochlores can be envisioned as the approach to a phase transition (presumably to the antiferromagnetic Q = 0 state predicted by Gingras and den Hertog40) induced by the condensation of monopoles into static effective dipoles, but no experiments have yet observed such a phase boundary (Figure 3). The application of physical pressure might be an avenue of further exploration of the monopole-dipole transition.

One of the most intensely studied rare earth pyrochlores is arguably Tb2Ti2O7. Early reports of this compound indicated no sign of magnetic ordering down to 57 mK, as evidenced by magnetic susceptibility, neutron scattering experiments, muon spin relaxation measurements, and heat capacity.47,48 This is surprising given the strong antiferromagnetic exchange (θCW ∼ − 17.5(3) K) between the Ising-like spins of μeff = 9.6 μB. Extensive magnetic diffuse scattering observed at low temperatures suggests that the spins are ordered in a local antiferromagnetic arrangement upon each tetrahedron, but these clusters are dynamic at low temperatures, yielding a “co-operative paramagnet” in the limit of 0 K.49 Many theories have been proposed for the low temperature behaviour of Tb2Ti2O7, including those of a “quantum spin ice” state whereby the local spins are ordered in a two-in, two-out spin arrangement, but the network of spins does not find long ranged magnetic ordering due to quantum fluctuations.50,51

An understanding of the dynamic ground state of Tb2Ti2O7 is further complicated by the unusual and fragile response to applied pressure. At an applied pressure of 8.6 GPa (which corresponds to a 1% reduction of the lattice), Tb2Ti2O7 magnetically orders at mK temperatures. However, the application of chemical pressure, such as with the synthesis of Tb2Ge2O7 (prepared at 8 GPa of applied pressure at 1000 °C), which results in a 2% reduction of the lattice, leaves the system in a dynamic state at low temperature.25,52 While many of the physical properties of the Ge analogue are similar to Tb2Ti2O7 (a Curie–Weiss-like susceptibility with θCW = − 19.2(1) K and μeff = 9.87(3) μB, and a broad series of heat capacity anomalies at low temperatures), the nature of the dynamic state seems to be quite different. Evidence for ferromagnetic clusters coexisting with the spin-ice clustering noted in Tb2Ti2O7 has been recently seen in polarized elastic neutron scattering experiments down to 100 mK (see Figure 6).25 Clearly, future study is required to elucidate the nature of the spin-liquid response observed.

FIG. 6.

The magnetic diffuse scattering of Tb2Ti2O7 as a function of temperature. The broad features which appear by 3.5 K are due to short-ranged magnetic ordering and are similar to Tb2Ti2O7. The rapid upturn near Q = 0 which appears at 100 mK is due to ferromagnetic clusters.25 

FIG. 6.

The magnetic diffuse scattering of Tb2Ti2O7 as a function of temperature. The broad features which appear by 3.5 K are due to short-ranged magnetic ordering and are similar to Tb2Ti2O7. The rapid upturn near Q = 0 which appears at 100 mK is due to ferromagnetic clusters.25 

Close modal

The XY pyrochlores Er2Ti2O7 and Y b2Ti2O7 have piqued the interest of the condensed matter community due to the role that quantum fluctuations play in stabilizing the ground state. Both of these systems have Kramers ions of Y b3+ and Er3+ which, when split by the local crystal-fields, result in the lowest-lying doublet having pseudo-S = 1/2 properties, with the moment lying predominantly in an XY plane which is perpendicular to the [111] direction.20 While the size of the moment is quite different in both materials (and thus the role of quantum fluctuations varies), the Er and Yb titanates are still studied as models of pyrochlore Hamiltonians, and are providing insight into the prevalence of the so-called “order-by-disorder” mechanism in frustrated systems.5 

In the case of Er2Ti2O7, a magnetic phase transition occurs at TN = 1.2 K as evidenced by a lambda-like feature in the specific heat, the appearance of magnetic Bragg peaks, and a cusp in the DC susceptibility.19,20,53 Early experiments on this material suggested that the ordered state, usually described by the irreducible representations ψ2 or ψ3, is selected by the quantum order-by-disorder mechanism.54 Within this scenario, quantum fluctuations drive the resulting spin configuration at low temperatures. For some time, the magnetic structure of this compound was ambiguous, with ψ2 and ψ3 having virtually indistinguishable powder neutron diffraction patterns. Recent neutron polarimetry experiments, coupled with sophisticated numerical work to model the thermodynamic properties within the ordered state, have verified that the non-coplanar ψ2 state is indeed the likely magnetic structure, and that order-by-disorder cannot be ignored as an essential ingredient of XY pyrochlore magnetism.58 

Despite this progress, however, there are many unanswered questions surrounding the physics of Er2Ti2O7. For example, many theoretical magnetic phase diagrams exist in the literature of XY pyrochlores in general, by Gingras55 and Moessner.56 These diagrams are extremely sensitive to approximations made in estimating the relevant J constants (J+−, J+−+−, Jz+−, Jzz) and the role of long-range dipolar interactions. Recent theoretical work suggests that Palmer-Chalker correlations are important for all XY systems (as for the case of Er2Sn2O7, which fails to order at low temperatures and is considered a spin-liquid candidate).57 It is clear that other model XY systems are needed to resolve these differences. This was the motivation behind the initial study of Er2Ge2O7.

Er2Ge2O7 was synthesized under 8 GPa of pressure and at a temperature of 1000 °C using a Walker press.23 High quality samples have been reported with ordered A/B sublattices and clean x-ray and neutron diffraction refinements. With the much smaller lattice compared to the titanate (a = 9.8767(3) Å with a = 10.0727 Å for Er2Ti2O7), Er2Ge2O7 is an excellent candidate to study the effects of increased nearest neighbour exchange.23 However, caution is needed in direct comparisons of J constants, as the local Er3+ environment exhibits distortion as a function of applied chemical pressure. One can parameterize this as ρ = z/y, a ratio of the nearest neighbour Er-O1 and Er-O2 distances. For a perfect pyrochlore under little strain, this is close to unity, but under strain the local environment distorts axially to provide two relevant distances (with ρ ∼ 0.85 for the germanate, signifying an axial distortion with shorter Er-O2 bonds). The overall effect is a shift of all J values towards stronger antiferromagnetic exchange, but a detailed determination of these constants is still a work in progress. However, initial heat capacity experiments indicate that there is an enhanced TN temperature of 1.41 K, illustrating the enhanced exchange, and a low temperature entropy release of Rln2 as expected by the ground state doublet.23 Preliminary neutron scattering experiments have confirmed this ordering, but there is some ambiguity with respect to the antiferromagnetic structure (ψ2 or ψ3 as shown in Figure 7) that needs to be resolved by polarized neutron diffraction measurements. The ordered moment is 3.2(1) μB as determined by refinements of either structure.59 A determination of the proper structure would be a crucial test of the “order-by-disorder” mechanism in an XY pyrochlore with markedly enhanced exchange interactions.55 

FIG. 7.

(a) Heat capacity measurements on Er2Ge2O7. The best fits to the data above TN = 1.41 K are to two crystal field doublets above the ground state doublet. The first two excited doublets at energies of 6.4 meV and 7.4 meV agree well with the crystal field scheme of Er2Ti2O7 (b) The integrated Cmagnetic/T entropy release which yields Rln2 at low temperatures. (c) DC susceptibility of Er2Ge2O7, with the inset demonstrating high temperature Curie-Weiss behaviour. (d) Schematic of the two possible magnetic structures ψ2 and ψ3 which belong to the irreducible representation Γ5.23,59

FIG. 7.

(a) Heat capacity measurements on Er2Ge2O7. The best fits to the data above TN = 1.41 K are to two crystal field doublets above the ground state doublet. The first two excited doublets at energies of 6.4 meV and 7.4 meV agree well with the crystal field scheme of Er2Ti2O7 (b) The integrated Cmagnetic/T entropy release which yields Rln2 at low temperatures. (c) DC susceptibility of Er2Ge2O7, with the inset demonstrating high temperature Curie-Weiss behaviour. (d) Schematic of the two possible magnetic structures ψ2 and ψ3 which belong to the irreducible representation Γ5.23,59

Close modal

Y b2Ti2O7 is also an easy-plane XY-system, albeit with a completely different magnetic ground state than Er2Ti2O7. The Curie-Weiss temperature of 0.59 K suggests net ferromagnetic interactions, with a paramagnetic moment size of 3.34 μB.20 Below 0.2 K, there is a transition as evidenced by a cusp in the heat capacity and a dramatic change in the fluctuation rate (by more than three orders of magnitude) as seen by muon spin relaxation and Mossbauer experiments.60 Despite this, there is little evidence of long-range order seen from elastic neutron scattering below the transition. Early polarized neutron scattering experiments suggested that there could be a small ferromagnetic moment from residual intensity developing upon the nuclear Bragg peaks, but this neither account for the entropy released as calculated from heat capacity nor does it explain the significant change in the spin dynamics at low temperatures.61 After many years of experiments backed by theory papers, a picture has slowly emerged of a “quantum spin-ice” state developing at low-temperatures—a ground state with local two-in, two-out spin configurations on each tetrahedra in zero applied magnetic field, but with no long-range frozen order.62 The dynamic character of the ground state has been seen in persistent low temperature spin fluctuations from μSR and elegant inelastic neutron scattering experiments.63,64 Applied fields lift the ground state degeneracy and induce magnetic ordering, which has been extensively modelled with relevant J-constants.65,66 Within the monopole picture of quasiparticle excitations, the “quantum spin-ice” state is composed of local “2-in, 2-out” spin configurations, but with only short-ranged order as the spins fluctuate between degenerate states.

Y b2Ge2O7 provides a natural route to explore the quantum spin ice phase diagram. The effect of chemical pressure, which reduces the lattice from 10.032 Å to 9.8306(6) Å, results in a phase with enhanced planar exchange constants (θCW = 0.9 K). However, the proximity to a magnetic ordered state out of the magnetic Coulomb liquid has only just been verified through sample property measurements.22 Preliminary experiments have shown evidence for ordering with a drop in the AC susceptibility at TN = 0.61 K (see Figure 8). The magnetic field dependence of the peak is very different than the titanate or stannate analogue—with an increase in TN as a function of applied fields as opposed to a decrease in TC in applied fields implying an antiferromagnetic ground state in the Ge system (as opposed to a ferromagnetic state). Neutron scattering experiments are needed to verify the magnetic structure. Even though the Curie-Weiss temperature is positive, the shift in the AC peak as a function of field is atypical of ferromagnetism, leading to the possibility that the ground state could be antiferromagnetic. Indeed, for Y b2Ti2O7, Thompson has proposed a Hamiltonian which includes JIsing = 0.76 K (for interactions along the [111] z-component of J), Jiso = 0.18 K for isotropic exchange, Jpd = − 0.26 K for pseudo-dipolar terms, and JDM = 0.25 K for the Dzyaloshinskii-Moriya interactions.67 The local distortions of the Y b3+ site under chemical pressure could easily shift these terms, in particular, the JIsing and Jiso terms. The dipolar interactions, which depend on 1/r3 interactions, would only change by a fraction of a Kelvin for the 4.6% reduction of the lattice from Y b2Ti2O7 to Y b2Ge2O7. The fragile nature of the ground state of Y b2Ti2O7 is clearly apparent.

FIG. 8.

(a) X-ray refinement of Y b2Ge2O7. The tiny peak near 2θ = 29° is due to a small amount of unreacted Y b2O3 precursor. (b) DC susceptibility measurements of Y b2Ge2O7 demonstrating Curie-Weiss behaviour. The inset shows the dependence of the lattice parameter as a function of chemical pressure for the Yb series. (c) χ’ measurements of Y b2Ti2O7 near TN (d) Field dependence of χ’ near TN (e) Field dependence of TN and TC of the known Yb pyrochlores.22 

FIG. 8.

(a) X-ray refinement of Y b2Ge2O7. The tiny peak near 2θ = 29° is due to a small amount of unreacted Y b2O3 precursor. (b) DC susceptibility measurements of Y b2Ge2O7 demonstrating Curie-Weiss behaviour. The inset shows the dependence of the lattice parameter as a function of chemical pressure for the Yb series. (c) χ’ measurements of Y b2Ti2O7 near TN (d) Field dependence of χ’ near TN (e) Field dependence of TN and TC of the known Yb pyrochlores.22 

Close modal

The Pb4+ species is the largest B-site cation allowed for the pyrochlore structure, with the RA/RB ratio = 1.36 for Gd2Pb2O7.26 Many of the representative members of the series show considerable A/B site mixing, ranging up to 5%-8% for several compounds. High pressure synthetic methods tend to promote site ordering; however, many of the plumbates can be produced under relatively low pressures of ∼2 GPa. Here, we report on the known magnetic plumbates in the literature, which includes Nd2Pb2O7, Pr2Pb2O7, and Gd2Pb2O7.

One of the distinct advantages of high pressure synthesis lies in ease of synthesizing the Nd3+ and Pr3+ species, which have no titanate analogues under ambient pressure techniques. The lattice parameter of Nd2Pb2O7 is enlarged from the next stable pyrochlore reported, Nd2Sn2O7 (10.834 Å as opposed to 10.567 Å), which leads to an enhanced Weiss temperature of −0.06 K compared to −0.31 K in the stannate. At low temperatures, evidence is noted for an isolated doublet from an anomaly in the heat capacity which integrates to an entropy release of Rln2 (Figure 9). This is expected for a Kramers ion with an isolated doublet (Nd3+ has J = 9/2, which is split into five doublets). DC magnetization measurements at low T show saturation of the moment in low fields to 1.26 μB which is only half of the expected effective moment of 2.47 μB. This is likely due to powder averaging effects and Ising anisotropy.

FIG. 9.

Cmagnetic of Nd2Pb2O7, which is indicative of a magnetic ordering transition near 0.41 K. In applied fields, the transition is not visible down to 0.35 K, but the broadening of the low-T Schottky anomaly from the ground state doublet is. The inset shows the integrated magnetic entropy release.26 

FIG. 9.

Cmagnetic of Nd2Pb2O7, which is indicative of a magnetic ordering transition near 0.41 K. In applied fields, the transition is not visible down to 0.35 K, but the broadening of the low-T Schottky anomaly from the ground state doublet is. The inset shows the integrated magnetic entropy release.26 

Close modal

A sharp peak in the heat capacity at TN = 0.41 K is suggestive of magnetic ordering in Nd2Pb2O7, much like in Nd2Sn2O7. However, very little is known about the magnetic structure. In Nd2Sn2O7, the Q = 0 antiferromagnetic structure below TN = 0.91 K has recently been confirmed via neutron scattering experiments. With the dramatic enhancement of the Weiss temperature in the plumbate, a different magnetic structure could very well be possible.

The ground state of Pr2Pb2O7 is currently a matter of debate in the literature. Like the Nd analogue, Pr2Ti2O7 does not exist due to the lanthanide contraction, and there is no titanate to reference. However, Pr2Sn2O7 has been studied extensively by the condensed matter community. Zhou first reported the existence of a “dynamic spin ice” state from their characterization work on Pb2Sn2O7.68 While the magnetic susceptibility did not show a transition down to mK temperatures (with a Weiss temperature of 0.3 K), the heat capacity displayed an anomaly which could not be fit to a standard Schottky functional form. The integrated entropy displayed a result that was inconsistent with the dipolar spin ice model. Elastic neutron scattering experiments down to 0.2 K displayed a characteristic spin-ice diffuse scattering profile, with slightly enhanced Q=0 scattering from ferromagnetic interactions.68 Inelastic neutron scattering experiments indicated that this ground state was dynamic in nature, as opposed to the spin-ices which have a frozen state on the typical neutron time scale. The interpretation of this result was that the local spins are stuck in a “two-in, two-out” ground state similar to a quantum spin ice, and fluctuating in a similar manner. However, detailed theory work to flesh out the J-constants is currently lacking. Another problematic issue is the non-Kramers nature of the Pr3+ ion, which may exist in a singlet state at lowest temperatures. This issue may also play a role in the “quantum spin ice” Pr2Zr2O7, which has properties similar to the stannate but can be grown in single crystal form. Issues of oxygen stoichiometry and site mixing due to the large Zr4+ cation are only now being resolved.69 

Pr2Pb2O7 has many similar properties to both the stannate and zirconate—a low temperature anomaly in the magnetic heat capacity which cannot be fit to a Schottky anomaly, an integrated magnetic entropy which falls short of the canonical spin-ice entropy, and significant single ion anisotropy as shown by a saturation of the moment in magnetization measurements at half of the expected value. At the time of writing, little is known about the ground state.26 

The origin of the unusual low temperature magnetism of the titanate Gd2Ti2O7 has remained a mystery for decades. At a first glance, this is a model Heisenberg system, with Gd3+ possessing a free-ion state of 8S7/2 and a moment of μ = 7.94 μB (consistent with the low temperature Curie-Weiss fits to the Gd2Ti2O7 susceptibility by Cashion,70 Raju,71 and Bramwell20). The Weiss temperature ranges from −11.8(4) K to −9.6(1) K depending upon the fitting range, but all of these results on powders are indicative of dominant antiferromagnetic exchange. Magnetic ordering has been confirmed via a cusp in the susceptibility at 1.1 K, and with the appearance of magnetic Bragg peaks observed from neutron scattering on isotopically selected samples. However, the magnetic structure is still a matter of controversy, with multi-k structures playing a role. In addition, the coexistence of magnetic diffuse scattering with the magnetic Bragg peaks implies phase separation or at the very least partial ordering coexisting with ordered regions.72,73 A unified scenario which is consistent with the experimental results is still lacking, although recent progress has been made with suggestions for a Palmer-Chalker ground state.

In an effort to understand and model the magnetism in Gd2Ti2O7, related samples have been synthesized and studied. Under ambient conditions, Gd2Zr2O7 and Gd2Hf2O7 have been prepared by Durand.74 Both samples order at temperatures slightly less than 1 K, but chemical disorder and site mixing are still an issue. Another example of an analogue to Gd2Ti2O7 has been recently prepared by Hallas with Gd2Pb2O7 via high pressure methods.26 With one of the smallest RA/RB ratios of any of the pyrochlores, site mixing is still expected to be an issue with the plumbate; recent estimates indicate A/B site mixing as high as 30% in some samples (as determined by x-ray diffraction).75 Similar to Gd2Ti2O7, Gd2Pb2O7 has an effective moment of 7.67(1) μB and a Weiss temperature of −7.4(1) K. The dominant antiferromagnetic exchange is diluted upon substitution with a larger cation (consistent with the other plumbates). There is a cusp in the low temperature DC susceptibility at TN = 0.8 K, along with a prominent lambda anomaly in the heat capacity, which when integrated, gives Rln8 expected for the entropy release of a S = 7/2 ion (Figure 10). However, the power-law of the magnetic heat capacity below TN varies as T3/2 as opposed to the expected T3 for an antiferromagnet. While it is tempting to associate this power-law with a ferromagnetic ground state, this has not been borne out yet with neutron scattering experiments, and this is unexpected with the large negative Weiss temperature (and for a simple Heisenberg model). It is clear that the Gd pyrochlores are not well understood, and systematic chemical pressure studies can play a key role in unravelling the character of the magnetic ground states.

FIG. 10.

Cmagnetic of Gd2Pb2O7, with the cusp at a magnetic transition temperature of 0.8 K. The integrated magnetic entropy release yields Rln8, as expected for the S = 7/2 Gd3+ ion.26 

FIG. 10.

Cmagnetic of Gd2Pb2O7, with the cusp at a magnetic transition temperature of 0.8 K. The integrated magnetic entropy release yields Rln8, as expected for the S = 7/2 Gd3+ ion.26 

Close modal

While the rare earth titanate pyrochlores have held the attention of condensed matter physicists for decades, it is only through recent breakthroughs, largely through new experimental and theoretical tools, that the magnetism in these model compounds is starting to be understood. Considerable progress can also be made with the synthesis of new quantum magnets under pressure, which serves as a tuning parameter for the relevant exchange parameters in the titanates. Various members of the germanate and plumbate series have been reviewed which have dramatically different ground states than their titanate counterparts. It is these compounds which will serve as crucial tests for theories that have been used for the titanates, and perhaps lead to further novel ground states within the pyrochlores. High pressure single crystal growth of these new species would also be desirable, but current methods, such as the floating zone approach which has been very successful for the titanates, typically cannot reach pressures greater than 1 GPa. The use of high pressure techniques could also be fruitful in obtaining titanates which cannot be synthesized at present, such as Nd2Ti2O7 and Pr2Ti2O7. Furthermore, new pyrochlores within the stability field should be able to be explored with high pressure techniques, such as those with transition metals upon the B-sites. There is ample phase space to explore, for example, the orbital physics of 4d and 5d transition metal oxides such as Mo and Os. Many of these studies are currently underway.

C. R. Wiebe is grateful for the financial support in the form of NSERC, CFI, CIFAR, and the Canada Research Chair program (Tier II). A. M. Hallas acknowledges support from the Vanier Scholar program (NSERC). This work is also made possible by various neutron scattering sources supported by the NSF, the DOE, EPSRC, STFC, and NRC.

The authors would like to express their gratitude to the many colleagues and mentors over the years who have encouraged and supported the study of geometric frustration: B. D. Gaulin, J. E. Greedan, G. M. Luke, A. S. Wills, M. J. P. Gingras, Y. J. Uemura, J. R. Stewart, H. D. Zhou, J. P. Attfield, R. J. Cava, P. Schiffer, J. S. Gardner, H. J. Silverstein, P. M. Sarte, A. J. Berlinsky, G. Ehlers, A. Z. Sharma, and J. S. Brooks.

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