Frustrated magnetism in 227 rare-earth iridium pyrochlores

Abstract

Rare-earth pyrochlore oxides A₂B₂O₇ provide a versatile platform for studying how geometrical frustration, strong spin–orbit coupling, and electronic correlations cooperate to generate unconventional magnetic and electronic states. Within this family, the iridium pyrochlores A₂Ir₂O₇ stand out due to the interplay between a 5d Ir sublattice and, in many members, a magnetic 4f A-site sublattice, giving rise to a hierarchy of coupled structural, magnetic, and transport phenomena. Recent theoretical proposals suggest that this interplay may support magnetic monopole-like excitations, both of spin-ice type and domain-wall origin, possibly carrying electric dipoles, thus enabling novel magnetoelectric responses. Although these excitations remain experimentally elusive, ongoing studies increasingly constrain their microscopic character. This review presents a unified framework linking lattice geometry, f-d exchange, and spin–orbit-driven anisotropies, summarises key experimental advances across the A₂Ir₂O₇ series, and outlines future research directions aimed at detecting and controlling magnetic monopole-like excitations through magnetic fields, electric fields, and lattice perturbations.

The A₂Ir₂O₇ iridium pyrochlores series exibit an interplay between a 5d Ir sublattice and, in many members, a magnetic 4f A-site sublattice, which gives rise to a hierarchy of coupled structural, magnetic, and transport phenomena that may support magnetic monopole-like excitations that can enable novel magnetoelectric responses. In this Review, the authors discuss a unified framework linking lattice geometry, f-d exchange, and spin–orbit-driven anisotropies, summarising key experimental advances and outlining future research directions aimed at detecting and controlling magnetic monopole-like excitations through magnetic fields, electric fields, and lattice perturbations.

Introduction

Geometrical frustration of magnetic moments

In 2021, the Nobel Prize in Physics acknowledged ‘groundbreaking contributions to our understanding of complex physical systems’, spanning topics from Earth’s climate science to the microscopic structure and dynamics of frustrated magnetic materials¹. Understanding the complexity of these systems—ranging from microscopic to macroscopic properties—is vital for their comprehension, predicting their behaviour and exploring prospective usage in advanced technologies.

Frustrated magnetism

Frustrated magnetism arises when magnetic exchange interactions within a material cannot all be simultaneously satisfied, resulting in complex and often highly degenerate ground states. Scientific interest in these systems emerged in the 1950s when antiferromagnetically coupled Ising moments on a triangular lattice exhibited behaviour markedly different from their ferromagnetic or bipartite antiferromagnetic isostructural counterparts². Since then, frustrated magnetic materials have attracted significant research attention, particularly those based on 3d transition metals. Conversely, materials containing heavier magnetic elements remained relatively underexplored until the recent recognition of strong spin-orbit coupling (SOC) effects in these systems. SOC introduces pronounced anisotropy in magnetic interactions, thereby enhancing magnetic frustration^3,4.

Frustrated systems can exhibit either magnetically ordered states at low temperatures or remain in disordered, frustrated states near absolute zero. Examples of these disordered states include classical and quantum spin liquids, spin ice, and spin glass phases^5–8.

Spin ice features a magnetic structure analogous to the hydrogen-bonded network in water ice⁶, harbouring phenomena such as Dirac strings and magnetic monopole-like excitations. Quantum spin liquids, defined by entangled spins without long-range magnetic ordering even at zero temperature, are predicted to support fractionalised quasiparticles, including Majorana fermions^9,10. Both spin ice and quantum spin liquids hold significant potential for (topological) quantum computing and other novel technological applications.

Geometrically frustrated lattices

Frustrated magnetism is inherently linked to geometrically frustrated crystallographic lattices. Among the most notable geometrically frustrated and quasi-frustrated structures are the triangular, square, hyper-Kagomé, and pyrochlore lattices, as illustrated in Fig. 1.

Fig. 1. Examples of geometrical frustration of antiferromagnetically coupled Ising moments on a triangle (a), a square (b), and a tetrahedron (c).

A question mark schematically indicates the impossibility of satisfying all interactions; however, it simultaneously applies to the moments on all vertices. See text for details.

A simple but fascinating example of a 2D frustrated system is the equilateral triangular lattice, where magnetic ions occupy the vertices and exhibit Ising-like magnetic moments governed by antiferromagnetic interactions (Fig. 1a). In this configuration, while two moments can align antiparallel, the third moment cannot simultaneously align antiparallel to both, generating intrinsic frustration. These lattices may consist of either edge-sharing or corner-sharing triangles, the latter forming a so-called Kagomé lattice. The triangular lattice comprises distinct, well-separated layers of triangles and stabilises in a 120-degree ground state configuration considering only nearest-neighbour antiferromagnetic interactions¹¹. When subjected to an external magnetic field, the moments in this lattice can adopt various field-induced magnetic phases, such as the 1/3-magnetisation plateau or the umbrella phase¹². Furthermore, recent theoretical studies suggest that anisotropic spin interactions on triangular lattice may stabilise quantum spin-liquid phases^13,14. Similarly, the Kagomé lattice, characterised by corner-sharing triangles, can host a variety of complex states, including superconductivity and charge-density-wave (CDW) order^15–17, as well as chiral magnetic order^18–20.

In square lattices, magnetic frustration arises from competing antiferromagnetic interaction pathways: one between nearest neighbours along the square’s edge and another between next-nearest neighbours along the square’s diagonal (Fig. 1b). The dominant interaction dictates the ground state—nearest-neighbour coupling favours Néel antiferromagnetic order, while a stronger next-nearest-neighbour interaction induces a columnar magnetic structure. Importantly, recent experiments have confirmed the existence of the theoretically predicted quantum spin-liquid state in spin-1/2 Heisenberg square-lattice antiferromagnets on the border between Néel and columnar phases^21–23.

Canonical examples of frustrated lattices in three-dimensional systems include the pyrochlore and hyper-Kagomé lattices. The hyper-Kagomé lattice, a 3D analogue of the 2D Kagomé lattice, consists of corner-sharing triangles arranged in three-dimensional space. Antiferromagnetic interactions within this structure generate significant 3D frustration among magnetic moments. Quantum spin-liquid states have been experimentally observed in several hyper-Kagomé materials^24,25. Recently, pioneering studies have begun to explore SOC-driven quantum phenomena in these systems²⁶.

The pyrochlore lattice, composed of corner-sharing tetrahedra (Fig. 1c), is discussed in detail below. When populated by magnetic ions, this structure can exhibit magnetic frustration in both isotropic Heisenberg and Ising antiferromagnetic limits. In the Ising case, antiferromagnetic interactions among Ising-like moments can stabilise a doubly degenerate ground state known as the all-in-all-out (AIAO) magnetic structure, where all four moments on a tetrahedron point either inward or outward^27,28. An illustration of this configuration is provided in Fig. 2a.

Fig. 2. Selected types of magnetic moment arrangements on the pyrochlore lattice.

Two tetrahedra with Ising moments along local [111] directions are shown: all-in-all-out (AIAO) structure (a); two-in-two-out (2I2O) structure (b); and three-in-one-out (3I1O) structure (c) are depicted.

In contrast, ferromagnetic interactions within the pyrochlore lattice lead to a ground state governed by the so-called ice rule²⁹. Analogous to the arrangement of hydrogen and oxygen ions in water ice, two magnetic moments on each tetrahedron point inward, and two point outward, forming the two-in-two-out (2I2O, Fig. 2b) configuration^7,30,31. This ground state exhibits higher degeneracy—sixfold compared to the twofold iso-energy degeneracy of the AIAO configuration—as all possible 2I2O arrangements within a tetrahedron are enabled.

An external magnetic field can lift this degeneracy by inducing magnetic ordering without disrupting lattice symmetry, leading to a first-order phase transition. Moreover, moment configurations can dynamically transition among themselves through thermal or quantum fluctuations, forming classical or quantum spin-liquid states³². Also magnetic monopole-like configuration, where three moments on the tetrahedron point inward and one points outward (3I1O or 1I3O, Fig. 2c), can be induced and manipulated by an external field³³.

A₂B₂O₇ oxides

The A₂B₂O₇ oxides (where A is a rare-earth element and B stands for a d- or p-electron element) constitute one of the most extensively studied classes of geometrically frustrated materials. This family—along with other notable systems such as perovskites, spinels, volborthite, and herbertsmithite—displays complex physical properties arising not only from the geometrically frustrated lattice that hosts A and/or B cations with intrinsic magnetic moments, but also from the interplay of electron-electron (Coulomb) interactions, spin-orbit coupling (SOC)—especially significant in heavy B elements—and crystal field effects acting on the cations. The delicate balance among these interactions can give rise to a range of conductive states and topologically non-trivial phases, such as the topological Mott insulator, topological band insulator, Weyl semimetal state, and Fermi arc surface states^34–37. The choice of magnetic A and B elements influences exchange magnetic interactions and f-d exchange couplings, leading to phenomena such as anisotropy-tuned magnetic ordering^7,38 and giant magnetoresistance³⁹. Additionally, a coexistence of antiferromagnetic all-in-all-out (AIAO) order alongside a Coulomb phase spin-liquid or spin-ice state has been proposed for several members with both A and B being magnetic elements^7,30.

Crystal structures

The A₂B₂O₇ oxides crystallise primarily in two cubic structures: the pyrochlore structure (Fd-3m, space group no. 227) and defect-fluorite structure (Fm-3m, space group no. 225), with an additional orthorhombic (Pnma, space group no. 62) structure also observed in several members⁴⁰. Focusing on the cubic variants, the defect-fluorite structure features a disordered lattice in which A and B cations occupy a shared Wyckoff 4a position, forming a frustrated face-centred cubic (fcc) sublattice, with oxygen anions arranged in 8c sites, surrounding the A/B cations in cubic coordination (occupied by seven oxygen atoms). An illustration of this structure is provided in Fig. 3b.

Fig. 3. Unit cells of cubic ordered pyrochlore (a) and disordered defect-fluorite structure (b).

Wyckoff positions of individual sites are indicated. The 4a position is shared by A and B cations. The 8c Wyckoff position of the defect-fluorite structure is occupied by oxygen from 7/8. The interconversions from pyrochlore to the defect-fluorite lattice are detailed in the text.

In contrast, the pyrochlore lattice (Fig. 3a) represents a fully ordered structure, where A and B cations occupy distinct sites (16 d and 16c, respectively), each forming a three-dimensional network of corner-sharing tetrahedra. This ordered structure establishes the pyrochlore lattice as a fundamental example of a 3D geometrically frustrated system. Moreover, when viewed along the cubic [111] direction, projections of the A and B cation layers can be visualised as a system composed of alternating Kagomé and triangular layers. In this structure, oxygen anions occupy two distinct Wyckoff positions (8b and 48 f), forming 8- and 6-coordinate cages around the A and B cations.

The differences and interconversion from the pyrochlore to the defect-fluorite structure can be analysed from different perspectives: cations that occupy two crystallographically distinct sites in the pyrochlore lattice (16 d and 16c) become disordered in the defect-fluorite lattice, merging into a single site (4a; Fig. 3b). In the defect-fluorite lattice, oxygen anions that occupy the 48 f position in the pyrochlore lattice move to a higher-symmetry position 8c. This positional shift leads to a disorder of oxygen anions previously in two distinct sites (48 f and 8b) and an ordered vacancy at the virtual 8a site in the pyrochlore lattice. These anions merge onto a single, 7/8-occupied 8c site in the defect-fluorite lattice. This shift also alters the cation coordination from an eight-coordinate (A-site) and six-coordinate (B-site) geometry in the pyrochlore lattice to an average seven-coordinate arrangement for A/B site in the defect-fluorite lattice.

Although both pyrochlore and defect-fluorite structures exhibit geometrical frustration, they differ in the nature of their exchange pathways and magnetic interactions. Furthermore, defects in the fluorite structure significantly impact the oxidation states of a part of cations, influencing the material’s electronic and magnetic properties (see, e.g., refs. ^42–44).

The long-range ordered crystal structures of A₂B₂O₇ oxides depend strongly on the size ratio of the A and B cations (r_A/r_B). For instance, A₂Zr₂O₇ zirconates crystallise in a cubic pyrochlore structure when the ratio falls within the range 1.48 < r_A/r_Zr < 1.63 for A = La–Gd. In contrast, when r_A/r_Zr ≤ 1.47 for A = Tb–Lu, the defect-fluorite structure is realised⁴⁵. Recently, there has been renewed interest in the disordered defect-fluorite structure in several A₂B₂O₇ oxides due to discrepancies between local structure data (total scattering, pair-distribution function (PDF) analysis) and average structure data (Rietveld refinement)^46–48. Such analyses across different length scales reveal that the defect-fluorite structure may actually comprise short-range orthorhombic or rhombohedral structural units within an averaged cubic framework. This local and long-range structural interplay is essential for understanding the unique magnetic, electronic, and material properties of A₂B₂O₇ oxides.

Indeed, a recent study on the disorder in defect-fluorite Tb₂Hf₂O₇ has uncovered intriguing physics, with the coexistence of the Coulomb phase and spin-glass behaviour⁴⁹. Despite the substantial structural disorder, the correlated phase aligns with the local structural principles of the pyrochlore lattice, exhibiting the characteristic power-law correlations.

The inherent disorder and reduced geometrical frustration in defect-fluorite structure compared to the pyrochlore lattice has led to underestimating the low-temperature electronic and magnetic properties in these A₂B₂O₇ compounds. However, instead of a sharp boundary between pyrochlore and defect-fluorite structures, there is likely a continuum of local disorder across the series. This suggests that pyrochlore-like correlated electron physics could exist even in nominal defect-fluorite structures.

Electronic properties

In addition to structural stability and geometrical frustration, the electronic and magnetic properties of A₂B₂O₇ compounds are determined by microscopic interactions, particularly electron correlations and SOC. SOC, which describes the interaction between an electron’s spin and its orbital angular momentum, has minimal effects in light elements but becomes increasingly significant in heavier elements, where it substantially impacts the material’s electronic properties and magnetic ground state. This trend is particularly evident among the d-elements: while SOC effects are usually weak in 3d elements, they become central for the heavier 4d and 5d elements. Concurrently, the d-orbitals become more spatially extended with increasing atomic number, reducing electron repulsion and weakening electron correlation effects. Consequently, SOC and electron correlations exhibit comparable strength in 5d elements, giving rise to complex phenomena such as Weyl semimetals³⁵, topological Mott insulators³⁷, axion insulators⁵⁰, spin-ice states with monopole-like excitations⁶, or spin-liquid phases⁵.

A strong SOC further enables the spin Hall effect (SHE) and inverse SHE, facilitating control over spin currents through charge currents and vice versa⁵¹. Crystallographic families with suitable symmetry and significant SOC include pseudo-cubic and planar perovskites^52,53, spinel-related structures⁵⁴, and for this study relevant A₂B₂O₇ pyrochlores, all of which have been proposed as hosts for these phenomena.

The considerations outlined above highlight why rare-earth pyrochlores represent one of the most versatile platforms for studying geometrical frustration, hosting a broad spectrum of competing interactions and emergent magnetic states. Within this wider family, the iridium pyrochlores A₂Ir₂O₇ occupy a particularly prominent position: the presence of 5d electrons on the Ir sublattice introduces strong SOC, anisotropic exchange pathways, and a delicate balance between electronic and magnetic energy scales. These characteristics make A₂Ir₂O₇ compounds especially sensitive to structural distortions and magneto-elastic effects, and thereby uniquely suited for exploring frustration-driven phenomena. The following section therefore focuses specifically on the rare-earth iridates and the physical mechanisms that distinguish them from other A₂B₂O₇ pyrochlores.

Taken together, these considerations motivate a focused examination of the rare-earth iridates. Against this broader background, the present review is organised around a central question: how the interplay between lattice geometry, local structural distortions, and spin–orbit–driven magnetism shapes the emergence of monopole-like excitations, domain-wall phenomena, and correlated electronic states in the A₂Ir₂O₇ family. Rather than compiling isolated observations, our aim is to synthesise current experimental and theoretical insights into a coherent framework that links structural motifs to magnetic textures and transport anomalies, with particular emphasis on the role of A-site magnetism and magnetic monopole-electric dipole coupling. This perspective provides the conceptual basis for the following sections.

Rare-earth A₂Ir₂O₇ iridates

The rare-earth iridates A₂Ir₂O₇ belong to a broad family of pyrochlore oxides A₂B₂O₇, which frequently exhibit complex and exotic electronic properties. These include, for instance, fragmentation of magnetic moments in Ho₂Ir₂O₇^7,30, or complex magnetic structures in Nd₂Ir₂O₇^55,56, Tb₂Ir₂O₇²⁸, and Yb₂Ir₂O₇²⁷, originating from interactions between the magnetic A and Ir sublattices. Additionally, spin-liquid and spin-ice states are governed by the geometrical frustration inherent to the pyrochlore lattice of these materials^5,6,57. The diversity of ground states observed in A₂Ir₂O₇ oxides stems from this geometrical frustration, as well as a delicate balance among exchange, dipolar, and spin-orbit interactions within the system.

The influence of SOC on the magnetic and electronic states, combined with often weak or intermediate electron Coulomb correlations in iridates^58–60, generates topologically non-trivial phases. Examples include topological Mott insulators, topological band insulators, Weyl semimetal states, and Fermi-arc surface states^60–63.

Robustness of pyrochlore structure in A₂Ir₂O₇

The pyrochlore structure (space group Fd-3m) represents a highly ordered cubic framework characterised by only two independent structural parameters: the lattice parameter a and the fractional oxygen coordinate x_48f at the 48f Wyckoff position (Fig. 3a, Table 1). The latter governs the trigonal distortion of oxygen polyhedra surrounding the rare-earth and iridium ions. Ir–O bond lengths and Ir–O–Ir bond angles, directly influenced by x_48f, critically affect the electronic and magnetic responses of these materials^64–66.

Table 1.

Structural parameters, Ir-sublattice ordering temperature, and crystal-field characteristics of A₂Ir₂O₇ compounds

A2Ir2O7	Y139	Pr71,175	Nd87,176	Sm135	Eu71	Gd136,177	Tb70,178	Dy7,72,108	Ho30,108	Er82,91,108	Tm67,82,108	Yb27,82,108	Lu92,108
rA3+	1.019	1.126	1.109	1.079	1.066	1.053	1.04	1.027	1.015	1.004	0.994	0.985	0.977
a (Å)	10.106	10.396	10.377	10.311	10.274	10.277	10.238	10.192	10.184	10.162	10.135	10.108	10.104
x48f	0.335	0.330	0.330	0.330	0.339	0.343	0.35	0.334	0.335	0.334	0.337	0.336	0.340
ϕ (deg)	129	132	131	131	126.7	125	121	129	129	129	128	128	126
TIr (K)	155	-	34	110	120	130	130	131	141	140	142	143	147
JA	0	4	4.5	2.5	0	3.5	6	7.5	8	7.5	6	3.5	0
B02 (meV)	-	n/l	n/l	n/l	-	n/l	34.94	32.05	32.00	29.36	18.09	36.30	-
B04 (meV)	-	n/l	n/l	n/l	-	n/l	36.12	38.38	30.25	42.48	30.98	32.25	-
B34 (meV)	-	n/l	n/l	n/l	-	n/l	244.19	267.11	210.02	258.56	195.04	343.13	-
B06 (meV)	-	n/l	n/l	n/l	-	n/l	4.71	8.09	5.56	4.70	5.58	5.26	-
B36 (meV)	-	n/l	n/l	n/l	-	n/l	-96.58	-116.17	-106.31	-71.16	-69.086	-127.83	-
B66 (meV)	-	n/l	n/l	n/l	-	n/l	91.11	66.02	85.49	116.82	105.07	78.463	-
E0 (meV)	-	0(d)	0(d)	0(d)	-	0(d)	0	0(d)	0	0(d)	0(s)	0(d)	-
E1 (meV)	-	14(s)	26(d)	11.73(d)	-	n/l	1.5(d)	29.5	21	5.581(d)	10.47(d)	77.78(d)	-
E2 (meV)	-	60(d)	42(d)	43.39(d)	-	n/l	10	37	25	8.684(d)	33.83(s)	111.35(d)	-
E3 (meV)	-	85(d)	57(d)	-	-	n/l	14	37.2	56	19.937(d)	36.73(d)	151.68(d)	-
E4 (meV)	-	105(s)	123(d)	-	-	-	35	43		64.149(d)	56.559(d)	-	-
E5 (meV)	-	120(s)	-	-	-	-		59		65.391(d)	72.768(s)	-	-
E6 (meV)	-	-	-	-	-	-		59.2		70.214(d)	73.868(d)	-	-
E7 (meV)	-	-	-	-	-	-		81		89.935(d)	89.239(s)	-	-
E8 (meV)	-	-	-	-	-	-		-		-	95.250(s)	-	-

Listed are: the ionic radius of the A³⁺ cation in eightfold coordination; the cubic lattice parameter a; the fractional oxygen coordinate x_48f of the symmetry-unrestricted 48f site (position (x_48f, 1/8, 1/8)); and the corresponding Ir-O-Ir bond angle ϕ of the pyrochlore Fd-3m unit cell at room temperature. The table further provides the Ir-sublattice ordering temperature T_Ir, the total angular momentum J_A of the A³⁺ ion, and the crystal-field parameters in Stevens notation together with the associated eigen-energies. For A = Tb, Dy, Ho, and Yb, the reported Wybourne parameters have been converted to Stevens notation. The notation ‘n/l’ used for Pr and Sm indicates that the crystal-field parameters are not listed in the corresponding references^135,175, while for Gd it denotes the atypical excitation scheme of the Gd³⁺ ion (L = 0), discussed in detail in Ref.¹³⁶. References corresponding to the values in each column are provided directly in the table.

In fact, magnetic ordering phenomena are closely coupled to local structural distortion, particularly through changes in x_48f. Two competing mechanisms appear to influence the ordering temperature of iridium sublattice T_Ir: (i) external pressure, which enhances the electronic bandwidth and favours a metallic state by suppressing magnetic order, and (ii) chemical substitution, which increases trigonal distortion and supports spin anisotropy, stabilising magnetic order.

Chemical pressure, introduced through progressive lanthanide substitution, results in a systematic reduction of the lattice parameter a, consistent with the expected lanthanide contraction. Simultaneously, the oxygen fractional coordinate x_48f increases monotonically across the series. These trends, previously reported in several studies^{27,28,30,67–72}, have been corroborated by uniform synthesis conditions, consistent experimental protocols, and standardised refinement procedures applied to multiple members of the A₂Ir₂O₇ series recently⁷³. The associated distortion of the IrO₆ octahedra, which intensifies with increasing x_48f, also exhibits temperature dependence: a slight increase in x_48f accompanies lattice contraction upon cooling. The strongest distortion is found in the Lu₂Ir₂O₇ end-member at low temperature. Extrapolation indicates that further compression - such as that achievable via Ir-site substitution - may shift x_48f towards ~0.375, approaching the defect-fluorite-type oxygen coordination limit^40,74. Simultaneously, the thermal expansion coefficient in A₂Ir₂O₇ is comparable to that in isostructural A₂Ti₂O₇ titanates^75,76.

No structural phase transitions were also detected under applied external pressure, underscoring the exceptional stability of the cubic pyrochlore framework up to at least 20 GPa⁷³. A systematic evolution of structural parameters with applied pressure is followed across the rare-earth series. The lattice parameter a contracts under pressure in a fashion analogous to that observed under chemical substitution. In contrast, the oxygen coordinate x_48f remains largely unaffected even at the highest applied pressures. This behaviour, documented for both light rare-earth iridates, such as Sm₂Ir₂O₇⁶⁹ and Eu₂Ir₂O₇⁷⁷, and heavier members⁷³, indicates that external pressure induces an essentially isotropic compression of the lattice, with negligible influence on the local bonding environment. Bulk modulus in A₂Ir₂O₇ increases systematically with A-site atomic number, reflecting the decreasing unit-cell volume^69,73,78 and approaching bulk modulus of A₂Zr₂O₇ zirconates⁷⁹.

Although the magnetic ordering temperature T_Ir increases markedly across the rare-earth series -most prominently between A = Pr and Sm (see phase diagram in Fig. 4) - this evolution is not accompanied by any abrupt changes in the lattice parameter or the oxygen coordinate x_48f, both of which vary only gradually. In Eu₂Ir₂O₇, however, anomalies in the Ir–O bond lengths and Ir–O–Ir bond angles have been observed near T_Ir⁸⁰, underscoring the potential relevance of subtle structural effects at the magnetic transition. For other members of the series, structural data in the vicinity of T_Ir and within the broader low-temperature range remain scarce or ambiguous⁷³.

Fig. 4. Phase diagram of the Ir sublattice ordering in A₂Ir₂O₇.

The evolution of the transition temperature (T_Iᵣ) as a function of the ionic radius of the rare-earth element A³⁺ is displayed. Distinct conducting states are schematically illustrated as well. The figure is adopted from ref. ⁶⁷, which was adapted from refs. ^{34, 81}.

A detailed understanding of pyrochlore phase stability and of high-pressure-induced local distortions is essential for a consistent interpretation of physical properties and pressure-dependent phenomena. Crucially, existing studies consistently demonstrate the structural robustness of the cubic framework and reveal systematic trends in both the macroscopic lattice parameter and the local octahedral geometry.

Iridium sublattice

The magnetic and conducting properties of A₂Ir₂O₇ can be analysed by separately examining the iridium and rare-earth sublattices, which naturally interact through f-d coupling or molecular fields. The strength of their exchange interactions and inter-sublattice coupling varies with temperature and depends on the intrinsic magnetic properties of the rare-earth element. The following sections address the two sublattices independently, highlighting their key characteristics while briefly noting the impact of the f-d coupling between them.

Magnetic ordering

The iridium sublattice orders magnetically at higher temperatures than the rare-earth sublattice in all studied iridates, except for Pr₂Ir₂O₇, which does not exhibit magnetic ordering of any of the two sublattices^62,81. The magnetic ordering of the Ir sublattice has been reported or proposed to be antiferromagnetic of the all-in-all-out (AIAO) type (see Fig. 2a). Studies documenting this type of ordering have primarily relied on neutron diffraction experiments on powder samples^27,28,70. However, these studies are limited by the small magnetic moment of Ir (estimated to be < 0.5 μ_B) and its relatively high neutron absorption cross-section. Although the community has largely accepted the AIAO structure of Ir sublattice, no clear or unambiguous magnetic signals in neutron data have been observed for, e.g., Er₂Ir₂O₇ and Tm₂Ir₂O₇ members^70,82.

Importantly, a recent neutron diffraction study on Lu₂Ir₂O₇, which contains non-magnetic Lu³⁺ ions, revealed several magnetic peaks consistent with the AIAO magnetic structure²⁷; thereby clearly demonstrating the ordering of only the magnetic Ir sublattice. Simultaneously, resonant elastic X-ray scattering (REXS), Raman scattering, or muon spin resonance strongly suggested the AIAO ordering of Ir sublattice also in several other iridates^55,83–86.

In addition to antiferromagnetic ordering, the application of a magnetic field along the <001> (<111 > ) direction is expected to overcome the exchange interactions, causing two (three) magnetic moments in AIAO tetrahedra to point inward, while the other two (one) point outward (Fig. 2b, c)^6,27. Consequently, an external field can induce spin-ice and magnetic monopole-like states due to competing interactions on the frustrated pyrochlore lattice.

Phase diagram

Although the heavy rare-earth A₂Ir₂O₇ members with A = Dy to Lu have been understudied, several recent reports^7,27,80,82, including the study of newly synthesised Tm₂Ir₂O₇ member⁶⁷, allowed completion of the phase diagram of the Ir sublattice magnetic ordering. The finalised phase diagram showing the dependence of the Ir sublattice magnetic ordering temperature (T_Ir) on the ionic radius of the rare-earth A³⁺ ion is presented in Fig. 4 and Table 1.

Pr₂Ir₂O₇ does not exhibit magnetic ordering down to low temperatures^81,87. Remarkably, a chiral spin-liquid state has been reported for this compound⁵⁷. All heavier members of the series reveal magnetic ordering at low temperatures. A steep increase in T_Iᵣ is observed from A = Pr (without ordering) through A = Nd (36 K) to A = Sm (120 K). Pm₂Ir₂O₇ has not been synthesised due to the challenges associated with radioactive Pm isotopes.

Several studies have attempted to bridge the gap between light A members by substituting Pr with Nd and Nd with Sm, respectively⁸⁸, or by additional doping, such as hole doping by substituting Eu with Sr⁸⁹ or Ir with Ru⁹⁰. A smooth evolution of T_Iᵣ has been observed from Pr₂Ir₂O₇ to Sm₂Ir₂O₇. For the heavier members, only a modest increase in T_Iᵣ was observed with further increasing atomic number (or decreasing ionic radius) of the rare-earth element⁶⁷. The T_Iᵣ values of the heaviest A members^91,92, including Tm₂Ir₂O₇⁶⁷, perfectly fit into the trend established by the lighter rare-earth iridates (Fig. 4).

Interpreting the phase diagram, it is apparent that the critical temperature T_Ir is independent of the magnetic and electronic properties of the rare-earth ions, at least for heavier members (A = Sm to Lu). Considering the T_Ir values of Lu₂Ir₂O₇ and Eu₂Ir₂O₇, which contain non-magnetic Lu and Eu ions^92,93, it is evident that magnetic coupling between the A and Ir sublattices is negligible at or above T_Ir. That is, the ordering temperature of the Ir sublattice is independent of the A-site ion and sublattice properties. Instead, the Ir sublattice ordering affects the rare-earth sublattice ordering below T_Ir, as demonstrated in Nd, Tb, Ho, and Yb iridates^27,28,30,55. In these compounds, the AIAO ordering of the Ir sublattice induces corresponding long-range AIAO ordering within the A sublattice. At lower temperatures, however, exchange interactions between the A moments prevail, resulting in different magnetic structures.

The evolution of T_Ir can be better understood from a structural perspective. Changes in the crystallographic lattice parameter a and the oxygen 48 f Wyckoff position free coordinate (x_48f, see Table 1) naturally lead to variations in interatomic distances and bond angles as a function of the A-ion radius. T_Ir increases with a reduction in the overall lattice size. The corresponding decrease in interatomic distances and the Ir-O-Ir bond angle influences the t_2g-block bandwidth of iridium, leading to larger 5d-orbital overlap with oxygen 2p orbitals^64,69, thereby increasing T_Ir.

The magnetic ordering temperature (T_Ir) and the general properties of A₂Ir₂O₇ can be thus tuned by altering structural parameters. This is achieved through A-site^77,94,95 or Ir-site^90,96,97 substitution and the application of external pressure. Unlike partial site substitution, pressure represents a clean approach to studying structural and physical properties of the material without introducing atomic disorder to the system. This is exemplified by studies on Sm₂Ir₂O₇⁷⁸ or recent studies on heavy rare-earth iridates⁷³. Of course, external and chemical pressure are expected to lead to contrasting changes in lattice parameters and trigonal distortion, affecting the bond angle, the iridium bandwidth, and magnetic moment anisotropy within the system.

Conductivity

Concomitant with the magnetic ordering of the Ir sublattice, A₂Ir₂O₇ compounds undergo a transition from a metallic, semi-metallic, or semiconducting state to an insulating state (MIT)^{34,35,62,67,81}. An illustration of these transitions is shown in Fig. 4. These magnetic and MIT transitions are strongly influenced by the tuning of Ir–O–Ir bond lengths and angles⁹⁰, which modulate the valence-electron bandwidth and, therefore, the effective correlation strength (see also Table 1). The bending vibrations of the Ir–O–Ir bonds are strongly coupled to a continuum composed of spin, charge, and orbital excitations via a confluence of all three mechanisms⁸³.

The conductive, as well as magnetic, properties of A₂Ir₂O₇ iridates are highly sensitive to sample quality^65,81,98,99. Significant differences in conductive behaviour have been reported even for samples with nominally identical stoichiometry, particularly when comparing polycrystalline and single-crystal forms^100–102. Notably, iridium deficiency/abundance shifts T_Ir to substantially lower temperatures¹⁰⁰.

The absolute resistivity values and their temperature-dependent variations have been found to differ by orders of magnitude depending on the specific sample^81,90,99,102. A growing body of evidence shows that even minor deviations from ideal A₂Ir₂O₇ stoichiometry exert a pronounced influence on electronic transport. In Eu₂₊ₓIr₂₋ₓO₇₋ᵧ, for example, Ir deficiency consistently drives the system towards more metallic behaviour: the MIT shifts to lower temperature, the low-temperature resistivity decreases, and the residual-resistivity ratio is markedly reduced as x increases^41,103. Simultaneously, a comparable trend has been reported for Ir-abundant samples¹⁰⁰, leaving nominally stoichiometric crystals the most insulating.

Oxygen off-stoichiometry is also believed to play a non-negligible role^104,105, although its quantitative determination is challenging; even sub-percent oxygen deficiency can significantly modify carrier density and suppress the transition. The oxygen vacancies are supposed to introduce extra electron carriers to the Ir 5d bands and potentially stabilise a metallic ground state^106,107. Overall, deviations on the order of a few percent in cation or anion balance already produce measurable, and sometimes dramatic, changes in the transport response of A₂Ir₂O₇.

The relationship between magnetic ordering and the MIT in A₂Ir₂O₇ iridates warrants further discussion. The Arrhenius thermal activation law can effectively describe electrical resistivity data collected well above T_Ir. The activation energies suggest that, in contrast to the light rare-earth iridates, the heavy A members are narrow-gap semiconductors above T_Ir^81,100,108 and show no apparent dependence on the atomic number of the A-site ion (see the phase diagram in Fig. 4 and Table 1).

An interplay between the Weyl semimetal (WSM) and Mott insulating states has been predicted for A₂Ir₂O₇ oxides⁶¹. Notably, members with heavy rare-earth elements, such as A = Yb, Lu¹⁰⁸, and Y¹⁰⁹, exhibit power-law behaviour consistent with the single-particle model for Weyl fermion scattering from a random Coulomb potential, described by the thermally screened charged impurities (TSCI) model¹¹⁰. However, the TSCI model alone fails to fully capture the complex behaviour of all rare-earth iridates.

To explain the (semi-)metal-insulator transition in these compounds, the Mott variable-range hopping (VRH) model, commonly applied to strongly disordered systems with localised charge-carrier states, has also been utilised^98,100,102. Despite the fully ordered pyrochlore structure, possible off-stoichiometry, often associated with charged impurities, justifies this approach. While the VRH model accurately describes intermediate-temperature behaviour, it breaks down at low temperatures.

Alternatively, the Slater insulator model has been considered. Unlike the Mott model, which attributes insulating behaviour to strong Coulomb interactions, the Slater mechanism associates it with periodic potential perturbations induced by commensurate magnetic ordering^111,112. This model successfully explains the insulating state of Cd₂Os₂O₇, a compound sharing structural and magnetic properties with A₂Ir₂O₇. The Slater mechanism predicts a gap opening at the Fermi level due to magnetic ordering, as observed in Cd₂Os₂O₇ just below its magnetic transition temperature^113,114.

However, the AIAO ordering of Os moments implies that the magnetic and crystallographic unit cells remain identical. As a result, the Slater mechanism must be considered without Brillouin zone-folding¹¹⁵. Despite this nuance, the prevailing view in the scientific community attributes the insulating behaviour of Cd₂Os₂O₇ to the Slater mechanism^113–115.

Heavy rare-earth iridates exhibit narrow-gap semiconducting behaviour in the paramagnetic regime above T_Ir. The emergence of a gap at or below T_Ir may be obscured in experimental data. Determining whether the MIT coincides with AFM ordering at T_Ir or occurs at lower temperatures remains challenging. Notably, MIT has never been observed above T_Ir, as demonstrated in studies of substituted iridates^89,90 and isostructural Cd₂Os₂O₇¹¹⁵. The Slater model predicts an MIT following the magnetic transition¹¹³, dictating it should occur below T_Ir. In contrast, the Mott model anticipates the MIT at or above T_Ir. Additionally, the continuous nature of the MIT supports the Slater mechanism^108,111, whereas the Mott model is connected with a first-order transition characterised by an abrupt resistivity change¹¹³.

Recent studies increasingly attribute the MIT in A₂Ir₂O₇ to the Slater mechanism, where AFM ordering drives the gap opening at the Fermi level, leading to the insulating state. A definitive conclusion remains, however, elusive due to the semiconducting nature of the paramagnetic state. The ongoing debate over Mott and Slater mechanisms in systems like Sr₂IrO₄^116–118, and the influence of Ir domain walls on transport properties discussed in following section offer additional perspectives for interpreting the conducting properties of A₂Ir₂O₇.

Antiferromagnetic domains and interfaces

The Ir magnetic moments in A₂Ir₂O₇ compounds are reported to spontaneously order in the all-in-all-out (AIAO) structure below T_Ir^{27,28,55,83–85}. Importantly, alongside the AIAO configuration, the time-reversal-symmetry-related all-out-all-in (AOAI) configuration is also realised. As these configurations are energetically equivalent, both are present in the bulk material below T_Ir, creating respective domains (Fig. 5). The interface between antiferromagnetic domains - the AIAO/AOAI interface - contains two types of magnetic moments: rotatable and frozen moments.

Fig. 5. Model of AIAO and AOAI domains and domains’ interface: 2D projection (a) and 3D scheme (b).

Non-zero total magnetisation at the interface is represented by green arrows. Simultaneously, these arrows indicate a possible electrical dipole associated with the magnetic monopole-like states.

Rotatable moments on domain boundaries are weakly coupled to the domains and can be influenced by an external magnetic field. In contrast, frozen uncompensated moments within a three-in-one-out (3I1O) or one-in-three-out (1I3O) arrangement form a net ferromagnetic moment at the domain boundary, which is strongly coupled to the domain structure⁴¹. Aligning and stabilising the antiferromagnetic domain interfaces in A₂Ir₂O₇ is achieved by applying a small magnetic field (~few mT). These interfaces, once stabilised at temperatures below T_Ir, remain robust against fields of several tesla^41,104,119, making these materials promising for magnetic recording applications.

Electronic and conductive properties are theorised to differ between AIAO/AOAI domains and their interfaces^120,121. AIAO domains exhibit strongly insulating behaviour, likely related to the Slater mechanism^41,122–124. In contrast, disturbed magnetic order at the domain interfaces may result in high metallic conductivity, as observed in Nd₂Ir₂O₇¹²¹. This disparity in conductivity between domain interiors and interfaces offers potential for reading magnetically encoded information via an external electric field³³. In summary, these materials may enable magnetic information recording by applying a small magnetic field during the transition through T_Ir. This information would be safeguarded by the magnetic structure, robust domains, and frozen interfaces against high magnetic fields below T_Ir, while reading could be achieved via an electric field due to the contrasting conductivity of the domains and interfaces.

The formation and stability of ferromagnetic interfaces in pyrochlore iridates remain a prominent research topic, paving the way for spintronic applications. The presence of domain interfaces, or rather related ferromagnetic component of magnetisation in antiferromagnetically ordered materials, was recently confirmed investigating single crystals of Lu₂Ir₂O₇ and Er₂Ir₂O₇¹⁰⁴. Cooling a sample through T_Ir in zero magnetic field results in randomly oriented domain interfaces, yielding zero total magnetisation. Conversely, cooling in a non-zero magnetic field aligns the domains and ferromagnetic interfaces, resulting in observable uncompensated magnetisation. This ferromagnetic signal persists even after the external (cooling) field is removed. Moreover, applying a high magnetic field (several T) below T_Ir has no permanent impact on the domains or interfaces, as the initial ferromagnetic signal reappears when the field is removed. This is due to the robustness of the domain structure, and therefore AIAO/AOAI interfaces, below T_Ir.

The rare-earth sublattice of magnetic A elements influences the Ir ordering and ferromagnetic interfaces, particularly at the lowest temperatures, as demonstrated for Er₂Ir₂O₇¹⁰⁴ and Ho₂Ir₂O₇³³. This interaction holds significant promise for next-generation spintronic devices. Recent magneto-resistivity studies on Ho₂Ir₂O₇ have demonstrated a strong coupling between magnetic monopoles (3I1O configuration) on the Ho sublattice and ordered Ir moments at low temperatures³³. By applying an external magnetic field, magnetic monopole-like states on the Ho sublattice are induced, which couple to the antiferromagnetic Ir domains and ferromagnetic interfaces. This mechanism allows the manipulation of antiferromagnetic domains through the rare-earth sublattice, bypassing the challenge of directly switching antiferromagnetic domains.

Magnetic domains, interfaces, and their associated magnetic and conductive properties have been studied using laboratory techniques such as ac-susceptibility, magneto-transport measurements, Hall effect, and atomic/magnetic force microscopy^{33,41,123,125,126}. However, advanced methods available at large-scale facilities are required for detailed, microscopic insights. These include coherent X-ray scattering, X-ray magnetic circular dichroism, X-ray polarisation-enhanced topography, and polarised resonant micro-diffraction X-ray imaging^127–129. This research direction on rare-earth iridates remains very active, offering promising prospects for further discoveries and potential technological applications.

Rare-earth sublattice

In addition to the Ir sublattice magnetism discussed in previous sections, the low-temperature physical properties of A₂Ir₂O₇ are significantly influenced by the electron configuration of the rare-earth elements. The rare-earth moments interact with the Ir molecular field; exhibit direct f-d coupling. Some A₂Ir₂O₇ members display Ir induced all-in-all-out (AIAO) magnetic order of the A sublattice, as demonstrated by neutron diffraction experiments in Nd₂Ir₂O₇⁵⁵ or Tb₂Ir₂O₇²⁸.

At low temperatures, the exchange interactions between rare-earth moments become more prominent, contributing a second component to the magnetic moments. The resulting ground-state magnetic structure may be a superposition of the Ir-induced AIAO component and the rare-earth exchange interactions magnetic component. For example, antiferromagnetic coupling between Ir and Sm moments and long-range Sm moment ordering has been inferred for Sm₂Ir₂O₇¹³⁰. In Yb₂Ir₂O₇, competition between AIAO and ferromagnetic ordering persists to the lowest temperatures²⁷.

Crystal field – single ion properties

Alongside long-range Ir order, Ir-induced A ordering, and A-A magnetic correlations, the magnetic properties of A₂Ir₂O₇ iridates are also shaped by the single-ion properties of the A cations. The crystal field (CF) influences the ground-state multiplet of these cations, lifting its degeneracy. In contrast to Ir⁴⁺, where the crystal field energetically dominates over SOC, the reverse holds for A³⁺ ions in a crystalline environment. Consequently, the physical properties of rare-earth compounds are primarily governed by crystal field effects, particularly in the paramagnetic state.

Crystal field schemes have been experimentally determined for most rare-earth A₂Ir₂O₇ members, including previously understudied heavy rare-earth compounds^{7,27,30,67,91} and the newly synthesised Tm₂Ir₂O₇⁸²; leaving no significant space for further studies or discussions.

The CF excitations in A₂Ir₂O₇ are well described within a standard crystal field model. A³⁺ ions occupy the Wyckoff position 16d in Fd-3m space group. The point group symmetry of the A position is therefore not cubic, but trigonal (point group D_3h) leading to lower degeneracy of energy levels.

A consistent systematics of crystal field schemes and parameters is observed across the A₂Ir₂O₇ (CF parameters are listed in Table 1), and more generally A₂B₂O₇ series. All A₂Ir₂O₇ compounds exhibit CF parameters of identical signs and comparable magnitudes. However, these values do not correlate with the expected contraction of the 4f radial wavefunction¹³¹. This discrepancy likely reflects only minor variations in hybridisation between the rare-earth 4f orbitals and the 2p states of the eight nearest-neighbour oxygen anions within the pyrochlore lattice. A similar trend has been reported for the related A₂Ti₂O₇ titanate series¹³². Almost identical CF schemes were observed in several pyrochlore members of the A₂B₂O₇ series mostly irrespective of the B element^91,132,133.

Comparison between macroscopic data such as magnetisation and specific heat, and respective values calculated from determined crystal field parameters and eigenenergies reveals good overall agreement within the whole A₂Ir₂O₇ series^67,91. In particular, the calculated Schottky contribution to the specific heat, based on experimentally determined CF excitation energies (details also in other publication by the author¹³⁴), closely matches the estimated magnetic contribution. The temperature dependence of the magnetic entropy change further supports this interpretation, reaching values consistent with the degeneracies and energy spacings of the CF levels.

Low-temperature magnetic properties

Although ordering of the Ir sublattice and the associated molecular field significantly influence the ground-state properties of A₂Ir₂O₇, low-temperature behaviour is also determined by exchange interactions between the rare-earth moments, typically mediated via superexchange through oxygen ligands. While the magnetic properties of most light rare-earth iridates have been extensively characterised^{34,55,124,135}, studies on their heavy counterparts have emerged only recently^{27,33,67,82,91,136}. Together, they have brought significant information on low-temperature characteristics of the whole rare-earth iridium series; although further studies are certainly demanded, especially using advanced microscopic methods to reveal the details of lowest-temperature magnetic orderings and (related) conducting properties.

Pr₂Ir₂O₇ remains metallic down to the lowest measured temperatures and does not exhibit long-range magnetic order. Instead, it realises a spin-liquid-like ground state, attributed to geometrical frustration and quantum fluctuations^57,137. This compound is often regarded as a model system for probing the interplay between spin–orbit coupling and electronic correlations in a metallic geometrically frustrated lattice.

Nd₂Ir₂O₇ exhibits an insulating ground state with AIAO magnetic ordering on both the Ir and Nd sublattices^86,124. The ordering of Nd³⁺ moments is induced via d-f exchange by the Ir sublattice, and the reduced ordered moment observed on Nd³⁺ suggests that quantum fluctuations persist even within the ordered phase⁵⁵. That is, an exchange coupling between Nd moments must play a role.

An AIAO-ordered insulating state is also adopted by Sm₂Ir₂O₇. In contrast to lighter members, it displays more robust electronic correlations, which are reflected in its transport and magnetic properties⁸⁴.

Eu₂Ir₂O₇ serves as a prototypical member for studying the Ir sublattice alone, since Eu³⁺ carries no magnetic moment (J = 0). The system exhibits a well-defined AIAO ordering of Ir moments below the magnetic transition temperature^93,100. Its clean magnetic environment makes it a valuable benchmark for separating 5d magnetism from 4f contributions in other iridates.

Quasi-isotropic exchange interactions between half-filled 4f ⁷ Gd³⁺ moments are featured in the Gd₂Ir₂O₇ member. The resulting magnetic ground state involves not only the canonical AIAO order, but also signatures of non-collinear arrangements, such as Palmer–Chalker-type correlations, indicating a competing magnetic landscape with enhanced complexity¹³⁶.

Tb₂Ir₂O₇ displays coupled magnetic ordering of Ir and Tb sublattices just below T_Ir. The Tb³⁺ moments align initially in the AIAO pattern, induced by the Ir ordering²⁸. A second magnetic component, with XY symmetry, emerges at lower temperatures (~10 K), reflecting independent rare-earth exchange interactions within the frustrated pyrochlore lattice.

Dy₂Ir₂O₇ hosts an unconventional magnetic state referred to as a ‘fragmented monopole crystal’, in which long-range antiferromagnetic order coexists with a Coulomb-phase spin liquid⁷. This fragmentation is driven by the competition between the local anisotropy of Dy³⁺ moments and the molecular field imposed by the Ir sublattice, likely a similar mechanism as in the Tb member.

Similarly, Ho₂Ir₂O₇ exhibits magnetic fragmentation: the Ho³⁺ spin-ice state is perturbed by the staggered internal field of the ordered Ir sublattice, leading to partial moment ordering¹³⁸. This makes the Ho₂Ir₂O₇ member a unique platform to study emergent magnetic monopole-like excitations in a correlated background.

In contrast, Er₂Ir₂O₇ does not show clear signatures of long-range magnetic order down to 0.3 K, as confirmed by the absence of magnetic Bragg peaks in neutron diffraction patterns⁷⁰. Nonetheless, broad anomalies in specific heat and bifurcation in magnetisation suggest the presence of, at least, short-range magnetic correlations at even lower temperatures. The entropy associated with the ground-state doublet is gradually recovered in applied fields, consistent with a Zeeman-split doublet ground state⁶⁷.

Tm₂Ir₂O₇ remains similarly ‘non-magnetic’ down to the lowest temperatures investigated. Its thermodynamic behaviour is governed solely by crystal-field excitations, in agreement with the singlet ground state of Tm³⁺ ^67,82. No signatures of magnetic ordering or strong correlations were observed in neutron diffraction data, and the specific heat closely resembles that of non-magnetic Lu₂Ir₂O₇.

Contrary, Yb₂Ir₂O₇ reveals a complex low-temperature magnetic behaviour. While weak AIAO order on the Ir sublattice is retained, additional ferromagnetic peaks develop below 1 K, indicating an unconventional magnetic ground state²⁷. Specific heat and entropy measurements confirm the presence of low-energy excitations and their field dependence, pointing to strongly anisotropic interactions between Yb³⁺ moments.

Lu₂Ir₂O₇, containing a fully non-magnetic Lu³⁺ ion, remains an important reference compound. It displays no rare-earth magnetic anomalies and serves to isolate the structural and Ir-derived magnetic contributions to thermodynamic observables⁶⁷, similar to the Eu member. Importantly, magnetic AIAO ordering of the Ir sublattice has been confirmed in this compound²⁷, supporting a unified picture of identical magnetic order of Ir sublattice across all A = Nd–Lu pyrochlore iridates.

To complete this brief overview of the rare-earth A₂Ir₂O₇ family, Y₂Ir₂O₇ should also be included as a representative non-magnetic A-site analogue. Like its Eu and Lu counterparts, Y₂Ir₂O₇ exhibits an all-in-all-out magnetic ground state driven by the Ir moments, accompanied by a metal-insulator transition^139,140. Its T_Ir is highest of the series (Fig. 4). Y₂Ir₂O₇ has been heavily studied theoretically and experimentally in the context of Weyl semimetal behaviour and possible Fermi-arc surface states^{109,141–143}. Moreover, a piezomagnetic response has recently been identified in this compound¹⁴⁴. This effect has been interpreted as a macroscopic manifestation of ferroic ordering of magnetic octupoles, while on the microscopic level it arises from strain-induced modifications of the g-tensor anisotropy and Dzyaloshinskii-Moriya interactions. This makes Y₂Ir₂O₇ more than a valuable reference compound for isolating the intrinsic properties of the Ir network in the absence of rare-earth magnetism and for benchmarking the evolution of magnetic and electronic characteristics across the entire A₂Ir₂O₇ series.

Magneto-conductive characteristics

In A₂Ir₂O₇ pyrochlores, the onset of all-in–all-out antiferromagnetic order on the Ir⁴⁺ sublattice is intimately linked to changes in electrical transport. See text above for a detailed overview. Light rare-earth members exhibit a pronounced metal-insulator transition (MIT)^81,135,145, whereas their heavier analogues display a transition from semiconducting or nonmetallic to fully insulating state^27,66,81,108. The formation of this insulating state has been ascribed to the opening of a gap at the magnetic ordering temperature T_Ir via a Slater-type mechanism^115,118. However, the low-temperature regime cannot be fully described within the Slater framework; a crossover to a Mott-like insulating state is anticipated upon further cooling below T_Ir.

Magnetoresistivity has emerged as a powerful tool to probe the intricate interplay between charge transport and magnetic degrees of freedom in these materials^{41,126,146,147}. Notably, these measurements have revealed the presence of AIAO/AOAI domain structures in the Ir sublattice and their coupling to the magnetic A³⁺ moments. In Eu₂Ir₂O₇, asymmetric magnetoresistance was observed upon cooling in a magnetic field, suggesting the emergence of an interfacial ferromagnetic component at domain boundaries^41,147. A similar phenomenon was reported in Ho₂Ir₂O₇, where domain-wall interactions with Ho³⁺ moments may even involve monopole-like excitations³³. In Sm₂Ir₂O₇, magnetoresistance measurements under pressure allowed for the mapping of a quantum critical point and the associated phase diagram, which remained hidden in ambient-pressure transport data¹³⁵.

Magnetoresistance measurements performed at low temperatures reveal a consistent decrease in electrical resistivity upon application of a magnetic field across the entire A₂Ir₂O₇ series¹⁰⁸. The effect is most pronounced for Dy₂Ir₂O₇ and Ho₂Ir₂O₇, where resistivity drops by over 80% already in an applied field of 2 T, while Lu₂Ir₂O₇ - bearing a nonmagnetic A-site - exhibits only a minor change of resistivity (~1%). Intermediate members fall within these bounds, with the magnitude of the effect broadly correlating with the magnetic moment of the A³⁺ cation. This trend highlights the pivotal role of exchange interactions between A and Ir ions, mediated via 4f-5d coupling, as well as the exchange interactions between rare-earth moments.

Two key experimental features were argued to support this interpretation¹⁰⁸: (i) the absence of hysteresis in magnetoresistance curves, indicating reversible field-induced effects, and (ii) a strong suppression of the magnetoresistive response with increasing temperature. Above ~15 K, the electrical resistivity becomes nearly field-independent, whereas below ~10 K, the interaction between rare-earth moments and the Ir sublattice becomes sufficiently strong to significantly influence transport properties.

These observations are consistent with a scenario in which the magnetic field perturbs the insulating state associated with all-in–all-out ordering of Ir moments, through the coupling with field-polarised A-site moments - possibly involving monopole-like excitations³³. The subsequent reorganisation of Ir domain boundaries leads to reduced carrier scattering and lower resistivity. The negligible magnetoresistance in nonmagnetic Lu₂Ir₂O₇ provides further support for the decisive role of f-electron magnetism in this mechanism.

We note that the contrast between Eu₂Ir₂O₇ and Lu₂Ir₂O₇ members, both containing formally non-magnetic A-site ions, remains unresolved. The asymmetric magnetoresistance observed in Eu₂Ir₂O₇ likely reflects compound-specific factors beyond the absence of 4f moments, such as subtle structural distortions, off-stoichiometry, or differences in Ir-domain dynamics. Current literature does not offer a unified microscopic explanation, and we therefore emphasise that the role of A-site magnetism should be viewed as an important but not exclusive factor governing magnetotransport in the A₂Ir₂O₇ series.

Simultaneously, detailed microscopic investigations remain lacking for many A members, particularly those focusing on rare-earth magnetic characteristics, leaving some of the prominent aspects of the field-induced transport response unresolved.

A₂Ir₂O₇ crystal growth

Given the growing interest in A₂B₂O₇ oxides in recent years, the demand for high-quality polycrystalline and, in particular, single-crystalline samples has substantially increased. Many earlier studies were severely limited by the poor quality of available specimens and the absence of single crystals, especially for compounds where B = Mn, Ir, Pt, or Pb. Considerable efforts have thus been devoted to developing reliable methods for synthesising high-quality and, crucially, stoichiometric samples.

Polycrystalline A₂B₂O₇ compounds are most commonly prepared via conventional solid-state reaction methods involving the constituent elements and/or their oxides¹⁴⁸. In contrast, single crystals are typically grown using the floating-hot-zone method, either from a pre-synthesised polycrystalline rod¹⁴⁹ or directly from a mixture of precursor materials^150–152. Many A₂B₂O₇ compounds exhibit congruent melting behaviour¹⁵³, which facilitates the growth of large single crystals. However, high melting or reaction temperatures associated with some A and B elements or their oxides impose substantial technical demands on laboratory equipment^151,154.

Alternative, melt-free synthesis routes such as sol-gel processing and co-precipitation have also been employed for selected A₂B₂O₇ phases¹⁵⁵. In certain cases, the formation of specific compositions requires chemical stabilisation during synthesis⁷¹. Another major challenge lies in the low vapour pressures of some components, which hinder reactions under ambient conditions. For example, the synthesis of A₂B₂O₇ compounds with B = Pb¹⁵⁶, Pt¹⁵⁷, V¹⁵⁸, or Mn¹⁵⁹ often necessitates high-pressure conditions within sealed reaction vessels. Hydrothermal synthesis methods, using aqueous solutions of precursors and a mineralising agent such as KOH, have also proven effective for producing some A₂B₂O₇ phases at comparatively low temperatures¹⁶⁰. To suppress component volatilisation during synthesis, appropriate solvents or fluxes - such as KF or CsCl - are commonly employed^70,71.

Establishing a robust and reproducible synthesis protocol for a specific A₂B₂O₇ compound remains a non-trivial task that requires substantial experimental experience and optimisation. In the case of A₂Ir₂O₇ iridates, polycrystalline samples -including small single crystals with typical edge lengths up to ~20 µm - have been synthesised by reacting stoichiometric or slightly off-stoichiometric mixtures of A₂O₃ and IrO₂ using conventional solid-state methods or flux-assisted growth^{71,100,101,122,125}.

To improve crystal size and quality, various flux materials have been tested, including NaCl, KCl, NaF, PbO₂, and PbF₂^67,161. These were combined with both stoichiometric and off-stoichiometric precursor mixtures. Among these, only KF flux yielded significant improvements in single crystal quality⁷¹, enabling the growth of bi-pyramidal crystals with edge lengths up to ~1 mm. However, this success was restricted to light rare-earth iridates, while heavier analogues such as Ho₂Ir₂O₇ yielded only smaller crystals³³.

More recently, relatively large single crystals of heavy A-site iridates (with edge lengths up to ~600 µm) have been successfully grown using PbF₂ flux¹⁰⁴. Representative examples are shown in Fig. 6, where well-faceted crystals with well-defined shapes are observed. Nevertheless, this method has notable drawbacks: small Pb inclusions were frequently detected in the crystal bulk, and partial substitution of the rare-earth ions by Pb was observed in some cases. These issues highlight the chemical aggressiveness of PbF₂ flux and underscore its limitations as a universal solvent for A₂Ir₂O₇ crystal growth.

Fig. 6. Single crystals of A₂Ir₂O₇ synthesised using PbF₂ flux.

Optical microscope images of Er₂Ir₂O₇ (a,b) and Nd₂Ir₂O₇ (c) single crystals. Backscattered electron (BSE) micrographs of Nd₂Ir₂O₇ (d,e), Ho₂Ir₂O₇ (f), Er₂Ir₂O₇ (g,h), and Lu₂Ir₂O₇ (i) single crystals. Adopted from ref. ¹⁷⁴.

Although substantial progress has been made, the growth of high-quality, inclusion-free, stoichiometric single crystals of A₂Ir₂O₇ remains an unresolved challenge. Alternative flux materials are urgently needed, as crystal quality is essential for uncovering the complex physical phenomena characteristic of these materials.

Thin films – concise overview

Although the primary focus of this review lies on bulk single crystals, where the most definitive insights into magnetic ordering, domain-wall behaviour, and transport mechanisms have been obtained, it is important to acknowledge that epitaxial thin films of A₂Ir₂O₇ have become an increasingly active and influential branch of research. Thin films offer experimental degrees of freedom not accessible in bulk, including substrate-induced strain, controlled epitaxial orientation, and engineered interfaces. These parameters enable systematic modification of Ir-O-Ir bond geometry, bandwidth, and magnetic anisotropies. At the same time, film growth introduces strain clamping, stoichiometric deviations, and enhanced defect densities, which often lead to behaviour distinct from that of bulk analogues. In line with the scope of this review, the following paragraphs therefore provide a concise overview rather than an exhaustive survey of thin-film work.

High-quality epitaxial films of several A₂Ir₂O₇ compounds have been stabilised using pulsed-laser deposition, sputtering, and reactive solid-phase epitaxy. (111)-oriented Nd₂Ir₂O₇ films grown on yttria-stabilised zirconia display clear pyrochlore ordering and coherent epitaxy, as demonstrated by XRD and STEM analyses^162,163. Substrate-induced strain modifies local Ir-O-Ir bond angles, producing measurable changes in electronic and magnetic properties. While this tuning parallels ‘chemical pressure’ trends in bulk A-site substitution, the magnitude and anisotropy of epitaxial strain often extend well beyond what is achievable in bulk materials, leading to sizeable modifications in, e.g., magnetotransport^145,146.

Transport studies frequently reveal behaviour that departs from bulk trends. Epitaxial Eu₂Ir₂O₇ films exhibit signatures of weak antilocalisation at low temperatures, interpreted as indicative of Weyl-type electronic states, and a crossover to weak localisation with increasing disorder¹⁶⁴. Related work demonstrates a pronounced dependence of resistivity and carrier density on oxygen partial pressure during growth¹⁰⁶, underscoring the strong sensitivity of thin films to stoichiometric variations. These tendencies qualitatively mirror known effects in bulk A₂Ir₂O₇, though their magnitude in films is often amplified.

Despite the small magnetic volume, several thin-film studies have reported responses that are suppressed or absent in bulk crystals. In (111)-oriented Eu₂Ir₂O₇ films, an intrinsic anomalous Hall effect with large coercivity but negligible net magnetisation emerges just below the onset of all-in-all-out order, consistent with a strain-stabilised Weyl regime¹⁶⁵. Pr₂Ir₂O₇ films show a spontaneous Hall effect persisting to higher temperatures than in bulk¹⁶⁶, suggesting that epitaxial strain may influence Ir magnetism, domain-wall energetics, or both. Such results highlight thin films as promising platforms for probing strain-enhanced topological responses and for exploring the robustness of magnetic monopole-related textures under controlled structural distortion.

Research on A₂Ir₂O₇ thin films continues to expand, particularly in the areas of heterostructures, superlattices, and field-tunable domain-wall engineering, and provides a complementary means of manipulating the same fundamental degrees of freedom that govern the bulk systems. For coherence, the present review acknowledges these developments while maintaining its central emphasis on bulk single crystals, where the clearest evidence for magnetic ground states and monopole-like excitations has been obtained.

Interplay of electric dipoles and magnetic monopole-like states in A₂Ir₂O₇: perspectives

The classical framework of Maxwell’s equations and field theory predicts the possibility of magnetic charges, or monopoles, capable of generating magnetic fields in analogy with electric charges. Moreover, the motion of such hypothetical monopoles would induce electric fields. Despite long-standing theoretical interest and considerable experimental efforts, no conclusive evidence for the existence of elementary magnetic monopoles has been established to date.

A shift in focus has recently occurred towards condensed matter systems, particularly geometrically frustrated magnetic materials, in which quasiparticles mimicking the behaviour of magnetic monopoles may emerge. Among these, spin-ice A₂B₂O₇ compounds might prove especially relevant^6,167. Magnetic moments are arranged on a pyrochlore lattice of corner-sharing tetrahedra and experience strong local constraints due to strong SOC, crystal field effects, exchange interactions and time-reversal symmetry. On such a lattice, elementary excitations may manifest as emergent magnetic monopole-like defects within an otherwise ordered or constrained spin configuration.

Beyond their magnetic character, these monopole-like states are predicted to interact with electric degrees of freedom. Specifically, theoretical studies propose that an electric dipole is inherently attached to each magnetic monopole-like excitation, reflecting the underlying symmetry-breaking and charge redistribution at the microscopic scale^6,168,169. In addition to acting as individual excitations, recent theoretical works have highlighted the potential for these monopole-like entities to form bound monopole-antimonopole pairs or so-called ‘fragmented’ magnetic states^170–172, further enriching the phase landscape of such systems.

In the pyrochlore lattice of general formula A₂B₂O₇ (Fig.3), magnetic ions at the vertices of tetrahedra are not free to orient arbitrarily. Instead, their directions are restricted by the crystal geometry and symmetry operations, especially those of the cubic space group and time-reversal constraints (Fig.1c). Magnetic exchange interactions between nearest neighbours lead to complex ground states, particularly under the influence of strong SOC, which introduces substantial anisotropy into the magnetic interactions^3,4. The interplay of geometric frustration, SOC, and crystal-field effects can stabilise unconventional magnetic states, such as classical and quantum spin liquids, spin-ice configurations (Fig. 2b), or fragmented magnetically ordered phases^5,6.

At the same time, the presence of heavy 4f or 5d elements on the A or B sites results in a comparable energy scale between SOC and electron–electron interactions (Coulomb repulsion). This delicate balance gives rise to a variety of emergent topological phenomena, including Weyl semimetals, topological Mott insulators, and axion insulators^35,37,50. These states exhibit nontrivial band topology, broken time-reversal or inversion symmetry, and novel magnetoelectric responses.

Khomskii’s intriguing proposal⁶ suggests that magnetic monopole-like states in spin-ice systems are intrinsically accompanied by electric dipoles. This attachment could provide a mechanism by which such exotic excitations may be manipulated via electric fields, potentially enabling novel magnetoelectric effects or even applications in quantum technologies. Although compelling, this idea remains largely unconfirmed, and only limited experimental evidence has been reported so far^168,169. Ongoing efforts should aim to critically examine and expand upon Khomskii’s proposal, also within the context of A₂Ir₂O₇ pyrochlore iridates, where strong SOC, electronic correlations, and geometrical frustration converge—on both magnetic sublattices—to give rise to complex magnetic and transport phenomena.

Several hypotheses are formed. First is that monopole-like states arise at interfaces between antiferromagnetic domains in AIAO (all-in–all-out) and AOAI (all-out–all-in) configurations, which are related by time-reversal symmetry and supported by the pyrochlore lattice geometry and SOC-induced anisotropy (Fig. 5). These domains are believed to form spontaneously at low temperatures in A₂Ir₂O₇ compounds, and their interfaces are proposed to host uncompensated moments arranged in three-in–one-out (3I1O) or one-in–three-out (1I3O) configurations, mimicking magnetic monopoles in spin ice⁴¹. The interface itself consists of both rotatable and frozen moments: while the former are weakly coupled to adjacent domains and susceptible to manipulation by weak external magnetic fields (on the order of a few millitesla), the latter remain protected by the bulk AFM order yet contribute a net ferromagnetic moment to the interface^104,119. The stabilisation of such interfaces, and the associated monopole-like states, has been partially demonstrated in low-field magnetisation studies and are predicted to persist even under strong applied fields.

In addition to their different magnetic structure, these interfaces may exhibit distinct electronic properties compared to the surrounding domains. The AIAO-ordered bulk is believed to exhibit Mott insulating behaviour, with the electronic gap intimately tied to the symmetry and arrangement of magnetic moments^{41,108,123,124}. At the domain walls, however, the disruption of long-range magnetic order could lead to enhanced electronic conductivity, as theoretically predicted and experimentally observed in selected compounds^120–122. This contrast between insulating domains and conductive interfaces may provide a transport signature of the presence of monopole-like excitations, particularly if they are accompanied by local electric dipoles as Khomskii proposed⁶.

A further hypothesis builds upon the magnetic interplay between the two sublattices of the pyrochlore structure. In many members of the A₂Ir₂O₇ family, the rare-earth (A-site) and Ir (B-site) sublattices both host magnetic moments, and their coupling - arising from f-d exchange interactions - can give rise to cooperative effects. Of course, this does not apply to members with non-magnetic A-site ions (A = Y, Eu, and Lu). It has been shown that the magnetic ordering of one sublattice can influence, or even stabilise, the ordering of the other. Specifically, the magnetic configuration of the Ir⁴⁺ sublattice plays a decisive role in determining the ground state of the rare-earth moments, potentially leading to nontrivial spin configurations such as fragmented states or spin-ice-like behaviour^30,172. In Ho₂Ir₂O₇, evidence of strong f-d coupling has emerged from anomalies in magnetisation and transport data, suggesting that monopole-like states in the Ho sublattice are correlated with the magnetic order of the Ir network³³. This mutual interaction enables the tuning of the Ir magnetic texture by controlling the Ho moments via applied magnetic fields¹⁷³; thereby providing an external handle for engineering and probing composite monopole-like states on both Ir and rare-earth sublattices.

From a broader perspective, it is also important to distinguish between two conceptually different types of monopole-like excitations anticipated in the pyrochlore iridates^33,173. First, theoretical considerations suggest that the rare-earth sublattice of insulating A₂Ir₂O₇ members may, under appropriate crystal-field and exchange conditions, support bulk 3I1O/1I3O defects analogous to the emergent monopoles of classical spin ice, albeit with significantly reduced mobility due to the presence of strong f–d exchange fields and the underlying AIAO background imposed by the Ir⁴⁺ network. Second, and more prominently evidenced experimentally, iridates host interfacial monopole-like states localised at AIAO/AOAI antiferromagnetic domain boundaries of the Ir sublattice. Their density is controlled not by thermal activation but by the domain microstructure, which is itself sensitive to cooling-field protocols and subtle crystallographic or stoichiometric variations.

Recent quantitative analyses in Lu₂Ir₂O₇¹⁰⁴ estimate ~10²–10³ AIAO/AOAI interfaces per μm³, corresponding to characteristic domain dimensions of 0.1–1 μm, consistent with micron-scale AFM domains directly observed in Nd₂Ir₂O₇¹²¹ and isostructural Cd₂Os₂O₇¹²⁸. These values imply an interfacial monopole density that is orders of magnitude lower and far more spatially confined than the bulk monopole population of classical spin ice, where a thermally activated Coulomb phase forms throughout the lattice. Consequently, the monopole-related response in iridates is expected to be weaker, anisotropic, and dominated by the nucleation, annihilation, and rearrangement of domain-wall configurations rather than by bulk defect dynamics. Nonetheless, the ability to manipulate domain populations through field-cooling provides an experimentally accessible route for probing these interfacial excitations and their potential coupling to electric dipoles.

A final consideration is the possibility of a genuine magneto-electric coupling associated with these monopole-like states. If the proposed attachment of an electric dipole is correct^6,168,169, then it should be possible not only to detect, but also to manipulate magnetic monopole-like excitations using electric fields or electric currents. Early magnetoelectric experiments on strongly insulating titanate pyrochlores (Dy₂Ti₂O₇ and Ho₂Ti₂O₇) have reported dielectric anomalies that may be linked to such coupling^168,169. However, the limited conductivity of these materials constrains their experimental accessibility. In contrast, A₂Ir₂O₇ iridates—with their small Mott gap and intrinsic 5 d magnetism—offer a more versatile platform. Their enhanced conductivity allows for current-driven investigations, while the magnetic Ir sublattice introduces an additional degree of freedom for tuning the system. Thus, if monopole-like excitations indeed carry electric dipoles in these materials, iridates may provide a realistic setting for their electric-field manipulation and eventual integration into functional devices.

Despite the conceptual richness and significant theoretical advances in this field, the proposed mechanisms remain largely hypothetical due to the persistent challenges in experimentally realising and identifying these exotic phases. A major obstacle lies in the limited availability of high-quality, homogeneous A₂Ir₂O₇ single crystals, which obscures intrinsic behaviour mapping and compromises reproducibility^84,122,124. Progress in this field thus critically depends on the synthesis of large, pure crystals and the application of advanced characterisation techniques combining magnetic, transport, and dielectric probes with microscopic synchrotron and neutron scattering experiments. Additionally, complementary first-principles calculations of electronic structure and spin dynamics are essential for interpreting experimental data and constructing a robust microscopic framework.

Summary and outlook

Magnetic monopoles, long predicted by field theory and Maxwell’s equations, remain experimentally elusive. However, geometrically frustrated spin-ice systems offer a route to realising monopole-like excitations, which, according to Khomskii’s proposal, may be accompanied by electric dipoles. This theoretical concept implies that such emergent quasiparticles could be manipulated via electric fields—potentially enabling novel magnetoelectric phenomena and applications. While initial indications have been reported in rare-earth A₂Ti₂O₇ titanates, conclusive experimental evidence is lacking.

Pyrochlore iridates A₂Ir₂O₇ represent a compelling platform to explore these ideas due to the confluence of strong SOC, electron correlations, and magnetic frustration. Their ground states exhibit AIAO-type antiferromagnetic order, with domains and domain interfaces that may stabilise uncompensated monopole-like configurations (3-in–1-out or 1-in–3-out). These interfacial states could couple to electric dipoles and exhibit distinct electronic transport properties. Moreover, the interplay between the magnetic rare-earth and Ir sublattices via f-d exchange enables external control of magnetic orderings, offering a pathway to stabilise or tune such monopole-like configurations. Continued advances in crystal growth, characterisation techniques, especially microscopic techniques, and first-principles calculations are necessary to establish the existence and properties of these monopole-like states, and to critically test Khomskii’s hypothesis in a realistic setting.

Author contributions

M.K. conceived the scope and structure of the review, conducted a comprehensive survey of the relevant literature, critically analysed and synthesised the published experimental and theoretical results, prepared all figures and tables, and wrote the manuscript.

Peer review

Peer review information

Communications Chemistry thanks the anonymous reviewers for their contribution to the peer review of this work.

Competing interests

The author declares no competing interests.