Introduction

Systems incorporating both spin–orbit coupling (SOC) and electron correlations, such as the members of the A2B2O7 family with A = rare-earth element and B = 4d or 5d transition metal, present a fertile area for research of novel exotic phases of matter1,2,3. Relativistic SOC by itself plays an essential role in the emergence of non-trivial topology in the band structures of these materials3,4. In particular, the A2Ir2O7 iridates make for an intriguing case by combining comparable strength of SOC and electron correlations, as well as f-d exchange coupling between rare-earth and Ir moments, and crystal field effects. Several non-trivial phases have been proposed to arise from these interactions, including antiferromagnetic Weyl semimetal5,6,7, topological Mott insulator7,8, axion insulator with non-collinear magnetic order5,9, spin-ice state with monopole-like excitations10,11, fragmented state12,13, or spin-liquid state14.

Complex electronic and magnetic states in A2Ir2O7 are also closely connected with their geometrically frustrated crystal lattice of the pyrochlore type. Pyrochlore structure is composed of two sublattices of A3+ and Ir4+ ions, forming interpenetrating networks of corner-sharing tetrahedra and eight- and six-coordinated oxygen cages around respective cations1,15. Focusing exclusively on the Ir sublattice, it has been repeatedly proposed and validated that it orders magnetically in the so-called all-in-all-out (AIAO and equi-energy AOAI) structure when the magnetic moments on vertices of the Ir tetrahedron point all in or out along local <111> directions16,17. With the exception of Pr2Ir2O7 without any signs of magnetic ordering18, heavier A2Ir2O7 display the magnetic transition at 30–140 K depending on the A ion1,3,15. Simultaneously with the magnetic ordering, a metal-to-insulator transition is observed19,20, and proposed to be induced by the antiferromagnetic ordering of the Ir sublattice21,22. Importantly, both the AIAO and energy-equivalent AOAI arrangements of magnetic moments could coexist in the lattice, forming the respective domains1,2,3. The interfaces between domains, AIAO/AOAI domain walls (DWs), seem to exhibit significantly different conductive properties compared to the interior of the domain23. It was demonstrated that insulating domains have relatively conductive domain walls with disturbed magnetic ordering. The perturbed ordering in the walls leads to uncompensated magnetic moments on the interfaces24,25, or even magnetic monopole-like excitations in the rare-earth sublattice induced by the f-d coupling11.

In the present paper, the magnetic response of the Ir sublattice to the applied magnetic field is discussed within the model of antiferromagnetic (AFM) domains and robust ferromagnetic (FM) domain walls in two representatives of the A2Ir2O7 family: Lu2Ir2O7 and Er2Ir2O7 single crystals. Solely Ir magnetism rules over the magnetic states in Lu2Ir2O7 with nonmagnetic Lu3+ ions, representing the model system for studying the Ir sublattice. Er2Ir2O7 properties are, in addition, influenced by the magnetic Er3+ sublattice, especially at low temperatures.

Single crystals of Lu2Ir2O7 and Er2Ir2O7 were synthesised for the first time, characterised by electron microscopy (Fig. 1e,f), X-ray diffraction methods, and investigated by means of magnetisation measurements. PbF2-flux method was used for sample synthesis, resulting in minor Pb contamination; details on sample synthesis and characterisation are provided in the “Methods” section and Supplementary Materials. Both single crystals reveal a bifurcation between zero-field-cooled (ZFC) and field-cooled (FC) magnetic susceptibility (Fig. 1a,b), tracking the magnetic ordering of the Ir4+ sublattice. The ordering temperature TIr is determined as an onset of the bifurcation: TIr = 128(3) K for Lu2Ir2O7 and TIr = 120(6) K for Er2Ir2O7. Previous studies of polycrystalline samples reported somewhat higher ordering temperatures of 140 K26,27,28 for Er2Ir2O7, and 147 K29 and 135 K30 for Lu2Ir2O7. The value of TIr in A2Ir2O7 iridates is strongly sample-dependent and can be connected with an Ir off-stoichiometry24,31, which is a common problem in synthesising these materials30,32,33,34. Smaller TIr of our single crystals is likely related to a slight Ir off-stoichiometry of up to 2% (minor Pb contamination of the samples should not affect the Ir sublattice substantially); see details in Supplementary Materials. Nevertheless, such a small Ir off-stoichiometry is considered to impact the physical properties of the material only slightly (TIr values close to the polycrystalline ones) compared to most of the previously reported A2Ir2O7 crystals.

Fig. 1
figure 1

Characterisation of Lu2Ir2O7 (top panels) and Er2Ir2O7 (bottom panels) single crystals. (a,b) Bifurcation between zero-field-cooled and field-cooled magnetic susceptibility with the zoomed-in high-temperature region in the insets (field applied along the [111] direction). (c,d) Field dependence of magnetisation at 2 K with field applied along the three principal crystallographic directions. Magnetisation along [110] was measured at several temperatures (insets). (e,f) Back-scattered electron (BSE) images of the synthesised single crystals.

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Focusing on the magnetic field dependence of magnetisation (Fig. 1c,d), the magnetisation does not saturate up to 7 T, nevertheless a clear saturation tendency is followed. Roughly one-third of the free-ion saturated magnetisation (1 µB for Ir4+ and 9 µB for Er3+ ions) is reached in 7 T. Further increase of magnetisation with increasing field is expected, likely reaching one-half of the free-ion value in the magnetic field well above 10 T. Such an evolution is perfectly in agreement with previous studies of both A2Ir2O7 single crystals and polycrystals11,26,35; being explained by strong geometrical frustration of the pyrochlore lattice and local magnetic moments anisotropy. The average magnetic anisotropy of the compound, measuring the magnetisation with a magnetic field applied along three principal crystallographic directions, is relatively low (Fig. 1c,d). The largest magnetisation is observed for a field applied along the [110] direction and the smallest one along the [111] direction in both Lu2Ir2O7 and Er2Ir2O7. The fact that the same anisotropy is revealed by both members strongly suggests only a minimum impact of the Er sublattice, which is expected to order in an easy-plane arrangement26. Indeed, an anomaly connected with magnetic correlations on the Er sublattice was reported at a much lower temperature of 0.6 K26. On the other hand, the impact of the Ir sublattice on the Er sublattice is demonstrated by a more substantial anisotropy (magnetisation in 7 T differs for [110] and [111] directions by 0.3 µB in Lu2Ir2O7 and 1.0 µB in Er2Ir2O7), likely reinforced by Ir-polarised Er moments. A crude explanation of the magnetic anisotropy in the system is based on the orientation of sublattice tetrahedra with respect to an applied magnetic field. The AIAO ordering of magnetic moments is characterised by the moments oriented parallel or antiparallel to local <111> directions. When a magnetic field is applied along this direction, it is naturally difficult to polarise the moments due to their antiferromagnetic exchange coupling on the geometrically frustrated tetrahedron. When a field is applied along another crystallographic direction, the system robustness to a field polarisation is lower as the vector of the magnetic field is not parallel to any direction of moments on the tetrahedra. No metamagnetic transition is observed in the data in a field up to 7 T. A standard temperature evolution of isothermal magnetisation in an antiferromagnet is followed (insets in Fig. 1c,d).

Ferromagnetic ordering in the investigated crystals is excluded based on the missing hysteresis loops measuring the magnetisation with an applied field of 0 → 7 → -7 → 0.2 T. However, a clear FM response is revealed (Fig. 2a,b) by measuring the samples with the following cooling protocol: before cooling the sample below the ordering temperature TIr, an external magnetic field is applied. The sample is cooled in this field (FC) down to low temperatures, and the field is removed. The additional magnetic signal induced by this cooling protocol is represented by a small shift of magnetisation Msh (insets of Fig. 2a,b). Importantly, this signal is present in any magnetic field, at least up to 7 T (-7 T); Msh is robust against the applied field (Fig. S1). No shift is observed in the ZFC regime. Msh is positive or negative depending on the sign of an applied cooling field and is symmetric around the ZFC value. That is, Msh measured in 7 T–FC is equal to − Msh measured in -7 T–FC. Such a FM contribution is ascribed to the disturbed magnetic ordering on the antiferromagnetic AIAO–AOAI interfaces, in agreement with some light rare-earth A2Ir2O724,36,37 and Cd2Os2O725. The domain wall model is characterised by the pinned magnetic moments on the interfaces between the AIAO and AOAI domains. The pinned moment is created upon the formation of the DW at TIr, resulting from uncompensated moments on the Ir tetrahedron (Fig. 3). One can describe this magnetic ordering as two-in-two-out (2I2O) or three-in-one-out (3I1O; alternatively 1I3O). In the ZFC regime, the DWs and related uncompensated moments are created randomly upon crossing the TIr. Therefore, the total net magnetisation is zero. The application of a magnetic field at low temperatures induces a standard response of an antiferromagnetic material without any field hysteresis of magnetic response. However, when a magnetic field is applied upon cooling (FC regime), crossing TIr, the domain walls are created in a way that has uncompensated magnetic moments oriented, preferably along the field direction. Once this ferromagnetic-like ordering on the DWs is formed, it is very robust against external magnetic field below TIr. The uncompensated moments on the interface are protected by the antiferromagnetic ordering of the AIAO and AOAI domains, similar to the exchange bias systems. Unless the antiferromagnetic ordering is disturbed, e.g. by increasing temperature above TIr, the pinned moments are extremely stable.

Fig. 2
figure 2

Ferromagnetic signal in Lu2Ir2O7 (top panels) and Er2Ir2O7 (bottom panels) single crystals. (a,b) Field hysteresis measured in different field-cooled protocols. No hysteresis is observed in the ZFC or FC data; however, a clear FM contribution to the magnetisation Msh accompanies the 7 T-cooled and -7 T-cooled data (see also the insets). (c,d) Temperature dependence of Msh measured along the three principal crystallographic directions in 0 T after cooling down in 7 T (see the scheme in b). Open symbols indicate values determined from the field measurements in (a,b). (e,f) Temperature dependence of Msh ([110] direction displayed) is compared with ΔM determined as the difference between ZFC and FC magnetisation in Fig. 1a,b and Fig. S2 (same units and scale for ΔM and Msh are used).

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Fig. 3
figure 3

Domain walls between AIAO and AOAI domains in the pyrochlore structure (cubic unit cell is outlined) induced by a magnetic field applied along the three principal crystallographic directions. The tetrahedra with compensated magnetic moments, all-in (red) or all-out (yellow), and uncompensated moments (grey) within the domain wall are depicted. The moments on the borders of the domain wall are drawn as pink arrows, in contrast to red arrows in the interior of domains. Total pinned magnetic moment (blue arrow) points perpendicular to the domain wall. (a) 2-in-2-out case, pinned moment along [100], (b) 3-in-1-out case, pinned moment along [111] and (c) combined 3-in-1-out and 1-in-3-out case to create the pinned moment along [110].

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Investigating magnetic moments within the DWs in detail, several measurement protocols are suggested and followed:

(i) field dependence of magnetisation on FC sample introduced above. Measuring the isothermal magnetisation at different temperatures, a temperature dependence of Msh is revealed (open symbols in Fig. 2c,d). Msh decreases with increasing temperature when approaching the ordering temperature TIr. We highlight that the Msh value remains constant on the field interval (-7, 7) T at any given temperature in Lu2Ir2O7. The same behaviour is observed in Er2Ir2O7, however, only at higher temperatures (≥ 20 K). At 2 K, the Msh is strongly influenced by the exchange interactions of the Er sublattice and develops with the applied magnetic field (Fig. S1). By increasing (decreasing) the field to 7 T (-7 T), Msh decreases, approaching zero in high field. Such an evolution can be attributed to the stronger exchange interactions between Er moments and their f-d coupling with Ir moments, including the pinned moments, in high magnetic field. Nonetheless, the reversibility of the processes must be emphasised; after removing the field, the Msh shift changes to its original value. The same evolution of Msh is observed for the magnetic field applied along all three principal crystallographic directions.

(ii) temperature dependence of magnetisation in zero field after field-cooling (the measurement scheme is introduced in the inset of Fig. 2d). That is, the ferromagnetic component induced in DWs by the FC protocol is followed (Fig. 2c,d). Importantly, the measured magnetisation corresponds to Msh and its temperature dependence follows the values of Msh determined from the field measurements (Fig. 2a,b). A similar value of Msh observed in Lu2Ir2O7 was previously reported for Eu2Ir2O7 single crystal24, while A = Eu, Lu polycrystals showed a few times weaker FM signal30,38. Although this can be expected based on the grain structure (DW cannot exist on the grain boundaries), the impact of the sample stoichiometry on the Msh value is too significant in previous reports to make conclusions on the strength of the FM component based solely on the crystallinity of the material. Only small anisotropy in Msh(T) is observed, in contrast to the isothermal magnetisation measurements in Fig. 1c,d. The most striking difference between Lu2Ir2O7 and Er2Ir2O7 members is observed at the lowest temperatures, where Msh is strongly enhanced in Er2Ir2O7. The impact of the Ir molecular field and exchange interactions between Er moments are proposed to play a significant role at these temperatures. Although the Er moments are not long-range ordered at 2 K (magnetic order is expected below 0.6 K26), the short-range correlations impact the magnetisation response. Moreover, they are influenced by the molecular field of the ordered Ir sublattice. The Er moments within the Ir AFM domains and within the FM DWs experience different effect of the f-d interaction. That is, the FM component in the Er sublattice could be induced at the Ir domain interfaces. Since these moments are not fully pinned, high magnetic field can diminish their contribution by (partially) rotating them, as mentioned in the previous paragraph. Notably, Msh is dependent on the magnitude of the cooling field; higher magnetic field leads to higher values of Msh (Fig. 2e).

(iii) a cooling-field evolution of a bifurcation between ZFC and FC magnetisation (Fig. 1a,b and Fig. S2). Subtracting the ZFC from the FC magnetisation (denoted as ΔM in Fig. 2e,f) reveals temperature evolution almost identical to the evolution of Msh discussed in the previous paragraph (see 0.01 T measurements in Fig. 2e). ΔM increases with increasing cooling field for both studied compounds. Similar values of ΔM and Msh, their temperature and field evolution and respective curve shapes strongly suggest that the bifurcation between ZFC and FC magnetisation below TIr is directly connected with the FM component Msh described by the domain interfaces in these materials. Increasing the cooling field (FC) results in an increase of Msh, suggesting a larger density (larger number or size/thickness) of DWs.

Domain wall structure (simple examples introduced in Fig. 3) is supposed to be strongly dependent on the direction of the applied magnetic field. In the presented examples, the net magnetic moment of the domains’ interface is aligned along the external field and perpendicular to the DW. Of course, more complex DWs with net magnetisation not necessarily perpendicular to the DW also exist. Nevertheless, these simple DW models can be sufficiently employed for rudimentary explanations of our experimental results. First, the total net magnetisation per one DW in one unit cell is calculated. The same value of 14.4% of the saturated moment is computed for the three types of DWs (see details of calculations in Supplementary Materials). That is, in the theoretical case of a perfect crystal with the same amount of DWs of a single type, Msh would be identical for the three principal crystallographic directions. Such estimation is in agreement with only very weak anisotropy of experimental Msh (Fig. 2c,d).

Secondly, the dimensions of the AFM domains are roughly estimated in Lu2Ir2O7, assuming only the Ir4+ ions’ magnetism. Considering the first type of DWs in Fig. 3a, the minimum distance between individual DWs along the net moment direction is estimated to be 0.064 µm (that is, 253 Ir layers) based on the measured value of Msh in 7 T–FC regimen (see Supplementary Materials for calculation details). Counting in also DWs parallel to the net magnetisation, simple cubic domains with dimensions of 0.19 × 0.19 × 0.19 µm3 are estimated. Previously, AFM domains were directly observed in Nd2Ir2O723 and Cd2Os2O739, with dimensions of microns and dozens of microns in ZFC, respectively. Importantly, the application of magnetic field during sample cooling below TIr reduced the number of domains, all the way to a single-domain state in the high-field-cooled regimes23,40. Such an observation contrasts with the increasing value of Msh with increasing cooling field (our work and similarly previous works24,25,30)—increasing value of Msh is explained by a higher concentration of DWs and, therefore, a higher number of domains. Following this explanation, the density of domains in Lu2Ir2O7 single crystal is calculated to be 146 AIAO/AOAI domain interfaces per µm3. Alternatively, increasing thickness of DWs (simply stacking more layers of tetrahedra on top of each other) can result in the same net magnetisation with bigger domain sizes compared to the presented monolayer model. Stacking an odd number of layers creates a conventional DW, while stacking an even number of layers creates a boundary between the same type of domains (two AIAO or two AOAI). That is, a single domain would possess the pinned moments. Of course, FM droplet model could also be considered: the FM component Msh would be attributed to the nonmagnetic Ir sublattice defects caused by off-stoichiometry30,38. However, previous studies of iridates’ off-stoichiometry showed a gradual decrease in net magnetisation with increasing off-stoichiometry24,31,41, disagreeing with the FM droplet model. Monoatomic droplets also create an uncompensated moment only along the < 111 > directions, that is, anisotropic FM contribution would be expected in single crystals.

In conclusion, the magnetic response of two representatives of A2Ir2O7 iridates, Lu2Ir2O7 and Er2Ir2O7 single crystals, was investigated and discussed within the framework of antiferromagnetic domains and ferromagnetic domain interfaces model. Ferromagnetic component of magnetisation was shown to manifest as (i) a shift in the value of isothermal magnetisation measured on the FC sample, (ii) a temperature evolution of magnetisation of the FC sample measured in zero magnetic field, and (iii) very importantly a bifurcation between ZFC and FC magnetisation below TIr. The three evolutions give the same value of the ferromagnetic component, proving the same origin of respective signals. Simultaneously, it was proved that the ferromagnetic component in A2Ir2O7 can be effectively investigated by any of the three methods. A model of pinned magnetic moments in domain walls between antiferromagnetic domains was employed to describe the measured ferromagnetic signal. The model allowed us to explain a bifurcation in magnetisation of antiferromagnetically AIAO ordered iridates, the negligible anisotropy of the ferromagnetic response, as well as to estimate the size and number of the antiferromagnetic domains in the studied single crystals. Robustness of the domain walls below ordering temperature was proved.

Methods

Single crystals of Lu2Ir2O7 and Er2Ir2O7 iridates were synthesised employing a lead-fluoride flux preparation method. The mixture of initial Lu2O3 (Er2O3), IrO2 and PbF2 was properly homogenised, inserted into a platinum crucible and reacted at 1100 °C in air in a resistance furnace. Further details on the synthesis process and its optimisation will be reported in our separate publication. Grown single crystals have an octahedral (bi-pyramidal) shape (Fig. 1e,f) with pronounced shiny facets perpendicular to the <111> crystallographic directions and with dimensions up to 0.5 × 0.5 × 0.5 mm3. The basic characterisation was done on multiple crystals from different batches. Laue X-ray diffraction confirmed the single-grain nature of larger crystals and was used for sample orientation. Electron microscopy with energy-dispersive X-ray (EDX) analysis revealed the stoichiometry of investigated crystals. Besides Lu (Er), Ir and O atoms, the EDX spectra indicated the presence of lead in the samples under the surface. Lead atoms from the PbF2 flux tended to mix with the rare-earth atoms in the pyrochlore structure. This was evident from a homogenous distribution of Ir in the 2D EDX maps while regions with more Pb showed smaller signal from Lu/Er and vice versa (Fig. S4). Overall, a small doping of lead was observed (up to 10% of rare-earth atoms). However, some crystals showed up to approximately 40% of rare-earth atoms being substituted by lead. Those crystals were excluded from further investigations. Within the experimental error of about 4%, an almost stoichiometric ratio of Ir and Lu(Er) + Pb was confirmed, the ratio of 0.98(4):1.02(4). The oxygen composition could not be reasonably determined as the EDX method is not sufficiently sensitive to lighter atoms.

Magnetisation measurements were performed using an MPMS7-XL instrument (Quantum Design), which employs a high-resolution superconducting quantum interference device (SQUID). The reciprocating sample option (RSO) was used to measure relatively weak magnetic signal from small single crystals. All measurements were done on the same sample of Lu2Ir2O7 and Er2Ir2O7, with a mass of 0.14 mg and 0.09 mg, respectively. The shape of the crystals and related demagnetisation factor were neglected. All relevant data were measured up to 150 K only. At higher temperatures (> 200 K), the magnetic signal of samples, especially Lu2Ir2O7, became unreliable. The signal of the sample environment (diamagnetic and paramagnetic signal of plastic holders and GE varnish) contributed significantly to the measured magnetisation.