Large magneto-thermoelectric effect on the verge of metal–insulator and topological transitionsin pyrochlore iridates

Thermoelectric application is of high practical importance for energy harvesting and solid-state cooling. To achieve the high efficiency of the thermoelectric conversion, much effort has been devoted to the enhancement of the Seebeck effect, i.e., the effective generation of an electric field longitudinal to a thermal gradient. Recently, the transverse electrical signal in a magnetic field, i.e., the Nernst effect, has also been attracting considerable attention because of its complementary orthogonal configuration of electric-field and thermal gradient directions to the Seebeck effect. In particular, the anomalous Nernst effect has been proven to yield a large signal due to the non-trivial geometry of the electronic bands, i.e., Berry curvature, as exemplified by a series of topological semimetals.^1–4 Among them, in the full-Heusler compound Co₂MnGa, the Nernst effect is particularly enhanced in the proximity to a quantum Lifshitz transition between type-I and type-II Weyl semimetal phases, where the density of states is markedly accumulated.³ In this sense, quantum materials, which can show phase transitions (Lifshitz transitions) among topological electronic phases, would be promising for high thermoelectric performance.

The pyrochlore iridate R₂Ir₂O₇ (R being rare-earth ions and Y) is a rare topological electron system that reveals various magnetic topological semimetallic phases due to the criticality of metal–insulator transition.^5,6 In this material, the paramagnetic metallic state is characterized by the zero-gap semiconductor hosting a quadratic band touching (QBT) at the Brillouin zone center,^7,8 which is one of the key features for topological electronic states as demonstrated for the cases of HgTe and RPtBi.^9–13 For instance, in the case of pyrochlore iridates, four pairs of Weyl points are expected to show up in the all-in all-out type magnetic order, where all four spins on the vertices of a tetrahedron point in or out of its center and hence only time-reversal symmetry, not inversion symmetry, is broken.¹⁴ Figure 1(a) shows the phase diagram of R₂Ir₂O₇. Nd₂Ir₂O₇ undergoes the transition from the paramagnetic metal to the antiferromagnetic insulator with the all-in all-out type magnetic configuration.^15,16 With increasing the Pr doping level x in (Nd_1−xPr_x)₂Ir₂O₇, the antiferromagnetic transition is systematically suppressed and eventually extinguished at around x ∼ 0.8.¹⁷ Pr₂Ir₂O₇ is a paramagnetic metal in the whole temperature range, except for the spin-ice like magnetic order of Pr moments at 0.6 K.¹⁸ Figure 1(b) shows the magnetic field dependence of resistivity for several compositions. For x = 0, the resistivity is relatively small after the zero-field cooling. However, as the magnetic field is applied, the resistivity abruptly increases at around 3 T. Returning to the zero field, the resistivity reaches more than two orders of magnitude larger than the initial value. This is due to the disappearance of the magnetic domain walls that host anomalous metallic states.^19,20 For x = 0.5, which is located near the quantum metal–insulator boundary, the resistivity markedly decreases by three orders of magnitude at 14 T, accompanied by the ferromagnetic-like 2-in-2-out or 3-in-1-out magnetic structures.^21–23 These transport properties are understood in the context of topological electronic states. In particular, the Hall effect shows remarkable field dependence across the transitions, indicative of the Berry curvature generation accompanied by the Fermi surface reconstruction.²³ The magneto-thermoelectric effect is quite sensitive to such band modulations because it is proportional to the energy derivative of the conductivities as given by the Mott relation.²⁴ However, the thermoelectric response of this family of materials has remained elusive.

In this Letter, we investigate thermoelectric effects related to the thermally induced and magnetic field-induced metal–insulator transitions in (Nd_1−xPr_x)₂Ir₂O₇. The temperature dependence of the Seebeck coefficient of x = 0 exhibits a prominent dip structure accompanied by an increase of the Nernst effect right below the magnetic transition temperature, possibly due to the emergence of Weyl nodes between the metal and insulator phases. The metallic domain walls are reversibly created and eliminated by the external field, causing a large hysteresis of the Seebeck coefficient. Furthermore, for the x = 0.5 compound, which undergoes the field-induced metal–insulator transition, the Nernst coefficient exhibits a sharp increase while the Seebeck coefficient changes its sign above the critical magnetic field. Remarkably, the Nernst signal is larger than those in ferromagnetic oxides, a possibility attributed to the topological nature of electrons.

High-quality polycrystals were synthesized under the high-pressure conditions.¹⁵ For the thermoelectric measurement, atemperature gradient (a few percent of each measurement temperature) was applied by Joule heating of a chip resistance (1 kΩ) and measured with Cernox thermometers and type-E thermocouples. The electrical voltage was measured through manganin wires that were attached to the sample by solder paste.

Figure 1(c) shows the temperature dependence of resistivity for the respective x. The resistivity of x = 0 (x = 0.5) sharply increases below the magnetic transition temperature T_N = 32 K (10 K), while that of x = 1 keeps metallic down to 2 K. The Seebeck coefficients S_xx for several x are displayed in Fig. 1(d). S_xx for x = 0 is negative at high temperatures and becomes almost zero at T_N. With decreasing temperature below T_N, S_xx sharply drops, reaches a minimum at 27 K, and abruptly increases with the sign reversal. A similar dip structure below T_N is also observed for x = 0.5. On the other hand, S_xx for x = 1 shows monotonic temperature dependence down to 2 K, as observed in a previous study.²⁶ It implies that the electronic structure changes dramatically with the magnetic order. Figure 1(e) shows the temperature dependence of the Nernst signal S_xy. General metals produce minimal S_xy because of the balance between the Hall current and off-diagonal Peltier current.²⁷ Actually, S_xy is almost negligible above 100 K for all samples. However, with lowering the temperature below 100 K, S_xy of x = 0 gradually increases toS_xy = 0.7 μV/K at T_N and 1 T. It is likely a normal Nernst effect since S_xy is proportional to the magnetic field [Fig. 1(f)]. The observed signal is very large compared to other oxides. For instance, it is less than 0.2 μV/K for high T_c cuprates Pr_2−xCe_xCuO₄.²⁸ The small Fermi energy or high electron mobility is known to yield large signals, empirically written as $Sxy=π3kBekBTεFμ$⁠, where k_B is the Boltzmann constant, ɛ_F is the Fermi energy, and μ is the carrier mobility.²⁹ Taking into account this relation and the Drude model 1/ρ_xx = enμ, we estimate the Fermi energy and mobility as 38 meV and 360 cm²/V s, respectively. Such a semimetallic character may reflect the QBT at the Γ point. More importantly, below T_N, S_xy increases further, reaching a maximum value of $∼0.9$ (μV/KT) at 27 K, at which S_xx takes the minimum as shown in Fig. 1(d). Eventually, S_xy monotonically decreases to zero at low temperatures.

Such a characteristic temperature dependence and large thermoelectric effect indicate the significant change of electronic structure below T_N. It is theoretically predicted that the four pairs of Weyl points are produced from the QBT by time-reversal symmetry breaking.³⁰ Given that the Weyl point acts as a source or sink of the Berry curvature in momentum space and hence affects the transverse signals, the observed S_xy maximum at 27 K is likely due to the emergence of Weyl points. With further lowering the temperature, on the other hand, the Weyl points immigrate along the [111] or its equivalent directions, causing the pair-annihilation at the Brillouin zone boundary,³¹ followed by the gap opening as reported in the spectroscopic study.^15,25 On the other hand, for x = 0.5, the increase of S_xy right below T_N is seemingly suppressed regardless of the similar modulation of electronic states. It is presumably due to the Kondo coupling between the itinerant Ir-5d electrons and the non-coplanar Pr-4f moments. As shown in Fig. 1(g), the Nernst signal of x = 1, which is not magnetically ordered above 2.5 K, starts to deviate from the linear field dependence below 12 K and eventually shows the negative broad dip structure at 2.5 K. It is attributable to the geometrical Nernst effect induced by the Kondo coupling.³² Therefore, we speculate that the Nernst signal enhanced by the emergence of the Weyl semimetal phase can be partially canceled by the negative contribution of the scalar spin chirality for x = 0.5 (T_N = 10 K). We note that S_xx does not appear to diverge at low temperatures despite the gap. Similar behaviors are also reported even in smaller R compounds, which show more insulating behavior or a larger charge gap.⁵ 

Next, we show the influence of the metallic domain walls on the Seebeck effect in Nd₂Ir₂O₇ (x = 0). Figure 2(a) shows the temperature dependence of the electrical conductivity σ. Here, we measured σ in two different ways. σ^ut is the conductivity measured as the temperature increases after cooling at zero magnetic field, i.e., in the untrained (multi-domain) state. On the other hand, σ^t is measured after the field cooling; first, the temperature is lowered under the magnetic field of 14 T, and then the field is turned off at 2 K. Namely, σ^t is regarded as the conductivity in the trained (single domain) state. These states are directly observed in real space; as systematically shown in Fig. 2(b), domain walls are present but cut by grain boundaries in the multi-domain state, whereas the domain is almost aligned and domain walls disappear in the trained state.²⁰ Both conductivities decrease significantly below T_N, but σ^ut is two orders of magnitude larger than σ^t at 2 K because of the existence of conductive domain walls. In a similar way, we obtain the Seebeck coefficient S^ut in the multi-domain state and S^t in the single domain state, as shown in Fig. 2(c). Both show a similar temperature dependence, but the S^t is larger than S^ut below 25 K; S^t reaches the maximum value of $∼200μ$V/K, whereas S^ut takes $∼100μ$V/K. To see the difference in S_xx between trained and untrained states, we plot S^ut − S^t as well in Fig. 2(b). S^ut − S^t takes a minimum (negative-maximum) value of −110 μV/K at around 17 K and gradually approaches zero at lower temperatures. This non-monotonic temperature dependence may reflect that the S_xx of domain walls becomes small at low temperatures like that of general metals, while at high temperatures the bulk conductivity is comparable to that of domain walls and hence the voltage drop is less likely to occur. It should be noted that the magnetic domain walls make no contribution to S_xx for 25 K $<T<TN$⁠, possibly implying that the electronic states trapped by domain walls (Fermi arc) depend on the bulk states. According to Ref. 31, as the pair of Weyl points approaches each other, the Fermi arc is elongated towards the zone boundary and eventually forms an open Fermi line after the pair annihilation.

In general, the electrical conductivity in a percolation system is described by the parallel circuit model. In the present case, it reads

where x is the volume fraction of magnetic domain walls and σ^DW is the electrical conductivity of domain walls. According to a previous study,¹⁹ σ^DW is almost independent of temperature. Therefore, we assume that σ^DW = 1250 S/cm, which is equal to the bulk conductivity right above T_N, and estimate x. The parallel circuit model can also be applied to the Peltier conductivity³⁶ 

We calculate α^DW by using Eqs. (3) and (4) and S^DW = α^DW/σ^DW, which is plotted in the black dashed line in Fig. 2(c). However, the temperature dependence of S^DW appears similar to that of the insulating bulk state, implying that the contribution of the domain walls is not properly extracted by this simple model. This may be attributed to: (i) the complex geometry of multiple domain walls interrupted by grain boundaries [see Fig. 2(b)]; (ii) the junctions of metallic domain walls and insulating bulks, which may cause inhomogeneous heat/voltage distributions. A microscopic study of such a nano-scale native metal–insulator heterostructure may be useful for future magneto-thermoelectric engineering.³⁷ 

Figure 2(d) shows the magnetic field dependence of S_xx at various temperatures. All measurements were taken after zero-field cooling (untrained state). At 32 and 27 K, right below T_N, the field dependence is very small and almost no hysteresis is observed. This is consistent with the temperature dependence of S^ut − S^t [Fig. 2(c)]. On the other hand, a large field dependence is observed below 25 K. For instance, at 17 K, S_xx increases rapidly from 85 to 150 μV/K as the magnetic field increases. As the field returns to zero, it becomes ∼100 μV/K larger than that in the initial multi-domain state. With a further decreasing field, S_xx shows the peak at around −3 T, rapidly decreases below −5 T, and saturates at around 150 μV/K at high fields. Similar hysteresis is also observed in the increasing field process. The observed magnetic field dependence can originate from the elimination and generation of magnetic domain walls across the switching of magnetic domains.

We now turn to the thermoelectric effect across the field-induced insulator-to-metal transition at x = 0.5. The temperature dependence of S_xx shows the sharp dip structure down to −21 μV/K below T_N ∼ 10 K as shown in the inset of Fig. 3(a). Figure 3(a) shows the magnetic field dependence of S_xx around T_N. At 15 K, S_xx increases slightly as the magnetic field increases. Below T_N, S_xx is negative at 0 T but increases rapidly toward a positive value with increasing field. At lower temperatures below 5.5 K, it shows a more complex field dependence [Fig. 3(b)]; S_xx shows an upturn at around 4 T and eventually overlaps the same curves at high fields, where the ferromagnetic-like 2-in-2-out or 3-in-1-out state is realized. The observed complex temperature/magnetic field variation can reflect the large modulation of the electronic structure. The optical conductivity at 0 T is shown in the inset of Fig. 3(b). The spectral weight below 30 meV shifts toward higher energies below T_N, consistent with the transport properties. At 8 K, the optical conductivity is somewhat linear with photon energy, as reported in the three-dimensional Dirac/Weyl semimetallic state.^33,35 Eventually, the small charge gap of ∼10 meV is fully open at 5 K. We speculate that the strong band renormalization occurs in proximity to the Mott transition,³⁴ leading to the accumulation of density of states near the Fermi energy and hence to the enhancement of S_xx.

The transverse electric signal against the thermal gradient, i.e., the anomalous Nernst effect, reflects the Berry curvature of electronic bands near the Fermi level,^1–4,38 and hence it provides us with another insight into phase transitions. Figures 3(c) and 3(e) show the magnetic field dependence of the temperature-normalized S_xy/T. At 15 K, it is almost linear with respect to the magnetic field, likely due to the normal Nernst contribution. On the other hand, at 5.5 K below T_N, it clearly decreases at small fields and jumps at around 3 T, at which S_xx rapidly increases. This feature is more pronounced in the magnetic field derivative d(S_xy/T)/dμ₀H shown in Fig. 3(d). While d(S_xy/T)/dμ₀H is almost constant at 15 K, it shows a peak at around 3–4 T at 5.5 K. This peak becomes sharper with a further decrease of temperature as shown in Figs. 3(e) and 3(f). It is presumably attributed to the modulation of electronic structures as discussed in S_xx. Notably, the observed S_xy/T above the critical field is large compared to general ferromagnetic oxides, as discussed below.

In order to clarify the correlation between the magneto-thermoelectric effect and phase transitions, we show the contour plot of the field derivative of S_xx and S_xy/T in Figs. 4(a) and 4(b), respectively. At zero magnetic field, S_xx sharply drops right below T_N as shown in the inset of Fig. 3(a) possibly because of the realization of all-in-all-out type Weyl semimetal state [Fig. 4(c)]. As mentioned earlier, as the temperature decreases, four pairs of Weyl points move toward the zone boundaries, followed by pair annihilation. Apparently, S_xx reaches the minimum (negative maximum) at around 6 K as shown in the yellow region [Fig. 4(a)], where the band renormalization may occur at the Mott transition. More importantly, the yellow region extends toward the lower temperature as the field is applied. Previous studies demonstrate that the external magnetic field changes the magnetic structure from the antiferromagnetic all-in-all-out configuration to the ferromagnetic 2-in-2-out or 3-in-1-out states, giving rise to the variation of Weyl nodes in the momentum space [Figs. 4(c)–4(e)].^21–23 We speculate that a similar mass renormalization occurs between the insulating and ferromagnetic-like semimetal states. Figure 4(b) is a contour plot of the field derivative of the Nernst effect d(S_xy/T)/dμ₀H. d(S_xy/T)/dμ₀H hardly appears in the region where the gap is open (blue region), but significantly increases across the field-induced transition, resulting in the large increase of S_xy exceeding 1 μV/K at 5.5 K just above the critical field (5 T), i.e., in the 2-in-2-out/3-in-1-out semimetal states. Representative values of anomalous Nernst effects in ferromagnetic oxides are S_xy = 0.03 μV/K at 5 T for SrRuO₃, 0.2 μV/K for (La, Sr)CoO₃,³⁹ and 0.65 μV/K for Nd₂Mo₂O₇.³² Generally, the mechanisms of enhanced Nernst effects are: (i) vortex motion or vortex-like excitations in superconductors;^40,41 (ii) unidentified order parameters in heavy fermion systems;⁴² and (iii) Berry curvature of electronic bands near the Fermi level.^2–4 For the present d electron system, we can clearly exclude the former two scenarios, and hence the last one based on the Berry curvature seems plausible. Moreover, the density of states accumulated by the Mott transition may amplify the thermoelectric signal that involves partially occupied bands near the Fermi level.³ These findings indicate that the electronic phase transition involving the topological states gives rise to the large magneto-thermoelectric effects, offering the future direction of thermoelectric materials exploration.

In summary, we investigate the magneto-thermoelectric properties of (Nd_1−xPr_x)₂Ir₂O₇, which undergoes the topological electronic transitions induced by thermal change and magnetic field. It is found that the Seebeck effect shows significant temperature dependence while the Nernst signal is enhanced below the magnetic transition temperature, possibly indicating that the Weyl nodes emerge between paramagnetic metal and antiferromagnetic insulator phases. In the latter phase, the magnetic domain walls cause the large magnetic-field hysteresis of the Seebeck coefficient, which is highly promising for the future application of nano-scale thermoelectric junctions. In proximity to the field-induced transitions for x = 0.5, both Seebeck and Nernst signals are markedly amplified, presumably due to the emergence of topological semimetals in ferromagnetic-like 2-in-2-out or 3-in-1-out states near the Mott criticality. These results demonstrate that the system near phase transitions would be favorable for next-generation topological thermoelectric materials.

This research was supported by JSPS/MEXT Grant-in-Aid for Scientific Research (Grant Nos. 21K13871 and 21K18813), CREST, JST (Grant No. JPMJCR16F1), and the Nippon Sheet Glass Foundation for Materials Science and Engineering.