We investigate the Seebeck and Nernst effects of pyrochlore iridium oxides (Nd_{1−x}Pr_{x})_{2}Ir_{2}O_{7} across the thermally induced and magnetic field-induced metal–insulator and topological transitions. Nd_{2}Ir_{2}O_{7} exhibits the salient temperature dependence of the Seebeck coefficient accompanied by the enhancement of the Nernst effect in the vicinity of the thermal magnetic transitions. Moreover, the Seebeck coefficient shows a remarkable magnetic field hysteresis with the differential magnitude reaching as large as 110 *μ*V/K, as the conductive magnetic domain walls are generated/annihilated by the external field. For *x* = 0.5, the Nernst signal increases rapidly across the field-induced metal–insulator transitions, exceeding the values reported in existing ferromagnetic oxides. These findings indicate that the thermoelectric effects increase significantly near the topological electronic phase transitions in strongly correlated systems, providing a new guideline for thermoelectric material design.

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 Co_{2}MnGa, 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 *R*_{2}Ir_{2}O_{7} (*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 *R*PtBi.^{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 *R*_{2}Ir_{2}O_{7}. Nd_{2}Ir_{2}O_{7} 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−x}Pr_{x})_{2}Ir_{2}O_{7}, the antiferromagnetic transition is systematically suppressed and eventually extinguished at around *x* ∼ 0.8.^{17} Pr_{2}Ir_{2}O_{7} 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−x}Pr_{x})_{2}Ir_{2}O_{7}. 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 to*S*_{xy} = 0.7 *μ*V/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 *μ*V/K for high *T*_{c} cuprates Pr_{2−x}Ce_{x}CuO_{4}.^{28} The small Fermi energy or high electron mobility is known to yield large signals, empirically written as $Sxy=\pi 3kBekBT\epsilon F\mu $, 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 cm^{2}/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 $\u223c0.9$ (*μ*V/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-5*d* electrons and the non-coplanar Pr-4*f* 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 Nd_{2}Ir_{2}O_{7} (*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 $\u223c200\mu $V/K, whereas *S*^{ut} takes $\u223c100\mu $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 *μ*V/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 conductivity^{36}

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 *μ*V/K as the magnetic field increases. As the field returns to zero, it becomes ∼100 *μ*V/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 *μ*V/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 *μ*V/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*)/*dμ*_{0}*H* shown in Fig. 3(d). While *d*(*S*_{xy}/*T*)/*dμ*_{0}*H* 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*)/*dμ*_{0}*H*. *d*(*S*_{xy}/*T*)/*dμ*_{0}*H* 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 *μ*V/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 *μ*V/K at 5 T for SrRuO_{3}, 0.2 *μ*V/K for (La, Sr)CoO_{3},^{39} and 0.65 *μ*V/K for Nd_{2}Mo_{2}O_{7}.^{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}_{−}_{x}Pr_{x})_{2}Ir_{2}O_{7}, 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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Kentaro Ueda**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Writing – original draft (lead). **Jun Fujioka**: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). **Naoya Kanazawa**: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). **Yoshinori Tokura**: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.