Strain-induced spontaneous Hall effect in an epitaxial thin film of a Luttinger semimetal

Significance

Weyl semimetals carry the promise of quantum electronic applications. Theoretical calculations have suggested that Luttinger semimetals recently found in iridium oxides may be a suitable group of materials where such topological phases, including Weyl semimetal state, may be found. For example, praseodymium-iridium oxide, ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$, is one such Luttinger semimetal that can be tuned by perturbations such as strain into a Weyl semimetal state. Despite theoretical predictions of Weyl semimetal states in ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ crystals, experimental proof remains elusive due to the difficulty of applying sufficient mechanical strain on single crystals. Our study provides strong experimental evidence that a Weyl semimetal state may indeed appear in strained pyrochlore iridium oxide films, opening a way to explore topological phases.

Abstract

Pyrochlore iridates have provided a plethora of novel phenomena owing to the combination of topology and correlation. Among them, much attention has been paid to ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$, as it is known as a Luttinger semimetal characterized by quadratic band touching at the Brillouin zone center, suggesting that the topology of its electronic states can be tuned by a moderate lattice strain and external magnetic field. Here, we report that our epitaxial ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films grown by solid-state epitaxy exhibit a spontaneous Hall effect that persists up to 50 K without having spontaneous magnetization within our experimental accuracy. This indicates that the system breaks the time reversal symmetry at a temperature scale that is too high for the magnetism to be due to Pr 4$f$ moments and must be related to magnetic order of the iridium 5$d$ electrons. Moreover, our analysis finds that the chiral anomaly induces the negative contribution to the magnetoresistance only when a magnetic field and the electric current are parallel to each other. Our results indicate that the strained part of the thin film forms a magnetic Weyl semimetal state.

In the field of condensed matter physics, there has been an intense search to find topologically nontrivial electronic phases in strongly correlated materials. To date, most research on topological electronic systems has been limited to weakly correlated materials where electronic correlations play a minor role. Among newly found topological phases, the Weyl semimetal phase has attracted the most attention. While a number of noncentrosymmetric materials have been identified as nonmagnetic Weyl semimetals (1, 2), only a few have been found for the magnetic version, including the antiferromagnetic metal ${\mathrm{M}\mathrm{n}}_3$Sn, which exhibits a large anomalous Hall effect (AHE) (3, 4). Historically, a magnetic Weyl system is the first Weyl semimetal predicted for condensed matter in 2012 by the seminal paper by Wan et al. (5). Notably, this prediction was specifically made for the pyrochlore iridates.

To search for novel topological phases in strongly correlated electron systems, the $5d$ electron systems such as iridates are particularly well suited because the strength of the Coulomb interaction and the spin–orbit coupling are of similar orders of magnitude (6), hypothetically giving rise to various types of topological phases (5, 7, 8). On the other hand, the topologically nontrivial electronic phases often exhibit novel transport phenomena. In particular, the noncollinear and/or noncoplanar spin textures in the geometrically frustrated magnets may host a large Berry curvature in the momentum space without magnetization, leading to a topological or unconventional AHE (3, 9–12).

Pyrochlore iridates $R_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ (where $R$ is a lanthanoid or yttrium) are interesting from both points of view, having the iridium $5d$ electron system and noncoplanar spin texture due to the geometrically frustrated magnetism in both $5d$ and $4f$ electron sectors (6, 9, 10, 13–22). While the ground state changes from a metal to an insulator as the $R^{3+}$ ionic radius decreases (6, 22), a thermal metal–insulator transition occurs as a function of temperature in almost all compounds in this series, except for $R$ = Pr, which has the largest ionic radius: ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ remains metallic down to the lowest temperatures (18). Unconventional magnetotransport is observed due to the coupling between the $5d$ conduction electrons and the frustrated $4f$ moments on the pyrochlore lattice (9, 10, 18, 19). As schematically indicated in Fig. 1A, the Pr $4f$ moments are located at the vertices of the pyrochlore lattice with an Ising anisotropy along the $⟨111⟩$ direction. Interestingly, a spin liquid state is found down to the lowest measurement temperature because a ferromagnetic coupling ($J\sim 0.7$ K) between the Ising $4f$ moments leads to a strongly frustrated “spin ice” correlation with a local 2-in–2-out structure (10, 18). Furthermore, the system shows a spontaneous Hall effect, i.e., AHE without spontaneous magnetization or external magnetic field between 0.3 K and 1.5 K (9, 10). The AHE is topological, originating from the spin chirality generated by a 2-in–2-out noncoplanar spin ice configuration below the spin ice correlation scale of $2J\sim 1.5$ K. This spin configuration is consistent with the observed anisotropic magnetotransport, i.e., the nonzero remnant Hall resistivity that reaches the maximum when a magnetic field is applied along the [111] direction (19), indicating that the cubic symmetry should be broken by the scalar chirality order of $4f$ moments in the spin liquid state.

Fig. 1.

Characters of pyrochlore $R_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ focusing on $R$ = Pr. (A) Unit cell of $R_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$. $R$ and Ir are drawn in blue and red, respectively, and each site forms a network of corner-sharing tetrahedra. Oxygen is omitted for simplification. (B) Magnetic and electronic structures of ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$: 2-in–2-out state on the Pr tetrahedra (Left); quadratic band touching of $5d$ electrons, which forms a Fermi node at the $\mathrm{\Gamma }$ point near $E_{\mathrm{F}}$ (Right).

The observation of the large spontaneous Hall effect without magnetization further indicates a nontrivial mechanism to induce large Berry curvature in the momentum space due to the time-reversal symmetry (TRS) breaking. Following the experimental observations of the Hall effect, ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ has been proposed and later confirmed to be a Luttinger semimetal, where a Fermi node is formed at the $\mathrm{\Gamma }$ point near the Fermi level ($E_{\mathrm{F}}$) by quadratic band touching of the doubly degenerate valence and conduction bands, as illustrated in Fig. 1B (20, 21). In comparison with Dirac and Weyl semimetals, which have the crossing of linearly dispersive bands, the Luttinger semimetal may have much stronger correlation effects, leading to non-Fermi liquid states and strong enhancement in the dielectric constant. In fact, recent optical conductivity measurements have found that the latter is indeed the case for the thin film that we discuss in this paper (23). In the stoichiometric ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ system, the charge neutrality locates $E_{\mathrm{F}}$ exactly at the node, but any off-stoichiometry may dope electrons (holes) in the conduction (valence) band. Hypothetically, this nodal state in ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ is sensitive to perturbations and may host various topological phases (20, 21, 24–27). For example, the Luttinger semimetal can be converted to a topological insulator or a Weyl metal by breaking the cubic symmetry and/or the TRS (20, 21). In contrast to vigorous theoretical studies, experimental observations of the topological phases have remained elusive.

Recent extensive studies on ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ have focused on bulk samples. In contrast, thin films offer a suitable platform for additional control over the crystal growth orientation and lattice deformation by compressive or tensile strain imposed by epitaxial lattice mismatch with a substrate. In this work, we demonstrate that (111)-oriented pyrochlore ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films can be epitaxially grown on yttria-stabilized zirconia (YSZ) (111) substrates. Given that the lattice constant of bulk ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ [10.394 Å (18)] is larger than that of YSZ ($2a=10.278$ Å), an epitaxial thin film can be compressively strained biaxially in the in-plane direction. We discuss that the resultant tensile strain along the surface normal [111] direction in fact induces a topologically nontrivial phase.

Results and Discussion

Crystal Structure Analysis.

Fig. 2A shows X-ray diffraction (XRD) patterns of as-grown and postannealed Pr-Ir-O/YSZ(111) films. The annealing procedure after deposition crystallizes the as-deposited amorphous film, leading to the formation of epitaxial ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ with the (111) orientation. The odd-numbered peaks confirm the pyrochlore phase as a superstructure of the fluorite structure. Additional characterization of the epitaxial growth and lattice relaxation is described in SI Appendix, Fig. S1.

Fig. 2.

Crystal structure analysis of ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$/YSZ(111). (A) $\theta -2\theta$ XRD patterns of as-grown (bottom of graph) and post-annealed (top of graph) Pr-Ir-O thin film on a YSZ(111) substrate. Thin film peaks are highlighted in blue. (B) Schematic illustration of STEM sampling and observation direction. A microwedge sample was cut from the central part of the Hall bar by focused ion beam milling. (C) Cross-sectional crystal structure of the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$/YSZ(111) interface (46). Pr, Ir, and Zr atoms are drawn in blue, red, and green, respectively. Oxygen is omitted. The model is slightly tilted around the $\left[11\overline{2}\right]$ direction to show the atomic arrangement in the depth direction. (D and E) Cross-sectional HAADF-STEM images of a single grain in the film, taken near the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ surface and the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$/YSZ interface, respectively. Pr, Ir, and Zr atom positions are shown in the Insets for the regions marked with the orange outlines. The colors correspond to those used in C. Yellow represents that Pr and Ir atoms are alternately arranged in the depth direction. Crystal axes are shown in D. (Scale bars: 3 nm.)

While reciprocal space analysis indicates that the epitaxial film is mostly relaxed, our microscopic cross-sectional imaging of individual grains in the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$/YSZ(111) film by high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) finds that some grains in the film are indeed strained. Fig. 2 B and C shows the STEM observation direction and a lattice model viewed along the STEM electron beam direction. The Pr and Ir atoms in the film and the Zr atoms in the substrate are responsible for the periodic contrast in Fig. 2 D and E since the intensity of a column of atoms in a HAADF-STEM image is proportional to the average atomic number in that column. This effect can be seen in the film part of the STEM images, where ⁷⁷Ir (red) columns are brighter than the ⁵⁹Pr (blue) columns. The contrast between the two columns in the STEM image shows that the atomic arrangement in the film matches the atomic arrangement expected for an ordered pyrochlore lattice illustrated in Fig. 2C. Insets in Fig. 2 D and E show diagonal parallel lines connecting neighboring atoms. At the interface, each line in the film continuously extends into the substrate, showing that the atom row spacings at the interface and at the grain surface are identical and there are no misfit dislocations at the interface in the STEM imaging area. The STEM image thus proves that some grains in the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ film are locked to the YSZ substrate, as is also suggested by the peak shoulder in the XRD reciprocal map (SI Appendix, Fig. S1). Such locked grains are coherently grown from the interface to their grain surfaces. The lattice mismatch of the film to YSZ is +1.15% and thus produces a tensile strain of 2.31% along the [111] direction when the film lattice is in-plane locked to YSZ.

Temperature Dependence of the Longitudinal Resistivity.

The temperature dependence of the longitudinal resistivity of a ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin film measured in zero magnetic field (${\rho }_{xx}\left(0\hspace{0.17em}\mathrm{T}\right)$) shows the metallic conductivity similarly to the bulk ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ (18) (Fig. 3A). Namely, it decreases on cooling from room temperature and shows an upturn after forming a minimum at $\sim 50$ K. The carrier density at 2 K is estimated to be 1.75$\times 10^{20}$ cm⁻³, suggesting that a slight off-stoichiometry causes hole doping in the valence band and shifts $E_{\mathrm{F}}$ by 18 meV (SI Appendix, Fig. S2). The carrier density is one order of magnitude smaller than the reported value for the single crystals (9), causing the larger resistivity than the value of the single crystals. On the other hand, the Hall mobility of our thin film is 12 ${\mathrm{c}\mathrm{m}}^2$/Vs (consistent with the mobility of 20–30 ${\mathrm{c}\mathrm{m}}^2$/Vs derived from the terahertz spectroscopy; SI Appendix, Fig. S3) and has the same order of magnitude as the one ($\sim$10 ${\mathrm{c}\mathrm{m}}^2$/Vs) for the single crystal (9), indicating the high quality of the film. A close look at the lowest temperature region in Fig. 3B shows that there is a small suppression below about 700 mK. This anomaly can be attributed to Q = (001) antiferromagnetic order of the Pr $4f$ moments (28).

Fig. 3.

Temperature dependence of the longitudinal resistivity of a ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin film. (A) Plot over a wide temperature range, measured at zero magnetic field. Inset shows a schematic illustration of the Hall bar geometry. The current $I$ flows along the $\left[1\overline{1}0\right]$ direction. (B) Enlargement of A below 1 K.

Hall Effect Measurements.

A striking feature is found in the magnetic field ($B$) dependence of the Hall resistivity (${\rho }_{xy}\left(B\right)$) shown in Fig. 4 A and B (SI Appendix, Fig. S4 for the data above 2 K). Similar to the bulk case, ${\rho }_{xy}\left(B\right)$ exhibits a hysteresis loop near-zero field. Significantly, however, a nonzero spontaneous Hall resistivity at zero field, ${\rho }_{xy}\left(0\right)$, persists up to $T_{\mathrm{H}}\sim 50$ K, much higher than the onset temperature ($T_{\mathrm{H}}\sim 1.5$ K) for the bulk. In addition, $T_{\mathrm{H}}\sim 50$ K coincides with the ${\rho }_{xx}\left(0\hspace{0.17em}\mathrm{T}\right)$ minimum temperature, below which the spontaneous Hall resistivity increases down to 700 mK (Fig. 4C). A slight reduction is seen below 700 mK most likely due to the Q = (001) order that sets in at around 700 mK.

Fig. 4.

Hall effect of a ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin film. (A and B) Magnetic field ($B$) dependence of the Hall resistivity (${\rho }_{xy}\left(B\right)$), measured at 1,037 and 9 mK, respectively. ${\rho }_{xy}\left(B\right)$ is defined as $\left[{\rho }_{xy}\left(B\right)-{\rho }_{xy}\left(-B\right)\right]/2$ to eliminate the ${\rho }_{xx}$ component. Insets show the magnified plot for the low field hysteresis part of ${\rho }_{xy}\left(B\right)$. Red and blue lines in A and B correspond to up and down sweeps of $B$, respectively. (C) Temperature dependence of the spontaneous Hall resistivity, i.e., the absolute value of ${\rho }_{xy}\left(B\right)$ at $B=0$ obtained after a field cycle. The orange open diamond indicates the temperature at which ${\rho }_{xx}\left(0\hspace{0.17em}\mathrm{T}\right)$ reaches a minimum. Spin configurations labeled in C are indicated for Pr $4f$ moments. $B$ is applied along the [111] direction in all measurements.

In contrast, no hysteresis was found in the magnetization curves within the experimental accuracy of $\sim$0.001 ${\mu }_{\mathrm{B}}$, as shown in SI Appendix, Fig. S5. Thus, in sharp contrast with the conventional AHE due to ferromagnetism, our ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin film exhibits a spontaneous Hall effect without magnetic field and at zero magnetization within our experimental accuracy. This indicates that the AHE must originate from the TRS breaking due to a noncoplanar or noncollinear spin texture in either a spin liquid state or antiferromagnetic order, similarly to bulk ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ (10). Interestingly, the onset temperature scale, $T_{\mathrm{H}}\sim 50$ K, is too high compared with the exchange coupling ($J\sim 0.7$ K) of Pr $4f$ moments (10). Thus, the spontaneous AHE should come from the magnetic order formed by Ir $5d$ electrons. In fact, the possible appearance of a magnetic order has been theoretically predicted for pyrochlore iridate thin films (29).

Possibility of the Weyl Semimetal Phase.

Next, we discuss the origin of the spontaneous Hall effect and its possible relation to a Weyl semimetal phase. Theoretically, in a pyrochlore iridate with a quadratic band touching such as ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$, the TRS must be broken for a magnetic Weyl semimetal phase to appear (5, 21). In addition, the breaking of the cubic symmetry is necessary for the Weyl semimetal phase to exhibit the AHE (30). The observation of the spontaneous Hall effect provides strong evidence for the macroscopic TRS breaking. As we discussed, the onset temperature scale ($T_{\mathrm{H}}\sim 50$ K) of the spontaneous Hall resistance is too high for Pr moments to form a long-range magnetic order or a chiral spin liquid phase. Therefore, it is most likely that the Ir $5d$ electrons are responsible for the TRS breaking. Recent calculations suggest that thin films may show a magnetic order near the surface or at an interface as the nearest neighbors are lost and the band becomes narrower (29). The most likely spin configuration is all-in–all-out, as this is the only spin order reported to date for pyrochlore iridates (31–34), and ${\mathrm{N}\mathrm{d}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$, which has a slightly smaller bandwidth than ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$, shows this type of spin order (31). Additionally, it is known that the all-in–all-out state of the Ir $5d$ moments is only weakly perturbed by hydrostatic pressure (34) and thus expected to be preserved under the lattice strain. As we mentioned above, for the system to exhibit the AHE, further breaking of the cubic symmetry is required (30). We thus argue that the strained parts of the film satisfy both preconditions for the magnetic Weyl semimetal phase to appear and moreover to exhibit the spontaneous Hall effect, i.e., the all-in–all-out spin configuration and strain along the [111] direction (21, 30).

As was discussed above, another magnetically ordered state is observed below 700 mK. This most likely comes from the Q = (001) order of Pr $4f$ moments, as seen in bulk samples (28). Since this state does not carry a net magnetization and does not macroscopically break TRS, an increase in the spontaneous Hall effect would not be expected. Moreover, this ordered state of Pr $4f$ moments is stabilized by ferromagnetic (RKKY type) coupling mediated through the nonmagnetic Ir $5d$ bands and thus should originate in the relaxed part of the film and not from the strained grains where antiferromagnetic correlation stabilizes the all-in–all-out order in $5d$ moments. However, as the relaxed part and the strained part are connected physically, the two interactions may compete to suppress the spontaneous Hall effect. This would be the reason why a slight suppression of the spontaneous Hall resistivity is found below 700 mK.

Negative Magnetoresistance Component Due to the Chiral Anomaly.

Our experimental results are consistent with the appearance of the magnetic Weyl semimetal phase in the strained part of the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ film. On the other hand, the majority of the thin film is relaxed and should have the Luttinger semimetal state. A simplified model estimates that the relaxed part should host the magnetic Weyl semimetal state under the magnetic field at least below $\sim$ 15 K (SI Appendix, Fig. S6). However, it is actually stabilized up to a much higher temperature of $\sim$70 K, as shown below. To confirm the existence of the field-induced Weyl semimetal state, here we examine the effects of the chiral anomaly using the longitudinal resistivity (35). Namely, the charge imbalance between the Weyl fermions with different chirality causes a negative (longitudinal) magnetoresistance (MR) through back scattering only when the magnetic field is parallel to the electric current, while the field perpendicular to the current leads to a positive (transverse) MR.

As shown in Fig. 5, we have performed the MR measurements at temperatures higher than 2 K where both $4f$ and $5d$ electrons in the relaxed part of the thin film have the paramagnetic states. In particular, as illustrated in Fig. 5A, we examine the longitudinal and transverse MR under the field along the axis crystallographically equivalent to [110], namely, respectively under $B\parallel \left[1\overline{1}0\right]$ and $B\parallel \left[\overline{1}\overline{1}0\right]$ with the fixed current direction along $\left[1\overline{1}0\right]$ (SI Appendix, Fig. S7). In both configurations, the uniform negative MR of about 5% is observed at 9 T, consistent with the previous work on the bulk ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ (19). The negative MR appears because with increasing the magnetic field, the macroscopic degeneracy of the spin ice 2-in–2-out configuration of the Pr moment is gradually lifted and the scattering by the fluctuating local Pr moments is suppressed. Very interestingly, MR curves measured at 2 K shown in Fig. 5B indicate that the negative MR is larger in the longitudinal case due to the additional negative response.

Fig. 5.

Additional contribution to the negative longitudinal MR for a ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin film. (A) Experimental configurations. For the longitudinal ($B\parallel I$) and transverse ($B\perp I$) cases, $B$ is applied along the $\left[1\overline{1}0\right]$ and $\left[\overline{1}\overline{1}0\right]$ direction, respectively. The current flows along the $\left[1\overline{1}0\right]$ direction in common. (B) MR curves as a function of $B$ measured at 2 K for $B\parallel I$ and $B\perp I$ configurations. The solid and dotted lines correspond to up and down sweeps of B, respectively. (C) Field dependence of the difference of the conductivity $\mathrm{\Delta }\sigma$ evaluated from the two MR curves in B. The fitting result with a quadratic function ($\mathrm{\Delta }\sigma =\gamma B^2$) is shown by a green curve. Only the up sweep of $B$ is shown. (D) Temperature dependence of $\gamma$ (black open circles) obtained by the fitting in C and SI Appendix, Fig. S10. Temperature dependence of mobility squared (${\mu }^2$), which is scaled to the value of $\gamma$ at 70 K, is also plotted by a blue curve. The original temperature dependence of $\mu$ is the one labeled with “average” in SI Appendix, Fig. S3. The red curve is a fitting result of the temperature dependence of $\gamma$ with $1/T^2$ function. Inset plots ${\gamma }^{-1}$ vs. $T^2$ (black solid circles) and shows the linear regression (an orange line, the correlation coefficient of 0.99), which verifies the $1/T^2$ dependence of $\gamma$.

Here, the MR data are analyzed assuming that the conductivity is composed of Ohmic ($\mathrm{\Delta }{\sigma }_{\mathrm{O}\mathrm{h}\mathrm{m}}$) and chiral anomaly ($\mathrm{\Delta }{\sigma }_{\mathrm{C}\mathrm{h}}$) contributions. $\mathrm{\Delta }{\sigma }_{\mathrm{O}\mathrm{h}\mathrm{m}}$ is expressed as $-\alpha B^2$ ($\alpha >0$) as it comes from the Lorentz force, and the transverse MR curves (SI Appendix, Fig. S8) indicate that the contribution of the Lorentz force is very small, specifically, less than 0.1% at 70 K. Regarding to $\mathrm{\Delta }{\sigma }_{\mathrm{C}\mathrm{h}}$, it is shown to have a quadratic dependence on $B$ (36): $\mathrm{\Delta }{\sigma }_{\mathrm{C}\mathrm{h}}=\beta B^2$ ($\beta >0$). These terms produces the corresponding component in both the longitudinal magnetoconductivity (${\sigma }_{\parallel }$) and the transverse magnetoconductivity (${\sigma }_{\perp }$). Namely, for the longitudinal case, $\mathrm{\Delta }{\sigma }_{\parallel }=\beta B^2$ because the Lorentz force is free. In contrast, in the transverse case, the corresponding component is $\mathrm{\Delta }{\sigma }_{\perp }=-\alpha B^2$ because $\mathrm{\Delta }{\sigma }_{\mathrm{C}\mathrm{h}}$ vanishes. Eventually, the difference of the conductivities between the longitudinal and transverse cases ($\mathrm{\Delta }\sigma ={\sigma }_{\parallel }-{\sigma }_{\perp }$) should be derived as $\mathrm{\Delta }\sigma =\left(\beta +\alpha \right)B^2\equiv \gamma B^2$. Note that the negative MR contribution owing to the field suppression of the spin fluctuations should be included in both ${\sigma }_{\parallel }$ and ${\sigma }_{\perp }$. However, the spin fluctuation term is canceled out in $\mathrm{\Delta }\sigma$ because it should not be dependent on the relative angle between $I$ and $B$. Supplementarily, while the negative longitudinal MR could be caused by inhomogeneous current distribution in the sample known as the current jetting effects (37, 38), such extrinsic contributions can be excluded by complementary experiments as shown in SI Appendix, Fig. S9.

Next, we compare the longitudinal and transverse MR obtained under a magnetic field along the equivalent crystallographic axis and estimate their difference $\mathrm{\Delta }\sigma$, as derived above. Fig. 5C shows the field dependence of $\mathrm{\Delta }\sigma$ evaluated from the two MR curves in Fig. 5B. As expected for the above, $\mathrm{\Delta }\sigma$ can be well fit by a quadratic function of $B$ at low fields. Similar comparisons of MR curves and the fitting analyses of $\mathrm{\Delta }\sigma$ at temperatures above 2 K are displayed in SI Appendix, Figs. S8 and S10, respectively. The temperature dependence of the fitting coefficient $\gamma$ ($\equiv \beta +\alpha$) is plotted in Fig. 5D, and $\gamma$ rapidly increases on cooling. Meanwhile, $\alpha$ describing the contribution of the Lorentz force should be proportional to the mobility squared (${\mu }^2$) in the low-magnetic-field region ($\alpha \propto {\mu }^2$). The temperature dependence of ${\mu }^2$ of ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films is derived by analyzing the experimental results of terahertz spectroscopy, as shown in SI Appendix, Fig. S3. By scaling the temperature dependence of ${\mu }^2$ to the value of $\gamma$ to a high-temperature value, for example, at 70 K, the temperature dependence of ${\mu }^2$ is plotted in Fig. 5D, revealing its very weak temperature dependence in sharp contrast to the one of $\gamma$. This indicates that the rapid increase of $\gamma$ on cooling should be governed by $\beta$, namely, nonzero ${\sigma }_{\mathrm{c}\mathrm{h}}$. Significantly, the temperature dependence of $1/\gamma$ is well fit to a linear function of $T^2$ over an extended region of temperature from 2 K to 70 K (Fig. 5D, Inset). Given that this $T^2$ dependence of ${\gamma }^{-1}$ is consistent with the theoretical expectation based on the chiral anomaly (39, 40), $\gamma$ should have the dominant contribution from the chiral anomaly at least up to 70 K. Thus, it is indeed the Weyl semimetal phase that induces the chiral anomaly in the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films and leads to the additional contribution to the negative longitudinal MR over an extended region of temperature between 2 and 70 K.

Planar Hall Effect.

As discussed above, the negative longitudinal MR provides the evidence for the chiral anomaly in the Weyl semimetals. On the other hand, recently, the planar Hall effect (PHE) has been theoretically predicted to arise from the chiral anomaly (41, 42) and experimentally demonstrated (43, 44) in Dirac and Weyl semimetals. For the measurements of the PHE, both the magnetic and the electric fields are applied in the same plane, as illustrated in Fig. 6A, which is different from the conventional Hall geometry. The planar Hall resistivity ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$ and the longitudinal anisotropic magnetoresistivity ${\rho }_{xx}$ in Dirac and Weyl semimetals are formulated as (41, 42).

$${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}=-\mathrm{\Delta }{\rho }_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}\sin \phi \cos \phi ,$$ [1]

$${\rho }_{xx}={\rho }_{\perp }-\mathrm{\Delta }{\rho }_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}{\cos }^2\phi .$$ [2]

Here, $\mathrm{\Delta }{\rho }_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}={\rho }_{\perp }-{\rho }_{\parallel }$ is the chiral resistivity originating from chiral anomaly, where ${\rho }_{\perp }$ and ${\rho }_{\parallel }$ are the resistivity when the electric and magnetic fields are perpendicular and parallel, respectively. $\phi$ is the angle between the electric and magnetic fields.

Fig. 6.

PHE in a ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin film. (A) A schematic illustration of the experimental geometry. The current $I$ flows along the $\left[1\overline{1}0\right]$ direction, and the magnetic field $B$ is applied in the sample plane. The angle between $I$ and $B$ ($\phi$) is changed by the in-plane rotation of the sample. (B and C) Angle ($\phi$) dependence of ${\rho }_{xx}$ and ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$, respectively, measured at 70 K and 9 T. The red and blue open circles are the experimental data, and the black solid lines are the fitting results using Eqs. 1 and 2 in the main text. As for the experimental data of ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$, the contribution of the normal Hall effect, which should come from the misalignment of the sample, is subtracted by taking the average of ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$ measured at positive and negative magnetic fields.

Fig. 6 B and C shows the angle ($\phi$) dependence of ${\rho }_{xx}$ and ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$ measured at 9 T. As Eqs. 1 and 2 are valid for a nonmagnetic/paramagnetic Dirac or Weyl semimetal state, the measurements were made at 70 K in the paramagnetic phase above the on-set temperature $T_{\mathrm{H}}=50$ K of the spontaneous Hall effect. It is also important to note that, as we discussed above, the negative MR due to the chiral anomaly is observed at 70 K at least up to 9 T, as shown in SI Appendix, Fig. S10. Both the measured ${\rho }_{xx}$ and ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$ show ${\mathrm{180}}^{○}$ periodicity and are well fit by Eqs. 1 and 2. $\mathrm{\Delta }{\rho }_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}$ estimated from the angular dependence of ${\rho }_{xx}$ and ${\rho }^{\mathrm{P}\mathrm{H}\mathrm{E}}$ is $\sim 0.1\hspace{0.17em}\mu \mathrm{\Omega }$ cm. Significantly, this is consistent with another estimate of $\mathrm{\Delta }{\rho }_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}$ based on the MR measurement, namely, $\mathrm{\Delta }{\rho }_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{l}}=\mathrm{\Delta }\sigma {\rho }_{\perp }{\rho }_{\parallel }\sim \mathrm{\Delta }\sigma {\left[{\rho }_{xx}\left(\mathrm{0}\hspace{0.17em}\mathrm{T}\right)\right]}^2\sim 0.1\hspace{0.17em}\mu \mathrm{\Omega }$ cm under 9 T at 70 K (SI Appendix, Fig. S10K). These results demonstrate the PHE and the anisotropic MR, as predicted by theory (41, 42), supporting the existence of the Weyl semimetal phase that induces the chiral anomaly under external magnetic field in the ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films. Incidentally, an anomalous planar Hall signal, which is odd upon the field reversal, has been reported in ${\mathrm{Z}\mathrm{r}\mathrm{T}\mathrm{e}}_5$ (45). It would be interesting to explore the possible anomalous signal in our ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films to further examine the nontrivial character of the Berry phase in the strained state.

Summary

To summarize, our first success in growing high-quality epitaxial thin film of ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ has allowed us to make the detailed transport measurements in the Luttinger semimetal state and to find strong evidence for the chiral anomaly by the negative longitudinal MR and the PHE, indicating that the Luttinger semimetal can be transformed into the Weyl semimetal state in a magnetic field. Furthermore, our observation of the spontaneous Hall effect below 50 K indicates the spontaneous formation of the Weyl semimatal state under zero field due to the combination of the magnetic order and the cubic symmetry breaking due to strain applied by the locking of the thin film to the substrate. The future device application of the high-quality thin film opens new avenues for the study of the novel phases associated with the Luttinger semimetal. For example, tuning the Fermi energy to the touching point by field-effect transistor would allow us to study the formation of the non-Fermi liquid phase and its instability.

Materials and Methods

Pyrochlore ${\mathrm{P}\mathrm{r}}_2{\mathrm{I}\mathrm{r}}_2{\mathrm{O}}_7$ thin films were fabricated on YSZ(111) single-crystal substrates using pulsed-laser deposition at room temperature, followed by solid-phase epitaxy. The film thickness was 100 nm, as measured with a stylus profilometer. The crystal structures of the samples were analyzed by XRD. Hall bars for transport measurements were fabricated by mechanical diamond milling. The transport measurements were done in a physical property measurement system (PPMS) and a top-loading dilution refrigerator (Kelvinox) above and below 1 K, respectively. Magnetization curves were obtained by using a superconducting quantum interference device magnetometer (MPMS). The details of the sample preparation and the characterization are described in SI Appendix.