Abstract
The magnetic fieldinduced changes in the conductivity of metals are the subject of intense interest, both for revealing new phenomena and as a valuable tool for determining their Fermi surface. Here we report a hitherto unobserved magnetoresistive effect in ultraclean layered metals, namely a negative longitudinal magnetoresistance that is capable of overcoming their very pronounced orbital one. This effect is correlated with the interlayer coupling disappearing for fields applied along the socalled Yamaji angles where the interlayer coupling vanishes. Therefore, it is intrinsically associated with the Fermi points in the fieldinduced quasionedimensional electronic dispersion, implying that it results from the axial anomaly among these Fermi points. In its original formulation, the anomaly is predicted to violate separate number conservation laws for left and righthanded chiral (for example, Weyl) fermions. Its observation in PdCoO_{2}, PtCoO_{2} and Sr_{2}RuO_{4} suggests that the anomaly affects the transport of clean conductors, in particular near the quantum limit.
Introduction
The magnetoconductivity or resistivity of metals under a uniform magnetic field μ_{0}H (μ_{0} is the permeability of free space) is highly dependent on the precise shape of their Fermi surface (FS) and on the orientation of the current flow relative to the external applied field H^{1,2}. This is particularly true for highpurity metals at low temperatures, whose carriers may execute many cyclotronic orbits in between scattering events. However, the description of the magnetoconductivity of real systems in terms of the Boltzmann equation including the Lorentz force, the electronic dispersion and realistic scattering potentials is an incredibly daunting task, whose approximate solutions can only be obtained through over simplifications. Despite the inherent difficulty in describing the magnetoresistivity of metallic or semimetallic systems, it continues to be a subject of intense interest. Indeed, in recent years, a number of new magnetoresistance phenomena have been uncovered. For example, although semiclassical transport theory predicts a magnetoresistivity ρ(μ_{0}H)∝(μ_{0}H)^{2}, certain compounds such as βAg_{2}Te display a linear, nonsaturating magnetoresistivity^{3}, which is ascribed to the quantum magnetoresistive scenario^{4}, associated with linearly dispersing Diraclike bands^{5}. However, in semimetals characterized by a bulk Dirac dispersion and extremely high electron mobilities such as Cd_{3}As_{2}, the linear magnetoresistivity develops a weak (μ_{0}H)^{2} term as the quality of the sample increases^{6}. Its enormous magnetoresistivity is claimed to result from the suppression of a certain protection against backscattering channels^{6}. The semimetal WTe_{2} was also found to display a very large and nonsaturating magnetoresistivity, which is ∝(μ_{0}H)^{2} under fields up to 60 T. This behaviour was ascribed to a nearly perfect compensation between the densities of electrons and holes^{7}. In recent times, a series of compounds were proposed to be candidate Weyl semimetals characterized by a linear touching between the valence and the conduction bands at several points (Weyl points) of their Brillouin zone^{8}. These Weyl points are predicted to lead to a pronounced negative magnetoresistivity for electric fields aligned along a magnetic field due to the socalled axial anomaly^{9,10}.
Here we unveil the observation of yet another magnetoresistive effect, namely a pronounced negative magnetoresistivity in extremely clean and nonmagnetic layered metals. We study the delafossitetype PtCoO_{2} and PdCoO_{2} compounds, which are characterized by a single FS sheet and, as with Cd_{3}As_{2}, can display residual resistivities on the order of a just few tenths of nΩ cm. Given its extremely low level of disorder, for specific field orientations along which the interlayer coupling vanishes, PdCoO_{2} can display a very pronounced positive magnetoresistivity that exceeds 550,000% for μ_{0}H≃35 T and for currents along the interlayer axis. Nevertheless, as soon as the field is rotated away from these specific orientations and as the field increases, this large orbital effect is overwhelmed by the emergence of a pronounced negative magnetoresistivity. For fields along the interlayer direction, a strong longitudinal negative magnetoresistivity is observed from μ_{0}H=0 T to fields all the way up to μ_{0}H=35 T. Very similar behaviour is observed in the PtCoO_{2} compound. For the correlated Sr_{2}RuO_{4}, the longitudinal negative magnetoresistivity effect is also observable but only in the cleanest samples, that is, those displaying the highest superconducting transition temperatures. We suggest that this effect might result from the axial anomaly between Fermi points in a fieldinduced, quasionedimensional electronic dispersion.
Results
Observation of an anomalous longitudinal magnetoresistivity
As shown in Fig. 1a, PdCoO_{2} crystallizes in the space group , which results from the stacking of monatomic triangular layers^{11}. The synthesis of PdCoO_{2} single crystals is described in the Methods section. According to band structure calculations^{12,13,14}, the Fermi level E_{F} is placed between the filled t_{2g} and the empty e_{g} levels with the Pd triangular planes dominating the conductivity and leading to its highly anisotropic transport properties. The reported room temperature inplane resistivity is just 2.6 μΩ cm, making PdCoO_{2} perhaps the most conductive oxide known to date^{15}. Figure 1b,c show the configuration of contacts used for measuring the longitudinal magnetoresistivity of all compounds. de Haas van Alphen measurements^{15} reveal a single, corrugated and nearly twodimensional FS with a rounded hexagonal crosssection, in broad agreement with both band structure calculations^{12,13,14} and angleresolved photoemission measurements^{16}. de Haas van Alphen yields an average Fermi wave vector m^{−1}or an average Fermi velocity m s^{−1} (where μ≃1.5 is the carrier effective mass^{15} in units of free electron mass). Recent measurements of interplanar magnetoresistivity ρ_{c}(μ_{0}H) reveal an enormous enhancement for fields along the direction, that is, increasing by ∼35,000% at 2 K under μ_{0}H=14 T, which does not follow the characteristic ρ(μ_{0}H)∝(μ_{0}H)^{2} dependence at higher fields^{17}. This behaviour can be reproduced qualitatively by semiclassical calculations, assuming a very small scattering rate^{17}. Most single crystals display inplane residual resistivities ρ_{ab0} ranging from only ∼10 up to ∼40 nΩ cm, which correspond to transport lifetimes ranging from \underset{~}{>} 20 down to ≃5.5 ps (e is the electron charge and n≃2.4 × 10^{28} m^{−3} (ref. 11)) or mean free paths ranging from ∼4 up to 20 μm (ref. 15). However, according to ref. 15, the quasiparticle lifetime extracted from the Dingle temperature becomes (in units of length) μm. Hence, the transport lifetime is larger than the quasiparticle lifetime by at least one order of magnitude, which is the hallmark of a predominant forward scattering mechanism (see ref. 18). For a magnetic field along c axis, when μ_{0}H\underset{~}{>}1 T; in contrast, when μ_{0}H>10 T. These estimations suggest the importance of the Landau quantization for understanding our observations over a wide range of fields up to μ_{0}H∼30 T.
As shown in Fig. 2a, the lowT magnetoresistivity or Δρ_{c}=(ρ_{c}−ρ_{0})/ρ_{0}, where ρ_{0} is the zerofield interplanar resistivity, decreases (up to ∼70%) in a magnetic field of 30 T oriented parallel to the applied current. Given that PdCoO_{2} is nonmagnetic and extremely clean (see Methods), this effect cannot be attributed to magnetic impurities. In addition, the magnitude of the observed magnetoresistivity cannot be explained in terms of weak localization effects^{19,20}. To support both statements, in Fig. 2b we show Δρ_{c} for a PdCoO_{2} single crystal as a function of H applied along the planar direction and for several temperatures T. In sharp contrast to results shown in Fig. 2a, as T decreases, Δρ_{c}(μ_{0}H) increases considerably, by more than three orders of magnitude when T<10 K, thus confirming the absence of scattering by magnetic impurities or any role for weak localization. In addition, it is noteworthy that Δρ_{c}∝(μ_{0}H)^{2} at low fields, which indicates that the interlayer transport is coherent at low fields^{21}. Figure 2c depicts a simple Kohler plot of the magnetoresistivity shown in Fig. 2b, where the field has also been normalized by ρ_{0}(T), which indicates unambiguously that the transverse magnetoresistive effect in PdCoO_{2} is exclusively orbital in character and is dominated by the scattering from impurities/imperfections and phonons^{1}.
The evolution of the longitudinal magnetoresistance with temperature is depicted in Fig. 3a. ρ_{c} is seen to decrease by a factor surpassing 60% for fields approaching 9 T and for all temperatures below 30 K. Figure 3b displays ρ_{c}(μ_{0}H)/ρ_{0} as a function of the angle θ between μ_{0}H and the c axis at a temperature T=1.8 K, for a third single crystal. For θ>10°, the pronounced positive magnetoresistance observed at low fields, due to an orbital magnetoresistive effect, is overpowered at higher fields by the mechanism responsible for the negative magnetoresistivity. This behaviour is no longer observed within this field range when θ is increased beyond ∼20°. Figure 3c shows a Kohler plot, that is, Δρ_{c}/ρ_{0} as a function of μ_{0}H normalized by ρ_{0}. As seen in Fig. 3c, all curves collapse on a single curve, indicating that a particular transport mechanism dominates even at high temperatures where phonon scattering is expected to be strong. The red line is a fit to (μ_{0}H)^{−1}, indicating that at lower fields.
Angular dependence of the anomalous magnetoresistive response
Fig. 4 shows the longitudinal magnetoresistance for fields and currents along the axis. For this orientation, the charge carriers follow open orbits along the axis of the cylindrical FS instead of quantized cyclotronic orbits. In contrast to Δρ_{c}/ρ_{0}, but similar to the longitudinal magnetoresistivity of ultraclean elemental metals^{1,2}, is observed to increase and saturate as a function of μ_{0}H. This further confirms that conventional mechanisms, for example, impurities, magnetism and so on, are not responsible for the negative longitudinal magnetoresistivity observed in Δρ_{c}/ρ_{0}.
Figure 5a shows ρ_{c} as a function of the angle θ between the field and the c axis, for three different field values: 8, 25 and 30 T. ρ_{c}(θ) displays the characteristic structure displayed by quasitwodimensional metals, namely a series of sharp peaks at specific angles called the Yamaji angles (where n is an integer, c is the interplanar distance and is the projection of the Fermi wave number on the conduction plane), for which all cyclotronic orbits on the FS have an identical orbital area^{22}. In other words, the corrugation of the FS no longer leads to a distribution of crosssectional areas, as if the corrugation has been effectively suppressed. As discussed below, in terms of the energy spectrum, this means that the Landau levels become nondispersive at the Yamaji angles^{18,23}; hence, one no longer has Fermi points. The sharp peak at θ=90° is attributed to coherent electron transport along small closed orbits on the sides of a corrugated cylindrical FS^{24,25}. The width of this peak Δθ, shown in Fig. 5b for several temperatures, allows us to estimate the interlayer transfer integral t_{c} (ref. 26),
assuming a simple sinusoidal FS corrugation along the k_{z} direction. Here, the interplanar separation is d=c/3, as there are three conducting Pd planes per unit cell, each providing one conducting hole and therefore leading to three carriers per unit cell. This value is consistent with our Halleffect measurements (not included here). The full width at half maximum of the peak at 90° is Δθ≃0.78° and E_{F} is given by eV; therefore, one obtains t_{c}=2.79 meV or ≃32.4 K. Figure 5c displays ρ_{c} as a function of μ_{0}H for two angles; the Yamaji angle θ_{n=1}=23.0° and θ=22.7°, respectively. As seen, ρ_{c}(μ_{0}H) for fields along θ_{n=1} displays a very pronounced positive magnetoresistance, that is, ρ_{c}/ρ_{0} increases by ∼550,000% when μ_{0}H is swept from 0 to 35 T. However, at μ_{0}H=35 T, ρ_{c}/ρ_{0} decreases by one order of magnitude as μ_{0}H is rotated by just ∼0.3° with respect to θ_{n=1}. Furthermore, as seen in Fig. 5d, at higher fields ρ_{c} displays a crossover from a very pronounced and positive to a negative magnetoresistance, resulting from a small increment in θ relative to θ_{n=1}. This is a very clear indication for two competing mechanisms, with negative magnetoresistivity overcoming the orbital effect when the orbitally averaged interlayer group velocity (or the transfer integral t_{c}) becomes finite at θ≠θ_{n}. We emphasize that for a conventional and very clean metal, composed of a single FS sheet, the magnetoresistivity should either be ∝(μ_{0}H)^{2} (ref. 21) or saturate as seen in quasitwodimensional metals close to the Yamaji angle^{27}, or in Fig. 2a,b for fields below ∼15 T. This is illustrated by the Supplementary Fig. 1 (see also Supplementary Note 1), which contrasts our experimental observations with predictions based on semiclassical transport models, which correctly describe the magnetoresistance of layered organic metals in the vicinity of the Yamaji angle. In contrast, as illustrated by the dotted red line in Fig. 5d, ρ_{c}(μ_{0}H) can be well described by the expression . Here, the ρ_{c}∝(μ_{0}H)^{−1} term describes the negative magnetoresistivity as previously seen in Fig. 3, whereas the ρ_{c}∝μ_{0}H term describes the nonsaturating linear magnetoresistance predicted and observed for systems close to the quantum limit^{3,4,5,28}. This expression describes ρ_{c}(μ_{0}H, θ) satisfactorily, except at the Yamaji angle where both terms vanish. In the neighbourhood of θ_{n}, the addition of a small ρ_{c}∝(μ_{0}H)^{2} term improves the fit, with its prefactor increasing as θ_{n} is approached. ρ_{c} also displays Shubnikov de Haas oscillations at small (and strongly θ dependent) frequencies, which were not previously detected in ref. 15. As discussed in ref. 29, these slow oscillations, observed only in the interlayer magnetoresistance of layered metals, originate from the warping of the FS. In Supplementary Fig. 2 (See also Supplementary Note 2), we show how these frequencies disappear when the group velocity vanishes at θ_{n}.
Significantly, this effect does not appear to be confined to PdCoO_{2}. Figure 6 presents an overall evaluation of the longitudinal magnetoresistance of isostructural PtCoO_{2}, whereas Supplementary Fig. 3 displays the observation of impuritydependent negative magnetoresistivity in the correlated perovskite Sr_{2}RuO_{4} (See also Supplementary Note 3). As shown in Fig. 6, PtCoO_{2} presents a pronounced negative longitudinal magnetoresistivity either for c axis or for μ_{0}H close to an Yamaji angle (j is the current density). It also presents a very pronounced and nonsaturating magnetoresistivy for fields applied along the Yamaji angle. For both systems, the magnetoresistivity does not follow a single power law as a function of μ_{0}H. In fact, as shown in Supplementary Fig. 4, at θ_{n} the magnetoresistivity of the (Pt,Pd)CoO_{2} system follows a (μ_{0}H)^{2} dependence for μ_{0}H≲15 T. At intermediate fields, ρ(μ_{0}H) deviates from the quadratic dependence, recovering it again at subsequently higher fields. As Kohler’s rule implies that Δρ/ρ_{0}∝(μ_{0}H/ρ_{0})^{2}, we argue that the observed increase in slope would imply a fielddependent reduction in scattering by impurities (see Supplementary Fig. 4 and Supplementary Note 4). The precise origin of this suppression in scattering remains to be identified. Nevertheless, the enormous and positive magnetoresistivity observed for fields along θ_{n} seems consistent with a simple scenario, that is, an extremely clean system(s) whose impurity scattering weakens with increasing magnetic field. In Sr_{2}RuO_{4}, the negative longitudinal magnetoresistivity is observed only in the cleanest samples and for angles within 10° away from the c axis. This compound is characterized by three corrugated cylindrical FS sheets, each leading to a distinct set of Yamaji angles, making it impossible to completely suppress the interplanar coupling at specific Yamaji angle(s).
Discussion
Negative magnetoresistivity is a common feature of ferromagnetic metals near their Curie temperature, or of samples having dimensions comparable to their electronic mean free path where the winding of the electronic orbits under a magnetic field reduces the scattering from the surface. It can also result from the fieldinduced suppression of weak localization or from the fieldinduced suppression of spinscattering/quantumfluctuations as seen in felectron compounds^{30}. None of the compounds described in this study are near a magnetic instability, nor do they contain significant amounts of magnetic impurities or disorder to make them prone to weak localization. The magnitude of this anomalous magnetoresistivity, coupled to its peculiar angular dependence, are in fact enough evidence against any of these conventional mechanisms. Below, we discuss an alternative scenario based on the axial anomaly, which in our opinion explains most of our observations.
The axial anomaly is a fundamental concept of relativistic quantum field theory, which describes the violation of separate number conservation laws of left and righthanded massless chiral fermions in odd spatial dimensions due to quantum mechanical effects^{31,32}. When threedimensional massless Dirac or Weyl fermions are placed under parallel electric and magnetic fields, the number difference between the left and the righthanded fermions is expected to vary with time according to the Adler–Bell–Jackiw formula^{9,33}
Here, n_{R/L} are the number operators for the right and the lefthanded Weyl fermions, with the electric and the magnetic field strengths respectively given by E and B. The Dirac fermion describes the linear touching of twofold Kramers degenerate conduction and valence bands at isolated momentum points in the Brillouin zone. By contrast, the Weyl fermions arise due to the linear touching between nondegenerate conduction and valence bands. The axial anomaly was initially proposed to produce a large, negative longitudinal magnetoresistance, for a class of gapless semiconductors, for which the lowenergy band structure is described by massless Weyl fermions^{10}. The reason for the negative magnetoresistance is relatively straightforward. The number imbalance due to axial anomaly can only be equilibrated through backscattering between two Weyl points. This involves a large momentum transfer Q_{W}. Quite generally the impurity scattering in a material can be modeled by a momentum dependent impurity potential V(Q), where Q is the momentum transfer between the initial and the final electronic states. If V(Q) is a smoothly decreasing function of Q (such as Gaussian or Lorentzian), the backscattering amplitude can be considerably smaller than its forward scattering counterparts (occurring with small Q around each Weyl point). Therefore in the presence of axial anomaly the transport lifetime can be considerably larger than the one in the absence of a magnetic field. Consequently the axial anomaly in the presence of parallel E and B fields can give rise to larger conductivity or smaller resistivity i.e., negative magnetoresistance. Recent theoretical proposals for Weyl semimetals^{34,35,36,37} followed by experimental confirmation^{38,39} have revived the interest in the experimental confirmation of the axial anomaly through efforts in detecting negative longitudinal magnetoresistivity^{40,41,42,43,44,45,46}. There are examples of threedimensional Dirac semimetals^{47,48,49}, which may be converted, through Zeeman splitting, into a Weyl semimetal. Examples include Bi_{1−x}Sb_{x} at the band inversion transition point between topologically trivial and nontrivial insulators^{42}, and Cd_{3}As_{2} (ref. 6).
In analogy with the predictions for the axial anomaly between Weyl points, here we suggest that our observations might be consistent with the emergence of the axial anomaly among the Fermi points of a fieldinduced, onedimensional electronic dispersion^{18}. In effect, in the presence of a strong magnetic field, the quantization of cyclotron motion leads to discrete Landau levels with onedimensional dispersion and a degeneracy factor eB/h, see Fig. 7a–c. Consider the lowenergy description of a onedimensional electron gas, in terms of the right and lefthanded fermions obtained in the vicinity of the two Fermi points. In the presence of an external electric field E, the separate number conservation of these chiral fermions is violated according to
where n_{R/L} corresponds to the number operators of the right and lefthanded fermions, respectively^{31,32}. Each partially occupied Landau level leads to a set of Fermi points and the axial anomaly for such a level can be obtained from equation 3, after multiplying by eB/h. Therefore, each level has an axial anomaly determined by equation (2). When only one Landau level is partially filled, we have the remarkable universal result for the axial anomaly described by Adler–Bell–Jackiw formula of equation (2). For a nonrelativistic electron gas, this would occur at the quantum limit. In contrast, this situation would naturally occur for Dirac/Weyl semimetals, when the Fermi level lies at zero energy, that is, the material has a zero carrier density. Figure 7b describes the situation for a quasitwodimensional electronic system on approaching the quantum limit, or when the interplanar coupling becomes considerably smaller than the inter Landau level separation (for example, in the vicinity of the Yamaji angle). We emphasize that the observation of a pronounced, linearinfield magnetoresistive component, as indicated by the fit in Fig. 5d, is a strong experimental evidence for the proximity of PdCoO_{2} to the quantum limit on approaching the Yamaji angle. Therefore, we conclude that the axial anomaly should be present in every threedimensional conducting system, on approaching the quantum limit. Explicit calculations indicate that the axial anomaly would only cause negative magnetoresistance for predominant forward scattering produced by ionic impurities^{18,50}. ρ(μ_{0}H)∝(μ_{0}H)^{−1} as observed here (Figs 3 and 5) would result from Gaussian impurities^{18}. As our experimental results show, PdCoO_{2} is a metal of extremely high conductivity, thus necessarily dominated by smallangle scattering processes and therefore satisfying the forward scattering criterion. In this metal the Landau levels disperse periodically as shown in Fig. 7b,c, depending on the relative strength of the cyclotron energy ħω_{c}=ħeB/μ with respect to the interlayer transfer integral t_{c}. The condition 4t_{c}>ħω_{c} is satisfied when μ_{0}H roughly exceeds 100 T. For this reason, Fig. 7c, with multiple partially occupied Landau levels, describes PdCoO_{2} for fields along the c axis or for arbitrary angles away from the Yamaji ones. Nevertheless, one can suppress the Fermi points by aligning the field along an Yamaji angle and this should suppress the associated axial anomaly. As experimentally seen, the suppression of the Fermi points suppresses the negative magnetoresistivity, indicating that the axial anomaly is responsible for it.
In summary, in very clean layered metals we have uncovered a very clear correlation between the existence of Fermi points in a onedimensional dispersion and the observation of an anomalous negative magnetoresistivity. The suppression of these points leads to the disappearance of this effect. This indicates that the axial anomaly and related negative magnetoresistivity would not be contingent on the existence of an underlying threedimensional Dirac/Weyl dispersion. Instead, our study in PdCoO_{2}, PtCoO_{2} and Sr_{2}RuO_{4}, which are clean metals with no Dirac/Weyl dispersion at zero magnetic field, indicates that the axial anomaly and its effects could be a generic feature of metal(s) near the quantum limit. Nevertheless, the detection of negative magnetoresistivity would depend on the underlying scattering mechanisms, that is, observable only in those compounds that are clean enough to be dominated by elastic forward scattering^{18,50}. In a generic metal with a high carrier density, it is currently impossible to reach the quantum limit; for the available field strength, many Landau levels would be populated, thus producing a myriad of Fermi points. In this regard, extremely pure layered metals such as (Pd,Pt)CoO_{2} are unique, as by just tilting the magnetic field in the vicinity of the Yamaji angle one can achieve the condition of a single, partially filled Landau level as it would happen at the quantum limit. An explicit analytical calculation of transport lifetime in the presence of axial anomaly due to multiple partially filled Landau levels is a technically challenging task. Therefore at present we do not have a simple analytical formula for describing the observed (μ_{0}H)^{−1} behavior of the negative magnetoresistance along the c axis (for magnetic field strengths much smaller than the one required to reach the quantum limit). Nevertheless, the suppression of this negative magnetoresistivity for fields precisely aligned along the Yamaji angles indicates unambiguously that the electronic structure at the Fermi level is at the basis for its underlying mechanism. The observation of (μ_{0}H)^{−1} behavior in the magnetoresistance around the Yamaji angle (when only one partially filled Landau level contributes) gives us the valuable insight that the anomaly induced negative magnetoresistance is quite robust irrespective of the number of partially filled Landau levels. However the determination of a precise functional form for the magnetoresistance in the presence of multiple partially filled Landau levels remains as a technical challenge for theorists. The situation is somewhat analogous to that of the Weyl semimetals, which are characterized by a number of Weyl points in the first Brillouin zone^{37}, and apparently with all Weyl points contributing to its negative longitudinal magnetoresistivity^{46}. Hence, our results suggest that the axial anomaly among pairs of chiral Fermi points may play a role in ultraclean systems even when they are located far from the quantum limit.
Finally, it is noteworthy that negative longitudinal magnetoresistivity is also seen in kish graphite at high fields, which is characterized by ellipsoidal electron and holelike FSs, on approaching the quantum limit and before the onset of a manybody instability towards a fieldinduced insulating densitywave ground state^{51}. As discussed in ref. 18, the axial anomaly on approaching the quantum limit may also play a role for the negative magnetoresistivities observed in ZrTe_{5} (ref. 52) and in α−(ET)_{2}I_{3} (ref. 53), indicating that this concept, which is the basis of our work, is likely to be relevant to a number of physical systems, in particular semimetals.
Methods
Crystal synthesis
Single crystals of PdCoO_{2} were grown by the selfflux method through the following reaction PdCl_{2}+2CoO→PdCoO_{2}+CoCl_{2} with starting powders of PdCl_{2} (99.999%) and CoO (99.99+%). These powders were ground for for up to 60 min and placed in a quartz tube. The tube was sealed in vacuum and heated up to 930 °C in a horizontal furnace within 2 h and subsequently up to 1,000 °C within 6 h, and then cooled down quickly to 580 °C in 1 or 2 h. The tube is heated up again to 700 °C within 2 h, kept at 700 °C for 40 h and then cooled down to room temperature at 40 °C h^{−1}. Single crystals, with sizes of approximately 2.8 × 1.3 × 0.3 mm^{3} were extracted by dissolving out CoCl_{2} with hot ethanol.
Singlecrystal characterization
These were characterized by powder Xray diffraction, energy dispersive Xray analysis and electron probe microanalysis. The powder Xray diffraction pattern indicated no impurity phases. In the crystals measured for this study, electron probe microanalysis indicated that the ratio of Pd to Co is 0.98:1, and that the amount of Cl impurities is <200 p.p.m.
Experimental setup
Transport measurements were performed by using conventional fourterminal techniques in conjunction with a Physical Properties Measurement System, a 18T superconducting solenoid and a 35T resistive magnet, coupled to cryogenic facilities such as ^{3}He systems and variable temperature inserts.
Additional information
How to cite this article: Kikugawa, N. et al. Interplanar couplingdependent magnetoresistivity in highpurity layered metals. Nat. Commun. 7:10903 doi: 10.1038/ncomms10903 (2016).
References
Pippard, A. B. Magnetoresistance in Metals: Cambridge Studies in Low Temperature Physics 2 Cambridge Univ. Press (1989).
Pippard, A. B. Longitudinal magnetoresistance. Proc. R. Soc. A A282, 464–484 (1964).
Lee, M., Rosenbaum, T. F., Saboungi, M. L. & Schnyders, H. S. Bandgap tuning and linear magnetoresistance in the silver chalcogenides. Phys. Rev. Lett. 88, 066602 (2002).
Abrikosov, A. A. Quantum linear magnetoresistance. Europhys. Lett. 49, 789793 (2000).
Zhang, W. et al. Topological aspect and quantum magnetoresistance of βAg2Te. Phys. Rev. Lett. 106, 156808 (2011).
Liang, T. et al. Ultrahigh mobility and giant magnetoresistance in the Dirac semimetal Cd3As2 . Nat. Mater. 14, 280–284 (2015).
Ali, M. N. et al. Large, nonsaturating magnetoresistance in WTe2 . Nature 514, 205–208 (2014).
Huang, S.M. et al. Theoretical discovery/prediction: Weyl semimetal states in the TaAs material (TaAs, NbAs, NbP, TaP) class. Nat. Commun. 6, 7373 (2015).
Bell, J. S. & Jackiw, R. A PCAC puzzle: π0→γγ in the σmodel. Nuovo Cimento A 60, 47–61 (1969).
Nielsen, H. B. & Ninomiya, M. The AdlerBellJackiw anomaly and weyl fermions in a crystal. Phys. Lett. B 130B, 389–396 (1983).
Takatsu, H. et al. Roles of highfrequency optical phonons in the physical properties of the conductive delafossite PdCoO2 . J. Phys. Soc. Jpn 76, 104701 (2007).
Eyert, V., Frésard, R. & Maignan, A. On the metallic conductivity of the delafossites PdCoO2 and PtCoO2 . Chem. Mater. 20, 2370–2373 (2008).
Seshadri, R., Felser, C., Thieme, K. & Tremel, W. Metalmetal bonding and metallic behavior in some ABO2 delafossites. Chem. Mater. 10, 2189–2196 (1998).
Kim, K., Choi, H. C. & Min, B. I. Fermi surface and surface electronic structure of delafossite PdCoO2 . Phys. Rev. B 80, 035116 (2009).
Hicks, C. W. et al. Quantum oscillations and high carrier mobility in the delafossite PdCoO2 . Phys. Rev. Lett. 109, 11640 (2012).
Noh, H. J. et al. Anisotropic electric conductivity of delafossite PdCoO2 studied by angleresolved photoemission spectroscopy. Phys. Rev. Lett. 102, 256404 (2009).
Takatsu, H. et al. Extremely large magnetoresistance in the nonmagnetic metal PdCoO2 . Phys. Rev. Lett. 111, 056601 (2013).
Goswami, P., Pixley, J. & Das Sarma, S. Axial anomaly and longitudinal magnetoresistance of a generic three dimensional metal. Phys. Rev. B 92, 075205 (2015).
Hikami, S., Larkin, A. I. & Nagaoka, Y. Spinorbit interaction and magnetoresistance in the twodimensional random system. Prog. Theor. Phys. 63, 707–710 (1980).
Bergmann, G. Weak localization in thin films a timeofflight experiment with conduction electrons. Phys. Rep. 107, 1–58 (1984).
Moses, P. & Mackenzie, R. H. Comparison of coherent and weakly incoherent transport models for the interlayer magnetoresistance of layered Fermi liquids. Phys. Rev. B 60, 7998 (1999).
Yamaji, K. On the angle dependence of the magnetoresistance in quasitwodimensional organic superconductors. J. Phys. Soc. Jpn 58, 1520–1523 (1989).
Kurihara, Y. A microscopic calculation of the angulardependent oscillatory magnetoresistance in quasitwodimensional systems. J. Phys. Soc. Jpn 61, 975–982 (1992).
Singleton, J. et al. Test for interlayer coherence in a quasitwodimensional superconductor. Phys. Rev. Lett. 88, 037001 (2002).
Hanasaki, H., Kagoshima, S., Hasegawa, T., Osada, T. & Miura, N. Contribution of small closed orbits to magnetoresistance in quasitwodimensional conductors. Phys. Rev. B 57, 1336–1339 (1998).
Uji, S. et al. Fermi surface and angulardependent magnetoresistance in the organic conductor (BEDTTTF)2Br(DIA). Phys. Rev. B 68, 064420 (2003).
Yagi, R., Iye, Y., Osada, T. & Kagoshima, S. Semiclassical interpretation of the angulardependent oscillatory magnetoresistance in quasitwodimensional systems. J. Phys. Soc. Jpn 59, 3069–3072 (1990).
Hu, J. & Rosenbaum, T. F. Classical and quantum routes to linear magnetoresistance. Nat. Mater. 7, 697–700 (2008).
Kartsovnik, M. V., Grigoriev, P. D., Biberacher, W., Kushch, N. D. & Wyder, P. Slow oscillations of magnetoresistance in quasitwodimensional metals. Phys. Rev. Lett. 89, 126802 (2002).
Zeng, B. et al. CeCu2Ge2: challenging our understanding of quantum criticality. Phys. Rev. B 90, 155101 (2014).
Peskin, M. E. & Schroeder, D. V. An Introduction to Quantum Field Theory AddisonWesley (1995).
Fujikawa, K. & Suzuki, H. Path Integrals and Quantum Anomalies Clarendon Press (2004).
Adler, S. Axialvector vertex in spinor electrodynamics. Phys. Rev. 177, 2426–2438 (1969).
Wan, X., Turner, A., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermiarc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
Xu, G., Weng, H., Wang, Z., Dai, X. & Fang, Z. Chern semimetal and the quantized anomalous Hall effect in HgCr2Se4 . Phys. Rev. Lett. 107, 186806 (2011).
Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).
Weng, H. M. et al. Weyl semimetal phase in noncentrosymmetric transitionmetal monophosphides. Phys. Rev. X 5, 011029 (2015).
Lv, B. Q. et al. Observation of Weyl nodes in TaAs. Nat. Phys. 11, 724–727 (2015).
Yang, L. X. et al. Weyl semimetal phase in the noncentrosymmetric compound TaAs. Nat. Phys. 11, 728–732 (2015).
Aji, V. AdlerBellJackiw anomaly in Weyl semimetals: application to pyrochlore iridates. Phys. Rev. B 85, 241101 (2012).
Son, D. T. & Spivak, B. Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88, 104412 (2013).
Kim, H.J. et al. Dirac versus Weyl fermions in topological insulators: AdlerBellJackiw anomaly in transport phenomena. Phys. Rev. Lett. 111, 246603 (2013).
Parameswaran, S. A., Grover, T., Abanin, D. A., Pesin, D. A. & Vishwanath, A. Probing the chiral anomaly with nonlocal transport in threedimensional topological semimetals. Phys. Rev. X 4, 031035 (2014).
Burkov, A. A. Chiral anomaly and diffusive magnetotransport in Weyl metals. Phys. Rev. Lett. 113, 247203 (2014).
Kim, K.S., Kim, H.J. & Sasaki, M. Boltzmann equation approach to anomalous transport in a Weyl metal. Phys. Rev. B 89, 195137 (2014).
Huang, X. C. et al. Observation of the chiralanomalyinduced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015).
Liu, Z. K. et al. Discovery of a threedimensional topological Dirac semimetal Na3Bi. Science 343, 864–867 (2014).
Neupane, M. et al. Observation of a threedimensional topological Dirac semimetal phase in highmobility Cd3As2 . Nat. Commun. 5, 3786 (2014).
Borisenko, S. et al. Experimental realization of a threedimensional Dirac semimetal. Phys. Rev. Lett. 113, 027603 (2014).
Argyres, P. N. & Adams, E. N. Longitudinal magnetoresistance in the quantum limit. Phys. Rev. 104, 900–908 (1956).
Fauqué, B. et al. Two phase transitions induced by a magnetic field in graphite. Phys. Rev. Lett. 110, 266601 (2013).
Li, Q. et al. Chiral magnetic effect in ZrTe5. Nat. Phys. (in the press).
Tajima, N., Sugawara, S., Kato, R., Nishio, Y. & Kajita, K. Effects of the zeromode landau level on interlayer magnetoresistance in multilayer massless Dirac fermion systems. Phys. Rev. Lett. 102, 176403 (2009).
Acknowledgements
We thank S. Das Sarma, V. Yakovenko, L. Balents, E. Abrahams and J. Pixley for useful discussions. The NHMFL is supported by NSF through NSFDMR1157490 and the State of Florida. N.K. acknowledges the support from the overseas researcher dispatch program at NIMS. P.M.C.R. and N.E.H. acknowledge the support of the HFMLRU/FOM, member of the European Magnetic Field Laboratory (EMFL). Y.M. is supported by the MEXT KAKENHI 15H05852. L.B. is supported by DOEBES through award DESC0002613.
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N.K. performed the measurements and analysed the data. A.K., E.S.C., D.G., R.B., J.S.B., S.U., K.S., T.T., P.M.C.R. and N.E.H. contributed to the collection of experimental data at high magnetic fields. L.B. provided scientific guidance and P.G. the theoretical interpretation. H.T., S.Y. and Y.M. synthesized and characterized the single crystals. Y.I. and M.N. performed electron probe microanalysis of the measured single crystals, to confirm their high degree of purity. P.G., N.H. and L.B. wrote the manuscript with the input of all coauthors.
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Supplementary Figures 14, Supplementary Notes 14 and Supplementary References (PDF 582 kb)
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Kikugawa, N., Goswami, P., Kiswandhi, A. et al. Interplanar couplingdependent magnetoresistivity in highpurity layered metals. Nat Commun 7, 10903 (2016). https://doi.org/10.1038/ncomms10903
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DOI: https://doi.org/10.1038/ncomms10903
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