Interplanar coupling-dependent magnetoresistivity in high-purity layered metals

Abstract

The magnetic field-induced 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 ultra-clean 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 so-called Yamaji angles where the interlayer coupling vanishes. Therefore, it is intrinsically associated with the Fermi points in the field-induced quasi-one-dimensional 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 right-handed chiral (for example, Weyl) fermions. Its observation in PdCoO₂ , PtCoO₂ and Sr₂ RuO₄ suggests that the anomaly affects the transport of clean conductors, in particular near the quantum limit.

ARTICLE

Received 13 Jan 2016 | Accepted 1 Feb 2016 | Published 29 Mar 2016

N. Kikugawa1,2, P. Goswami2,3, A. Kiswandhi2,w, E.S. Choi2, D. Graf2, R.E. Baumbach2, J.S. Brooks2, K. Sugii1,w,Y. Iida1, M. Nishio4, S. Uji1, T. Terashima1, P.M.C. Rourke5,w, N.E. Hussey6,7, H. Takatsu8,9,w, S. Yonezawa9,Y. Maeno9 & L. Balicas2

The magnetic eld-induced 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 ultra-clean 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 elds applied along the so-called Yamaji angles where the interlayer coupling vanishes. Therefore, it is intrinsically associated with the Fermi points in the eld-induced quasi-one-dimensional 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 right-handed chiral (for example, Weyl) fermions. Its observation in PdCoO2, PtCoO2 and Sr2RuO4 suggests that the anomaly affects the transport of clean conductors, in particular near the quantum limit.

1 National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan. 2 Condensed Matter Group, National High Magnetic Field Laboratory, Florida State University, 1800 East Paul Dirac Drive, Tallahassee, Florida 32310, USA. 3 Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742-4111, USA. 4 National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan. 5 H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK. 6 High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525 ED Nijmegen, Nijmegen, The Netherlands. 7 Institute of Molecules and Materials, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.

8 Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan. 9 Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. w Present addresses: Department of Physics, University of Texas at Dallas, Richardson 75080, USA (A.K.); Institute for Solid State

Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan (K.S.); Department of Energy and Hydrocarbon Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan (H. T.); National Research Council, Ottawa, Ontario, Canada K1A 0R6 (P.M.C.R.). Correspondence and requests for materials should be addressed to L.B. (email: mailto:[email protected]

Web End [email protected] ).

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Interplanar coupling-dependent magnetoresistivity in high-purity layered metals

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10903

The magnetoconductivity or -resistivity of metals under a uniform magnetic eld m0H (m0 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 ow relative to the external applied eld H1,2. This is particularly true for high-purity 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 simplications. Despite the inherent difculty in describing the magneto-resistivity of metallic or semi-metallic 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 semi-classical transport theory predicts a magnetoresistivity r(m0H)p(m0H)2, certain compounds such as b-Ag2Te display a linear, non-saturating magnetoresistivity3, which is ascribed to the quantum magnetoresistive scenario4, associated with linearly dispersing Dirac-like bands5. However, in semi-metals characterized by a bulk Dirac dispersion and extremely high electron mobilities such as Cd3As2, the linear magnetoresistivity develops a weak (m0H)2 term as the quality of the sample increases6. Its enormous magnetoresistivity is claimed to result from the suppression of a certain protection against backscattering channels6. The semi-metal WTe2 was also found to display a very large and non-saturating magnetoresistivity, which is p(m0H)2 under elds up to 60 T. This behaviour was ascribed to a nearly perfect compensation between the densities of electrons and holes7. In recent times, a series of compounds were proposed to be candidate Weyl semi-metals characterized by a linear touching between the valence and the conduction bands at several points (Weyl points) of their Brillouin zone8. These Weyl points are predicted to lead to a pronounced negative magnetoresistivity for electric elds aligned along a magnetic eld due to the so-called axial anomaly9,10.

Here we unveil the observation of yet another magnetoresistive effect, namely a pronounced negative magnetoresistivity in extremely clean and non-magnetic layered metals. We study the delafossite-type PtCoO2 and PdCoO2 compounds, which are characterized by a single FS sheet and, as with Cd3As2, can display residual resistivities on the order of a just few tenths of nO cm. Given its extremely low level of disorder, for specic eld orientations along which the interlayer coupling vanishes, PdCoO2 can display a very pronounced positive magnetoresistivity that exceeds 550,000% for m0HC35 T and for currents along the interlayer axis. Nevertheless, as soon as the eld is rotated away from these specic orientations and as the eld increases, this large orbital effect is overwhelmed by the emergence of a pronounced negative magnetoresistivity. For elds along the interlayer direction, a strong longitudinal negative magnetoresistivity is observed from m0H 0 T to elds all the way

up to m0H 35 T. Very similar behaviour is observed in the

PtCoO2 compound. For the correlated Sr2RuO4, 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 eld-induced, quasi-one-dimensional electronic dispersion.

ResultsObservation of an anomalous longitudinal magnetoresistivity. As shown in Fig. 1a, PdCoO2 crystallizes in the space group

R3m D53d

, which results from the stacking of monatomic

triangular layers11. The synthesis of PdCoO2 single crystals is described in the Methods section. According to band structure calculations1214, the Fermi level EF is placed between the lled t2g and the empty eg levels with the Pd triangular planes dominating the conductivity and leading to its highly anisotropic transport properties. The reported room temperature in-plane resistivity is just 2.6 mO cm, making PdCoO2 perhaps the most conductive oxide known to date15. Figure 1b,c show the conguration of contacts used for measuring the longitudinal magnetoresistivity of all compounds. de Haas van Alphen measurements15 reveal a single, corrugated and nearly two-dimensional FS with a rounded hexagonal cross-section, in broad agreement with both band structure calculations1214 and angle-resolved photoemission measurements16. de Haas van Alphen yields an average Fermi wave vector kF

p 9:5 109

m 1or an average Fermi velocity vF kF=m7:6 105 m s 1

(where mC1.5 is the carrier effective mass15 in units of free electron mass). Recent measurements of interplanar magneto-resistivity rc(m0H) reveal an enormous enhancement for elds along the 1 10

direction, that is, increasing by B35,000% at 2 K

under m0H 14 T, which does not follow the characteristic

r(m0H)p(m0H)2 dependence at higher elds17. This behaviour can be reproduced qualitatively by semi-classical calculations, assuming a very small scattering rate17. Most single crystals display in-plane residual resistivities rab0 ranging from only B10 up to B40 nO cm, which correspond to transport lifetimes ttrm=ne2rab0 ranging from \20 down to C5.5 ps (e is the

electron charge and nC2.4 1028 m 3 (ref. 11)) or mean free

paths vFttr ranging from B4 up to 20 mm (ref. 15). However,

according to ref. 15, the quasiparticle lifetime t extracted from the Dingle temperature becomes (in units of length) vFt 0:6 mm.

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 eld along c axis, octtr41 when m0H\1 T; in contrast, oct41 when m0H410 T. These estimations suggest the importance of the Landau quantization for understanding our observations over a wide range of elds up to m0HB30 T.

Pd

2eF=

a b

[11

0]

[110]

[001]

H

O

I

Co

c

V

H

[001]

[11

0]

[110]

V

l

Figure 1 | Crystal structure of PdCoO2 and conguration of electrical contacts. (a) Crystallographic structure of the delafossite PdCoO2 with Pd, Co and O atoms shown in green, blue and red, respectively.

(b) Conguration of contacts for measuring the interplanar longitudinal resistivity (rc), showing concentric contacts at the top and at the bottom surface of each hexagonal platelet-like crystal. (c) Conguration of contacts

for measuring the in-plane longitudinal resistivity r 1 10

for currents

owing along the 1 10

axis and elds applied along the same direction.

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a b c

0.0

H // [110]

Sample #1

H // [110]

126 K 190 K 295 K

c / 0

3,500

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2,500

2,000

1,500

1,000

500

0

H // [001]1.4 K

0H (T) 0H (T)

1.4 K4.2 K10 K20 K30 K40 K55 K69 K88 K

126 K 190 K 295 K

c / 0

4,000

3,000

2,000

1,000

0

1.4 K4.2 K10 K20 K30 K40 K55 K69 K88 K

0.2

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Sample #1

Sample #1

0.6

0

10

20

30

0

10

20

30

0

1

2

3

4

0H / 0 (106T / cm)

Figure 2 | Negative longitudinal and colossal orbital magnetoresistance of PdCoO2. (a) Normalized interplanar magnetoresistivity Drc/r0

(rc(m0H) r0)/r0, where r0 is the resistivity at zero eld, for a PdCoO2 single crystal and as a function of m0H j

k axis at T 1.4 K. The very pronounced

negative longitudinal magnetoresistance arising in the presence of cyclotron motion in the ab plane is noteworthy. (b) Drc(m0H)/r0 as a function of m0H applied along the 1 10

direction and for several temperatures T, describing positive transverse magnetoresistance. At T 1.4 K, Drc surpasses 350,000%

under a eld H 35 T. (c) Kohler scaling of the transverse positive magnetoresistance Drc(m0H). It is noteworthy that (i) all data collapse on a single curve

as a function of m0H/r0 and (ii) at low elds Drc(m0H)/r0p(m0H/r0)2 as expected for classical orbital magnetoresistance.

1.2

1.0

0.8

0.6

0.4

0.2

010 5 0 5 10

a b c

2

2 20 10 0 10 20

Sample #2

= 20.5

18.5

15.5

H // I // c Fit to 1/H

104

6

Sample #2

4

c / 0

c(cm)

2

Sample #3

c / 0

1.0

0.8

0.6

0.4

1.0 0.5 0 0.5 1.0

300 K 200 K 100 K70 K60 K50 K

40 K30 K20 K

10 K4.2 K1.8 K

105

6

13.5

10.5

5.5 0

4

0H (T)

0H (T) 0H / 0 (106 T / cm)

Figure 3 | Anomalous magnetoresistive response of PdCoO2. (a) Interlayer resistivity rc normalized by its zero-eld value r0 as a function of the external eld m0H and for m0H parallel to current I (itself parallel to the sample interlayer c-axis) and for several temperatures T. It is noteworthy that the very pronounced negative magnetoresistivity, that is, rc/r0 decreases by a factor 460% when sweeping the eld from 0 to 5 T. It is also worth noting that this effect disappears when the T approaches and/or surpasses B200 K. (b) rc as a function of m0H from a third crystal at T 1.8 K and for several angles y

between m0H and the c axis. It is noteworthy how the negative magnetoresistivity observed at low elds is progressively suppressed as y increases, becoming strongly positive. Nevertheless, the mechanism leading to the negative magnetoresistivity is observed to overpower the orbital one at higher elds and higher angles. (c) Kohler plot for all the temperature-dependent rc/r0. Red line is a t of Drc/r0 to (m0H) 1.

As shown in Fig. 2a, the low-T magnetoresistivity or Drc (rc r0)/r0, where r0 is the zero-eld interplanar

resistivity, decreases (up to B70%) in a magnetic eld of 30 T oriented parallel to the applied current. Given that PdCoO2 is

non-magnetic 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 effects19,20. To support both statements, in Fig. 2b we show Drc for a PdCoO2 single crystal as a function of H applied along the 1 10

planar direction

and for several temperatures T. In sharp contrast to results shown in Fig. 2a, as T decreases, Drc(m0H) increases considerably, by more than three orders of magnitude when To10 K, thus conrming the absence of scattering by magnetic impurities or any role for weak localization. In addition, it is noteworthy that Drcp(m0H)2 at low elds, which indicates that the interlayer transport is coherent at low elds21. Figure 2c depicts a simple Kohler plot of the magnetoresistivity shown in Fig. 2b, where the eld has also been normalized by r0(T), which indicates unambiguously that the transverse magnetoresistive effect in PdCoO2 is exclusively orbital in character and is dominated by the scattering from impurities/imperfections and phonons1.

The evolution of the longitudinal magnetoresistance with temperature is depicted in Fig. 3a. rc is seen to decrease by a factor surpassing 60% for elds approaching 9 T and for all temperatures below 30 K. Figure 3b displays rc(m0H)/r0 as a

function of the angle y between m0H and the c axis at a temperature T 1.8 K, for a third single crystal. For y410, the

pronounced positive magnetoresistance observed at low elds, due to an orbital magnetoresistive effect, is overpowered at higher elds by the mechanism responsible for the negative magnetoresistivity. This behaviour is no longer observed within this eld range when y is increased beyond B20. Figure 3c shows a

Kohler plot, that is, Drc/r0 as a function of m0H normalized by r0. 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 t to (m0H) 1, indicating that r 1csc pm0H

at lower elds.

Angular dependence of the anomalous magnetoresistive response. Fig. 4 shows the longitudinal magnetoresistance r 1 10

m0H

=r0 for elds and currents along the 1 10

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 Drc/r0, but similar to the longitudinal magnetoresistivity of ultra-clean elemental metals1,2, r 1 10

m0H

=r0 is

observed to increase and saturate as a function of m0H. This

further conrms that conventional mechanisms, for example, impurities, magnetism and so on, are not responsible for the negative longitudinal magnetoresistivity observed in Drc/r0.

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Figure 5a shows rc as a function of the angle y between the eld and the c axis, for three different eld values: 8, 25 and30 T. rc(y) displays the characteristic structure displayed by quasi-two-dimensional metals, namely a series of sharp peaks at

specic angles yn arctanp n 1=4

=ckjjF called the Yamaji

angles (where n is an integer, c is the interplanar distance and kjjF is the projection of the Fermi wave number on the conduction plane), for which all cyclotronic orbits on the FS have an identical orbital area22. In other words, the corrugation of the FS no longer leads to a distribution of cross-sectional 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 non-dispersive at the Yamaji angles18,23; hence, one no longer has Fermi points. The sharp peak at y 90 is

attributed to coherent electron transport along small closed orbits on the sides of a corrugated cylindrical FS24,25. The width of this peak Dy, shown in Fig. 5b for several temperatures, allows us to estimate the interlayer transfer integral tc (ref. 26),

Dy

2kFtcd

EF -tc

DyEF

5

4

3

2

1

0

10 5 0 5 10

1.8 K10 K20 K30 K40 K

50 K70 K 100 K

[110]/ 0

2kFd 1

assuming a simple sinusoidal FS corrugation along the kz 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 Hall-effect measurements (not included here). The full width at half maximum of the peak at 90 is DyC0.78 and EF is given by 2k2F=2m2:32 eV;

therefore, one obtains tc 2.79 meV or C32.4 K. Figure 5c

displays rc as a function of m0H for two angles; the Yamaji

H // I // [1 1 0]

0H (T)

Figure 4 | Longitudinal magnetoresistance for elds along the planes. In-plane longitudinal resistivity r 1 10

normalized by its zero eld value r0 as a function of the eld applied along the 1 10

direction, for a PdCoO2 single

crystal and for several temperatures. The absence of negative magnetoresistivity is noteworthy.

0.03

0.02

0.01

0 100

50

0

50

a b

0.04

0.03

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0.01

0 93

92

91

90

89

88

87

[110]30 T25 T8 T

[001]

Sample #1

1.4 K

4.2 K10 K20 K30 K40 K55 K

[110]

0H = 35 T

c(cm)

c(cm)

(degree) (degree)

69 K88 K 126 K 190 K 295 K

Sample #1

c d

c/ 0

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0

T = 1.8 K

Sample #3

= 21.50 T = 1.8 K

= 23.00

22.70

c/ 0

Sample #3

0H (T)

0H (T)

Figure 5 | Angular magnetoresistance oscillations for a PdCoO2 single crystal. (a) Interplanar magnetoresistivity rc for a PdCoO2 single crystal as a function of the angle y between the [001] interplanar direction and the external eld m0H. The pronounced peaks observed as a function of y are the so-called Yamaji-effect peaks22. (b) Interlayer coherence peak observed for elds nearly along the interplanar direction, which indicates an extended FS along the interlayer direction24. From the width Dy of the peak at half maximum, one can estimate the value of the interlayer transfer integral tc 2.79 meV from

equation (1). (c) Interplanar resistivity rc as a function of m0H at T 1.8 K and for two angles, that is, the Yamaji value yn1 23.0 and y 22.7. It is

noteworthy how the pronounced positive magnetoresistivity observed at yn1 is strongly suppressed when m0H is rotated by just B0.3, leading to

magnetoresistance saturation. (d) rc as a function of m0H under T 1.8 K and for y 21.5. It is noteworthy how rc, after increasing by several orders of

magnitude, displays negative magnetoresistivity at higher elds, thus indicating a clear competition between the orbital and another mechanism, which suppresses the magnetoresistivity. Dotted red line corresponds to a t of rc1=s 1c s0 am0H b=m0H

1.

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angle yn 1 23.0 and y 22.7, respectively. As seen, rc(m0H)

for elds along yn 1 displays a very pronounced positive

magnetoresistance, that is, rc/r0 increases by B550,000% when m0H is swept from 0 to 35 T. However, at m0H 35 T, rc/r0

decreases by one order of magnitude as m0H is rotated by just B0.3 with respect to yn 1. Furthermore, as seen in Fig. 5d, at

higher elds rc displays a cross-over from a very pronounced and positive to a negative magnetoresistance, resulting from a small increment in y relative to yn 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 tc) becomes nite at yayn. We emphasize that for a conventional and very clean metal, composed of a single FS sheet, the magnetoresistivity should either be p(m0H)2 (ref. 21) or saturate as seen in quasi-two-dimensional metals close to the Yamaji angle27, or in

Fig. 2a,b for elds below B15 T. This is illustrated by the Supplementary Fig. 1 (see also Supplementary Note 1), which contrasts our experimental observations with predictions based on semi-classical 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, rc(m0H) can be well described by the expression rc m0H

s 1c s0 am0H b=m0H

1. Here, the

rcp(m0H) 1 term describes the negative magnetoresistivity as previously seen in Fig. 3, whereas the rcpm0H term describes the non-saturating linear magnetoresistance predicted and observed for systems close to the quantum limit35,28. This expression describes rc(m0H, y) satisfactorily, except at the Yamaji angle where both terms vanish. In the neighbourhood of yn, the

addition of a small rcp(m0H)2 term improves the t, with its pre-factor increasing as yn is approached. rc also displays Shubnikov de Haas oscillations at small (and strongly y 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 yn.

Signicantly, this effect does not appear to be conned to PdCoO2. Figure 6 presents an overall evaluation of the long-itudinal magnetoresistance of isostructural PtCoO2, whereas Supplementary Fig. 3 displays the observation of impurity-dependent negative magnetoresistivity in the correlated

perovskite Sr2RuO4 (See also Supplementary Note 3). As shown in Fig. 6, PtCoO2 presents a pronounced negative longitudinal magnetoresistivity either for j H

k k c axis or for m0H close to an

Yamaji angle (j is the current density). It also presents a very pronounced and non-saturating magnetoresistivy for elds applied along the Yamaji angle. For both systems, the magnetoresistivity does not follow a single power law as a function of m0H. In fact, as shown in Supplementary Fig. 4, at yn the magnetoresistivity of the (Pt,Pd)CoO2 system follows a (m0H)2 dependence for m0Ht15 T. At intermediate elds, r(m0H)

deviates from the quadratic dependence, recovering it again at subsequently higher elds. As Kohlers rule implies that Dr/r0p

(m0H/r0)2, we argue that the observed increase in slope would imply a eld-dependent reduction in scattering by impurities (see Supplementary Fig. 4 and Supplementary Note 4). The precise origin of this suppression in scattering remains to be identied. Nevertheless, the enormous and positive magnetoresistivity observed for elds along yn seems consistent with a simple scenario, that is, an extremely clean system(s) whose impurity scattering weakens with increasing magnetic eld. In Sr2RuO4, 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 specic Yamaji angle(s).

DiscussionNegative 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 eld reduces the scattering from the surface. It can also result from the eld-induced suppression of weak localization or from the eld-induced suppression of spin-scattering/quantum-uctuations as seen in f-electron compounds30. None of the compounds described in this study are near a magnetic instability, nor do they contain signicant 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 eld theory, which describes the violation of separate number conservation laws of left- and right-handed massless chiral fermions in odd spatial dimensions due to quantum mechanical effects31,32. When three-dimensional massless Dirac or Weyl fermions are placed under parallel electric and magnetic elds, the number difference between the left and the right-handed fermions is expected to vary with time according to the AdlerBellJackiw formula9,33

@t nR nL

e2EB

6

5

PtCoO2 T =~ 0.35 K

c / 0

4

3

150 = n=1 = 22.8

c / 0

200

100

50

00 10 20 30

2

1

00 10 20 30 0H (T)

= 17

= 0

Figure 6 | Negative longitudinal magnetoresistance in PtCoO2.

Interplanar resistivity rc normalized by its zero-eld value r0 for a PtCoO2 single crystal at a temperature T 0.35 K and as a function of the magnetic

eld m0H applied along two angles with respect to the c axis, respectively y 0 (pink line) and 17 (blue line). Dashed magenta line describes the

smoothly varying background. Inset: rc/r0 as a function of m0H applied along the rst Yamaji angle yn 22.8.

2p2 2 : 2

Here, nR/L are the number operators for the right- and the left-handed Weyl fermions, with the electric and the magnetic eld 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 low-energy band structure

0H (T)

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a

is described by massless Weyl fermions10. 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 QW. 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 nal 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 eld. Consequently the axial anomaly in the presence of parallel E and B elds can give rise to larger conductivity or smaller resistivity i.e., negative magnetoresistance. Recent theoretical proposals for Weyl semi-metals3437 followed by experimental conrmation38,39 have revived the interest in the experimental conrmation of the axial anomaly through efforts in detecting negative longitudinal magnetoresistivity4046. There are examples of three-dimensional Dirac semi-metals4749, which may be converted, through Zeeman splitting, into a Weyl semi-metal. Examples include Bi1 xSbx at the band inversion transition point between

topologically trivial and nontrivial insulators42, and Cd3As2 (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 eld-induced, one-dimensional electronic dispersion18. In effect, in the presence of a strong magnetic eld, the quantization of cyclotron motion leads to discrete Landau levels with one-dimensional dispersion and a degeneracy factor eB/h, see Fig. 7ac. Consider the low-energy description of a one-dimensional electron gas, in terms of the right- and left-handed fermions obtained in the vicinity of the two Fermi points. In the presence of an external electric eld E, the separate number conservation of these chiral fermions is violated according to

@t nR nL

eEp ; 3 where nR/L corresponds to the number operators of the right- and left-handed fermions, respectively31,32. 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 lled, we have the remarkable universal result for the axial anomaly described by AdlerBellJackiw formula of equation (2). For a non-relativistic electron gas, this would occur at the quantum limit. In contrast, this situation would naturally occur for Dirac/Weyl semi-metals, when the Fermi level lies at zero energy, that is, the material has a zero carrier density. Figure 7b describes the situation for a quasi-two-dimensional 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, linear-in-eld magnetoresistive component, as indicated by the t in Fig. 5d, is a strong experimental evidence for the proximity of PdCoO2 to the quantum limit on approaching the Yamaji angle. Therefore, we conclude that the axial anomaly should be present in every three-dimensional 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 impurities18,50. r(m0H)p(m0H) 1 as observed here (Figs 3 and 5) would result from Gaussian impurities18. As our experimental results show, PdCoO2 is a metal of extremely high conductivity, thus necessarily dominated by small-angle 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 :oc :eB/m with respect to the

interlayer transfer integral tc. The condition 4tc4:oc is satised when m0H roughly exceeds 100 T. For this reason, Fig. 7c, with multiple partially occupied Landau levels, describes PdCoO2 for elds along the c axis or for arbitrary angles away from the Yamaji ones. Nevertheless, one can suppress the Fermi points by aligning the eld along an Yamaji angle and this should suppress the associated axial anomaly. As experimentally seen, the suppression of the Fermi points suppresses the

E = (n+1/2) hc+ h2k2z /2m

L2

L1

R1 R2

EF

E(n,k Z)

0

kZ

b

E = (n+1/2) hc+ 2tccos(kzd)

(n+7/2) hc

(n+5/2) hc (n+3/2) hc

(n+1/2) hc

E(n,k Z)

L

R

EF

EF

2/d

kZ

kZ

c

E(n,k Z)

Figure 7 | Field-induced electronic dispersion for metals of different dimensionality. (a) The dispersion of Landau levels for a conventional three-dimensional metal placed in an external magnetic eld applied along the z-direction. Owing to the underlying parabolic dispersion, each Landau level disperses quadratically as a function of kz, the momentum component along the applied eld. Each partially occupied Landau level intersects the

Fermi energy EF at two Fermi points, as indicated by the red dots. In the vicinity of the two Fermi points located at kz kF,n for the n-th partially

lled Landau level, the quasiparticles disperse linearly with opposite group velocities v,n :kF,n/m where m is the effective mass. The signs of

the group velocity respectively dene the chirality of the right- and the left-moving one-dimensional fermions. (b) In contrast, for quasi-two-dimensional metals the Landau levels possess a periodic dispersion relation as a function of kz, owing to the tight binding term 2tc cos(kzd), with interlayer hopping strength and spacing, respectively, given by tc andd. Within the rst Brillouin zone dened as p/dokzop/d, each partially

lled Landau level again gives rise to a pair of one-dimensional fermions of opposite chirality around the Fermi points. The situation depicted here corresponds to 4tco:oc, or when only one Landau level is partially lled.

(c) Landau levels for 4tc4:oc or when multiple Landau levels are partially occupied and each of them gives rise to a pair of chiral fermions.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10903 ARTICLE

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 one-dimensional 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 three-dimensional Dirac/Weyl dispersion. Instead, our study in PdCoO2, PtCoO2 and Sr2RuO4, which are clean metals with no Dirac/Weyl dispersion at zero magnetic eld, 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 scattering18,50. In a generic metal with a high carrier density, it is currently impossible to reach the quantum limit; for the available eld 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)CoO2 are unique, as by just tilting the magnetic eld in the vicinity of the Yamaji angle one can achieve the condition of a single, partially lled 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 lled Landau levels is a technically challenging task. Therefore at present we do not have a simple analytical formula for describing the observed (m0H) 1 behavior of the negative magnetoresistance along the c axis (for magnetic eld strengths much smaller than the one required to reach the quantum limit). Nevertheless, the suppression of this negative magnetoresistivity for elds 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 (m0H) 1 behavior in the magnetoresistance around the Yamaji angle (when only one partially lled Landau level contributes) gives us the valuable insight that the anomaly induced negative magnetoresistance is quite robust irrespective of the number of partially lled Landau levels. However the determination of a precise functional form for the magnetoresistance in the presence of multiple partially lled Landau levels remains as a technical challenge for theorists. The situation is somewhat analogous to that of the Weyl semi-metals, which are characterized by a number of Weyl points in the rst Brillouin zone37, and apparently with all Weyl points contributing to its negative longitudinal magnetoresistivity46. Hence, our results suggest that the axial anomaly among pairs of chiral Fermi points may play a role in ultra-clean 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 elds, which is characterized by ellipsoidal electron- and hole-like FSs, on approaching the quantum limit and before the onset of a many-body instability towards a eld-induced insulating density-wave ground state51. As discussed in ref. 18, the axial anomaly on approaching the quantum limit may also play a role for the negative magnetoresistivities observed in ZrTe5 (ref. 52) and in a (ET)2I3 (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 semi-metals.

Methods

Crystal synthesis. Single crystals of PdCoO2 were grown by the self-ux method through the following reaction PdCl2 2CoO-PdCoO2 CoCl2 with starting

powders of PdCl2 (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 approximately2.8 1.3 0.3 mm3 were extracted by dissolving out CoCl2 with hot ethanol.

Single-crystal characterization. These were characterized by powder X-ray diffraction, energy dispersive X-ray analysis and electron probe microanalysis. The powder X-ray 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 o200 p.p.m.

Experimental setup. Transport measurements were performed by using conventional four-terminal techniques in conjunction with a Physical Properties Measurement System, a 18-T superconducting solenoid and a 35-T resistive magnet, coupled to cryogenic facilities such as 3He systems and variable temperature inserts.

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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 NSF-DMR-1157490 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 HFML-RU/FOM, member of the European Magnetic Field Laboratory (EMFL). Y.M. is supported by the MEXT KAKENHI 15H05852. L.B. is supported by DOE-BES through award DESC0002613.

Author contributions

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 elds. L.B. provided scientic 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 conrm their high degree of purity. P.G., N.H. and L.B. wrote the manuscript with the input of all co-authors.

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How to cite this article: Kikugawa, N. et al. Interplanar coupling-dependent magnetoresistivity in high-purity layered metals. Nat. Commun. 7:10903doi: 10.1038/ncomms10903 (2016).

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