Interplay between strong correlations and electronic topology in the underlying kagome lattice of Na2/3CoO2

Electronic topology in metallic kagome compounds is under intense scrutiny. We present transport experiments in Na2/3CoO2 in which the Na order differentiates a Co kagome sub-lattice in the triangular CoO2 layers. Hall and magnetoresistance (MR) data under high fields give evidence for the coexistence of light and heavy carriers. At low temperatures, the dominant light carrier conductivity at zero field is suppressed by a B-linear MR suggesting Dirac like quasiparticles. Lifshitz transitions induced at large B and T unveil the lower mobility carriers. They display a negative B MR due to scattering from magnetic moments likely pertaining to a flat band. We underline an analogy with heavy Fermion physics. Introduction: The influence of frustration of exchange on the magnetic properties of transition metal compounds has been investigated thoroughly in spin structures. The Herbertsmithite compound, whose Cu sites are ordered in a two-dimensional kagome structure, is considered as a good reference for a Quantum Spin Liquid, as no spin ordering has been detected in its ground state (1). Attempts to synthesize doped metallic states in this kagome system by chemical substitutions have been so far unsuccessful (2). Those were motivated by a search for superconductivity in the phase diagram but also by an analogy with the honeycomb lattice such as that of graphene in which Dirac points are located (3). Local Density Approximation (LDA) calculations for a single orbital per site in weakly correlated kagome lattices also show the presence of Dirac or Weyl points as well as flat electronic bands (4). One wonders how such topological states would evolve in the presence of strong correlations. This led intensive searches for correlated kagome compounds e.g. in 3d stannite materials like FeSn (5), CoSn (6) or Co3Sn2S2 (7). Here, we present an alternative approach based on the Na cobaltate compounds NaxCoO2 where the Co atoms are ordered on a triangular lattice. The originality of this system has been revealed by earlier NMR/NQR experiments. It was shown that the Na located between the CoO2 layers (Fig. 1a) displays distinct orderings depending on the Na content (8) (9). The electrostatic incidence of the Na ionic order induces a charge disproportionation of the Co sites (10). This has been evidenced in great detail in the case of the x=2/3 phase (11) (12) (13) in which a subset of Co sites remain in a Co state with filled non-magnetic t2g orbitals (Fig. 1b) while the complementary set of Co sites are ordered in a kagome sub-lattice having delocalized charge carriers (Fig. 1d). LDA calculations indicate that the Na order yields the minimum energy state for this compound (14), while LDA+U computations (15) demonstrate that a large coulomb interaction U is required to induce the disproportionation of the Co sites. Fig. 1. (a) Structure of an isolated CoO2 layer. (b) Splitting of the Co 3d levels induced by the crystal field in NaxCoO2 for x=1. (c) Differentiation for x=2/3 of the Co sites induced by the ordered stacking of Na above and below the CoO2 layer. (d) 2D arrangement of the two types of sites in the Co plane (11). The Co NMR shift data gave evidence that Co1a and Co1b (yellow and brown) are non-magnetic Co sites with filled t2g levels as in (b). The 3d orbitals of the Co2a and Co2b sites (light and dark blue) arranged in a kagome sub-lattice are both nearly identically involved at the Fermi level(10) (16). (e) The Na NMR shift K (left scale) monitors the spin susceptibility on those sites (associated with the hole doping of the correlated electronic bands of their kagome structure). It displays a Curie-Weiss like T dependence (17) below 30 K concomitantly with the anomalous increase in magnitude of the Hall constant RH (right scale) (18). Early experiments performed on samples with Na content near x=2/3 have indeed revealed singularly large values of the magnetic susceptibility (17) (19), specific heat(20) (21) and of T dependence of the resistivity (22) which are obvious signs of strong electronic correlations. NMR shift measurements on the various Na and Co nuclear sites have given evidence that the local spin susceptibility  of the Co2a and Co2b sites forming the kagome sub-lattice displays a large increase below 30 K (Fig. 1e), with respect to phases with a different Na content. Meanwhile, SR experiments do not provide evidence for static magnetic order down to 0.1 K(23). Hence the local  reaches a constant value only below T=1.5 K(17). We have recently synthesized(24) high quality single crystals of the Na ordered Na2/3CoO2 phase with large residual resistivity ratios RRR=R(300 K)/R(1.5 K)~200. A change in the sign of the Hall-effect at 200 K was found to be followed below 30 K by a reproducible and unexpected large increase(18) in its negative magnitude (Fig. 1e). These modifications of the electronic properties indicate that the reconstruction of the Fermi Surface (FS) which occurs already above 200 K is followed by a large increase in electronic carrier mobility below 30 K. Those results therefore underscore the need to perform detailed low T band structure (BS) and FS studies on this specific kagome lattice material. As quantum oscillations (QOs) were discovered in uncontrolled quality samples(25) we have done high field measurements on our high-quality single crystals. We did not find QOs immediately in this phase but disclosed a series of new unexpected behaviors in the transport data. We shall detail hereafter that an applied field of ~30 T in the ground state induces a change in the sign of the Hall effect, implying a major change in the electronic properties. We shall then underline the contrasting behavior of the magnetoresistance (MR) which changes sign and field dependence for increasing T. The negative B dependence of the MR attributable to heavy carriers will be assigned to paramagnetic spin scattering in analogy to similar observations in the Heavy Fermion compounds. Comparisons with the multiband BS known for kagome compounds will lead us to suggest that the B linear MR of the mobile carriers could be associated with Dirac/Weyl linear dispersing bands. In this peculiar metallic kagome compound the band structure topology therefore retains singular behaviors in the presence of strong correlations. Experimental results Hall effect: Transport data was taken on a series of distinct samples and in two different high magnetic field facilities as detailed in the Supplemental Material (26). The small residual resistivity (~2 ·cm) measured in our single crystals proves the low level of disorder for this x=2/3 phase. In preliminary studies done above 2 K the Hall resistivity xy was found to be linear in B below 9 T(18). But, when searching for QO at T=0.35 K under high magnetic fields at the Maglab in Tallahassee, we found that xy goes through a minimum at ~15 T, and becomes positive above 30 T as displayed in Fig. 2a. This drastic non-linear behavior only moderately changes when the temperature is increased up to T=5 K. Then xy progressively becomes linear in B and remains negative above 30 K within the experimental field range. Our extensive data set for the Hall constant RH (26) are summarized in the Fig. 2b. It provides evidence that RH is nearly T and B independent up to 1.5 K and 12 T but abruptly changes its sign above ~30 T and levels off at a positive value at the highest fields Fig. 2. (a) Hall resistivity xy as a function of B=0H at various T taken under DC applied fields H for sample TalD2 and under pulsed fields for sample TlsD agrees perfectly. (b) The full dataset described in (26), section II are displayed as a 3D plot of the Hall constant RH versus B and T (log scale). RH becomes positive under large fields for T<10 K, while the low field behavior of Fig. 1e is seen to saturate below 1.5 K at its lower negative value (26). This abrupt sign change of RH beyond 30 T implies a sharp reduction in electronic carrier density and/or mobility. So, assuming a Landé factor g=2, a Zeeman energy gBB~2 meV is sufficient to switch the transport from electron to hole-like carriers. On the contrary, the low field RH remains negative up to 200 K so that the electrons remain dominant well beyond 30 K. These results demonstrate that thermal and Zeeman energies have distinct incidences on the transport. Magnetoresistance: In the first search for Shubnikov de Haas effect at T=0.35 K we did not find any indication for QOs at high frequencies which would be associated with large FS pockets. But as shown in Fig. 3a an unexpectedly large positive MR was detected (~100% at 10 T and 800 % at 45 T). At T=0.35 K it exhibits an initial linear in field behavior, while above 5 K it becomes negative and follows a B dependence. Fig.3 (a) Selected (B) curves taken from 0.35K to 1.5K for TalA sample and from 1.5K to 10.5K for TlsD. Details are available in (26), sec. II. (b) d(B)/dB displays maxima at 12T below 1.5K and at 20T below 5K. Insets: in (a) high T variation of the negative coefficient k2 of the B2 MR including data above 10K detailed in (26), sec. II. (b) Low T variation of the linear coefficient k1 of the MR (see also (26), sec. V). Other unusual behaviors also appear as the large T variation of B,T displayed for B=0 T disappears up to 1.5 K for fixed field B>12 T and even up to 5 K for fixed B>20 T. Those two fields also appear as slight steps in (B) seen as maxima in the d(B)/dB curves of Fig. 3b. There the maximum at 20 T is seen for T up to 5 K, while the one at 12 T disappears above 1.5 K. These derivative curves were found to be astonishingly reproducible on distinct sample batches (26). The similar drastic loss of the ground state conductivity xx which occurs beyond 20 T or beyond 5 K is illustrated in the 3D representation shown in Fig. 4. There  is shown in a logarithmic scale versus B and T, with the latter also on a log scale. The corresponding linear representation of  is display