Spin-orbit coupling induced semi-metallic state in the 1/3 hole-doped hyper-kagome Na3Ir3O8

The complex iridium oxide Na3Ir3O8 with a B-site ordered spinel structure was synthesized in single crystalline form, where the chiral hyper-kagome lattice of Ir ions, as observed in the spin-liquid candidate Na4Ir3O8, was identified. The average valence of Ir is 4.33+ and, therefore, Na3Ir3O8 can be viewed as a doped analogue of the hyper-kagome spin liquid with Ir4+. The transport measurements, combined with the electronic structure calculations, indicate that the ground state of Na3Ir3O8 is a low carrier density semi-metal. We argue that the semi-metallic state is produced by a competition of the molecular orbital splitting of t2g orbitals on Ir3 triangles with strong spin-orbit coupling inherent to heavy Ir ions.


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The realization of a quantum spin liquid is a long-sought dream in condensed matter physics, where exotic phenomena like fractional excitations or unconventional superconductivity nearby are anticipated 1 . The most promising arena for a spin liquid is a geometrically frustrated lattice based on a triangular motif. Antiferromagnetically interacting spins on such a frustrated lattice cannot simultaneously satisfy all magnetic bonds and, as a result, macroscopic degeneracy could remain in the ground state. To date the experimental efforts have put forward several candidates such as transition metal oxides with a kagome lattice 1,2 and organic Mott insulators with a triangular lattice 1,3 .
The recent discoveries of candidate materials include Na 4 Ir 3 O 8 which emerged as the first candidate for a three-dimensional quantum spin liquid 4 . In this complex oxide, Ir atoms with a localized S = 1/2 moment (or more likely close to J eff = 1/2 moment, see below) form a cornersharing network of triangles in three-dimensions, called "hyper-kagome" lattice. All the Ir sites and the Ir-Ir bonds are equivalent, rendering the hyper-kagome lattice magnetically frustrated. Indeed, Na 4 Ir 3 O 8 exhibited no magnetic ordering down to 2 K despite the strong antiferromagnetic interaction inferred from the Curie-Weiss temperature θ W of -650 K. This discovery triggered intensive experimental and theoretical surveys on this compound, including proposals for the presence of a spin Fermi surface 5 .
In tandem with the discovery of Na 4 Ir 3 O 8 , complex Ir 4+ oxides have recently emerged as a novel playground for physics of strong spin-orbit coupling (SOC). The SOC of Ir 4+ is as large as λ SO ~ 0.6 eV, reflecting the heavy atomic mass. λ SO is not small at all as compared with the other parameters dominating the electronic states, such as the inter-site hopping t, the Coulomb repulsion U and the crystal field splitting ∆. The interplay of the large SOC with the other parameters leads to the formation of unprecedented electronic phases. In the layered perovskite Sr 2 IrO 4 , for example, the spin and orbital degrees of freedom are intimately entangled to produce J eff = 1/2 states, giving rise to a novel spin-orbital Mott insulator 6 . The magnetic coupling of J eff = 1/2 moments in such spinorbital Mott insulators can be distinct from those of the spin-dominant moments in 3d oxides, as exemplified by the possible Kitaev spin-liquid proposed for honeycomb iridates 7 . Such unique magnetic couplings in iridates would make the mystery of the possible spin-liquid state of Na 4 Ir 3 O 8 even more intriguing 8 .
The insulating state of Na 4 Ir 3 O 8 is marginally stabilized by a modest U with the help of strong SOC. Such weak Mottness, implying the close proximity to a metallic state, has been discussed to play a vital role in realizing the spin-liquid ground state 9 , along with organic triangular spin liquids 1 . It should not be very difficult to bring Na 4 Ir 3 O 8 to a metallic state by carrier-doping or by applying pressure. Exotic superconductivity at the critical boarder to a spin liquid might be anticipated in analogy with the organic systems 10 and is worthy for exploration.
Despite such intriguing outlooks in Na 4 Ir 3 O 8 , not much progress has been achieved in the critical investigation of the spin-liquid like state including the role of spin-orbit coupling, largely due 3 to the lack of single crystals. During the course of attempting to grow single crystals of Na 4 Ir 3 O 8 , we obtained single crystals of Na 3 Ir 3 O 8 , a B-site ordered spinel. The crystal structure is distinct from that of Na 4 Ir 3 O 8 , a spin liquid candidate, but shares the same Ir-O hyper-kagome network. Na 3 Ir 3 O 8 therefore can be viewed as a doped hyper-kagome spin liquid. We report here that Na 3 Ir 3 O 8 has a semi-metallic ground state produced by the strong spin-orbit coupling.

Results
Crystal Structure Analysis. X-ray diffraction analysis and detailed refinement of the crystal structure was performed on Na 3 Ir 3 O 8 single crystals grown by a flux method (see Methods).
Satisfactory refinement was obtained with space groups P4 1 32 or P4 3 32. Accordingly, the crystal under investigation turned out to be a racemic twin (see Supplementary). The refined structural parameters are listed in Table 1. The structure shown in Fig.1a can be viewed as an ordered spinel, an intimately related but distinct structure to that of polycrystalline Na 4 Ir 3 O 8 . Rewriting the chemical formula of 1/2 Na 3 Ir 3 O 8 as Na(Na 1/4 , Ir 3/4 ) 2 O 4 , in correspondence with that of spinel AB 2 O 4 , is convenient to understand the structure. Na(2) in Table 1  is as large as ~100 cm 2 /V·s and, clearly, the disorder is not the dominant factor of the poorly metallic behavior. The carrier number estimated from the Hall constant is of the order of 10 19 cm -3 at 5 K, 4 which is too small to be accounted as a simple 1/3 hole-doped Mott insulator with 2/3 electrons. The rapid increase of Hall coefficient with temperature despite the metallic behavior of resistivity, more than one order of magnitude from 5 K to 300 K, indicates the coexistence of two different types of carriers. This implies that Na 3 Ir 3 O 8 is either a semi-metal or a very lightly doped narrow gap semiconductor. Unexpectedly, the ground state of a doped hyper-kagome appears to be very close to a band insulator.
The low temperature specific heat of Na 3  Electronic Structure Calculation. Ab-initio electronic structure calculation using a fully relativistic LMTO code 12 revealed that Na 3 Ir 3 O 8 is a compensated semi-metal due to the interplay of periodic potential and SOC, which is consistent with the experimental observation described above. Figure 4 depicts the electronic state around the Fermi energy where t 2g orbitals of Ir have a dominant contribution. In iridates, the 5d electrons are accommodated into the t 2g manifolds due to large t 2g -5 e g crystalline field splitting. In spite of the non-integer Ir 5d 4.67 filling with 14 t 2g electrons per 3 Ir in a formula unit, a gap of 0.2 eV opens within t 2g bands in the scalar-relativistic calculation neglecting SOC as shown in Fig. 4a. A similar gap separating t 2g bands was also found for Na 4 Ir 3 O 8 and explained by strong p-d hopping 13 . In this case, due to higher Na content the t 2g bands above the gap are partially filled.
The strong SOC of Ir, in reality, splits the conduction and the valence bands substantially and a negative band gap is enforced. Bands calculated by including SOC are shown in Fig. 4c

Discussion
The most striking effect of SOC in Na 3 Ir 3 O 8 is the closure of scalar relativistic gap. This is in stark contrast to the case of SrIrO 3 , where the LDA calculation without SOC yields a metal rather than a band insulator. SOC reconstructs metallic t 2g bands and leads to the semi-metallic state due to the band split 15 . An inspection of orbitally resolved densities of Ir 5d states reveals that only two of three t 2g states of each Ir ion contribute to the unoccupied t 2g bands whereas the third t 2g orbital is completely filled, in the scalar-relativistic calculation shown in Fig. 4b. The energy of the filled orbital is lower because the average Ir-O distance in the plane containing it is somewhat larger than in the planes of the other two orbitals. However, the 14-to-4 ratio of the number of occupied and unoccupied t 2g bands implies that the gap cannot be simply caused by local distortions of IrO 6 octahedra. We may understand the gap formation by considering Ir 3 triangular molecules which are the basic structural unit of the hyper-kagome network. Each Ir 3 molecule in Na 3 Ir 3 O 8 accommodates an even number of t 2g electrons when 1/3 hole is doped to 5d 5 Ir 4+ , 14 electrons for 18 quasi-6 molecular orbitals, which could make the system a band insulator.
SOC splits five-fold degenerate d-orbitals into j = 5/2 and 3/2 characters. In an isolated octahedron, SOC split states are mixed due to the crystalline field, and form a lower lying quartet with j = 5/2 and 3/2 admixture (J eff = 3/2 state), a doublet with pure j = 5/2 character (J eff = 1/2 state), and upper doubly-degenerate e g orbitals 16 . Ir 5d 4.67 configuration indicates that E F locates within the J eff = 1/2 manifolds, which can be expressed as the equal superpositions of three t 2g orbitals. Such J eff = 1/2 picture was also reported for Na 4 Ir 3 O 8 13 . An analysis of the orbital character of the t 2g bands for Na 3 Ir 3 O 8 , depicted in Fig. 4d, shows that the bands below -0.5 eV have predominantly J eff = 3/2 character. The bands just below the E F have an almost pure J eff = 1/2 character, indicating that SOC entangles the three t 2g orbitals in an equal weight unlike the scalar relativistic calculation described above. On the other hand, the unoccupied bands have significant admixture of J eff = 3/2 states. This implies rather strong hopping between J eff = 1/2 and J eff = 3/2 states, which is likely inherent to the edge-sharing network of IrO 6 octahedra 17,18 . In summary, we successfully synthesized single crystals of hyper-kagome iridate Na 3 Ir 3 O 8 .
Unlike the putative spin-liquid Na 4 Ir 3 O 8 , Na 3 Ir 3 O 8 was found to be a semi-metal produced by spinorbit coupling. The semi-metallic Na 3 Ir 3 O 8 could be an intriguing platform to test the possible nontrivial topological effects, associated with the frustrated and chiral geometry of the lattice and the negative gap produced by SOC. The presence of chirality is particularly unique to the hyper-kagome iridate as compared with the other complex iridium oxides. If single crystals with a sizable size of domains could be grown, we might have a chance to capture topological effects related to the chirality. In the obtained crystals, we in fact found a very small amount of insulating crystals besides the majority of metallic ones. The X-ray diffraction indicated that the insulating crystals also had the Na 3 Ir 3 O 8 stoichiometry, and the spectroscopic measurements showed no difference between the metallic and insulating crystals. Judging from the poor quality of the insulating crystals, we suspect that the insulating crystals include domains of the Na 4 Ir 3 O 8 phase and there might be a temperature window where Na 4 Ir 3 O 8 phase is stabilized during the crystal growth. However, the growth of Na 4 Ir 3 O 8 single crystal is a future perspective and out of the scope of this article.
Horizontally scattered X-rays were analyzed by a diced and spherically-bent Si(844) crystal. The total energy resolution was 70 meV. The energy of the incident X-ray was tuned at 11.214 keV, which corresponds to the L 3 -edge of Ir.     The two spectra were normalized by the intensity of the high energy tail above 6 eV. The blue and pink solid lines around zero energy show the normalized elastic signals independently measured with σ-polarized incident X-ray.   (2)