The realization of quantum spin liquid is a long-sought dream in condensed matter physics, where exotic phenomena such as spinon Fermi surface, fractional excitations or unconventional superconductivity upon doping are anticipated. A prime candidate in the recent discoveries is Na4Ir3O8 . It is one of the very first candidates for a three-dimensional quantum spin liquid, where Ir4+ ions form a corner-sharing network of triangles in three-dimensions, called a "hyper-kagome" lattice. All the Ir sites and the Ir-Ir bonds are equivalent, rendering the hyper-kagome lattice magnetically frustrated. Indeed, Na4Ir3O8 exhibits no magnetic ordering down to 2 K despite the strong antiferromagnetic interaction inferred from the Curie-Weiss temperature ΘW ≈ –650 K.
We recently succeeded in synthesizing a new form of hyper-kagome iridate Na3Ir3O8, which shares the same Ir-O hyper-kagome network with Na4Ir3O8. Na3Ir3O8 therefore can be regarded as a doped hyper-kagome spin liquid. We found that Na3Ir3O8 in fact has a semi-metallic ground state produced by a subtle competition of strong spin-orbit coupling of Ir4+ and molecular orbital splitting of t2g electrons in an Ir3 triangular unit .
Single crystals of Na3Ir3O8 were grown by a flux method. The crystal structure shown in Fig. 1(a) can be recognized as an ordered spinel. Rewriting the chemical formula of 1/2 Na3Ir3O8 as Na(Na1/4, Ir3/4)2O4, in correspondence with that of spinel AB2O4, is useful to understand the structure. One of two Na sites, Na(2), corresponds to the tetrahedral A-site of spinel structure, and the other (Na(1)) corresponds to 1/4 of the pyrochlore B sub-lattice. The remaining 3/4 of the pyrochlore B sub-lattice sites are occupied by Ir atoms. All the Ir sites are equivalent and form the same hyper-kagome network as in Na4Ir3O8.
The composition of Na3Ir3O8 corresponds to Ir valence of 4.33+. Considering that Na4Ir3O8 is an Ir4+ (5d5) Mott insulator, Na3Ir3O8 may be viewed as 1/3 hole-doped hyper-kagome spin liquid. Indeed, the obtained single crystals were found to show metallic behavior of resistivity as shown in Fig. 2(a), in marked contrast to the spin liquid Na4Ir3O8. The magnitude of resistivity is relatively large as a metal, ≈1 mΩcm at 5 K. The Hall coefficient of Na3Ir3O8 indicates that the poorly metallic behavior originates from a low carrier concentration. The Hall coefficient shown in Fig. 2(b) is negative and its magnitude is orders of magnitude larger than those of typical metals, indicating that a very low density of electrons dominates the transport. The carrier number estimated from the Hall constant is of the order of 1019 cm–3 at 5 K. The Hall coefficient shows a rapid decrease, more than one order of magnitude, with increasing temperature from 5 K to 300 K, despite the metallic behavior of resistivity. This very likely implies the coexistence of two different types of carriers at least at high temperatures, suggesting that Na3Ir3O8 is either a semi-metal or a degenerate narrow gap semiconductor. Unexpectedly, the ground state of a doped hyper-kagome appears to be very close to a band insulator.
Na3Ir3O8 is indeed found to be a semi-metal by ab initio electronic structure calculation. Figure 3 depicts the electronic states around the Fermi energy where t2g orbitals of Ir have a dominant contribution. The 5d-electrons are accommodated into the t2g manifold due to the large t2g – eg crystal field splitting. The number of d-electrons per Ir atom is non-integer, 4.67 (Ir4.33+). In the unit formula with 3 Ir atoms, we have even number of electrons, 14 (=3×4.67) t2g electrons. In the scalar-relativistic calculation neglecting SOC, shown in Fig. 3(a), a well-defined gap of 0.2 eV can be seen within the t2g bands. 14 t2g electrons fill up all the bands below the 0.2 eV gap and the system is a band insulator.
The presence of gap in the scalar-relativistic calculation can be understood as a formation of molecular orbits in the Ir3 triangules, the basic unit of hyper-kagome lattice. The dominant hopping process between Ir t2g states in the hyper-kagome lattice is the one between the nearest neighbor Ir d-orbitals via oxygen p orbitals. There are two oxygen sites, O1 or O2, forming the IrO6 octahedra, and the Ir–O bond lengths for those two sites are very different (Ir–O1: 2.053 Å, Ir–O2: 1.976 or 1.978 Å). Because of the contrasting Ir–O bond lengths, the hopping via O2 is expected to be significantly stronger than via O1. Let us define for each Ir site its own local frame with the x- and y-axis lying in the plane of the IrO4 square with two more distant O1 ions (inset Fig. 1(b)). Among the three t2g orbitals of Ir, the dxy orbital should be energetically stabilized by the crystal field associated with the longer Ir–O1 distance. This can be clearly recognized in the orbitally resolved density of states shown in Fig. 3(b).
Both dyz and dzx have the hopping path to t2g orbitals of the neighboring Ir atoms via O2 but it is spatially limited. dzx on Ir(1) can hop only to dzx on Ir(2) or Ir(3), whereas dyz only to dyz on Ir(4) or Ir(5), meaning that for each of the two orbitals the hopping paths via O2 are restricted to one of the two Ir3 triangles sharing common Ir(1). In this simplified picture considering only the hopping via O2 2p-orbitals, the dzx states form molecular orbitals on the Ir(1)Ir(2)Ir(3) triangle, namely, two degenerate bonding and one anti-bonding molecular orbitals. The dyz on Ir(1) participates in molecular orbitals on the Ir(1)Ir(4)Ir(5) triangle. Note that dzx and dyz orbitals are nearly orthogonal to each other, which means that those molecular orbitals which belong to different triangles do not interact and are almost localized. Considering the energy levels of the Ir3 molecule with 3×3 = 9 t2g orbitals, 3 dxy orbitals have the lowest energy. The other two, dyz and dzx, orbitals form 4 bonding and 2 anti-bonding orbitals, as schematically shown in Fig. 3(c). With 14 electrons per Ir3 triangle, 3 dxy and 4 bonding molecular orbitals are fully filled and the splitting of bonding and anti-bonding molecular orbitals gives rise to an energy gap.
Strong SOC of Ir, in reality, splits the molecular orbits and produces the semi-metallic ground state. The band structure calculated with SOC is shown in Fig. 3(d). In the presence of realistic SOC, a pair of unoccupied t2g bands, colored in magenta, bends down near the R point and becomes degenerate with the conduction bands at the R point. This pronounced SOC effect results in the overlap of the pair of conduction band and the valence band colored in red, and creates two electron-like Fermi surfaces around the R point and four hole-like Fermi surfaces near the Γ point.
The split of molecular orbits originates from the formation of spin-orbital entangled Jeff = 1/2 state. In a cubic environment SOC splits three-fold degenerate t2g d-orbitals into 2:1 ratio of lower Jeff = 3/2 (admixture of j = 5/2 and 3/2 characters) and higher Jeff = 1/2 states (purely j = 5/2 character), which accommodate 12 and 6 electrons respectively for the Ir3 triangle, as shown in Fig. 3(f). In the strong SOC limit, Jeff = 3/2 states are fully filled up and Jeff = 1/2 states are 1/3 filled. The splitting into Jeff = 3/2 and higher Jeff = 1/2 states can be seen in the relativistic calculation with SOC shown in Figs. 3(d), (e). Since the SOC mixes all three t2g states, dzx and dyz forming two distinct molecular orbitals and dxy, the 0.2 eV gap within the molecular orbits is suppressed. The semi-metallic state of Na3Ir3O8 is therefore marginally formed by the competition between the molecular orbital split and strong SOC of Ir.