Superatomic crystal emerging in transition-metal oxides: Molybdenum hollandite K${}_2$Mo${}_8$O${}_{16}$

Density-functional-theory-based electronic structure calculations are carried out to elucidate the origins of the observed electronic properties of molybdenum hollandite K${}_2$Mo${}_8$O${}_{16}$. We find that the Mo${}_4$ cluster in the double Mo chains behaves as a “superatom,” a hypothetical big atom with a single composite molecular orbital, and that the system can be regarded as a solid of the superatoms condensed into a simple monoclinic structure with four superatoms per unit cell, thereby yielding four energy bands near the Fermi level at half filling. Based on an effective model proposed, we argue that K${}_2$Mo${}_8$O${}_{16}$ is a Mott insulator with one electron per superatom, which exhibits strongly frustrated antiferromagnetic spin correlations in the superatomic crystal.

Figure 1

Schematic representation of the superatomic crystal for K${}_2$Mo${}_8$O${}_{16}$. (a) Hollandite-type crystal structure viewed down the $c$ axis, (b) simple monoclinic lattice as a superatomic crystal representing K${}_2$Mo${}_8$O${}_{16}$, (c) Mo${}_4$ cluster (large shaded circle) as a superatom in the double chains, and (d) local coordinate axes $(x,y,z)$ and three $t_{2g}$ orbitals in the double chain. The lattice constants and bond lengths are shown in units of Å in (b) and (c), respectively.

Figure 2

Calculated DOS for K${}_2$Mo${}_8$O${}_{16}$ in a wide range of energy. The vertical line represents the Fermi level.

Figure 3

Calculated orbital-decomposed PDOS for the $t_{2g}$ orbitals of (a) Mo(1), (b) Mo(2), (c) Mo(3), and (d) Mo(4) in K${}_2$Mo${}_8$O${}_{16}$. The vertical line in each panel represents the Fermi level.

Figure 4

Schematic representations of the electron distributions in the Mo${}_4$ cluster of K${}_2$Mo${}_8$O${}_{16}$. In (a), three $t_{2g}$ orbitals, i.e., $d_{xy}$, $d_{yz}$, and $d_{zx}$, are represented by the six lobes, where two of the four lobes for each of the three $t_{2g}$ orbitals are drawn (Ref. [27]), so that the overlap of the neighboring orbitals can be grasped visually. The shorter bonds are indicated by thicker lines. There are nine electrons in this cluster; the large arrow indicates one electron, while the small arrow indicates 0.5 electrons, for a total of eight bonding electrons. The remaining electron occupies the composite molecular orbital near the Fermi level, which is primarily comprised of the “nonbonding” components of the four $d_{yz}$ orbitals. In (b), the energy-level scheme of the Mo${}_4$ cluster is drawn schematically with the energy ranges in the GGA band structure. Left: Two pairs of the bonding and antibonding $d_{xy}$ orbitals. Right: A pair of the bonding and antibonding $d_{zx}$ orbitals and two nonbonding $d_{zx}$ orbitals. Middle: Four nondegenerate molecular orbitals stemming from the four $d_{yz}$ orbitals, the second lowest of which lies at the Fermi level.

Figure 5

Calculated band dispersions for K${}_2$Mo${}_8$O${}_{16}$. The Brillouin zones with the definitions of the $\mathit{k}$ points used are shown in the inset.

Figure 6

Illustration of the maximally localized Wannier functions for the Mo${}_4$ cluster comprised of (a) Mo(1) and Mo(2) ions as well as (b) Mo(3) and Mo(4) ions. The colors distinguish the $\pm$ signs of the wave function. The bonding nature of the two $d_{yz}$ orbitals and small contributions of the $2p_x$ and $2p_y$ orbitals of O ions are clearly visible.

Figure 7

Tight-binding band structures (solid curves) for the four bands of K${}_2$Mo${}_8$O${}_{16}$ compared to the GGA results (dashed curves). Hopping integrals between the superatoms used are summarized in Ref. [29].

Figure 8

Schematic representation of the arrangements of the Mo${}_4$ clusters (or superatoms) in the lattice. The on-site energies of the superatoms are defined as ${\varepsilon }_i$ with $i=1$ for the superatoms comprised of Mo(1) and Mo(2), and as ${\varepsilon }_i$ with $i=2$ for the superatoms comprised of Mo(3) and Mo(4). The hopping parameters between the superatoms are defined as $t$ and $t^{\prime }$ as indicated.

Figure 9

Calculated electron and hole distributions near the Fermi level in K${}_2$Mo${}_8$O${}_{16}$. In (a), we show the electron distribution in the energy window between $-0.38$ eV and $E_{\mathrm{F}}=0$ eV, and in (b), we show the hole distribution in the energy window between $E_{\mathrm{F}}=0$ and $0.26$ eV. An isovalue of 0.008 electrons/Bohr${}^3$ is used.

Figure 10

Calculated mean-field phase diagram of the effective model for K${}_2$Mo${}_8$O${}_{16}$. There appear four phases: ionic band-insulator (BI), ionic paramagnetic semimetal (PM), antiferromagnetic metal (AFM), and antiferromagnetic Mott insulator (AFI) phases. The arrangements of the electron spins are also shown schematically.

Figure 11

Schematic illustration of the PDOS in the AFI phase of our effective model. The strong ionicity and exchange splitting shown in (a) yield the characteristic PDOS shown in (b). The vertical lines represent the Fermi level.