First-principles study of localized and delocalized electronic states in crystallographic shear phases of niobium oxide

Crystallographic shear phases of niobium oxide form an interesting family of compounds that have received attention both for their unusual electronic and magnetic properties, as well as their performance as intercalation electrode materials for lithium-ion batteries. Here we present a first-principles density-functional theory study of the electronic structure and magnetism of H-${\mathrm{Nb}}_2{\mathrm{O}}_5$, ${\mathrm{Nb}}_{25}{\mathrm{O}}_{62}$, ${\mathrm{Nb}}_{47}{\mathrm{O}}_{116}$, ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$, and ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$. These compounds feature blocks of niobium-oxygen octahedra as structural units, and we show that this block structure leads to a coexistence of flat and dispersive energy bands, corresponding to localized and delocalized electronic states. Electrons localize in orbitals spanning multiple niobium sites in the plane of the blocks. Localized and delocalized electronic states are both effectively one-dimensional and are partitioned between different types of niobium sites. Flat bands associated with localized electrons are present even at the GGA level, but a correct description of the localization requires the use of $\mathrm{GGA}+U$ or hybrid functionals. We discuss the experimentally observed electrical and magnetic properties of niobium suboxides in light of our results, and argue that their behavior is similar to that of $n$-doped semiconductors, but with a limited capacity for localized electrons. When a threshold of one electron per block is exceeded, metallic electrons are added to existing localized electrons. We propose that this behavior of shear phases is general for any type of $n$-doping, and should transfer to doping by alkali metal (lithium) ions during operation of niobium oxide-based battery electrodes. Future directions for theory and experiment on mixed-metal shear phases are suggested.

Figure 1

(a) Idealized (left) and locally distorted (right) crystal structure of ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$. The tetrahedral site is shown as a black dot in the idealized structure. Crystal structures of (b) H-${\mathrm{Nb}}_2{\mathrm{O}}_5$, (c) monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$, (d) orthorhombic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$, (e) ${\mathrm{Nb}}_{25}{\mathrm{O}}_{62}$, and (f) ${\mathrm{Nb}}_{47}{\mathrm{O}}_{116}$. Light and dark colored blocks are offset by half of the lattice parameter perpendicular to the plane of the page. Unit cells are outlined in black.

Figure 2

Spin-polarized band structure and electronic density of states of ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$ ($\mathrm{PBE}+U$, $U_{\mathrm{eff}}=4$ eV). Up and down spins colored in red and blue. High symmetry points are marked on slices through the first Brillouin zone. The flat bands below the Fermi level (dashed line) represent localized states.

Figure 3

Spin-summed projected density of states ($\mathrm{PBE}+U$) for ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$. Fermi level is indicated by the dashed line. (a) PDOS for central (gold) and peripheral (green) niobium sites. (b) DOS projected for sites in different blocks, demonstrating separate localization of electrons in $3\times 3$ and $3\times 4$ blocks. Contributions from sites are proportional to the shaded area.

Figure 4

(a) Spin density plot of ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$. Niobium and oxygen shown in dark blue and orange, respectively. Purple and light blue represent up and down spin density, respectively. The rectangles outline the $3\times 4$ and $3\times 3$ blocks. Spin density isosurface drawn at a value of 0.03 $e^{-}/{\AA }^3$. (b) Kohn-Sham orbitals associated with localized states (flat bands in Fig. 2) in $3\times 4$ and $3\times 3$ blocks, different phases of the orbitals shown in yellow and light green.

Figure 5

$\mathrm{\Gamma }\to Z$ segment of the band structure of ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$ calculated with different levels of theory. PBE, $\mathrm{PBE}+U$ ($U_{\mathrm{eff}}=4.0$ eV), and HSE06, from left to right. Only one spin component is shown for clarity.

Figure 6

Band structure and density of states of monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$ ($\mathrm{PBE}+U$). Fermi level indicated by a dashed line. Up and down spins colored in red and blue, respectively. Flat and dispersive bands are present, with strong similarity to those in ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$.

Figure 7

Projected density of states ($\mathrm{PBE}+U$) for central (gold) and peripheral (green) niobium sites in monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$. Central niobium sites contribute to the density of occupied states only in a narrow window that is associated with the flat bands.

Figure 8

Spin density plot (a) and orbitals associated with localized (c) and delocalized (b) and (d) states in monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$. Spin density (a) is predominantly located on the central niobium sites, and results from the occupation of localized states (c). Delocalized states have no contribution from the central niobium sites.

Figure 9

Band structure ($\mathrm{PBE}+U$) (a) and spin density (b) of orthorhombic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$. $Y={\mathbf{a}}^{*}/2$. The orthorhombic and monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$ polymorphs show a strong similarity in their band structure and spin density distribution [cf. Figs. 6 and 8].

Figure 10

Summed charge densities from bands in (a) ${\mathrm{Nb}}_{25}{\mathrm{O}}_{62}$ and (b) ${\mathrm{Nb}}_{47}{\mathrm{O}}_{116}$. Selected empty and filled localized states within blocks are framed by rectangles. The same (conventional) unit cell as in Fig. 1 is shown for ${\mathrm{Nb}}_{25}{\mathrm{O}}_{62}$, but a smaller primitive cell for ${\mathrm{Nb}}_{47}{\mathrm{O}}_{116}$.

Figure 11

Band structure of H-${\mathrm{Nb}}_2{\mathrm{O}}_5$ (PBE). Valence and conduction bands are colored in orange and blue, respectively. Flat and dispersive bands are present above the Fermi level (dashed line) similar to those in ${\mathrm{Nb}}_{22}{\mathrm{O}}_{54}$ and ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$, but are unoccupied.

Figure 12

Spin density plot of lithiated H-${\mathrm{Nb}}_2{\mathrm{O}}_5$. A single lithium is located in the middle of the smaller block, inducing a localized state.

Figure 13

Schematic of band structure for (a) filling fraction $\nu \le$ 1 $e^{-}$/block and (b) $\nu >$ 1 $e^{-}$/block. O $2p$ and Nb $4d$ dominated bands are colored in orange and blue, respectively. Fermi level is indicated by a dashed line, ${\mathbf{k}}_{\perp }$ designates reciprocal space vector associated with the real space direction perpendicular to the block plane. The relative position of flat and dispersive bands changes with the filling fraction $\nu$.

Figure 14

(a) Possible spin-spin interactions $J_i$ in monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$ along crystallographic $a$, $b$, and $c$ directions. (b) Different spin arrangements in monoclinic ${\mathrm{Nb}}_{12}{\mathrm{O}}_{29}$. AFM along $c$ (top left), fully FM (top right), AFM along $a$ (bottom left), AFM along $b$ (bottom right). White and gray blocks are offset by 0.5 $b$ throughout.