First-Principles Study of Localised and Delocalised Electronic States in Crystallographic Shear Phases of Niobium Oxide

TL;DR

This study uses first-principles calculations to analyze the electronic structure and magnetism of niobium oxide shear phases, revealing coexistence of localized and delocalized states influenced by their block structures.

Contribution

It provides a detailed theoretical understanding of electronic localization and magnetic properties in niobium oxide shear phases, highlighting the importance of advanced functionals for accurate descriptions.

Findings

Flat bands indicate localized electrons in niobium oxide structures.

Delocalized electrons contribute to metallic behavior beyond a certain doping threshold.

Localized and delocalized states are effectively one-dimensional and site-dependent.

Abstract

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-Nb$_2$O$_5$, Nb$_{25}$O$_{62}$, Nb$_{47}$O$_{116}$, Nb$_{22}$O$_{54}$, and Nb$_{12}$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 localised and delocalised electronic states. Electrons localise in orbitals spanning multiple niobium sites in the plane of the blocks. Localised and delocalised electronic states are both effectively one-dimensional and are partitioned between different types of niobium sites. Flat bands associated with localised electrons are present even at the GGA level, but a correct description of the localisation requires the use of 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 behaviour is similar to that of $n$-doped semiconductors, but with a limited capacity for localised electrons. When a threshold of one electron per block is exceeded, metallic electrons are added to existing localised electrons. We propose that this behaviour 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.

Figures

Tables

Table 1. Table 1: Structural properties of niobium (sub)oxides. Experimental and DFT optimised lattice parameters a 𝑎 a , b 𝑏 b and c 𝑐 c are given in Å angstrom \mathrm{\SIUnitSymbolAngstrom} . Structural optimisations with DFT+U were performed with a U eff subscript 𝑈 eff U_{\mathrm{eff}} value of 4.0 eV on niobium d 𝑑 d -orbitals. N e − subscript 𝑁 superscript 𝑒 N_{e^{-}} denotes number of electrons introduced by doping. Difference between experimental and DFT space group choices related to ordering of tetrahedral sites (see text).

Compound	O/Nb	$N_{e^{-}}/$block	Space group	Source	$a$	$b$	$c$	$\beta$	Block size(s)
H-\ceNb2O5	2.500	0	$P2/m$	Expt. Kato (1976)	$21.153$	$3.823$	$19.356$	${119.80}^o$	$3\times 4,3\times 5$
			$P2$	PBE	$21.433$	$3.841$	$19.614$	${119.85}^o$
\ceNb25O62	2.480	$1/2$	$C2$	Expt. Cava et al. (1991a)	29.78	3.825	21.14	${94.7}^o$	$3\times 4$
				PBE	$30.224$	$3.84$	$21.44$	${95.0}^o$
\ceNb47O116	2.468	$2/3$	$C2$	Expt. Cava et al. (1991a)	57.74	3.823	21.18	${105.3}^o$	$3\times 3,3\times 4$
				PBE	$58.43$	$3.84$	$21.44$	${105.3}^o$
\ceNb22O54	2.455	1	$P2/m$	Expt. McQueen et al. (2007)	15.749	3.824	17.852	${102.029}^o$	$3\times 3,3\times 4$
			$P2$	PBE	$15.931$	$3.842$	$18.036$	${102.06}^o$
			$P2$	PBE+U	$15.935$	$3.836$	$18.061$	${101.99}^o$
m-\ceNb12O29	2.417	2	$A2/m$	Expt. Waldron et al. (2004)	15.695	3.831	20.723	${113.103}^o$	$3\times 4$
				PBE	$15.903$	$3.832$	$20.967$	${113.1}^o$
				PBE+U	$15.885$	$3.837$	$20.950$	${113.09}^o$
o-\ceNb12O29	2.417	2	$Cmcm$	Expt. McQueen et al. (2007)	3.832	20.740	28.890	-	$3\times 4$
				PBE	$3.833$	$20.955$	$29.241$	-
				PBE+U	$3.836$	$20.961$	$29.204$	-

Full text

I Introduction

Transition metal oxides form a fascinating class of compounds with interesting electronic, magnetic, and crystallographic structures. The phase diagram of niobium oxide is especially rich, with a large number of reported phases for \ceNb2O5 Kato and Tamura (1975); Kato (1976); Schäfer et al. (1966), in addition to \ceNbO and \ceNbO2. The high-temperature \ceNb2O5 polymorph (H-\ceNb2O5) can be regarded as the parent compound of a family known as crystallographic shear (or Wadsley–Roth) phases Roth and Wadsley (1965a, b). In these phases, niobium is present in octahedral coordination, but the Nb/O ratio of \ceNb2O5 prevents the formation of purely corner-sharing octahedra. Instead, the structure must include some amount of edge-sharing connections between octahedra. The crystal structures of these compounds consequently consist of blocks of corner-sharing octahedra of size $n\times m$ that are connected to neighbouring blocks via crystallographic shear planes of edge-sharing connections. In the direction perpendicular to the $n\times m$ plane the units connect infinitely, and tetrahedrally coordinated ions are sometimes present to fill voids in the structure. By reduction of \ceNb2O5, small amounts of Nb4+ can be incorporated, and a series of \ceNb2O_5-δ compounds form. These suboxides include \ceNb25O62, \ceNb47O116, \ceNb22O54, and two polymorphs of \ceNb12O29 with different crystal symmetries (Fig. 1, Table 1). The metal-oxygen octahedra in these compounds are strongly distorted due to a combination of electrostatic repulsion between transition metal ions and the second-order Jahn–Teller effect Kunz and Brown (1995); Bersuker (2006). Niobium sites in the center of the block are less distorted than those at the periphery. The structural principle of blocks as the building unit, as introduced by Wadsley and Roth, also applies to phases in the \ceTiO2-\ceNb2O5 and \ceWO3-\ceNb2O5 phase diagrams Roth and Wadsley (1965b).

The fully oxidised parent compound \ceNb2O5 is a wide bandgap insulator. Low concentrations of valence electrons are introduced through $n$-type doping to form the \ceNb2O_5-$\delta$ phases. This reduction changes the crystal structure, but the structural motif of the blocks is retained, which makes the niobium suboxides an excellent series of phases to study the interplay between charge state and crystal structure. Magnetic susceptibility measurements show that all \ceNb2O_5-$\delta$ phases are paramagnetic, with the number of localised moments increasing with $\delta$ Cava et al. (1991a); Ruscher and Nygren (1991). Spin interactions are antiferromagnetic and their strength increases with the level of reduction, as indicated by their Curie–Weiss constants. However, only the monoclinic \ceNb12O29 phase is found to exhibit long-range antiferromagnetic order, with an ordering temperature of $12\text{\,}\mathrm{K}$ Cava et al. (1991b); Andersen et al. (2005). Electrical conductivity measurements show that all \ceNb2O_5-$\delta$ phases show thermally-activated transport, except for \ceNb12O29, which is metallic down to $2\text{\,}\mathrm{K}$ Ruscher et al. (1988); Cava et al. (1991a). Both electrical and optical measurements indicate that the electron transport in the \ceNb2O_5-$\delta$ phases is effectively one-dimensional along the block columns Rüscher (1992). Despite the evidence for localised electrons, single crystal X-ray diffraction studies on \ceNb22O54 and o-\ceNb12O29 have not found evidence for charge ordering McQueen et al. (2007). Additional studies have been performed on \ceNb12O29 indicating the presence of localised as well as delocalised electrons Ruscher et al. (1988); Lappas et al. (2002); Cheng et al. (2009).

Despite their interesting physical properties, the niobium suboxides have not been investigated with first-principles methods, and the relationship between the level of reduction and the electronic and magnetic properties remains unclear. Two previous first-principles studies have examined the two polymorphs of \ceNb12O29, with rather different conclusions regarding the electronic structure, despite their very strong structural similarity Fang et al. (2014); Lee and Pickett (2015). Additionally, there is new interest in crystallographic shear phases due to their excellent performance as electrode materials in batteries Griffith et al. (2018); Li et al. (2018), and the electronic structure of the suboxides is likely to be similar to that of other shear phases.

In this article, we study six different niobium (sub)oxides using density-functional theory calculations and establish common principles governing their electronic and magnetic behaviour. As the 4$d$ band in these materials is highly complex, we first examine \ceNb22O54 in detail, and then present results on \ceNb12O29, \ceNb47O116, \ceNb25O62, and H-\ceNb2O5. We show that all of these structures host flat and dispersive bands, which correspond to localised and delocalised electronic states. Each block can host a single localised state in the block plane that is spread over multiple niobium sites. Delocalised states are present along the shear planes. These results are independent of the $n$-type doping, and alkali metal doped (lithiated) shear phases show similar features to the suboxides, which has implications for their use as battery electrodes. We discuss experimental studies of electrical and magnetic properties of the suboxides in terms of a consecutive filling of localised and delocalised states. Finally, based on these results, we suggest new directions for theory and experiment.

II Methods

All density-functional theory calculations were performed using the planewave DFT code CASTEP Clark et al. (2005) (version 18.1). Pseudopotentials including Nb $4s$, $4p$, $4d$ and $5s$, O $2s$ and $2p$, and Li $1s$ and $2s$ states were used for all calculations. Calculations using hybrid functionals employed norm-conserving pseudopotentials Hamann et al. (1979), all other calculations were performed using Vanderbilt ultrasoft pseudopotentials Vanderbilt (1990). Crystal structures were obtained from the Inorganic Crystal Structure Database Hellenbrandt (2004) (ICSD) when available. The structure of \ceNb47O116 was constructed as described in Ref. Cava et al., 1991a as a unit cell intergrowth of \ceNb25O62 and \ceNb22O54 since no crystallographic data, other than the lattice parameters, was available. The space groups of H-\ceNb2O5 and \ceNb22O54 are reported as both $P2$ and $P2/m$ in the literature Kato (1976); Cava et al. (1991a); McQueen et al. (2007). These two space group choices differ only in the full or partial occupancy of the tetrahedral site. For modelling purposes, the ion on the tetrahedral site has to be ordered, resulting in space group $P2$. Atomic positions and lattice parameters of the structures were relaxed using the gradient-corrected Perdew–Burke–Ernzerhof (PBE) functional Perdew et al. (1996), until the maximum force on any atom was smaller than $0.01$ eV/$\mathrm{\SIUnitSymbolAngstrom}$. The calculations used a planewave kinetic energy cutoff of 800 eV for ultrasoft pseudopotentials, and 900 eV for norm-conserving pseudopotentials, unless otherwise stated. The Brillouin zone was sampled with a Monkhorst–Pack grid Monkhorst and Pack (1976) finer than $2\pi\times 0.03$ $\mathrm{\SIUnitSymbolAngstrom}$-1. Lattice parameters obtained from the structural relaxations are listed in Table 1, and agree very well with the experimental values. Crystallographic information files (CIF) of the PBE optimised structures are available in the Supplemental Material of this article 111See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.99.075151 for crystallographic information files (CIF) of the structures dealt with in this article.. All electronic structure calculations were performed for antiferromagnetic spin arrangements in the conventional unit cells, as antiferromagnetic spin interactions are observed experimentally Cava et al. (1991a).

Semilocal density functionals suffer from self-interaction error, which can be alleviated by the use of DFT+U. For calculations in this work, the DFT+U implementation in CASTEP Dudarev et al. (1998) was used, which defines an effective $U$ value $U_{\mathrm{eff}}=U-J$. A value of $U_{\mathrm{eff}}=4.0$ eV was chosen for the Nb $d$ orbitals, in line with other studies on niobium oxides that employed similar implementations of DFT+U within planewave codes Nahm and Park (2008). The results presented herein are mostly insensitive to the exact value of the $U_{\mathrm{eff}}$ parameter if it lies in the range 3–5 eV, even though the value of the bandgap does depend on the choice of the $U_{\mathrm{eff}}$ parameter. The structures of \ceNb22O54 and the \ceNb12O29 polymorphs were additionally optimised with PBE+U, and the results are listed in Table 1. PBE and PBE+U lattice parameters agree closely, and PBE+U bandstructure and density of states calculations for compounds other than \ceNb22O54 and \ceNb12O29 were performed on PBE optimised structures.

Hybrid functionals are another way to correct the self-interaction error of semilocal functionals. The range-separated HSE06 functional Heyd et al. (2003) was used to calculate the bandstructure for \ceNb22O54. Due to the significant additional expense incurred by the use of hybrid functionals, the computational parameters for the calculations of bandstructures at the HSE06 level are coarser. The unit cell of \ceNb22O54 contains 610 valence electrons, but since the cell is rather short in one particular dimension and extended in the other two, one cannot use only the $\Gamma$-point in the Brillouin zone (BZ) sampling. Instead, a $1\times 5\times 1$ $\Gamma$-centered $\mathbf{k}$-point grid was used in the HSE06 self-consistent field calculations for \ceNb22O54.

Bandstructure calculations were performed for high-symmetry Brillouin zone directions according to those obtained from the SeeK-path package Hinuma et al. (2017), which relies on spglib Togo and Tanaka (2018). A spacing between $\mathbf{k}$-points of $2\pi\times 0.025$ $\mathrm{\SIUnitSymbolAngstrom}$-1 was used. Density of states calculations were performed with a grid spacing of $2\pi\times 0.01$ $\mathrm{\SIUnitSymbolAngstrom}$-1, and the results were postprocessed with the OptaDOS package Morris et al. (2014), using the linear extrapolative scheme Pickard and Payne (1999, 2000). The c2x Rutter (2018) utility and VESTA Momma and Izumi (2011) were used for visualisation of wavefunction and density data. Data analysis and visualisation was performed with the matador Evans package.

III Results

III.1 \ceNb22O54

\ce

Nb22O54 crystallises in space group $P2/m$ McQueen et al. (2007), and shows an ordered mixture of $3\times 3$ and $3\times 4$ blocks of octahedra, in addition to a tetrahedral site (Figure 1a). Assuming an ionic model, the compound can be described as $(\mathrm{Nb}^{5+})_{20}(\mathrm{Nb}^{4+})_{2}(\mathrm{O}^{2-})_{54}$, with two $4d$ electrons per 22 Nb atoms (1 $e^{-}$ per block, Table 1).

The $\mathbf{a}$ and $\mathbf{c}$ lattice vectors of \ceNb22O54 are longer than $\mathbf{b}$, which is perpendicular to the block plane (Fig. 1, Table 1). The Brillouin zone (BZ) therefore has one long (along $\mathbf{b}^{\ast}$) and two short directions. The PBE+U spin-polarised bandstructure and electronic density of states (DOS) of \ceNb22O54 show a large gap between the valence and conduction bands, which are of oxygen $2p$ and niobium $4d$ character, respectively (Fig. 2). Two fully occupied flat bands (one for each spin) lie within the band gap, leading to the peaks in the DOS below the Fermi level. The flat bands have a very small one-dimensional dispersion, as evidenced by the shapes of the corresponding peaks in the DOS, and represent localised states. In addition to the flat bands, a set of dispersive bands exists just above the Fermi level, which show the largest dispersion along $\mathbf{b}^{\ast}$. The separation between the flat bands and the rest of the conduction states is smallest at special points lying in the $\mathbf{a}^{\ast}$-$\mathbf{c}^{\ast}$ plane of $\Gamma$ ($Y$, $A$, $B$), and largest in the parallel plane at the BZ boundary ($Z$, $C$, $D$, $E$). Due to this pattern, the dispersive bands are also effectively one-dimensional.

With 12 inequivalent niobium sites in the unit cell of \ceNb22O54, site-resolved projected densities of states (PDOS) are complicated and difficult to interpret. More insight is gained by summing PDOS for sets of sites. Figure 3a shows the projection onto different types of niobium sites within the structure, which are classified as central and peripheral, depending on where they sit within the block. We note two things: (1) Both peripheral and central niobium sites contribute to the localised states, even though the contribution of the central sites is greater given the ratio of the two; and (2) only peripheral niobium sites contribute to the unoccupied density of states above the Fermi level (until 0.5 eV above), the contribution from the central sites is exactly zero. The PDOS resolved by block in Fig. 3b demonstrates that one localised state is contained in the $3\times 4$ block, and the other, lower energy one, in the $3\times 3$ block. Both blocks contribute roughly equally to the density of unoccupied conduction states.

Spin density in \ceNb22O54 is predominantly located on the central niobium sites (Fig. 4a), which also dominate the relevant states as seen from the PDOS (Fig. 3a). One spin is located in each block, and the spin arrangement is antiferromagnetic between the two blocks. However, the ferromagnetic arrangement is only marginally higher in energy (less than 1 meV), indicating very weak spin interactions that are likely a result of the long (nm) magnetic interaction lengths. Kohn–Sham orbitals that are occupied by these localised electrons span the entire block, but only have contribution from niobium sites in the same block (Fig. 4b). The flat dispersion is a result of the very weak face-on overlap ($\delta$-overlap) between these orbitals along $\mathbf{b}$. Both localised orbitals are similar in appearance, despite the different sizes of the blocks. This suggests that the presence of these states is a general feature of block-type structures.

The results presented above were obtained from PBE+U ($U_{\mathrm{eff}}=4$ eV) calculations. The $\Gamma\to Z$ segment of the \ceNb22O54 bandstructure computed with HSE06 and PBE is compared to the PBE+U result in Fig. 5. Only the up-spin component is shown, which is associated with the localised electron in the $3\times 3$ block. The bandstructure looks similar for all functionals, and importantly, the relevant feature of localised states, i.e. the flat bands, are present even at the PBE level. However, PBE places the flat bands within the dispersive conduction bands, and both are partially occupied, while both DFT+U and the HSE06 functional place the flat bands below the other conduction bands. PBE also places the opposite spin partner of the localised state in the other block much lower in energy than either PBE+U or HSE06. The precise placement of the flat bands depends on the $U$ value, but in the tested range of 2-5 eV the flat bands are placed below the conduction bands, and the gap between them increases by approximately 200 meV per increase in $U$ by 1 eV. The degree of electron localisation depends on the presence of a gap between flat and dispersive bands. PBE implies metallic behaviour with localised electrons, while HSE06 and PBE+U show full localisation of the electrons. A major difference between the HSE06 and PBE or PBE+U calculations is the size of the gap between valence and conduction bands, which is larger by approximately 1.2 eV for HSE06 compared to PBE+U. The spin density and Kohn-Sham orbitals were plotted from the output of PBE+U calculations, but we note that the results from PBE and HSE06 are visually indistinguishable from the PBE+U results.

III.2 \ceNb12O29

\ce

Nb12O29 is more reduced than \ceNb22O54 and hosts two $4d$ electrons per 12 niobium sites (i.e. 2 per block, Table 1). The two \ceNb12O29 polymorphs are structurally similar, and only differ in the long-range arrangement of the blocks; in the monoclinic polymorph the blocks form a ribbon along $a$, while in the orthorhombic structure the blocks zig-zag along $c$ (Fig. 1).

The bandstructure of monoclinic \ceNb12O29 shows two flat bands (one for each spin), which lead to two peaks in the DOS (Fig. 6). The shape of the real-space unit cell results in a Brillouin zone with two short and one long dimension, and the bandstructure path segments are similar to those in \ceNb22O54. The bands for both spins lie exactly on top of each other due to the symmetry of the crystal structure, even though there is a spatial separation of spins (Fig. 8a). The flat bands coexist with more dispersive conduction bands, which show a dispersion which is largest in the $\mathbf{b}^{\ast}$ direction, making them effectively one-dimensional. Independent of the position of the flat bands, the larger number of electrons per block requires that some of the electrons fill dispersive conduction bands. This indicates a structural capacity for localised electrons. In \ceNb12O29 flat and dispersive bands are interspersed, while in \ceNb22O54, the flat bands lie below the rest of the $d$-bands (cf. Fig. 2). Similar to \ceNb22O54, the central Nb sites contribute exclusively to the occupied density of states in a narrow region that is associated with the flat bands (Fig. 7). The remainder of the conduction states involve contributions from the peripheral sites.

Kohn-Sham orbitals of the localised states (Fig. 8c) look remarkably similar to those in \ceNb22O54, and are predominantly made up of Nb $d$-orbitals lying within the plane of the block. Electrons occupying these localised states are responsible for the non-zero spin density (Fig. 8a). Orbitals associated with dispersive bands (Fig. 8b,d) are made up of $d_{xy}$ and $d_{yz}$ atomic orbitals that are parallel to the crystallographic shear planes. The band dispersion along $\Gamma\to Z$ is explained by a reduction of in-phase overlap of the constituent atomic orbitals along the real-space $\mathbf{b}$ direction. The fact that the contributing atomic orbitals are parallel to the crystallographic shear planes and overlap face-on (Fig. 8b,d), rather than end-on, can be understood from a crystal field argument. For a transition metal ion in an ideal octahedral crystal field, the $t_{2g}$ orbitals form a degenerate set. The \ceMO6 octahedra in shear phases, however, are far from ideal. When the degeneracy of the $t_{2g}$ orbitals is lifted by a distortion, those $d$-orbitals that do not overlap with any $\sigma$-type ligand orbitals will be lowest in energy and contribute to the low-energy $d$-bands.

The bandstructure of the orthorhombic \ceNb12O29 polymorph (Fig. 9) looks very similar to that of the monoclinic polymorph. In fact, the results on the monoclinic polymorph presented above are transferable to the orthorhombic one. Experimentally, both polymorphs are metallic and show antiferromagnetic spin interactions, but only the monoclinic polymorph shows long-range magnetic order Andersen et al. (2005). The magnetic susceptibility of m-\ceNb12O29 can be fit using the Bonner-Fisher form, possibly indicating one-dimensional magnetism Lappas et al. (2002). The differences are clearly subtle, and the small energy differences (10–20 K, around 1 meV) make comparisons using density-functional theory total energy differences difficult. However, the picture of the electronic structure of \ceNb12O29 that emerges is clear: for both polymorphs, conductivity and local moment magnetism are provided by different sets of electrons. Our conclusions on the orthorhombic polymorph are broadly in line with the first-principles study of Lee and Pickett Lee and Pickett (2015). Those authors also found a coexistence of localised and delocalised electrons, with the localised spin residing in a large orbital dominated by the central niobium sites of the blocks, with delocalised electrons forming another subset. Our results as well as experimental studies using heat capacity measurements Cheng et al. (2009) and $\mu$SR spectroscopy Lappas et al. (2002) establish the presence of localised magnetic electrons in m-\ceNb12O29. We note that a previous study suggested the presence of itinerant moments in m-\ceNb12O29 on the basis of GGA calculations Fang et al. (2014). However, the high density of states at the Fermi level that was described to be the reason for the itinerant magnetism in fact arises from the flat band representing a localised state.

III.3 \ceNb25O62 and \ceNb47O116

The compounds \ceNb25O62 and \ceNb47O116 are less reduced than \ceNb22O54 and host less than one electron per structural block unit (Fig. 1, Table 1). The mutually occurring localised and delocalised electronic states that were found above for \ceNb22O54 and monoclinic \ceNb12O29 are also present in \ceNb25O62 and \ceNb47O116. Localised states in blocks of the same size are nearly degenerate, and since only a fraction of the localised states is occupied (less than 1 electron per block), it is very difficult in a first-principles calculation to localise the electrons within a specific block. This could be done if the occupation of particular bands was constrained. Similarly, since the energy of the localised states depends on their occupation, judging the relative position of dispersive and localised states in these two compounds is very difficult. Charge densities for the localised states in the \ceNb47O116 and \ceNb25O62 are shown in Fig. 10. It seems very likely that both \ceNb25O62 and \ceNb47O116 possess only localised electrons, occupying a fraction of these localised states. Since \ceNb47O116 is a unit cell level intergrowth of \ceNb22O54 and \ceNb25O62, and \ceNb22O54 shows complete localisation of electrons, it is very likely that electrons should also fully localise in \ceNb47O116, at least in those parts of the structure that derive from \ceNb22O54.

III.4 H-\ceNb2O5

H-\ceNb2O5 is the high-temperature phase of niobium pentoxide, and crystallises in space group $P2/m$ (Table 1, Fig. 1). As the parent compound of the crystallographic shear structures, its electronic structure provides a reference. However, since it is fully oxidised, all niobium ions have a $d^{0}$ configuration and there are no electrons occupying the conduction band.

The bandstructure of H-\ceNb2O5 shows the presence of flat and dispersive bands (Fig. 11), similar to those found for the other shear phases above. However, the relative position of these bands depends on whether or not they are occupied. Doping by alkali metal ions is one way to introduce electrons into the conduction band, and in the particular case of H-\ceNb2O5 this has a practical relevance. Transition metal oxides in general, and the niobium-based oxide shear phases of this work, are used as electrodes within lithium-ion batteries. Like oxygen removal, lithium intercalation is a method to $n$-dope the material. Similar behaviour can often be observed from charge doping and ion insertion, for example in \ceNa_xWO3Walkingshaw et al. (2004). The H-\ceNb2O5 phase has been studied extensively for lithium-ion battery applications and it is closely related to other shear phases that have been examined for the same purpose Griffith et al. (2016, 2018). Inserting a single lithium per unit cell into the middle of the $3\times 4$ block results in a localised state similar to those present in the niobium suboxides (Fig. 12). Note that the electron is entirely localised within the $3\times 4$ block, with the $3\times 5$ block remaining empty. Oxygen removal (as in the suboxides above) and lithium intercalation (examined here) clearly result in similar electronic structure features.

IV Discussion

Our results establish that the presence of defect-like flat bands and metallic conduction states is an innate feature of block-type structures. This coexistence arises due to the two different types of niobium sites present in the crystal structures; the central \ceNbO6 octahedra are purely corner-shared, the distance between niobium atoms is larger and orbital overlap is reduced. This isolation results in localised electronic states, while along the crystallographic shear planes, where Nb-Nb distances are smaller and orbitals overlap more strongly, delocalised states are present. Each block can host one localised electron that is, rather unusually, spread over multiple niobium sites. This spread over multiple sites explains why single crystal X-ray diffraction studies on \ceNb22O54 and o-\ceNb12O29 do not show the presence of charge ordering McQueen et al. (2007), despite the detection of localised electrons by magnetic measurements Cava et al. (1991a). As the electronic structure features are ultimately a result of the blocks as structural units, the same principles are likely to apply to other crystallographic shear phases in the \ceWO3-\ceNb2O5 and \ceTiO2-\ceNb2O5 phase diagrams.

Regarding possible electronic conduction mechanisms in the niobium suboxides, the relevant quantities are the filling fraction $\nu$ (number of $e^{-}$ per block), and the energy gap between the flat and dispersive bands. For filling fractions of less than one, only localised states are filled (\ceNb25O62 and \ceNb47O116, Fig. 10) and electrons can hop from one filled block to another empty one by a polaron hopping mechanism. The hopping process will have an activation energy. However, in \ceNb22O54 all localised states are filled and this hopping mechanism becomes impossible. With no metallic conduction electrons present, thermal excitation from the defect-like flat bands into the dispersive conduction bands might provide the dominant mechanism, as illustrated in Fig. 13a. This mechanism is reminiscent of doped semiconductors, and the activation energy associated with this process will depend on the separation between the flat and dispersive bands (cf. Figs 5, 13a). Thermal excitation from flat into dispersive bands is also possible in \ceNb25O62 and \ceNb47O116, and could coexist with a polaron hopping mechanism. Finally, in the case of \ceNb12O29, all localised states are filled, but metallic conduction states are also partially filled (Fig. 13b). The result is metallic conductivity, which does not require thermal activation. Both Rüscher et al. Ruscher et al. (1988) and Cava et al. Cava et al. (1991b, a) have studied the conductivity of \ceNb2O_5-$\delta$ compounds and observed that all phases except \ceNb12O29 exhibit thermally activated conductivity. Rüscher et al. also noted the effectively one-dimensional electron transport properties along the block columns Rüscher (1992), which are consistent with the calculated band dispersions. In addition, the experiments of Cava et al. show semiconducting electronic behaviour for \ceNb25O62 from 4–300 K; \ceNb47O116 and \ceNb22O54 exhibit semiconducting (thermally activated) conductivity from 0–250 K and from 0–100 K, respectively Cava et al. (1991a). Beyond those temperatures, there is a metallic-like range of temperatures in which conductivity decreases again. Given this change in the temperature dependence of the conductivity from semiconducting to metallic, the flat bands associated with localised electrons are likely shallow donor levels (Fig. 13). The complex interplay between electron localisation and delocalisation in the suboxides is more similar to phenomena occurring in semiconductors on $n$-type doping, but distinctly different from metal-insulator transitions in transition metal oxides Imada et al. (1998). Our results suggest that a similar phenomenon of crossover from localised to metallic conduction could occur on lithium doping of H-\ceNb2O5, which might be observed with electrochemical, spectroscopic, or magnetic measurements.

Magnetic susceptibility measurements on the suboxides show that the number of localised moments increases with the degree of reduction Ruscher and Nygren (1991); Cava et al. (1991a). However, under the assumption that $g=2$, the number of moments calculated from the measurements is smaller than the number of introduced electrons Ruscher and Nygren (1991); Cava et al. (1991a). For \ceNb12O29, this is consistent with one delocalised and one localised electron, but for the remaining suboxides this apparent reduction in the number of local moments is unexpected. Our first-principles results are consistent with complete localisation of electrons until a threshold is exceeded, and therefore all moments should be seen. Deviations in the $g$ value of the electrons might explain the discrepancy. As the electrons in these suboxides are well-localised, electron paramagnetic resonance (EPR) spectroscopy could provide some insight into the nature of the electronic states and the $g$ values. For \ceNb22O54 in particular, the different shapes of the magnetic orbitals could be used to detect electrons occupying specific blocks. Another possibility is that localised electrons contributing magnetic moments coexist with magnetically inactive electrons in all suboxides, not just \ceNb12O29. However, we see no evidence for this in our calculations, and the thermally activated conductivity of \ceNb22O54, \ceNb47O116 and \ceNb25O62 seems inconsistent with the presence of magnetically inactive (Pauli-paramagnetic) metallic electrons.

Long-range antiferromagnetic order is observed only in the monoclinic \ceNb12O29 phase below 12 K, all other niobium suboxides are paramagnetic Cava et al. (1991b, a). The Curie–Weiss constants of \ceNb2O_5-$\delta$ are in the range of $024\text{\,}\mathrm{K}$ ($02\text{\,}\mathrm{meV}$), and indicate antiferromagnetic interactions that become stronger with increasing degree of reduction Cava et al. (1991a). In \ceNb25O62 and \ceNb47O116, some of the localised states are empty (cf. Fig. 10), and the magnetic lattice is not fully filled. Independent of the strength of interaction, if not all spins have neighbours to couple with, or there is some randomness in the distribution of the spins, long-range magnetic order is unlikely to emerge. With first-principles calculations it is very difficult to address the question of why only m-\ceNb12O29 orders, but \ceNb22O54 and o-\ceNb12O29 do not, since the energy differences between different magnetic states are very small. However, we can discuss the possible spin-spin interactions simply based on the shape and orientation of the magnetic orbital within the crystal structure. We will focus in particular on monoclinic \ceNb12O29, but similar considerations apply to the other suboxides. The magnetic orbital lies within the plane of the block. The two closest distances (two neighbours) between spins (two nearest neighbours) are along the block columns, with a separation of about 3.8 $\mathrm{\SIUnitSymbolAngstrom}$ ($J_{b}$, Fig. 14a bottom). By symmetry, the interaction with spins in the four next-nearest neighbouring blocks along $c$, that are offset by 0.5 $b$, has to be the same ($J_{c}$, distance 10.6 $\mathrm{\SIUnitSymbolAngstrom}$, Fig. 14a bottom). In addition to that, each block is connected to two blocks on the same level in monoclinic \ceNb12O29 along the $a$ direction ($J_{a}$, distance 15.9 $\mathrm{\SIUnitSymbolAngstrom}$, Fig. 14a top), and four others offset by 0.5 $b$ along $a$ (distance 15.3 $\mathrm{\SIUnitSymbolAngstrom}$, Fig. 14a top). Different spin arrangements are easily obtained from DFT calculations (spin densities are shown in Fig. 14b), but the energy differences between them are very small (few meV), and change significantly with the level of theory (PBE or PBE+U). Energy differences of a few meV are consistent with the interaction strengths obtained experimentally. The lowest energy magnetic ordering found in our calculations is antiferromagnetic along the $c$ direction (Fig. 14b, top left).

V Conclusion

We have shown that the electronic structure features common to $n$-doped crystallographic shear phases include (1) effectively one-dimensional flat and dispersive bands corresponding to localised and delocalised electronic states (2) electron localisation in orbitals spanning the block planes, and (3) the partition of localised and delocalised states between central and peripheral niobium sites. Structural block units are also present in \ceWO3-\ceNb2O5 Roth and Wadsley (1965c) and \ceTiO2-\ceNb2O5 Wadsley (1961a, b) phases, and many of these mixed-metal shear phases have been explored as lithium-ion battery electrodes Griffith et al. (2018); Guo et al. (2014). The principles laid out in this work are likely transferable to these compounds, and are important for the interpretation of spectroscopic and electrochemical data.

The niobium suboxides show a transition from localised to delocalised electrons, but it is much smoother than commonly observed for metal-insulator transitions in transition metal oxides. In fact, our results portray the suboxides to be closer to $n$-doped semiconductors, but with a limited capacity for localised electrons. Once a filling threshold is exceeded, delocalised metallic electrons are simply added to existing localised electrons. This process is likely to occur in heavily lithium-doped shear phases during battery operation. Similarly, the experimentally observed crossover from localised to delocalised electronic behaviour in \ceWO_3-x Salje and Güttler (1984) might have the same underlying mechanism, as \ceWO_3-x phases also exhibit some amount of crystallographic shear. More broadly, the niobium suboxides are an elegant example of the interplay between crystal and electronic structure, and the balance between electron localisation and delocalisation in oxides of an early transition metal.

Acknowledgements.

The authors would like to thank Bartomeu Monserrat and Ieuan Seymour for useful discussions. We acknowledge the use of Athena at HPC Midlands+, which was funded by the EPSRC on grant EP/P020232/1, in this research via the EPSRC RAP call of spring 2018. C.P.K. thanks the Winton Programme for the Physics of Sustainability and EPSRC for financial support. K.J.G. thanks the Winston Churchill Foundation of the United States and the Herchel Smith Foundation. K.J.G. and C.P.G. also thank the EPSRC for funding under a programme grant (EP/M009521/1). The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Material files.