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Disentangling the Effects of Inter- and Intra-Octahedral Distortions on the Electronic Structure in Binary Metal Trioxides Woosun Jang, Jongmin Yun, Taehun Lee, Yonghyuk Lee, and Aloysius Soon J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b11295 • Publication Date (Web): 25 Jan 2018 Downloaded from http://pubs.acs.org on January 27, 2018

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The Journal of Physical Chemistry

Disentangling The Effects of Inter- and Intra-Octahedral Distortions on The Electronic Structure in Binary Metal Trioxides Woosun Jang,†,‡ Jongmin Yun,†,‡ Taehun Lee,† Yonghyuk Lee,† and Aloysius Soon∗,† †Department of Materials Science and Engineering, Yonsei University, Seoul 03722, Korea ‡Contributed equally to this work E-mail: [email protected]

Abstract Recently, a sub-class of binary perovskite-structured metal trioxides, such as WO3 and MoO3 , have been propounded for many key optoelectronic applications due to their proper band edge positions and appropriate band gap sizes. Unlike their superclass perovskites, the structure-property relationship for these binary metal trioxides is less apparent, given that they suffer from much larger structural deformities within the octahedra. In this work, by using first-principles density-functional theory calculations and atomistic scale models, we examine the internal and external distortions of WO3 and MoO3 polymorphs. We then compare our results with conventional polyhedral distortion descriptors and finally use a refined data set of different perovskite-structured oxides to establish and demonstrate how these binary metal trioxides operate with a different structure-property relationship from the conventional oxide perovskites.

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Keywords Density-functional theory; Binary metal trioxides; Octahedral distortions; Band gap engineering

1 Introduction Perovskite-structured materials (ABX3 ) are long known for their outstanding physiochemical properties and have been exploited for a wide range of key application areas such as heterogeneous catalysis, 1–3 opto-electronics, 4 fuel cells, 5–8 and multiferroics. 9,10 The versatility of these functional materials originates from the complex association of their crystal and electronic structures. 11,12 In particular, the so-called intra- and inter-octahedral distortions have been suggested as key structural factors that may be used to design and determine the electronic structure of these ABX3 functional materials. 13–15 Adopting the same ABX3 archetype crystal structure, the halide-based (hybrid) perovskite materials have rose to flame recently for their impressive track records of everincreasing solar cell efficiencies (of more than 20 %), and being cheap and simple to produce. 16–18 In fact, it was recently reported that steric engineering can be an effective means to tune the optical band gaps in metal-halide perovskites, 14 and a non-trivial role of octahedral tilting in organohalide perovskites on its optoelectronic structure exists. 15 This leads one to believe that the structure-property relationship for these proper conventional ABX3 perovskites may well be rationalized by examining the inter-octahedral distortions within the rigid framework of close-to-ideal octahedra (i.e. no strong structural distortions within the octahedra). It is then clear that a more straightforward, tilting-only design rule can be applied to exploit their optoelectronic properties (especially the band gap energy, Eg ) for future key energy applications. 14,15 As a sub-class of ABX3 perovskites, binary metal trioxides (i.e. BO3 where the A cation

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site in perovskite is vacant, as in MoO3 and WO3 ) are gaining the limelight for their roles in important energy technologies such as in photocatalysis and carrier transport layer in organic devices. 19–23 Similar to their ternary oxide counterparts, these binary metal trioxides are also highly-tunable, in the sense that their electronic structure also shows a strong dependence on their explicit atomic arrangements and specific crystal structures. 24,25 Unlike their superclass ABX3 perovskites, the structure-property relationship for these binary metal trioxides is less apparent, given that they suffer from a much larger structural deformities within the octahedra, resulting in strongly coupled intra- and inter-octahedral distortions. Thus, recent structure-property rules outlined in Refs. 14,15 can no longer work nor be applied for these binary trioxides with tilted non-ideal BO6 octahedral networks. It is also not clear how strongly the intra- and inter-octahedral distortions are coupled and which will actually have a dominant effect on their desired material property (e.g. the Eg ). It is therefore our aim in this work to use first-principles calculations and a simple, straightforward atomistic scale model to examine and study the disentanglement of the coupled intra- and inter-octahedral distortions in perovskite-related binary oxides (exemplified by all known polymorphs of WO3 and MoO3 ). We then compare our results with conventional polyhedral distortion descriptors (e.g. quadratic elongation index), and finally use a refined data set of 35 different perovskite-structured oxides (as obtained from the Materials Project database 26 ) to establish and demonstrate how these binary metal trioxides operate with a different (but coherent) structure-property relationship from the conventional proper oxide perovskites.

2 Methods 2.1 Computational details All density-functional theory (DFT) calculations are performed using the Vienna Ab initio Simulation Package (VASP). 27–29 The van der Waals (vdW) corrected semilocal generalized 3



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gradient approximations (GGA) to the exchange-correlation (xc) functional, optB88 30 is used for all geometry relaxations to obtain the optimized structural parameters. The projectoraugmented wave (PAW) method 31 is adopted to describe the ion-electron interactions, and a planewave basis-set with a kinetic energy cutoff of 500 eV is used to expand the Kohn-Sham orbitals for all DFT calculations. The valence electron configurations for Mo, W, and O atoms are taken as 4s2 4p6 4d5 5s1 , 5p6 5d5 6s1 , and 2s2 2p4 , respectively. Furthermore, a Γcentered k-point mesh with a grid spacing of at least 0.3 Å−1 is used to sample the Brillouin zone. All DFT calculations have been tested for convergence with respect to the kinetic energy cutoff and k-point grid where the total energies, forces, and external pressure varying less than 20 meV, 0.02 eVÅ−1 , and 0.5 kbar, respectively. For all self-consistent field calculations, the total energy convergence threshold is kept at 10−5 eV. After obtaining the optimized geometries, to afford a more accurate description of the electronic band structure of these oxides, the hybrid DFT xc functional due to Heyd, Scuseria, and Ernzerhof (HSE06) is employed. 32,33 Using the GULP code, 34 the Madelung potentials are obtained under the point charge model using an Ewald summation technique. 35 Here, the electrostatic Madelung potential only depends on the charges of the ions and their separations following the equation:

V0 =

N ∑ qi r i=1 0−i

,

(1)

where V0 , qi , and r0−i denotes the Madelung potential, ion charges, and ion separations, respectively.

2.2

Scale model: An approximate model for binary metal oxides

To decouple the effects from the inter- and intra-octahedral distortions observed in the binary metal oxides, WO3 and MoO3 , we have adopted a simple (but tractable) atomistic scale model

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for both oxides. We first start from a cubic unit cell containing one MO6 octahedron (where its octahedral volume is set to that of the optimized β-MoO3 and γ-WO3 structures). This is taken as the ideal non-distorted scale model where the MO6 octahedra are connected at all vertices. We note that this first approximation is not completely unfounded as the optimized β-MoO3 and γ-WO3 structures can, in fact, be understood as distorted cubic structures. We emphasize that this starting ideal cubic non-distorted structure is free from either inter- nor intra-octahedral distortions, which makes this scale model a good starting point to study the individual contributions of the inter- and intra-octahedral distortions to the electronic band gap energy of the WO3 and MoO3 polymorphic structures. Next, we consider three possible contributions to general intra-octahedral distortions, i: (1) distortions due to having metal cations displaced far from the center of the octahedron, iM , (2) distortions due to shape of the octahedron, ishape , and finally (3) distortions due to the volume (i.e. size) of the octahedron, ivol . To study iM , while keeping the O atoms at the vertices fixed, we displace the metal cation (M) away from the central position towards the edge of the octahedron. This is in line with previous reports where the most commonly observed off-centered positions for Mo and W cations are near the edges of the octahedron. 36 To model ishape , we apply both compressive and tensile strain to the atomistic scale model on the z-direction while keeping the octahedral volume as a constant. This naturally determines the lattice parameters in both the x- and y-directions while we vary that in the z-direction, distorting the overall shape of the octahedron in this scale model. Likewise, to account for ivol , we isotropically expand and compress the lattice constant of our cubic scale model to reflect the changes to the volume or size of the octahedra. In this way, it will allow us to independently survey and study the individual contributions – iM , ishape , and ivol – to the intra-octahedral distortions in our binary trioxides of interest. After isolating the intra-octahedral distortions, we now turn to the inter-octahedral distortions. Here, we can also decouple individual contributions, and identify two key factors: (1) distortions due to the angle tilting of the octahedra, Itilt and (2) distortions due to the

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way how the individual MO6 are connected (i.e. via the vertex, edge, or face of the octahedron), Iconnect . Given that the latter (Iconnect ) is not straightforward to discuss (e.g. may involve changes to the stoichiometry of the system in question), at the first instance, we temporarily neglect its contribution to the inter-octahedral distortions. Thus, in this work, we focus on Itilt and design our tilt angle system based on the Glazer notation. 37 In particular, we choose the simplest a+ b+ c+ tilt system. Adapting to Ref. 14, we first consider the ideal cubic MO6 octahedron with the center metal atom placed at the origin, and six vertex O atoms located at (±1, 0, 0) d, (0, ±1, 0) d, and (0, 0, ±1) d respectively, where the d denotes the ideal M-O distance. To model Itilt , we follow the general concept of Euler angles (i.e. θ, ϕ, and ψ), as done for the previous work on halide perovskites in Ref. 14. Here, θ is the tilt angle from the c-axis, and ϕ denotes the precession angle to the apical c-axis, while ψ is the spinning angle with the spinning axis of octahedral apical direction (c-axis). With following the Ref. 38, rotation matrix can be defined as: 



cos ψ cos ϕ − cos θ sin ϕ sin ψ  cos ψ sin ϕ + cos θ cos ϕ sin ψ   sin θ sin ψ

− sin ψ cos ϕ − cos θ sin ϕ cos ψ − sin ψ sin ϕ + cos θ cos ϕ cos ψ sin θ cos ψ

sin θ sin ϕ   − sin θ cos ϕ   cos θ

After constructing the tilted octahedron, we transpose the octahedron to build a (2 × 2 × 2) supercell with three mirror planes perpendicular to the three orthogonal unit cell axes crossing over the corner-shared vertex oxygen atoms.

2.3 Generic measurement indices for polyhedral distortions To quantitatively determine the extent of intra-octahedral distortion (i) within a single polyhedron in a given crystal structure, one can define a bond length-based index (i.e. the quadratic elongation, ⟨λ⟩) and a bond angle-based index (i.e. the bond angle variance, σ 2 ,

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in in

◦2

). 39 Mathematically, they can be expressed as: 1∑ ⟨λ⟩ = n i=1 n

( )2 li l0

(2)

,

1 ∑ (ϕi − ϕ0 )2 m − 1 i=1 m

σ2 =

,

(3)

where n is the coordination number of the central M atom (i.e., n = 6 refers to an octahedral environment), li is the distance from the central M atom to the ith coordinating atom, and l0 is the the center-to-vertex distance of an ideal polyhedron of the same volume in Equation 2, and m, ϕi and ϕ0 in Equation 3 denotes the number of bond angles, ith bond angle, and the ideal bond angle of the regular polyhedron.

2.4 Continuous shape measurement method Another means to capture the structural deviations of various binary and ternary oxide structures is to employ the continuous shape measurement methodology (CShM). 36,40 Within the CShM formalism, a vector, ⃗qi from given a distorted structure which have N vertices is first defined and then another vector, p⃗i which originates from a structure within a constrained symmetry and the given shape. By using these two vectors, CShM allows one to derive the S value which then indicates the degree of distortion for a closed system with the following equation:  2 |⃗ q − p ⃗ | i   i=1 i × 100 , S = min  ∑N 2 q − ⃗ q | |⃗ i 0 i=1 

∑N

(4)

where p⃗i and ⃗qi are the vectors of atomic coordinates of distorted and symmetric constrained structure in the system which have N vertices, and ⃗q0 is the vector derived from the atomic position of the geometric center. The obtained S expresses the degree of agreement for a structure Q in shape with respect 7



to perfect symmetric structure, P . In short, when S is close to 0, no polyhedral distortion is present in the structure (when compared to the target symmetry). Vice versa, if calculated S is a large value, severe structural distortions are present, deviating from the ideal symmetry and shape.

2.5 Tilt angle measurements in strongly distorted octahedra For cases where the crystal structure suffers from a large structural distortions within the interconnected polyhedrons, strongly coupled intra- and inter-octahedral distortions may prevail (as in the cases of MoO3 and WO3 ). Geometrically, referring to Figure 1, we measure three different tilt angles: the θMVM , θCVC , and θVVV . Specifically, we survey all pairs of adjacent octahedra (within periodic boundary conditions) and calculate every possible tilt angles between these two octahedra. We find that the measured values do not deviate significantly for each tilt angle type, and thus supporting the use of a statistically averaged value for each tilt angle per crystal structure.

3 Results and discussion 3.1 Polymorphic expressions in WO3 and MoO3 The basic building block in perovskites (ABX3 ) and perovskite-related materials are BX6 octahedra, with the A cation in cuboctahedral site. In the case of the binary metal trioxides (where the A cation is missing), how the MO6 octahedra are packed and connected dominates, and the origins behind the relation between MO6 octahedral distortions and their physiochemical properties is non-trivial and complex. The polymorphic expressions in these binary metal trioxides are determined by the spatial network of connecting MO6 octahedra (as a basic building block), and are notably energetically competitive. 25,41,42 In particular, up to eight different polymorphic phases of WO3 (see Figure S1 of the Supporting Information) have been reported – namely, the cubic (cub), tetragonal (tet), 8


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Figure 1: (Color online) (a) The band edge alignment between different polymorphic phases of MoO3 (shaded in blue) and WO3 (shaded in red), taken with reference to the vacuum potential. Details of the band alignment approach is outlined in the Supporting Information. (b) Schematic figure showing a network of highly-distorted corner-sharing octahedra in binary metal trioxides. V, C, and M denotes the vertex O anion, centroid of octahedron, and metal cation inside the octahedron. orthorhombic (ort), triclinic (tri), low-temperature monoclinic (ε), room-temperature monoclinic (γ), and two hexagonal (H1 and H2) phases, with the γ phase most widely studied. 41–43 Less well-studied, MoO3 may crystallize in four different polymorphs (see Figure S2 of the Supporting Information), i.e. the orthorhombic (α), 44,45 hexagonal (h), 46–48 and two monoclinic (II 49–51 and β 52–54 ) phases. For WO3 , depending on the polymorphic phase, the HSE06 calculated Eg values can range from 1.60 to almost 3.00 eV, while for that of MoO3 , the corresponding range is from 2.80 to 4.12 eV (largest for h-MoO3 , with the Eg variation plotted in Figure 1(a) and tabulated in Table S1 of the Supporting Information). In these structures with polyhedral networks, it is known that the local electrostatic potential each atom experiences (i.e. Madelung potential) may influence the absolute band edge positions. 35,55 In Figure S3, following the argument in Reference 35, the variation in the Madelung potentials of the metal cation and oxygen

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anion for WO3 and MoO3 polymorphs are plotted. In general, the variation trend of the band edge positions (with the exception of H1- and H2-WO3 ) is established. For the H1and H2-WO3 phases, the slight deviation may be attributed to the different coordination environment of the oxygen anion and thus the metal-oxygen orbital hybridization. However, as depicted in Figure 1(b), the octahedral network in these binary metal trioxides undergo large intra- and inter-octahedral distortions. The extent of how these distortions in their octahedral networks may dictate the physiochemical properties of each polymorph is still largely unexplored. To simply disentangle the complex contributions to octahedral distortions in these pseudoperovskite-related materials (including binary trioxides of W and Mo, where both intra- and inter-octahedral distortions can be strongly coupled), we propose that their physiochemical properties, X (e.g. the band gap energy, Eg ) can be expressed as X = f (I, i) ∼ = f (Itilt , Iconnect ) + f (iM , ishape , ivol ), where I and i denotes the contributions to the inter- and intra-octahedral distortions, respectively. Here, Itilt and Iconnect are inter-octahedral distortion factors due to the geometrical tilt about the connection vertex of two octahedra and that due to way how the individual octahedra are interconnected (i.e. either via the vertex/corner, edge, or face of the octahedron), respectively. Likewise, iM , ishape , and ivol are intra-octahedral distortion factors due specifically to metal off-center displacement, shape, and volume (or size) changes upon distortion, respectively. To understand the contributions of these individual components for the various polymorphs of both WO3 and MoO3 , we now turn to our simple binary metal trioxide atomistic scale model which allows us to start with the ideal non-distorted cubic for both oxides. Here, the word scale model (or scale similitude), indicates an artificial model which is designed to explain certain phenomena or properties observed in the complex system. The concept of scale model is commonly used in many fields of engineering, since it allows one to provide a physical representation of the system of interest while maintaining accurate relationships

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between all important aspects of the model. Here, the advantage of this approximate model allows one to break down the complexity of the real system to demonstrate some behavior or property of the original without directly involving the complex system.

3.2 Examining inter-octahedral distortions with scale models We separate the contributions from inter-octahedral distortions into mainly two effects: Itilt and Iconnect , where in this work we will focus predominantly on the former while neglecting the latter effect at the first instance. As shown in Figure 2(a), to study Itilt , we modulate the tilt angle of the octahedra, θVVV (which is simply the oxygen vertex-to-vertex-to-vertex angle) in our atomistic scale models from 180 to ∼ 130 ◦ while keeping the minimum dO−O not shorter than that observed in the optimized WO3 and MoO3 structures). For each Itilt model, using hybrid HSE06 functional, we calculated the electronic Eg for each tilted structure. Here, we can see that, based on our scale model, the change in the Eg is found to be quite small (∆Eg ≤ 0.15 eV), reflecting that distortions from geometrical tilts about the connection vertex of two octahedra is rather small and may be negligible. This weak dependence on the inter-octahedral tilt distortions is rather interesting as it is in contrast to what has been reported for organo-halide perovskites 14,15 where the inter-octahedral distortions greatly modifies their optical band gap energies. To understand the possible origin of this weak dependence, we turn to the electronic density-of-states (DOS, calculated using the hybrid HSE06 xc functional) for the MoO3 scale model (as plotted in Figure 2(a) for θVVV = 180, 157, and 145 ◦ ). Given that the center metal atom is bound to 6 neighbouring O atoms within an octahedral ligand field, we can also plot the orbital contributions to the DOS by grouping the dz2 and dx2 −y2 as the Mo 4d eg states, while the dxy , dyz , and dxz as Mo 4d t2g states. For θVVV = 180 ◦ , our ideal non-distorted cubic structure clearly shows that valence band and conduction band edges are mainly composed the O 2p and Mo 4d t2g states, respectively. Upon tilting, in the valence band region, there is still a major contribution from the O 11



Figure 2: (Color online) (a) Calculated band gap energies (Eg , in eV) of the scale model (triangle: MoO3 ; circle: WO3 ) as a function of V-V-V inter-octahedral tilt angle (θVVV ). (b) Calculated Eg as a function of shape distortion. (c) Calculated Eg as a function of volume changes. (d) Calculated Eg as a function of how far the metal cation is displaced from the center of the ideal cubic octahedron towards the edge. Dotted vertical lines denote the ideal cubic model limit. The corresponding projected density-of-states for selected MoO3 scale model structures in each case are presented in the left panel. Red, purple, and blue shaded regions denote the O 2p, Mo 4d eg , and Mo 4d t2g states, respectively.

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2p states, while deeper in the conduction band region, a clear further splitting of the Mo 4d eg and Mo 4d t2g states is noticed. The O 2p states also overlap more locally with the Mo 4d t2g states. We find that overall DOS is modified by the inter-octahedral tilts (especially in the conduction band Mo 4d states), however these changes are found much deeper in both the valence and conduction bands rather than the states near the band edges, and thus unraveling the weak dependence of the Eg on the inter-octahedral tilt distortions. We note that the DOS for the WO3 scale model gives qualitatively the same results as the MoO3 scale model for all distortion contributions, and thus is not shown.

3.3 Examining intra-octahedral distortions with scale models Moving on the the intra-octahedral distortions, using the scale model, we explore three contributions from intra-octahedral distortions, namely ishape , ivol , and iM , sequentially. For many classes of materials composed of polyhedral units, it is common to find geometrical shape distortions that contributes to the overall stability of the material. One of the most common distortion type is that of the Jahn-Teller effect, which resolves degeneracies in electronic levels by breaking certain crystal symmetries, resulting in a lowering of the total energy of the system. 56,57 To study the ishape contributions, we mimic the Jahn-Teller effect by applying both compressive and tensile strain (up to ±6 % of the original shape) to the atomistic scale model along the z-direction while keeping the original equilibrium octahedral volume fixed (see the atomic structure inserts in Figure 2(b)). For each shaped-distorted structure, we calculate the Eg and plot this in Figure 2(b). Here, we also find that this Jahn-Teller-type shape distortion has minimal impact on the Eg values (∆Eg ≤ 0.3 eV). On this note, we do observe that Eg decreases almost monotonically for both compressive and tensile strains. While inspecting the corresponding calculated DOS in Figure 2(b), we find almost no significant change to the overall DOS (except for the some delocalization of Mo 4d eg states deep in the valence band region and minor renormalization of the O 2p states). Similar to the Itilt contributions, the weak dependence of Eg on ishape is 13



also due to the unaffected band edge states upon shape distortion. Referring now to Figure 2(c), we find that for the next contribution – ivol , a slightly larger influence on the Eg variation is noted. Here, we simply vary the volume of the MO6 octahedron in the atomistic scale model isotropically (for ∼ ±18 % of the original volume) and find that the Eg value monotonically decreases with increasing octahedron volume for ∆Eg of up to 0.35 eV. From the calculated DOS presented in Figure 2(c), this weak correlation is again clear from the unchanged band edge states while the largest alteration to the DOS occurs deeper in the bands, especially so for Mo 4d eg states. Gathering both orthogonal intra-octahedral distortion (ishape and ivol ) and inter-octahedral tilting (Itilt ) effects, based on our simple atomistic scale model, we can deduce that ∆Eg does seem to have a weak dependence on all these orthogonal and tilt distortion effects. Now, we turn to the last non-orthogonal intra-octahedral distortion – iM , the distortions due to having metal cations displaced far from the center of the octahedron. Based on our scale model, we examine the iM effect by moving the central metal cation in finite steps towards the edge of the octahedron, and calculate the electronic Eg value for each distorted structure. In Figure 2(d), as the off-center displacement of the metal cation increases, for both the scale models of WO3 and MoO3 , the Eg dramatically increases and then saturates to a constant value (i.e., ∼ 4.25 and 3.32 eV in WO3 and MoO3 , respectively). The maximum ∆Eg upon cation off-center displacement is approximately 2.7 eV in both MoO3 and WO3 . This large increase in ∆Eg is strongly corroborated in the changes to the calculated DOS, as shown in Figure 2(d). Though not much variation to the filled valence bands are found, a direct up-shifting to higher energies for the conduction band states is found. The Mo 4d t2g and O 2p states are now more localized upon a larger iM effect, and the splitting of Mo 4d eg states is observed (which results in a small overlapping of the Mo 4d eg and O 2p states). Focusing only on the band edge states after iM distortion, the O 2p states are still the major contribution to the valence band edge states, while the contribution of the Mo 4d eg and O 2p states to the conduction band edge states increases. One may rationalize the

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the anisotropic overlapping between metal cation 4d orbitals and O 2p orbitals increases as the metal cation moves toward the octahedral edge position. Moreover, this non-orthogonal intra-distortion will also be somewhat responsible for the breaking of octahedral crystal field to a distorted tetrahedral field, which is then reflected in the splitting and lowering of the Mo 4d eg energy levels. Collectively, with our simplified binary metal trioxide atomistic scale models, we can now disentangle the individual contributions of inter- and intra-octahedral distortions to the electronic band gap energies of these pseudo-perovskite oxides, where the largest dominant factor comes from iM , the off-center metal cation intra-distortion (and not the speculated inter-octahedron tilts as seen in proper pervoskites 14,15 ).

3.4 Generalized angle-based descriptors and polyhedral measurement indices Concentrating now on the dominant influence on the metal cation off-center distortions within the octahedron (iM ), it will be timely to move away from the atomistic scale models and survey this effect in the various polymorphic expressions for both WO3 and MoO3 (cf. Figures. S1 and S2). To compare and contrast our results for both WO3 and MoO3 , we have also turned to the Materials Project database 26 to extract the structural and Eg information for oxide perovskites (where available). Details for this data extraction is reported in our Supporting Information. Thus, this results in a pool of 46 oxide materials with perovskiterelated structures (including the binary metal trioxides examined in this work). To take into account the importance of iM in studying the structural variations in the oxides with an octahedral network, we define three angle-based descriptors: (1) M-V-M angle (θMVM ), (2) C-V-C angle (θCVC ), and (3) V-V-V angle (θVVV ). The M-V-M angle measures the angle between center metal cations in two adjacent octahedra and the sharing vertex O atom (M1 -V2 -M2 in Figure 1(b)), the C-V-C angle is then the angle between two geometric centroids of neighboring octahedra (which explicitly takes the iM non-orthogonal 15



Figure 3: (Color online) Relations between various generalized angle descriptors (cf. Figure 1) for the polymorphs of WO3 and MoO3 , and other perovskite-structured oxides: 26 (a) θMVM versus θCVC , (b) θCVC versus θVVV , (c) θMVM versus θVVV , and (d) the distortion of MO6 octahedron versus that of the O6 octahedron, as measured by the continuous shape measurement method (CShM). 36,40 The size of markers indicates the magnitude of the band gap energies, Eg while the dotted and dot-dashed lines are meant to guide the eye. intra-octahedral distortion effect into account) and the sharing vertex O atom (C1 -V2 -C2 in Figure 1(b)), and lastly, the V-V-V angle is defined as the angle between sharing vertex O atoms and their diagonal vertices in adjacent octahedra (V1 -V2 -V3 in Figure 1(b)). Here, the geometric centroid of the octahedron is simply calculated by numerically averaging the positions of six vertex oxygen atoms. The non-tilted structures is expected to possess the inter-octahedral distortion angle descriptor of 180◦ . From these definitions, our angle-based descriptors are general and flexible enough to include cases where only inter-octahedron tilts are dominant (i.e. with almost no iM effect, such as for conventional proper perovskites 15,58,59 ) to situations where intra-octahedral 16


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distortions become severe (as in the binary metal trioxides considered in this work). In passing, for conventional proper perovskites, owing to the close-to-ideal geometry of the octahedron (with negligible iM effect), θMVM and θVVV are often used as a standard descriptor for studying octahedral distortions. 15,58,59 Other useful structural descriptors commonly found in the literature are also used as a benchmark to test if our generalized angle-based descriptors can successfully capture the expected features of the structure (distortion)-property relations. Namely, we have calculated the quadratic elongation index, ⟨λ⟩ 39 (cf. Equation 2), bond angle variance index, σ 2 39 (cf. Equation 3, given in

◦2

), and lastly using the continuous shape measurement method

(CShM) 36,40 to quantify polyhedral distortions (cf. Equation 4). By and large, these polyhedral measurement methods focus predominantly on intra-octahedral distortions (with on explicit considerations for inter-octahedral distortions, where our generalized angle descriptors may appear superior). Now, we proceed to compare the structure (distortion)-property relations for this group of perovskite-related oxide materials. Firstly, we plot the relations between various generalized angle descriptors – θMVM versus θCVC in Figure 3(a), θCVC versus θVVV in Figure 3(b) and θMVM versus θVVV in Figure 3(c). In all cases, the size of markers are weighted by the magnitude of the band gap energies, Eg . The diagonal trend line in the plot of θMVM versus θCVC allows us to distinguish the extent of metal cation off-center intra-octahedral distortions (iM ) in the presence of inter-octahedral tilts, while that in the plot of θCVC versus θVVV then expresses the severity of both ishape and ivol in the presence of Itilt . The plot of θMVM versus θVVV then truly measures the full effect of iM , ishape , and isize including Itilt . By virtue of our definition, oxide structures without strong intra-octahedral distortions will fall nicely on the diagonal linear line where θMVM ∼ = θCVC ∼ = θVVV . This happens to be the clear case for most of the oxide perovskites where only the inter-octahedral tilts are thought to be the most important. Most of our calculated binary MoO3 polymorphs fall out of the diagonal trend lines, largely suggesting that the iM effect for MoO3 polymorphs can

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be very large and critical. In contrast, the WO3 polymorphs tend to exhibit a much smaller iM effect (but not negligible). We have also employed another sensitive polyhedral shape measurement approach using the CShM 36,40,60,61 and plotted the relation between MO6 octahedron distortions and the O6 octahedron distortions in Figure 3(d), weighting the markers by the magnitude of Eg . Within the CShM approach, the volume of the octahedron is normalized and the explicit influence of inter-octahedral tilts are not considered (i.e. ivol and Itilt are not included explicitly). Here, it is now clear that our generalized angle-based descriptors can capture the complex interplay of both inter- and intra-octahedral distortions in both binary trioxides of W and Mo, where it is apparent that the Eg of the MoO3 polymorphs shows a stronger reliance of iM while that of WO3 is much weaker. We have also unequivocally established and demonstrated that by using both our atomistic scale models and generalized angle descriptors, the binary metal trioxides do operate with a different (but coherent) structure (distortion)-property relationship as compared to the conventional proper oxide perovskites – i.e. intra-octahedral distortions versus inter-octahedral distortions, respectively.

3.5 Establishing the structure-property relationship in WO3 and MoO3 Lastly, to afford a clearer “structure-property” (i.e. Eg versus distortion) relation map, 36,62,63 we proceed to plot the Eg versus ⟨λ⟩ and σ 2 in Figures 4(a) and 4(b), respectively. Here, we can finally understand that in the absence of large intra-octahedral distortions (e.g. in many of the conventional oxide perovskites), they lie very close to the ⟨λ⟩ = 1 and σ 2 = 0 lines. The further they deviate from these lines the larger their intra-octahedral distortions. It is also interesting to note that these intra-octahedral distortion (only) descriptors are generally proportional to ∆Eg for the binary metal trioxides where intra-distortions are dominant, while the other oxide perovskites show no clear trend due to their lack of appreciable intraoctahedral distortions. 18


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Figure 4: (Color online) Calculated band gap energies (Eg , in eV) as a function of the quadratic elongation (⟨λ⟩) and bond angle variance (σ 2 , in ◦ 2 ) indices. Data points for the MoO3 and WO3 polymorphs are denoted by blue triangles and red circles, respectively. Data points for the various perovskite-structured oxides obtained from the Materials Project 26 are depicted by grey diamonds. Vertical dotted line is meant to guide the eye. We find that the calculated values of ⟨λ⟩ for the various MoO3 polymorphs range from 1.032 to 1.070, while that for WO3 polymorphs take values from 1.000 to 1.020. This implies that the octahedral networks in MoO3 undergo a more severe internal distortion than those in WO3 , corroborating with our atomistic scale models and generalized angle descriptor data. This is in line with previous works that have reported that the average off-center displacements of Mo6+ cation may be predicted to be more severe than that of W6+ cation, due to relatively higher effective electronegativity. 36,64 Furthermore, following Reference 57, we also calculate the covalency metric (Cd,p ), and plot the variation in ⟨λ⟩ and Eg as a function of Cd,p in Figure S4 of the Supporting Information. We find that both ⟨λ⟩ and Eg are directly proportional to Cd,p , implying that the covalency in these binary metal oxides (i.e. more overlap and hybridization between the metal d and oxygen p orbitals) increases as the octahedral distortion becomes more severe. Interestingly, this trend is somewhat opposite to what was observed in Reference 57 for proper perovskites. We tentatively attribute this opposite behavior to the differences in the atomic structures as well as the distortion index used in this work, where the distortion

19



behavior and the character of chemical bonding in proper perovskites and the binary metal trioxides can be different. Noting these differences, we argue that our distortion analysis method may be more general, highlighting the usefulness of the distortion indices in this work. In addition, the role of the metal cation in bond covalency may be approached from the point of view of effective electronegativity, as already discussed in Reference 36. From the proposed trend study in a large set of MO6 -containing compounds in Reference 36, a clear predictive trend of the MO6 octahedral distortion as a function of effective electronegativity was suggested and this clearly is also reflected in our work (see Figure S4 in Supporting Information).

4 Conclusion From our atomistic scale models and generalized angle descriptors, we disentangled two different types of octahedral distortions present in binary metal trioxides, namely the intraand inter-octahedral distortions.

Different from their superclass pervoskites, the inter-

octahedral distortion is found to influence the electronic structure nominally, while the intra-octahedral distortion, especially the metal off-center, dictates the electronic structure dominantly. Based on these observations, we can now appreciate the physical insights and origins of these deviations (or not) for these binary metal oxide materials. We envisage that this suit of general structural analysis tools will largely benefit our understanding of the structure-property relations in other octahedrally-connected functional materials. For instance, our models can be extended to investigate and explore alloys of binary oxide materials in an octahedral network (e.g. Wx Mo1−x O3 alloys for electrochromic smart window applications). 65–68

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Supporting Information Available Supporting Information.

The crystal structures of different WO3 and MoO3 poly-

morphs; calculated ionization potential, electron affinity, electronic band gap, lattice constants, quadratic elongation, bond angle variance of WO3 and MoO3 polymorphs; the information of perovskites from the Materials Project database; calculated Madelung potential of metal cation and oxygen anion in WO3 and MoO3 ; calculated quadratic elongation and electronic band gap as a function of covalency metric.

Acknowledgement We gratefully acknowledge support by Samsung Research Funding Center of Samsung Electronics under Project Number SRFC-MA1501-03. Computational resources have been provided by the KISTI supercomputing center (KSC-2017-C3-0008) and the Australian National Computational Infrastructure (NCI).

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Disentangling the Effects of Inter- and Intra-Octahedral Distortions on

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