1. Introduction

In the family of transition metal oxides, tungsten oxide has attracted great attention due to its distinctive opto-and electrochromic properties as well as excellent gas sensing performances. It is considered a promising material for a multitude of potential applications including semiconductor gas sensors, solar-energy devices, photocatalysts, field emitters, and transparent thin-film electrochromic displays^[1-3]. In recent years, various nanostructured materials such as nanowires, nanorods, nanotubes, and nanobelts, have been experimentally proved to show enhanced opto-and electrical properties due to their unique structural characteristics^[4]. Considerable progress has been made in the experimental studies about preparation and properties of tungsten oxide nanostructures, especially one-dimensional nanowires^{[3, 5, 6]}.

In addition to the experiments, theoretical investigations through atomistic modeling of nanostructural materials are also necessary, so as to give intrinsic properties of sample under ideal conditions, and to propose the fundamental understanding of the physical properties behind the observation at the nanoscale^[7-9]. Some theoretical results about bulk WO$_{3}$ have been reported. For instance, Yakovkin et al.^[10] simulated the optimized structure of the bulk WO$_{3}$ (001) surface with termination of (1 $\times$ 1) O, (1 $\times$ 1) WO$_{2}$, $c$(2 $\times$ 2) O using density functional theory (DFT) and predicted the potential role of surface relaxation in formation of crystalline nano-size clusters of WO$_{3}$. Based on the hybrid functions of plane wave and localized basis sets, Wang et al.^[11] conducted a systematic study about the structural and electronic properties in all phases of WO$_{3}$. However, despite the few pursuable examples of the bulk WO$_{3}$, to date systematical and theoretical results on WO$_{3}$ nanowires are seldom reported. In particular, there is no available structural model for WO$_{3}$ nanowire to be proposed, which is virtually necessary and important to theoretically investigate the optical, electrochromic and gas sensing properties.

In this work, aiming to propose a reasonable structural model of a WO$_{3}$ nanowire, the dependence of the electronic properties, especially the band structure, the gap and the density of states (DOS) on the surface structure of WO$_{3}$ nanowires (surface termination with WO$_{2}$ surface, bare and hydrogenated oxygen monolayer) was first investigated in the framework of first-principles DFT calculations. Then the effects of wire size and orientation on the electronic bandgaps were studied. It is found that the O-terminated WO$_{3}$ nanowire with hydrogenation can exhibit a reasonable indirect bandgap of 2.340 eV, which is 0.257 eV wider than bulk WO$_{3}$, and the surface structure dependence, electronic properties also clearly depend on the wire size and orientation.

2. Modeling and calculation

2.1 Construction of WO$_{3}$ nanowires model

According to the experimental results^[12-14], WO$_{3}$ nanowires usually grow preferentially along the $b$-or $c$-axis, and at room temperature, the WO$_{3}$ structure is monoclinic ($\gamma$-WO$_{3}$ phase with space group P21/n($C_{\rm 2h}^5))$. In this work,[001]-and [010]-oriented WO$_{3}$ nanowire with different surface structure and different size were modeled and calculated. The initial model of crystalline WO$_{3}$ nanowire was constructed by selecting all the W and O atoms that fall within a virtual prism placed in bulk WO$_{3}$ monoclinic crystal (room-temperature $\gamma$-WO$_{3}$ phase with lattice parameters of $a$ $=$ 7.297 Å, $b$ $=$ 7.539 Å, $c$ $=$ 7.688 Å and $\beta$ $=$ 90.91$^\circ$^[14]). All W and O atoms falling outside this virtual prism were removed. The direction of the prism was chosen to produce wires with [001] and [010] growth direction. Thus, termination of the WO$_{3}$ nanowire surface can be accomplished by either a WO$_{2}$ plane or an oxygen monolayer. Figure 1(a) and 1(b) show the top views of the initial O-and WO$_{2}$-terminated WO$_{3}$ nanowires with the smallest size along the [001] orientation and Figs. 1(c) and 1(d) along the [010] orientation. For the O-terminated WO$_{3}$ nanowire as shown in Figs. 1(a) and 1(c), there was one dangling bond in each O atom, which can be passivated by a H atom^[15]. In the present study, both bare and hydrogenated WO$_{3}$ nanowires terminated by an oxygen monolayer are discussed respectively. The latter was constructed through auto-updating H atoms onto the hanging bond of the surface oxygen. When the W atoms were exposed on the surface of the nanowire, auto-updating of hydrogen was not able to realize and the nanowire was actually terminated by a "bare" WO$_{2}$ surface (Figs. 1(b) and 1(d)).

A super cell method was employed where each wire was periodically repeated along the growth direction. To eliminate the interaction between neighboring wires, the square supercell was set to be sufficiently large with a vacuum of 10 Å in thickness^[16]. In fact, it is found that a 5-Å separation between two closest facets is sufficient to make the interaction negligible^[17]. The initial geometries were then relaxed to their closest. After full structural relaxation, the band structure and density of states were calculated for all the as-constructed nanowire models.

2.2 Computational details

The structural optimizations, band structure, and electronic density of states were preformed by using the first principles calculations based on density functional theory (DFT). All calculations were carried out with CASTEP (Cambridge Sequential Total Energy Package) code in Materials Studio of Accelrys Inc. The ultra-soft pseudo-potential and the widely used local density approximation (LDA) with the exchange correlation functional parameterized by Ceperley and Alder (CA-PZ) were adopted. The Brilloiouin zone $k$-point sampling was performed using a 1 $\times$ 1 $\times$ 5 Monkhorst-Pack (MP) mesh, and the cutoff energy of plane waves is set to 340 eV. The convergence criteria for structural optimization were set to be medium quality with the tolerance for self-consistent field (SCF), energy, maximum force, and maximum displacement of 2.0 $\times$ 10$^{-6}$ eV/atom, 2.0 $\times$ 10$^{-5}$ eV/atom, 0.05 eV/Å and 2.0 $\times$ 10$^{-3}$ Å, respectively.

3. Results and discussion

The band structures of WO$_{3}$ nanowires corresponding to the previously mentioned three surface structures shown in Figs. 2(b)-2(d) revealed their sensible dependence on the terminator, i.e., surface structure of the nanowire. The WO$_{2}$-terminated nanowire (Fig. 2(b)) showed a direct bandgap (at the Q point) of 0.212 eV and larger dispersion of the conduction band (CB) than the O-terminated nanowires. Figure 2(c) indicated that the nanowire with termination of hydrogenated oxygen monolayer had an indirect bandgap, with the valence-band maximum (VBM) located at the Q point and the conduction-band minimum (CBM) located at the G point. The calculated bandgap for this kind of hydrogenated nanowire was 2.340 eV. However, the bare O-terminated nanowire (Fig. 2(d)) exhibited metallic characteristic. It can be seen that the bandgap was inconspicuous and the conduction band and valence band almost confuse.

As the dimensions of WO$_{3}$ are reduced from the bulk to one-dimensional nanowire geometry, quantum confinement will result in the increase of the bandgap of the material^[18]. This has been confirmed by the published experimental results. The as-prepared ultrafine tungsten oxide nanowires in the diameter range of 1-4 nm showed much larger bandgap than that in bulk form (3.36 and 2.62 eV respectively)^{[19, 20]}. However, Figures 2(a) and 2(b) indicate that the bandgap of the WO$_{2}$-terminated WO$_{3}$ nanowire was markedly narrower than bulk WO$_{3}$. Figures 3(a)/3(b) and 3(c)/3(d) respectively show the top view/side view of the initial and optimized [001]-oriented WO$_{3}$ nanowires terminated by WO$_{2}$ surface. It can be seen that, as a result of structural optimization, the O atoms on WO$_{2}$ surface tend to shift outward, resulting in the W-O bond length changing from 1.895 to 1.866 Å and the W-W bond length changing from 3.705 to 2.577 Å. These changes caused the redistribution of energy states. The total DOS and partial DOS (PDOS) of O-2p and W-5d states of the WO$_{2}$-terminated WO$_{3}$ nanowires are shown in Fig. 4. The total DOS shown in Fig. 4(a) indicate that there existed a much high density of states at the Fermi level, which were responsible for the gap narrowing. A comparison of PDOS of O-2p and W-5d states for WO$_{2}$-terminated nanowire as well as W-5d states for bulk WO$_{3}$ revealed the reason of the appearance of surface states at Fermi level for the nanowire. When WO$_{3}$ was terminated by WO$_{2}$ surface, the W-5d states moved toward valence band (Fig. 4(b)) and resulted in the relatively high electronic states at Fermi level. That is, the surface states were mainly induced by 5d states of W atoms at WO$_{2}$ surface. Therefore, it is just the "bare" WO$_{2}$ surface that leads to the substantially narrow gap for a WO$_{2}$-terminated WO$_{3}$ nanowire.

Figure 5 shows the side view of the initial and optimized O-terminated [001] WO$_{3}$ nanowire, and (a, b) and (c, d) respectively correspond to the hydrogenated and bare nanowire. For the WO$_{3}$ nanowire terminated by hydrogenated oxygen monolayer, structural optimization resulted in a substantial relaxation of the structure. As shown in Fig. 5(b), the H atoms tended away from W atoms because of the electrostatic repulsion between H and W ions. Such a surface relaxation leads to a much stable structure. Figure 6 shows the DOS and PDOS of the WO$_{3}$ nanowire terminated by hydrogenated oxygen monolayer. Differing from the WO$_{2}$-terminated nanowire, the O-terminated WO$_{3}$ nanowire with hydrogenation showed a bandgap of 2.340 eV, which is 0.257 eV wider than bulk WO$_{3}$. This result is expected and can be well fitted to quantum confinement effect confirmed by previous experimental investigations^[18-20]. Nevertheless, the experimental bandgap of 3.36 eV reported for a tungsten oxide nanowire^[19] is larger than our calculated value. The reason for such a disagreement is the well-known shortcoming of the theory frame of DFT-LDA to underestimate bandgaps^[21].

As seen from the DOS shown in Fig. 6(a), as the dimensions of WO$_{3}$ are reduced from the bulk to a nanowire geometry, quantum confinement mainly results in the conduction band edge, which was mainly contributed by the 5d orbits of W atoms as shown in Fig. 6(b), shifts to high energy and thereby leads to the effective gap enlargement of the O-terminated nanowire. The 1s orbits of H atoms had little contribution to both conduction and valence band edges.

When the dangling bonds of the surface oxygen atoms were not passivated, a radically different band structure as shown in Fig. 2(d) was obtained. The metallic characteristic of WO$_{3}$ nanowire based on this structural model can be understood from the change of electronic states in close vicinity to the Fermi level originating from the optimized nanowire structure. As it is seen from a structural comparison between the initial and optimized bare O-terminated WO$_{3}$ nanowires (Figs. 5(c) and 5(d)), the optimization resulted in the formation of O$=$O between two adjacent slabs. The formed O$=$O structure can contribute a very large density of states just at the Fermi level, as illustrated in Fig. 7 which shows the PDOS of an O$=$O structure. Therefore, according to the calculated results described above, it is necessary to terminate the dangling bonds on the surface of the O-terminated WO$_{3}$ nanowire in order to avoid the formation of O$=$O structure. As illustrated in Fig. 2, the WO$_{3}$ nanowire with termination of hydrogenated oxygen monolayer showed a reasonable electric band structure with an indirect bandgap of 2.340 eV, which is well fit to quantum confinement effect in nanowire.

To further investigate the effects of the wire size and orientation on the electronic properties of a nanowire, the band structure of the hydrogenated oxygen monolayer-terminated WO$_{3}$ nanowires with different sizes and orientations ([001] and [010] respectively) were calculated. Figure 8 shows the results obtained from [001]-oriented nanowires. Experimentally, the general diameter of as-synthesized tungsten oxide nanowire is about several nanometers and the minimum value ever reported is 1 nm^{[6, 19]}. In this work, the bare and stoichiometric WO$_{3}$ nanowire studied are respectively made up of 40, 82 and 144 atoms (W and O) in a unit cell, and the corresponding nanowire sizes are 0.9, 1.7 and 2.5 nm, respectively. Figures 8(a), 8(b) and 8(c) show the three types of nanowire structures ([001] orientation). The WO$_{3}$ nanowire has undergone a transition from an indirect to direct bandgap with the increase of nanowire size, which is related to the different magnitude of the energy shift for CBM and VBM produced by quantum confinement for each point in the band structure^[18]. The WO$_{3}$ nanowire made up of 40 atoms has an indirect bandgap, while the bandgaps of the nanowires with 82 and 144 atoms translate to direct (at the G point).

As shown in Figs. 2 and 8, variations in nanowire surface structure and size can affect the bandgap obviously. In addition, the bandgap of nanowire also depends on its orientation. Figure 9 further compares the calculated bandgaps of the [001]-and [010]-oriented WO$_{3}$ nanowires with different sizes. As a reference, the bandgap of the bulk WO$_{3}$ (2.083 eV) is also indicated by the broken line in the figure. It can be seen that the [010]-oriented WO$_{3}$ nanowires showed larger bandgaps than that of bulk WO$_{3}$, and the calculated bandgap increases as the size of the nanowire decreases due to the stronger quantum confinement effect in thinner wires. This observed trend shows generally good agreement with the experimental data^{[19, 20]}. Still, the trend in the size dependence of the bandgap agrees well with those theoretical studies in Refs. [18, 22]. It has been found that the strong quantum confinement effect generally occurs when the size of the crystal is reduced to much smaller than the Bohr radius for the material (about 3 nm for WO$_{3}$^[23])^[24]. As a result, the bandgap variation of the nanowire with size beyond 1.7 nm is no longer apparent, as shown in Fig. 9.

However, it also can be seen from Fig. 9 that, for the same wire size, the [001]-oriented nanowire shows much smaller bandgap than the [010]-oriented one, even than bulk WO$_{3}$, indicating that other effects relating to the orientation of nanowire also might have remarkable contributions to the bandgap change. It has been reported that the surface effect can dominate the bandgap change of AlN nanowire and then result much smaller gap value than that of bulk solid^[25]. For the [001]-oriented monoclinic WO$_{3}$ nanowire, there exists an additional periodicity of alternately long and short O-O separations perpendicular to the growth direction of the nanowire, which introduced by tilting of the WO$_{6}$ octahedral^[26]. Structural optimization can result in the obvious change of the long and short O-O bond length and then the corresponding inclination change of the WO$_{6}$ octahedral. The formed surface reconstructed structure, which is directly related to the periodicity degree of O-O separations (i.e. wire size) and the orientation of WO$_{3}$ nanowire, is assumed to greatly affect the bandgap^[18], although the quantum confinement effect is very important in determining the bandgap of thin nanowire ($n$ $=$ 40). Figure 10 shows the DOS of the [001]-and [010]-oriented nanowires with $n$ $=$ 40. Clearly, it is the shift of the conduction band edge to low energy as well as the shift of valence band edge to high energy that leads to the band narrowing of the [001]-oriented WO$_{3}$ nanowire.

4. Conclusions

The electronic structure of WO$_{3}$ nanowires with various types of termination (WO$_{2}$ surface, bare oxygen surface, and hydrogenated oxygen surface), different orientation, and various diameters were investigated by using the first principles calculation of plane-wave ultra-soft pseudo-potential technology based on DFT. The electronic structures of the WO$_{3}$ nanowires, including their electronic density of states, band structures, and bandgaps are found to depend sensitively on the surface structure. Surface termination with hydrogenated oxygen monolayer is expected to be the most reasonable structure for a WO$_{3}$ nanowire. The wire size and the orientation-resulted surface reconstructed structure can affect the bandgap of the WO$_{3}$ nanowire significantly. Nanowires with smaller diameters show larger bandgaps due to a much stronger quantum confinement effect. The special surface reconstructed structure of [001]-oriented WO$_{3}$ nanowire, differing from the one with [010] growth direction, can narrow the bandgap by modifying the band edges.