Electronic and Structural Properties of WO3: A Systematic Hybrid DFT

ARTICLE pubs.acs.org/JPCC

Electronic and Structural Properties of WO3: A Systematic Hybrid DFT Study Fenggong Wang, Cristiana Di Valentin,* and Gianfranco Pacchioni Dipartimento di Scienza dei Materiali, Universit
a di Milano-Bicocca, Via R. Cozzi 53, 20125 Milano, Italy

bS Supporting Information ABSTRACT: Various hybrid functionals combined with both plane wave and localized basis sets have been used for a systematic study of the structural and electronic properties of all phases of WO3. It is found that hybrid functionals work at least as well as the standard DFT/GGA functional in predicting lattice constants and equilibrium volumes. However, the adoption of hybrid functionals has the advantage to considerably improve the Kohn Sham band gap which is always severely underestimated by the standard DFT calculations. The HSE06 functional in combination with a plane wave basis set describes well the band gap of WO3, while the B3LYP functional associated with a localized basis set slightly overestimates it. The band gap can be made fully consistent with experiment by fixing the amount of Hartree Fock exchange in the hybrid functional to 15%.

1. INTRODUCTION Hydrogen production through water splitting by a photocatalytic or photoelectrochemical process is one of the most attractive methods, being environmentally friendly.1 3 For optimal exploitation of solar energy, the optically visible-range spectra must be used, corresponding to a band gap of the semiconductor photocatalyst around 2.0 eV. Furthermore, the band edge positions must match the water-splitting potential.1,4 Besides the possible application in a variety of technologies, including electrochromic materials,5 gas sensing,6 photoelectrodes,7,8 and other optoelectrical devices, tungsten oxide (WO3) has attracted a lot of interest as an n-type oxide semiconductor.9 It is well-known for its photosensitivity,10 12 good electron transport properties,13 and stability against photocorrosion14,15 in acidic aqueous solutions. Its band gap is smaller than that of some other semiconductors but still too high to realize a sufficiently large absorption of the solar spectrum;16 its conduction band minimum (CBM) is too low compared to the hydrogen evolution reaction redox potential.17 Experimentally, the band gap of WO3 has been measured by optical absorption, with values ranging from 2.5 to 3.2 eV.5,18 23 Also some photocurrent24 26 and UPS27 measurements are reported in the literature with values ranging from 2.6 to 3.2 eV. It is clear that the structure has an important effect on the electronic properties, including the band gap. The simplest structure of WO3 is cubic, as for ReO3,28 and is composed of corner-sharing regular octahedrons, occurring with incorporation of impurities.29 As a function of temperature, several distortions are possible: low-temperature (LT) monoclinic structure from 140 to 50 °C,30,31 triclinic structure from 50 to 17 °C,30,32 35 followed by a roomtemperature (RT) monoclinic structure from 17 to 330 °C.32,33,36,37 Above 330 °C up to 740 °C, WO3 becomes orthorhombic,38 41 and above 740 °C it assumes a tetragonal structure.39,41,42 r 2011 American Chemical Society

Density functional theory (DFT) calculations are useful in studying the structural and electronic properties of oxide semiconductors. Recently, there have been numerous theoretical studies on WO3, including DFT and Hartree Fock (HF) calculations.1,43 50 However, most of these studies are limited to the simple cubic system, except for three DFT studies43,44,50 where the most common phases have been considered. It is common practice, although theoretically not fully justified, to estimate the semiconductor band gap using the single-particle Kohn Sham eigenvalues. Studies on WO3 are mainly based on standard DFT or HF calculations,47 which severely underestimate or overestimate, respectively, the band gap values. As far as the band gap problem is concerned, hybrid exchange-correlation functionals including a mixture of HF and DFT exchange terms represent a practical, although not perfect, solution to reproduce the experimental band gap.51,52 Only fundamental gaps obtained from UPS or from electrochemical measurements should be directly compared to computed interband energies. The first comparison of course relies on the validity of Koopmans’ theorem. Up to now, to the best of our knowledge, no comprehensive study of the various phases of WO3 using hybrid functionals has been reported. To properly describe WO3 and its potential applications, it is useful to perform a systematic study of its properties by means of hybrid DFT methods. In the present work, three kinds of hybrid functionals, including the popular B3LYP,53,54 PBE0,55,56 and the more recent HSE0657 59 functionals, with both localized atomic orbital and plane wave basis sets, are used to study the structural and electronic Received: February 1, 2011 Revised: March 18, 2011 Published: April 04, 2011 8345

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Table 1. Relative Stabilities of Various Structures by Different Methods

a

B3LYP

PBE0

PW91

structure

(gto)a (meV)

(gto)a (meV)

(gto)a (meV)

RT monoclinic simple cubic

0 243

0 253

2 185

tetragonal

4

5

30

LT monoclinic

7

3

3

triclinic

6

6

0

orthorhombic

6

13

1

“gto” stands for gaussian type orbitals.

properties of WO3. Standard DFT/GGA calculations are also performed for comparison. We will see below that, compared with the standard DFT/GGA, hybrid functionals successfully increase the band gap for all the structures without losing accuracy in predicting the lattice parameters. The paper is organized as follows: in Section 2 the adopted computational methods and parameters are described in detail. The results and related discussion are reported in Section 3. Finally, some conclusions are presented in Section 4.

2. COMPUTATIONAL DETAILS The calculations were carried out within the linear combination of atomic orbitals (LCAO) approach combined with hybrid functionals, as implemented in CRYSTAL0660 code. Two kinds of hybrid functionals, including PBE055,56 and B3LYP53,54 composed of the Becke three-parameter exchange functional and the LYP correlation functional, were adopted. The corresponding percentage of exact HF exchange in PBE0 and B3LYP is 25% and 20%, respectively. The all-electron Gaussian-type basis set 8-411(d1) was adopted for oxygen, which was proven to work well for highly ionic compounds.61 For the heavy tungsten atom, we adopted the well-assessed effective core pseudopotential (ECP)62,63 techniques. The small-core ECP derived by Hay and Wadt was chosen,62 leaving only the 5p, 5d, and 6sp valence electrons to be treated explicitly. For the tungsten valence shell electrons, we used a modified Hay Wadt double-ζ basis set.62 Here the innermost s shell states and the most diffuse outer s and p functions were deleted, while the second innermost s shells were uncontracted and the exponents of the remaining outermost p and d basis function optimized to 0.20. (see Supporting Information) These modifications were necessary to adapt the basis set, conceived for molecular calculations, to calculations with periodic boundary conditions. For each WO3 phase, the corresponding unit cell was fully optimized with respect to both atomic coordinates and cell parameters. The cutoff limits in the evaluation of Coulomb and exchange series appearing in the self-consistent field (SCF) equation for periodic systems were set to 10 7 for Coulomb overlap tolerance, 10 7 for Coulomb penetration tolerance, 10 7 for exchange overlap tolerance, 10 7 for exchange pseudo-overlap in the direct space, and 10 14 for exchange pseudo-overlap in the reciprocal space. The condition for the SCF convergence was set to 10 6 au on the total energy difference between two subsequent cycles. The number of k-points was chosen in such a way that the results of the numerical integration were well converged, and thus the reciprocal space was sampled according to a regular sublattice with a shrinking factor (input IS) equal to or larger

than 4 (more than 27 k-points) in the sampling of the irreducible Brillouin zone (BZ) for different structures. The density of states (DOS) have been computed using the same k-points (or more) as that for geometry optimizations. The Kohn Sham eigenvalues were computed on each k point of the mesh and used to estimate the band gap. The gradients with respect to atomic coordinates and lattice parameters were evaluated analytically. The equilibrium structure was determined by using a quasiNewton algorithm with a BFGS Hessian updating scheme.64 Convergence in the geometry optimization process was tested on the root-mean-square (rms) and the absolute value of the largest component of both the gradients and nuclear displacements. For all atoms, the thresholds for the maximum and the rms forces were set to 0.00045 and 0.00030 au and those for the maximum and the rms atomic displacements to 0.00180 and 0.00120 au, respectively. HSE57,58 hybrid functional calculations have also been performed based on Kohn Sham theory65 and the projectoraugmented wave (PAW)66,67 pseudopotential as implemented in the VASP68,69 code. For W, the 5p, 5d, and 6s states were treated as valence states, and the Perdew, Burke, and Ernzerhof (PBE) functional with a core radius of 2.5 au was used. For O, the standard PBE functional with a core radius of 1.5 au was used. In the HSE calculations, the exchange functional is separated into long-range and short-range parts. In the short-range part the exact HF exchange is mixed with PBE exchange, while in the long-range part the exchange functional is essentially described by PBE. In the present work, the screening parameter μ = 0.2 Å 1 was used for the semilocal (GGA) exchange as well as for the screened nonlocal exchange as suggested for the HSE06 functional.58 The cutoff energy for the plane wave basis was 600 eV for all the calculations, and the BZ integrations were performed by using the second-order Methfessel Paxton method.70 Both the lattice parameters (the volume and shape) and the atomic coordinates were fully optimized until all the forces on each ion were less than 0.02 eV/Å. When carrying out structural optimizations, we adopted Monkhorst Pack71 kpoint sampling mesh containing at least eight k-points according to the symmetries. For example, a 2  2  2 Γ-centered mesh was used for the RT monoclinic cell. However, for the DOS calculations, a more accurate k-mesh containing at least 27 kpoints was used for all structures. To compare with previous results, calculations were also performed with the standard DFT/PW91 functional72 using both localized and plane wave basis set approaches.

3. RESULTS AND DISCUSSION As mentioned above, tungsten oxide has many different structures under different temperature conditions. In this section, we will give the detailed results and discussion for all the listed phases. The relative stabilities (Table 1) show that the simple cubic phase is the least stable and that the differences for all the other phases are tiny, whatever functional is used. This is similar to what was previously obtained with other studies based on the GGA functional.43,50 3.1. Room-Temperature Monoclinic Structure. RT monoclinic structure WO3 with space group P21/n is the most common and stable phase of tungsten oxide from 17 to 330 °C. Figure 1(a) shows the 2  2  2 supercell. The unit cell consists of 8 W and 24 O atoms and contains eight oxygen corner-sharing octahedrons in a slightly distorted cubic arrangement 8346

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Figure 1. Structural models of various structures of WO3: (a) roomtemperature monoclinic; (b) simple cubic; (c) tetragonal; (d) lowtemperature monoclinic; (e) triclinic; and (f) orthorhombic structures. The large green and small red spheres represent W and O atoms, respectively. All the models shown contain 256 atoms.

Table 2. Calculated Lattice Parameters, Equilibrium Volume, and Band Gap of Room-Temperature Monoclinic WO3 by Various DFT Methods† a basis‡

EXP36

b

c

β

V

gap

(Å) (Å) (Å) (°) (Å3/cell) 7.31 7.54 7.69 90.9

type§

(eV) )

functional

53.0

2.6 3.2

56.7

1.73

ID

1.31

PD

56.3

0.90

D

56.5

1.19

D

pw (PAW) 7.56 7.80 7.84 90.1

57.8

1.36

D

PW91 HSE06

gto pw

7.45 7.67 7.88 90.6 7.39 7.64 7.75 90.3

56.3 54.7

1.57 2.80

D D

B3LYP

gto

7.44 7.73 7.91 90.2

56.8

3.13

D

PBE0

gto

7.33 7.60 7.80 90.6

54.4

3.67

D

RPBE43

pw

LDA1

pw

7.38 7.47 7.63

PW9144

pw

7.55 7.62 7.83 90.2

PW91

pw (US)

7.50 7.73 7.80 90.3

PW91

ID

†

)

The experimental values and results from the literature are also shown for comparison. ‡ “US” stands for “ultrasoft pseudopotentials”, “PAW” for “PAW pseudopotentials”, “pw” for “plane wave”, and “gto” for “gaussian type orbitals”. § “ID” stands for “indirect”, “D” for “direct”, and “PD” for “pseudodirect” band gap, respectively. See refs 24 27.

(β angle between 90.2° and 90.9°). It can be considered as a deformed perovskite ABO3 structure, where the A ions are missing1,73 and thus suitable for interstitial doping. To test the validity of our computational approach, first we fully optimized the structure with both the standard DFT/GGA and hybrid functionals. Table 2 lists the optimized lattice parameters, equilibrium volume, and the corresponding band gap. Also shown for comparison are previous results from the literature using standard pseudopotential plane wave DFT methods. First of all, the structural properties are satisfactorily described by all methods when compared with the experiment. The present equilibrium volume by the PW91 ultrasoft (US) pseudopotential plane wave method is almost the same as in previous studies,44 although the specific lattice constants are slightly different. Not surprisingly, PW91 calculations produce a larger volume than experimentally determined due to the general trend of GGA enhancing interatomic separations. The use of

Figure 2. Band structure and density of states (DOS) of roomtemperature monoclinic WO3 by different kinds of functionals including (a) PW91, (b) B3LYP, and (c) PBE0, respectively.

PAW pseudopotentials leads to an additional slight increase of the lattice parameters. Compared with the plane wave PW91, the localized basis set (same functional) results are slightly better, except for the lattice constant in the c direction. From another point of view, this result indicates that our localized basis set is good enough in predicting WO3 lattice parameters. As one would expect, previous LDA calculations provided the best description of the lattice constants within all considered standard DFT functionals. Nevertheless, it is important to observe that the three kinds of hybrid functionals, i.e., HSE06, B3LYP, and PBE0, predict at least as good results as the standard DFT/GGA. Figure 2 shows the band structure and the DOS of RT monoclinic WO3 using three different functionals, including both standard DFT/PW91 and hybrid functionals. At first sight, all methods provide qualitatively similar electronic structures. Both the top of the valence band (VB) and the bottom of the conduction band (CB) locate at the Γ point, although the values at the Z point are similar. All three kinds of functionals predict a direct band gap for RT monoclinic WO3 in disagreement with the experiment and previous revised PBE (RPBE) calculations.43 However, a direct band gap was also reported in a GGA study,44 where it was argued that the distinction based on the experiment data could be artificial; i.e., under suitable circumstances a direct band gap can give rise to experimental properties that can be interpreted as due to an indirect band gap. Considering that band gaps obtained with hybrid functionals are close to the experiment and that the energy difference between direct Γ to Γ and indirect Z to Γ band gaps is very small (∼0.01 eV), one cannot rule out a direct band gap for this material. Also, similarly to what was already reported,43 8347

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Figure 3. (a) Band gap dependence on the percentage of exact Hartree Fock exchange in the B3LYP hybrid functional. (b) The bad gap as a function of volume by the standard B3LYP hybrid functional.

the valence band is relatively flat compared to the conduction band, which indicates a large effective mass for the holes in the first BZ. The high dispersive characteristic of the conduction bands along the corresponding symmetry points indicates a low effective mass of electrons in the system, which may lead to interesting electronic behavior. As shown in the right panel of Figure 2, the sharp rise of the valence-band DOS near the Fermi level is consistent with the flat bands, whereas the conduction-band DOS edge is less sharp. As expected in a highly ionic metal oxide semiconductor, the valence band of WO3 largely consists of the O 2p states, while the CB has predominantly W 5d character.1 When one considers the electronic properties of WO3, all the standard DFT calculations severely underestimate the band gap. Since the band gap and the positions of the band edges are very important for the description of the photocatalytic or photoelectrochemical process,4 it is reasonable to question whether this serious underestimation of the band gap affects the description of electronic properties or not. All three hybrid functionals give a much larger band gap than standard GGA (Table 2), though the band gap nature (direct or indirect) is the same. The HSE06 hybrid functional with a plane wave basis (Eg = 2.80 eV) works best in reproducing the experimental value, followed by the B3LYP hybrid functional with a localized basis set (Eg = 3.13 eV), whereas the PBE0 functional overestimates the band gap (Eg = 3.67 eV). To summarize, compared with the well-known shortcomings of standard DFT calculations, hybrid functionals increase the band gap and better reproduce experimental data. Since the HSE06 functional associated with a plane wave basis set is computationally rather expensive compared to the B3LYP functional with a localized basis set, we varied the percentage of exact HF exchange in the B3LYP hybrid functional to obtain more accurate band gap values. Figure 3(a) clearly shows that the band gap increases with the amount of HF exchange. The predicted band gap is 2.73 eV for a HF contribution of 15%, showing an improvement with respect to the standard B3LYP functional. As shown in Figure 3(b), the gap is volume dependent and increases under moderate compression conditions. As the volume decreases from 55.1 to 49.1 Å3/cell, the band gap increases from 3.27 to 3.78 eV. On the contrary, if the volume further decreases to 47.2 Å3/cell, then the band gap slightly decreases to 3.72 eV. This phenomenon can be understood as the material behaves in an essentially ionic fashion because of the competing effects of the “band broadening” and “gap opening” due to the different shift of the conduction and valence bands under compression.44 The band gap is direct for volumes larger than 51.1 Å3/cell but becomes indirect under further compression. The top of the valence band is set along the paths of Γ B and Y Γ for systems

Table 3. Calculated Lattice Parameters, Equilibrium Volume, and Band Gap of the Simple Cubic WO3 by Various DFT Methods† functional

basis‡

EXP74

a (Å)

V (Å3/cell)

gap (eV)

type§

3.77

53.7

3.78

54.0

7.7

ID

pw

55.0

0.69

ID

PW9144

pw

56.3

0.4

ID

LDA45

FP-LMTO

3.78

0.3

ID

PW91

pw (US)

3.84

56.5

0.41

ID

PW91 PW91

pw (PAW) gto

3.84 3.83

56.6 56.3

0.56 0.55

ID ID

HSE06

pw

3.80

54.7

1.67

ID

B3LYP

gto

3.82

55.6

1.89

ID

PBE0

gto

3.79

54.4

2.25

ID

HF47

gto

RPBE43

†

Results from previous results are also shown for comparison. ‡ “US” stands for “ultrasoft pseudopotentials”, “PAW” for “PAW pseudopotentials”, “pw” for “plane wave”, and “gto” for “gaussian type orbitals”. § “ID” stands for “indirect” band gap.

with volume 49.1 and 47.2 Å3/cell, respectively. This suggests that compression leads to a qualitative change of the electronic structure. 3.2. Simple Cubic Structure. The simple cubic structure (space group Pm-3m) of WO3 was often reported to occur for high fractions of intercalated material because of the large stabilization induced by impurities.29 However, there is at least one report in the literature of WO3 polycrystalline powder with simple cubic structure.74 Furthermore, it is worth considering as a good model for the distorted structures. The simple cubic structure contains the basic structural characteristics, e.g., the corner-sharing octahedron, but ignores the distortion. In the cubic phase, the W metal atom is surrounded by a regular octahedron of six nearestneighbor O atoms; the unit cell contains only a single formula unit WO3. Figure 1(b) shows the 4  4  4 supercell of the simple cubic WO3 which contains the same number of atoms as the RT monoclinic 2  2  2 supercell described above. During the structural optimization only the lattice constant (a) is optimized without any atomic relaxation because all the internal coordinates are unambiguously determined by internal symmetries, leaving only a single degree of freedom. Table 3 shows the optimized lattice constants, equilibrium volumes, and band gaps using various functionals. We also list previous HF and standard DFT results for comparison. Again, the present PW91 equilibrium volumes agree quite well with previous PW91 calculations,44 despite the use of a different pseudopotential. 8348

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Figure 4. Band structure and density of states (DOS) by the standard B3LYP hybrid functional for various structures of WO3: (a) room-temperature monoclinic; (b) simple cubic; (c) tetragonal; (d) low-temperature monoclinic; (e) triclinic; and (f) orthorhombic structures.

Still, they are larger than the HF or the RPBE volumes because of the overestimation of lattice constants with the GGA/PW91 functional. Hybrid functionals give slightly smaller lattice constants and equilibrium volumes, while HSE06 and PBE0 equilibrium volumes best fit with experiments.74 Most of the previous methods, including the LDA fullpotential linear muffin-tin orbitals (FP-LMTO),45 RPBE43 and PW91,44 give a small band gap for the cubic structure, while, not surprisingly, the HF method systematically overestimates it with respect to the other methods. Here, band gap values of 1.67, 1.89, and 2.25 eV, respectively, are obtained with HSE06, B3LYP, and PBE0 hybrid functionals, suggesting that a considerably smaller band gap is associated to the simple cubic phase compared to the RT monoclinic one. Figure 4(b) shows the electronic structure of the simple cubic WO3 (B3LYP functional with a localized basis set). For comparison, the band structure and DOS of RT monoclinic structure are also shown. The CB is mainly composed of the 5d states of W, while the VB mainly originates from the 2p states of O with the onset of p d hybridization from ∼1 eV below the VBM. The bandwidths of CB and VB, 5 and 8 eV, respectively, are much larger than that of the RT monoclinic structure. The band structure is in qualitative agreement with previous results,44 except for the VB bandwidth due to the different treatment of exchange and correlation (B3LYP vs PW91) and the consequent different volume. The band gap of the simple cubic WO3 is indirect, in contrast with the RT monoclinic structure. The VBM locates at the R point and the CBM at the Γ point. Though the present valence bandwidth agrees well with the FP-LMTO calculations,45 a different VBM at the M point was found. From

the band structure plot one can see that the highest state of the VB from M to R is rather flat, leading to a tiny energy difference between M and R points, which may be the reason for different VBM points using different DFT methods. Moreover, the lowest CB state is also dispersionless along the X Γ direction, suggesting a difficult conduction along this direction. The B3LYP band structure is similar to the HF one,47 except for VBM positions. As shown in Figure 5(b), the band gap is only slightly volume dependent and increases moderately with decreasing volume. The band structure, with an indirect band gap from R to Γ, does not change qualitatively upon compression. 3.3. Tetragonal Structure. The tetragonal WO3 phase is slightly distorted from the simple cubic structure. Its space group is P4/nmm, and the unit cell contains two formula units (8 atoms). Figure 1(c) shows the 4  4  2 supercell of tetragonal WO3 which includes 256 atoms as for the RT monoclinic and the simple cubic supercells. Two sets of inequivalent oxygen atoms are present in the deformed octahedron, i.e., two axial and four equatorial oxygen atoms. In the equatorial x y plane all W O bond lengths are equal, while in the axial z direction the W O bond lengths alternate in an “antiferroelectric” fashion.44 For the optimization of the structure, the lattice constants, shape, and atoms coordinates are relaxed under the constraint of the internal symmetry operations. The lattice constants and equilibrium volumes (Table 4) are similar to those of ref 44 obtained when using the same PW91 method. As for the RT monoclinic structure, the PAW pseudopotential gives slightly larger lattice constants and volume than the ultrasoft 8349

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Figure 5. Band gap as a function of volume by the standard B3LYP hybrid functional for different structures of WO3: (a) room-temperature monoclinic; (b) simple cubic; (c) tetragonal; (d) low-temperature monoclinic; (e) triclinic; and (f) orthorhombic structures.

Table 4. Calculated Lattice Parameters, Equilibrium Volume, and Band Gap of the Tetragonal WO3 by Various DFT Methods† functional

basis‡

EXP42

a

c

V

gap

(Å)

(Å)

(Å3/cell)

(eV)

5.25

3.92

54.0

Table 5. Calculated Lattice Parameters, Equilibrium Volume, and Band Gap of the Low-Temperature Monoclinic WO3 by Various DFT Methods†

type§

functional

basis‡

EXP30

a

b

c

β

(Å)

(Å)

(Å)

(°)

V

gap

(Å3/cell) (eV) type§

5.28 5.16 7.66 91.8

52.1

5.37 5.31 7.81 90.8 5.36 5.32 7.83 90.6

55.8 55.8

PW91

pw (PAW) 5.40 5.37 7.84 90.4

56.8

1.57 D

PD

PW91

gto

55.4

1.76 D

0.54

PD

HSE06

pw

5.31 5.27 7.67 91.2

53.6

3.14 D

0.52

PD

B3LYP

gto

5.35 5.29 7.89 90.7

55.8

3.33 D

1.71

PD

PBE0

gto

5.29 5.19 7.79 91.2

53.5

3.84 D

57.6

1.85

PD

56.1

2.28

PD

47

HF RPBE43

gto pw

PW9144

pw

5.36

3.98

57.2

PW91

pw (US)

5.36

3.98

57.1

PW91

pw (PAW)

5.36

4.01

57.6

PW91

gto

5.35

4.04

57.8

HSE06

pw

5.31

3.97

55.9

B3LYP

gto

5.32

4.03

PBE0

gto

5.28

4.02

55.5 57.2

8.10 0.66 0.40

D ID

PW9144 PW91

pw pw (US)

D

†

The experimental values and results from the literature are also shown for comparison. ‡ “US” stands for “ultrasoft pseudopotentials”, “PAW” for “PAW pseudopotentials”, “pw” for “plane wave”, and “gto” for “gaussian type orbitals”. § “ID” stands for “indirect”, “D” for “direct”, and “PD” for “pseudodirect” band gap, respectively.

pseudopotential. The PW91 functional with a localized basis set also works well in predicting the structural properties. It is not surprising that the three hybrid functionals considered predict lattice constants and volumes of the same quality (B3LYP) or even better (PBE0 and

5.35 5.27 7.86 91.1

1.35 D

†

The experimental values and results from the literature are also shown for comparison. ‡ “US” stands for “ultrasoft pseudopotentials”, “PAW” for “PAW pseudopotentials”, “pw” for “plane wave”, and “gto” for “gaussian type orbitals”. § “D” stands for “direct” band gap.

HSE06) than the standard DFT/PW91 functional. Again, PBE0 and HSE06 show the best agreement with the experiment. All the standard DFT calculations give rather small band gaps, with the largest value being 0.66 eV for RPBE.43 The band gap of tetragonal WO3 predicted by HSE06, B3LYP, and PBE0 hybrid functionals, 1.71, 1.85, and 2.28 eV, respectively, is similar to that computed for the simple cubic structure due to the fact that there 8350

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Table 6. Calculated Lattice Parameters, Equilibrium Volume, and Band Gap of the Triclinic WO3 by Various DFT Methods† functional

basis‡

EXP32

a (Å)

b (Å)

c (Å)

R (°)

β (°)

γ (°)

V (Å3/cell)

7.31

7.53

7.69

88.8

90.9

90.9

52.9

7.54

7.64

7.84

89.7

90.2

90.2

PW9144

pw

RPBE43

pw

PW91

pw (US)

7.64

7.64

7.78

89.8

90.1

PW91

pw (PAW)

7.71

7.69

7.78

89.9

PW91

gto

7.44

7.67

7.85

gap (eV)

type§

56.5 56.7

1.34

D

90.1

56.8

1.44

D

90.0

90.1

57.7

1.65

ID

89.3

90.5

90.6

55.9

1.63

D

HSE06

pw

7.44

7.58

7.75

89.4

90.3

90.3

54.6

2.94

D

B3LYP PBE0

gto gto

7.42 7.33

7.71 7.59

7.89 7.78

89.5 89.2

90.3 90.6

90.3 90.7

56.5 54.1

3.17 3.67

D D

† The experimental values and results from the literature are also shown for comparison. ‡ “US” stands for “ultrasoft pseudopotentials”, “PAW” for “PAW pseudopotentials”, “pw” for “plane wave”, and “gto” for “gaussian type orbitals”. § “ID” stands for “indirect” and “D” for “direct” band gap, respectively.

is only a small deformation from the simple cubic to the tetragonal structure. Figure 4(c) shows the B3LYP band structure and DOS for the tetragonal WO3. The VBM locates at the Z point, while the CBM is at the Γ point; the conduction and valence bands are quite flat along the Γ Z path, and the energy difference between Γ and Z points for the highest VB is rather small. Thus, the tetragonal band gap should be pseudodirect. This is why previous calculations gave different results as shown in Table 4. The VB width is smaller than for the cubic structure but larger than for the RT monoclinic structure. On the contrary, the CB width is more or less the same as for the simple cubic structure. The VB is overall flatter than for the cubic phase, leading to two sharp peaks in the valence-band DOS plot. The whole band structure is in qualitative agreement with previous GGA45 and HF47 results, but the B3LYP functional provides an intermediate and more reliable band gap value. As shown in Figure 5(c), the band gap remains largely unaffected by compression as for the cubic structure. 3.4. Low-Temperature (LT) Structure. The low-temperature monoclinic structure with space group Pc is another distorted structure of WO3. Its unit cell contains four formula units composed of 4 W and 12 O atoms, respectively. Figure 1(d) shows the 2  2  4 supercell of LT monoclinic WO3 containing 256 atoms. Compared with the tetragonal structure, the oxygen octahedron is much more distorted with the W atom moving off the center (Figure 1(d)). The PW91 results agree well with previous calculations, despite slightly different lattice constants (Table 5). It seems a general trend that for WO3 the PAW pseudopotential gives a slightly larger equilibrium volume than the ultrasoft pseudopotential. As in the other structures, PBE0 and HSE06 hybrid functionals best describe the lattice constants and equilibrium volume if compared with the experiment, whereas the B3LYP functional provides slightly larger values. As expected, DFT/GGA with both plane wave and localized basis sets predicts too small band gaps, an aspect which is corrected by hybrid functionals. With HSE06, B3LYP, and PBE0, we compute a band gap of 3.14, 3.33, and 3.84 eV, respectively. Note that HSE06 gives a smaller band gap than B3LYP despite the fact that the contribution of HF exchange is larger. All the functionals give a direct band gap with both CBM and VBM located at the Γ point. Figure 4(d) shows the B3LYP band structure and DOS for LT monoclinic structure which does not present significant differences from that of the other phases. Similar to the RT monoclinic structure, there are three toroidal shapes in the band structure near Γ and Z and along the C E path, respectively. The band gap of the LT monoclinic structure is slightly larger than for the RT one. Moreover,

Table 7. Calculated Lattice Parameters, Equilibrium Volume, and Band Gap of the Orthorhombic WO3 by Various DFT Methods† functional

basis‡

EXP38

a (Å) b (Å) c (Å) V (Å3/cell) gap (eV) type§ 7.24

7.57

7.75

RPBE43

pw

PW91

pw (US)

7.53

7.76

PW91

pw (PAW) 7.55

PW91

gto

HSE06

pw

7.46

7.70

7.80

B3LYP PBE0

gto gto

7.48 7.42

7.75 7.71

7.94 7.84

7.52

53.9 56.8

1.50

D

7.83

57.2

1.09

D

7.80

7.87

57.9

1.31

D

7.77

7.91

57.8

1.35

D

56.0

2.57

D

57.5 56.1

2.89 3.35

D D

†

The experimental values and results from the literature are also shown for comparison. ‡ “US” stands for “ultrasoft pseudopotentials”, “PAW” for “PAW pseudopotentials”, “pw” for “plane wave”, and “gto” for “gaussian type orbitals”. § “D” stands for “direct” band gap.

both the lowest CB and the highest VB are rather flat along the Γ Z path with a very small energy difference between Γ and Z. As for the RT structure, the band gap increases under compression. 3.5. Triclinic Structure. The unit cell of the triclinic structure (space group P-1) contains 8 W and 24 O atoms. There is also a long short W O bond length alternation along all three directions, inducing a deformation of the oxygen octahedron. The 2  2  2 supercell of triclinic WO3 containing 256 atoms is shown in Figure 1(e). Table 6 shows the calculated results of triclinic WO3. Also in this case HSE06 and PBE0 give the lattice constants and equilibrium volumes in better agreement with experiment, followed by the PW91 functional combined with a localized basis set, the B3LYP hybrid functional, and the PW91 functional with ultrasoft pseudopotential. As usual, the PW91 and RPBE functionals give a quite small band gap which increases to 2.94, 3.17, and 3.67 eV when using HSE06, B3LYP, and PBE0 hybrid functionals, respectively. The B3LYP band gap is direct with both CBM and VBM at the Γ point (Figure 4(e)). The highest VB is very flat along the Γ Z path, and the energy difference between the extremes is very small. This is why the PAW pseudopotential gives an indirect band gap. As for RT and LT monoclinic structures, B3LYP results show that the band gap is volume dependent, increasing with volume compression. 3.6. Orthorhombic Structure. Orthorhombic WO3 (space group of Pmnb) also contains a distorted oxygen octahedron. The unit cell consists of 24 O and 8 W atoms. Figure 1(e) shows 8351

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The Journal of Physical Chemistry C the 2  2  2 supercell (256 atoms) for this phase. Unlike the triclinic structure, here the long short W O bond length alternation is only in the y and z directions, as a consequence of the W atom moving away from its equilibrium position. As shown in Table 7, the PW91 and B3LYP functionals slightly overestimate the lattice constants and the equilibrium volumes, while HSE06 and PBE0 better predict the structural properties. The HSE06, B3LYP, and PBE0 band gap is 2.57, 2.89, and 3.35 eV, respectively. This is smaller than that obtained at the same level for the two monoclinic and the triclinic structures, while it is larger than that of the cubic and tetragonal ones. All calculations indicate that the band gap is direct at the Γ point. Figure 4(f) shows the B3LYP band structure and DOS. As in the triclinic structure, the highest VB is quite flat along the Γ Z path. As a result, the indirect energy gap from Z to Γ is only slightly larger than the direct band gap at the Γ point. Also, in this case the VB top is flat, producing a sharp valence-band DOS. As for the cubic and tetragonal structures, the band gap is weakly volume dependent. However, here the B3LYP band gap slightly decreases upon compression. This odd behavior suggests that the “band broadening” contribution compensates the “gap opening” effect under compression.

4. CONCLUSIONS In this work, we have systematically investigated the structural and electronic properties of various phases of tungsten oxide by using hybrid functionals combined with both plane wave and localized basis sets. Standard DFT/GGA calculations are also performed for comparison. We find that the hybrid functionals HSE06 and PBE0 give lattice constants and equilibrium volumes of WO3 in better agreement with the experiment than the standard DFT/GGA functionals. At the same time, the B3LYP hybrid functional in combination with a localized basis set gives results of similar quality as the PW91 functional regarding structural parameters. Most important, the introduction of a portion of exact HF exchange in hybrid functionals systematically improves the description of the band gap which is always severely underestimated by standard DFT methods. The HSE06 hybrid functional associated with a plane wave basis set works best in predicting the band gap of WO3, followed by B3LYP with a localized basis set, while PBE0 tends to overestimate this quantity. For the RT monoclinic structure, the effect of varying the exact HF exchange percentage in the B3LYP hybrid functional has also been investigated. With 15% of HF exchange, the band gap (2.73 eV) is very close to the experiment. An interesting outcome of this hybrid functional investigation is the observation that the band gap value is strongly dependent on the structure. Considering B3LYP values, it ranges from 1.85 eV for tetragonal to 3.33 eV for LT monoclinic structures. A similar trend was previously observed with GGA calculations,50 although in that work the energy range goes from 0.61 to 1.71 eV. On the basis of the present study, we can conclude that B3LYP, associated with a localized basis set approach, represents an acceptable compromise between computational cost and accuracy, for both structural and electronic properties of WO3. While an ad-hoc functional containing only 15% of HF exchange would provide a Kohn Sham band gap in closer agreement with experiment, the existence of a large database of computational results for semiconducting oxides based on the standard B3LYP functional is an additional reason to adopt this method for future studies to allow a direct comparison of electronic structures. This

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conclusion could be probably generalized to other materials, but certainly it lays solid foundation for further work on more complex WO3 involving systems.

’ ASSOCIATED CONTENT

bS

Supporting Information. Basis set information for the W atom. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work is supported by CARIPLO Foundation through an Advanced Materials Grant 2009. Regione Lombardia and CILEA Consortium, through a LISA Initiative (Laboratory for Interdisciplinary Advanced Simulation), are also gratefully acknowledged. ’ REFERENCES (1) Huda, M. N.; Yan, Y.; Moon, C.-Y.; Wei, S.-H.; Al-Jassim, M. M. Phys. Rev. B 2008, 77, 195102. (2) Khaselev, O.; Turner, J. A. Science 1998, 280, 425. (3) Kudo, A.; Miseki, Y. Chem. Soc. Rev. 2009, 38, 253. (4) Bak, T.; Nowotny, J.; Rekas, M.; Sorrell, C. C. Int. J. Hydrogen Energy 2002, 27, 991. (5) Granqvist, C. G. Sol. Energy Mater. Sol. Cells 2000, 60, 201. (6) Smith, D. J.; Vatelino, J. F.; Falconer, R. S.; Wittman, E. L. Sens. Actuators, B 1993, 13, 264. (7) Miller, E. L.; Marsen, B.; Cole, B.; Lunn, M. Electrochem. SolidState Lett. 2006, 9, G248. (8) Miller, E. L.; Paluselli, D.; Marsen, B.; Rocheleau, R. E. Sol. Energy Mater. Sol. Cells 2005, 88, 131. (9) Cole, B; Marsen, B.; Miller, E.; Yan, Y.; To, B.; Jones, K.; AlJassim, M. M. J. Phys. Chem. C 2008, 112, 5213. (10) Deb, S. K. Appl. Opt. 1969, 3, 192. (11) Deb, S. K. Philos. Mag. 1973, 27, 801. (12) Deb, S. K. Sol. Energy Mater. Sol. Cells 2008, 92, 245. (13) Berek, J. M.; Sienko, M. J. J. Solid State Chem. 1970, 2, 109. (14) Butler, M. A.; Nasby, R. D.; Quinn, R. K. Solid State Commun. 1976, 19, 1011. (15) Scaife, D. E. Solar Energy 1980, 25, 41. (16) Green, M.; Hussain, Z. J. Appl. Phys. 1991, 69, 7788. (17) Desilvestro, J.; Gratzel, M. J. Electroanal. Chem. 1987, 238, 129. (18) Kharade, R. R.; Mane, S. R.; Mane, R. M.; Patil, P. S.; Bhosale, P. N. J. Sol-Gel Sci. Technol. 2010, 56, 177. (19) Marsen, B.; Cole, B.; Miller, E. L. Sol. Energy Mater. Sol. Cells 2007, 91, 1954. (20) Gesheva, K.; Szekeres, A.; Ivanova, T. Sol. Energy Mater. Sol. Cells 2003, 76, 563. (21) Bange, K. Sol. Energy Mater. Sol. Cells 1999, 58, 1. (22) Bamwenda, R. G.; Sayama, K.; Arakawa, H. J. Photochem. Photobiol. A 1999, 122, 175. (23) Gonzalez-Borrero, P. P.; Sato, F.; Medina, A. N.; Baesso, M. L.; Bento, A. C.; Baldissera, G.; Persson, C.; Niklasson, G. A.; Granqvist, C. G.; Ferreira da Silva, A. Appl. Phys. Lett. 2010, 96, 061909. (24) Koffyberg, F. P.; Dwight, K.; Wold, A. Solid State Commun. 1979, 30, 433. (25) Hodes, G.; Cahen, D.; Manassen, J. Nature 1976, 260, 313. (26) Di Quarto, F.; Di Paola, A.; Sunseri, C. Electrochim. Acta 1981, 26, 1177. (27) Bringans, R. D.; Hochst, H.; Shanks, H. R. Phys. Rev. B 1981, 24, 3481. 8352

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