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Electronic and optical properties of anatase TiO2 nanotubes
Computational Materials Science 48 (2010) 854–858
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Electronic and optical properties of anatase TiO2 nanotubes Faruque M. Hossain a,*, Alexander V. Evteev a, Irina V. Belova a, Janusz Nowotny b, Graeme E. Murch a a
University Center for Mass and Thermal Transport in Engineering Materials, Priority Research Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia b Solar Energy Technologies, School of Natural Sciences, University of Western Sydney, Penrith South DC, NSW 1797, Australia
a r t i c l e
i n f o
Article history: Received 12 November 2009 Received in revised form 7 April 2010 Accepted 8 April 2010
Keywords: DFT Electronic structure Optical properties Nanotubes
a b s t r a c t The structural, electronic and optical properties of anatase TiO2 nanotubes are investigated using pseudopotential density-functional theory (DFT) calculations. Band structure and density of states (DOS) show discrete energy levels at the top of the valence band and immediately below the Fermi level. This groundwork of electronic structure calculations predicts a possible band gap modification of the TiO2 nanotube structure compared to the bulk. We observe significant electronic structural differences with the change in dimensions (radius) of the nanotube. The photon energy dependent imaginary part of the dielectric function further indicates the exact optical transitions from occupied valence bands to unoccupied conduction bands. All allowed optical transitions determine the actual optical band gap of anatase TiO2 nanotubes, which is higher than the direct band gap at the C point of the band structure. The result reveals that the band gap of TiO2 nanotube is not only dependent on the tube radius but also on other parameters such as the tube wall thickness and atomic arrangement in the wall of the tube. This result also shows the considerable optical anisotropy along the two axial directions of the nanotube. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Titanium dioxide (TiO2) has received special attention as the prime candidate material for photo-electrochemical water-splitting and other photo-catalytic applications [1–4]. It is also promising for its chemical stability in aqueous environments and under high energy illumination [3]. However, because of its rather large band gap, 3.0–3.2 eV (for the rutile and anatase phases respectively), TiO2 lacks sensitivity to visible light which is necessary for high performance under solar illumination. This is especially the case for the generation of hydrogen from water to reach commercial efficiencies [3]. Hence, considerable effort has been made to increase the visible light sensitivity through band gap modification [3,4] and/or increasing the active area exposure to light. Recently, interest has been directed towards nanostructures because of the large surface area exposable to sunlight, the high surface-to-volume ratio, non-toxicity, the unique photochemical and photo-physical properties and also for their electron-transport properties. In particular, TiO2 nanostructures have received special attention for their potential photo-catalytic activity [5–8], their biomedical applications, their Li+ and H2 storage properties [9–11], their potential in rechargeable batteries and catalyst sup-
* Corresponding author. Address: School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia. Tel.: +61 3 8344 8744; fax: +61 3 9347 4783. E-mail address: [email protected] (F.M. Hossain). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.04.007
ports in gas sensors [12,13], their functional behaviour as biological coatings [14,15] and their self-cleaning properties [1,16]. By determining the band structure, the few theoretical calculations [17–21] on anatase TiO2 nanotubes have shown that the nature of the band gap changes with nanotube dimension. All band structure calculations have shown that the band gap of an anatase TiO2 nanotube decreases with tube radius. This contrasts incidentally with the behaviour of TiO2 nanoparticles where the band gap increases with decreasing radius (the size quantization effect) [22]. In principle, the band gap in band structure calculations is estimated by taking the energy difference between the HOMO (highest occupied molecular orbitals) in the valence band and the LUMO (lowest unoccupied molecular orbitals) in the conduction band. Although, this energy estimation is the first step towards identifying the actual optical band gap of the structure, it does not take into account all the allowed energy states governed by the selection rules. Therefore, to provide a more accurate prediction of the optical band gap, the exploration of the optical matrix elements in the valence and conduction bands, that take part in the transition process, is required. In this paper, we present results of first-principles calculations of the variations of electronic and optical properties of anatase TiO2 nanotubes with structural and dimensional changes. The variations of electronic and optical properties are assessed by the band structure and the frequency-dependent imaginary part of the dielectric function respectively. This work provides a bench-mark for further ab initio calculations to find the best optimum dimension
F.M. Hossain et al. / Computational Materials Science 48 (2010) 854–858
and orientation of a TiO2 nanotube-based structure for effective and efficient technological applications.
2. Structure of TiO2 nanotubes The initial single-walled anatase TiO2 nanotube models were made by rolling up one (1 0 1) layer of the anatase structure along 0 1 direction. We obtained one (1 0 1) layer of the anatase the ½1 structure by placing the anatase motif unit (two TiO2 formula units) at each lattice point of a single (1 0 1) crystallographic plane of the body centered tetragonal Bravais lattice of the anatase structure. The atomic positions in the anatase motif unit with respect to each lattice point can be chosen (in units of lattice parameters) as follows: (0, 0, 0) – Ti; (0, 0.5, 0.25) – Ti; (0, 0.5, 0.042) – O; (0.5, 0, 0.042) – O; (0, 0, 0.208) – O; (0, 0, 0.208) – O. Thus the atoms within the anatase motif unit are situated in six different atomic planes parallel to the (1 0 1) crystallographic plane. Furthermore, the (1 0 1) crystallographic plane contains both lattice points of different types within the unit cell of the body centered tetragonal Bravais lattice. Therefore, one (1 0 1) layer of the anatase structure, which was rolled up, contained six atomic planes. The layer has the centered rectangular 2D Bravais lattice with 12 atoms (four titanium and eight oxygen atoms) in the unit cell 0 1 and with the basis vectors a2D and b2D along [0 1 0] and ½1 directions, respectively, of the body centered tetragonal Bravais lattice of the anatase structure. Two nanotubes were obtained by rolling up the layer in ways that the chiral vectors (6, 0) = 6a2D and (12, 0) = 12a2D become the circumferences of the first and second nanotubes, respectively. The lattice parameters of anatase a = b = 3.776 Å and c = 9.486 Å were taken to build a periodic TiO2 nanotube model structure with a periodicity L = 10.21 Å along 0 1. Initially made for the purposes of geometrical the tube axis ½1 optimization the (6,0) and (12,0) nanotubes had radii r = 3.776 Å and r = 7.295 Å, respectively, and were of identical length L = 10.21 Å. The radii of the TiO2 nanotube models were deterP P mined as r ¼ i mi qi = i mi , where mi is the mass of atom i and qi is the radial distance from the nanotube axis to the position of atom i. We will designate the small radius (r = 3.776 Å) nanotube as ‘P’ and the large radius (r = 7.295 Å) nanotube as ‘Q’ for the remainder of the description of the results. A total of 24 and 48 formula units of TiO2 were taken for the construction of the ‘P’ and ‘Q’ nanotube unit cells respectively. A three-dimensional (3D) periodicity is imposed to simulate the nanotubes with finite length in this calculation by considering a tetragonal supercell with a dimension 20 20 c Å3 and 30 30 c Å3 for ‘P’ and ‘Q’ respectively. The parameter c = 10.21 Å is considered as the 1D periodic length for both ‘P’ and ‘Q’ nanotubes. The center of the end face of nanotube is placed at the origin (0, 0, 0) and the nanotube axis is placed along the caxis of the 3D tetragonal box in order to impose the radial and the axial periodicity along X, Y and Z directions respectively. Two 3D space group symmetry of tetragonal crystal system (P42 mc and P4 mm for ‘P’ and ‘Q’ respectively) were imposed based on the different elements of symmetry of the nanotubes. Since the ‘P’ and ‘Q’ nanotubes have different chiral vectors (6, 0) and (12, 0), there exist different elements of symmetry for ‘P’ and ‘Q’ nanotubes. In particular, the axial point group of the ‘P’ nanotube is 6 mm (principal axis is 6-fold symmetry axis), while the axial point group of the ‘Q’ nanotube can be written as 12 mm (principal axis is 12-fold symmetry axis). Taking into account that both nanotubes have the same translation periodicity along the axial direction, the 1D line symmetry groups [23,21] of the ‘P’ and ‘Q’ nanotubes can be represented as P126mc and P2412mc, respectively. The 126 and 2412 screw axes (i.e., rotation for p/6 and p/12, respectively, around the nanotubes axes followed by translation of 1/2 of the
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lattice vector along the axes), and glide planes c (we use c by convention in the international symbol notation, because the glide planes contain the principal axes of the groups) appear as a result of products of the axial point groups (6 mm and 12 mm) and translation periodicity along the axial direction of the nanotubes.
3. Method of calculation Since the quantum mechanical DFT calculations are very computationally intensive for structural optimization for a system of many atoms and, in addition, a previous study on optimization using the DFT-based tight-binding method (DFTB) [24] showed that the optimization does not change the Ti–O bond length significantly, classical molecular mechanics simulation was performed in this study for the structural optimization. After structural optimization, ab initio quantum mechanical methods were used for electronic structure and optical properties calculations. In the molecular mechanics based calculation, the universal force field (UFF) method [25] was applied to model the interaction between the atoms in the structure. A cascaded optimization algorithm of the steepest descent, adjusted basis set Newton–Raphson (ABNR), and quasi-Newton was used in the process of structural relaxation. The structures were allowed to relax by setting up the convergence criteria for threshold energy change, maximum force, maximum pressure, and maximum displacement of 2.0 105 kcal/mol, 0.001 kcal/mol/Å, 0.001 GPa, and 1.0 105 Å, respectively. The energy band structure, density of states (DOS), and optical properties calculations were performed using the quantum mechanical based ab initio plane wave (PW) pseudopotential density-functional theory (DFT) method and were carried out using the CASTEP software code [26,27]. The Vanderbilt ultrasoft pseudopotential (USP) method [28] with a plane wave basis set cut-off energy of 340.0 eV was used in this calculation. The Monkhorst–Pack (MP) [29] mesh of 2 2 4 was used on the nanotube structure that produces two irreducible k-points in the Brillouin zone (BZ). The effects of exchange and correlation energy were treated within the generalized gradient approximation (GGA) [30]. The optical properties were calculated by selecting the appropriate dipole transition matrix elements in the valence band and conduction band. The dielectric function eðxÞ ¼ e1 ðxÞ þ ie2 ðxÞ is connected with the interaction of photons with electrons and determines the linear response of the system to an electromagnetic wave and is determined by the matrix elements. The imaginary part e2(x) of the dielectric function is thus calculated from the matrix elements of the momentum operator between the occupied and the unoccupied wave functions within the selection rules. This represents the probability of real transitions between the occupied (ground state) wvk and empty (excited state) wck wave functions (electronic states) and is given by [31]:
e2 ðxÞ ¼
4p2 e2 X v jhw j~ pjwck ij2 dðEv k Eck hxÞ m2 x2 V v ;c;k k
ð1Þ
where e is the elementary charge, m is the electron mass, V is the volume of the unit cell, hx is the energy of the incident photon, Evk and Eck are the quasi-particle energies (approximated to the GGA eigenvalues) of occupied valence band states and empty conduction band states respectively, ~ p is the momentum operator, pjwck i are the momentum matrix elements for the direct and hwvk j~ transition between valence band and conduction band. The calculation was performed by taking the polarization direction of the electric field component of the electromagnetic wave (the incident photon) parallel and perpendicular to the nanotube axis for the optical anisotropy check.
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‘P’ nanotube decreases from r = 3.776 Å to r = 3.535 Å. Similarly, the radius of the ‘Q’ nanotube decreases from r = 7.295 Å to r = 6.356 Å. Figs. 1 and 2 show the cross-sectional and side view of the optimized ‘P’ and ‘Q’ nanotubes respectively.
4. Results and discussion 4.1. Structural properties of TiO2 nanotubes At the end of the rolling up and optimization process, both structures’ inner and outer walls were terminated with oxygen atoms. The wall thickness was determined by the distance between the outer wall and the inner wall oxygen atoms. The optimized wall thickness for ‘P’ and ‘Q’ nanotubes are 2.74 Å and 2.63 Å respectively. The lengths of both nanotubes increase to L = 10.51 Å for the ‘P’ and to L = 10.64 Å for ‘Q’. The radius of the
A
B
Y
Z
X
Fig. 1. (A) Cross-sectional view and (B) side view of anatase TiO2 nanotube formed 0 1 direction. The geometrically optimized by rolling up a (1 0 1) layer along the ½1 length and diameter of the tube are 10.51 Å and 3.535 Å, respectively. Tetragonal crystal system with P42mc group symmetry was considered.
A
B
Y
Z
X
Fig. 2. (A) Cross-sectional view and (B) side view of anatase TiO2 nanotube formed 0 1 direction. The geometrically optimized by rolling up a (1 0 1) layer along the ½1 length and diameter of the tube are 10.64 Å and 6.356 Å, respectively. Tetragonal crystal system with P4mm group symmetry was considered.
4.2. Electronic band structure and density of states of TiO2 nanotubes Fig. 3A and B shows the energy band structure along high symmetry directions in the BZ and corresponding partial density of states (DOS) for the TiO2 nanotubes ‘P’ and ‘Q’ respectively. Partial DOS clearly shows the contribution of oxygen and titanium to the valence and conduction bands. Similar to bulk TiO2, the valence band and conduction bands of the nanotube mainly consist of O2p and Ti3d states respectively. Depending on the nanotube radius and atomic arrangement in the nanotube wall, the valence band and conduction band modify in different manners to change the band gap. In order to perform a detailed analyses of the band gap modification, the band structures were enlarged at three high symmetry k-points C(0, 0, 0), Z(0, 0, 1/2), and R(0, 1/2, 1/2) in the BZ of tetragonal unit cell as shown in Fig. 4A for ‘P’ and in Fig. 4C for ‘Q’. In addition, the orbital iso-surfaces (0.001 e/Å3) of the highest occupied molecular orbital (HOMO) at the top of the valence band and the lowest unoccupied molecular orbital (LUMO) at the bottom of the conduction band were constructed to show the contribution of atomic orbitals in band edges as shown in Fig. 4B for ‘P’ and in Fig. 4D for ‘Q’. It clearly illustrates that the direct electronic band gap at the C point for ‘P’ and ‘Q’ nanotubes are 1.46 eV and 1.75 eV, respectively. Similarly, the direct band gap at the Z point for the ‘P’ and ‘Q’ nanotubes are 1.78 eV and 2.08 eV respectively. This result of the electronic structure calculations is consistent with previous theoretical predictions [17–21], which noted the proportionate change of band gap with the anatase TiO2 nanotube radius. Moreover, our enlarged view of the band structure [Fig. 4A and C] indicates two different values of the energy gap at different locations within the nanotube. The fractional coordinates of symmetry points C(0, 0, 0) and Z(0, 0, 1/2) in the BZ of tetragonal unit cell lie on the nanotube axis as mentioned in Section 2. Both nanotubes (‘P’ and ‘Q’) show a 0.3 eV energy gap lowering from the middle point to the end face of the TiO2 nanotube of length L = 10.21 Å. This result clearly indicates that particular attention on the orientation of anatase TiO2 nanotubes should be made during the fabrication process in order to get better photo-catalytic activity. In addition, we analyze the volume visualization of the HOMO (topmost energy state of the valence band) – the LUMO (bottom most energy state of conduction band) iso-surfaces of both ‘P’
Fig. 3. Energy band structure of the anatase TiO2 nanotube along high symmetry directions in the Brillouin zone (BZ) for (A) tube length, L = 10.51 Å and tube radius r = 3.535 Å (B) tube length, L = 10.64 Å and tube radius r = 6.356 Å. Dashed line at zero energy represents the Fermi level.
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wall of O atoms and that most of the LUMO contribution comes from the in-wall Ti atoms. On the other hand, the HOMO of nanotube ‘Q’ in Fig. 4D shows that the major electronic contribution comes from the inner wall of O atoms with a partial contribution from the outer wall of O atoms and LUMO contribution remains from the in-wall Ti atoms. Hence, we observe that significant occupied and unoccupied states are available for the more concentric (less hollow space) ‘P’ nanotube to help narrow the band gap compared to the less concentric (more hollow space) ‘Q’ nanotube. However, all electronic information explained so far is not enough to test the actual photo-reactivity of anatase nanotube structures. We need to explore further the transition matrix element based optical transition energies between occupied valence band states and unoccupied conduction band states. 4.3. Optical properties
Fig. 4. Enlarged view of the energy states in the band structure shown in Fig. 3 taken for three high symmetry points located at C(0, 0, 0), Z(0, 0, 1/2) and R(0, 1/ 2, 1/2) for (A) tube length, L = 10.51 Å and tube radius r = 3.535 Å and for (c) tube length, L = 10.64 Å and tube radius r = 6.356 Å. (B) Orbital iso-surfaces (0.001 e/Å3) of highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) corresponding to HOMO–LUMO states indicated in (A). (D) Orbital iso-surfaces (0.001 e/Å3) of HOMO and LUMO corresponding to HOMO–LUMO states indicated in (C). Iso-surfaces are presented only for the first quadrant and for cross-section view of tubes.
and ‘Q’ nanotubes with identical iso-values (electron density of 0.001 e/Å3). This HOMO–LUMO analysis provides a more detailed electronic understanding of the engineered band gap modification for the nanotube structure. The HOMO of nanotube ‘P’ in Fig. 4B shows that the entire electronic contribution comes from the inner
The frequency and dimension dependent optical properties of anatase TiO2 nanotubes were obtained by calculating the optical transition matrix elements between occupied eigenstates in the valence bands and unoccupied eigenstates in the conduction bands. Fig. 5 shows the optical properties of two nanotubes ‘P’ and ‘Q’ with identical tube-length (L) but a different tube-radius (r) calculated within the photon energy range of 0–8 eV. The imaginary parts of the dielectric function e2(x) were evaluated by polarizing the incoming light along two directions of the tube (longitudinal [along the X-axis in Fig. 1A] and transverse [along the Z-axis in Fig. 1B]). Fig. 5A and B shows transition peaks arising from allowed band to band transitions for nanotubes ‘P’ and ‘Q’, respectively. The optoelectronic spectra along the tube axis (Z-axis) and perpendicular to the tube axis (X-axis) for the two selected radii show a significant change in the fundamental absorption edges (upper valence band to lower conduction band) and optical anisotropy. It is clear from the spectra that the optical band gap (absorption edge) along the tube axis shifts with the changes of tube radius and with the change of atomic arrangement in the tube-wall. The shift of the fundamental absorption edge of 0.9 eV to a lower energy occurs for doubling-up of the tube radius. This result obviously is not the relationship between band gap and tube radius that we obtained and discussed in the band structure, but represents an actual optical response of the nanotube structure. It seems here that the calculated results obtained from the band structure and the optical properties give different estimations of the energy gap of nanotube. However, detailed analysis provides us with a quantum level solution as we carefully investigate the band structure in Fig. 3 for both ‘P’ and ‘Q’ nanotubes. Fig. 3A for ‘P’ shows a doubly degenerate energy state just below the Fermi level, which is completely isolated from the top of the valence band and there exists a considerable energy gap. On the other hand, Fig. 3B for ‘Q’
Fig. 5. The frequency-dependent imaginary part of the dielectric function, e2(x), evaluated for the two directions of incoming light w.r.t. tube axis for (A) tube length, L = 10.51 Å and tube radius r = 3.535 Å and for (B) tube length, L = 10.64 Å and tube radius r = 6.356 Å.
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shows that two doubly degenerate energy states are placed just below the Fermi level and are very close to the valence band. In optical properties calculation, these two states become a part of the valence band. The selection rules of optical transitions treat these two states as allowed energy states. Therefore, the fundamental absorption edge occurs at a lower energy due to the participation of these two states. Whereas, the isolated state in Fig. 3A for ‘P’ is energetically far from the valence band. The selection rules of optical transitions never treat this state as an allowed energy state. As a result, the optical transition occurs only from the valence band to the conduction band without any participation of the isolated energy state. Hence, the fundamental absorption edge occurs at a higher energy compared to ‘Q’. In previous theoretical calculations and our enlarged view analysis [shown in Fig. 4] considered this isolated energy state level as the top of the valence band. The above quantum mechanics based analysis provides a solid understanding of the electronic structure and demonstrates the importance of tube wall thickness on the optical properties of the nanotube structure. Further investigation on the optical response by changing the polarization of light in a particular direction gives an interesting feature. We find a noticeable difference in absorption spectra that occurs depending on the polarization direction (whether in radial or axial) of the electric field of the incoming electromagnetic wave (light). The emission probability (intensity) decreases for the tube of larger radius entirely over the 2–6 eV spectral energy range and the emission intensity shows a significant anisotropy. These structural dependent optical properties are required to identify ways of designing a suitable nanostructured surface that would be more efficient and sensitive to light and absorbing species. Our anisotropy result indicates that normal oriented nanotubes would show a better optical absorption compared to nanotubes that are inclined to the horizontal surface/substrate.
provide an understanding of the actual optical transition and electronic band gap of such hollow (single walled) nanostructures of TiO2. An observed optical anisotropy gives further evidence of how the orientation of nanotubes can vary the photo-catalytic activity. In conclusion, a proper design and combination of lower band gap and higher surface area exposable to light make this nano-structured material a potentially important candidate for use in efficient photo-catalytic devices.
5. Conclusion
[26]
We have investigated the energy band structure, density of states and optical properties of anatase TiO2 nanotubes using a first-principles DFT calculation. We found noticeable electronic structural differences with the change in dimensions (radii) and possible band gap lowering compared to bulk TiO2. These results
[27] [28] [29] [30] [31]
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Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Electronic and optical properties of anatase TiO2 nanotubes Faruque M. Hossain a,*, Alexander V. Evteev a, Irina V. Belova a, Janusz Nowotny b, Graeme E. Murch a a
University Center for Mass and Thermal Transport in Engineering Materials, Priority Research Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia b Solar Energy Technologies, School of Natural Sciences, University of Western Sydney, Penrith South DC, NSW 1797, Australia
a r t i c l e
i n f o
Article history: Received 12 November 2009 Received in revised form 7 April 2010 Accepted 8 April 2010
Keywords: DFT Electronic structure Optical properties Nanotubes
a b s t r a c t The structural, electronic and optical properties of anatase TiO2 nanotubes are investigated using pseudopotential density-functional theory (DFT) calculations. Band structure and density of states (DOS) show discrete energy levels at the top of the valence band and immediately below the Fermi level. This groundwork of electronic structure calculations predicts a possible band gap modification of the TiO2 nanotube structure compared to the bulk. We observe significant electronic structural differences with the change in dimensions (radius) of the nanotube. The photon energy dependent imaginary part of the dielectric function further indicates the exact optical transitions from occupied valence bands to unoccupied conduction bands. All allowed optical transitions determine the actual optical band gap of anatase TiO2 nanotubes, which is higher than the direct band gap at the C point of the band structure. The result reveals that the band gap of TiO2 nanotube is not only dependent on the tube radius but also on other parameters such as the tube wall thickness and atomic arrangement in the wall of the tube. This result also shows the considerable optical anisotropy along the two axial directions of the nanotube. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Titanium dioxide (TiO2) has received special attention as the prime candidate material for photo-electrochemical water-splitting and other photo-catalytic applications [1–4]. It is also promising for its chemical stability in aqueous environments and under high energy illumination [3]. However, because of its rather large band gap, 3.0–3.2 eV (for the rutile and anatase phases respectively), TiO2 lacks sensitivity to visible light which is necessary for high performance under solar illumination. This is especially the case for the generation of hydrogen from water to reach commercial efficiencies [3]. Hence, considerable effort has been made to increase the visible light sensitivity through band gap modification [3,4] and/or increasing the active area exposure to light. Recently, interest has been directed towards nanostructures because of the large surface area exposable to sunlight, the high surface-to-volume ratio, non-toxicity, the unique photochemical and photo-physical properties and also for their electron-transport properties. In particular, TiO2 nanostructures have received special attention for their potential photo-catalytic activity [5–8], their biomedical applications, their Li+ and H2 storage properties [9–11], their potential in rechargeable batteries and catalyst sup-
* Corresponding author. Address: School of Physics, The University of Melbourne, Parkville, VIC 3010, Australia. Tel.: +61 3 8344 8744; fax: +61 3 9347 4783. E-mail address: [email protected] (F.M. Hossain). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.04.007
ports in gas sensors [12,13], their functional behaviour as biological coatings [14,15] and their self-cleaning properties [1,16]. By determining the band structure, the few theoretical calculations [17–21] on anatase TiO2 nanotubes have shown that the nature of the band gap changes with nanotube dimension. All band structure calculations have shown that the band gap of an anatase TiO2 nanotube decreases with tube radius. This contrasts incidentally with the behaviour of TiO2 nanoparticles where the band gap increases with decreasing radius (the size quantization effect) [22]. In principle, the band gap in band structure calculations is estimated by taking the energy difference between the HOMO (highest occupied molecular orbitals) in the valence band and the LUMO (lowest unoccupied molecular orbitals) in the conduction band. Although, this energy estimation is the first step towards identifying the actual optical band gap of the structure, it does not take into account all the allowed energy states governed by the selection rules. Therefore, to provide a more accurate prediction of the optical band gap, the exploration of the optical matrix elements in the valence and conduction bands, that take part in the transition process, is required. In this paper, we present results of first-principles calculations of the variations of electronic and optical properties of anatase TiO2 nanotubes with structural and dimensional changes. The variations of electronic and optical properties are assessed by the band structure and the frequency-dependent imaginary part of the dielectric function respectively. This work provides a bench-mark for further ab initio calculations to find the best optimum dimension
F.M. Hossain et al. / Computational Materials Science 48 (2010) 854–858
and orientation of a TiO2 nanotube-based structure for effective and efficient technological applications.
2. Structure of TiO2 nanotubes The initial single-walled anatase TiO2 nanotube models were made by rolling up one (1 0 1) layer of the anatase structure along 0 1 direction. We obtained one (1 0 1) layer of the anatase the ½1 structure by placing the anatase motif unit (two TiO2 formula units) at each lattice point of a single (1 0 1) crystallographic plane of the body centered tetragonal Bravais lattice of the anatase structure. The atomic positions in the anatase motif unit with respect to each lattice point can be chosen (in units of lattice parameters) as follows: (0, 0, 0) – Ti; (0, 0.5, 0.25) – Ti; (0, 0.5, 0.042) – O; (0.5, 0, 0.042) – O; (0, 0, 0.208) – O; (0, 0, 0.208) – O. Thus the atoms within the anatase motif unit are situated in six different atomic planes parallel to the (1 0 1) crystallographic plane. Furthermore, the (1 0 1) crystallographic plane contains both lattice points of different types within the unit cell of the body centered tetragonal Bravais lattice. Therefore, one (1 0 1) layer of the anatase structure, which was rolled up, contained six atomic planes. The layer has the centered rectangular 2D Bravais lattice with 12 atoms (four titanium and eight oxygen atoms) in the unit cell 0 1 and with the basis vectors a2D and b2D along [0 1 0] and ½1 directions, respectively, of the body centered tetragonal Bravais lattice of the anatase structure. Two nanotubes were obtained by rolling up the layer in ways that the chiral vectors (6, 0) = 6a2D and (12, 0) = 12a2D become the circumferences of the first and second nanotubes, respectively. The lattice parameters of anatase a = b = 3.776 Å and c = 9.486 Å were taken to build a periodic TiO2 nanotube model structure with a periodicity L = 10.21 Å along 0 1. Initially made for the purposes of geometrical the tube axis ½1 optimization the (6,0) and (12,0) nanotubes had radii r = 3.776 Å and r = 7.295 Å, respectively, and were of identical length L = 10.21 Å. The radii of the TiO2 nanotube models were deterP P mined as r ¼ i mi qi = i mi , where mi is the mass of atom i and qi is the radial distance from the nanotube axis to the position of atom i. We will designate the small radius (r = 3.776 Å) nanotube as ‘P’ and the large radius (r = 7.295 Å) nanotube as ‘Q’ for the remainder of the description of the results. A total of 24 and 48 formula units of TiO2 were taken for the construction of the ‘P’ and ‘Q’ nanotube unit cells respectively. A three-dimensional (3D) periodicity is imposed to simulate the nanotubes with finite length in this calculation by considering a tetragonal supercell with a dimension 20 20 c Å3 and 30 30 c Å3 for ‘P’ and ‘Q’ respectively. The parameter c = 10.21 Å is considered as the 1D periodic length for both ‘P’ and ‘Q’ nanotubes. The center of the end face of nanotube is placed at the origin (0, 0, 0) and the nanotube axis is placed along the caxis of the 3D tetragonal box in order to impose the radial and the axial periodicity along X, Y and Z directions respectively. Two 3D space group symmetry of tetragonal crystal system (P42 mc and P4 mm for ‘P’ and ‘Q’ respectively) were imposed based on the different elements of symmetry of the nanotubes. Since the ‘P’ and ‘Q’ nanotubes have different chiral vectors (6, 0) and (12, 0), there exist different elements of symmetry for ‘P’ and ‘Q’ nanotubes. In particular, the axial point group of the ‘P’ nanotube is 6 mm (principal axis is 6-fold symmetry axis), while the axial point group of the ‘Q’ nanotube can be written as 12 mm (principal axis is 12-fold symmetry axis). Taking into account that both nanotubes have the same translation periodicity along the axial direction, the 1D line symmetry groups [23,21] of the ‘P’ and ‘Q’ nanotubes can be represented as P126mc and P2412mc, respectively. The 126 and 2412 screw axes (i.e., rotation for p/6 and p/12, respectively, around the nanotubes axes followed by translation of 1/2 of the
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lattice vector along the axes), and glide planes c (we use c by convention in the international symbol notation, because the glide planes contain the principal axes of the groups) appear as a result of products of the axial point groups (6 mm and 12 mm) and translation periodicity along the axial direction of the nanotubes.
3. Method of calculation Since the quantum mechanical DFT calculations are very computationally intensive for structural optimization for a system of many atoms and, in addition, a previous study on optimization using the DFT-based tight-binding method (DFTB) [24] showed that the optimization does not change the Ti–O bond length significantly, classical molecular mechanics simulation was performed in this study for the structural optimization. After structural optimization, ab initio quantum mechanical methods were used for electronic structure and optical properties calculations. In the molecular mechanics based calculation, the universal force field (UFF) method [25] was applied to model the interaction between the atoms in the structure. A cascaded optimization algorithm of the steepest descent, adjusted basis set Newton–Raphson (ABNR), and quasi-Newton was used in the process of structural relaxation. The structures were allowed to relax by setting up the convergence criteria for threshold energy change, maximum force, maximum pressure, and maximum displacement of 2.0 105 kcal/mol, 0.001 kcal/mol/Å, 0.001 GPa, and 1.0 105 Å, respectively. The energy band structure, density of states (DOS), and optical properties calculations were performed using the quantum mechanical based ab initio plane wave (PW) pseudopotential density-functional theory (DFT) method and were carried out using the CASTEP software code [26,27]. The Vanderbilt ultrasoft pseudopotential (USP) method [28] with a plane wave basis set cut-off energy of 340.0 eV was used in this calculation. The Monkhorst–Pack (MP) [29] mesh of 2 2 4 was used on the nanotube structure that produces two irreducible k-points in the Brillouin zone (BZ). The effects of exchange and correlation energy were treated within the generalized gradient approximation (GGA) [30]. The optical properties were calculated by selecting the appropriate dipole transition matrix elements in the valence band and conduction band. The dielectric function eðxÞ ¼ e1 ðxÞ þ ie2 ðxÞ is connected with the interaction of photons with electrons and determines the linear response of the system to an electromagnetic wave and is determined by the matrix elements. The imaginary part e2(x) of the dielectric function is thus calculated from the matrix elements of the momentum operator between the occupied and the unoccupied wave functions within the selection rules. This represents the probability of real transitions between the occupied (ground state) wvk and empty (excited state) wck wave functions (electronic states) and is given by [31]:
e2 ðxÞ ¼
4p2 e2 X v jhw j~ pjwck ij2 dðEv k Eck hxÞ m2 x2 V v ;c;k k
ð1Þ
where e is the elementary charge, m is the electron mass, V is the volume of the unit cell, hx is the energy of the incident photon, Evk and Eck are the quasi-particle energies (approximated to the GGA eigenvalues) of occupied valence band states and empty conduction band states respectively, ~ p is the momentum operator, pjwck i are the momentum matrix elements for the direct and hwvk j~ transition between valence band and conduction band. The calculation was performed by taking the polarization direction of the electric field component of the electromagnetic wave (the incident photon) parallel and perpendicular to the nanotube axis for the optical anisotropy check.
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‘P’ nanotube decreases from r = 3.776 Å to r = 3.535 Å. Similarly, the radius of the ‘Q’ nanotube decreases from r = 7.295 Å to r = 6.356 Å. Figs. 1 and 2 show the cross-sectional and side view of the optimized ‘P’ and ‘Q’ nanotubes respectively.
4. Results and discussion 4.1. Structural properties of TiO2 nanotubes At the end of the rolling up and optimization process, both structures’ inner and outer walls were terminated with oxygen atoms. The wall thickness was determined by the distance between the outer wall and the inner wall oxygen atoms. The optimized wall thickness for ‘P’ and ‘Q’ nanotubes are 2.74 Å and 2.63 Å respectively. The lengths of both nanotubes increase to L = 10.51 Å for the ‘P’ and to L = 10.64 Å for ‘Q’. The radius of the
A
B
Y
Z
X
Fig. 1. (A) Cross-sectional view and (B) side view of anatase TiO2 nanotube formed 0 1 direction. The geometrically optimized by rolling up a (1 0 1) layer along the ½1 length and diameter of the tube are 10.51 Å and 3.535 Å, respectively. Tetragonal crystal system with P42mc group symmetry was considered.
A
B
Y
Z
X
Fig. 2. (A) Cross-sectional view and (B) side view of anatase TiO2 nanotube formed 0 1 direction. The geometrically optimized by rolling up a (1 0 1) layer along the ½1 length and diameter of the tube are 10.64 Å and 6.356 Å, respectively. Tetragonal crystal system with P4mm group symmetry was considered.
4.2. Electronic band structure and density of states of TiO2 nanotubes Fig. 3A and B shows the energy band structure along high symmetry directions in the BZ and corresponding partial density of states (DOS) for the TiO2 nanotubes ‘P’ and ‘Q’ respectively. Partial DOS clearly shows the contribution of oxygen and titanium to the valence and conduction bands. Similar to bulk TiO2, the valence band and conduction bands of the nanotube mainly consist of O2p and Ti3d states respectively. Depending on the nanotube radius and atomic arrangement in the nanotube wall, the valence band and conduction band modify in different manners to change the band gap. In order to perform a detailed analyses of the band gap modification, the band structures were enlarged at three high symmetry k-points C(0, 0, 0), Z(0, 0, 1/2), and R(0, 1/2, 1/2) in the BZ of tetragonal unit cell as shown in Fig. 4A for ‘P’ and in Fig. 4C for ‘Q’. In addition, the orbital iso-surfaces (0.001 e/Å3) of the highest occupied molecular orbital (HOMO) at the top of the valence band and the lowest unoccupied molecular orbital (LUMO) at the bottom of the conduction band were constructed to show the contribution of atomic orbitals in band edges as shown in Fig. 4B for ‘P’ and in Fig. 4D for ‘Q’. It clearly illustrates that the direct electronic band gap at the C point for ‘P’ and ‘Q’ nanotubes are 1.46 eV and 1.75 eV, respectively. Similarly, the direct band gap at the Z point for the ‘P’ and ‘Q’ nanotubes are 1.78 eV and 2.08 eV respectively. This result of the electronic structure calculations is consistent with previous theoretical predictions [17–21], which noted the proportionate change of band gap with the anatase TiO2 nanotube radius. Moreover, our enlarged view of the band structure [Fig. 4A and C] indicates two different values of the energy gap at different locations within the nanotube. The fractional coordinates of symmetry points C(0, 0, 0) and Z(0, 0, 1/2) in the BZ of tetragonal unit cell lie on the nanotube axis as mentioned in Section 2. Both nanotubes (‘P’ and ‘Q’) show a 0.3 eV energy gap lowering from the middle point to the end face of the TiO2 nanotube of length L = 10.21 Å. This result clearly indicates that particular attention on the orientation of anatase TiO2 nanotubes should be made during the fabrication process in order to get better photo-catalytic activity. In addition, we analyze the volume visualization of the HOMO (topmost energy state of the valence band) – the LUMO (bottom most energy state of conduction band) iso-surfaces of both ‘P’
Fig. 3. Energy band structure of the anatase TiO2 nanotube along high symmetry directions in the Brillouin zone (BZ) for (A) tube length, L = 10.51 Å and tube radius r = 3.535 Å (B) tube length, L = 10.64 Å and tube radius r = 6.356 Å. Dashed line at zero energy represents the Fermi level.
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wall of O atoms and that most of the LUMO contribution comes from the in-wall Ti atoms. On the other hand, the HOMO of nanotube ‘Q’ in Fig. 4D shows that the major electronic contribution comes from the inner wall of O atoms with a partial contribution from the outer wall of O atoms and LUMO contribution remains from the in-wall Ti atoms. Hence, we observe that significant occupied and unoccupied states are available for the more concentric (less hollow space) ‘P’ nanotube to help narrow the band gap compared to the less concentric (more hollow space) ‘Q’ nanotube. However, all electronic information explained so far is not enough to test the actual photo-reactivity of anatase nanotube structures. We need to explore further the transition matrix element based optical transition energies between occupied valence band states and unoccupied conduction band states. 4.3. Optical properties
Fig. 4. Enlarged view of the energy states in the band structure shown in Fig. 3 taken for three high symmetry points located at C(0, 0, 0), Z(0, 0, 1/2) and R(0, 1/ 2, 1/2) for (A) tube length, L = 10.51 Å and tube radius r = 3.535 Å and for (c) tube length, L = 10.64 Å and tube radius r = 6.356 Å. (B) Orbital iso-surfaces (0.001 e/Å3) of highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) corresponding to HOMO–LUMO states indicated in (A). (D) Orbital iso-surfaces (0.001 e/Å3) of HOMO and LUMO corresponding to HOMO–LUMO states indicated in (C). Iso-surfaces are presented only for the first quadrant and for cross-section view of tubes.
and ‘Q’ nanotubes with identical iso-values (electron density of 0.001 e/Å3). This HOMO–LUMO analysis provides a more detailed electronic understanding of the engineered band gap modification for the nanotube structure. The HOMO of nanotube ‘P’ in Fig. 4B shows that the entire electronic contribution comes from the inner
The frequency and dimension dependent optical properties of anatase TiO2 nanotubes were obtained by calculating the optical transition matrix elements between occupied eigenstates in the valence bands and unoccupied eigenstates in the conduction bands. Fig. 5 shows the optical properties of two nanotubes ‘P’ and ‘Q’ with identical tube-length (L) but a different tube-radius (r) calculated within the photon energy range of 0–8 eV. The imaginary parts of the dielectric function e2(x) were evaluated by polarizing the incoming light along two directions of the tube (longitudinal [along the X-axis in Fig. 1A] and transverse [along the Z-axis in Fig. 1B]). Fig. 5A and B shows transition peaks arising from allowed band to band transitions for nanotubes ‘P’ and ‘Q’, respectively. The optoelectronic spectra along the tube axis (Z-axis) and perpendicular to the tube axis (X-axis) for the two selected radii show a significant change in the fundamental absorption edges (upper valence band to lower conduction band) and optical anisotropy. It is clear from the spectra that the optical band gap (absorption edge) along the tube axis shifts with the changes of tube radius and with the change of atomic arrangement in the tube-wall. The shift of the fundamental absorption edge of 0.9 eV to a lower energy occurs for doubling-up of the tube radius. This result obviously is not the relationship between band gap and tube radius that we obtained and discussed in the band structure, but represents an actual optical response of the nanotube structure. It seems here that the calculated results obtained from the band structure and the optical properties give different estimations of the energy gap of nanotube. However, detailed analysis provides us with a quantum level solution as we carefully investigate the band structure in Fig. 3 for both ‘P’ and ‘Q’ nanotubes. Fig. 3A for ‘P’ shows a doubly degenerate energy state just below the Fermi level, which is completely isolated from the top of the valence band and there exists a considerable energy gap. On the other hand, Fig. 3B for ‘Q’
Fig. 5. The frequency-dependent imaginary part of the dielectric function, e2(x), evaluated for the two directions of incoming light w.r.t. tube axis for (A) tube length, L = 10.51 Å and tube radius r = 3.535 Å and for (B) tube length, L = 10.64 Å and tube radius r = 6.356 Å.
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shows that two doubly degenerate energy states are placed just below the Fermi level and are very close to the valence band. In optical properties calculation, these two states become a part of the valence band. The selection rules of optical transitions treat these two states as allowed energy states. Therefore, the fundamental absorption edge occurs at a lower energy due to the participation of these two states. Whereas, the isolated state in Fig. 3A for ‘P’ is energetically far from the valence band. The selection rules of optical transitions never treat this state as an allowed energy state. As a result, the optical transition occurs only from the valence band to the conduction band without any participation of the isolated energy state. Hence, the fundamental absorption edge occurs at a higher energy compared to ‘Q’. In previous theoretical calculations and our enlarged view analysis [shown in Fig. 4] considered this isolated energy state level as the top of the valence band. The above quantum mechanics based analysis provides a solid understanding of the electronic structure and demonstrates the importance of tube wall thickness on the optical properties of the nanotube structure. Further investigation on the optical response by changing the polarization of light in a particular direction gives an interesting feature. We find a noticeable difference in absorption spectra that occurs depending on the polarization direction (whether in radial or axial) of the electric field of the incoming electromagnetic wave (light). The emission probability (intensity) decreases for the tube of larger radius entirely over the 2–6 eV spectral energy range and the emission intensity shows a significant anisotropy. These structural dependent optical properties are required to identify ways of designing a suitable nanostructured surface that would be more efficient and sensitive to light and absorbing species. Our anisotropy result indicates that normal oriented nanotubes would show a better optical absorption compared to nanotubes that are inclined to the horizontal surface/substrate.
provide an understanding of the actual optical transition and electronic band gap of such hollow (single walled) nanostructures of TiO2. An observed optical anisotropy gives further evidence of how the orientation of nanotubes can vary the photo-catalytic activity. In conclusion, a proper design and combination of lower band gap and higher surface area exposable to light make this nano-structured material a potentially important candidate for use in efficient photo-catalytic devices.
5. Conclusion
[26]
We have investigated the energy band structure, density of states and optical properties of anatase TiO2 nanotubes using a first-principles DFT calculation. We found noticeable electronic structural differences with the change in dimensions (radii) and possible band gap lowering compared to bulk TiO2. These results
[27] [28] [29] [30] [31]
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