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Anatase(001) 3 ML Nanotubes, The First TiO2 Nanotube With Negative Strain Energies: A DFT Prediction Anna Maria Ferrari,*,† D enes Szieberth,† Claudio M. Zicovich-Wilson,‡ and † Raffaella Demichelis †
Dipartimento di Chimica IFM, Universit a di Torino and NIS -Nanostructured Interfaces and Surfaces Centre of Excellence, Via P. Giuria 7, 10125 Torino, Italy, and ‡Universidad Autonoma del Estado de Morelos-Cuernavaca- Mexico
ABSTRACT Nanotubes created from a three monolayer thick anatase(001) layer were investigated by DFT calculations in the 33-66 Å diameter range. Strain energies were found to be negative in the whole diameter range, indicating the stability of the nanotubes relative to the flat layer, providing a possible explanation for the rolling-up of the nanotubes. The strain energy curve was shown to exhibit a minimum close to the experimentally observed TiO2 nanotube diameters. Band gaps of the nanotubes were found to be close to that of the flat layer and more than 1 eV with respect to bulk anatase. SECTION Nanoparticles and Nanostructures
A
and the film undergoes a transition that involves the shift of the upper and the lower part of the film perpendicularly to each other,15 resulting in a structure halfway between anatase and lepidocrocite (Figure 1c). The aim of this work is to investigate TiO2 nanotubes created from 3 ML anatase(001) films by means of periodic density functional (DFT) calculations. The computation of nanotubes having large unit cells (540 atoms in case of the (60,0) nanotubes) was made possible by the full exploitation of the helical rototranslational symmetry of these structures by employing the periodic CRYSTAL09 code.16 The symmetry is used for the automatic generation of the structures, for the calculation of one and two-electron integrals (only the irreducible part of the Fock matrix is calculated then rotated by the symmetry operators of the point group to generate the full Fock matrix), and for the diagonalization of the Fock matrix, where each irreducible representation is treated separately.3,17 A development version of the code was used, where the restriction on the number of symmetry operators was relieved from the previous constraint of 48. The hybrid PBE018 functional was used. All calculations have been performed using an all-electron 8-411G(d) Gaussian-type basis set19 for O and a Hay-Wadt small core ECP20 with a 411-31 [3sp2d] basis set for valence electrons19 for Ti atoms. The reciprocal space was sampled according to a regular sublattice determined by the shrinking factor 6 6 (four independent k-points in the irreducible part of the Brillouin zone). Because the strain originating from the curvature of the nanotubes is expected to affect the reconstruction of the anatase(001) film, we have performed a detailed investigation of
lthough TiO2 nanotubes are widely investigated in the scientific literature, both their atomic-level structure and the mechanism of their formation from layers is still ambiguous. (See refs 1 and 2 and references therein). Besides the trititanate and lepidocrocite structures, anatase layers are also considered as possible constituents of the tube walls. One of the possible driving forces for the rolling up of a layer into a nanotube can be the asymmetry of the structure on the two sides of the layer. Among the examples of this phenomena are the naturally occurring tubular crystals chrysotile3 and imogolite,4,5 the latter being the first example of a structure that shows negative strain energies at diameters accessible by ab initio modeling methods. Titanate (hydrated TiO2) tubes are thought to roll up as a consequence of the different hydration state or different ionic environment on the two layer sides.6 All hitherto investigated pure TiO2 structures however show uniformly positive strain energies,7-10 indicating that the nanotubes are less stable than the corresponding flat layers at all diameters. (In the case of nanotubes derived from (001) anatase monolayer,11 a negative strain energy has been reported, but this film is not representative of any feasible titania structure.) Anatase(001) layers are one of the possible wall structures found in TiO2-based nanotubes.1,12 The unreconstructed anatase(001) surface has a squared unit cell, and its main features are the rows of two coordinate oxygen atoms (O2c). Ultrathin anatase(001) layers are known to display spectacular reconstructions. The two TiO2 layers thick (2 ML) anatase(001) film rearranges into lepidocrocite.13,14 In the case of the three TiO2 layers thick (3 ML) anatase film, the O2c rows on the bottom and top surfaces are running perpendicularly to each other, creating a directional asymmetry on the two sides of the sheet (Figure 1a); the rearrangement to the thermodynamically more stable lepidocrocite is hindered,
r 2010 American Chemical Society
Received Date: August 19, 2010 Accepted Date: September 9, 2010 Published on Web Date: September 13, 2010
2854
DOI: 10.1021/jz101184f |J. Phys. Chem. Lett. 2010, 1, 2854–2857
pubs.acs.org/JPCL
Figure 1. Anatase(001) 3 ML film: (1a) regular 001 surface; (1b) shifted along only one direction; (1c) shifted along both directions. See also the text for details.
Figure 2. (N, 0) and (0, N) anatase(001) 3 ML nanotubes.
(Figure 2) have not been considered because they were always found to be significantly less stable than the corresponding (N,0) tubes of approximately the same size. (For instance, the energy difference is 0.52 eV/TiO2 in the case of the (24,0) tubes.) The (N,0) nanotubes correspond to a chiral angle equal to zero and therefore to a zigzag structure according to usual nanotube nomenclature. Selected energetic and geometrical parameters of the nanotubes are collected in Table 1. Interestingly, at the smallest (D < 38 Å) diameters, the structure of the nanotube walls proved to be different from that of the flat sheet: the optimized geometry of the nanotube wall is more similar to what is shown on Figure 1b: whereas the external and internal layers are shifted from their bulk positions in the circumferential direction, they remain unshifted in the axial direction. This phenomenon can be explained by tracking the bond lengths most affected by the strain exerted by the rolling up of the nanotubes. As the distance between the Ti1 and Ti2 atoms increases with the increasing curvature (see labeling of the atoms in Figure 2 and
the transformation between the 1a and 1c forms of the trilayer. The unreconstructed slab (bond lengths were optimized, but the original symmetry of the bulk was preserved) is characterized by a square cell (a = b = 3.59 Å) and a film energy with respect to bulk anatase (ΔEfilm) of 0.70 eV/TiO2. The 1a-1c transformation can be conceived as a two-step process. Shifting the top layer in the direction of the arrow indicated on Figure 1b, we got a structure characterized by a rectangular unit cell (a = 3.60 Å; b = 3.48 Å) and ΔEfilm = 0.65 eV/TiO2, and it is therefore 0.05 eV/TiO2 more stable than 1a. Shifting the bottom layer as well perpendicularly to the previous direction results in 1c, the preferred structure of the flat anatase(001) 3 ML sheet. 1c is characterized by a square cell considerably shrunk with respect to the unreconstructed 1a (a = b = 3.41 Å) and ΔEfilm = 0.56 eV/TiO2. The energy gain of the relaxation 1b-1c is 0.09 eV, and the one connected to the overall process 1a-1c is 0.14 eV/TiO2. Nanotubes in the 33-66 Å diameter range (rollup vectors (24,0)-(60,0)) were constructed from the anatase(001) 3 ML slab, and the structures were optimized. (0,N) nanotubes
r 2010 American Chemical Society
2855
DOI: 10.1021/jz101184f |J. Phys. Chem. Lett. 2010, 1, 2854–2857
pubs.acs.org/JPCL
Table 1. Rollup Indexes (N), Diameters (D [Å], Calculated As the Average of the Diameters Measured at the O2c Atoms on the Internal and on the External Surface), Strain Energies (Es [eV/TiO2], Bond Lengths (d [Å]), Torsional Angles between the Ti1-Ti2-Ti3 Plane and O4 (ω [deg]), and Band Gaps (Eg, [eV]) of Anatase(001) 3 ML Nanotubes N
D
Es
dTi1-O4 dTi2-O4 dTi1-Ti2
dO4-Ti3
ω
Eg
28 32.9 -0.098
2.468
1.848
4.270
1.916
32 37.1 -0.112
2.324
1.877
4.146
1.940
0.0 5.0
33 38.0 -0.115 34 38.4 -0.120
2.291 2.072
1.886 2.037
4.120 4.022
1.946 1.960
0.0 5.0 8.7 5.3
36 40.6 -0.132
2.044
2.031
3.972
1.972
10.3 5.3
0.0 5.0
40 44.5 -0.148
2.016
2.009
3.898
1.990
12.3 5.3
44 48.8 -0.155
1.997
1.993
3.845
2.002
13.5 5.3
48 52.9 -0.159
1.982
1.979
3.802
2.012
14.4 5.3
54 59.4 -0.160
1.967
1.963
3.752
2.023
15.8 5.3
60 65.8 -0.157
1.955
1.952
3.712
2.033
16.5 5.3
¥
1.884
1.884
3.409
2.086
24.0 5.2
0
Figure 4. Nanoribbon model of an anatase(001) 3 ML nanotube wall.
the nanotube diameter for anatase(001) 3 ML nanotubes, although the flatness of the curve in the 48-66 Å diameter region hints at a wide preferred range around these diameters. These nanotube sizes also agree with the experimentally observed nanotube diameters measured as 4 to 5 nm on the internal and 9 to 10 nm on the external nanotube surfaces;1,12 in addition, these structures also fit the observations of a larger stability of the anatase(001) terminations in nanostructures21 and with the X-ray diffraction experiments that suggest the presence of a majority of five-coordinated Ti atoms.1,22 At the minimum of the strain energy curve, ΔEfilm þ Es = 30 eV/TiO2 describes the stability of the nanotube with respect to the bulk anatase. The exceptional stability of curved structures is usually explained by the different spatial requirements of the top and bottom layers. Because the bidimensional lepidocrocite cell has significantly different unit cell parameters in the two directions (a = 2.980 Å, b = 3.740 Å), the two perpendicular lepidocrocite-like layers that build up the anatase(001) 3 ML slab create a strain that causes the rolling-up of the nanotube. Although the strain cannot be relieved by the conversion to the thermodynamically more stable lepidocrocite structure,15 the curvature of the nanotube walls allows the recovery of 0.14 eV/TiO2, a significant part of the film energy difference of 0.28 eV/TiO2 of the two flat structures. In addition, considering that the curvature of the lepidocrocite nanotubes further reduced the stability of the tubes with respect to the anatase(001) structure, at diameters of ∼50 Å (still observed experimentally), the energy difference is
