Published on Web Date: September 13, 2010

r 2010 American Chemical Society 2854 DOI: 10.1021/jz101184f |J. Phys. Chem. Lett. 2010, 1, 2854–2857

pubs.acs.org/JPCL

Anatase(001) 3 ML Nanotubes, The First TiO2 NanotubeWith Negative Strain Energies: A DFT PredictionAnna Maria Ferrari,*,† D�enes Szieberth,† Claudio M. Zicovich-Wilson,‡ andRaffaella 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) layerwere investigated by DFT calculations in the 33-66 Å diameter range. Strainenergies were found to be negative in the whole diameter range, indicating thestability of the nanotubes relative to the flat layer, providing a possible explanationfor the rolling-up of the nanotubes. The strain energy curvewas shown to exhibit aminimum close to the experimentally observed TiO2 nanotube diameters. Bandgaps of the nanotubes were found to be close to that of the flat layer andmore than1 eV with respect to bulk anatase.

SECTION Nanoparticles and Nanostructures

A lthough TiO2 nanotubes arewidely investigated in thescientific literature, both their atomic-level structureand the mechanism of their formation from layers is

still ambiguous. (See refs 1 and 2 and references therein).Besides the trititanate and lepidocrocite structures, anataselayers are also considered as possible constituents of the tubewalls.

One of the possible driving forces for the rolling up of alayer into a nanotube can be the asymmetry of the structureon the two sides of the layer. Among the examples of thisphenomena are the naturally occurring tubular crystals chrys-otile3 and imogolite,4,5 the latter being the first example of astructure that shows negative strain energies at diametersaccessible by ab initio modeling methods. Titanate (hydratedTiO2) tubes are thought to roll up as a consequence of thedifferent hydration state or different ionic environmenton thetwo layer sides.6 All hitherto investigated pure TiO2 structureshowever show uniformly positive strain energies,7-10 indicat-ing that the nanotubes are less stable than the correspondingflat layers at all diameters. (In the case of nanotubes derivedfrom (001) anatase monolayer,11 a negative strain energyhas been reported, but this film is not representative of anyfeasible titania structure.)

Anatase(001) layers are one of the possible wall structuresfound in TiO2-based nanotubes.1,12 The unreconstructedanatase(001) surface has a squared unit cell, and its mainfeatures are the rows of two coordinate oxygen atoms (O2c).Ultrathin anatase(001) layers are known to display spectacularreconstructions. The two TiO2 layers thick (2ML) anatase(001)film rearranges into lepidocrocite.13,14 In the case of thethree TiO2 layers thick (3 ML) anatase film, the O2c rows onthe bottom and top surfaces are running perpendicularly toeach other, creating a directional asymmetry on the twosides of the sheet (Figure 1a); the rearrangement to thethermodynamically more stable lepidocrocite is hindered,

and the film undergoes a transition that involves the shift ofthe upper and the lower part of the film perpendicularly toeach other,15 resulting in a structure halfway between ana-tase and lepidocrocite (Figure 1c).

The aim of this work is to investigate TiO2 nanotubescreated from 3 ML anatase(001) films by means of periodicdensity functional (DFT) calculations. The computation ofnanotubes having large unit cells (540 atoms in case of the(60,0) nanotubes) was made possible by the full exploitationof the helical rototranslational symmetry of these structuresby employing the periodic CRYSTAL09 code.16 The symmetryis used for the automatic generation of the structures, for thecalculation of one and two-electron integrals (only the irre-ducible part of the Fock matrix is calculated then rotatedby the symmetry operators of the point group to generate thefull Fock matrix), and for the diagonalization of the Fockmatrix, where each irreducible representation is treatedseparately.3,17 A development version of the code was used,where the restriction on the number of symmetry operatorswas relieved from the previous constraint of 48. The hybridPBE018 functional was used. All calculations have beenperformed using an all-electron 8-411G(d) Gaussian-typebasis set19 for O and a Hay-Wadt small core ECP20 with a411-31 [3sp2d] basis set for valence electrons19 for Ti atoms.The reciprocal space was sampled according to a regularsublattice determined by the shrinking factor 6 6 (fourindependent k-points in the irreducible part of the Brillouinzone).

Because the strain originating from the curvature of thenanotubes is expected to affect the reconstruction of the ana-tase(001) film, we have performed a detailed investigation of

Received Date: August 19, 2010Accepted Date: September 9, 2010

r 2010 American Chemical Society 2855 DOI: 10.1021/jz101184f |J. Phys. Chem. Lett. 2010, 1, 2854–2857

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the transformation between the 1a and 1c forms of thetrilayer. The unreconstructed slab (bond lengths were opti-mized, but the original symmetry of the bulk was preserved)is characterized by a square cell (a= b= 3.59 Å) and a filmenergy with respect to bulk anatase (ΔEfilm) of 0.70 eV/TiO2.The 1a-1c transformation can be conceived as a two-stepprocess. Shifting the top layer in the direction of the arrowindicated on Figure 1b, we got a structure characterized by arectangular 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 stablethan 1a. Shifting the bottom layer as well perpendicularly tothe previous direction results in 1c, the preferred structure ofthe flat anatase(001) 3 ML sheet. 1c is characterized by asquare cell considerably shrunk with respect to the unrecon-structed 1a (a= b=3.41 Å) andΔEfilm = 0.56 eV/TiO2. Theenergy gain of the relaxation 1b-1c is 0.09 eV, and the oneconnected 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) 3MLslab, and the structures were optimized. (0,N) nanotubes

(Figure 2) have not been considered because they werealways found to be significantly less stable than the corre-sponding (N,0) tubes of approximately the same size. (Forinstance, the energy difference is 0.52 eV/TiO2 in the case ofthe (24,0) tubes.) The (N,0) nanotubes correspond to a chiralangle equal to zero and therefore to a zigzag structureaccording to usual nanotube nomenclature. Selected ener-getic and geometrical parameters of the nanotubes arecollected in Table 1.

Interestingly, at the smallest (D<38 Å) diameters, thestructure of the nanotube walls proved to be different fromthat of the flat sheet: the optimized geometry of the nanotubewall is more similar to what is shown on Figure 1b: whereasthe external and internal layers are shifted from their bulkpositions in the circumferential direction, they remain un-shifted in the axial direction. This phenomenon can beexplained by tracking the bond lengths most affected bythe strain exerted by the rolling up of the nanotubes. As thedistance between the Ti1 and Ti2 atoms increases with theincreasing curvature (see labeling of the atoms in Figure 2 and

Figure 1. Anatase(001) 3ML film: (1a) regular 001 surface; (1b) shifted along only one direction; (1c) shifted along both directions. See alsothe text for details.

Figure 2. (N, 0) and (0, N) anatase(001) 3 ML nanotubes.

r 2010 American Chemical Society 2856 DOI: 10.1021/jz101184f |J. Phys. Chem. Lett. 2010, 1, 2854–2857

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the corresponding bond distances in Table 1), to preserve theO4-Ti1 and O4-Ti2 bonds, the O4 oxygens, which in thestructure on Figure 1c are situated between the planes of theTi atoms, move closer to the Ti1-Ti2-Ti3 planes. (See the ωtorsion angles in Table 1.) At extremely small diameters(D = 38 Å, N = 33), the O4 oxygen atoms move into theTi1-Ti2-Ti3 planes, and the structure of the nanotube wallsbecomes analogous to 1b. Because the increasing Ti1-Ti2distances cannot be further compensated by the changing oftheω torsion angle, with the further decrease in the diameter,theO4-Ti1 andO4-Ti2bonds cannot bothbepreserved; theTi1-O4bondbreaks, whereas the Ti2-O4distancebecomessignificantly shorter. (See corresponding distances in Table 1.)

Strain energies (ES = ETiO2

nanotube - ETiO2

slab, where ETiO2

nanotube isthe energy of one TiO2 unit in the nanotube and ETiO2

slab is theenergy of one TiO2 unit in the flat slab) were also calculated.Unexpectedly, at all considered diameters, the nanotubesdisplay negative strain energies, indicating that thenanotubesare more stable than the original flat sheet. At diameterssmaller than 38 Å, the conversion of the nanotube wall fromstructure 1c to 1b can be monitored by the change in thesteepness of the strain curve. The strain energy curve showsa minimum at D ≈ 57 nm (Figure 3). The presence of aminimum in the strain energycurve indicates a preference for

the nanotube diameter for anatase(001) 3 ML nanotubes,although the flatness of the curve in the 48-66 Å diameterregion hints at a wide preferred range around these diam-eters. These nanotube sizes also agree with the experimen-tally observed nanotube diameters measured as 4 to 5 nmon the internal and 9 to 10 nm on the external nanotubesurfaces;1,12 in addition, these structures also fit the observa-tions of a larger stability of the anatase(001) terminations innanostructures21 and with the X-ray diffraction experimentsthat suggest the presence of a majority of five-coordinatedTi atoms.1,22 At the minimum of the strain energy curve,ΔEfilm þ Es = 30 eV/TiO2 describes the stability of the nano-tube with respect to the bulk anatase.

The exceptional stability of curved structures is usuallyexplained by the different spatial requirements of the top andbottom layers. Because the bidimensional lepidocrocite cellhas significantly different unit cell parameters in the twodirections (a=2.980 Å, b=3.740 Å), the two perpendicularlepidocrocite-like layers that build up the anatase(001) 3 MLslab create a strain that causes the rolling-up of the nanotube.Although the strain cannot be relieved by the conversion tothe thermodynamicallymore stable lepidocrocite structure,15

the curvature of the nanotube walls allows the recovery of0.14 eV/TiO2, a significant part of the filmenergydifferenceof0.28 eV/TiO2 of the two flat structures. In addition, consider-ing that the curvature of the lepidocrocite nanotubes fur-ther reduced the stability of the tubes with respect to theanatase(001) structure, at diameters of ∼50 Å (still observedexperimentally), the energy difference is <0.1 eV/TiO2.

The behavior of thesenanotubes can be illustratedwith thehelp of an 1D nanoribbonmodel created from the same slab.(See Figure 4.) The ∼18 Å wide ribbon was cut from ananatase(001) 3 ML sheet; the dangling bonds at the edgeswere saturated with H atoms. The geometry optimization ofthis ribbon resulted in a curved geometry, characterized by acurve corresponding to a nanotube of∼60 Å diameter, whichcoincides well with the minimum of the strain energy curve.This ribbon also adopts a 1c-like structure, with the bottomand top layers shifted from each other in both directions.

The band gaps of the nanotubes possessing differentdiameters were also calculated. (See Table 1.) It is apparentthat the band gap is not changing significantly with thediameter in either the 1b or in the 1c cases, even if there isa jump of 0.3 eV (from 5.0 to 5.3 eV) between the two struc-tures. Both values are close to that of the flat slab (5.2 eV)and

Table 1. Rollup Indexes (N), Diameters (D [Å], Calculated As theAverage of the Diameters Measured at the O2c Atoms on theInternal and on the External Surface), Strain Energies (Es [eV/TiO2],Bond Lengths (d [Å]), Torsional Angles between the Ti1-Ti2-Ti3Plane 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 0.0 5.0

32 37.1 -0.112 2.324 1.877 4.146 1.940 0.0 5.0

33 38.0 -0.115 2.291 1.886 4.120 1.946 0.0 5.0

34 38.4 -0.120 2.072 2.037 4.022 1.960 8.7 5.3

36 40.6 -0.132 2.044 2.031 3.972 1.972 10.3 5.3

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

¥ 0 1.884 1.884 3.409 2.086 24.0 5.2

Figure 3. Strain energy (Es) of anatase(001) 3 ML nanotubesversus the nanotube diameter D [Å].

Figure 4. Nanoribbon model of an anatase(001) 3 ML nanotubewall.

r 2010 American Chemical Society 2857 DOI: 10.1021/jz101184f |J. Phys. Chem. Lett. 2010, 1, 2854–2857

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are similar to that of lepidocrocite TiO2 layers (5.4 eV). Theband gap of bulk anatase obtained with the same computa-tional setup is 4.3 eV.

In conclusion, we have shown that nanotubes constructedfrom the anatase(001) layers with threemonolayer thicknesspossess a negative strain energy, being more stable than thecorresponding flat slabs. This behavior is hitherto uniqueamong TiO2-related nanotubes and might contribute tothe understanding of the creation of the tubular structuresfrom the flat TiO2 layers. The strain energy curve displays aminimum close to the experimentally observed nanotubediameters. The unusual stability of these nanotubes wasexplained with the directional asymmetry between the twosurfaces of the anatase(001) 3 ML layers.

AUTHOR INFORMATION

Corresponding Author:*To whom correspondence should be addressed. E-mail: anna.ferrari@unito.it.

ACKNOWLEDGMENT We acknowledge the CINECA for theavailability of high-performance computing resources of theCINECA-INSTM for cofinanced Key Project.

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