ARTICLE pubs.acs.org/JPCC
Surface Effect and Band-Gap Oscillation of TiO2 Nanowires and Nanotubes T. He,† Z. S. Hu,‡ J. L. Li,† and G. W. Yang*,† †
State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, Nanotechnology Research Center, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong, China ‡ Research Center of Functional Materials, Kaifeng University, Kaifeng 475004, Henan, China ABSTRACT: We have presented a comprehensive investigation for the geometric, energetic, and electronic structures of the rutile TiO2 nanowire (NW) and nanotube (NT) by the firstprinciples calculations accompanied with the crystal orbital overlap population and the wave function analysis. Taking into account the surface interactions, we found that the geometry and stability of TiO2 NWs and NTs greatly depend on the surface structure with the size decreasing, which demonstrates the forming shape and structure. The competition between the surface-bonding effect and the quantum-confinement effect induces the oscillation behavior and the directindirect transition of the band gap of TiO2 NWs and NTs. These findings not only provide a new insight into the fundamental understanding of TiO2 nanostructures but also provide useful information to design TiO2 nanostructures for different applications.
1. INTRODUCTION Titanium dioxide (TiO2) is a versatile material in a broad field of solar-energy-related applications.14 In photoelectronic conversion, TiO2 can act as either carriers in dye-sensitized solar cells (DSSCs)57 or reactors in the water splitting and hydrogen generation.3,8 Recently, the exhaustion of fossil fuel and the urgent need for renewable power have stimulated abundant studies on the photoelectrochemical application of TiO2. As is well-known, the surface is the main reacting portion of solar-ray capture and photoelectric conversion. To increase the conversion power of TiO2, it is crucial to increase the surface-to-bulk ratio by decreasing the materials' size to the nanometer scale. Therefore, TiO2 nanowires (NWs) and nanotubes (NTs) are two promising nano-TiO2, which have been prepared by a solgel process,9 electrochemical deposition,10 electrophoretic deposition,11 thermal evaporation,12 or hydrothermal synthesis7,13 and have been widely used in photocatalysis and photovoltaics.7,1323 Generally, high electron mobility, excellent electronhole separation power, and long-distance transport ability are unique advantages of one-dimensional (1D) semiconductor nanostructures compared with that of bulk.7,1322 Thus, theoretical focus is always paid on surface structure,2426 cross section,27 termination,28 and quantum-confinement effect29 of 1D TiO2 nanostructures. Therefore, understanding the relationship of structure, for example, size, orientation, and morphologies, with the electronic properties of TiO2 nanomaterials is essential for the possible manipulating of photosensitivity through structure control.30 On the other hand, when the size of TiO2 is reduced to several nanometers, the effect of the surface on their mechanical and electronic behaviors will be amplified.31 It has been found that the surface morphology, stability, and equilibrium shape of chemically r 2011 American Chemical Society
prepared TiO232 and other nanocrystals33,34 are strongly affected by their surface conditions. Another important fact is that the experimentally prepared NWs always have squarelike cross sections due to the perfect tetragonal lattice.13,19,35,36 The parallel surfaces may introduce the notable surfacesurface interaction in the NWs with small diameters, because of the close distance between surfaces. Moreover, TiO2 NTs possess an upgraded surface-to-bulk ratio, where the surface interaction may be more significant. However, there have been little theoretical studies on the surface effect of TiO2 NWs and NTs. Thus, it is necessary to provide a comprehensive insight into the relationship between surface and electronic structure of TiO2 NWs and NTs. In this paper, we systematically studied the geometric, energetic, and electronic properties of the rutile TiO2 NW and NT and revealed their surface dependence of geometric and electronic structures.
2. METHOD AND COMPUTATION First-principles calculations are carried out in the framework of density functional theory (DFT).37,38 Norm-conserving pseudopotentials3941 and the generalized gradient approximation (GGA) exchange-correlation functional proposed by Perdew, Burke, and Ernzerhof (PBE)42,43 are adopted. The electron wave functions are expanded using a double-ζ basis set plus polarization functions.44 Numerical integrals are performed on a real space grid with an equivalent cutoff of 150 Ry. The Brillouin zone is sampled by a 1 1 8 k-mesh following the MonkhorstPack scheme.45 A lateral vacuum space of at least 20 Å is kept to exclude the Received: April 25, 2011 Revised: June 8, 2011 Published: June 22, 2011 13837
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Figure 2. (a) Surface energy (Esurf) and (b) energy band gap (Egap) of NW001 (black squares), NW110 (red circles), and NW100 (blue triangles) with varying diameters (D). The inset numbers refer to parameter N. In Egap, solid (or hollow) shapes represent the direct (or indirect) band gaps. The Egap of bulk TiO2 is plotted as a dashed line.
characters with DFT-GGA. It implies the effectiveness of our scheme in predicting the reliable structure and electronic properties of TiO2 nanostructures.
3. RESULTS AND DISCUSSION
Figure 1. Top and side views of TiO2 nanowires along [001] (a, b), [110] (c, d), and [100] (e, f) directions. Titanium (or oxygen) atoms are plotted as big blue (or small red) balls. The structures with odd and even N are shown in the left and the right, respectively.
interaction of images. The total energy is calculated in a precision of 104 eV. All atoms are fully relaxed under the conjugate gradient (CG) algorithm without any symmetric constraints. An equilibrium structure is obtained when the HellmannFeynman force is less than 0.02 eV/Å. The wave functions with numerical basis and all computational settings are integrated in SIESTA code.4648 Our scheme gives the equilibrium lattice parameters of a = 4.573 Å and c = 2.965 Å of bulk rutile TiO2, agreeing well with the experimental data (a = 4.594 Å and c = 2.959 Å).49 The band gap of rutile TiO2 is calculated to be 2.23 eV, closer to the experimental value (3.0 eV)50 than that of the plane-wave basis DFT-GGA method (∼1.7 eV),26 but sharing the same energy dispersion
3.1. TiO2 NWs. Three growth orientations are found in experimentally prepared TiO2 NWs, that is, the [001],7,13,1820,51 [110],12,21,35,52,53 and [100]36,53 directions. We calculated the stable structures of these three NWs, as shown in Figure 1. All NWs have squarelike cross sections, agreeing with experiment.13,19,35,36 The most stable facet groups are tested for each direction, and it is found that the NW in the [001] direction (NW[001]) with four {110} surfaces, the NW in the [110] direction (NW[110]) with two {110} and two {001} surfaces, and the NW in the [100] direction (NW[100]) with two {100} and two {001} surfaces are preferable. Additionally, we can see that, by changing the diameter of the NWs, the NW in each direction can be classified into two distinct categories, which can be distinguished by the number of Ti atom rows (N) (NWN) in the side length, as shown in Figure 1. The energetic stability of the NWs is studied by calculating the surface energy (Esurf), defined as
Esurf ¼ ðENW μTiO2 nTiO2 Þ=S
ð1Þ
where ENW and μTiO2 are the total energy and number of TiO2 units in the NW, respectively. μTiO2 refers to the chemical potential of a TiO2 unit in crystal TiO2, and S is the surface area measured from the outmost atoms. The surface energy separates the energy contribution of the surface from that of the core in the 13838
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Figure 3. Energy band and crystal orbital overlap population (COOP) of (a) NW[001] , (b) NW[001] , (c) NW[110] , (d) NW[110] , (e) NW[100] , 5 6 5 6 7 and (f) NW[100] . The dashed-dottted line denotes the Fermi level (EF). 8 Average COOPs of surface Ti 3dTi 3d and Ti 3dO 2p orbitals are plotted in blue solid and red dashed lines, respectively.
NW and is superior to the formation energy (Eform) or binding energy (Ebind), which unreasonably decays to zero when the diameter becomes large. The {110} surface is the most stable surface in the rutile TiO2 compared with other low-index surfaces.54 It is expected that involving the {110} surface should decrease the Esurf of the NWs. In fact, this is confirmed by our calculated Esurf hierarchy of NW[001] < NW[110] < NW[100], as shown in Figure 2, where the NW[001] with four equivalent {110} surfaces is the most stable, whereas the NW[100] with no {110} surface is the least preferable, which agrees well with experiments.7,13,1820,36,51,53 The Esurf of the NWs with varying N (NWN) between N = 2m and N = 2m + 1 (m is a positive integer) shows an oscillation behavior when the diameter (D) is smaller than 24 Å. From Figure 1, we can see that the {110} surface in NW[001] and NW[110] contains two types of Ti atoms; one is five-coordinated (Ti5c), which relaxes inward, and another
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is six-coordinated (Ti6c), relaxing outward. When the diameter of the NWs is small, the parallel {110} surfaces become close to each other. Thus, the surface relaxation from one side will conflict with that from the opposite side when they have countering Ti5c (or Ti6c), as shown in Figure 1a,c. Actually, it increases the Esurf of and NW[110] with an odd N value. On the other hand, NW[001] N N if the alignment of Ti5c (or Ti6c) in two surfaces is staggered (N = 2m), as in Figure 1b,d, the surface relaxation is fully allowed, and the Esurf is decreased. Here, we can see the effect of the surface interaction on the stability of the NWs. The Esurf oscillation in NW[100] is marginal because the surface interaction is between the four-coordinated Ti atoms (Ti4c) and the dangling O atoms in distant corners shown in Figure 1e,f. The Esurf of NW[100] decreases with the D increasing, which shows the less stability of the {001} and {100} facets in NW[100]. We study the energy band structures of these NWs above. Their band-gap value (Egap) of varying diameters is plotted in Figure 2b. The Egap of bulk TiO2 is shown as a dashed line as reference. We find clearly the oscillation behavior of Egap between NW2m and NW2m+1, which converges to a constant value (∼2.1 eV) in NW[001] and NW[110] when the diameter is increasing. NW2m+1 has a locally lower Egap than that of NW2m. We also find the directindirect band gap switching between NW2m and NW2m+1 for NW[001] and NW[100], as shown in Figure 2b. These oscillation behaviors should be correlated with the surface interaction inducing changing surface state levels in the band gap. Meanwhile, we plot the energy band structures of NW2m and NW2m+1 in Figure 3, where the vacuum energy levels are lined-up for direct comparison. The band-gap oscillation is mostly due to the changes of the conduction band minimum (CBM) level, which is lower in energy in NW2m+1 than in NW2m, as shown in Figure 3. However, the valence band maximum (VBM) changes little for all three NWs, giving rise to the Egap oscillation. Why are the behaviors of the conduction and valence bands so different in TiO2 NWs? We calculate their crystal orbital overlap population (COOP)55,56 around the band gap, as shown in Figure 3. The CBM has a large amount of Ti 3dTi 3d bonding characteristics. A small ratio of Ti 3dO 2p bonding states are also found in NW[001] and NW[110]. These Ti 3d-related bonding states at the bottom of the conduction band distribute in the subsurface of NW[001] and NW[110], or around two corners of NW[100], as shown in the wave function isosurface pictures of Figure 4. The bonding states overlap in NW2m+1, while being isolated in NW2m. We have known that surface counteracting gives rise to face-to-face [110] Ti5c in NW[001] 2m+1 and NW2m+1. Therefore, these low-coordinated Ti atoms provide the surface states (of Ti 3d bonding characters), which rehybridize with each other when they counteract. This rehybridization produces new bonding orbitals under the bottom of the conduction band and results in the downshift of the CBM. In NW[100], the corner Ti4c provides the surface states that rehybridize [100] in Figure 4e). These when two Ti4c corners are close (NW2m+1 orbital rehybridizations can be characterized as new bonding between surface states. Accordingly, we call it the surface-bonding effect (SBE), which competes with the quantum-confinement effect (QCE). In contrast, the isolation and nonbonding of surface states in NW2m is responsible for the upshift of the CBM. On the other hand, from the wave function of the VBM, as shown in Figure 5, we can see that the highest occupied orbitals (with O 2p character) spread along the wire axis either in the center (NW[001] and NW[110]) or at the corners (NW[100]), where the lattice relaxation is quenched and the surface states interaction is weak. 13839
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Figure 4. Wave function isosurfaces of the conduction band minimum (CBM) of (a) NW[001] , (b) NW[001] , (c) NW[110] , (d) NW[110] , (e) 5 6 5 6 [100] , and (f) NW . NW[100] 7 8
It thus explains the smaller shift of the VBM. It should be noted that the effective mass of an electron (m*) plays a major role in the change of the band-gap excitation property when the conduction band shifts. The lighter the m*, the faster the conduction band shifts, in the band folding picture.57 For example, in bulk TiO2, the m* of the bottom of the conduction band at the Γ-point along the constraint direction is calculated to be 0.105 m0 (m0 is the electron rest mass), much lighter than that at about half ΓZ (0.25 m0). The bottom of the conduction band at the Γ point shifts faster and gives rise to the directindirect transition in NW[001] (Figure 3a,b). Effective mass theory also works in the other two NWs. Therefore, we conclude that surface states interaction is an intrinsic property in NWs, which results in the surface dependence of the structure and properties. 3.2. TiO2 NTs. TiO2 crystal NTs have a larger surface-to-bulk ratio than that of NWs with similar diameters, which makes
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Figure 5. Wave function isosurfaces of the valence band maximum (VBM) of (a) NW[001] , (b) NW[001] , (c) NW[110] , (d) NW[110] , (e) 5 6 5 6 [100] [100] NW7 , and (f) NW8 .
them suitable for dye molecule anchoring. An intuitive deduction is that NTs may have a more complicated surface interaction due to the coexistence of outer and inner surfaces. Figure 6 shows the optimized crystal NTs along the [001], [110], and [100] directions. We introduce another wall thickness parameter, T, into the NTs, which is defined as the number of Ti atom rows between two surfaces, similar to N, as illustrated in Figure 1. T is the second degree of freedom to tune the surface interaction in NTs. The surface energy (Esurf) of NTs is calculated with considering both outer and inner surface areas (S). The Esurf of NTs with varying N and T (NTN,T) is shown in Figure 7. The Esurf of a NW is also plotted as a reference. Compared with the NW, about 13840
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Figure 8. Egap of (a) NT001, (b) NT110, and (c) NT100 with varying N and T. The inset number refers to parameter N. The value of a thick NW is plotted as a dashed line. Solid, hollow, and half-filled shapes represent direct, indirect, and zero band gaps, respectively.
Figure 6. Top views of TiO2 crystal nanotubes of (a) NT[001] 8,2 , (b) [110] [110] [100] [100] NT[001] 8,3 , (c) NT8,2 , (d) NT8,3 , (e) NT10,2 , and (f) NT10,3 . Titanium (or oxygen) atoms are plotted as big blue (or small red) balls.
Figure 9. E gap and fitted lines versus diameter (D) of (a) NW001 , (b) NW110, and (c) NW100 . A square (or triangle) represents the NWN with N = 2m (or N = 2m + 1). Solid and hollow shapes represent direct and indirect band gaps, respectively.
Figure 7. Esurf of (a) NT001, (b) NT110, and (c) NT100 with varying N (inset number) and T (legend). The value of a thick NW is plotted as a dashed line.
three-quarters of NT[001] have lower surface energy than that of the NW, as shown in Figure 7a. It is attributed to that the inner
{110} surface has higher coordination than that of the outer {110} surface. As shown in Figure 7, a T-dependent oscillation of Esurf is found in NT[001] and NT[110], which bears the resemblance to the N-dependent Esurf oscillation in NWs. These results imply that the Esurf oscillation comes from the interaction of outer and inner surfaces. The band-gap (Egap) values of all NTs are calculated and shown in Figure 8. Egap oscillates with varying diameter or wall thickness. The band gap of NTN,2m+1 is smaller than that of 13841
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Table 1. Fitted Parameters of eq 2a Egap(∞) (eV)
R (eV 3 Åβ)
Egap(∞)* (eV)
β
γ (eV 3 Åδ)
δ
NW[001] 2m
2.114
2.884
13.5
1.348
0
NW[001] 2m+1 NW[110] 2m
2.114 2.128
2.884 2.898
13.5 5.559
1.348 1.427
41.957 0
1.657 0
NW[110] 2m+1
2.128
2.898
5.559
1.427
6.619
1.358
NW[100] 2m
0.398
1.168
13.752
0.94
0
NW[100] 2m+1
0.461
0.309
32.042
0.963
42.458
0
0 1.431
The largest standard error for Egap(∞), R, β, γ, and δ is 0.002 eV, 0.2 eV 3 Å1.348, 0.01, 0.3 eV 3 Å1.657, and 0.005, respectively. Egap(∞)* corresponds to Egap(∞) plus an increment of the difference between experimental and calculated band gaps of bulk TiO2. a
NTN,2m. Nearly half of NT[001] and all of NT[110] have a smaller band gap than that of the NW form. The Egap of NT[100] N,2 becomes zero, which comes from the strong Coulomb attraction between being O and Ti4c corners. It explains the Esurf of NT[100] N,2 anomalously higher than that of NT[100] N,3 (Figure 7c). Because there is an inner surface, the bonding/nonbonding of surface states works between outer and outer surfaces, and outer and inner surfaces, simultaneously, which results in the both N and the T dependence of Egap oscillation in NTs. It should be noted that pure DFT may give quite different results of TiO2 surface chemistry compared with DFT+U,58 where pure DFT shows limitations in the description of localized states around surface defects, for example, the oxygen vacancy. To see whether the localization is present in the carved NT structures, we performed GGA+U calculations using an effective Coulomb repulsion (Ueff = U J) varied from 3.4 to 5.8 eV26,58,59 and found no qualitative change in the variation trends of Egap. Moreover, the band-gap excitation properties, that is, direct/indirect band gap or metallicity in NTs, are well coincident with those from GGA, which means that the electronic properties of pure DFT are intrinsic and reliable. Thus, our finding of the wall-thickness dependence of Egap in TiO2 NTs shows another significant opportunity in tuning their band gap. 3.3. Relationship between SBE and QCE. Both the surfacebonding effect (SBE) and the quantum-confinement effect (QCE) work on the band-gap evolution of NWs and NTs. What is the relationship between them is a central issue. Now we probe into the contribution of both SBE and QCE to the development of Egap, using an equation Egap ðDÞ ¼ Egap ð∞Þ + ΔQCE ðDÞ + ΔSBE ðDÞ
ð2Þ
where Egap(∞) is the gap value of NWs/NTs with a large diameter and ΔQCE and ΔSBE are the gap variation due to QCE and SBE, respectively. ΔQCE(D) = R 3 Dβ and ΔSBE(D) = γ 3 Dδ. In most cases, ΔQCE(D) only differs for NWs with a different growth direction, and the SBE (ΔSBE(D)) is only present in NW2m+1. The SBE will be quenched (ΔSBE(D) = 0) when the wire diameter (D) approaches infinite. Moreover, the Egap(∞) of NW2m and NW2m+1 should be identical because of the same facets. One exception is NW[100], where the divergence of Egap between NW2m[100] and NW2m+1[100] increases when diameter is increasing (see Figure 2). The Egap(∞) and ΔQCE(D) of NW2m[100] and NW2m+1[100] must be different, which is mostly due to the O and Ti4c corners introducing directional distortions, changing the cross-sectional shapes and building different crystal potentials to NW2m[100] and NW2m+1[100]. On the basis of these preconditions, we study the contributions of SBE and QCE simultaneously through least-squares fitting of Egap(D) in eq 2. The well-converged trends of Egap(∞) shown in
Figure 9 and the preconditions discussed above both increase the accuracy of fitted Egap(D) and reduce the fitting error of the finite diameter range. Meanwhile, we plot the Egap(D) curves in Figure 9 and list the best fitted parameters along with the standard errors in Table 1. The cooperation of QCE and SBE has the same mechanism in NTs. From Table 1, it is clear that SBE is always competing with QCE, for their opposite signs of γ and R in ΔSBE(D) and ΔQCE(D), respectively. NW[001] and NW[110] have close Egap(∞) values (2.114 and 2.128 eV), slightly smaller than the Egap of bulk TiO2 (2.24 eV). These results thus show that the surface states will be always present in the band gap of NW[110] and NW[110], irrespective of the wire diameter. We also find that NW2m[100] has a very small Egap(∞) value (0.398 eV). The negative Egap(∞) of NW2m+1[100] with a infinite diameter (see Table 1) means that the band gap is closed (Egap(D) = 0) when D is beyond a critical value (66 Å). However, considering that our calculated Egap of bulk TiO2 (2.23 eV) is smaller than the experiment value (3.0 eV) by 0.77 eV, due to the underestimation of the band gap in DFT, it may be more appropriate to add the same increment to Egap(∞) to deduce a more realistic band gap.60 In this contribution, the band gaps of thick NWs should be finite positive values that are all smaller than that of the bulk TiO2 (see Table 1). The band-gap narrowing accompanied with bandgap oscillation indicates that NWs and NTs are very promising in solar energy collection and photosensor applications.
4. CONCLUSION We have performed a comprehensive DFT-based first-principles calculation to study the geometric, energetic, and electronic properties of the rutile TiO2 NWs and NTs with the [001], [110], and [100] growth directions. NW001 is more favorable with a lower surface energy. It was found that the band gap oscillates between NW2m and NW2m+1, and the crystal orbital overlap population (COOP) and electron wave function analysis showed that the bonding of surface states forms in NW2m+1. Wall thickness is the second factor in tuning the surface energy and band-gap values in the crystal TiO2 NTs. The bonding of surface states between outer and inner surfaces and that between outer and outer surfaces both contribute to band-gap narrowing in NTs. The band-gap revolution of NWs and NTs can be well addressed by considering the surface-bonding effect and quantum-confinement effect. Surface effect is an essential aspect in exploring photocapture devices based on TiO2 nanomaterials. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 13842
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