Strain effects on the electronic and transport properties of TiO₂ nanotubes

Abstract

TiO₂ nanotubes are promising materials for photocatalysis, solar cells and lithium ion batteries. Using first-principles calculations, we found that strain is an efficient way to improve the electronic properties and transport properties of TiO₂ nanotubes. For our modeled nanotubes, the armchair (12,12) nanotube shows a higher Young's modulus and Poisson ratio than its zigzag counterpart (12,0) nanotube due to the different orientations of Ti–O bond topologies. An increase in axial compressive strain leads to a progressive decrease in the band gap for the armchair nanotube. Moreover, there is an indirect-to-direct band gap transition at a compressive strain of about 6% in the (12,12) nanotube. For both armchair and zigzag nanotubes, holes uniformly have a larger effective mass than electrons, in addition to the (12,0) TiO₂ nanotube with compressive strain. It is found that the hole mobility is higher than its electron counterpart for the (12,12) nanotube, whereas the electron mobility is higher than its hole counterpart for the (12,0) nanotube with compressive strain. Our results highlight the tunable electron transport properties of TiO₂ nanotubes that are promising for interesting applications in optoelectronic applications.

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1. Introduction

One-dimensional semiconductors such as titania nanotubes (TiO₂ NTs) have attracted attention as an electrode for optoelectronic applications, including solar energy conversion,¹ photocatalysis,² and sensors,³ because of their exceptional physical and chemical properties. The large surface area of the nanotubes enables efficient light harvesting, maximizing the amount of photogenerated rated charge. At the same time, an ordered and strongly interconnected nanotube offers the potential for improved electron transport leading to higher photoefficiencies.^4–7 The highly ordered architecture allows for improved charge separation and charge transport, with a calculated quantum efficiency of over 80% for incident photons with energies larger than that of the titania band gap. However, traditional TiO₂ has two critical limitations. The first is that exciton creation, the electron–hole pair (e–h⁺) precursor, is achieved only with UV light, rendering the use of solar irradiation inefficient because of the wide band gap of TiO₂ (3.0 eV for rutile⁸ and 3.2 eV for anatase⁹). Second, the large band-gap oxide semiconductors such as TiO₂ often have short exciton diffusion lengths, so it is mainly the carriers generated within the space charge layer that contribute to the photocurrent.¹⁰ Therefore, in order to satisfy the special needs in some fields, various methods have been developed to investigate the potential enhancement of light conversion and electron transport.

External strain has an important effect on the electronic structures of low-dimensional materials. Strain induces shifts in the intrinsic interatomic distance. The resulting structure deformation significantly modifies the electronic properties of the nanomaterial. The impact of tensile and compressive strain on TiO₂ NTs is fundamentally important since the nanomaterial responds to strain differently than their bulk counterparts.^11–15 Currently little work has been done on this concerted effect of strain. Therefore, it is necessary to understand the effect of strain in the transport properties of TiO₂ NTs, as a first step to exploit the controlled strain to manipulate their electronic properties.

Here, we study the strain-induced changes of the transport properties of zigzag and armchair TiO₂ nanotubes by density functional theory. In our simulations, the band structure and carrier effective masses of TiO₂ nanotubes can be effectively modulated by the strain, depending on the nanotube type and strain type, see Fig. S1 in ESI.† For armchair nanotubes, the transition from an indirect band gap to a direct gap at the compressive stain of about 6% for the armchair (12,12) nanotube makes the nanotubes suitable for optoelectronic applications. The electron effective mass becomes larger both in armchair and zigzag nanotubes when the compressive strain increases. For (12,0) zigzag nanotubes, the electron effective mass is larger than holes with compressive strain and possesses a larger mobility than holes.

2. Computational methods and models

The tubular structures were constructed by wrapping an anatase-(101) TiO₂ (space group I4₁/amd) monolayer along a chiral vector. As for the carbon nanotubes,¹⁶ the chirality of TiO₂ nanotubes can be described in terms of two integer indices along the primitive two-dimensional (2D) lattice vectors. Our calculations were performed for “armchair” (12,12) and “zigzag” (12,0) TiO₂ nanotubes, as shown in Fig. 1, as models, respectively. Both nanotubes contain 24 TiO₂ units, namely, 72 atoms, and the periodic direction is along the c-axis. The total variation in energies (ΔE) for the larger TiO₂ tubes is lower than that of the smaller ones with ε = 0% strain (see Fig. S2 in ESI†). The spin-polarized calculations were performed using density-functional theory as implemented in the DMol³ module (ref. 17) and the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE–GGA)¹⁸ for the exchange correlation. An all-electron double numerical atomic orbital augmented by d-polarization functions (DNPs) was used as the basis set. The self-consistent field (SCF) procedure was used with a convergence threshold of 10⁻⁶ a.u. on the energy and electron density. In the radial directions, a vacuum space of 15 Å was kept to avoid mirror interactions. Atomic relaxation was performed until the change of total energy was less than 10⁻⁵ a.u. and all the forces on each atom were smaller than 0.002 a.u. Å⁻¹, which was sufficient to obtain relaxed structures. For nanotubes, Monkhorst–Pack¹⁹ k-point meshes of 1 × 1 × 6 were used for the structure relaxation, whereas a denser mesh of 1 × 1 × 20 was used to calculate the band structure and density of states (DOS). Moreover, in order to understand the origin of the changes in the band gap and effective mass with strain, the composition of the frontier orbital was analyzed. Therefore, the calculated projected density of states (PDOS) of (12,0) and (12,12) nanotubes was performed using the Vienna Ab initio Simulation Package (VASP)²⁰ at the DFT-PBE level, and the specific setting for VASP (see ESI†).

Fig. 1 (a) Armchair (12,12) TiO₂ nanotube and (b) zigzag (12,0) TiO₂ nanotube. The Ti and O atoms are shown as gray and red spheres, respectively. Arrows indicate the directions of applied tensile (the dash line) or compressive (the solid line) strain.

3. Results and discussion

The optimized values of the nanotube diameter and lattice constant along the z direction are summarized in Table 1. For the armchair (12,12) nanotube, the lattice constant is 2.67 Å, slightly smaller than the value of 2.68 for the anatase-(101) monolayer. On the contrary, the zigzag (12,0) is contracted with a lattice constant of 5.46 Å, compared to a value of 5.43 Å for the anatase-(101) monolayer.

Table 1 Tube diameter (measured from the Ti atoms), lattice constant along the z-direction, Young's modulus and Poisson ratio of (12,12) and (12,0) TiO₂ nanotubes

	(12,12)	(12,0)
Diameter (Å)	20.67	11.91
Lattice constant (Å)	2.67	5.46
Young's modulus (GPa)	121	113
Poisson ratio	0.35	0.31

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The compressive and tensile strains were applied through decreasing and increasing the lattice constants along the axial direction of the armchair or zigzag TiO₂ nanotubes, respectively. Considering the structural stability, we calculated the increase of total energy of the TiO₂ nanotubes as a function of isotropic strain, as shown in Fig. 2(a). The total variation in energies (ΔE) for the larger TiO₂ nanotubes (armchair (12,12) or zigzag (12,0)) is lower than that of the smaller ones (armchair (8,8) or zigzag (8,0)). This result is not surprising, since the rolling of a monolayer into a smaller tube causes more strain energy than a larger one. Moreover, the armchair (12,12) nanotube becomes more energetically favorable than the zigzag (12,0) when the isotropic tensile strain ε > 4%.

Fig. 2 (a) The total variations in energy (ΔE) of TiO₂ nanotubes versus isotropic strain. (b) Nanotube radius (measured from Ti atoms) with respect to axial unit cell length (c) and (d) Ti–O bond lengths for the (12,0) and (12,12) TiO₂ nanotubes with respect to the applied strain along the tube axis. > indicates that the bond is roughly 60° (for zigzag type) or 30° (for armchair type) to the tube axis. // indicates that the bond is parallel to the tube axis (only applicable for the zigzag type). ⊥ indicates that the bond is perpendicular to the tube axis (only applicable for the armchair type).

The Young's modulus, which describes the resistance of the nanotube to stretch, can be calculated as

where V₀ = 2πR₀L₀δ is the volume of the unstressed tube, R₀ and L₀ are the radius and the unit cell length of the unstrained nanotube, and δ is the thickness of the tube wall. The calculated values of the Young's modulus are 121 and 113 GPa for (12,12) and (12,0) nanotubes, respectively. Similar to MoS₂ nanotubes,²¹ the calculated Young's modulus of both armchair and zigzag TiO₂ nanotubes (121 and 113 GPa) are smaller than that of bulk TiO₂ (261 GPa). The calculated bulk modulus (B₀) value for TiO₂ is 180 GPa, which is in good agreement with the experimental results (179 ± 2 GPa)²² and previous theoretical calculations (177 GPa).²³ Moreover, for zigzag (12,0) and armchair (12,12) nanotubes, the zigzag nanotubes have a smaller Young's modulus than their armchair counterparts.

The nanotube radius with respect to the axial unit cell length is shown in Fig. 2(b). The Poisson ratio was obtained using the ratio of transverse contraction and axial elongation. In Table 1, the armchair TiO₂ nanotube has a larger Poisson ratio than its zigzag counterpart, which was found to be 0.35 and 0.31. The smaller value of Poisson ratio of (12,0) nanotube indicates that its diameter is more resistant to the tube elongation, owing to the topologies of the Ti–O bonds of different orientation, as shown in Fig. 1. For TiO₂ nanotubes, Ti–O bonds are not equivalent, and these bonds will be deformed when a strain is applied to the nanotubes. The lengths of various Ti–O bonds with respect to the external compressive/tensile strain are shown in Fig. 2(c) and (d). For the zigzag nanotube, the Ti–O bond is 60° to the tube axis, and the bond length decreases with tube elongation. Moreover, for the Ti–O bonds parallel to the tube axis, their bond length increase with tube elongation. For the armchair nanotube, the Ti–O bonds at 30° to the tube axis are increased in accordance with the tube elongation. However, the Ti–O bonds that are perpendicular to the tube axis are decreased in length, which reflects the fact that the armchair nanotubes are shrinking in diameter under a tensile strain. From the above analysis, armchair nanotubes store more energy than zigzag nanotube at the same strain, which is consistent with the above Young's modulus and Poisson ratio analysis results.

Accompanied with the isotropic strain, the band structure generally changes with the deformation of tubular structures, as shown in Fig. 3. Density functional theory is known to usually underestimate band gaps for oxide semiconductors. The DFT+U method introduces an effective Hubbard U to take into account the d electrons of transition metals to improve the estimation of the band gap. The DFT+U improved the calculated band gap, and the trend predicted by DFT+U is consistent with that by GGA (see Fig. S3 in ESI†). Herein, we are interested in the effect introduced by strain. Thus, we used a DFT method instead of DFT+U for our calculations. In Fig. 3(b) and (e), the (12,0) zigzag nanotube exhibits a direct band gap, whereas the (12,12) armchair nanotube possesses a nonzero indirect gap. As shown in Fig. 3(c) and (d), the conduction band minimum (CBM) moves toward the Fermi level, which indicates that the band gap is significantly reduced. For the (12,0) zigzag nanotube, the edge of VBM and CBM is both still at the G-point (ε = −6%) and exhibit a moderate direct band gap. For the (12,12) armchair nanotube, the edge of the CBM at the midpoint along the G–Z symmetric line (ε = 0%) moves to the G-point (ε = −6%), signifying an indirect-to-direct band gap transition with 6% compressive strain. This suggests that under strain, the band edges obviously change towards the Fermi level, and these will be easily transformed into the armchair TiO₂ nanotube from the indirect-to-direct semiconductor, which will largely constrain the application of TiO₂ nanotubes to an optical device. In addition, the required isotropic strain for the transition between the direct band-gap and indirect band-gap becomes smaller with increasing tube diameter. For the (8,8) armchair nanotube, an indirect-to-direct band gap transition occurs with 8% compressive strain (see Fig. S4 in ESI†).

Fig. 3 Electronic band structures for TiO₂ nanotubes. The (a)–(c) (12,0) and (d)–(f) (12,12) TiO₂ nanotubes are at a strain of ε = −6% (compressive strain), 0%, and 6% (tensile strain), respectively. The positions of the conduction band minimum (CBM) and valence band maximum (VBM) are indicated by the arrows. The dashed line denotes the Fermi level.

The carrier mobility is a critical parameter that dictates the electronic application of the TiO₂ nanotubes. For one-dimensional systems, the carrier mobility can be obtained from the calculated band structures using the following definition of the effective mass

where ε(k) denotes the band dispersion that is easily obtained by fitting a parabola to the band structure. It is apparent that the strain will change the band curvature (Fig. 3), leading to changes in the effective masses of the charge carrier. The effective masses with respect to strain are shown in Fig. 4. In general, for both the armchair and zigzag nanotubes, the holes uniformly have a larger effective mass than the electrons. For the (12,0) nanotube, it is obvious that the electron/holes effective mass is quite sensitive to both the compressive and tensile strain with a value of roughly m_e. When the compressive strain increases, the holes effective mass increases drastically and reaches a maximum value of m_e in the (12,0) nanotube. However, the electron effective mass is smaller than the holes with strain and possesses a larger mobility than the holes. For the case of the (12,12) nanotube, the effective masses of both the holes and electrons are tunable under strain, and the electron effective mass decreases continuously when the tensile strain increases. Moreover, the electron effective mass becomes larger in the armchair nanotube when the compressive strain increases.

Fig. 4 Carrier effective masses of (a) (12,0) and (b) (12,12) TiO₂ nanotube.

In order to understand the origin of the changes in the band gap and effective mass with strain, the composition of the frontier orbital was analyzed. The calculated projected density of states (PDOS) of the (12,0) and (12,12) nanotubes are shown in Fig. 5 and 6. For Ti (Fig. 5(a)), the main contribution to the CBM is from the two states, d_xz and d_yz, whereas the d_z² state is relatively suppressed in the (12,0) nanotube. However, the CBM of the (12,12) nanotube shown in Fig. 5(b) is predominantly derived from the d_z² states of Ti with a weak contribution from the d_xy/d_x²−y² states, which differs from the CBM of the (12,0) nanotube. For O (Fig. 6(a) and (b)), the p_x and p_y orbitals are degenerate in energy, and the main contribution to the VBM of the (12,0) and (12,12) nanotubes are p_x/p_y and p_z, respectively. The d_z² of Ti and p_z of O are both singly degenerate orbitals.

Fig. 5 Projected density of states of (a) Ti atom in the (12,0) nanotube without strain, (b) Ti atom in the (12,12) nanotube without strain, (c) Ti atom in the (12,0) nanotube under a 6% tensile strain and (d) Ti atom in the (12,12) nanotube under a 6% tensile strain. (e) Ti atom in the (12,0) nanotube under a 6% compressive strain and (f) Ti atom in the (12,12) nanotube under a 6% compressive strain.

Fig. 6 Projected density of states of (a) O atom in the (12,0) nanotube without strain, (b) S atom in the (12,12) nanotube without strain, (c) O atom in the (12,0) nanotube under a 6% tensile strain and (d) O atom in the (12,12) nanotube under a 6% tensile strain. (e) O atom in the (12,0) nanotube under a 6% compressive strain and (f) O atom in the (12,12) nanotube under a 6% compressive strain.

The axial tensile/compressive strain will increase/decrease the axial length while decrease/increase the diameter of the nanotube, leading to different changes in the orbital shapes of the in-plane and out-of-plane orbitals. As carrier effective mass is determined by the band shape, the tensile/compressive strain results in quite different effects on the effective mass of in-plane and out-of-plane dominated bands. Spatially, the d_xz and d_yz orbitals are the in-plane orbitals, whereas the d_z² orbitals are the out-of-plane orbitals. The CBM of the (12,0) nanotube is mainly contributed by the in-plane d_xz and d_yz orbitals, while the CBM of the (12,12) nanotube is predominately derived from the out-of-plane d_z² orbitals. The changes in the nanotube length have an obvious effect on the orbital energies of the out-of-plane orbitals. Thus, the out-of-plane d_z² orbitals are more sensitive to the changes in nanotube length than the in-plane d_xz and d_yz orbitals. As a consequence, the band gap of the (12,12) nanotube is more sensitive to the tubular contraction than the (12,0) nanotube, as shown in Fig. 3(d) and (a). This explains the interesting phenomenon that the band gap of the (12,12) nanotube vanishes at a 6% compressive strain, while for the (12,0) nanotube, it does not vanish. Moreover, the electrons effective mass (corresponding to the CBM) of the (12,12) nanotube decreases dramatically with increasing tensile strain because the main contribution to the CBM is from the states, d_z². The O atoms play an important role in the VBM states, and the VBM of the (12,12) nanotube is predominately derived from the p_z orbitals. Under axial tensile strain, the main contribution to the VBM of the armchair nanotube is the p_x/p_y orbitals, which differs from the VBM of the (12,12) nanotube. This difference explains why the edge of the VBM at the G-point (0% tensile strain) moves to the midpoint along the G–Z symmetric line (6% tensile strain). Comparing with the O atoms in the inner shell and outer shell, we found that only O atoms in the outer shell contribute to the edge states, while the contributions from the inner O atoms are almost negligible (Fig. 1). This difference is caused by the symmetry breaking effect where the O atoms in the outer shell gain slightly more electrons than those in the inner shell transferred from the Ti atoms. This effect is more pronounced for TiO₂ nanotubes with a smaller diameter.

4. Conclusions

In this study, the mechanical properties and the strain effects on the electronic properties of TiO₂ nanotubes were systematically studied. Under uniaxial strain, the band gap and carrier effective masses are sensitive to strain, which can be efficiently tuned. For the armchair TiO₂ nanotube, holes have larger effective masses than electrons, thus they possess a smaller mobility than electrons in armchair nanotubes. Zigzag nanotubes possess a tunable direct band gap, suggesting that they could be used for optoelectronic devices. On the other hand, although armchair nanotubes have an indirect band gap, there is an indirect-to-direct band gap transition at a tensile strain of 6%. These phenomena can be understood by the changes in the compositions of the frontier orbitals in TiO₂ nanotubes. We hope that our theoretical results can pave the way for the potential applications of TiO₂ nanotubes in optoelectronic and semiconductor devices.

Acknowledgements

This study was supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20150480), the National Basic Research Program of China (973 Program, Grant No. 2012CB932400), the National Natural Science Foundation of China (Grant No. 91233115, 21273158, and 21673149), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Additional support came from the Fund for Innovative Research Teams of Jiangsu Higher Education Institutions, Collaborative Innovation Center of Suzhou Nano Science and Technology.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra13427k

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