where d is the nanotube diameter and n is an integer. At the same time, the wave vector along the tube axis (kN) remains continuous. Due to this quantisation, the energy bands that are produced by the surfaces in the k-space in a 2-D case, are reduced to a set of sub-bands. This is illustrated in Figure 3.9a, where the valence and conduction bands of a 2-D semiconductor are schematically represented by two paraboloids. The energy spectrum of nanotubes in this case is a set of parabolas (as shown in Figure 3.9b), which are formed from cross-sections of the 2-D bands by a set of parallel planes, each of which corresponds to a different value of the quantised vector k> and is oriented perpendicular to the k>-vector. The separation of the subbands along both the k> and energy axes depends on the nanotube diameter, increasing when d decreases.

Figure 3.9 The transformation of the electron band structure of nanosheet semiconductors accompanying the formation of nanotubes. Band diagrams of a) a 2-dimensional nanosheet and b) quasi-1-dimensional nanotubes are shown. Diagram c) shows the energy density of states for nanosheets (G2D) and nanotubes (G1D). EG1D and EG2D are the band gaps of 1-D and 2-D structures, respectively (see Equation 3.11), and kx and ky are the wave vectors.

Figure 3.9 The transformation of the electron band structure of nanosheet semiconductors accompanying the formation of nanotubes. Band diagrams of a) a 2-dimensional nanosheet and b) quasi-1-dimensional nanotubes are shown. Diagram c) shows the energy density of states for nanosheets (G2D) and nanotubes (G1D). EG1D and EG2D are the band gaps of 1-D and 2-D structures, respectively (see Equation 3.11), and kx and ky are the wave vectors.

Within the effective mass model, the energy spectrum of 2-D TiO2 sheets can be described by:

where the 'plus' and 'minus' signs correspond to the conduction and valence bands, respectively, EG is the energy gap, hP is the reduced Planck's constant; me and mh are the effective masses of electrons and holes, respectively. The electronic band structure of a TiO2 nanotube, can be obtained from this relation by zone-folding and is given by a series of quasi 1-D sub-bands with different indices, n, as seen in Figure 3.9b:

Eg h

This transition from a 2-D to a quasi 1-D energy spectrum has a dramatic effect on the energy density of states. In the 2-D case, the density of states, G2D = meih/nhp, has a constant value54,55 for energies outside the energy gap, as indicated in Figure 3.9c. In the quasi 1-D case, however, the density of states in each sub-band is given by:

This diverges at the band edge En(0) leading to van Hove singularities.54 The resulting density of state is formed by a series of sharp peaks with long overlapping tails, as shown in Figure 3.9c. The energy gap between the valence and conductance bands in the quasi-1-D case is larger than that in the parental 2-D material, and this difference increases with decreasing diameter of the nanotube.

The change in the energy gaps after rolling of a nanosheet to a nanotube can be expressed as:

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