# A Review of Double-Walled and Triple-Walled Carbon Nanotube Synthesis and Applications
## 1 Introduction and Motivation
### Single-Walled Carbon Nanotubes  
SWNTs can be simply understood from a fundamental nano-science standpoint on the basis of their parent material graphene, the material that led to their discovery, and which was awarded the Nobel Prize in 2010 [2; 3; 4; 5; 6]. As depicted in Figure 1a, departing from the origin O for graphene using the direct space basis vectors \(\mathbf{a_{1}}\) and \(\mathbf{a_{2}}\), the chiral vector (\(\mathbf{C_{h}}\)) is generated by displacing the vector n times in the \(\mathbf{a_{1}}\) direction and m times in the \(\mathbf{a_{2}}\) direction, forming the \((n,m)\) chiral vector \((4,2)\). Next, a stripe whose width is equal to the \(\mathbf{C_{h}}\) size is \"cut\", and by joining both ends of the chiral vector, one obtains a cylinder, which is nothing but the \((4,2)\) carbon nanotube itself. Each carbon nanotube has its very own \((n,m)\) index, and through this \((n,m)\) indexing, one may understand everything regarding the nanotube structure, as for example, its mechanical, electronic, vibrational and thermal structures [2; 3; 4]. The \((n,m)\) index will, therefore, also define whether a carbon nanotube is metallic or semiconducting. Additionally, due to the carbon nanotube one-dimensional character, both the electronic and the vibrational structures of CNTs present Van Hove singularities (VHS), and these VHS enhance many phenomena involving electrons, phonons and their mutual interactions, which are often called many-body interactions [2; 4; 5; 7; 8; 9; 10]. These VHS have a high density of states, and they result from the quantum confinement observed in the circumferential direction of the nanotubes [2; 4; 5; 7; 8]. [PERSON]'s group was one of the first groups to measure the Van Hove singularities in CNTs using the scanning tunneling microscopy (STM) technique, as shown in Figure 1b[11]. From a theoretical point of view, there are no doubts about how unique nanotubes are, and if one could specifically synthesize a particular nanotube with a particular set of \((n,m)\) values, one could certainly utilize these \((n,m)\) numbers to build more efficient products, which are designed for a specific commercial application. As an example, semiconducting SWNTs have been used to fabricate field-effect transistors (FET) [12; 13; 14]. As demonstrated by [PERSON] \(et\)\(al.\)[12], optical emissions can be electrically induced from FETs (see Figure 2a,b). However, semiconducting SWNTs with different \((n,m)\) indexes will present different bandgaps, and these different bandgaps make the  
Figure 1: **(a)** Illustration of the graphene lattice, which serves as a template for carbon nanotube construction. The vectors \(\mathbf{a_{1}}\) and \(\mathbf{a_{2}}\) are the basis vectors of graphene in direct space. As illustrated, by displacing the lattice vector n times in the \(\mathbf{a_{1}}\) direction and m times in the \(\mathbf{a_{2}}\) direction, the chiral vector \(\mathbf{C_{h}}\) is formed, which together with the orthogonal vector \(\mathbf{T}\) forms the unit cell of the carbon nanotube. The indices of the chiral vector \((n,m)\) define the indices of the carbon nanotube. Namely, the rectangle AOB’ is the unit cell in real space for the \((4,2)\) tube, in which one chiral vector component is displaced four times in the \(\mathbf{a_{1}}\) direction and two times in the \(\mathbf{a_{2}}\) direction. The stripe formed (see dotted lines), which has a length equal to the size of \(\mathbf{C_{h}}\) forms the carbon nanotube when it is rolled up. Figure adapted with permission from [4], Copyright Elsevier, 2006. **(b)** Van Hove singularities (VHS) obtained from scanning tunneling microscopy (STM) measurement, as shown in the upper curve, and theoretical calculations of the VHS for the \((13,7)\) and the \((12,6)\) carbon nanotubes. The vertical dashed lines indicate the VHS in the experimental curve. Reproduced with permission from [11], Copyright American Physical Society, 1999. FET light emissions happen at distinct wavelengths.