# New Generation of Electrochemical Sensors Based on Multi-Walled Carbon Nanotubes
## 3 MWCNT Electrochemical Properties
\[T=\frac{(2n_{2}+n_{1})a_{1}-(2n_{1}+n_{2})a_{2}}{d_{k}} \tag{5}\]  
where \(d_{k}\) ranging from \(d\) (when \(n_{1}-n_{2}\) and \(d_{k}\) is not a multiple of \(3d\)) to \(3d\) (when \(n_{1}=n_{2}\) and \(d_{k}\) is a multiple of \(3d\)) for chiral tubulenes, considering \(d\) the greatest common divisor for the coefficients \(n_{1}\) and \(n_{2}\)[6; 15; 23; 27]. In addition, a translation vector (\(\tau\)) is also necessary to delimit the unitary cell of the nanotube. It can be represented by Equations (6)-(8),  
\[\tau =t_{1}a_{1}+t_{2}a_{2} \tag{6}\] \[t_{1} =\frac{(2n_{2}+n_{1})}{d_{r}}\] (7) \[t_{2} =\frac{-(2n_{1}+n_{2})}{d_{r}} \tag{8}\]  
knowing that \(d_{r}\) (\(d\)-\(3d\)) is the greatest common divisor of the respective coefficients. In summary, the limits of the 1D nanotube unit cell (plane segment \(OAB^{\prime}BO\)) are demarcated by the vectors \(C_{h}\) and \(T\), whereas \(a_{1}\) and \(a_{2}\) define the area of the 2D graphene unit cell [6; 27]. This sequence of equations is very important for the development of electrochemical sensors because it allows to determine the number of electric and phononic bands in the carbon nanostructures, i.e., the magnitude of the electrical conduction through the nanomaterial. The influence of the band structure on the nanotube conduction bands is defined in the context of the Fermi velocity (\(v_{F}\)). Figure 3.0  
where \(\hbar\) is the reduced Planck constant, \(k_{F}\) is the Fermi wavevector, and \(m\) represents the mass of the nanostructure under study. These values are not well-defined in cases where the Fermi surface is non-spherical, but it is estimated to be \(v_{F}\sim 10^{6}\) ms\({}^{-1}\) for semiconductors or even higher for conductors [27]. This provision corroborates the observation recorder in the density of states diagram for SWCNT with different chirality, since at the Fermi energy level (\(E_{F}\)) there is a finite electronic transition for metallic structures (first van Hove optical transition), but a zero-band gap for semiconductors (second van Hove optical transition). A general description of the band structures and density of states for metallic and semiconductor nanotube is presented in Figure 3.1 Thus, if we imagine that MWCNT are composed by several coaxial SWCNT, it might be expected that they are not strictly a 1D-type conductor [7; 27]. Likewise, when the nanotubes have different chirality, the band structure undergoes alterations, and the resultant coupling between the electronic states at the \(E_{F}\) and phonon modes may cause superconductivity [4; 7; 27]. Despite the similarity with metallic materials, the carrier density in carbon nanotubes (1D quantum wires) is much lower and novel physical electron-electron phenomena (such as spin-charge separation and suppression of the electron tunneling density of states) should be considered to justify the possible discrepancies, as predicted by [PERSON] liquid theory [23; 26; 27; 28]. Some electrochemical studies showed a lower \(R_{tc}\) for MWCNT containing structural defects and/or traces of metallic impurities (catalyst nanoparticles), so that the current flow occurs mainly through the outer most nanotube cylinder [26; 27; 28]. Only one of the concentric tubes needs to exhibit this behavior for the overall electronic properties to be essentially metallic-low quantum capacitance values [7; 27]. It is believed that if \(d_{t}\) is smaller than the elastic mean free path, the one-dimensional ballistic transport predominates, but the opposite characterizes a two-dimensional diffusive current flow [7; 26; 27; 28]. In addition, nanotubes with larger \(d_{t}\) have a greater density of structural defects, besides facilitating their modification and functionalization, which are interesting requirements for the development of electroanalytical devices [2; 6; 7; 8; 9]. The mean length of the nanotubes also has a significant effect on the electron-transfer rate; they are inversely proportional properties [22; 24; 27].