Introduction
Given the above coordinate choice for graphene unit cell, the band structure of a (N,0) CNT, obtained from the tight-binding model of graphene (Castro Neto et al. 2009), is expressed as:
\[\begin{equation} \label{eq:band-structure} E = \pm t_0 \sqrt{f(k_x, m)} = \pm t_0 \sqrt{1+4\cos\bigg{(}\frac{3k_xa}{2}\bigg{)}\cos\bigg{(}\frac{\pi m}{N}\bigg{)}+4\cos^2\bigg{(}\frac{\pi m}{N}\bigg{)} } \end{equation}\]
where \(t_0 = 2.5eV\) is the tight-binding coupling constant and m is an index that runs from \(0\) to \(N-1\).
Analytical expression for band-gap
CNT is one of those nice systems where the band-gap can be expressed analytically. In \(\eqref{eq:band-structure}\), consider the bands with a \(+\) sign. Then the band-minimum, \(E_{min_m}\) for a given band index, \(m\), occurs when \[\begin{equation} \partial_{k_x}f(k_x, m)=0 \implies \frac{3k_xa}{2} \in \{0,\pi\} \end{equation}\]
while the minimum of the set \(E_{min_m}\) across all bands, as \(m\) is varied is obtained when the quadratic equation in \(\cos(\frac{\pi m}{N})\) is minimized: \[\begin{equation} \begin{split} \partial_{\cos(\pi m/N)}f(k_x, m)\bigg|_{\frac{3k_xa}{2} \in \{0,\pi\}} &= 0 \\ \implies \cos\bigg(\frac{\pi m}{N}\bigg) &= -\frac{1}{2}\cos\bigg(\frac{3k_xa}{2}\bigg)\bigg|_{\frac{3k_xa}{2} \in \{0,\pi\}} \\ &= \pm \frac{1}{2} \end{split} \end{equation}\] which means \(\cos\bigg(\frac{\pi m}{N}\bigg)\) needs to be as close to \(\pm\frac{1}{2}\) as possible. This is exactly possible whenever \(N = 3m\), then band-gap is exactly 0 and we call such a CNT metallic. Though in this article, we will consider only CNTs with a bandgap so let us restrict ourselves to when \(N=3m-1\), i.e \(N=8, 11, 14, 17\) etc.
Given a value of \(N\), \(m\) will take values in the set \(\{0, 1, 2, \ldots, N-1\}\). Let \[m_0 := \operatorname*{arg\,min}_{m} \left| m - \frac{N}{3} \right|.\] The band-minima is then written as (pairing the value of \(m_0\) with \(\frac{3k_x a}{2} = \pi\)): \[\begin{equation} \label{eq:bandgap} E_g = 2t_0\sqrt{1-4\cos\bigg{(}\frac{\pi m_0}{N}\bigg{)}+4\cos^2\bigg{(}\frac{\pi m_0}{N}\bigg{)} } . \end{equation}\]
Analytial expression for Density of States
Good point to obtain an expression for the density of states near the band-minimum. First, we need to show the following approximation holds near the band-minimum:
\[\begin{equation} \label{eq:dkx-dE} \frac{d}{dE} \bigg{(}\frac{3k_x a}{2}\bigg{)} \approx \frac{|E|}{t_0\sqrt{E^2-(E_g/2)^2}} \end{equation}\]
In section 1.1 we discovered that a neighbourhood of band minimum comprises of \(m_0\) with \(\frac{3k_x a}{2} \approx \pi\). So equation \(\eqref{eq:band-structure}\) around band-minimum can be rewritten as a taylor expansion around \(\frac{3k_x a}{2} = \pi\): (Hereafter, \(k_x\) refers to the small-signal expansion around \(k_x = \frac{2\pi}{3a}\)) \[\begin{equation} E \approx \pm t_0 \sqrt{1+4\cos\bigg{(}\frac{\pi m_0}{N}\bigg{)}\bigg{(}-1-\frac{1}{2}\bigg{(}\frac{3k_xa}{2}\bigg{)}^2\bigg{)}+4\cos^2\bigg{(}\frac{\pi m_0}{N}\bigg{)} } \end{equation}\] Plugging in the value of \(E_g\) from \(\eqref{eq:bandgap}\), we get:
\[\begin{equation} E \approx \pm t_0 \sqrt{\bigg{(}\frac{E_g}{2t_0}\bigg{)}^2-\bigg{(}2\cos\bigg{(}\frac{\pi m_0}{N}\bigg{)}\bigg{)}\bigg{(}\frac{3k_xa}{2}\bigg{)}^2 } \end{equation}\]
Since \(m_0\) takes \(\cos(\frac{\pi m_0}{N})\) to be as close to \(\frac{1}{2}\) as possible, for a big enough N, we can approximate the above equation as:
\[\begin{equation} E \approx \pm t_0 \sqrt{\bigg{(}\frac{E_g}{2t_0}\bigg{)}^2-\bigg{(}\frac{3k_xa}{2}\bigg{)}^2 } \end{equation}\] Rerrangment gives:
\[\begin{equation} \label{eq:kx-of-E} \frac{3k_xa}{2} \approx \pm \frac{1}{t_0}\sqrt{E^2 - \bigg{(}\frac{E_g}{2}\bigg{)}^2} \end{equation}\]
\(\eqref{eq:dkx-dE}\) follows from differentiating \(\eqref{eq:kx-of-E}\) with respect to \(E\).