6.20: Analytic approximations for the bandstructure of graphene and carbon nanotubes
- Page ID
- 52375
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Since the conduction properties of graphene are dominated by electrons occupying states at or near the K points, it is convenient to linearize the energy at \({\bf{\kappa = k+K}}\).
The exact tight binding solution from Equation (6.18.23) is:
\[ \varepsilon = \alpha \pm \beta \sqrt{3+ 2cos({\bf{k\cdot\tilde{a_{1}}}})+2cos({\bf{k\cdot\tilde{a_{2}}}})+2cos({\bf{k\cdot(\tilde{a_{1}}-\tilde{a_{2}})}})} \nonumber \]
We substitute \(\bf{k = K+\kappa}\) and expand the \(cos({\bf{K+\kappa}})\) terms as a Taylor series to second order in \(\kappa\). This yields:
\[ \varepsilon = \alpha \pm \beta \sqrt{3 +2cos({\bf{K\cdot\tilde{a_{1}}}})+2cos({\bf{K\cdot\tilde{a_{2}}}})+2cos({\bf{K\cdot(\tilde{a_{1}}-\tilde{a_{2}}}}))+2{\bf{\kappa \cdot \tilde{a_{1}}}}sin({\bf{K\cdot \tilde{a_{1}}}})+2{\bf{\kappa \cdot \tilde{a_{2}}}}sin({\bf{K\cdot \tilde{a_{2}}}})+2{\bf{\kappa \cdot (\tilde{a_{1}}}-\tilde{a_{2}})}sin({\bf{K\cdot (\tilde{a_{1}}}-\tilde{a_{2}})})-({\bf{\kappa \cdot \tilde{a_{1}}}})^{2}cos({\bf{K\cdot\tilde{a_{1}}}})-({\bf{\kappa \cdot \tilde{a_{2}}}})^{2}cos({\bf{K\cdot\tilde{a_{2}}}})-({\bf{\kappa \cdot (\tilde{a_{1}}-\tilde{a_{2}})}})^{2}cos({\bf{K\cdot(\tilde{a_{1}}-\tilde{a_{2}})}})} \nonumber \]
Next, we note some identities:
\[ cos({\bf{K\cdot\tilde{a_{1}}}}) = cos({\bf{K\cdot\tilde{a_{2}}}}) = cos({\bf{K\cdot(\tilde{a_{1}}-\tilde{a_{2}})}}) = -\frac{1}{2} \nonumber \]
\[ sin({\bf{K\cdot\tilde{a_{1}}}}) = -sin({\bf{K\cdot\tilde{a_{2}}}}) = -sin({\bf{K\cdot(\tilde{a_{1}}-\tilde{a_{2}})}}) \nonumber \]
From these identities Equation (6.19.2) reduces to
\[ \varepsilon = \alpha \pm \beta \sqrt{\frac{1}{2}({\bf{\kappa \cdot \tilde{a_{1}}}})^{2}+\frac{1}{2}({\bf{\kappa \cdot \tilde{a_{2}}}})^{2}+\frac{1}{2}({\bf{\kappa \cdot (\tilde{a_{1}}}-\tilde{a_{2}})})^{2}} \nonumber \]
Solving this (see the Problem Set) gives the approximate dispersion relation for graphene:
\[ \varepsilon = \alpha \pm \frac{3}{2}\beta |\kappa| \nonumber \]
Since the speed of the charge carrier is given by the group velocity: \(v = \hbar^{-1} \partial \varepsilon/\partial k\), we get
\[ v = \frac{3}{2}\frac{\beta a_{0}}{\hbar} \nonumber \]
For \(a_{0} = 1.42\AA\) and \(\beta = 2.5\ eV\), \(v = 10^{6}\ m/s\)
Now, for carbon nanotubes, the periodic boundary condition on the circumfrence is
\[ {\bf{\kappa + K}\cdot w} = 2\pi l, l \in \mathbb{Z} \nonumber \]
Let's consider each K point in turn:
For \({\bf{K}}= \left( \frac{4\pi}{3\sqrt{3}a_{0}}, 0 \right)\)
\[ {\bf{(\kappa + K})\cdot w} = {\bf{\kappa \cdot w + K \cdot}}(n {\bf{\tilde{a_{1}}}}+m{\bf{\tilde{a_{2}}}}) \\
= {\bf{\kappa \cdot w}} + n{\bf{K \cdot}} \left( -\frac{\sqrt{3}}{2}a_{0}, \frac{3}{2}a_{0} \right)+m{\bf{K \cdot}} \left( \frac{\sqrt{3}}{2}a_{0}, \frac{3}{2}a_{0} \right) \\
= {\bf{\kappa \cdot w}} + \frac{2\pi}{3} (m-n) \nonumber \]
Rearranging gives:
\[ {\bf{\kappa \cdot w}} = 2\pi l + 2\pi \frac{(n-m)}{3} \nonumber \]
For \({\bf{K}}= \left( \frac{2\pi}{3\sqrt{3}a_{0}}, \frac{2\pi}{3a_{0}} \right)\)
\[ {\bf{\kappa \cdot w}}= 2\pi l + 2\pi \frac{(2n+m)}{3} \\
=2\pi l + 2\pi \frac{(3n-(n-m))}{3} \\
\equiv 2\pi l - 2\pi \frac{(n-m)}{3} \nonumber \]
For \({\bf{K}}= \left( -\frac{2\pi}{3\sqrt{3}a_{0}}, \frac{2\pi}{3a_{0}} \right)\)
\[ {\bf{\kappa \cdot w}}= 2\pi l + 2\pi \frac{(n+2m)}{3} \\
=2\pi l + 2\pi \frac{((n-m)+3m))}{3} \\
\equiv 2\pi l + 2\pi \frac{(n-m)}{3} \nonumber \]
The other K points follow by symmetry, and we can conclude that
\[ {\bf{\kappa}}_{\perp} = \frac{2\pi}{|{\bf{w}}|} \left( l + \frac{(n-m)}{3} \right) \nonumber \]
where we have separated \(\kappa\) into two components parallel, \(\kappa_{\parallel}\), and perpendicular, \kappa_{\perp} to the tube axis. From Equation (6.19.6) we get
\[ \varepsilon = \alpha \pm \frac{3\beta a_{0}}{d} \sqrt{\left( l + \left( \frac{n-m}{3} \right) \right)^{2}+ \left( \frac{\kappa_{\parallel}d}{2} \right)^{2}} . \nonumber \]
where the tube circumference is \(|{\bf{w}}| = \pi d\). Interestingly, Eq. (6.98) predicts that tubes are metallic when \([(n-m)/3] \in \mathbb{Z}\). Assuming that n and m are generated randomly, we expect that 1/3 of tubes should be metallic. Indeed, this seems to be the case in practice. Note also that for semiconducting tubes the band gap is inversely proportional to the tube diameter.
