Graphene Equation LaTeX Source

\documentclass[aps,12pt,prb]{revtex4}

\begin{document}

\title{Graphene Band Structure}

\author{R. B. Laughlin}

\affiliation{Stanford}

\date{17 May 08}

\maketitle

Near neighbors:

\begin{displaymath}
{\bf r}_1 = b
 \left[
\begin{array}{r}
\sqrt{3}/2 \\
1/2
\end{array} \right]
\; \; \; \; \; \; \; \; 
{\bf r}_2 = b
\left[
\begin{array}{r}
-\sqrt{3}/2 \\
1/2
\end{array} \right]
\; \; \; \; \; \; \; \; 
{\bf r}_3 = b
\left[
\begin{array}{r}
0 \\
-1
\end{array} \right]
\end{displaymath}

\begin{displaymath}
{\bf R}_{\ell m} = \ell {\bf R}_1 + m {\bf R}_2
\end{displaymath}

Primitive translations:

\begin{displaymath}
{\bf R}_1 = {\bf r}_1 - {\bf r}_3 = b \sqrt{3}
\left[
\begin{array}{r}
1/2 \\
\sqrt{3}/2
\end{array} \right]
\; \; \; \; \; \; \; \; \; \; \;  
{\bf R}_2 = {\bf r}_2 - {\bf r}_3 = b \sqrt{3}
\left[
\begin{array}{r}
- 1/2 \\
\sqrt{3}/2
\end{array} \right]
\end{displaymath}

Reciprocal lattice:

\begin{displaymath}
{\bf G}_{\ell m} = \ell {\bf G}_1 + m {\bf G}_2
\end{displaymath}

Reciprocal lattice generators:

\begin{displaymath}
{\bf R}_j \cdot {\bf G}_k = 2 \pi \; \delta_{jk}
\end{displaymath}

\begin{displaymath}
{\bf G}_1 = \frac{4 \pi}{3 b}
\left[
\begin{array}{r}
\sqrt{3}/2 \\
1/2
\end{array} \right]
\; \; \; \; \; \; \; \; \; \; \;  
{\bf G}_2 = \frac{4 \pi}{3 b}
\left[
\begin{array}{r}
\sqrt{3}/2 \\
-1/2
\end{array} \right]
\end{displaymath}

Special brillouin zone points:

\begin{displaymath}
{\bf \Gamma} =
 \left[
\begin{array}{c}
0 \\
0
\end{array} \right]
\; \; \; \; \; \; \; \; 
{\bf K} = \frac{4 \pi}{3 \sqrt{3}b}
\left[
\begin{array}{c}
1 \\
0
\end{array} \right]
\; \; \; \; \; \; \; \; 
{\bf M} = \frac{2 \pi}{3 b}
\left[
\begin{array}{c}
\sqrt{3}/2 \\
1/2
\end{array} \right]
\end{displaymath}

Band structure:

\begin{displaymath}
E_q^\pm = \pm | t \sum_\nu \exp(i {\bf q} \cdot {\bf r}_\nu) |
\end{displaymath}

\end{document}