The purpose of this post is to calculate the energy band structure of the famous \(2\)-D material graphene. This of course is a monolayer of carbon \(\text C\) atoms arranged in a hexagonal “honeycomb” lattice. Sheets of graphene stacked on each other are called graphite.
Triangular Lattices, Brillouin Zone, Dirac Points
The first thing to notice is that this hexagonal lattice is non-Bravais because the \(\text C\) atoms are not in identical environments. As usual, this is resolved by viewing it as the convolution of a Bravais triangular lattice \(\Lambda_{\Delta}\) with a motif of \(2\)-carbon atoms dropped at each lattice point of \(\Lambda_{\Delta}\) (strictly speaking, this \(\Lambda_{\Delta}\) that I’m calling “triangular” is also confusingly often called hexagonal in crystallography). Therefore, we use the following primitive lattice vectors \(\textbf x_1,\textbf x_2\) which \(\text{span}_{\textbf Z}(\textbf x_1,\textbf x_2)=\Lambda_{\Delta}\):
Note that the lengths of the primitive lattice vectors are related to the inter-carbon atom spacing \(a\approx 1.4\) Å by \(|\textbf x_1|=|\textbf x_2|=\sqrt 3a\) and that the area of the spanning parallelogram is \(|\textbf x_1\times\textbf x_2|=\frac{3\sqrt{3}}{2}a^2\). Of course we also have a reciprocal Bravais lattice \(\Lambda^*_{\Delta}\) which is also triangular and spanned by reciprocal lattice vectors \(\textbf k_1,\textbf k_2\in\Lambda^*_{\Delta}\):
where now \(|\textbf k_1|=|\textbf k_2|=\frac{4\pi}{3a}\) and \(|\textbf k_1\times\textbf k_2|=\frac{8\pi^2}{3\sqrt{3}a^2}\) by the defining criterion \(\textbf x_{\mu}\cdot\textbf k_{\nu}=2\pi\delta_{\mu\nu}\). The first Brillouin zone is hexagonal and henceforth we reduce all other Brillouin zones to this:
where the side lengths of the Brillouin zone hexagon are \(\frac{4\pi}{3\sqrt{3}a}\) and thus the area of the Brillouin zone is \(\frac{8\sqrt 3\pi^2}{a^2}\). It turns out for graphene that the corners of the Brillouin zone hexagon are especially interesting points, called the Dirac points \(\textbf k_{\text{Dirac}}^{\pm}\) of graphene. Despite the fact that the Brillouin zone hexagon has \(6\) vertices, there are only really \(2\) physically distinct Dirac points as labelled because the other Dirac points are manifestly connected to them by a reciprocal lattice vector in \(\text{span}_{\textbf Z}(\textbf k_1,\textbf k_2)=\Lambda^*_{\Delta}\) and so are identified modulo the reduced zone scheme.
Physics of Graphene (Tight-Binding)
Each carbon \(\text C\) atom in graphene has valence \(Z=1\) from donating its one \(2p_3\) atomic orbital electron \(e^-\) into a collective \(\pi\) orbital delocalized over the entire graphene sheet. We’ll model these electrons as being tightly bound to carbon \(\text C\) atoms but free to tunnel/hop around the graphene lattice. In general, the tight-binding/hopping Hamiltonian is:
\[H=H(\Lambda_{\Delta},\mathcal M,t_0,t_1,t_2,…)=-\sum_{n=0}^{\infty}t_n\sum_{\textbf x\in\Lambda_{\Delta}}\sum_{\textbf x’\in\Lambda_{\Delta}*\mathcal M:|\textbf x’-\textbf x|_1=n}|\textbf x\rangle\langle\textbf x’|+|\textbf x’\rangle\langle\textbf x|\]
where \(\mathcal M\) is the \(2\)-carbon atom motif at each lattice point in \(\Lambda_{\Delta}\), \(t_n\in\textbf R\) are hopping energy amplitudes for \(n\)-th nearest neighbours, \(|\textbf x’\rangle\langle\textbf x|=(|\textbf x\rangle\langle\textbf x’|)^{\dagger}\) are mutual adjoints which physically permits bidirectional tunneling and mathematically ensures \(H^{\dagger}=H\) is Hermitian, and \(|\textbf x-\textbf x’|_1\) is a taxicab-like metric on the triangular Bravais lattice \(\Lambda_{\Delta}\) which makes it into a metric space, defined as the shortest number of hops between the two points \(\textbf x,\textbf x’\in\Lambda_{\Delta}\) (in graph theoretic terms, this is commonly called the graph geodesic distance between \(\textbf x\) and \(\textbf x’\)). In practice, it is common to restrict to only \(n=0\) (the on-site energy) and \(n=1\) (nearest neighbour) hopping, thereby ignoring all hopping across \(n\geq 2\) atoms. For graphene, the nearest-neighbour hopping energy amplitude turns out to be \(t_1\approx 2.8\text{ eV}\). With this simplification, the tight-binding Hamiltonian becomes (via a resolution of the identity):
\[H\approx -2t_01-t_1\sum_{\textbf x\in\Lambda_{\Delta}}\sum_{\textbf x’\in\Lambda_{\Delta}*\mathcal M:|\textbf x’-\textbf x|_1=1}|\textbf x\rangle\langle\textbf x’|+|\textbf x’\rangle\langle\textbf x|\]
Since the \(n=0\) term \(-2t_01\) is just proportional to the identity \(1\), it doesn’t affect any of the physics so we can ignore it henceforth. For graphene, there are of course \(3\) nearest neighbour carbon atoms \(\textbf x’\in\Lambda_{\Delta}*\mathcal M\) for each carbon atom at position \(\textbf x\in\Lambda_{\Delta}\):
\[H=-t_1\sum_{\textbf x\in\Lambda}|\textbf x\rangle\langle\textbf x+\frac{2}{3}\textbf x_1-\frac{1}{3}\textbf x_2|+|\textbf x\rangle\langle\textbf x-\frac{1}{3}\textbf x_1+\frac{2}{3}\textbf x_2|+|\textbf x\rangle\langle\textbf x-\frac{1}{3}\textbf x_1-\frac{1}{3}\textbf x_2|\]
As usual, we now declare that we wish to solve \(H|E\rangle=E|E\rangle\). For this we’ll simply make a Wannier-type ansatz of the \(H\)-eigenstate \(|E\rangle\) as a sum of plane waves modulated by
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