Phy5670/Phonon in Graphene

Structure of Graphene

Carbon atoms in graphene are constructed on a honeycomb lattice, which is shown in Fig. 1. The circle and solid point together combine into a two-point basis in the Bravais lattice. The carbon-carbon distance is $a~1.42{\mathrm {\AA} }$. The primitive vectors are

${\vec {a_{1}}}={\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {3}{2}}a{\hat {y}}$

${\vec {a_{2}}}=-{\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {3}{2}}a{\hat {y}}$

Then the reciprocal lattice parameters can by generated by using ${\vec {b_{i}}}\cdot {\vec {a_{j}}}=2\pi \delta _{ij},(i,j=1,2)$

Failed to parse (unknown function "\fact"): {\displaystyle \vec{b_1}=\frac{2\pi}{a}(\frac{\sqrt{3}}{3}\hat{x}+\fact{1}{3}\hat{y})}

Failed to parse (unknown function "\fact"): {\displaystyle \vec{b_2}=\frac{2\pi}{a}(-\frac{\sqrt{3}}{2}a\hat{x}+\fact{3}{2}a\hat{y})}