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 ${\vec {b_{i}}}\cdot {\vec {a_{j}}}=2\pi \delta _{ij},(i,j=1,2)$

${\vec {b_{1}}}={\frac {2\pi }{a}}({\frac {\sqrt {3}}{3}}{\hat {x}}+{\frac {1}{3}}{\hat {y}})$

${\vec {b_{2}}}={\frac {2\pi }{a}}(-{\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {3}{2}}a{\hat {y}})$