Question

Problem 4: What are the TWO lattice vectors and ONE basis vector to describe 2D material of graphene? (Many cases possible)

Answer 1

Answer :-

Graphene is a single atomic layer of graphite and defined as a one-atom-thick planar sheet of sp² -bonded carbon atoms densely packed in a honeycomb crystal lattice. This single layer structure makes graphene a two-dimensional (2D) system.​​​​​​

The honeycomb lattice of graphene in real space is spanned by two lattice vectors expressed as −

aj = (+3a/2, 0/2), az = (v3a/2, -a/2)

giving the relations,

a.a = 72.= a

a.az = a/2

These lattice vectors form an angle of 60^$\degree$ between them and are of magnitude ,

a = = 2 = acc * V3 = 1.42 * V3 = 2.46 Å

the lattice constant, where is the carbon-carbon bond length in graphene.

Acc = 1.464

The ONE basis vector of the lattice is,

a1 = 3a(1,0)

Another basis vector can be,

a2 = a(V3/2,3/2)

   

a a2

fig : The honeycomb lattice of graphene. The unit cell defined by vectors $a_{1}$ and $a_{2}$ containing to the two atoms belonging to sublattices A (blue) and B (red) is highlighted in light blue.

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