Graphene Brillouin Zone and Electronic Energy Dispersion

show Brillouin zone

π-

*

π

-band

3D

usual

extend Brillouin zone

1

1.5

2

3

4

show more detailed dispersion at:

full BZ

M-saddle

K-linear

K-points

3D view

default

above

front

right

add BZ and points

cut segment

change calculation parameters

π-bonding (transfer) integral

t

-3

on-site energy

ϵ

0

overlap integral

s

0.13

show: mesh

box

axes

picture quality:

low

medium

high

color scheme:

Rainbow

Graphene is a single layer of carbon atoms densely packed in a honeycomb lattice. This carbon allotropes and is the first known example of a truly two-dimensional (2D) crystal. This Demonstration considers the construction of the Brillouin zone (BZ)

π

-bands electronic dispersion relations for a 2D honeycomb crystal lattice of graphene under the tight binding (TB) approximation. Plots are shown for the electron energy dispersion for

π

and

⋆

π

-bands in the first and extended Brillouin zones as contour plots at equidistant energies and as pseudo-3D representations for the 2D structures. Conventional representation of the energy dispersion relations along the lines between the high symmetry points

K-Γ-M-K

of the first Brillouin zone is shown if you select the "usual" Brillouin zone button. You can also see the details of the dispersion curves at the hyperbolic

M

-point (van Hove saddle point), and also around the

K

-points with the linear energy dispersion for the two

π

-bands (Dirac electrons), selecting "

M

-saddle" and "

K

-points" buttons, respectively.