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Graphene Brillouin Zone and Electronic Energy Dispersion

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) -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 and -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 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 -point (van Hove saddle point), and also around the -points with the linear energy dispersion for the two -bands (Dirac electrons), selecting "-saddle" and "-points" buttons, respectively.

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Graphene is a single layer of carbon atoms densely packed in a honeycomb lattice. The energy structure of crystals depends on the interactions between orbitals in the lattice. The tight binding approximation (TB) neglects interactions between atoms separated by large distances—an approximation that greatly simplifies the analysis. In solid-state physics, the TB model [1] calculates the electronic band structure using an approximate set of wavefunctions constructed by superposition of localized atomic orbitals.
This Demonstration calculates and plots the tight-binding (TB) electronic band structure of graphene as the 2D hexagonal carbon crystal. The band structure of graphene is obtained from the TB approximation including only first-nearest-neighbor carbon-carbon interactions of -orbitals of a single honeycomb graphite sheet. This is given by a simple analytical relation, derived by diagonalization of the Bloch Hamiltonian for the diatomic graphene unit cell [2]:
, with ,
with the phase factor
.
In this expression, is the TB hopping parameter (overlap integral between -orbitals), is the on-site energy parameter, is the overlap parameter, is the lattice parameter of graphene, the and indices stand for valence and conduction bands, respectively, and represents the 2D wavevector components along the and directions in the 2D Brillouin zone of graphene. The parameters and are expressed in electron-volt units (eV), whereas is given in nondimensional units. At the special points and of the graphene BZ the valence and conduction bands cross at the Fermi level (energy at 0 eV).
The traditional presentation of an electronic dispersion along the lines in Brillouin zone is given by selecting "usual" under "show Brillouin zone". Checking "show mesh" gives the distribution of allowed states inside the bands representing the density of states function. Check "add BZ and points" to see the location of high-symmetry points in BZ.
Setter bars under "show more detailed dispersion" let you compare the lattice electron behavior at the hyperbolic saddle -point and quasi-linear (Dirac) dispersion curves at the -points of BZ.
Snapshot 1: traditional representation of an electronic dispersion relation for the graphene along the lines of the first Brillouin zone
Snapshot 2: pseudo-3D energy dispersion for the two -bands in the first Brillouin zone of a 2D honeycomb graphene lattice
Snapshot 3: constant energy contours for the -valence band and the first Brillouin zone of the graphene
Snapshot 4: same constant energy contours for the -conduction band in an extended Brillouin zone
Snapshot 5: pseudo-3D energy dispersion for the -conduction band at the saddle -point (van Hove saddle point)
Snapshot 6: pseudo-3D near-linear energy dispersion for the two -bands near -points (Dirac electrons)
References:
[1] C. Kittel, Solid State Physics, Hoboken, NJ: John Wiley and Sons, Inc., 1996.
[2] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes, London: Imperial College Press, 1998.
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