"Nanoscale physics"

Approximation of a simple rectangular lattice for a conduction electron in graphene
A.D. Suprun1, L.V. Shmeleva1
1 Department of Theoretical Physics, Faculty of Physics, Taras Shevchenko National University of Kyiv. Volodymyrska Street, 64/13, Kyiv-01601, Ukraine. 
E-mail: lshmel@univ.kiev.ua

The dispersion law for a conduction electron in graphene (normal branch): 
,
which is simultaneously Hamiltonian with respect to the wave momentum , is rather complicated for constructing the general analytical dynamics of an electron.  This is due to the fact that the dispersion law can not be represented in a typical form for crystals with a simple rectangular lattice.  Namely, in the form of two terms summands, each of which contains only one component of the wave vector :  or . Due to this, one can find in the analytical form only one of the seven dynamic characteristics of the conduction electron: velocity.
 As a result of previous researches it was determined that there is a region in the -space of the conduction band of the graphene [1] in which it is possible to construct an effective Hamiltonian:
.
This Hamiltonian allows us to find all the dynamic characteristics to a sufficiently high accuracy. As a result, the energy range of the injection of the electron was estimated.  This estimate was made possible by the generalized Louis de Broglie relation between the components of the wave and mechanical momenta.  To determine the components of the mechanical momentum, in addition to the velocity, the effective mass of the electron was determined. The mechanical Hamiltonian, the mechanical Lagrangian and the wave Lagrangian were determined also.

1. Suprun A. D., Shmeleva L V. Features of the Generalized Dynamics of Quasiparticles in Graphene // Nanoscale Research Letters. - 2017. - 12. P. 187.
http://dx.doi.org/10.1186/s11671-017-1940-0 .