NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2013, 4 (6), P. 725-746
SEMICLASSICAL ANALYSIS OF TUNNELING THROUGH A SMOOTH POTENTIAL BARRIER AND LOCALIZED STATES IN GRAPHENE MONOLAYER WITH MASS GAP
V. V. Zalipaev – Krylov State Research Institute, St. Petersburg, Russia; v.zalipaev@lboro.ac.uk
C. M. Linton – Loughborough University, Loughborough, UK; c.m.linton@lboro.ac.uk
We present a semiclassical analysis of Dirac electron tunnelling in a graphene monolayer with mass gap through a smooth potential barrier in the ballistic regime. This 1D scattering problem is formulated in terms of a transfer matrix and treated in the WKB approximation. For a skew electron incidence this WKB approximation deals, in general, with four turning points. Between the first and the second, and the third and the fourth, turning points two tunnelling domains are observed. Scattering through a smooth barrier in graphene resembles scattering through a double barrier for the 1D Schr¨odinger operator, i.e. a Fabry-Perot resonator. The main results of the paper are WKB formulas for the entries of the barrier transfer matrix which explain the mechanism of total transmission through the barrier in a graphene monolayer with mass gap for some resonance values of energy of a skew incident electron. Moreover, we show the existence of modes localized within the barrier and exponentially decaying away from it and its behaviour depending on mass gap. There are two sets of energy eigenlevels, complex with small imaginary part and real, determined by a Bohr-Sommerfeld quantization condition, above and below the cut-off energy. It is shown that total transmission through the barrier takes place when the energy of the incident electron coincides with the real part of one of the complex energy eigenlevels. These facts were confirmed by numerical simulations performed using the finite element method (COMSOL).
Keywords: Graphene monolayer with mass gap, high-energy eigenstates, semiclassical approximation, generalized Bohr-Sommerfeld quantization condition.
PACS 03.65.-w, 03.65.Sq, 04.20Ha, 04.30Db