Hybrid matrix method for stable numerical analysis of the propagation of Dirac electrons in gapless bilayer graphene superlattices

Abstract

Abstract. Gapless bilayer graphene (GBG), like monolayer graphene, is a material system with unique properties, such as anti-Klein tunneling and intrinsic Fano resonances. These properties rely on the gapless parabolic dispersion relation and the chiral nature of bilayer graphene electrons. In addition, propagating and evanescent electron states coexist inherently in this material, giving rise to these exotic properties. In this sense, bilayer graphene is unique, since in most material systems in which Fano resonance phenomena are manifested an external source that provides extended states is required. However, from a numerical standpoint, the presence of evanescentdivergent states in the eigenfunctions linear superposition representing the Dirac spinors, leads to a numerical degradation (the so called Ωd problem) in the practical applications of the standard Coefficient Transfer Matrix (K) method used to study charge transport properties in Bilayer Graphen based multi-barrier systems. We present here a straightforward procedure based in the hybrid compliance-stiffness matrix method (H) that can overcome this numerical degradation. Our results show that in contrast to standard matrix method, the proposed H method is suitable to study the transmission and transport properties of electrons in GBG superlattice since it remains numerically stable regardless the size of the superlattice and the range of values taken by the input parameters: the energy and angle of the incident electrons, the barrier height and the thickness and number of barriers. We show that the matrix determinant can be used as a test of the numerical accuracy in real calculations.

Fano resonances in bilayer graphene superlattices

“…However, as we have shown 46 – 48 this method has numerical instabilities and it is not suitable for the computation of the transmission properties of BGSLs. We have also shown that a better and reliable option is the hybrid matrix method 42 , 43 , 45 . The hybrid matrix method relies on writing the Dirac-like equation for bilayer graphene as an ordinary second order differential equation system of the Sturm-Liouville form 43 .…”

“…The wave functions and dispersion relation in the barrier region can be obtained by solving the following Dirac-like equation: where the Hamiltonian is given as 32 , 33 , here q x and q y are the quasiparticle wavevectors along the x and y directions respectively; m is the band effective mass with a value of 0.035 m 0 , being m 0 the bare electron mass 29 – 31 , 45 ; and V ( x ) = V 0 represents the strength of the electrostatic potential.…”

“…With the hybrid matrices of the well ( H w ) and barrier ( H b ) we can find the hybrid matrix of the unit cell of the superlattice ( H uc ) and with it the hybrid matrix of the entire superlattice H ( x R ; x L ), for more details see ref. 45 .…”

Enhancement of the thermoelectric properties in bilayer graphene structures induced by Fano resonances

“…In the case of Fano resonances, it is known that they are related to the chiral nature of electrons in bilayer graphene 33 , 34 , 38 . In particular, they arise due to the chiral matching between electron states inside and outside electrostatic barriers at oblique incidence 35 – 38 , 40 . On its side, hybrid resonances are the result of the coupling between Fano resonances and resonant states in double and superlattice barrier structures 38 , 40 .…”

A novel stiffness model for a wall-climbing hexapod robot based on nonlinear variable stiffness

“…21 Several different performance indices have been proposed of stiffness evaluation, including minimum stiffness, maximum stiffness, and determinant of stiffness matrix. [22][23][24] The determinant of stiffness matrix cannot distinguish specific stiffness values in a certain direction. 25,26 The minimum and maximum stiffness [27][28][29] can reflect variation range of the stiffness value of the mechanism, and the corresponding eigenvector directions of the minimum and maximum stiffness of mechanism represent the minimum and maximum stiffness direction, respectively.…”