Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac...

Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac lines

by Michaël Barbier, Panagiotis Vasilopoulos, and François M. Peeters

Philosophical Transactions AVolume 368(1932):5499-5524

December 13, 2010

©2010 by The Royal Society

(a) A one-dimensional potential barrier of height Vb and width Wb. (b) A single unit of a potential well next to a potential barrier.

Michaël Barbier et al. Phil. Trans. R. Soc. A 2010;368:5499-5524


(a) Contour plot of the transmission through a single barrier with μ = 0, Wb = L and ub = 10.



(a) Conductance G versus strength P of a δ-function barrier in single-layer graphene; the conductance is independent of the energy.



(a) Four different regions for a single unit of figure 1b with ub = 24, uw = 16, Wb = 0.4 and Ww = 0.6.



The lowest conduction band of the spectrum of graphene near the K point (a,b) in the absence of SL potential and (c,d) in its presence with u = 4π.



The spectrum of graphene near the K point (a) in the absence of an SL and (b) in its presence with u = 4.5π.



The group velocity components vy and vx at the Dirac point j = 0 (shown, respectively, by the solid and the double dotted-dashed curve), and at the extra Dirac points j = 1,2,3 (shown,

respectively, by the dotted-dashed and the dashed curves) as a function ...



Conductivities (a) σxx and (b) σyy, versus Fermi energy for an SL on single-layer graphene with u = 4π and 6π shown by, respectively, the dashed and solid curves.



(a) Schematics of Kronig–Penney SL on single-layer graphene.



(a) Spectrum for a Kronig–Penney SL with P = 0.4π.



Four different types of band alignments in bilayer graphene.



(a) Contour plot of the transmission for the potential of figure 1b in bilayer graphene with Wb = Ww = 40 nm, Vb = −Vw = 100 meV and zero bias.



Contour plot of the transmission through a single barrier in (a,b), for width Wb = 50 nm, and through double barriers in (c–f) of equal widths Wb = 20 nm that are separated by Ww = 20 nm.



Two-terminal conductance of four equally spaced barriers versus energy for Wb = Ww = 10 nm and different SL types I–IV.



(a) Bound states of the antisymmetric potential profile (type IV) with bias Δw = −Δb = 200 meV.



Lowest conduction and highest valence band of the spectrum for a square SL with period L = 20 nm and Wb = Ww = 10 nm.



Conductivities (a) σxx and (b) σyy versus Fermi energy for the four types of SLs with L = 20 nm and Wb = Ww = 10 nm, at temperature T = 45 K; .



Single-layer and bilayer graphene superlattices: collimation, additional Dirac points and Dirac...

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