Electronic structure of graphene beyond the linear dispersion regime

S. R. Power and M. S. Ferreira

Phys. Rev. B 83, 155432 – Published 18 April 2011

Abstract

Among the many interesting features displayed by graphene, one of the most attractive is the simplicity with which its electronic structure can be described. The study of its physical properties is significantly simplified by the linear dispersion relation of electrons in a narrow range around the Fermi level. Unfortunately, the mathematical simplicity of graphene electrons is limited only to this narrow energy region and is not very practical when dealing with problems that involve energies outside the linear dispersion part of the spectrum. In this communication we remedy this limitation by deriving a set of closed-form analytical expressions for the real-space single-electron Green function of graphene which is valid across a large fraction of the energy spectrum. By extending to a wider energy range the simplicity with which graphene electrons are described, it is now possible to derive more mathematically transparent and insightful expressions for a number of physical properties that involve higher energy scales. The power of this new formalism is illustrated in the case of the magnetic (RKKY) interaction in graphene.

Received 1 November 2010

DOI:https://doi.org/10.1103/PhysRevB.83.155432

Authors & Affiliations

S. R. Power and M. S. Ferreira^*

School of Physics, Trinity College Dublin, Dublin 2, Ireland

^*ferreirm@tcd.ie

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Issue

Vol. 83, Iss. 15 — 15 April 2011

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Images

Figure 1
Schematic of the graphene lattice with the zigzag (armchair) direction denoted by the blue (red) arrow marked “Z” (“A”). The filled and hollow circles represent carbon atoms on each of the two sublattices that compose the graphene lattice.Reuse & Permissions
Figure 2
Constant energy plots of the function $E_{+} (k⃗)$ in reciprocal space for a few different energies. Horizontal and vertical axes are rescaled as $k_{x} a / 2$ and $k_{y} a \sqrt{3} / 2$ , respectively, so they are plotted as dimensionless quantities. The rectangular shaded area delimits the first Brillouin zone over which the integrals for the armchair direction case are taken. Constant energy plots for two specific energies, $E = 0.7 | t |$ (solid) and $E = 1.8 | t |$ (dashed), are drawn with thicker lines. The corresponding stationary wave vectors $q̃$ for these energies are highlighted with arrows. Note that the sign of $q̃$ , shown as positive here for simplicity, must be chosen according to the conditions outlined in the text.Reuse & Permissions
Figure 3
$G_{Δ} (E)$ as a function of energy (in units of $| t |$ ) for the case of $Δ = 10 \sqrt{3} a$ . Top (Bottom) panel shows the real (imaginary) part of the Green function. Lines correspond to the results evaluated by Eq. (11), whereas points are the result of brute-force numerical calculations of Eq. (3).Reuse & Permissions
Figure 4
Fraction $F_{1 %}$ of the bandwidth for which the error between the analytical and brute force numerical results is below $1 %$ , plotted as a function of the separation $Δ$ (in units of $a$ ). The circles (triangles) correspond to separations in the armchair (zigzag) direction.Reuse & Permissions
Figure 5
The constant energy surfaces of the function $E_{+} (k⃗)$ as before with the Brillouin zone for the zigzag case now illustrated by the shaded area. The thick lines once more refer to constant energy plots for $E = 0.7 | t |$ (solid) and $E = 1.8 | t |$ (dashed). The corresponding stationary wave vectors $q̃$ for these energies are highlighted with arrows. Note that the sign of $q̃$ , shown as positive (solid) or negative (dashed) only for simplicity, must be chosen according to the conditions outlined in the text. For clarity the arrows are shifted slightly in the vertical direction, but it should be noted that all the stationary points occur at $k_{y} = 0$ .Reuse & Permissions
Figure 6
$G_{Δ} (E)$ as a function of energy (in units of $| t |$ ) for the case of $Δ = 20 a$ . Top (Bottom) panel shows the real (imaginary) part of the Green function. Lines correspond to the results evaluated by Eq. (17), whereas points are the result of brute-force numerical calculations of Eq. (3).Reuse & Permissions

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