Green's functions for a graphene sheet and quantum dot in a normal magnetic field

Abstract

This paper is concerned with the derivation of the retarded Green's function for a two-dimensional graphene layer in a perpendicular magnetic field in two explicit, analytic forms, which we employ in obtaining a closed-form solution for the Green's function of a tightly confined magnetized graphene quantum dot. The dot is represented by a δ(2)(r)-potential well and the system is subject to Landau quantization in the normal magnetic field.

Landau quantized dynamics and spectra for group-VI dichalcogenides, including a model quantum wire

“…5 Of course, the determination of the Green's functions in these cases also provides detailed information about the pertinent carrier propagation characteristics, including the Peierls phase factor and orbit curvature, as well as the energy eigenstate spectra.…”

“…1,2 The relativistic Green's function in a magnetic field was most elegantly first determined by Schwinger, 3 and interesting variants for particular Dirac-like materials have been advanced by Rusin and Zawadski, 4 as well as the author. 5 This paper is focused on yet another important representation for the Landau-quantized Green's function for the group-VI dichalcogenides, particularly in direct time representation, whose behavior is more transparent in the low field limit in comparison with that of the eigenfunction expansion. Moreover, it facilitates the analysis of the role of a quantum wire, modeled here by a one-dimensional δ(x)-potential profile, enabling the explicit determination of the wire Green's function and its Landau quantized Dirac-like spectrum.…”

Aspects of the theory of graphene

“…The requirement of gauge invariance leads to (Horing 1965 13) where the factor ↔ G (r − r ; t − t ) is spatially translationally invariant and gauge invariant, satisfying the equation (Horing & Liu 2009) 14) while the factor C (r, r ) embodies all non-spatially translationally invariant structure and all gauge dependence as…”

“…where L z is the angular momentum operator, we note that L Z G (R, T ) = 0 (Horing 1965;Horing & Liu 2009). Therefore, equation (2.18) may be written in the form 21) which is readily recognizable as Green's function equation for an isotropic twodimensional harmonic oscillator (with the impulsive Dirac d-function driving term having its source point at the origin) in position-frequency representation.…”

“…Therefore, equation (2.18) may be written in the form 21) which is readily recognizable as Green's function equation for an isotropic twodimensional harmonic oscillator (with the impulsive Dirac d-function driving term having its source point at the origin) in position-frequency representation. Its retarded solution in position-'time' (t) representation may be written as (Horing 1965;Horing & Liu 2009) (h + (t) is the Heaviside unit step function)…”