Two-point Green functions of free Dirac fermions in single-layer graphene ribbons with zigzag and armchair edges

Green function technique is a very efficient theoretical tool for the study of dynamical quantum processes in many-body systems. For the study of dynamical quantum processes in graphene ribbons it is necessary to know two-point Green functions of free Dirac fermions in these materials. The purpose of present work is to establish explicit expressions of two-point Green functions of free Dirac fermions in single-layer graphene ribbons with zigzag and armchair edges. By exactly solving the system of Dirac equations with appropriate boundary conditions on the edges of graphene ribbons we derive formulae determining wave functions of free Dirac fermions in above-mentioned materials. Then the quantum fields of free Dirac fermions are introduced, and explicit expressions of two-point Green functions of free Dirac fermions in single-layer graphene ribbons with zigzag and armchair edges are established.


Introduction
The discovery of graphene by Geim, Novoselov et al [1][2][3][4] has stimulated the development of a new multidisciplinary area of science and technology of graphene-based nanomaterials [5,6]. Recently a new approach to the theoretical study of these nanomaterials as well as to the electromagnetic interaction processes in single-layer graphene using mathematical tools of quantum field theory was proposed [7,8]. In particular, a comprehensive study on two-point Green functions of Dirac fermions in Dirac fermion gas of an infinitely large graphene single layer was performed [7]. The purpose of present work is to study two-point Green functions of Dirac fermions in the Dirac fermion gas of graphene ribbons with zigzag and armchair edges. It was known that hexagonal crystalline lattice of graphene comprises two interpenetrating sublattices with triangular symmetry [4]. Throughout the work following notations and conventions will be used.
The distance between two nearest carbon atoms in the hexagonal graphene lattice is denoted a. Then the distance between two nearest vertices in each triangular sublattice is a a 3 . 0 = Denote a 1 and a 2 the translation vectors of the triangular crystalline sublattice, and b 1 and b 2 those of its reciprocal sublattices We chose the xOy Cartesian coordinate system as follows: Ox axis is parallel to the direction of the length of the ribbon, while Oy axis is parallel to that of its width. Then for the graphene ribbon with zigzag edges we have the crystalline | Vietnam Academy of Science and Technology Advances in Natural Sciences: Nanoscience and Nanotechnology Adv. Nat. Sci.: Nanosci. Nanotechnol. 7 (2016)  Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. lattice structure represented in figure 1(a) and the reciprocal lattice represented in figure 1(b), while for that with armchair edges the crystalline lattice structure is represented in figure 2(a) and the reciprocal lattice is represented in figure 2 For the simplicity we shall limit our study to the case of a Dirac fermion gas with the Fermi energy level E F =0 and at the vanishing absolute temperature T=0. The extension to other cases is straightforward.
In section 2 we study the quantum fields of Dirac fermions in graphene ribbon with zigzag edges, and the subject of section 3 is the study of quantum fields of Dirac fermions in graphene ribbon with armchair edges. Conclusion and discussion are presented in section 4.  figure 1(b)).

Graphene ribbon with zigzag edges
where two components τ 1 and τ 2 of the 2×2 vector matrix t are the Pauli matrices and rewrite Dirac equations in the form Dirac equations (7) and (8) must be invariant with respect to the translations along the Ox axis which do not change the graphene ribbon crystalline lattice as a whole. These translations form a group called the translational symmetry group of this crystalline lattice. According to the Bloch theorem [9] eigenfunctions of Dirac equations (7) and (8) In [4] it was shown that by solving Dirac equations (7) and (8) one obtains eigenvalues ε determined by relation with some real constants λ and eigenfunctions (9), where y Due to the boundary condition (11) between the constants According to formula (12) Dirac equations (7) and (8) have two common eigenvalues , In the first case with k, In both case we have Let us now study the conditions determining the values of the parameter λ. From boundary condition (10) for function (18) we derive following algebraic equation while from the same boundary condition (10) for function (19) we obtain another one e k k 24 In [4] it was noted that whenever k is positive (k>0), equation (23) for λ has real solutions corresponding to surface waves propagating near the edges of the graphene ribbon. Similarly, whenever k is negative (k<0), equation (24) for λ also has real solutions corresponding to surface waves propagating near the edges of the graphene ribbon.
Thus we have demonstrated that Dirac equations (7) and ) the corresponding eigenfunctions are It is easy to verify that u y v y dy v y u y dy 0.     By means of standard calculations [7] it is straightforward to derive following explicit expression of two-point Green functions of free Dirac fermions in a single-layer graphene ribbon with zigzag edges

Wave functions of free Dirac fermions
In the case of graphene ribbon with armchair edges the quantum states of Dirac fermions with wave vectors near both inequivalent Dirac points K and K¢ must be simultaneously taken into account, so that wave functions of stationary states are two orthogonal and normalized linear combinations e e r r r 1 2 , and e e r r r 1 2 , In [4] it was shown that functions

Conclusion
For the future application in the study of dynamical quantum processes in single-layer graphene ribbons with zigzag and armchair edges, we have derived formulae determining the wave function of free Dirac fermions in these materials, taking into account appropriate boundary conditions at the edges of the ribbons. Then these wave functions were used for construction of quantum fields of free Dirac fermions, and explicit expressions of their two-point Green functions were established.
In the present work we have considered the simplest case of the free Dirac fermion gas with Fermi level E F =0 at vanishing absolute temperature T=0. A lot of works should be done to extend obtained results to more general cases such as the free Dirac fermion gas with Fermi level E F ≠0 and at non-vanishing absolute temperature T≠0 as well as to the case of non-equilibrium free Dirac fermion gas. Moreover, the study of the interaction between the electromagnetic field and Dirac fermions in graphene ribbons would be also interesting scientific subject with practical applications.