Theory of photon–electron interaction in single-layer graphene sheet

The purpose of this work is to elaborate the quantum theory of photon–electron interaction in a single-layer graphene sheet. Since the light source must be located outside the extremely thin graphene sheet, the problem must be formulated and solved in the three-dimensional physical space, in which the graphene sheet is a thin plane layer. It is convenient to use the orthogonal coordinate system in which the xOy coordinate plane is located in the middle of the plane graphene sheet and therefore the Oz axis is perpendicular to this plane. For the simplicity we assume that the quantum motions of electron in the directions parallel to the coordinate plane xOy and that along the direction of the Oz axis are independent. Then we have a relatively simple formula for the overall Hamiltonian of the electron gas in the graphene sheet. The explicit expressions of the wave functions of the charge carriers are easily derived. The electron–hole formalism is introduced, and the Hamiltonian of the interaction of some external quantum electromagnetic field with the charge carriers in the graphene sheet is established. From the expression of this interaction Hamiltonian it is straightforward to derive the matrix elements of photons with the Dirac fermion–Dirac hole pairs as well as with the electrons in the quantum well along the direction of the Oz axis. (paper)