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.


Introduction
The discovery of graphene with extraordinary physical properties by Geim and Novoselov [1][2][3][4] has opened a new era in the development of condensed matter physics and materials science as well as many fields of high technologies. Right after this discovery the graphene-based optoelectronics has emerged. Xia et al [5] have explored the use of zero-bandgap large-area graphene field effect transistor as ultrafast photodetector. One year later Xia et al [6] have reported again the use of photodetector based on graphene. A broad-band and high-speed waveguide-integrated electroabsorption modulator based on monolayer graphene has been demonstrated by Liu et al [7]. In [8] Wang et al have demonstrated a graphene/silicon-heterostructure waveguide photodetector on silicon-on-insulator material. An ultrawideband complementary metal-oxide semiconductor-compatible graphene-based photodetector has been fabricated by Muller et al [9]. In [10] Englund et al have demonstrated a waveguide integrated photodetector etc. At the present time the research on graphene photodetectors in still developing [11][12][13].
In all above-mentioned research works the theoretical reasonings on the light-graphene interaction were limited to the case when the light waves propagate inside very thin graphene layer. However, in the study of the photon-electron interaction in a thin graphene sheet, the light waves always must be sent from the sources located outside the graphene sheet. Therefore the theoretical problem of photon-electron interaction in graphene layers must be formulated and solved as a problem in the three-dimensional physical space. This is the content of the present work.
In the subsequent section 2 the physical model of the electron gas in a single-layer graphene sheet is formulated and the notations are introduced. In particular, the overall Hamiltonian of the free electron gas in a graphene sheet with some thickness d, which may be extremely small but must be finite, is presented, and the explicit expressions of the wave functions of charge carrier are derived. In section 3 the electron-hole formalism, convenient for the application to the study of the electron-hole pair photo-excitation, is introduced. The theory of the interaction of an external quantum electromagnetic field with charge carriers in a graphene sheet is elaborated in section 4. The explicit expressions of the matrix elements of the photon-electron interaction in the graphene sheet are derived in section 5. The conclusion and discussion are presented in section 6.

Physical model of the electron gas in a singlelayer graphene sheet
Consider a single graphene sheet as a plane slab of a semiconducting material with a very small but finite thickness such that the xOy coordinate plane is parallel to the graphene sheet surface and located in its middle, while the Oz axis is perpendicular to the graphene surface. It was known [14] that each graphene single layer is a two-dimensional (2D) lattice of carbon atoms with the hexagonal structure (figure 1), and the first Brillouin zone (BZ) in the reciprocal lattice of the graphene 2D lattice has two corners at two points K and K¢ (figure 2).
Suppose that the quantum motion of electrons along any direction parallel to the xOy coordinate plane and that along the direction of the Oz axis are independent. Then the electron quantum field z t r, , ( ) y is decomposed in terms of the twocomponent wave functions r of Dirac fermions with momenta k close to the corner K or K¢ of the first BZ and the k-dependent energies E as well as in terms of the wave functions f z i ( ) of electrons with energies ε i in the potential well along the Oz axis. For the simplicity we assume that this potential well has a great depth and therefore wave functions f z i ( ) must vanish at the boundary of the potential well.
Since each corner K or K¢ is an extreme point of the electron distribution cones and the electron momenta k are always close to K or K , ¢ the electron quantum field of corresponding Dirac fermions. Although the graphene sheet may be infinitely large, for the simplicity of the reasoning during the quantization procedure we impose on the wave functions of Dirac fermions the periodic boundary conditions in a square with the large side L (figure 3).
The overall Hamiltonian of free electron gas (without mutual electron-electron Coulomb interaction) has following expression     1  2  d d  , ,  i  , , , , i , , , 5 m is the effective mass of electron in the potential well, and F n is the effective speed of massless Dirac fermion. From expression (3) and (4) of the Hamiltonians H K 0 and H K 0 ¢ we derive the Dirac equations determining two-component wave functions r It can be shown [14] that for each momentum k there exist two eigenvalues of each of two equations (7) and (8) The quantum fields z t r, , K ( ) y and z t r, , K ( ) y ¢ have following expansions in terms of wave functions r ,

Electron-hole formalism in graphene
As usual, for the study of grahene as a semiconductor we work in the electron-hole formalism. We shall use following short notations The destruction and creation operators of holes are defined as follows: for all indices I and J, meaning that in the ground state there does not exist any Dirac fermion above the Fermi level as well as any hole of Dirac fermion on or below the Fermi level.
In the sequel the hole of Dirac fermion on or below the Fermi level will be shortly called Dirac hole. The energies of Dirac  From the Heisenberg equations of motion (14) for the fields z t r, , K ( ) y and z t r, , K ( ) y ¢ it follows the Heisenberg equation of motion for the field z t r, , and similar equation with K K¢  .

Photon-electron interaction in graphene
The overall Hamiltonian H G of the single-layer graphene sheet interacting with the transverse electromagnetic field A(r, z, t)