Features of the Interaction of Light with Graphene Nanostructures and Transition Metal Dichalcogenides
The aim of this work is to describe the optical spectra of the quasi-energies of charge carriers in a Dirac material. To achieve this goal, we solve the problem of finding the energy spectrum near the edge of the energy zones of grapheme ( ) and near the edge of the dichalcogenide energy zone (MoS2) ( ) irradiated by an electromagnetic field with a high photon energy and various intensities I, where the field is linearly polarized along the Y, X axis; The solution to this problem provides many zones by which it can be determined whether the material is a conductor, semiconductor or insulator.
S. М. R. H. Hussein 1, 2
Samara National Research University named after Academician S. P. Korolev, Samara, Russia
University of Karbala, Karbala, Iraq
The aim of this work is to describe the optical spectra of the quasi-energies of charge carriers in a Dirac material. To achieve this goal, we solve the problem of finding the energy spectrum near the edge of the energy zones of grapheme () and near the edge of the dichalcogenide energy zone (MoS2) () irradiated by an electromagnetic field with a high photon energy and various intensities I, where the field is linearly polarized along the Y, Х axis; The solution to this problem provides many zones by which it can be determined whether the material is a conductor, semiconductor or insulator.
Keywords: graphene, photon energy, transition metal dichalcogenides, optical spectrum
Received on: 18.03.2020
Accepted on: 13.05.2020
The transition metals are 38 elements in groups from the 3rd to the 12th periodic table. Monolayers of transition metal dichalcogenides (TMDC) are two-dimensional (2D) semiconductors. They are widely used in electronics and optics. Numerous applications in electronics are due to the properties of the direct forbidden zone (in contrast to indirect forbidden zones in three-dimensional layered crystals of TMDCs). Monolayer TMDCs have already been implemented in field effect transistors, logic devices, as well as in optoelectronic devices. The discovery of graphene, a monolayer of carbon atoms with a linear (Dirac) dispersion of electrons [1–3], initiated studies of a new class of artificial nanostructures known as Dirac materials.
Although graphene itself is characterized by a gapless energy spectrum of electrons, the creation of Dirac materials with a wide forbidden gap between the valence and conducting bands (Dirac materials with a gap) holds great promise for the development of optoelectronic devices. Monolayers of these materials effectively absorb and generate light. These qualities are important for creating optoelectronic elements based on 2D materials from transition metal dichalcogenides (MX2): tunable optical microresonators for thin-film VCSEL lasers and detectors. Therefore, great efforts were directed towards their creation. This determines the relevance of studies of the optical properties of nanostructures based on graphene and MX2.
The energy spectrum of the electrons of the material is described by a parabolic function near the edges of the bands, but turns into a linear Dirac dispersion if the wide forbidden gap disappears. Thus, the electronic properties of Dirac materials with a gap substantially depend on the size of the gap and, therefore, this determines their prospects for nanoelectronic and optoelectronic applications [4–6].
Despite the fact that the structures of graphene and MoS2 have been the focus of attention for a long time, a mathematical quantum theory for calculating their energy spectrum in optical applications has not been developed. Since the electronic structure of Dirac materials is fundamentally different from traditional structures of materials in a condensed state, the well-known theory of weak coupling of electrons with atoms of matter cannot be directly applied to Dirac materials with a gap. Moreover, it should be noted that Dirac materials with a gap are currently considered as the basis for a new generation of optoelectronic devices. At the moment, there is the task of completing the theory of communication. To solve it, a graphene layer grown on a hexagonal boron nitride substrate is considered [7, 8].
When studying the interaction of laser radiation with the studied objects, we use the term «strong electromagnetic field». This means that the photon energy of the light field incident on the structure under study should be much larger than the band gap in the source material.
Solving the problem of determining eigenvalues by the Cholesky method
To calculate the optical spectra of charge carriers of the nanostructures under study and their analysis, it is necessary to find the eigenvalues of the Hamiltonian. We use the standard Cholesky method, since in this method the index m, when decomposing a function Ψ into basic functions, has the same value both in a cylindrical system and in a spherical coordinate system.
Both coordinate systems use the same and azimuthal coordinate ϕ. Since in this case the symmetry is cylindrical and all quantum points lie on the axis of the system, there is no need to use the addition theorem for cylindrical functions.
We need to find the eigenvalues E (k) of the Hamiltonian with eigenfunctions Ψ
We represent the function Ψ in the form of expansion in basis functions (in the general case, not orthogonal) n1, m1, p1 = 1, 2, 3….
where Φ is the eigenfunction of the Hamiltonian. Thus, we can write equation (2) in the following form:
where S is the scalar product of thw functions. The solution to this equation is obtained only approximately. The formulas obtained can be used to calculate the spectrum of charge carriers in an electromagnetic field both in the case of graphene and dichalcogenides. Let us reduce the solution of this equation to the usual eigenvalue problem. We write the equation in matrix form:
Taking into account the contribution of inelastic interaction of light with structures
When solving the problem, it is necessary to take into account the contribution of inelastic interaction of light with the structures under study. Let a linearly polarized electromagnetic field fall perpendicular to the monolayer of transition metal dichalcogenide and graphene (Fig. 1).
In contrast to , in this work, the spectrum of charge carriers is analyzed based on the solution of the equation:
Let us analyze the contribution of inelastic interactions. To do this, let us consider the equation
Let us use the expressions of the matrix elements of the stationary Hamiltonian from [9, 10] and the expression for the Hamiltonian from .
where is the boundary of the zone; , , is the electronic wave vector in the plane of the layer, Δg is the band gap, γ is a constant proportional to the Fermi velocity of the quasiparticles; and is a spin-orbit splitting of the valence and conduction bands, respectively; s = ±1 is a spin index; τ = ±1 is a valley index corresponding to K and K′, respectively , δnn’ is the Kronecker delta, J is the Bessel function of the first kind.
The block matrix of the Hamiltonian has the form
Let us write the matrix taking into account = –1, 0, 1
After we have reduced the multidimensional matrix to the usual square one, we can use the standard functions that are available in mathematical packets.
The matrix H can be represented as
where the diagonal matrix has the form:
Calculation results in the presence of inelastic interactions for graphene and chalcogenide (MoS2)
First of all, we consider the application of the developed theory to graphene with a gap, assuming that , in all derived expressions. The electron dispersion in gap graphene is shown in Fig. 2 and 3 for special cases of linearly and circularly polarized gauge fields. In the absence of an incident field, the electron dispersion is isotropic in the graphene plane (see solid blue lines in Figs. 2 and 3).
However, a linearly polarized field violates the equivalence of the X, Y axes [see equation (5)]. As a result of electron anisotropy, dispersion appears along the wave vectors kx and ky (see the dotted green and red lines in Figs. 2 and 3).
Unlike polarization of chalcogenide (MoS2) ( meV, meV), a circularly polarized gauge field does not cause anisotropy in the plane [see equation (6)]. However, electron dispersion differs significantly for polarizations clockwise and counterclockwise (see dotted lines in Figs. 4 and 5).
An analysis of the obtained graphical dependences shows that the band gap and the energy spectrum near the edge of the graphene zones change with changing coordinates along the X, Y axes and the intensity of the incident light field I. However, the band gap and the energy spectrum near the edge of the dichalcogenide zones do not change with coordinates along the X, Y axes. This means that the energy spectrum of the dichalcogenide remains constant, even when the intensity changes in the range of 0–15 kW / cm2. The calculation results are intended for the use of graphene and MoS2 nanostructures in the manufacture of optical and semiconductor devices.
A method has been developed for calculating optical spectra (spectrum of charge carriers) in a linear array of quantum dots in a linear lattice for both graphene and MoS2.
New equations for describing inelastic interactions of electromagnetic fields with graphene carriers and dichalcogenic transition metals.
Based on the results of studying the energy spectrum and energy gaps, it can be determined whether the material is a conductor, semiconductor or dielectric. This is important for many applications in electronics and optics, which are associated with the values of the energy zones of the material and the values of their energy gaps.
The results of the analysis of the graphs showed that the band gap and the energy spectrum near the edge of the graphene regions change with a change in the coordinate of the Y, X axes and the intensity of the probe radiation. But at the same time, the band gap and the energy spectrum near the edge of the MoS2 regions do not change when the coordinates along the X, Y axes and the intensity of the probe radiation change. This means that the energy spectrum of MoS2 remains constant, even if the intensity of the probe radiation varies (in the range of 0–15 kW / cm2).
The results showed that MoS2 is a promising material for creating elements of modern optoelectronics with fixed characteristics. Our results coincide with those of other authors [9–11], who investigated the possibility of controlling the properties of graphene by changing the axes or the intensity of the incident radiation. These properties of the material studied determine their promise for optoelectronics. Both MoS2 and graphene have application prospects for creating tunable optical microresonators that are part of optoelectronic devices: thin-film VCSEL lasers and detectors.
Hussain Safaa Mohammed Ridha Hussain, graduate student, Samara National Research University named after Academician S. P. Koroleva (443086, Samara, Russia); University of Karbala (56001, Karbala, Iraq). E‑mail: firstname.lastname@example.org.