Digital Designing and Parameter Optimization for Plasmonic Circuits

The use of numerical procedures and algorithms in designing plasmonic circuits based on two-dimensional materials and semiconductor quantum dots is considered. Such mathematical approaches as the basis for a system of automated calculations of semiconductor spherical quantum dot parameters are described, along with the finite-difference–time-domain approach for the full-wave electromagnetic simulation near a surface of nanostructured graphene.


INTRODUCTION
The problems of creating and using software for calculating hybrid schemes of resonant interaction between individual emitters and an electromagnetic field are a weak point of all available commercial software tools. Such problems include schemes of interaction in which the quantum emitter (an atom, molecule, quantum dot (QD), or quantum wire (QW)) is subjected to the combined effects of a free electromagnetic field and the localized electromagnetic waves. These waves (surface plasmon polaritons, or SPPs) arise in the waveguide interfaces of materials with strongly differing dielectric constants (metal/dielectric). Their advantage over freely spreading waves is the possibility of spatial localization at scales on the order of nanometers, which notably exceed the wavelength of light. On the other hand, the characteristic frequencies of these excitations can correspond to the terahertz, infrared, or even optical ranges, allowing us to achieve unprecedented rates of response and nanometer scales of devices based on them.
The aim of this work was to develop and use mathematical models for designing plasmonic circuits based on semiconductor quantum dots and nanostructured graphene. A database of characteristics of A3B5 and A2B6 QDs, including the ceiling energies of the valence band, the energies of the bottom of the conduction band, the bandgap width, and other parameters, was thus created. The developed mathematical means allowed automated computation of, e.g., energy structures, frequencies and dipole moments, and rates of relaxation.
The next stage was to develop a finite-differencetime-domain (FDTD) approach for simulating the propagation of SPPs in plasmonic waveguides based on conducting structures as prospective media for the use of interstitial links in plasmonic circuits. A mathematical model based on the FDTD approach was thus obtained for simulation of plasmon-polariton waves in 2D graphene structures. The developed algorithms were tested by comparing the results from this work and data on simulating the distribution of SPPs in graphene schemes [1,2]. There was full agreement between our techniques and commercial software, and practical works by foreign authors based on the original code.

MODEL FOR CALCULATING THE POSITIONS OF ENERGY LEVELS FOR SEMICONDUCTOR QDs WITH SPECIFIC SIZES
The size dependences of a limited number of the energy levels of a hole in the valence band of a QD can be found using the relation [3][4][5][6] (1) where is the ceiling energy of the valence band of a QD (in eV); is charge C of an electron; is the effective mass of the hole; is the QD radius (m); is the Planck constant; is the main quantum number of the hole; is the orbital quantum number for the hole, and denotes the roots of a spherical Bessel equation of the first kind. The size dependences of energy levels of the electron in the conduction band of a QD can be calculated using the relation [5,6] (2) where is the energy of the bottom of the conduction band of a QD (eV); is the effective mass of the electron; is the main quantum number for the electron, is the orbital quantum number of the electron; and denotes the roots of a spherical Bessel equation of the first kind. The frequency of the intraband transition of the hole can be estimated as [7]: (3) where and are set in an arbitrary manner within the custom range and the parameter obeys rule of selection Based on the frequency of transition, the interband transition wavelength is determined as Frequency of the intraband (electronic) transition can be calculated via the relation [7] (4) where and are set in an arbitrary manner within the range of interest and the parameter obeys rule of selection Using the frequency of transition, we can determine the interlevel transition wavelength as The interband frequency of transition is thus defined as [7] (5) The relevant rule of selection is k′ − l = 0, and the wavelength of the appropriate transition has the form According to [8], the dipole moment of the intraband transition in a QD can be found as (6) With a triple integral in spherical coordinates, this parameter can be calculated as where is the dielectric constant of the QD; is the radius vector determining the distance of a charge carrier to the center of the QD; is the azimuth angle; is the polar angle; are the wave functions of initial and final levels of the transition, respectively; is the QD radius; and is the conjugated wave function of the initial level of the transition. According to [9], the wave function for a spherical QD is determined by the expression are spherical Bessel functions of the first order and the l-th and (l + 1)-th orders, respectively; are the Bessel functions of the first order of the (l + ½)-th and (l + 3/2)-th orders, respectively; are spherical functions; is the n-th root of the Bessel function of the l-th order; are the main, orbital, and quantum numbers of the particle, respectively. Substituting formulas (8) for the two different levels in expression (7), we obtain (9) where and are the main, orbital, and the magnetic quantum numbers of the initial and final levels of the transition. The spherical functions in general form are written as (10) where is the azimuth angle; is the solid angle; are the radial functions in the form (11) where are the associated Legendre polynomials [10], expressed as Below, we assume the magnetic quantum number is m = 0. Then The dipole momentum of the interband transition can be estimated using the relation [11] where is the frequency of transition; is the distance between levels and and is the spinorbital splitting. The energies, frequencies, and dipole moments of transitions in the QD were calculated using formulas (1)- (14) combined into a single software module [12]. The energies of interband transition at semiconductor CdSe QD radii from 3 to 6 nm in particular are plotted in Fig. 1. The dependences obtained using the analytical expressions match the one reported in [6]. For QD radii of less than 3 nm, it does not agree with the results from the best approximation in [6]. This is because we must allow for the Coulomb interaction in the strong confinement mode when where is the Bohr radius of an exciton.

MATHEMATICAL MODEL OF GRAPHENE CONDUCTIVITY AND THE FINITE-DIFFERENCE-TIME-DOMAIN APPROACH
The basis for the formation of interstitial bonds in plasmonic circuits can be either metallic and graphene waveguides (the latter would seem to be more promising). Graphene [13] is a 2D material with a hexagonal lattice of carbon atoms one atom thick. A unique feature of such structure is the ability to induce high-temperature conductivity in a pure material [14] and its doped modifications [15], which allows graphene to be considered as the basis for integrated circuits of the future. The benefits of graphite as the basis for plasmonic circuits are its ability to achieve marginal localization of an electromagnetic wave at scales of several nanometers [16]. The electric conductivity of graphene at frequency can be described in the context of the Kubo formalism [17] and is defined as where is the chemical potential, is the speed of charge carrier (electron) scattering, T = 300 K is the temperature, and k is the Boltzmann constant. The first term in Eq. (16) describes the contribution from the intraband conductivity; the second is the result of the interband conductivity. The formation of SPPs on a graphene-dielectric surface is possible if where imaginary part of conductivity moves from positive to negative values. The existence of SPPs at 1.55 μm is thus possible when the chemical potential of graphene is at least 0.5 eV. With conductivity = , the dielectric permittivity of graphene is defined as (17) where is the dielectric permittivity of a dielectric with a graphene sheet, and is the graphene's thickness (usually assumed as ~ 1 nm). The formation of SPPs in a graphene sheet is possible only under condition The relative permittivity of a graphene sheet in a vacuum is determined by the equation [1] ( 18) where and is the dielectric constant of air. At the same time, an SPP wave forming in a single sheet is characterized by the wave vector (19) where At , and a source frequency of , the SPP wavelength is while effective mean free path = is 396 μm.
The formation of SPPs on graphene was simulated using the FDTD method for the Maxwell equations [1] (20a) where and are the relative dielectric constant of the medium and the speed of light, D is the flow density, E is the electric field strength, H is the magnetic field strength, and μ is the magnetic permittivity of the medium. The use of normalized parameters The discretization of the Maxwell equations using the central derivative approach and step where is the spatial dimension (Fig. 2), results in (23a) The formation and propagation of surface electromagnetic waves were preliminarily simulated using the magnetic dipole as a source: The FDTD approach was initially calibrated with respect to the results obtained in [1,2,16]. A full-wave electromagnetic simulation with source (27) was then carried out at wavelengths of 2, 4, and 8 μm for a structure of two graphene sheets. The formation of SPPs in  Fig. 3. For convenience, an online application was developed in which the user's side is represented by the builder of a layout of twodimensional plasmon schemes, while the block for numerical calculation and simulation of the electromagnetic field is on the side of the server. However, the developed approaches and their implementation in the present form do not allow full-scale simulation of the resonant interaction between the QD and the SPPs in graphene; i.e., simulation for several QDs near graphene will be the topic of future studies. At the same time, a version of the application configured for simulating thin films of arbitrary conductive materials allowed us to carry out a full-wave electromagnetic simulation of the formation processes and reveal the features of control of SPPs in planar devices based on nanostructured gold films. In the future, it is planned to extend the existing application for the possibility of three-dimensional simulation and analysis of both individual localized plasmon-polariton structures [18] and entire arrays based on them [19].

CONCLUSIONS
The mathematical models and numerical algorithms needed for designing functional devices based on 2D materials and semiconductor quantum dots were considered. Models for calculating the main optical characteristics of semiconductor A3B5 and A2B6 QDs were developed in particular. The numerical model underlying the finite-difference-timedomain method was obtained and used to simulate the propagation of plasmon-polariton modes in plasmonic structures based on graphene. The correctness of the above algorithms was verified by comparing the results from simulation and different experimental and theoretical works.