Numerical Simulations of Optical Multilayer Structure with an Embedded Octahedron Nanocrystals Using a FEM Based Approach Grazia Lo Sciuto Department of Engineering Roma Tre University Rome, Italy glosciuto@dii.unict.it Abstract—This theoretical study describes the optical multi- The principal aspects of the fabrication of Si nanocrystals in layer structure models leading to investigate Surface Plasmon thin SiO2 layers with the Si substrate have been considered Polariton (SPP) to an extern applied magnetic field at visible in [3]. The Si nanocrystals are widely used for a number of wavelenght using a 3D-FEM based on Comsol Multiphysics software. The layered structures include the amorphous silicon solid state electronic devices, such as solar cells, solid state c-Si octahedral nanocrystals on the interface SiO2/dielectric were photosensors and thin film transistor for liquid crystal displays. investigated to enhance the SPP intensity. Third-generation photovoltaic devices are realized in silicon Keywords-Nanoplasmonics, FEM, Surface Plasmon Polariton nanocrystals (Si-NCs) embedded in a dielectric matrix [4], [5]. In amorphous materials, the bond lenghts and number vary I. I NTRODUCTION slightly for different atoms in the lattice. The coordination The rapid evolution of electronics and telecommunications of bonds of an atom with its closest neighbours is almost has been possible by the presence of innumerable high the same as in the corresponding crystalline material, but is performance devices based on silicon technology. The new gradually lost with more distant neighbours. Thus amorphous applications of microelectronics, photovoltaics and photonics Si is a direct band semiconductor unlike crystalline Si, and refer to effort to optimize the optical components. On the other has a high absorption coefficient with a deposition process hand the complexity of the junction solar cell stems from the applicable at low temperature. delicate balance that exists between the different properties of The most numerous defect type, crucial for use fo a-Si in the materials used and the geometric structure of the cell itself. solar cells, is a dangling bond decreasing the charge carriers Therefore several parameters affect the solar cell conversion lifetime and mobility. To improve this situation, hydrogen efficiency. Consequently, understanding of interaction between (H) is incorporated into a-Si during fabrication. Hydrogenated the incident EM waves and materials is fundamental to define amorphous silicon a − Si : H solar cells are a low-cost an accurate analysis on effects due to the presence of the alternative to bulk crystalline Si cells, offering a larger ab- nanostructures and electronic equipment. sorption coefficient across the solar radiation spectrum. Thus, The development of nanoplasmonic has been a topic of an a − Si : H film of thickness of 500 nm absorbs sufficient increasing interest in recent years. This recent progress has sunlight to enable efficient solar cell operation, compared to been possible as a result of advances in nanofabrication tech- thicknesses of several tens to hundreds of microns that are nology [1],[2]. Plasmonic nanoparticles are of great interest for required for bulk crystalline Si devices. light trapping in thin-film silicon solar cells, Ag nanoparticles However, the high defect densities typically present in can provide light-trapping performance, through excitation of a−Si : H thin films limit the typical minority carrier diffusion charge carriers, comparable to state-of-the-art random textures lengths to 100 nm3 , consequently, a − Si : H solar cells are in n − i − p amorphous silicon solar cells. The excellent light generally fabricated using even thinner a − Si : H layers, trapping is a result of strong light scattering and low parasitic resulting in reduced absorption of incident solar radiation. absorption of self-assembled Ag nanoparticles embedded in However, the significantly reduced thickness of their silicon the back reflector. Infact, the characteristic length scale of layer makes it more difficult for them to absorb sunlight. the structures necessary to manipulate and generate surface Therefore the capability of measuring the change of phase plasmon polaritons (SPPs) in the visible and near-infrared by the reflection of polarized light on a surface or layer region of the optical spectrum is in the nanometre regime. The structure allows a typical sensitivity of less than one nanometer plasmonic structures are used to increase optical absorption for the layer thickness and for the refractive index. The and the power conversion effeciency in thin-film solar cells. thickness, homogeneity and interface qualities of the layers can be measured directly, whereas the properties related to Copyright c 2016 held by the author. the nanocrystal structure (like the crystallinity, the nanocrystal size or the density of the layer) can be obtained indirectly 65 using proper optical models. From numerical simulation it The distance over which the wave drops to 1/e its original is possible create a relation between the optical properties value is known as the penetration or skin depth. Therefore it is as λspp with thickness of metal in a multilayer structure. the expression for the penetration depth by taking the square The external electric field effects on spectra and decay of root of the expression for k 2 and reinserting it into the plane photoluminescence as well as on absorption spectra were wave solution E ~ = E~0 ei(Re[~k]·~x−ωt) e−Im[~k]·~x . It follows that measured for CdSe nanoparticles in a polymethyl methacrylate 1 the 1/e distance is δp = Im[k] . For the metals as silver, gold film by Takakazu Nakabayashi and al. [6]. and copper the electric field of optical waves falls to 1/e of The major aim of the deposition of amorphous Si nanopar- its initial value in a few nm. ticels on the SiO2 film is to improve solar light harvesting Surface plasmons represent coupling of an electromagnetic and lead to increased efficiencies due to excitation of surface field to the kinetic motion of free charge carriers. Surface plas- plasmons polaritons. In this paper, the excitation of the SPP is mons exist at the boundary between dielectric and conductor. due to the combination of the amorphous silicon nanocrystal Oscillation of surface charge density σ is the source of the related to the presence of SiO2 /air and the exiting external electric fields. A discontinuity of the normal component of electric field source with wavelenght spanning from 300 to 700 the external electric field at the boundary of dielectric and nm. The numerical calculations and simulations for the reso- conductor with dielectric functions εd and εc , respectively: lution of electromagnetic field have been developed to solve Maxwell’s equation with the Finite Element Method (FEM) Ez1 − Ez0 = 4πσ (5) using the commercial software packages Comsol Multiphysics. This theoretical work used a 3D-FEM modeling based on where Ez1 and Ez0 are the normal components of electric Comsol Multiphysics software to investigate the SPP in a mul- field in the conductor and dielectric respectively. The wave tilayer structure of SiO2 interfaces with dielectric substrate function for a traveling charge density wave is: containing the embedded small octahedral amorphous silicon nanocrystals. σ(x, t) = σ0 ei(Kx x−ωt) (6) The multilayer structures have a very important role in Renewable energies integration in electric generation systems Kx is the wave vector along the boundary. The charge to promote their use and then the economic development and oscillations are coupled with external electric field (Ex , Ez ), growth of rural areas and less developed countries have to be which has components normal to the surface and in the considered so as their management problems [7], [8]. propagation direction, and the transverse magnetic field (H). The SPP is a p-polarized electromagnetic wave because its II. T HEORY: M ODEL D RUDE FOR S URFACE P LASMON electric field vector E lies in the plane (x, z) defined by the P OLARITON surface normal and the propagation vector while the magnetic Metals have different optical properties as dielectric func- field vector H is perpendicular to this plane. The wave function tion compared to semiconductors and dielectric due to their for the normal component of the electric field is electronic band structure. In the Drude model Maxwells equations describe an electromagnetic wave in a medium of Ez = A ei(Kx x+Kz z−ωt) (7) conductivity σ, and net zero charge. where Kz is mostly imaginary. This causes exponential decay from interface, making SPPs evanescent waves. The energy ρf ree density can exceed that of the incident radiation that excites O·E =  the SPP. The wave vectors Kx and Kz are related according ∂B to the following: O×E =− ∂t O·B =0 (1) Kx2 + Kzd,zc 2 = εd,c (ω/c) 2 (8) ∂J O×B =µ + µE where εd,c is the complex dielectric function of the dielectric ∂t In the absence of external charge and current densities, the or conductor, respectively. The dispersion relation for the non- curl equations can be combined to yield the wave equation: radiative SPP mode can be derived by applying Maxwells equations together with the continuity conditions for E and O2 E + k02 E = 0 (2) H. For p-polarized oscillations (Ey , Hx = Hz = 0), boundary conditions yield, obtaining the surface plasmon wave vector This equation has plane wave solutions with complex wave- r vectors εd εc KSP P (ω) = (ω/c) (9) εd + εc ~ = E~0 ei(~k·~x−ωt) E (3) which describe propagation and damping, respectively. Ac-   cording to the Drude model for metals, the electrons in a 2 2 2 iσω conductor behave like an ideal gas, and the real part of the k = µεω + iµσω = µε ω + (4) ε dielectric constant for a conductor varies as 66 ωp2 εc (ω) = 1 − (10) ω2 where the plasma frequency is: N e2 ωp2 = (11) mo N is free electron density, m and e, the mass and charge of electron, and 0 the permittivity of free space. The field intensity of SPPs also decreases exponentially both in con- ductor and dielectric. The dispersion relationship using this approximation elucidates an important physical phenomena. Below the plasma frequency is imaginary and waves are attenuated as they enter the metal. Above the plasma frequency becomes real allowing for traveling waves. These traveling waves are waves in the electron plasma and at the plasma frequency they are completely longitudinal. The quantum of a plasma oscillation is known as a plasmon. Surface plasmon polaritons (SPP’s), can occur at any interface. The basic structure of a propagating SPP is of an evanescent wave, decaying exponentially in intensity normal to the interface, and oscillating in the direction of propagation. Traditionally, Fig. 1. 3D Octahedron of amorphous silicon (a-Si) nanocrystals modeling to find the solutions that constitute confined surface waves one assumes that a wave exists at the boundary and solves for the appropriate boundary conditions. The wavelength of the SPP is defined by: 2π λSP P = (12) Re[KSP P ] LSP P is the SPP propagation length, physically the energy dissipated through the metal heating and it is the propagation distance. Where: 1 LSP P = (13) Im[KSP P ] III. M ODEL DEVELOPMENT This theoretical work used a 3D-FEM modeling based on Comsol Multiphysics software to investigate the SPP in a multilayer structure of SiO2 interfaces with dielectric (air) sub- strate containing the embedded small octahedral amorphous silicon nanocrystals. Fig. 2. 3D Multilayer structure with embedded amorphous silicon (a-Si) The numerical procedure for FEM allows the approximate nanocrystals modeling solutions of partial differential equations (PDE) over a model with specified boundary conditions. It is thereby a procedure that may be used to solve many different kind of problems centered, deposited and embedded on the SiO2 surface (inter- in physics. A three-dimensional model of the structure is face) of multilayer structure. The modes of SPP excitation created and constraints and parameters are applied on each depends on the size and shape of a-Si nanoparticles. The subdomain and boundary, defining the necessary expressions nanocrystals mostly have an lenght octahedral shape of 140 for the incident wave, and setting the optical properties of the nm with optical property. Octahedron has a pyramid on the different domain. The finished models are then exported to the top and a pyramid on the bottom, it is a square bipyramid program Comsol Multiphysics chosen as environment for the in any of three orthogonal orientations. The SiO2/dielectric application tool. Thus, parasolid geometries are modified in multilayer structure is shown in the fig. 2 COMSOL Multiphysics to add the original CAD design the The thickness of the dielectric/SiO2 is fixed at 300nm. The external physical effects to simulate such as the optical effects. dimension of the domain is 600 nm x 600 nm. The interesting model consists of octahedrons nanoparticles In the system the incoming electric field is a TM-polarized 67 Parameter Value Unit εair 8.8590e-12 F/m nair 1 nSiO2 1.52 kSiO2 1e-5 λ Varies from 300 to 700 nm plane wave, in order to excite the SPP associated to the nanoparticels. It was considered the contribution of surface plasmon-polariton in the model with two different layers, air and the insulator SiO2 (Silicon dioxide) substrate, where are embedded the amorphous silicon’s nanocrystals (a-Si) in the centre of structure.Amorphous silicon has distinct ad- vantages such as high refractive index, low absorption loss at telecommunication wavelengths of 1550 nm , capability of low-temperature (200-400C) plasma-enhanced chemical vapor deposition (PECVD) on almost any substrates, and even possibility for active modulation and detection. Recently, Fig. 3. Extinction coefficient of a-Si nanocrystals it has emerged as an important material for integrated Si photonics. A very low propagation loss of 2 − 3dB/cm at 1550 nm has been reported fora − Si : H wire waveguides, which is comparable to the crystalline Si counterparts with the same dimensions.For this work the refractive index and the extinction coefficient of SiO2 are respectively 1.52 and 1e − 5 as shown in the below table the optical and electrical data used for the SPP analysis: The a − Si nanocrystals have excellent optical and electric properties, including a high index refractive and extinction coefficient at different wavelenghts in and near the visible part of the spectrum. It is very common to find the description of the optical properties of solids in terms of the index of refraction √ n. The general relationship of n and epsilon is n = ε. When the radiation passes through a medium, some part of it will always be attenuated,taken into account the complex refractive index: n = n + ik where the real part n is the refractive index and indicates the phase velocity, while the imaginary part k is the ”extinction coefficient”. The Fig. 4. Refractive index of a-Si nanocrystals refractive index of a−Si nanocrystals, as in all cases,decreased monotonically with increasing wavelength: These surface polaritons are induced and generated by confirm the role of the octahedron photonic crystal in the electromagnetic radiation emitting in the visible region of the coupling mechanism as shown in the Fig. 5. The information spectrum, using different wavelenghts. The boundary condition about the SPP has been established by simulations for the chosen for the surface of dielectric and SiO2 is perfectly interaction the nanocrystals with the electromagnetic field. matched layer (PML) , meaning that is an artificial absorbing layer for wave equations. The PML is used to limit the IV. C ALCULATION AND R ESULTS reflections from kind of open, free-space, boundaries. For We have investigated the excitation of the SPP in an boundaries the conditions for the perfect electric conductor extension of the simple metal surface at visible frequencies in is given by: a-Si nanocrystals in two layer system.The method of analyses for SPP’s is essentially the same of heterostructure or single n̂ × E = 0 (14) flat surface , however, because of the additional interface the In model the Cartesian axis system are chosen in such way dispersion relation becomes more complex.It is possible to that the z-axis is normal to the xy plane of the layers. The excite surface polaritons of amorphous Si (a-Si) at frequencies electromagnetic wave is assumed normal to the xy layer. The at which a-Si has a positive real component of the permittivity surface polariton modes are found at the interface SiO2/air and a large imaginary component. These modes on films of a- where one of the face of amorphous silicons nanocystals sits Si have similar or even superior characteristics to those on gold on interface of the substrate SiO2/air. Numerical simulations films, with longer propagation lengths and similar confinement 68 -28 x 10 6.5 Electric Field E=(0,1,0) 6 5.5 5 Magnetic Field 4.5 4 3.5 3 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 Wavelength (nm) -7 x 10 Fig. 6. Magnetic Field vs wavelenght calculated in d0 (see Fig. 5) d3 d4 d5 Fig. 5. Multilayer structure with amorphous silicon (a-Si) nanocrystals d6 to the thin film. The result obtained from FEM calculations are the complex description of the fields in terms of Hz . The magnetic field decreases when the wavelenght is in- creased in the range from 300 to 700 nm. Carefully studying the magnetic field can reveal that the peak is shifting to lower wavelenght, maximum energy is at 300 nm at different magnetic field calculated on the interface nanocrystals/SiO2. Systematic work has illustrated considering the interaction of the SPP with the nanocrystal on interface of SiO2. Surface Plasmons are a result of the mutual-coupling be- Fig. 7. Magnetic Field vs wavelenght calculated in d1, d2, d3, d4, d5, d6 (see Fig. 5) tween photons and collectively oscillating electrons at the dielectric/amorphuos silicon interface. The electromagnetic energy of the surface plasmons are nicely confined in the vicin- ity of the interface.It is calculated the magnetic field across (EM) waves that propagate along metallic nano-structure. different directions respect to the nanocrystals to provide the The materials and technologies play an important role in the interactions between the nanocrystals and SiO2 surface. This development of plasmonics. approach is consisted to calculate and relate the magnetic field at different wavelenght and directions on surface SiO2. The For this reason, in the literature there are a large variety of electromagnetic waves of SPP are detected travelling along optimization techniques by using soft computing techniques vertical, horizontal and diagonal directions of nanocrystals [9] in order to improve the conversion efficiency of solar cells positions as function of different wavelenghts throught the [10], [11], [12], [13]. interface SiO2/dielectric as shown in the Fig. 6 and Fig. 7. This study describes the implementation and development The intensity of SPP is influenced by nanoparticels distance of the application tool Comsol Multiphysics for the simulation and disposition. Applying the magnetic field, both the real and of the mechanisms and the interaction of the SPP with the a- the imaginary part of the SPP wavevector KSP P are modified. Si nanocrystals on interface of SiO2/dielectric. To obtain a high efficiency the models are initially created within Comsol V. C ONCLUSION Multiphysics where it is easy to create a geometry, apply The plasmonic applications represent an interesting attrac- boundary conditions, define the necessary expressions, e.g. tion due to the confinement of signals and light in structures for the incident wave, and set the optical properties of the with the possibility to work in the optical near field, localizing different domains. 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