=Paper=
{{Paper
|id=Vol-1730/p11
|storemode=property
|title=Numerical Simulations of Optical Multilayer Structure with an Embedded Octahedron Nanocrystals Using a FEM Based Approach
|pdfUrl=https://ceur-ws.org/Vol-1730/p11.pdf
|volume=Vol-1730
|authors=Grazia Lo Sciuto
|dblpUrl=https://dblp.org/rec/conf/system/Sciuto16
}}
==Numerical Simulations of Optical Multilayer Structure with an Embedded Octahedron Nanocrystals Using a FEM Based Approach
==
https://ceur-ws.org/Vol-1730/p11.pdf
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. We show the interactions between EM and
the surface plasmon polaritons known as electromagnetic a-Si nanocrystals located on layer of nanostructure.
69
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