In this paper the authors optimize the geometry of
a solar cell with metallic nanodisks by employing a finite element
code to solve the light scattering problem from the cell and
genetic algorithm optimization

Aiello, Giovanni

Alfonzetti, Salvatore

Chiarello, Viviana

Salerno, Nunzio

DSpace at University of West Bohemia

I-53Solar Cell Optimization  by means of Metallic Nanodisks  G. Aiello,  S. Alfonzetti, V. Chiarello and  N. Salerno Dip. di Ingegneria Elettrica, Elettronica e Informatica, Università di Catania,  Viale A. Doria, 6, Catania, Italy   Abstract— In this paper the authors optimize the geometry of a solar cell with metallic nanodisks by employing a finite element code to solve the light scattering problem from the cell and genetic algorithm optimization. Index Terms— Photovoltaic cells, Nanoparticles, Genetic Algorithms, Scattering, Computational electromagnetics, Finite element methods. I. INTRODUCTION Recently several researchers have demonstrated that the insertion of metallic nanoparticles within or near PV layers may improve significantly the efficiency of the solar cells. In this paper the authors optimize the geometry of a thin solar cell by employing an FEM code to solve the light scattering problem from the cell and GA (genetic algorithm) optimization.  II. FEM ANALYSIS OF LIGHT SCATTERING Consider the thin film solar cell depicted in Fig. 1, in which metallic nanodisks, having radius R and high h, are placed near a PV layer, having a thickness t. The particles centers are regularly placed at the nodes of a rectangular grid exhibiting the same grid step 2d along the x- and y-axis. To simplify the optimization the particles are assumed to have a given volume V=πr2h, so that only one degree of freedom specifies their shape. A monochromatic electromagnetic plane wave at optical frequency is incident normally on this system. For the sake of simplicity we assume that the wave is E-polarized along the x-axis.  For this electromagnetic scattering problem the Helmholtz vector equation holds:        ( ) 0EkE r201r =ε−×∇µ×∇ −             (1)  where µr and εr are the relative magnetic permeability and electrical permittivity, respectively and k0 is the free-space wavenumber, given by:         000k µεω= ,           (2)  with ω  being the angular frequency and µ0 and ε0 the free-space permeability and permittivity, respectively.  2r     t4d  h  Fig. 1.  Solar cell with metallic nanodisk.    For symmetry reasons the analysis can be restricted to the domain 0<x<d, 0<y<d, by imposing homogeneous Dirichlet boundary conditions on the x=0 and x=d planes and homogeneous Neumann ones on the y=0 and y=d planes. On the z direction the domain is truncated by means of two PML layers, one over and one under the cell. The analysis domain is discretized by means of  tetrahedral edge finite elements. At optical frequencies the metallic nanoparticles give rise to plasmons oscillations, which are taken into account by modeling the metal by means of a complex relative electric permittivity (Drude model) [1], given by:      )(j)(222p22222pr ν+ωωων−ν+ων+ω−ω−=ε          (3)  where ωp is the plasma frequency of the free electrons and ν is the relaxation frequency, the values of which are experimentally determined [2]. Note that at the optical frequencies of interest, the real part of the relative electric permittivity is negative. Assuming a unitary relative magnetic permeability, the FEM analysis is easily performed [3, 4]. In postprocessing the following non dimensional quantity is computed:        2incjoidSWWW)t,d,r(f−−=              (4)  where Wi is the flux of the Poynting vector through a square surface over the cell, Wo is the same flux through a square surface under the cell, Wj is the Joule loss in the metallic particle and Sinc is the Poynting vector module of the incident wave. I-54III. OPTIMIZATION BY GENETIC ALGORITHMS By employing the quantity in (4) as objective function to be maximized, a stochastic optimization is started by assuming the following data:  Metal of the nanoparticles:   aurum Nanoparticle volume   V = 25000 nm3 Light Wavelengtht:   λ = 900 nm PV layer permittivity:   εr = 9 PV layer permeability:  µr = 1   The geometrical parameters to be optimized are assumed to vary in the following ranges:        25 nm  < r <   90 nm          100 nm  < d < 500 nm         5 nm  < t < 100 nm   The stochastic optimization was pursued by means of Genetic Algorithms (GAs), which are search algorithms which simulate the random evolution of populations of biological entities [5, 6]. These algorithms have proven to be very efficient in optimizing electromagnetic devices [7, 8] both for low and high frequency applications.  In this paper GAs have been used with the following characteristics [9, 10]:   Population size:  P=80 Number of generations:  Ng=30 Representation:  binary (not Gray encoding) Binary lengths:  Na=6, Nd=8, Nt=6          for a total of N=20 bits Selection:  tournament selection with elitism Crossover type: two-point crossover Crossover probability at generation k:       )1Ng/()1k(4.03.0Pck −−+=  Mutation probability at generation k:       )1Ng/()1k(04.005.0Pmk −−−=    As far as representation is concerned, the three geometric variables a, d, and t, were coded into binary strings of six, eight and six bits, respectively, giving a total of N=20 bits. The reproduction process, which randomly creates a new generation from the old one, was chosen by tournament selection with a shuffling technique to choose random pairs for mating. The crossover process, by means of which individuals exchange chromosomes from one generation to the other, was two-point crossover with a probability Pc linearly varying from 0.3 to 0.7 while optimization goes on.  The mutation process, by means of which some random flips in the chromosomes of an individual are made, was employed with a probability Pm linearly decreasing from 0.05 to 0.01 while optimization proceeds. The evolution was halted after 30 generations, reaching an optimal function value fmax= 0,51, in relation to the following parameter configuration:       nm10t,nm115d,nm41r ===   The computations were performed by means of ELFIN, a large FEM code developed by the authors for electromagnetic CAD research [11].  More details and example will be provided in the full paper.   ACKNOWLEDGEMENTS  This work has been partially supported by the MIUR (the Italian Ministry for University and Scientific Research).  REFERENCES  [1] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, 1983 [2] P.B. Johnson and R. W. Cristy, “Optical constants of the noble metals”, Phys. Rev. B, vol 6, pp. 4370-4379, 1972. [3] S. Alfonzetti, G. Borzì, "Finite element solution to electromagnetic scattering problems by means of the Robin boundary condition iteration method," IEEE Transactions on Antennas and Propagations, vol. 50, n. 2, pp. 132-140. February 2002. [4] G. Aiello, S. Alfonzetti, V. Brancaforte, V. Chiarello, N. Salerno, "Applying FEM-RBCI to the analysis of plasmons in metallic nanoparticles", International Journal of Applied Electromagnetics and Mechanics (IJAEM), vol. 39, n. 1-4, pp. 13-20, 2012. [5] J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975.  [6] D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning, Addison Wesley, 1989. [7] S. Russenschuck, "Synthesis, inverse problems and optimization in computational electromagnetics," Int. J. Numer. Modelling: Electronic Networks, Devices and Fields, vol. 9, pp. 45-57, January-April 1996. [8] P. G. Alotto, et alii, "Stochastic algorithms in electromagnetic optimization," IEEE Transactions on Magnetics, vol. 34, pp. 3674-3684, Sept. 1998.  [9] S. Alfonzetti, E. Dilettoso, N. Salerno, "A proposal for a universal parameter configuration for genetic algorithm optimization of electromagnetic devices," IEEE Transactions on Magnetics, vol. 37, n. 5, pp.3208-3211, September 2001. [10] E. Dilettoso, N. Salerno, "A self-adaptive niching genetic algorithm for multimodal optimization of electromagnetic devices", IEEE Transactions on Magnetics, vol. 42, n. 4, pp. 1203-1206, April 2006. [11] G. Aiello, S. Alfonzetti, G. Borzì, and N. Salerno, "An overview of the ELFIN code for finite element research in electrical engineering", in Software for Electrical Engineering Analysis and Design, A Konrad & C. A. Brebbia (ed.), WIT Press, Southampton, 1999.  

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Solar Cell Optimization by means of Metallic Nanodisks

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Solar Cell Optimization by means of Metallic Nanodisks

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