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A performance analysis of planar and radial pn junction based photovoltaic solar cells Ska-Hiish, Manuel Dave Hokororo 2013

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A Performance Analysis of Planar and Radial pn JunctionBased Photovoltaic Solar CellsbySka-Hiish Dave Hokororo ManuelB. Business Administration, Thompson Rivers University, 2006B. Applied Science, The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)October 2013? Ska-Hiish Dave Hokororo Manuel, 2013AbstractIn this thesis, the performance of both planar and radial pn junction based pho-tovoltaic solar cells are examined for a broad range of materials. The materialsconsidered include silicon, gallium arsenide, germaniun, indium nitride, and gal-lium nitride. The photovoltaic solar cell performance model of Kayes et al. [B.M.Kayes, H.A. Atwater, and N.S. Lewis, Journal of Applied Physics, volume 97, pp.14302-1-11, 2005], is employed for the purposes of this analysis. Three solar cellperformance metrics, evaluated using the et al., model are considered in this anal-ysis: (1) the short-circuit current density, (2) the open-circuit voltage, and (3) theefficiency. The results suggest that while planar pn junction based photovoltaic so-lar cells are sensitive to trapping concentration levels, the radial pn junction basedphotovoltaic solar cells are relatively insensitive to trapping concentrations. Thissuggests that in certain cases, such as when there are materials with high con-centration of traps, radial pn junction based photovoltaic solar cell offer inherentadvantages over their planar counterparts.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The air-mass spectrum . . . . . . . . . . . . . . . . . . . . . . . 92.3 The operation of a pn junction solar cell . . . . . . . . . . . . . . 122.4 Planar and radial photovoltaic solar cell configurations . . . . . . 15Chapter 3: Planar and Radial Photovoltaic Solar Cell PerformanceModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17iii3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Key assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Planar photovoltaic solar cell current density model . . . . . . . . 193.3.1 The diffusion length in the p-type region . . . . . . . . . . 233.3.2 The diffusion length in the n-type region . . . . . . . . . . 243.3.3 The depletion region . . . . . . . . . . . . . . . . . . . . 263.4 Radial photovoltaic solar cell current density model . . . . . . . . 273.4.1 Diffusion length in the p-type region . . . . . . . . . . . . 293.4.2 Diffusion length in the n-type region . . . . . . . . . . . . 323.4.3 The depletion region . . . . . . . . . . . . . . . . . . . . 333.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Chapter 4: Solar Cell Material Parameters . . . . . . . . . . . . . . . 364.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 The spectral dependence of the optical absorption coefficient . . . 394.2.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.2 Gallium arsenide . . . . . . . . . . . . . . . . . . . . . . 504.2.3 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.4 Indium nitride . . . . . . . . . . . . . . . . . . . . . . . 544.2.5 Gallium nitride . . . . . . . . . . . . . . . . . . . . . . . 564.3 Recombination velocities . . . . . . . . . . . . . . . . . . . . . . 584.4 Other semiconductor properties . . . . . . . . . . . . . . . . . . . 604.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Chapter 5: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63iv5.2 Current density, open-circuit voltage, and efficiency . . . . . . . . 645.2.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.2 Gallium arsenide . . . . . . . . . . . . . . . . . . . . . . 735.2.3 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . 815.2.4 Indium nitride . . . . . . . . . . . . . . . . . . . . . . . 895.2.5 Gallium nitride . . . . . . . . . . . . . . . . . . . . . . . 975.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Chapter 6: Conclusions and Possible Future Work . . . . . . . . . . . 107References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Appendix A: Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . 113A.1 Radial P-N Junction . . . . . . . . . . . . . . . . . . . . . . . . . 113A.2 P-N Junction Depletion Region . . . . . . . . . . . . . . . . . . . 125A.3 P-N Junction Depletion . . . . . . . . . . . . . . . . . . . . . . . 127A.4 Planar P-N Junction . . . . . . . . . . . . . . . . . . . . . . . . . 128A.5 Silicon Absorption Depth Class . . . . . . . . . . . . . . . . . . . 140A.6 Silicon Absorption Depth Implementation . . . . . . . . . . . . . 146vList of TablesTable 4.1 Parameters used in the model of Geist [14] in order to determinethe spectral dependence of the extinction coefficient associatedwith c-Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Table 4.2 Parameters used in the model of Adachi [15] in order to deter-mine spectral dependence of the real and imaginary parts of thedielectric function associated with c-Si. . . . . . . . . . . . . . 44Table 4.3 The c-Si optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental resultsof Palik [16]. R is the correlation coefficient. The coefficientsare dimensionless. . . . . . . . . . . . . . . . . . . . . . . . . 48Table 4.4 The GaAs optical absorption polynomial coefficients, determinedfrom polynomial least-squares fit to the experimental results ofPalik [16]. R is the correlation coefficient. The coefficients aredimensionless. . . . . . . . . . . . . . . . . . . . . . . . . . . 50viTable 4.5 The Ge optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental resultsof Palik [16]. R is the correlation coefficient. The coefficientsare dimensionless. . . . . . . . . . . . . . . . . . . . . . . . . 52Table 4.6 The InN optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental resultsof Bhuiyan [17]. R is the correlation coefficient. The coeffi-cients are dimensionless. . . . . . . . . . . . . . . . . . . . . 54Table 4.7 The GaN optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental resultsof Muth [18]. R is the correlation coefficient. The coefficientsare dimensionless. . . . . . . . . . . . . . . . . . . . . . . . . 56Table 4.8 Other relevant semiconductor material properties associated withSi, GaAs, Ge, InN, and GaN based photovoltaic solar cells. . . 60Table 5.1 The semiconductor cell thickness and diffusion length geome-tries for planar Si, GaAs, Ge, InN, and GaN based photovoltaicsolar cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Table 5.2 The cell thickness and diffusion length geometries for radial Si,GaAs, Ge, InN, and GaN based photovoltaic solar cells. . . . . 105Table 5.3 The cell thickness and diffusion length geometries for planarand radial Si, GaAs, Ge, InN, and GaN based photovoltaic solarcells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106viiList of FiguresFigure 1.1 Global electrical energy generation by energy source. . . . . . 2Figure 1.2 NREL?s best photovoltaic solar cell efficiency chart, as of 2013. 5Figure 2.1 The AM0 and AM1.5G solar spectral irradiance distributions. 10Figure 2.2 The pn junction based photovoltaic solar cell with the p-typeregion on the left and the n-type region on the right. In theplanar mode, the photon flux enters from the right to the left.In the radial mode, the photon flux is from the top to the bottom. 13Figure 2.3 A representative cross section of a planar pn junction basedphotovoltaic solar cell and a radial photovoltaic solar cell, withthe n-type region, the p-type region, and the photon flux direc-tions clearly indicated. For the planar case, the photon fluxis parallel to the diffusion path, while for the radial case, thediffusion path is orthogonal to the photon flux. . . . . . . . . 16Figure 3.1 The planar pn junction based phovotovotlaic solar cell dimen-sions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22viiiFigure 3.2 The radial pn junction based phovotovotlaic solar cell dimen-sions with the photon flux direction indicated. . . . . . . . . . 30Figure 4.1 The intensity of the normally incident beam of light as a func-tion of the penetration depth, x, within in a homogeneous ma-terial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 4.2 The spectral dependence of the extinction coefficient, Eqs. (4.3),associated with c-Si as a function of the wavelength, in nanome-tres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 4.3 The real and imaginary parts of the dielectric function associ-ated with c-Si as a function of the photon energy using Eqs.(4.7) through to (4.15). . . . . . . . . . . . . . . . . . . . . . 46Figure 4.4 The spectral dependence of the optical absorption coefficientassociated with c-Si. The empirical models of Geist [14] andAdachi [15] are employed. Experimental data from Palik [16]is also depcited. There is a discontinuity at 3.0 eV, as there is adiscontinuity between the two empirical models. The one be-low 3.0 eV is from Geist [14] and the one above 3.0 eV is fromAdachi [15]. The implemented equations fit the experimentaldata set rather well. . . . . . . . . . . . . . . . . . . . . . . . 47Figure 4.5 The optical absorption coefficient associated with c-Si. Thepolynomial fit, and the experimental data from Palik [16], aredepicted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49ixFigure 4.6 The optical absorption coefficient associated with GaAs. Thepolynomial fit and the experimental data from Palik [16] aredepicted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.7 The optical absorption coefficient associated with Ge. Thepolynomial fit and the experimental data from Palik [16] aredepicted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.8 The optical absorption coefficient associated with InN. Thepolynomial fit and the experimental data from Bhuiyan [17]are depicted . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.9 The optical absorption coefficient associated with GaN. Thepolynomial fit and the experimental data from Muth [18] aredepicted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 4.10 The photon flux of the AM1.5G and the fitted spectral depen-dencies associated with the optical absorption coefficient asso-ciated with Si, GaAs, Ge, InN, and GaN. . . . . . . . . . . . 62Figure 5.1 The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-Siphotovoltaic solar cell in the planar configuration. . . . . . . . 66Figure 5.2 The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the diffu-sion length, for a pn junction based c-Si photovoltaic solar cellin the radial configuration. . . . . . . . . . . . . . . . . . . . 67xFigure 5.3 The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-Si photo-voltaic solar cell in the planar configuration. . . . . . . . . . . 69Figure 5.4 The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length,for a pn junction based c-Si photovoltaic solar cell in the radialconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 5.5 The efficiency as a function of the device thickness and thediffusion length for a pn junction based c-Si photovoltaic solarcell in the planar configuration. . . . . . . . . . . . . . . . . . 71Figure 5.6 The efficiency as a function of the device thickness and theradius, where the radius is equal to the diffusion length, fora pn junction based c-Si photovoltaic solar cell in the radialconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 5.7 The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-GaAs photovoltaic solar cell in the planar configuration. . . . 74Figure 5.8 The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the diffu-sion length, for a pn junction based c-GaAs photovoltaic solarcell in the radial configuration. . . . . . . . . . . . . . . . . . 75Figure 5.9 The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-GaAs photo-voltaic solar cell in the planar configuration. . . . . . . . . . . 77xiFigure 5.10 The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length,for a pn junction based c-GaAs photovoltaic solar cell in theradial configuration. . . . . . . . . . . . . . . . . . . . . . . 78Figure 5.11 The efficiency as a function of the device thickness and thediffusion length for a pn junction based c-GaAs photovoltaicsolar cell in the planar configuration. . . . . . . . . . . . . . . 79Figure 5.12 The efficiency as a function of the device thickness and theradius, where the radius is equal to the diffusion length, for apn junction based c-GaAs photovoltaic solar cell in the radialconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.13 The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-Gephotovoltaic solar cell in the planar configuration. . . . . . . . 82Figure 5.14 The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the dif-fusion length, for a pn junction based c-Ge photovoltaic solarcell in the radial configuration. . . . . . . . . . . . . . . . . . 83Figure 5.15 The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-Ge photo-voltaic solar cell in the planar configuration. . . . . . . . . . . 85Figure 5.16 The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length,for a pn junction based c-Ge photovoltaic solar cell in the radialconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 86xiiFigure 5.17 The efficiency as a function of the device thickness and thediffusion length for a pn junction based c-Ge photovoltaic solarcell in the planar configuration. . . . . . . . . . . . . . . . . . 87Figure 5.18 The efficiency as a function of the device thickness and theradius, where the radius is equal to the diffusion length, fora pn junction based c-Ge photovoltaic solar cell in the radialconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 5.19 The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-InN photovoltaic solar cell in the planar configuration. . . . . 90Figure 5.20 The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the dif-fusion length, for a pn junction based c-InN photovoltaic solarcell in the radial configuration. . . . . . . . . . . . . . . . . . 91Figure 5.21 The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-InN photo-voltaic solar cell in the planar configuration. . . . . . . . . . . 93Figure 5.22 The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length,for a pn junction based c-InN photovoltaic solar cell in the ra-dial configuration. . . . . . . . . . . . . . . . . . . . . . . . 94Figure 5.23 The efficiency as a function of the device thickness and thediffusion length for a pn junction based c-InN photovoltaic so-lar cell in the planar configuration. The donor and acceptorconcentration are too high to make this high efficiency realistic. 95xiiiFigure 5.24 The efficiency as a function of the device thickness and theradius, where the radius is equal to the diffusion length, fora pn junction based c-InN photovoltaic solar cell in the radialconfiguration. The donor and acceptor concentration are toohigh to make this high efficiency realistic. . . . . . . . . . . . 96Figure 5.25 The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-GaN photovoltaic solar cell in the planar configuration. . . . . 98Figure 5.26 The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the diffu-sion length, for a pn junction based c-GaN photovoltaic solarcell in the radial configuration. . . . . . . . . . . . . . . . . . 99Figure 5.27 The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-GaN photo-voltaic solar cell in the planar configuration. . . . . . . . . . . 101Figure 5.28 The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length,for a pn junction based c-GaN photovoltaic solar cell in theradial configuration. . . . . . . . . . . . . . . . . . . . . . . 102Figure 5.29 The efficiency as a function of the device thickness and thediffusion length for a pn junction based c-GaN photovoltaicsolar cell in the planar configuration. . . . . . . . . . . . . . . 103xivFigure 5.30 The efficiency as a function of the device thickness and theradius, where the radius is equal to the diffusion length, for apn junction based c-GaN photovoltaic solar cell in the radialconfiguration. . . . . . . . . . . . . . . . . . . . . . . . . . . 104xvNotationThis notation covers the list of symbols used in this document.Jn electron current densityJp hole current densityJdr depletion region current densityJsc short-circuit current densityVoc open-circuit voltage? efficiency?n life-time of the minority carrier for the electrons injected into the p-typeregion?p life-time of the minority carrier for the holes injected into the n-typeregionLn p-type region electron diffusion lengthLp n-type region hole diffusion lengthxvinp electron concentration in the p-type regionNa acceptor concentrationNd donor concentrationxviiGlossaryASTM American Society for Testing and MaterialsEHP electron-hole pairsAM air-mass spectrumNREL National Renewable Energy LaboratoryNCPV National Center for PhotovoltaicsDOE Department of EnergySi siliconGe germaniunGaAs gallium arsenideInN indium nitrideGaN gallium nitridexviiiAcknowledgmentsFirst, I would like to thank my academic supervisor, Dr. Stephen O?Leary,for the technical guidance, dedicated support, and endless patience. Without hissupport, this work would not be possible. I would also like to thank Dr. KennethChau and Dr. Jonathan Holzman for serving on the committee. Their valuable timeand constructive comments are appreciated.I would like to thank the Neskonlith Indian Band for the financial support dur-ing my years of studies. Special thanks to my wife, son, parents, family, andthe Secwepemc people for their continual support and unwavering love during myyears of education.xixChapter 1IntroductionIn the 20th Century, fossil fuels served as the dominant source of electricalenergy. Unfortunately, this has resulted in diminished supplies and environmentaldegradation. Electrical energy may be generated through hydroelectric, nuclear,coal, oil, natural gas, and renewable energy sources. Figure 1.1 illustrates theglobal annual electrical energy generation, from 1971 through to 2009, from coaland peat, oil, natural gas, nuclear, hydroelectric, and other energy sources. Dueto the increase in the demand for electrical heating in developed countries, andelectrification programs in traditionally rural areas, the global demand for electricalenergy has increased, on average, by 3.6% per year, from 1971 through to 2009 [1].It is clear that the demand for electrical energy will continue to increase in thecoming years.The future demand for electrical energy can be met through: (1) efforts atimproving efficiencies and conservation, and (2) increased electrical energy gen-eration. Many forms of electrical energy generation lead to social, economic, andenvironmental side-effects. In particular, fossil fuel supplied electrical energy gen-1CHAPTER 1. INTRODUCTIONFigure 1.1: Global electrical energy generation by energy source.Source: Organization for Economic Co-operation and Development [1],2CHAPTER 1. INTRODUCTIONeration leads to green house gas (GHG) emissions. Currently, the Earth?s atmo-sphere is believed to have the highest concentration of GHGs for the past threemillion years, with 400 parts per million of GHGs; the pre-industrialization (1750)GHG concentration is suspected to be around 280 parts per million [2, 3]. In ad-dition, fossil fuels are a non-renewable energy source, and depleted supplies areleading to increased costs. This has led people to look for other sources of electri-cal energyRenewable forms of electrical energy are expected to contribute significantly tofuture electrical energy supplies. Renewable energy generation is site specific, andmany different types of renewable forms of energy will be required in the future.Currently used renewable energy sources include wind, solar, tidal, geothermal,and hydroelectric. Many such sources of electrical energy in use today are in actualfact indirect forms of solar energy: (1) hydroelectric generated electrical energy isderived from solar evaporation, (2) fossil fuels are an aggregation of energy fromthe Sun over millions of years, and (3) wind is created through solar atmosphericheating. This study concentrates on the direct conversion of energy from the Sun?sphotons into electrical energy through the use of pn junction based photovoltaicsolar cells.At present, there are four basic types of photovoltaic solar cells that are inuse: (1) single-junction crystalline, (2) multi-junction crystalline, (3) thin-film, and(4) emerging. While crystalline silicon (c-Si) based photovoltaic solar cells offerhigher efficiencies, they are expensive to manufacture, as they are made out of c-Siwafers. Unfortunately, inexpensive materials for photovoltaics, such as thin-filmsilicon (Si), possess relatively high concentrations of impurities, leading to shortdiffusion lengths. A short diffusion length reduces the absorption depth over which3CHAPTER 1. INTRODUCTIONsolar generated photons may contribute to the photoelectric effect. An added com-plication is the fact that the diffusion length decreases as the temperature increases.This has motivated researchers to consider new types of electronic materials andnew types of geometric configurations, beyond the traditional planar pn junctionbased photovoltaic solar cell configuration. This thesis explores how the perfor-mance of a pn junction based photovotlaic solar cell is shaped by the selection ofelectronic material employed and by the selection of solar cell geometry utilized.Before investigating further, a brief historical overview of the development ofthe photovoltaic solar cell is provided. Becquerel is credited with discovering thephotovoltaic effect in 1839 [4]. The first c-Si based photovoltaic solar cell wasdeveloped by Chapin et al. at Bell Telephone Laboratories in 1954 [?]. The firstmodern use of photovoltaic solar cells was to provide electrical energy for spacevehicles and satellite communications systems in the 1950s [5]. In the 1970s, itwas recognized that photovoltaic solar cells may also be considered for residentialand commercial applications. The developments that have occurred since that timeinclude efficiency increases, cost reductions, and the use of alternative electronicmaterials and geometric configurations. Much of the work that has occurred sincethe 1990s was prompted by the Kyoto Protocol [6].The National Center for Photovoltaics (NCPV), an organization hosted by theNational Renewable Energy Laboratory (NREL) in Golden, Colorado, is a UnitedStates Department of Energy (DOE) initiative aimed at helping the United Statesbased photovoltaics industry [7]. The NCPV is a central resource for United Statesresearch, development, deployment, and outreach related to photovoltaic technol-ogy. The NCPV maintains a record of research cell efficiencies, and the latestphotovoltaic solar cell efficiency chart, published by NREL in 2013, is provided in4CHAPTER 1. INTRODUCTIONFigure 1.2: NREL?s best photovoltaic solar cell efficiency chart, as of 2013.Source: NREL?s National Center for Photovoltaics [7].5CHAPTER 1. INTRODUCTIONFigure 1.2.The efficiency chart depicted in Figure 1.2 illustrates the various categories ofphotovoltaic solar cell. Five basic photovoltaic solar cell categories are suggested:(1) single-junction c-Si based cells, (2) single-junction crystalline gallium arsenide(c-GaAs) based cells, (3) multi-junction based cells, (4) thin-film based cells, and(5) emerging technology based cells. Multi-junction based photovoltaic solar cellspossess the highest efficiencies, some approaching 44%. The more common c-Sibased photovoltaic solar cells, however, have efficiencies of around 25%, whilesingle-junction c-GaAs based cells have efficiencies approaching 29%. This thesisfocuses on single-junction crystalline based photovoltaic solar cells.Traditionally, photovoltaic solars cells were prepared using the planar geome-try. Most photovoltaic solar cells in use today are based on the planar pn junction.Recently, however, radial photovoltaic solar cells, i.e., nano-rod based photovoltaicsolar cells, have been shown to offer performance advantages when contrasted withthat of their planar counterparts. This thesis explores the physics and materialproperties of both the planar and the radial pn junction based photovoltaic solarcell. The electronic materials considered in this thesis include: Si, GaAs, germa-niun (Ge), indium nitride (InN), and gallium nitride (GaN).This thesis is organized in the following manner. Chapter 2 contains a descrip-tion of the air-mass spectrum, this providing the input to a photovoltaic solar cell,i.e., it provides a quantitative measure of the distribution of energy received fromthe Sun by a photovoltaic solar cell. The background, on how a photovoltaic solarcell converts solar energy from the solar flux into electrical energy, is also providedin this chapter. Then, Chapter 3 presents performance models, for both the planar6CHAPTER 1. INTRODUCTIONand the radial photovoltaic solar cell configurations, these models being used inthe subsequent analysis. Chapter 4 tabulates the material parameters required ofthese models, for the different electronic materials considered in this analysis, i.e.,Si, GaAs, Ge, InN, and GaN. The results, corresponding to both the planar andthe radial phovoltaic solar cell cases, for the materials considered in the analysis,are then featured in Chapter 5. Finally, Chapter 6 offers the conclusions of thisanalysis, with some suggestions for possible future work.7Chapter 2Background2.1 OverviewWithin a photovoltaic solar cell, electron-hole pairs (EHP) are generated, re-combined, and extracted. EHP generation arises as a consequence of the absorptionof photons from the Sun. EHP recombination occurs when the generated electronsand holes recombine, usually through a trapping level, i.e., a drifting electron isfirst trapped and then recombined with a hole or a drifting hole is first trappedand then recombined with an electron. A recombined EHP cannot contribute to thephotoelectric effect. EHP extraction, however, occurs when the generated electronsand holes reach the external contacts of the device, i.e., they reach the external con-tacts before they have a chance to recombine. It is through EHP extraction that thephotoelectric effect manifests itself.This chapter provides the background information needed for this thesis. Ini-tially, the solar irradiance air-mass distribution, this providing the input to a pho-tovoltaic solar cell, is described. Then, the operation of a pn junction based pho-8CHAPTER 2. BACKGROUNDtovoltaic solar cell is qualitatively discussed. Finally, the introduction of a newtype of pn junction photovoltaic solar cell geometry, the radial pn junction basedphotovoltaic solar cell, is presented, its advantages being extolled.This chapter is organized in the following manner. In Section 2.2, the air-massspectrum is introduced. Then, in Section 2.3, the operational characteristics ofa pn junction based photovoltaic solar cell is presented. Finally, a new type ofpn junction based photovoltaic solar cell geometry, the radial pn junction basedphotovoltaic solar cell, is introduced in Section 2.4, a comparison with the moretraditional planar pn junction based photovoltaic solar cell being provided.2.2 The air-mass spectrumThe solar spectral irradiance distribution provides for how the energy receivedby the solar cell, i.e., the solar flux, is distributed over the wavelengths in the elec-tromagnetic spectrum. The American Society for Testing and Materials (ASTM),in concert with the photovoltaics industry, government, and developmental labo-ratories, has created the ASTM G-173-03 document, which contains a number ofstandard terrestrial solar spectral irradiance distributions. These include: (1) theair-mass zero (AM0) solar spectral irradiance distribution, and (2) the air-massone-point-five (AM1.5G) solar spectral irradiance distribution. AM0 correspondsto the spectral irradiance distribution received at the edge the Earth?s atmosphere,i.e., the Sun?s rays did not propagate through the Earth?s atmosphere. AM1.5G,however, corresponds to the case in which the Sun?s rays had to propagate throughone and half atmospheres, i.e., an angle of incidence of 48.2?. In Figure 2.1, thesetwo solar spectral irradiance distributions are depicted.Typically, the AM spectrum is specified as a function of the wavelength. For9CHAPTER 2. BACKGROUNDFigure 2.1: The AM0 and AM1.5G solar spectral irradiance distributions.10CHAPTER 2. BACKGROUNDthe proposed analysis, however, this spectrum must be converted to a function ofthe photon energy instead. The air-mass spectrum (AM) spectrum comes in unitsof energy per meter squared per nanometre as a function of the wavelength, ? , i.e.,AMW (? )[Wm2nm]. (2.1)For the AM spectrum, to be specified in terms of the photon energy, the units needto be converted to the number of photons per cm2 per second per eV, i.e.,[photonscm2eVs]. (2.2)The energy of one photon may be described using Planck?s constant, h, and thespeed of light, c, i.e.,E =hc? . (2.3)Thus, the number of photons per Joule can be expressed asphotonsJoule=1nmhc? . (2.4)As a consequence, the AM spectrum, in units of number of photons per cm2 persecond per eV, may be given asAMeV = AMW1nmhc?1002eV. (2.5)11CHAPTER 2. BACKGROUND2.3 The operation of a pn junction solar cellThe pn junction is a common semiconductor based device configuration oftenused in photovoltaic device applications. Fundamentally, the pn junction is formedfrom two semiconductors, one that is donor doped and the other that is acceptordoped. In a pn junction, three distinct regions are formed: (1) a p-type region, (2) adepletion region, and (3) an n-type region. Figure 2.2 illustrates these distinctiveregions, and provides some insights into how pn junction based photovoltaic solarcells operate.The p-type region is doped with an acceptor concentration, Na, and the n-typeregion is doped with a donor concentration, Nd . The p-type region doping results ina minority electron concentration and the n-type region doping results in a minor-ity hole concentration. Minority carrier concentrations may be described using theFermi-Dirac distribution. The Fermi-Dirac distribution describes the thermal equi-librium occupancy of the electrons and holes within the various electronic states.The depletion region is between the p-type region and the n-type region. Itresults in an internal voltage between x2 and x3, which creates drift forces in thesolar cell. The depletion region is formed through concentration gradients in thep-type region and the n-type region. The n-type region has a large concentration ofelectrons and only a few holes. The p-type region has a large concentration of holesand only a few electrons. The differences in the hole and electron concentrationsin the p-type region and the n-type region create electron and hole concentrationgradients. The concentration gradients create diffusion forces for the holes andthe electrons. The hole diffusion forces act from the p-type region to the n-typeregion and the electron diffusion forces act from the n-type region to the p-type re-12CHAPTER2.BACKGROUNDN-RegionP-regionEDiffusionEcEvnp0 pn0EcEvDiffusionDriftDriftnp0 Minortiy Carrier Concentration pn0 Minortiy Carrier ConcentrationElectrons HolesDiffusion Length Diffusion LengthPhotonPhotonDepletion-RegionWnWpPhotonx1x2x3x4EHP GenerationEHP GenerationEHP GenerationCharge Density (Neutral)-eNaeNdCharge Density Due to Unneutralized Impurity Ions+ -Figure 2.2: The pn junction based photovoltaic solar cell with the p-type region on the left and the n-type region on theright. In the planar mode, the photon flux enters from the right to the left. In the radial mode, the photon flux isfrom the top to the bottom.13CHAPTER 2. BACKGROUNDgion. However, diffusion forces are counteracted by the drift forces that are presentaround the metallurgical junction. The electron concentration builds up on the leftside of the depletion region and the hole concentration builds up on the right sideof the depletion region. The concentration build-up creates a charge density, whichresults in an internal electric field from the n-type region to the p-type region withinthe depletion region. An internal electric field results in an internal voltage bias anddrift forces [8].The pn junction has two different diffusion lengths: (1) the p-type region elec-tron diffusion length, Ln, and (2) the n-type region hole diffusion length, Lp. Thephotons radiate into the p-type, n-type, and depletion regions and generate EHPs.If an EHP is generated within the p-type region or within the n-type region and,the generation of this EHP is within the diffusion length of either the p-type orthe n-type regions, diffusion forces carry the free generated carriers, which are ei-ther holes or electrons, to the depletion region. If the EHP is generated outsideof the diffusion length, then the generated EHP recombines and the photon gen-erated EHP will not contribute to the resultant photocurrent. The diffusion lengthis a function of the life-time of the minority carrier for the holes injected into then-type region, ?p, and the life-time of the minority carrier for the electrons injectedinto the p-type region, ?n. Once the EHPs charges reach the depletion region,the internal electric field carries these charge carriers across the depletion region,which leads to a drift current, and the generated EHPs contribute to the resultantphotocurrent [9].14CHAPTER 2. BACKGROUND2.4 Planar and radial photovoltaic solar cellconfigurationsThe planar and radial photovoltaic solar cell geometric configurations havesimilar pn junction solar cells characteristics, but have different geometric con-figurations. In a typical planar solar cell configuration, the n-type region is on thetop while the p-type region is on the bottom. The photon flux enters through then-type region, as shown in Figure 2.3. The n-type is typically on top because ofthe diffusion length is generally very short compared to the p-type diffusion length.The photon flux is from the top to the bottom of the figure. The performance of theplanar photovoltaic solar cell is dependent on its diffusion length. Increasing thecell thickness will not increase the corresponding efficiency.In radial solar cells, however, the pn junction is configured into two concentriccylinders, with the p-type region being the interior cylinder and the n-type regionbeing the exterior cylinder, as shown in Figure 2.3. Radial solar cells maybe fabri-cated through the use of vapour-liquid-solid growth, and Si nanowire based photo-voltaic solar cells have been reported as early as 1978 [10]. The radial photovoltaicsolar cell allows the photon flux to be orthogonal to the diffusion path, therebymaking radial solar photovoltaic cells more independent of the diffusion length,allowing for its use in low diffusion length and low optical absorption materials.15CHAPTER 2. BACKGROUNDFigure 2.3: A representative cross section of a planar pn junction based pho-tovoltaic solar cell and a radial photovoltaic solar cell, with the n-typeregion, the p-type region, and the photon flux directions clearly indi-cated. For the planar case, the photon flux is parallel to the diffusionpath, while for the radial case, the diffusion path is orthogonal to thephoton flux.16Chapter 3Planar and Radial PhotovoltaicSolar Cell Performance Models3.1 OverviewThe traditional pn junction based planar c-Si single-junction photovoltaic solarcell offers a number of distinct advantages. This is due, in large measure, to theexperience that has been acquired from the microelectronics industry in handling c-Si. The c-Si substrates, upon which such solar cells may be fabricated, are widelyavailable. The processing and fabrication techniques, through which such cellsmay be fabricated and subsequently refined, are well known and well established.The resultant solar cells are efficient and effective. Unfortunately, they are alsoexpensive. This has motivated researchers to explore the use of other materials andgeometries.This chapter describes the photovoltaic solar cell performance models that willbe used for the purposes of this analysis, for both the planar and the radial photo-17CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSvoltaic solar cell geometry cases. These models provide the basis for the subse-quent analysis. The key assumptions underlying these modeling analyzes will beemphasized as the models are developed. Figure 2.2 illustrates the basic featuresassociated with an abrupt pn junction based photovoltaic solar cell, including theband gap diagram, photon flux direction, and diffusion path. The models that aredeveloped here focus on the current density found within the regions of such a so-lar cell. The three regions considered are: (1) the n-type region, (2) the depletionregion, and (3) the p-type region.This chapter is organized in the following manner. In Section 3.2, the key as-sumptions, underlying these models, are described. The planar photovoltaic solarcell model is then developed in Section 3.3. In Section 3.4, the radial photovoltaicsolar cell model is then featured. Finally, the results from this chapter are summa-rized in Section 3.5.3.2 Key assumptionsThe key assumptions, underlying the performances models presented withinthis chapter, are divided into the following three assumption categories: (1) as-sumptions common to the planar and radial cases, (2) assumptions for the planarcase only, and (3) assumptions for the radial case only [11].1. Assumptions common to the planar and radial cases(a) The pn junction is abrupt.(b) The recombination that occurs is due to Shockley-Read-Hall recombi-nation from a single-trap at mid-gap.(c) Auger recombination is negligible.18CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELS(d) The AM1.5G solar spectral irradiance distribution is employed for thepurposes of this analysis.2. Assumptions for the planar case only(a) The n-type region is on the top.(b) The photon flux is incident on the top.3. Assumptions for the radial case only(a) The n-type region is the outer component of the rod.(b) The photon flux is incident to the top of the rod.(c) The surface-recombination that occurs within a radial pn junction basedphotovoltaic solar cell is exactly the same as the surface-recombinationthat occurs in the planar case.(d) The current density is purely radial.3.3 Planar photovoltaic solar cell current density modelThe total current density, Jtot as a function of wavelength, is the summationof the hole current density, Jp, the electron current density, Jn, and the depletionregion current density, Jdr, i.e., [12]Jtot(? ) = Jp(? )+ Jn(? )+ Jdr(? ). (3.1)The equations, Eqs. (3.1) through to Eqs. to (3.10), describe the equations used todetermine the total current density [12]. The total photocurrent density, obtained19CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSfrom the solar spectrum, may be expressed as the integration over the AM spec-trum, i.e.,J = q? ?0?(? )[1?R(? )]SR(? )d? , (3.2)where ? denotes the photon flux, i.e., usually AM0 and AM1.5G, R denotes thereflection coefficient, q denotes the charge of an electron, and where SR representsthe spectral response, i.e.,SR(? ) = Jtotq?(? )[1?R(? )] . (3.3)The electron and hole current density equations are derived in the followingsubsections. The one-dimensional, minority carrier, low-injection, steady-statecontinuity equations for EHP generation, EHP recombination, and diffusion forces,for the electron current density in the p-type region, isGn????EHP Generation?np?np0?n? ?? ?EHP Recombination?1qdJndx? ?? ?Current Density Differential= 0. (3.4)The electron concentration in the p-type region, np, is described by the Fermi-Diracdistribution for the non-degenerate case, i.e.,np = ni exp(Fn?EikT), (3.5)where Fn denotes the electron quasi-Fermi energy level, Ei is the intrinsic Fermilevel, T is the temperature, ni is the intrinsic electron concentration, ? is carrierlifetime, and k is the Boltzman constant.The minority carrier, low-injection, one-dimensional, steady-state continuity20CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSequations for EHP generation, EHP recombination, and diffusion forces, for thehole current density in the n-type region, isGp?pn? pn0?p?1qdJpdx= 0. (3.6)The hole concentration in the n-type region, pn, may also be described through theFermi-Dirac distribution. For the non-degenerate case, i.e.,pn = pi exp(Ei?FpkT), (3.7)where Fp is the quasi-Fermi hole energy level, Ei is the intrinsic Fermi energy level,and pi is the intrinsic hole concentration.The EHP generation rate, at a distance x from the edge of the phovoltaic solarcell surface, as shown in Figure 3.1, may be expressed asG(? ,x) = ?(? )?(? )[1?R(? )]e??(? )x. (3.8)The current density for the electrons in the p-type regionJn = q?nnp?+qDndnpdx, (3.9)while the current density for the holes in the n-type regionJp = q?ppn??qDpdpndx, (3.10)where ? is the carrier mobility and D is the diffusion coefficient.21CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSFigure 3.1: The planar pn junction based phovotovotlaic solar cell dimen-sions.22CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELS3.3.1 The diffusion length in the p-type regionCombining the low-injection, minority carrier expression of Eq. (3.4) with thecurrent density expression of Eq. (3.9), results in the hole continuity and currentdensity differential equation, i.e.,Dnd2npdx2+?(? )F(? )[1?R(? )]e??x?np?np0?n= 0. (3.11)The general solution for np?np0, where np0 denotes the equilibrium electron con-centration in the p-type region, will take both a homogeneous, nhp, and a particularsolution, npp, to the differential equation, i.e.,np = nhp+npp. (3.12)The general solution for the hole current density may thus be expressed asnp?np0 = Acosh(xLn)+Bsinh(xLn)? c3e??x, (3.13)where A and B are unknown coefficients, L is the diffusion length, andc3 =(?F(1?R)?n?2L2n?1). (3.14)Subject to the following boundary conditionsnp?np0 = 0|x=x j+W , (3.15)andSp(np?np0) =?Dnd(np?np0)dx????x=H, (3.16)23CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSwhere x j+W represents the dividing line between the depletion region and the p-type region and H corresponds to the edge of p-type region; recall Figure 3.1. Sdenotes the surface recombination velocity This yieldsnp?np0 =(?F(1?R)?n?2L2n?1)e??(x j+W )(cosh(x? (x j +W )Ln)+ e??(x?(x j+W ))?sinh(x? (x j +W )Ln)[LnSnDn(e??H?+ cosh(H ?Ln))+?Lne??H?+ sinh(H ?Ln)]LnSnDnsinh(H?(x j+W )Ln)+ cosh(H?(x j+W )Ln)?? .(3.17)Solving for the electron current densityJn = qDn(dnpdx)????x=x j+W, (3.18)the total electron current density may thus be expressed asJn =(??(1?R)Ln?2L2n?1)e??(x j+W )???Ln?[LnSnDn(e??H?+ cosh(H ?Ln))+?Lne??H?+ sinh(H ?Ln)]LnSnDnsinh(H?(x j+W )Ln)+ cosh(H?(x j+W )Ln)?? . (3.19)3.3.2 The diffusion length in the n-type regionCombining the minority carrier, low-injection expression of Eq. (3.6) with thecurrent density expression of Eq. (3.10), results in the hole continuity and current24CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSdensity differential equation [12], i.e.,Dpd2pndx2? ?? ?Diffusion+?(? )F(? )[1?R(? )]? ?? ?EHP Generatione??x?pn? pn0?p? ?? ?EHP Recombination= 0. (3.20)The general solution for pn? pn0, where pn0 denotes the equilibrium hole concen-tration in the n-type region, will take both a homogeneous, phn, and a particularsolution, ppn , to the differential equation, i.e.,pn = phn+ ppn . (3.21)The general solution for the hole concentration may thus be expressed aspn? pn0 = Acosh(xLn)+Bsinh(xLn)? c1e??x, (3.22)where A and B are unknown coefficients andc1 =(?F(1?R)?n?2L2n?1). (3.23)The specific solution needs to be solved for, subject to the following boundaryconditions, i.e.,d(pn? pn0)dx= Sp(pn? pn0)????x=0, (3.24)pn? pn0 = 0|x=x j . (3.25)25CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSThis yieldspn? pn0 = c1??c2(sinh(x j?xLp))+ e??x j(cosh(xLn)+ SpLpDp sinh(xLn))SpLpDpsinh(x jLn)+ cosh(x jLn) ? e??x?? ,(3.26)wherec2 =(LpSpDp+Lp?). (3.27)Solving for the hole current density, i.e.,Jp =?qDp(dpndx)????x=x j, (3.28)the following hole current density expression may be obtainedJp =(q?LpF(1?R)?2L2p?1)??c2? e??x j(cosh(x jLp)+ SpLpDp sinh(x jLp))SpLpDpsinh(x jLn)+ cosh(x jLn) ??Lpe??x j?? .(3.29)3.3.3 The depletion regionIn the depletion region, photons can contribute to the photoelectric effect. Thefield in the depletion region may be strong, and if this is the case, the generatedEHPs are swept out of this region before they can recombine [12]. FollowingKayes et al. [11], it is found thatJdr =?q?0e??1x1(1? e??1x2(V )??2x3(V )). (3.30)26CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELS3.4 Radial photovoltaic solar cell current density modelAccording to Kayes et al. [11], the total current density corresponding to aradial pn junction based photovoltaic solar cell, at distance z from the surface, asshown in Figure 3.2, may be expressed asJp(z) = (Jp0 +Jn0 )(eqV/kT ?1)?Jpl ?Jnl ?Jdep,pg (V )?Jdep,ng (V )+Jdepr (V ), (3.31)whereJp0 =?2n0LDnL2p?5? 21I1?5I0? 21, (3.32)Jp0 =?2p0LDpL2p?2? 21(f1K1(?2)? f2I1(?2)f1K0(?2)+ f2I0(?2)), (3.33)Jpl =?2q?0L2nL2p?5? 21I1?5I0?5(1? e??6), (3.34)Jnl =?2q?0?2? 21[K1(?2)( f1??4I0(?2))? I1(?2)( f2 +?4K0(?2))f1K0(?2)+ f2I0(?2)], (3.35)Jdep,pg (V ) =?2q?0d22 ? x24R2(1? e??6), (3.36)Jdep,ng (V ) =?2q?0(d2 + x2)2R2(1? e??3), (3.37)andJdepr (V ) =?2qLUmax(r2 + r1)2R2, (3.38)where In(?) and Kn(?) are modified Bessel functions, of the first and second kind,respectively, with the Bessel integer, n, being set to either 0 or 1. The geometricparameters x, d, and r are illustrated in 3.2. The dimensionless parameters are27CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSdefined as?1 =RLp, (3.39)?2 =R? x1Lp, (3.40)?3 = ?1L, (3.41)?4 =LpSpDp, (3.42)?5 =x4Ln, (3.43)?3 = ?2L, (3.44)f1(?1,?4) = I1(?1)+?4I0(?1), (3.45)andf2(?1,?4) = K1(?1)+?4K0(?1). (3.46)In addition,Umax =ni??n?psinh(qV2kT), (3.47)r1(V ) = r(V )?x2(V )+ x3(V ))2?, (3.48)r2(V ) = r(V )+x2(V )+ x3(V ))2?, (3.49)r(V ) = x4 +log(Nani,p)log(NaNdni,pni,n)(x2(V )+ x3(V )), (3.50)and? = pikTq(Vbi?V ), (3.51)28CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSwhereVbi?V =qNd2?2(d2 + x2)2 log(d2 + x2d2)+qNd4?n[d22 ? (d2 + x22)]+qNa?px24 log(x4d2)+qNa4?p(d22 ? x24) (3.52)andVbi =kTqlog(NaNdni,pni,n). (3.53)3.4.1 Diffusion length in the p-type regionThe continuity equation for the p-type region may be expressed as?2n??n?L2n=? 2n?? r2 +1r?n?? r ?n?L2n=??2?0Dne??2x, (3.54)wheren? = np?np0, (3.55)andDn =kTq?n, (3.56)r denoting the radial distance from the centre of the pn junction based photovoltaicsolar cell. This equation must be solved for subject to the following boundaryconditions:n?(0) = finite, (3.57)29CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSFigure 3.2: The radial pn junction based phovotovotlaic solar cell dimensionswith the photon flux direction indicated.30CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSandn?(x4) = n0(eqV/kT ?1), (3.58)where x4 denotes the centre of the radial solar cell. The total current density maythus be expressed asJp =2x4? L0 Jp(z)dzR2, (3.59)whereJp(z) = ?qDn?n?? r????r=x4. (3.60)The general solution, including the homogeneous and particular solutions, maybe expressed asn? = n?h+n?p = c1I0(r)+ c2K0(r)+??0L2pDpe??z, (3.61)where c1 and c2 are the unknown constant coefficients. When subjected to bound-ary conditions, i.e., the boundary conditions specified in Eqs. (3.57) and (3.58), itis concluded thatn?(r) =??n0(eqV/kT ?1)???0L2pDpe??xI0(x4Lp)?? I0(r)+??0L2pDpe??z. (3.62)Thus, it is concluded thatJp =??2x4qDnLpR2n0(eqV/kT ?1)I0(x4LpI1(x4Lp)+2x4qDnLpR2???0L2pDp(1? e??L)I0(x4Lp?? I1(x4Lp).(3.63)31CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELS3.4.2 Diffusion length in the n-type regionThe continuity equation for the n-type region may be expressed as?2p??p?L2p=? 2p?? r2 +1r? p?? r ?p?L2p=??2?0Dpe??2x. (3.64)This equation must be solved for subject to the boundary conditionsp?(R? x1) = p0(eqV/kT ?1), (3.65)andSpp?(R) = ?Dp? p?? r????r=R, (3.66)R denoting the radius of the radial solar cell; recall Figure 3.2. ? denotes the photonflux. The total current density in the n-type region may be expressed asJn =2(R? x1)? L0 Jn(z)dzR2, (3.67)whereJn(z) = qDp?n?? r????r=x4. (3.68)Thus, the general solution may be expressed asp?(x) = A2I0(xLp)+B2K0(xLp)+??0L2pDpe??z, (3.69)where A2 and B2 are unknown constant coefficients and I0(?) and K0(?) are modifiedBessel functions, of the first and second kind, respectively, with the Bessel integerof 0. The x, r, and d are illusrated in 3.2. When the boundary conditions are32CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSapplied, i.e., Eqs. (3.65) and (3.66), it may be concluded thatJn =2(R? x1)?M?LpR2(p0(eqV/kT ?1)L f2 +?0L2pDp(1? e??L)(f2 +K0(R? x1Lp)))I1(x4Lp)+2(R? x1)?M?LpR2(p0(eqV/kT ?1)L f1 +?0L2pDp(1? e??L)(f1 + I0(R? x1Lp)))K1(x4Lp),(3.70)where ?M? denotes the determinant of the matrixM =???cosh(x j+WLn)sinh(x j+WLn)Sn cosh(HLn)+ DnLn sinh(HLn)Sn sinh(HLn)+ DnLn cosh(x j+WLn)??? . (3.71)3.4.3 The depletion regionIn the depletion region, EHP generation and recombination occurs. FollowingKayes et al. [11], it is found thatJdep,pg (V ) = q?0(1? e?2L)d22 ? x24R2, (3.72)and thatJdep,ng (V ) = q?0(1? e?2L)(d2 + x2)2? x24R2. (3.73)Thus, it may be concluded thatJdep,pg (V ) =?2q(?F(1?R)?n?2L2n?1)d22 ? x24R2(1? e??6), (3.74)Jdep,ng (V ) =?2q(?F(1?R)?n?2L2n?1)(d2 + x2)2?d22R2(1? e??3), (3.75)33CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELSandJdr ep(V ) =?qLUmaxr22? r21R2. (3.76)3.5 SummaryAnalytical expressions for the current density have been obtained for the casesof both the planar and the radial pn junction based photovoltaic solar cell. For thecases of the planar and radial photovoltaic solar cell geometries, there are threefactors that contribute to the performance: (1) EHP generation, (2) EHP recombi-nation, and (3) EHP extraction. EHP generation is due to the photon flux from theSun and EHP recombination is due to the trap centre densities. EHPs generatedoutside of the diffusion length do not contribute to the photoelectric effect. Thediffusion length is a function of the trap centre density. For the planar solar cellcase, the diffusion path is parallel with the photon flux, while for the radial solarcell case, the diffusion path is orthogonal with the photon flux. Thus, for the radialsolar cell, the results are found to be relatively insensitive to the photon flux. Inaddition, EHP generation is dependent upon the photon absorption depth.These models, that were developed for the different types of pn junction basedphotovoltaic solar cells, require the following input parameters.? the spectral dependence of optical absorption coefficient, ?? the effective density of states for the conduction band, Nc? the effective density of states for the valence band, Nv? the recombination velocity, vs? the acceptor concentration, ND34CHAPTER 3. PLANAR AND RADIAL PHOTOVOLTAIC SOLAR CELLPERFORMANCE MODELS? the donor concentration, NA? the electron mobility, ?n? the hole mobility, ?p? the electron surface recombination velocity, Sn? the hole surface recombination velocity, Sp? the electron conductivity, ?n? the hole conductivity, ?p,and? the thermal velocity, vth.In the next chapter, these materials properties are specified for all of the mate-rials considered in the analysis, i.e., Si, GaAs, Ge, InN, and GaN.35Chapter 4Solar Cell Material Parameters4.1 OverviewA variety of electronic materials and geometries have been considered for usein pn junction based photovoltaic solar cells. Initially, Si was considered for usein photovoltaic solar cells [13]. Si offers a number of advantages and it is abun-dantly available, Si being a staple of the microelectronics industry. Si is an indirectenergy gap semiconductor, with an energy gap of 1.12 eV. GaAs, however, a III-V compound semiconductor, has also been used for photovoltaic solar cell deviceapplications. GaAs is a direct energy gap semiconductor, with an energy gap of1.424 eV. GaAs is less temperate sensitive than Si owing to its wider energy gap.InN, a compound III-V semiconductor with a direct energy gap of 0.65 eV, hasbeen considered for photovotlaic device applications. The low energy gap asso-ciated with InN allows for the absorption of low energy photons. GaN, anothercompound III-V direct energy gap semiconductor with a wide energy gap of 3.4eV, has also been considered for the use in photovotlaic device applications, al-36CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSthough its energy gap is within the ultraviolet range.This thesis is focused upon determining how the performance of such a photo-voltaic solar cell is influenced by the selection of material and geometry employed.The following materials are considered in this analysis: Si, GaAs, Ge, InN, andGaN. Each material considered possesses unique material properties, which willresult in different performance characteristics. The material characteristics neededin order to use the performance models proposed in Chapter 3, are presented in thischapter. Specifically, for each material considered: (1) the spectral dependence ofthe optical absorption coefficient, (2) the recombination velocities, and (3) a va-riety of other relevant semiconductor material properties, are specified. For eachmaterial considered, the performance of both the planar and the radial pn junctionbased photovoltaic solar cell configuration will be considered. In this chapter, therelevant material properties, corresponding to the evaluation of the performance ofa photovoltaic solar cell, are identified for each material considered in this analysis.This chapter is organized in the following manner. In Section 4.2, models forthe spectral dependence of the optical absorption coefficient, corresponding to thedifferent materials consider in this analysis, are specified. Then, in Section 4.3,recombination velocities, associated with the different materials considered in thisanalysis, are tabulated. A tabulation of other semiconductor properties, relevant tothis analysis, are presented in Section 4.4. Finally, this chapter is summarized inSection 4.537CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.1: The intensity of the normally incident beam of light as a functionof the penetration depth, x, within in a homogeneous material.38CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.2 The spectral dependence of the optical absorptioncoefficientConsider a beam of light incident upon a slab of material, as depicted in Fig-ure 4.1. As the beam propagates through the material, the intensity of this beamwill attenuate. Assuming that the material is homogeneous, an exponential attenu-ation in the beam intensity occurs, i.e.,I(x) = I0 exp(??x), (4.1)where I(x) denotes the intensity of the beam within the material at x, I0 representsthe intensity of the beam at the surface itself (within the material), and ? is theoptical absorption coefficient. The optical absorption coefficient depends upon thewavelength of the light being considered, and this dependence is of crucial impor-tance for this analysis; in material studies, the dependence of ? on the wavelengthof the beam is often used for the quantitative characterization of the optical re-sponse of a given material. Quantitatively, the optical absorption coefficient maybeexpressed as? = 4pik? , (4.2)where ? denotes the wavelength of the light and k is the extinction coefficient.Both ? and k are optical functions, and the spectral dependence of these functionsis of crucial importance in the characterization of a given material.In this section, models for the optical functions associated with the electronicmaterials considered in this analysis, i.e., Si, GaAs, Ge, InN, and GaN, are intro-duced. As Si is the most well known semiconductor, a variety of models for thespectral dependence of its optical functions are considered. Ultimately, an elemen-tary polynomial fit is employed for all other materials considered.39CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.2.1 SiliconThe optical response of c-Si is well known. A number of empirical modelsfor the optical functions associated with it have been proposed, including that ofGeist [14], which is applicable for photon energies below 3.38 eV, and that ofAdachi [15], which is applicable for photon energies above 3.38 eV. These modelsare used in the subsequent analysis.The spectral dependence of the absorption coefficient below 3.38 eVGeist [14] characterizes the spectral dependence of the extinction coefficientassociated with c-Si, k, using Eqs. (4.3) through to (4.6), the parameters being de-picted in Table 4.1. The resultant spectral dependence of k is depicted in Figure 4.2.That is,k(hv) =f (hv)4pi[1.23985 ?10?4cmeVhv][g(hv)hv], (4.3)whereg(hv) =C3[2L(hv? ?3)]N+dNhv+2?i=11?i=?1Ci, j[L(hv? ?1 + jd?i)]2, (4.4)f (hv) = exp{K4L(hv? ?4)+K5L(hv? ?5)}, (4.5)andL(x1) =(x1 + |x1|)2eV, (4.6)40CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSwhere hv refers to the corresponding photon energy, Eq. (4.6) being a function ofits arbitrary input, i.e., x1.Table 4.1: Parameters used in the model of Geist [14] in order to determinethe spectral dependence of the extinction coefficient associated with c-Si.Parameter Value?r 1.09969 eVd?1 0.0583148 eVC1,?1 5030.02 cm?1eVC1,+1 483.916 cm?1eVd?2 0.0220161 eVC2,?2 1634.30 cm?1eVC2,+2 79.4079 cm?1eV?3 1.40985 eVC3 1046.08 cm?1eVN 0.394122dN 1.23084 (h??)?1?4 2.15396 eVK4 0.210102?5 2.15396 eVK5 1.0368141CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.2: The spectral dependence of the extinction coefficient, Eqs. (4.3),associated with c-Si as a function of the wavelength, in nanometres.42CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSThe spectral dependence of the real and imaginary parts of the dielectricfunctions between 3.38 and 5.32 eVAdachi [15] characterizes the spectral dependence of the real and imaginaryparts of the dielectric function associated with c-Si, for photon energies between3.38 and 4.26 eV, using Eqs. (4.7) through to (4.15), the corresponding parametersbeing specified in Table 4.2. That is, Adachi [15] takes the real part of the dielectricfunction?1(h??) =?B1??21 ln(1???21 ), (4.7)and the imaginary part?2(h??) = piB1??21 , (4.8)where?1 =h??E1, (4.9)where h?? refers to the corresponding photon energy. Adachi [15] uses ? to rep-resent the radial frequency, but in this analysis, h?? is used to denote the photonenergy. These real and imaginary parts of the dielectric function are also opticalfunctions. The spectral dependence of ? is related to ?1(h??) and ?2(h??) through?(h??) = 4pi?((?21 (h??)+ ?22 (h??))1/2? ?1(h??)2)1/2. (4.10)43CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSTable 4.2: Parameters used in the model of Adachi [15] in order to determinespectral dependence of the real and imaginary parts of the dielectric func-tion associated with c-Si.Parameter ValueE1 3.38B1 5.52?[E1/E1 +?1] 7.47E2 0.05C 3.01v 0.127F 3.51?[E2] 0.04Ea 5.32Ca 0.21va 0.08944CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSIn contrast, Adachi [15] suggestes that the real part of the dielectric function,between 4.26 and 5.32 eV, may be alternatively expressed as?1(h??) =C(1??22 )(1???22 )+?22v2, (4.11)and?2(h??) =C?2v(1???22 )+?22v2, (4.12)where?2 =h??E2. (4.13)Beyond 5.32 eV, the real part of the dielectric function may be shown to be?1(h??) =?F??22 ln(1??2c11??22), (4.14)and?2(h??) = piF??22 H(1??2)H(?c1?1), (4.15)where?c1 =h??E1. (4.16)Figure 4.2 illustrates the spectral dependence of the real part of the dielectric func-tion for c-Si for photon energies below 3.38 eV. Figure 4.3 illustrates the spectraldependence of the dielectric function for Si for photon energies between 3.38 and5.32 eV. Figure 4.4 illustrates the spectral dependence of the optical absorption45CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.3: The real and imaginary parts of the dielectric function associatedwith c-Si as a function of the photon energy using Eqs. (4.7) through to(4.15).46CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.4: The spectral dependence of the optical absorption coefficient as-sociated with c-Si. The empirical models of Geist [14] and Adachi [15]are employed. Experimental data from Palik [16] is also depcited. Thereis a discontinuity at 3.0 eV, as there is a discontinuity between the twoempirical models. The one below 3.0 eV is from Geist [14] and the oneabove 3.0 eV is from Adachi [15]. The implemented equations fit theexperimental data set rather well.47CHAPTER 4. SOLAR CELL MATERIAL PARAMETERScoefficient using the models the of from Geist [14] and Palik [16].The empirical expressions for the optical functions associated with c-Si, de-vised by Geist [14] and Adachi [15], while essentially agreeing with the results ofexperiment, are rather cumbersome and unwieldy. A least-squares polynomial fitto the experimental results of Palik [16], i.e.,? = f (h??)5 + e(h??)4 +d(h??)3 + c(h??)2 +bh??+a. (4.17)is shown to be just as effective. The coefficients, corresponding to this least-squarespolynomial fit, are depicted in the Table 4.3. The resultant fit is shown in Fig-ure 4.5, alongside the experimental results of Palik [16]. For legacy reasons, theexpressions of Geist [14] and Adachi [15] are employed for the subsequent c-Sianalyzes. For all other materials, however, a polynomial fit approach is adoptedwith the correlation coefficient.Table 4.3: The c-Si optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental results of Pa-lik [16]. R is the correlation coefficient. The coefficients are dimension-less.Range (eV) R f e d c6,4.25 0.996 1.44106 ?3.77107 3.94108 ?2.051094.25,3.1 0.999 0 4.16106 ?5.87107 3.111081.1,3.1 0.993 3.61104 ?3.41105 1.27106 ?2.31106Range (eV) b a6,4.25 5.29109 ?5.451094.25,3.1 ?7.29108 6.381081.1,3.1 2.06106 ?7.1610548CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.5: The optical absorption coefficient associated with c-Si. The poly-nomial fit, and the experimental data from Palik [16], are depicted.49CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.2.2 Gallium arsenideFor the case GaAs, a sixth-order polynomial is least-squares fit with the cor-responding experimental optical absorption spectrum, these results being from Pa-lik [16]. The coefficients resulting from the fit are tabulated in Table 4.4. The fittedspectrum, and the corresponding experimental spectrum, are depicted in Figure 4.6.The optical absorption spectrum fitting equation is? = g(h??)6 + f (h??)5 + e(h??)4 +d(h??)3 + c(h??)2 +bh??+a. (4.18)Table 4.4: The GaAs optical absorption polynomial coefficients, determinedfrom polynomial least-squares fit to the experimental results of Pa-lik [16]. R is the correlation coefficient. The coefficients are dimen-sionless.Range (eV) R f e d c b a4.94,3.94 1 0 0 1.26?106 1.77?107 8.07?107 1.21?1083.94,2.92 0.93 6.2?106 1.0?108 7.05?108 2.37?109 3.96?109 2.63?1092.92,1.5 1 1.4?106 1.4?107 5.71?107 1.15?108 1.16?108 4.59?10750CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.6: The optical absorption coefficient associated with GaAs. Thepolynomial fit and the experimental data from Palik [16] are depicted51CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.2.3 GermaniumFor the case Ge, a sixth-order polynomial is least-squares fit with the corre-sponding experimental optical absorption spectrum, these results being from Pa-lik [16]. The coefficients resulting from the fit are tabulated in Table 4.5. The fittedspectrum, and the corresponding experimental spectrum, are depicted in Figure 4.7.The optical absorption spectrum fitting equation is? = g(h??)6 + f (h??)5 + e(h??)4 +d(h??)3 + c(h??)2 +bh??+a. (4.19)Table 4.5: The Ge optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental results of Pa-lik [16]. R is the correlation coefficient. The coefficients are dimension-less.Range (eV) R d c b a4.94,4.38 1 5.98 ?105 9.45 ?106 4.99 ?107 8.96 ?1074.38,2.3 1 4.75?108 1.18?109 1.53?109 8.22?1082.3,1.85 1 2.54?1011 3.84?1011 3.09?1011 1.03?10111.85,1 1 9.16?104 4.48?105 6.06?105 2.58?105Range (eV) R g f e4.38,2.3 1 6.22?105 1.27?107 1.07?1082.3,1.85 1 1.54?109 1.87?1010 9.45?101052CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.7: The optical absorption coefficient associated with Ge. The poly-nomial fit and the experimental data from Palik [16] are depicted53CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.2.4 Indium nitrideFor the case InN, a fourth-order polynomial is least-squares fit with the cor-responding experimental optical absorption spectrum, these results being fromBhuiyan [17]. The coefficients resulting from the fit are tabulated in Table 4.6.The fitted spectrum, and the corresponding experimental spectrum, are depicted inFigure 4.8. The optical absorption spectrum fitting equation is? = e(h??)4 +d(h??)3 + c(h??)2 +bh??+a. (4.20)Table 4.6: The InN optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental results ofBhuiyan [17]. R is the correlation coefficient. The coefficients are di-mensionless.Range (eV) R e d c b a0.52,2.1 1 95,782 539,978 1,049,229 743,538 182,29454CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.8: The optical absorption coefficient associated with InN. The poly-nomial fit and the experimental data from Bhuiyan [17] are depicted55CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.2.5 Gallium nitrideFor the case GaN, a sixth-order polynomial is least-squares fit with the cor-responding experimental optical absorption spectrum, these results being fromMuth [18]. The coefficients resulting from the fit are tabulated in Table 4.7. Thefitted spectrum, and the corresponding experimental spectrum, are depicted in Fig-ure 4.9. The optical absorption spectrum fitting equation is? = g(h??)6 + f (h??)5 + e(h??)4 +d(h??)3 + c(h??)2 +bh??+a. (4.21)Table 4.7: The GaN optical absorption polynomial coefficients, determinedfrom the polynomial least-squares fit to the experimental results ofMuth [18]. R is the correlation coefficient. The coefficients are dimen-sionless.Range (eV) R g f e d c3.03,3.326 0.98 2.46?109 4.65?1010 3.66?1011 1.54?1012 3.64?10123.326,3.36 0.94 0 0 2.30?1011 3.07?1012 1.54?10133.36,3.589 0.99 0 0 0 1.10?107 1.16?1083.589,6.44 1 0 0 0 0 1.51?103Range (eV) b a3.03,3.326 4.59?1012 2.41?10123.326,3.36 3.43?1013 2.87?10133.36,3.589 4.04?108 4.72?1083.589,6.44 5.23?104 7.14?10456CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.9: The optical absorption coefficient associated with GaN. The poly-nomial fit and the experimental data from Muth [18] are depicted57CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.3 Recombination velocitiesHigh defect densities result in large trap centre densities and elevated recombi-nation velocities. Defects are divided into four basic categories: (1) point, (2) one-dimensional, (3) planar, and (4) volume defects. Point defects are substitutional,vacancies, interstitials, and Frenkel defects [19]. For elemental semiconductors,vacancies are typically point defects. Substitutional defects are impurities that arenot intrinsic to the semiconductor type. Interstitials are defects between the dif-ferent lattice points. One-dimensional defects are line defects, edge dislocations,and screw dislocations. Line defects are dislocations with a displacement of atomsalong a line. Planar defects occur owing to: (1) rotations of the lattice, and (2)surface or interface defects between two different types of materials, or the sametype of material but with different orientations.Within a pn junction, EHP recombination will occur when there is a distur-bance in the thermal equilibrium. At recombination centers or traps, a recombina-tion of electrons and holes will occur. There are four cases for EHP recombination[20]:1. The trap centre is occupied by a hole and the electron recombines with thishole.2. The trap centre is occupied by a hole and the electron is emitted to the va-lence band.3. The trap centre is occupied by an electron, and the trapped electron is emittedto the conduction band.4. The trap centre is occupied by an electron and a valence band hole recom-58CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSbines with this electron.Under low injection conditions, the recombination process may be used to de-termine the minority-carrier lifetime for the n-type and the p-type semiconductorcases. This process leads to?p =1?pvthNt, (4.22)where Nt is the concentration of bulk trapping centres per unit area. The minority-lifetime carrier is related to the diffusion length, which was discussed in Chapter 2,by the square root of the product of the diffusion constant and the minority carrierlifetime, i.e.,Ld ??D?. (4.23)Rearranging Eq. (4.23), and substituting Eq. (4.22) into it, leads toL2d =D?pvthNt, (4.24)where vs = ?pvthNt denotes the bulk recombination velocity. Therefore, the diffu-sion length, which is important for photocurrent power generation, is related to therecombination trap centre concentration through Eq. (4.25), i.e.,L2d ?1Nt. (4.25)The surface recombination velocitySp = ?pvthNst , (4.26)where Nst is the concentration of surface trapping centres per unit area.59CHAPTER 4. SOLAR CELL MATERIAL PARAMETERS4.4 Other semiconductor propertiesOther relevant semiconductor material parameters, used in this analysis, arepresented in Table 4.8. The temperature is assumed to be 300 k for each material.The surface recombination, thermal velocity, and conductivity are assumed to beequal for each material. In the same manner as Atwater and el. [11], the mobility,intrinsic density, and the density of states are given by the Ioffe-Physico-TechnicalInstitute [21].Table 4.8: Other relevant semiconductor material properties associated withSi, GaAs, Ge, InN, and GaN based photovoltaic solar cells.Description Si GaAs Ge InN GaNT (K) 300 300 300 300 300Nc (cm?3) 2.8?1019 4.7?1017 1.04?1017 9.15?1018 2.2?1017Nv (cm?3) 1.04?1019 7?1018 6?1018 5.2?1017 4.6?1020ni (cm?3) 1.4?1010 1.8?106 1.4?1010 1?109 6.2?105NA,ND (cm?3) 1?1018 1?1017 2.4?1013 1?1017 1?1017?n (cm2/V-s) 270 5000 620 600 400?p (cm2/V-s) 95 320 320 300 100Sn (cm s?1) 1?105 1?105 1?105 1?105 1?105?n (cm2) 1?10?15 1?10?15 1?10?15 1?10?15 1?10?15vth (cm s?1) 1?107 1?107 1?107 1?107 1?107?R 11.9 13.1 16 13.1 9.64.5 SummaryThis chapter provides the spectral dependence of the optical absorption coef-ficients corresponding to the various materials considered in this analysis. Figure4.10 illustrates the AM spectrum, with the absorption coefficient as a function ofthe photon energy for the different materials considered. The materials consideredhave optical absorption coefficients that exhibit a wide range of values, materials60CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSwith higher absorption coefficients at lower photon energy levels offering greaterefficiencies. This chapter also covered the recombination centre densities and otherrelevant semiconductor material properties.61CHAPTER 4. SOLAR CELL MATERIAL PARAMETERSFigure 4.10: The photon flux of the AM1.5G and the fitted spectral depen-dencies associated with the optical absorption coefficient associatedwith Si, GaAs, Ge, InN, and GaN.62Chapter 5Results5.1 OverviewThe electronic material employed and the geometric configuration selected willultimately the shape the performance of a photovoltaic solar cell. In Chapter 3, ex-pressions for the current density associated with the planar and radial pn junctionbased solar cells, were developed. In Chapter 4, the material parameters corre-sponding to the different materials being consider in this analysis, i.e., Si, GaAs,Ge, InN, and GaN, were specified. Using the models developed in Chapter 3, andthe material parameters presented in Chapter 4, in this chapter, the performance ofpn junction based planar and radial photovoltaic solar cells is examined.This chapter is organized in the following manner. In Section 5.2, the perfor-mance of planar and radial based pn junction photovoltaic solar cells are examined,for the materials considered in this analysis, i.e., Si, GaAs, Ge, InN, and GaN.Then, in Section 5.3, a critical comparison between the different materials and ge-ometries considered, is provided. Finally, the results obtained within this chapter63CHAPTER 5. RESULTSare summarized in Section 5.4.5.2 Current density, open-circuit voltage, and efficiencyThe performance results that are presented are the short-circuit current density,the open-circuit voltage, and the efficiencies, for both the radial and the planarcases for the materials being consider in this analysis, i.e., Si, GaAs, Ge, InN, andGaN. The short-circuit current density, the open-circuit voltage, and the efficiencyare presented as functions of the cell thickness, L, and the diffusion length, Ln, forthe planar cases, and as functions of the radius, R, and the cell thickness, L, forthe radial cases. The total current density, J, as a function of the voltage bias, wasdetermined for each forward bias voltage, and summed over each wavelength in theAM1.5G solar spectral irradiance distribution. The short-circuit current density isdetermined by setting the voltage bias to zero. The open-circuit voltage, however,is determined by finding the bias at which the total current is zero. The efficiency ?is the maximum power output divided by the maximum power input. Appendix Acontains the Matlab code needed for these performance evaluations [22]. For eachmaterial considered, both the planar and the radial geometries were considered.Trap density distributions associated with two special cases are considered inthis analysis: (1) the recombination lifetime is assumed to be constant in the p-type region, the n-type region, and the depletion region, and (2) the trap densitywithin the p-type and the n-type regions are consistent throughout the material,corresponding the recombination lifetime being set to 6 ?s. Atwater et al. [11]assumed either cases could be realized in practice depending on the fabricationprocess. For the radial case, the radius is set to be the diffusion length, and, asdiscussed in Section 4.3, the diffusion length is a function of the trap centre density64CHAPTER 5. RESULTSand the recombination life-time. The diffusion lengths for each material in thisanalysis is not necessarily physical feasible but is used of comparison proposes.5.2.1 SiliconFigure 5.1 depicts the dependence of the short-circuit current density on thecell thickness and the diffusion length for the case of the planar pn junction basedc-Si photovoltaic solar cell. The planar thickness, L, ranges from 1 to 100 ?m andthe diffusion length ranges from 0.5 to 500 ?m. In Figure 5.2, the short-circuitcurrent density is depicted as a function of the cell radius and the diffusion lengthfor the case of the radial pn junction based c-Si photovoltaic solar cell. Consistentwith Kayes et al. [11], it is seen that for the planar case, the short-circuit currentdensity is dependent on the diffusion length, while the short-circuit current densitycorresponding to the radial nano-rod case is essentially independent of the diffu-sion length. The maximum short-circuit current density is approximately 0.033and 0.038 mA/cm2, for the planar and radial cases, respectively. The planar caseexhibits its maximum short-circuit current density at a cell thickness of 100 ?mand a diffusion length of 500 ?m. The radial case, however, exhibits its maxi-mum short-circuit current density at a cell thickness of 500 ?m and is essentiallyindependent of the radius. The short-circuit current density is independent of thedepletion region trap density, for both the planar and the radial cases.Figure 5.3 illustrates the dependence of the open-circuit voltage on the cellthickness and the diffusion length for the case of the planar pn junction based c-Si photovoltaic solar cell. The planar cell thickness and the diffusion length havethe same ranges as that considered earlier in Figure 5.1. The planar case has amaximum open-circuit voltage of approximately 0.6 V at a diffusion length of65CHAPTER 5. RESULTSFigure 5.1: The short-circuit current density as a function of the device thick-ness and the diffusion length for a pn junction based c-Si photovoltaicsolar cell in the planar configuration.66CHAPTER 5. RESULTSFigure 5.2: The short-circuit current density as a function of the device thick-ness and the radius, where the radius is equal to the diffusion length, fora pn junction based c-Si photovoltaic solar cell in the radial configura-tion.67CHAPTER 5. RESULTS500 ?m. Figure 5.4 illustrates the dependence of the open-circuit voltage on thecell thickness and the diffusion length for the case of the radial pn junction basedc-Si photovoltaic solar cell. The radial solar cell thickness and the diffusion lengthhave the same ranges as that considered earlier in Figure 5.2. The radial case has amaximum Voc of 0.7 V at a cell thickness of 500 ?m. The open-circuit voltage forthe planar case is dependent on the diffusion length, while the open-circuit voltageis found to be relatively insensitive to the diffusion length for the radial case. Adecreasing depletion region trap density increases the open-circuit voltage, for boththe planar and radial cases.Figure 5.5 illustrates the dependence of the efficiency, ? , on the cell thicknessand the diffusion length for the case of the planar pn junction based c-Si photo-voltaic solar cell. The planar cell thickness and the diffusion length have the sameranges as that considered earlier in Figure 5.1. For the planar case, an efficiencyof approximately 16.6% at a cell thickness of 100 ?m and a diffusion length of500 ?m is observed. The efficiency for the planar solar cells is independent ofthe depletion region trap density. Figure 5.6 illustrates the dependence of the effi-ciency, ? , on the cell thickness and the diffusion length for the case of the radialpn junction based c-Si photovoltaic solar cell. The radial cell thickness and thediffusion length have the same ranges as that considered earlier in Figure 5.2. Theradial case, however, has an efficiency of approximately 17.3% at a cell thicknessof 500 ?m and a diffusion length of 500 ?m. The efficiency for the radial solarcell is strongly dependent on the depletion region trap density.68CHAPTER 5. RESULTSFigure 5.3: The open-circuit voltage as a function of the device thickness andthe diffusion length for a pn junction based c-Si photovoltaic solar cellin the planar configuration.69CHAPTER 5. RESULTSFigure 5.4: The open-circuit voltage as a function of the device thickness andthe radius, where the radius is equal to the diffusion length, for a pnjunction based c-Si photovoltaic solar cell in the radial configuration.70CHAPTER 5. RESULTSFigure 5.5: The efficiency as a function of the device thickness and the dif-fusion length for a pn junction based c-Si photovoltaic solar cell in theplanar configuration.71CHAPTER 5. RESULTSFigure 5.6: The efficiency as a function of the device thickness and the radius,where the radius is equal to the diffusion length, for a pn junction basedc-Si photovoltaic solar cell in the radial configuration.72CHAPTER 5. RESULTS5.2.2 Gallium arsenideFigure 5.7 depicts the dependence of the short-circuit current density on thecell thickness and the diffusion length for the case of the planar pn junction basedc-GaAs photovoltaic solar cell. The planar thickness, L, ranges from 0.5 to 50 ?mand the diffusion length ranges from 0.5 to 100 ?m. In Figure 5.8, the short-circuitcurrent density is depicted as a function of the cell radius and the diffusion lengthfor the case of the radial pn junction based c-GaAs photovoltaic solar cell. Con-sistent with Kayes et al. [11], it is seen that for the planar case, the short-circuitcurrent density is dependent on the diffusion length, while the short-circuit cur-rent density corresponding to the radial nano-rod case is essentially independent ofthe diffusion length. The maximum short-circuit current density is approximately0.0285 and 0.0015 mA/cm2, for the planar and radial cases, respectively. The pla-nar case exhibits its maximum short-circuit current density at a cell thickness of50 ?m and a diffusion length of 100 ?m. The radial case, however, exhibits itsmaximum short-circuit current density at a cell thickness of 50 ?m and is essen-tially independent of the radius. The short-circuit current density is independent ofthe depletion region trap density, for both the planar and the radial cases.Figure 5.9 illustrates the dependence of the open-circuit voltage on the cellthickness and the diffusion length for the case of the planar pn junction based c-GaAs photovoltaic solar cell. The planar cell thickness and the diffusion lengthhave the same ranges as that considered earlier in Figure 5.7. The planar case hasa maximum open-circuit voltage of approximately 0.985 V at a diffusion length of100 ?m. Figure 5.10 illustrates the dependence of the open-circuit voltage on thecell thickness and the diffusion length for the case of the radial pn junction based73CHAPTER 5. RESULTSFigure 5.7: The short-circuit current density as a function of the device thick-ness and the diffusion length for a pn junction based c-GaAs photo-voltaic solar cell in the planar configuration.74CHAPTER 5. RESULTSFigure 5.8: The short-circuit current density as a function of the device thick-ness and the radius, where the radius is equal to the diffusion length, fora pn junction based c-GaAs photovoltaic solar cell in the radial config-uration.75CHAPTER 5. RESULTSc-GaAs photovoltaic solar cell. The radial solar cell thickness and the diffusionlength have the same ranges as that considered earlier in Figure 5.8. The radialcase has a maximum Voc of 1.05 V at a cell thickness of 100 ?m. The open-circuitvoltage for the planar case is dependent on the diffusion length, while the open-circuit voltage is found to be relatively insensitive to the diffusion length for theradial case. A decreasing depletion region trap density increases the open-circuitvoltage, for both the planar and radial cases.Figure 5.11 illustrates the dependence of the efficiency, ? , on the cell thick-ness and the diffusion length for the case of the planar pn junction based c-GaAsphotovoltaic solar cell. The planar cell thickness and the diffusion length have thesame ranges as that considered earlier in Figure 5.7. For the planar case, an effi-ciency of approximately 25.4% at a cell thickness of 50 ?m and a diffusion lengthof 100 ?m is observed. The efficiency for the planar solar cells is independent ofthe depletion region trap density. Figure 5.12 illustrates the dependence of the ef-ficiency, ? , on the cell thickness and the diffusion length for the case of the radialpn junction based c-GaAs photovoltaic solar cell. The radial cell thickness and thediffusion length have the same ranges as that considered earlier in Figure 5.8. Theradial case, however, has an efficiency of approximately 22% at a cell thickness of50 ?m and a diffusion length of 10 ?m. The efficiency for the radial solar cell isstrongly dependent on the depletion region trap density.76CHAPTER 5. RESULTSFigure 5.9: The open-circuit voltage as a function of the device thickness andthe diffusion length for a pn junction based c-GaAs photovoltaic solarcell in the planar configuration.77CHAPTER 5. RESULTSFigure 5.10: The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length, for apn junction based c-GaAs photovoltaic solar cell in the radial configu-ration.78CHAPTER 5. RESULTSFigure 5.11: The efficiency as a function of the device thickness and the dif-fusion length for a pn junction based c-GaAs photovoltaic solar cell inthe planar configuration.79CHAPTER 5. RESULTSFigure 5.12: The efficiency as a function of the device thickness and the ra-dius, where the radius is equal to the diffusion length, for a pn junctionbased c-GaAs photovoltaic solar cell in the radial configuration.80CHAPTER 5. RESULTS5.2.3 GermaniumFigure 5.13 depicts the dependence of the short-circuit current density on thecell thickness and the diffusion length for the case of the planar pn junction based c-Ge photovoltaic solar cell. The planar thickness, L, ranges from 0.2 to 100 ?m andthe diffusion length ranges from 0.2 to 100 ?m. In Figure 5.14, the short-circuitcurrent density is depicted as a function of the cell radius and the diffusion lengthfor the case of the radial pn junction based c-Ge photovoltaic solar cell. Consistentwith Kayes et al. [11], it is seen that for the planar case, the short-circuit currentdensity is dependent on the diffusion length, while the short-circuit current densitycorresponding to the radial nano-rod case is essentially independent of the diffu-sion length. The maximum short-circuit current density is approximately 0.0475and 0.048 mA/cm2, for the planar and radial cases, respectively. The planar caseexhibits its maximum short-circuit current density at a cell thickness of 100 ?mand a diffusion length of 100 ?m. The radial case, however, exhibits its maxi-mum short-circuit current density at a cell thickness of 100 ?m and is essentiallyindependent of the radius. The short-circuit current density is independent of thedepletion region trap density, for both the planar and the radial cases.Figure 5.15 illustrates the dependence of the open-circuit voltage on the cellthickness and the diffusion length for the case of the planar pn junction based c-Ge photovoltaic solar cell. The planar cell thickness and the diffusion length havethe same ranges as that considered earlier in Figure 5.13. The planar case has amaximum open-circuit voltage of approximately 0.56 V at a diffusion length of100 ?m. Figure 5.16 illustrates the dependence of the open-circuit voltage on thecell thickness and the diffusion length for the case of the radial pn junction based81CHAPTER 5. RESULTSFigure 5.13: The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-Ge pho-tovoltaic solar cell in the planar configuration.82CHAPTER 5. RESULTSFigure 5.14: The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the diffusionlength, for a pn junction based c-Ge photovoltaic solar cell in the ra-dial configuration.83CHAPTER 5. RESULTSc-Ge photovoltaic solar cell. The radial solar cell thickness and the diffusion lengthhave the same ranges as that considered earlier in Figure 5.14. The radial case has amaximum Voc of 0.62 V at a cell thickness of 100 ?m. The open-circuit voltage forthe planar case is dependent on the diffusion length, while the open-circuit voltageis found to be relatively insensitive to the diffusion length for the radial case. Adecreasing depletion region trap density increases the open-circuit voltage, for boththe planar and radial cases.Figure 5.17 illustrates the dependence of the efficiency, ? , on the cell thicknessand the diffusion length for the case of the planar pn junction based c-Ge photo-voltaic solar cell. The planar cell thickness and the diffusion length have the sameranges as that considered earlier in Figure 5.13. For the planar case, an efficiencyof approximately 21.2% at a cell thickness of 100 ?m and a diffusion length of100 ?m is observed. The efficiency for the planar solar cells is independent of thedepletion region trap density. Figure 5.18 illustrates the dependence of the effi-ciency, ? , on the cell thickness and the diffusion length for the case of the radialpn junction based c-Ge photovoltaic solar cell. The radial cell thickness and thediffusion length have the same ranges as that considered earlier in Figure 5.14. Theradial case, however, has an efficiency of approximately 22% at a cell thickness of100 ?m and a diffusion length of 100 ?m. The efficiency for the radial solar cellis strongly dependent on the depletion region trap density.84CHAPTER 5. RESULTSFigure 5.15: The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-Ge photovoltaic solarcell in the planar configuration.85CHAPTER 5. RESULTSFigure 5.16: The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length, for a pnjunction based c-Ge photovoltaic solar cell in the radial configuration.86CHAPTER 5. RESULTSFigure 5.17: The efficiency as a function of the device thickness and the dif-fusion length for a pn junction based c-Ge photovoltaic solar cell in theplanar configuration.87CHAPTER 5. RESULTSFigure 5.18: The efficiency as a function of the device thickness and the ra-dius, where the radius is equal to the diffusion length, for a pn junctionbased c-Ge photovoltaic solar cell in the radial configuration.88CHAPTER 5. RESULTS5.2.4 Indium nitrideFigure 5.19 depicts the dependence of the short-circuit current density on thecell thickness and the diffusion length for the case of the planar pn junction basedc-InN photovoltaic solar cell. The planar thickness, L, ranges from 0.5 to 50 ?mand the diffusion length ranges from 0.2 to 50 ?m. In Figure 5.20, the short-circuitcurrent density is depicted as a function of the cell radius and the diffusion lengthfor the case of the radial pn junction based c-InN photovoltaic solar cell. Con-sistent with Kayes et al. [11], it is seen that for the planar case, the short-circuitcurrent density is dependent on the diffusion length, while the short-circuit cur-rent density corresponding to the radial nano-rod case is essentially independent ofthe diffusion length. The maximum short-circuit current density is approximately0.053 and 0.054 mA/cm2, for the planar and radial cases, respectively. The planarcase exhibits its maximum short-circuit current density at a cell thickness of 50 ?mand a diffusion length of 50 ?m. The radial case, however, exhibits its maximumshort-circuit current density at a cell thickness of 1000 ?m and is essentially in-dependent of the radius. The short-circuit current density is independent of thedepletion region trap density, for both the planar and the radial cases.Figure 5.21 illustrates the dependence of the open-circuit voltage on the cellthickness and the diffusion length for the case of the planar pn junction basedc-InN photovoltaic solar cell. The planar cell thickness and the diffusion lengthhave the same ranges as that considered earlier in Figure 5.19. The planar casehas a maximum open-circuit voltage of approximately 1.15 V at a diffusion lengthof 50 ?m. Figure 5.22 illustrates the dependence of the open-circuit voltage on thecell thickness and the diffusion length for the case of the radial pn junction based c-89CHAPTER 5. RESULTSFigure 5.19: The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-InN pho-tovoltaic solar cell in the planar configuration.90CHAPTER 5. RESULTSFigure 5.20: The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the diffusionlength, for a pn junction based c-InN photovoltaic solar cell in the ra-dial configuration.91CHAPTER 5. RESULTSInN photovoltaic solar cell. The radial solar cell thickness and the diffusion lengthhave the same ranges as that considered earlier in Figure 5.20. The radial case hasa maximumVoc of 1.21 V at a cell thickness of 50 ?m. The open-circuit voltage forthe planar case is dependent on the diffusion length, while the open-circuit voltageis found to be relatively insensitive to the diffusion length for the radial case. Adecreasing depletion region trap density increases the open-circuit voltage, for boththe planar and radial cases.Figure 5.23 illustrates the dependence of the efficiency, ? , on the cell thick-ness and the diffusion length for the case of the planar pn junction based c-InNphotovoltaic solar cell. The planar cell thickness and the diffusion length have thesame ranges as that considered earlier in Figure 5.19. For the planar case, an effi-ciency of approximately 55.1% at a cell thickness of 50 ?m and a diffusion lengthof 50 ?m is observed. The donor and acceptor concentration are too high to makethis high efficiency realistic. The efficiency for the planar solar cell case is inde-pendent of the depletion region trap density. Figure 5.24 illustrates the dependenceof the efficiency, ? , on the cell thickness and the diffusion length for the case ofthe radial pn junction based c-InN photovoltaic solar cell. The radial cell thicknessand the diffusion length have the same ranges as that considered earlier in Figure5.20. The radial case, however, has an efficiency of approximately 55.2% at a cellthickness of 1000 ?m and a diffusion length of 250 ?m. The efficiency for theradial solar cell is strongly dependent on the depletion region trap density.92CHAPTER 5. RESULTSFigure 5.21: The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-InN photovoltaicsolar cell in the planar configuration.93CHAPTER 5. RESULTSFigure 5.22: The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length, for a pnjunction based c-InN photovoltaic solar cell in the radial configuration.94CHAPTER 5. RESULTSFigure 5.23: The efficiency as a function of the device thickness and the dif-fusion length for a pn junction based c-InN photovoltaic solar cell inthe planar configuration. The donor and acceptor concentration are toohigh to make this high efficiency realistic.95CHAPTER 5. RESULTSFigure 5.24: The efficiency as a function of the device thickness and the ra-dius, where the radius is equal to the diffusion length, for a pn junctionbased c-InN photovoltaic solar cell in the radial configuration. Thedonor and acceptor concentration are too high to make this high effi-ciency realistic.96CHAPTER 5. RESULTS5.2.5 Gallium nitrideFigure 5.25 depicts the dependence of the short-circuit current density on thecell thickness and the diffusion length for the case of the planar pn junction basedc-GaN photovoltaic solar cell. The planar thickness, L, ranges from 1 to 500 ?mand the diffusion length ranges from 1 to 500 ?m. In Figure 5.26, the short-circuitcurrent density is depicted as a function of the cell radius and the diffusion lengthfor the case of the radial pn junction based c-GaN photovoltaic solar cell. Con-sistent with Kayes et al. [11], it is seen that for the planar case, the short-circuitcurrent density is dependent on the diffusion length, while the short-circuit cur-rent density corresponding to the radial nano-rod case is essentially independent ofthe diffusion length. The maximum short-circuit current density is approximately0.0016 and 0.0017 mA/cm2, for the planar and radial cases, respectively. The pla-nar case exhibits its maximum short-circuit current density at a cell thickness of500 ?m and a diffusion length of 500 ?m. The radial case, however, exhibits itsmaximum short-circuit current density at a cell thickness of 500 ?m and is essen-tially independent of the radius. The short-circuit current density is independent ofthe depletion region trap density, for both the planar and the radial cases.Figure 5.27 illustrates the dependence of the open-circuit voltage on the cellthickness and the diffusion length for the case of the planar pn junction based c-GaN photovoltaic solar cell. The planar cell thickness and the diffusion lengthhave the same ranges as that considered earlier in Figure 5.25. The planar case hasa maximum open-circuit voltage of approximately 2.66 V at a diffusion length of500 ?m. Figure 5.28 illustrates the dependence of the open-circuit voltage on thecell thickness and the diffusion length for the case of the radial pn junction based97CHAPTER 5. RESULTSFigure 5.25: The short-circuit current density as a function of the devicethickness and the diffusion length for a pn junction based c-GaN pho-tovoltaic solar cell in the planar configuration.98CHAPTER 5. RESULTSFigure 5.26: The short-circuit current density as a function of the devicethickness and the radius, where the radius is equal to the diffusionlength, for a pn junction based c-GaN photovoltaic solar cell in theradial configuration.99CHAPTER 5. RESULTSc-GaN photovoltaic solar cell. The radial solar cell thickness and the diffusionlength have the same ranges as that considered earlier in Figure 5.26. The radialcase has a maximum Voc of 2.7 V at a cell thickness of 500 ?m. The open-circuitvoltage for the planar case is dependent on the diffusion length, while the open-circuit voltage is found to be relatively insensitive to the diffusion length for theradial case. A decreasing depletion region trap density increases the open-circuitvoltage, for both the planar and radial cases.Figure 5.29 illustrates the dependence of the efficiency, ? , on the cell thick-ness and the diffusion length for the case of the planar pn junction based c-GaNphotovoltaic solar cell. The planar cell thickness and the diffusion length havethe same ranges as that considered earlier in Figure 5.25. For the planar case, anefficiency of approximately 3.88% at a cell thickness of 500 ?m and a diffusionlength of 500 ?m is observed. The efficiency for the planar solar cells is indepen-dent of the depletion region trap density. Figure 5.30 illustrates the dependence ofthe efficiency, ? , on the cell thickness and the diffusion length for the case of theradial pn junction based c-GaN photovoltaic solar cell. The radial cell thicknessand the diffusion length have the same ranges as that considered earlier in Figure5.26. The radial case, however, has an efficiency of approximately 3.55% at a cellthickness of 500 ?m and a diffusion length of 500 ?m. The efficiency for the radialsolar cell is strongly dependent on the depletion region trap density. The dopingconcentrations for GaN are too high resulting in unrealistic efficiencies.100CHAPTER 5. RESULTSFigure 5.27: The open-circuit voltage as a function of the device thicknessand the diffusion length for a pn junction based c-GaN photovoltaicsolar cell in the planar configuration.101CHAPTER 5. RESULTSFigure 5.28: The open-circuit voltage as a function of the device thicknessand the radius, where the radius is equal to the diffusion length, for apn junction based c-GaN photovoltaic solar cell in the radial configu-ration.102CHAPTER 5. RESULTSFigure 5.29: The efficiency as a function of the device thickness and the dif-fusion length for a pn junction based c-GaN photovoltaic solar cell inthe planar configuration.103CHAPTER 5. RESULTSFigure 5.30: The efficiency as a function of the device thickness and the ra-dius, where the radius is equal to the diffusion length, for a pn junctionbased c-GaN photovoltaic solar cell in the radial configuration.104CHAPTER 5. RESULTS5.3 ComparisonsEach of the materials considered in this analysis have different characteristics,which results in a different short-circuit current density, open-circuit voltage, andefficiency characteristics. In addition, the solar cell thickness and diffusion lengthselection are considered for each material, as these parameters shape the perfor-mance characteristics. Tables 5.1 and 5.2 summarizes the planar and radial basedsolar cell performance characteristics, respectively.Table 5.1: The semiconductor cell thickness and diffusion length geometriesfor planar Si, GaAs, Ge, InN, and GaN based photovoltaic solar cells.Description Si GaAs Ge InN GaNMin. Cell Thickness (?m) 1 0.4 0.3 1 1Max. Cell Thickness (?m) 63.1 100 10 100 631Min. Diffusion Length (?m) 0.1 0.4 0.1 1 1Max. Diffusion Length (?m) 100 10 100 100 1000Table 5.2: The cell thickness and diffusion length geometries for radial Si,GaAs, Ge, InN, and GaN based photovoltaic solar cells.Description (?m) Si GaAs Ge InN GaNMin. Cell Thickness (?m) 1 0.1 0.01 1 1Max. Cell Thickness (?m) 631 63.1 3.2 1000 1000Min. Diffusion Length (?m) 0.2 0.1 0.1 0.1 0.3Max. Diffusion Length (?m) 100 10 100 1000 100The various material and geometry selections result in different short-circuitcurrent densities, open-circuit voltages, and efficiencies. These results are summa-rized in Table 5.3.In general, the short-circuit current density is independent of the trap densityin the diffusion length and is also independent of the depletion region trap density.105CHAPTER 5. RESULTSTable 5.3: The cell thickness and diffusion length geometries for planar andradial Si, GaAs, Ge, InN, and GaN based photovoltaic solar cells.Description Si GaAs Ge InN GaNPlanar Jsc (mA/cm2) 0.033 0.0285 0.0475 0.053 0.0016Radial Jsc (mA/cm2) 0.038 0.0015 0.048 0.054 0.0017Planar Voc (V) 0.6 0.985 0.56 1.15 2.66Radial Voc (V) 0.7 1.05 0.62 1.21 2.7Planar Efficiency (%) 16.6 25.4 21.2 55.1 3.88Radial Efficiency (%) 17.3 22 22 55.2 3.55The short-circuit current density for radial solar cells are dependent on the cellthickness because more photons are absorbed into the material. The short-circuitcurrent density plateaus at the maximum photon absorption depth.For both the radial and planar cases, the open-circuit voltage increases withdecreasing trap densities. The open-circuit voltage, for the planar case, is indepen-dent of the cell thickness, but a function of the diffusion length. For the radial case,however, the open-circuit voltage has a maximum at the minimum cell thicknessand maximum radius.5.4 SummaryThis chapter provides results and comparisons for each material in the planarand radial photovoltaic solar cell configurations. In general, radial solar cells areindependent of the diffusion length while planar solar cells are dependent on thediffusion length. However, the different geometries, for each type of photovoltaicsolar cell, provides different results. For Si, Ge, and InN, the radial solar cells effi-ciencies are marginally better than those associated with their planar photovoltaicsolar cells counterparts.106Chapter 6Conclusions and Possible FutureWorkThe photovoltaic solar cell performance model of Kayes et al. [11] was em-ployed for the purposes of this analysis. This analysis contribution is that extendsthe Kayes et al. [11] from Si and GaAs to Ge, InN, and GaN. Three solar cellperformance metrics, evaluated using the Kayes et al. [11] model were consideredin this analysis: (1) the short-circuit current density, (2) the open-circuit voltage,and (3) the efficiency. The results suggest that while planar pn junction basedphotovoltaic solar cells are sensitive to trapping concentration levels, the radial pnjunction based photovoltaic solar cells are relatively insensitive to trapping concen-trations. This suggests that in certain cases, such as when there are materials withhigh concentration of traps, radial pn junction based photovoltaic solar cell offeran inherent advantage.The model of Kayes et al. [11] has a number of fundamental limitations. These107CHAPTER 6. CONCLUSIONS AND POSSIBLE FUTURE WORKinclude: (1) the assumption that the photon flux is incident from the top of theradial solar cell, (2) the assumption that the current density is purely radial, and(3) the assumption that the material properties for the radial case are equivalent tothose of the planar case. The material properties for the radial case may, in fact,differ significantly from those of the planar case. Limitations, common to both theplanar and radial cases, include: (1) a neglect of Auger recombination, and (2) aneglect of temperature effects.As noted, there is a lot of room for improvement in this analysis. An improve-ment in the models would be a more advanced treatment of the incident photon flux,including consideration of internal reflections generated from the space in and be-tween each cell. In addition, the role that implanting metallic spheres between thecells plays in enhancing their performance should be examined. The propagation oflight in metallic nanowire arrays could also be considered [23]. Finite-difference-time-domain and frequency-dependent-time-domain solutions can be applied tosolve for the scattering and absorption occurring within complex geometries, suchas those within an array of nanorods. Further research will have to be conductedon the light trapping within Si based nanowire solar cells [24].Furthermore, different geometries can also be explored, including sphericalphotovoltaic solar cells. The silicon spherical solar cell is an option for low costphotovoltaic electrical energy generation, as they do not require the electronicgrade Si wafers demanded of planar photovoltaic solar cells, while the crystal qual-ity and the purity of such spherical cells can be improved [25]. The spherical Sisolar cell?s detailed material and electronic characterization has been performedby Gharghi [25], with experimental results on crystalline impurities and struc-ture [26].108CHAPTER 6. CONCLUSIONS AND POSSIBLE FUTURE WORKAnother major improvement in this analysis would be the material characteri-zation of radial solar cells. For the purposes of this analysis, it has been assumedthat the planar and radial material properties are similar. So, material properties,such as the surface recombination velocity, could be re-evaluated. Demichel etal. [27] has conducted research on the surface recombination velocity within sili-con nanowires, finding this to be quite distinct from the associated with c-Si.An additional improvement may be to consider the thermal effects on the cur-rent density, open-circuit voltage, and efficiencies. Many semiconductor materialproperties and characteristics are dependent upon temperature. The most importantcharacteristic is the dependence of the diffusion length on temperature. Recall thatthe diffusion length is dependent on the mobility, and mobility is temperature de-pendent. The radial photovoltaic solar cell is relatively independent of the diffusionlength, while the planar solar cell is dependent on the diffusion length. Therefore,radial solar cells should exhibit performance characteristics that are more temper-ature independent than planar solar cells.109References[1] OECD, ?Oecd factbook 2011-2012: Economic, environmental and socialstatistics,? Aug. 2013.[2] A. 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Gharghi, ?Spherical silicon for photovoltaic applications: Material, mod-eling and devices,? IEEE TRANSACTIONS ON ELECTRON DEVICES,,vol. 53, 2006.111REFERENCES[27] O. Demichel, V. Calvo, A. Besson, P. Noe, B. Salem, N. Pauc, F. Oehler,P. Gentile, and N. Magnea, ?Propagration of light in metallic nanowire ar-rays,? The American Physical Society, vol. 65, 2003.112Appendix AMatlab CodeThis would be any supporting material not central to the dissertation.A.1 Radial P-N Junction1 %Author: Brendan Kayes2 %Title: Radial pn Junction, Wire Array Solar Cells3 %Publisher: California Institute of Technology4 %Location: Pasedena California5 %Year: 200967 %Modifying Author: Ska - Hiish Manuel (SM)8 %Year: 20139 %Institution: University of British Columbia - Okanagan1011 %page 1221213 format short g14 warning off MATLAB:fzero:UndeterminedSyntax15 warning off MATLAB:divideByZero %#ok<RMWRN>16 clear all17 clc1819 %%%% Start Modified - SM %%%%20 [type, sheets] = xlsfinfo('materialProperties.xls')21 [smarts2Num, smarts2Txt, smarts2Raw] = ...xlsread('materialProperties.xls', 'materialProperties', ...'B8:I2008');22 %[smarts2Num, smarts2Txt, smarts2Raw] = ...xlsread('matProperties', 'matProp', 'A2:C44');113APPENDIX A. MATLAB CODE23 [parameters2Num, parameters2Txt, parameters2Raw] = ...xlsread('materialProperties.xls', 'Parameters', 'C4:H18');24 %Si 1, GaAs 2, Ge 3, GaN 4, InN Cu1489 5, InN CuW251 625 materialNumber = 52627 parameters = parameters2Num(:, materialNumber)28 alphaColumn = parameters(1)29 size(smarts2Num)30 smarts2Num(any(smarts2Num(:, alphaColumn) == 0, 2), :) = ...[]; %used to remove the zeros from alpha31 size(smarts2Num)32 E = smarts2Num(:, 1);33 flux = smarts2Num(:, 2);34 alpha = smarts2Num(:, alphaColumn);35 %%%% End Modified - SM %%%%3637 n = length(E) - 13839 dE = (E(2:(n + 1)) - E(1:n));40 alpha2 = alpha(2:(n + 1));41 alpha1 = alpha2; % alpha1 and alpha2 are the absorption ...coefficients of the n - and p - type materials, ...respectively42 gammaA = flux(2:(n + 1));43 %%gammaA = flux(1:(n));44 gammaC = gammaA; % gammaA and gammaA are the photon fluxes ...incident on the top of the n - and p - type ...quasineutral regions, respectively45 matrix = [];4647 %%%% Start Modified - SM %%%%48 %silicon lengths49 %lengths = [1e - 4, 2e - 4, 5e - 4, 1e - 3, 2e - 3, 5e - 3, ...1e - 2, 2e - 2, 5e - 2, 1e - 1]; % lengths is a vector ...containing the various cell thicknesses that will be ...considered, in cm50 resolution = 10;51 switch materialNumber52 case 1 %Si53 Lpower = - 4.054 Rpower = - 4.855 lengths = logspace(Lpower, - 1.2, resolution);56 R = logspace(Rpower, - 2, resolution);57 materialSymbol = 'Si'58 case 2 %GaAs59 Lpower = - 5.060 Rpower = - 5.061 lengths = logspace(Lpower, - 2.2, resolution);62 R = logspace(Rpower, - 3, resolution);114APPENDIX A. MATLAB CODE63 materialSymbol = 'GaAs'64 case 3 %Ge65 Lpower = - 666 Rpower = - 5.067 lengths = logspace(Lpower, - 3.5, resolution);68 R = logspace(Rpower, - 2, resolution);69 materialSymbol = 'Ge'70 case 4 %GaN71 Lpower = - 4.072 Rpower = - 4.573 lengths = logspace(Lpower, - 1, resolution);74 R = logspace(Rpower, - 2, resolution);75 materialSymbol = 'GaN'76 %R = logspace( - 5, - 2, 40); This works but this ...an eblow at R ? = 4.877 case 5 %InN Cu148978 Lpower = - 4.079 Rpower = - 5.080 lengths = logspace(Lpower, - 1, resolution);81 R = logspace(Rpower, - 1, resolution);82 materialSymbol = 'InNCu1489'83 case 6 %InN CuW25184 Lpower = - 485 Rpower = - 586 lengths = logspace(Lpower, - 1, resolution);87 R = logspace(Rpower, - 3, resolution);88 materialSymbol = 'InNCuW251'89 end90 Rpowermin = 0.01 * 10?(floor(Rpower)) / 1e-6; %in um91 Lpowermin = lengths(1) / 1e-3; %in mm92 %GaAs lengths93 %lengths = [2e - 5, 5e - 5, 1e - 4, 2e - 4, 5e - 4, 1e - 3, ...2e - 3, 5e - 3, 1e - 2];94 %%%% END Modified - SM %%%%9596 %page 1239798 thing = length(lengths);99 %dope = [1, 10, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8] * 5e12;100 %muele = [1500, 1500, 1500, 1500, 1000, 400, 150, 100, 100];101 %muhol = [500, 500, 500, 400, 300, 150, 60, 50, 50]; %these ...lines indicate the maximum102 %electron and hole mobilities possible in Si at various ...doping levels, as103 %given by [52], for reference104 %max Na = [>1e20, >1e20, 9e19, 4e19, 2e19, 9e18, ...4e18, 2e18, 8e17, 3e17]105 %vs Ln = [1e - 5, 2e - 5, 5e - 5, 1e - 4, 2e - 4, 5e ...- 4, 1e - 3, 2e - 3, 5e - 3, 1e - 2]115APPENDIX A. MATLAB CODE106 % = > mun = [95, 95, 95, 100, 110, 130, 160, 195, ...300, 540] %these lines indicate the maximum electron ...diffusion length possible in Si at various doping ...levels, as given by [53], for reference107108 %%%% Start Modified - SM %%%%109 %Define constants110 q = parameters(2);111 kB = parameters(3);112 T = parameters(4);113 %Silicon data114 %Eg = 1.1115 Nc = parameters(5);116 Nv = parameters(6);117 ni = parameters(7);118 dopants = parameters(8);119 Na = dopants;120 mun = parameters(9);121 Nd = dopants;122 mup = parameters(10);123 %%%% END Modified - SM %%%%124125 %page 124126127 Dn = kB * T / q * mun;128 Dp = kB * T / q * mup;129130 %%%% Start Modified - SM %%%%131 Sn = parameters(11);132 sigman = parameters(12);133 sigmap = sigman;134 vth = parameters(13);135 %%%% End Modified - SM %%%%136137 if ni?2 ? Na * Nd138 break139 end140 np0 = ni?2 / Na;141 pn0 = ni?2 / Nd;142 Vbi = kB * T / q * log(Na * Nd / ni?2)143144 %%%% Start Modified - SM %%%%145 epsilon0 = parameters(14);146 epsilonn = parameters(15) * epsilon0; epsilonp = epsilonn; ...epsilons = epsilonn;147 %%%% End Modified - SM %%%%148149 N = length(R);150 count = 1:N;116APPENDIX A. MATLAB CODE151 for thingee = 1:thing %% different lengths152 L = lengths(thingee) * ones(1, N);153 aspectratio = L./ R;154 V = linspace(0, 95 / 100 * Vbi); %the program assumes ...that Voc < 0.95 Vbi155 NN = length(V);156157 %page 125158159 x1 = zeros(NN, N);160 x2 = zeros(NN, N);161 x3 = zeros(NN, N);162 x4 = zeros(NN, N);163 J0A = zeros(NN, N);164 J0C = zeros(NN, N);165 J0 = zeros(NN, N);166 JlAperE = zeros(n, N, NN);167 JlCperE = zeros(n, N, NN);168 JlA = zeros(NN, N);169 JlC = zeros(NN, N);170 Jl = zeros(NN, N);171 Jrdep = zeros(NN, N);172 JdepAperE = zeros(n, N, NN);173 JdepCperE = zeros(n, N, NN);174 JdepA = zeros(NN, N);175 JdepC = zeros(NN, N);176 Jgdep = zeros(NN, N);177 J = zeros(NN, N);178 bias = zeros(NN, N);179 VBI = zeros(NN, N);180 Voc = zeros(1, N);181 Jsc = zeros(1, N);182 A1A = zeros(NN, N);183 A1C = zeros(NN, N);184 A2A = zeros(NN, N);185 A2C = zeros(NN, N);186 B1A = zeros(NN, N);187 B2A = zeros(NN, N);188 trapDensity = zeros(NN, N);189 for i = 1:N %% Different Radii190191 %page 126192193 disp(round(((thingee - 1) / thing + i / (thing * ...N)) * 100)) %a counter, so that the user ...doesn't think the program has crashed194 for ii = 1:NN %% Different Voltages195 Ln = R(i); %setting radius = minority ...electron diffusion length117APPENDIX A. MATLAB CODE196 Nr = Dn / (Ln?2 * sigman * vth);197 taun0 = 1 / (sigman * Nr * vth);198 taup0 = 1 / (sigmap * Nr * vth);199 taun = taun0;taup = taup0;200 %taun0 = 1e - 6; taup0 = taun0; %activate this ...line to set the depletion region minority ...electron lifetime tn0 separately from the ...quasineutral region minority electron ...lifetime tn (similarly for holes)201 trapDensity(ii, i) = taun0;202 Lp = sqrt(taup * Dp);203 windowwidth = 1e-6; %minimum emitter thickness, ...in cm204 bias(ii, i) = V(ii);205 [x1(ii, i), x2(ii, i), x3(ii, i), x4(ii, i), ...VBI(ii, i)] = depletionregion(Nd, Na, R(i), ...epsilonp, epsilonn, ni, bias(ii, i), ...windowwidth);206 if bias(ii, i)>VBI(1, i)207 break208 end209 d1(ii, i) = x1(ii, i) + x2(ii, i);210 d2(ii, i) = x3(ii, i) + x4(ii, i);211 %define dimensionless variables212 beta1(ii, i) = R(i) / Lp;213 beta2(ii, i) = (R(i) - x1(ii, i)) / Lp;214 beta3 = alpha1 * L(i);215 beta4 = Lp * Sn / Dp;216 beta5(ii, i) = x4(ii, i) / Ln;217218 %page 127219220 beta6 = alpha2 * L(i);221 f1(ii, i) = besseli(1, beta1(ii, i)) + beta4 * ...besseli(0, beta1(ii, i));222 f2(ii, i) = besselk(1, beta1(ii, i)) - beta4 * ...besselk(0, beta1(ii, i));223224 JlnperE(:, i, ii) = - 2 * q * gammaA * ...beta2(ii, i) / (beta1(ii, i))?2 * ( - ...besseli(1, beta2(ii, i)) * (f2(ii, i) + ...beta4 * ...225 besselk(0, beta2(ii, i))) + besselk(1, ...beta2(ii, i)) * (f1(ii, i) - ...226 beta4 * besseli(0, beta2(ii, i)))) / (f1(ii, i) ...* ...227 besselk(0, beta2(ii, i)) + f2(ii, i) * ...besseli(0, beta2(ii, i))).* (1 - exp( - ...beta3)).* dE;118APPENDIX A. MATLAB CODE228229 Jln(ii, i) = sum(JlnperE(:, i, ii));230231 J0n(ii, i) = 2 * L(i) * q * pn0 / taup * ...beta2(ii, i) / (beta1(ii, i))?2 * ...232 (f2(ii, i) * besseli(1, beta2(ii, i)) - f1(ii, ...i) * ...233 besselk(1, beta2(ii, i))) / (f1(ii, i) * ...besselk(0, beta2(ii, i)) + ...234 f2(ii, i) * besseli(0, beta2(ii, i)));235236 JlpperE(:, i, ii) = 2 * q * gammaC * Ln?2 / ...Lp?2 * beta5(ii, i) / ...237 (beta1(ii, i))?2 * besseli(1, beta5(ii, i)) / ...238 besseli(0, beta5(ii, i)) .* (1 - exp( - beta6)) ....* dE;239240 Jlp(ii, i) = sum(JlpperE(:, i, ii));241242 Jlp(ii, i) = - Jlp(ii, i);243244 J0p(ii, i) = - 2 * L(i) * q * np0 * Dn / Lp?2 ...* beta5(ii, i) / (beta1(ii, i))?2 * ...245 besseli(1, beta5(ii, i)) / besseli(0, beta5(ii, ...i));246247 JgdepnperE(:, i, ii) = - q * gammaA * ((d2(ii, ...i) + x2(ii, i))?2 - ...248 (d2(ii, i))?2) / (R(i))?2 .* (1 - exp( - ...beta3)).* dE;249250 Jgdepn(ii, i) = sum(JgdepnperE(:, i, ii));251252 JgdeppperE(:, i, ii) = - q * gammaC * ((d2(ii, ...i))?2 - (x4(ii, i))?2) / ...253 (R(i))?2 .* (1 - exp( - beta6)) .* dE;254255 Jgdepp(ii, i) = sum(JgdeppperE(:, i, ii));256257 U(ii, i) = ni / sqrt(taun0 * taup0) * sinh(q * ...bias(ii, i) / (2 * kB * T));258 kappa(ii, i) = pi * kB * T / (q * (VBI(ii, i) - ...bias(ii, i)));259260 %page 128261262 r(ii, i) = x4(ii, i) + (log(Na / ni)) / (log(Nd .../ ni) + log(Na / ni)) * (x2(ii, i) + ...263 x3(ii, i));119APPENDIX A. MATLAB CODE264 r2(ii, i) = r(ii, i) + (x2(ii, i) + x3(ii, i)) ...* 1 / 2 * kappa(ii, i);265 r1(ii, i) = r(ii, i) - (x2(ii, i) + x3(ii, i)) ...* 1 / 2 * kappa(ii, i);266267 Jrdep(ii, i) = - q * L(i) * U(ii, i) * (r2(ii, ...i)?2 - r1(ii, i)?2) / (R(i))?2;268 end269 end270 Jgdep = Jgdepn + Jgdepn;271272 Jgdep = [Jgdep;zeros(NN - length(Jgdep), N)];273274 Jl = Jln + Jlp;275 Jl = [Jl;zeros(NN - length(Jl), N)];276 J0 = J0n + J0p;277 J0 = [J0;zeros(NN - length(J0), N)];278279 J = (J0 .* (exp(q * bias / (kB * T)) - 1) - Jl - Jgdep ...+ Jrdep);280 sizeJ0 = size(J0);281 sizeBias = size(bias);282 sizeJ1 = size(Jl);283 sizeJgdep = size(Jgdep);284 sizeJrdep = size(Jrdep);285 sizeJ = size(J);286 temp = [J(1, :);zeros(NN - 1, N)];287288 J(bias == 0) = 0;289 J = J + temp;290 P = J .* bias;291 P(P<0) = 0;292 [maxP, posn] = max(P);293 eff = maxP / (.1) * 100;294 Jsc = J(1, :);295 b = sum(J>0);296297 for j = 1:N;298 if b(j) 6= 0299 if b(j) == 100300 Voc(j) = 0;301302 %page 129303304 elseif bias(b(j) + 1, j) == 0305 Voc(j) = 0;306 else307 Voc(j) = (bias(b(j), j) + bias(b(j) + 1, ...j)) / 2;120APPENDIX A. MATLAB CODE308 end309 else310 Voc(j) = 0;311 end312 end313 P;314 Voc = Voc;315 Jsc = Jsc;316 ff = maxP ./ (Voc .* Jsc);317 arshort = aspectratio(ff<0.9999&ff>0);318 Lshort = L(ff<0.9999&ff>0);319 Rshort = R(ff<0.9999&ff>0);320 Vocshort = Voc(ff<0.9999&ff>0);321 Jscshort = Jsc(ff<0.9999&ff>0);322 ffshort = ff(ff<0.9999&ff>0);323 effshort = eff(ff<0.9999&ff>0);324 %disp(' Aspect Ratio L (cm) R (cm) Jsc (A / cm?2) Voc ...(V) ff Efficiency(%)')325 %disp([aspectratio', L', R', Jsc', Voc', ff', eff'])326 matrix = [matrix; arshort', Lshort', Rshort', ...effshort', Jscshort', Vocshort', ffshort'];327 end328 t1 = num2str(Lp, 4);329 t2 = num2str(Ln, 4);330 t3 = num2str(Sn, 4);331 t6 = num2str(mup, 4);332 t7 = num2str(mun, 4);333334 %page 130335336 t8 = num2str(Na, 4);337 t9 = num2str(Nd, 4);338 t10 = num2str(windowwidth, 4);339 t11 = num2str(taun0, 4);340 t12 = num2str(taup0, 4);341 name = sprintf('Radial Silicon matrix Ln = %s, Lp = %s, ...Sn = %s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s.txt', t2, t1, t3, t11, t12, t8, t9, ...t10);342 %save(name, 'matrix', ' - ASCII', ' - TABS')343 Rmin = min(matrix(:, 3))344 Rmax = max(matrix(:, 3))345 R = R((R ? Rmin)&(R ? Rmax));346 nR = length(R);347 Lmin = min(matrix(:, 2))348 Lmax = max(matrix(:, 2))349 L = lengths((lengths ? Lmin)&(lengths ? Lmax));350 nL = length(L);351 index = cumsum(ones(1, nR));121APPENDIX A. MATLAB CODE352 for i = 1:nL353 temp = matrix(matrix(:, 2) == L(i), 4);354 temp = temp';355 tempJsc = matrix(matrix(:, 2) == L(i), 5);356 tempJsc = tempJsc';357 tempVoc = matrix(matrix(:, 2) == L(i), 6);358 tempVoc = tempVoc';359 tempff = matrix(matrix(:, 2) == L(i), 7);360 tempff = tempff';361 posR = (min(matrix(matrix(:, 2) == L(i), 3)) == R) * ...index';362 matrix2(i, :) = [zeros(1, posR - 1), temp, zeros(1, nR ...- length(temp) - posR + 1)];363 matrixJsc(i, :) = [zeros(1, posR - 1), tempJsc, ...zeros(1, nR - length(tempJsc) - ...364 posR + 1)];365366 %page 131367368 matrixVoc(i, :) = [zeros(1, posR - 1), tempVoc, ...zeros(1, nR - length(tempVoc) - ...369 posR + 1)];370 matrixff(i, :) = [zeros(1, posR - 1), tempff, zeros(1, ...nR - length(tempff) - posR + 1)];371 end372 matrix2 = [L', matrix2];373 matrix2 = [0, R;matrix2];374 matrix2(:, 2);375 matrixJsc = [L', matrixJsc];376 matrixJsc = [0, R;matrixJsc];377 matrixVoc = [L', matrixVoc];378 matrixVoc = [0, R;matrixVoc];379 matrixff = [L', matrixff];380 matrixff = [0, R;matrixff];381 matrix(:, 4:6);382 maxeff = num2str(max(max(matrix2)), 3);383 [junk, row] = max(matrix2);384 [junk, column] = max(max(matrix2));385 row = row(column);386 name2 = sprintf('Radial Silicon Eff Ln = %s, Lp = %s, Sn ...= %s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', ...387 t2, t1, t3, t11, t12, t8, t9, t10, maxeff);388 name3 = sprintf('Radial Silicon Jsc Ln = %s, Lp = %s, Sn ...= %s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', ...389 t2, t1, t3, t11, t12, t8, t9, t10, maxeff);390 name4 = sprintf('Radial Silicon Voc Ln = %s, Lp = %s, Sn ...= %s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...122APPENDIX A. MATLAB CODEwindow width = %s, maxeff = %s.txt', ...391 t2, t1, t3, t11, t12, t8, t9, t10, maxeff);392393 %page 132394395 name5 = sprintf('Radial Silicon ff Ln = %s, Lp = %s, Sn = ...%s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', t2, t1, t3, t11, ...t12, t8, t9, t10, maxeff);396 % save(name2, 'matrix2', ' - ASCII', ' - TABS')397 % save(name3, 'matrixJsc', ' - ASCII', ' - TABS')398 % save(name4, 'matrixVoc', ' - ASCII', ' - TABS')399 % save(name5, 'matrixff', ' - ASCII', ' - TABS')400401 %%%% Start Modified - SM %%%%402403 h5 = figure(5)404 %surf(log10(R), log10(L), matrix2(2:(nL + 1), 2:(nR + 1)))405 surf(R, L, matrix2(2:(nL + 1), 2:(nR + 1)))406 set(gca, 'yscale', 'log')407 set(gca, 'xscale', 'log')408 xlabel('R (cm)')409 ylabel('L (cm)')410 zlabel('Efficiency (%)')411 name2a = sprintf('%s Radial cell Eff. with Ln = %s, Na = ...%s, Nd = %s, Sn = %s, maxEff = %s', materialSymbol, ...t2, t8, t9, t3, maxeff);412 title(name2a)413 % set(gca, 'XTickLabel', {'0.01', '0.1', '1', '10'})414 % set(gca, 'YTickLabel', {'0.001', '0.01', '0.1', '1'})415 fileName = sprintf('Eff%sRadial', materialSymbol)416 saveas(h5, fileName, 'png')417418 h6 = figure(6)419 %surf(log10(R), log10(L), matrixJsc(2:(nL + 1), 2:(nR + 1)))420 surf(R, L, matrixJsc(2:(nL + 1), 2:(nR + 1)))421 set(gca, 'yscale', 'log')422 set(gca, 'xscale', 'log')423 %axis 'tight'424 xlabel('R (cm)')425 ylabel('L (cm)')426 zlabel('Jsc')427 name2b = sprintf('%s Radial cell Jsc with Ln = %s, Na = ...%s, Nd = %s, Sn = %s, maxEff = %s', materialSymbol, ...t2, t8, t9, t3, maxeff);428 title(name2b)429 %set(gca, 'XTick', [floor(Rpower), floor(Rpower) + 1, ...floor(Rpower) + 2, floor(Rpower) + 3])123APPENDIX A. MATLAB CODE430 %set(gca, 'XTickLabel', {Rpowermin * 1, Rpowermin * 10, ...Rpowermin * 100, Rpowermin * 1000})431 %set(gca, 'YTickLabel', {'0.001', '0.01', '0.1', '1'})432 %set(gca, 'XMinorTick', 'on', 'YMinorTick', 'on')433 fileName = sprintf('Jsc%sRadial', materialSymbol)434 saveas(h6, fileName, 'png')435436 h7 = figure(7)437 %surf(log10(R), log10(L), matrixVoc(2:(nL + 1), 2:(nR + 1)))438 surf(R, L, matrixJsc(2:(nL + 1), 2:(nR + 1)))439 set(gca, 'yscale', 'log')440 set(gca, 'xscale', 'log')441 xlabel('R (cm)')442 ylabel('L (cm)')443 zlabel('Voc')444445 %page 133446447 name2c = sprintf('%s Radial cell Voc with Ln = %s, Na = ...%s, Nd = %s, Sn = %s, maxEff = %s', materialSymbol, ...t2, t8, t9, t3, maxeff);448 title(name2c)449 % set(gca, 'XTickLabel', {'0.01', '0.1', '1', '10'})450 % set(gca, 'YTickLabel', {'0.001', '0.01', '0.1', '1'})451 fileName = sprintf('Voc%sRadial', materialSymbol)452 saveas(h7, fileName, 'png')453454 h8 = figure(8)455 %surf(log10(R), log10(L), matrixff(2:(nL + 1), 2:(nR + 1)))456 surf(R, L, matrixJsc(2:(nL + 1), 2:(nR + 1)))457 set(gca, 'yscale', 'log')458 set(gca, 'xscale', 'log')459 xlabel('R (cm)')460 ylabel('L (cm)')461 zlabel('ff')462 name2d = sprintf('%s Radial cell ff with Ln = %s, Na = %s, ...Nd = %s, Sn = %s, maxEff = %s', materialSymbol, t2, ...t8, t9, t3, maxeff);463 title(name2d)464 fileName = sprintf('Ff%sRadial', materialSymbol)465 saveas(h8, fileName, 'png')466 disp('Completed')467 %%%% End Modified - SM %%%%468469 beep470 disp('Completed')124APPENDIX A. MATLAB CODEA.2 P-N Junction Depletion Region1 %Author: Brendan Kayes2 %Title: Radial pn Junction, Wire Array Solar Cells3 %Publisher: California Institute of Technology4 %Location: Pasedena California5 %Year: 200967 %Modifying Author: Ska - Hiish Manuel (SM)8 %Year: 20139 %Institution: University of British Columbia - Okanagan1011 %page 1331213 function [x1, x2, x3, x4, Vbi] = depletionregion(Nd, Na, R, ...epsilonp, epsilonn, ni, ...14 bias, windowwidth) % depletionregion.m %calculates the ...thickness of each region of the pn junction in the ...cylindrical geometry, and the built in voltage, ...given certain dopant densities and a wire of a ...given radius and minimum emitter thickness, at a ...given bias15 q = 1.60219e-19;16 kB = 1.3807e-23;17 T = 300;18 Vbi = kB * T / q * log(Na * Nd / ni?2);19 if bias ? Vbi20 x1 = 0;21 x2 = 0;2223 %page 1342425 x3 = 0;26 x4 = 0;27 Vbi = 0;28 %disp( 3 )29 elseif bias<Vbi30 x3guess = sqrt(2 * Nd * epsilonn * epsilonp * (Vbi ...- bias) / (q * Na * (Nd * epsilonn + Na * ...31 epsilonp)));32 x3old = 0;33 x3new = sqrt(2 * Nd * epsilonn * epsilonp * (Vbi - ...bias) / (q * Na * (Nd * epsilonn + Na * ...34 epsilonp)));35 %disp( 4 )36 while abs(x3old - x3new)>1e-937 x3old = x3new;125APPENDIX A. MATLAB CODE38 %%x3new = fzero(@depletion, x3old, [], Vbi - ...bias, 0, epsilonp, epsilonn, Na, Nd);39 x3new = fzero(@(x)depletion(x, Vbi - bias, 0, ...epsilonp, epsilonn, Na, Nd), x3old);40 end41 x3max = x3new;42 x2max = sqrt(Na / Nd * x3max?2 + x3max?2) - x3max;43 %VBI = q * Na / (4 * epsilonp) * x3max?2 + q * Nd / ...(4 * epsilonn) * x3max?2 - ...44 %q * Nd / (4 * epsilonn) * R?2 + q * Nd / (2 * ...epsilonn) * R?2 * log(R / x3max);45 if (x2max + x3max) ? R46 x4 = 0;47 x1 = 0;48 x2 = x2max * R / (x2max + x3max);49 x3 = x3max * R / (x2max + x3max);50 Vbi = (q * Na / (4 * epsilonp) * x3?2 - q * Nd .../ (4 * epsilonn) * ((x3 + x2)?2 - x3?2) + ...51 q * Nd / (2 * epsilonn) * (x3 + x2)?2 * ...(log((x3 + x2) / x3)));5253 %page 1355455 %disp( 5 )56 elseif (x2max + x3max)<R57 x4old = 1;58 x4new = 0;59 while abs(x4old - x4new)>1e-960 x4old = x4new;61 x3 = fzero(@depletion, x3guess, [], Vbi - ...bias, x4old, epsilonp, epsilonn, Na, Nd);62 %%x3 = fzero(@(x) depletion(x, Vbi - ...bias, 0, epsilonp, epsilonn, Na, Nd), ...x3guess);63 x2 = sqrt(Na * ((x3 + x4old)?2 - x4old?2) / ...Nd + (x3 + x4old)?2) - x3 - x4old;64 if x3 ? (R - windowwidth)65 x1 = R - x2 - x3;66 x4new = 0;67 elseif x3<(R - windowwidth)68 x1 = max(0, windowwidth - x2);69 x4new = max(0, R - x1 - x2 - x3);70 end71 end72 x4 = x4new;73 else74 disp('error 2')75 end76 else126APPENDIX A. MATLAB CODE77 disp('error 3')78 endA.3 P-N Junction Depletion1 %Author: Brendan Kayes2 %Title: Radial pn Junction, Wire Array Solar Cells3 %Publisher: California Institute of Technology4 %Location: Pasedena California5 %Year: 200967 %Modifying Author: Ska - Hiish Manuel (SM)8 %Year: 20139 %Institution: University of British Columbia - Okanagan101112 %page 1351314 function output = depletion(x3, V, x4, epsilonp, epsilonn, ...Na, Nd) % depletion.m15 %fzero(@depletion, x3old, [], Vbi - bias, 0, epsilonp, ...epsilonn, Na, Nd);16 %is used to calculate the value of x3 that self - ...consistently gives the correct voltage drop across the ...pn junction, in the cylindrical geometry1718 %page 1361920 q = 1.60219e-19;21 d2 = x3 + x4;22 x2 = sqrt(Na * (d2?2 - x4?2)/Nd + d2?2) - d2;23 if x4 > 024 output = V - (q * Na/(4 * epsilonp) * (d2?2 - x4?2) ...- q * Na/(2 * epsilonp) * x4?2 * ...25 log(d2/x4) - q * Nd/(4 * epsilonn) * ((d2 + x2)?2 - ...d2?2) + q * Nd/(2 * epsilonn) * ...26 (d2 + x2)?2 * (log((d2 + x2)/d2)));27 elseif x4 == 028 output = V - (q * Na/(4 * epsilonp) * d2?2 - q * ...Nd/(4 * epsilonn) * ((d2 + x2)?2 - d2?2) + ...29 q * Nd/(2 * epsilonn) * (d2 + x2)?2 * (log((d2 + ...x2)/d2)));30 end127APPENDIX A. MATLAB CODEA.4 Planar P-N Junction1 %Author: Brendan Kayes2 %Title: Radial pn Junction, Wire Array Solar Cells3 %Publisher: California Institute of Technology4 %Location: Pasedena California5 %Year: 200967 %Modifying Author: Ska - Hiish Manuel (SM)8 %Year: 20139 %Institution: University of British Columbia - Okanagan1011 %page 1361213 clear all14 clc15 format short g16 warning off MATLAB:divideByZero1718 %%%% Start Modified - SM %%%%19 [type, sheets] = xlsfinfo('materialProperties.xls');20 [smarts2Num, smarts2Txt, smarts2Raw] = ...xlsread('materialProperties.xls', 'materialProperties', ...'B7:I2008');21 %[smarts2Num, smarts2Txt, smarts2Raw] = ...xlsread('matProperties', 'matProp', 'A2:C44');22 [parameters2Num, parameters2Txt, parameters2Raw] = ...xlsread('materialProperties.xls', 'Parameters', 'C4:H18');23 %Si 1, GaAs 2, Ge 3, GaN 4, InN Cu1489 5, InN CuW251 624 materialNumber = 4252627 parameters = parameters2Num(:, materialNumber)28 alphaColumn = parameters(1)29 size(smarts2Num)30 smarts2Num(any(smarts2Num(:, alphaColumn) == 0, 2), :) = ...[]; %used to remove the zeros from alpha31 size(smarts2Num)32 E = smarts2Num(:, 1);33 flux = smarts2Num(:, 2);34 alpha = smarts2Num(:, alphaColumn);35 %%%% END Modified - SM %%%%3637 n = length(E) - 138 %isthereadepletionregion = input('Depletion Region? (1 = ...yes, 0 = no) ');128APPENDIX A. MATLAB CODE39 isthereadepletionregion = 1; %gives the user the explicit ...option of turning off depletion region recombination4041 %page 1374243 %L = [1e - 4, 2e - 4, 5e - 4, 1e - 3, 2e - 3, 5e - 3, 1e - ...2, 2e - 2, 5e - 2, 1e - 1]; % L is a vector containing ...the various cell thicknesses that will be considered, ...in cm4445 %%%% Start Modified - SM %%%%46 resolution = 40;47 switch materialNumber48 case 1 %Si49 Lpower = - 5.0;50 Rpower = - 5.0;51 L = logspace(Lpower, - 2.2, resolution);52 Lns = logspace(Rpower, - 2.0, resolution);53 materialSymbol = 'Si'54 case 2 %GaAs55 Lpower = - 4.4;56 Rpower = - 4.4;57 L = logspace(Lpower, - 2.0, resolution);58 Lns = logspace(Rpower, - 3.0, resolution);59 materialSymbol = 'GaAs'60 case 3 %Ge61 Lpower = - 4.5;62 Rpower = - 5.0;63 L = logspace(Lpower, - 3, resolution);64 Lns = logspace(Rpower, - 2, resolution);65 materialSymbol = 'Ge'66 case 4 %GaN67 Lpower = - 4.0;68 Rpower = - 4.0;69 L = logspace(Lpower, - 1.2, resolution);70 Lns = logspace(Rpower, - 1, resolution);71 materialSymbol = 'GaN'72 %R = logspace( - 5, - 2, 40); This works but this ...an eblow at R ? = 4.873 case 5 %InN Cu148974 Lpower = - 4;75 Rpower = - 4;76 L = logspace(Lpower, - 2, resolution);77 Lns = logspace(Rpower, - 2.0, resolution);78 materialSymbol = 'InNCu1489'79 case 6 %InN CuW25180 Lpower = - 4;81 Rpower = - 4;82 L = logspace(Lpower, - 2, resolution);129APPENDIX A. MATLAB CODE83 Lns = logspace(Rpower, - 2.0, resolution);84 materialSymbol = 'InNCuW251'85 end86 %%%% End Modified - SM %%%%8788 %%L = [1e - 2, 2e - 2, 5e - 2, 1e - 1];89 matrix2 = L';90 matrixJsc = L';91 matrixVoc = L';92 matrixff = L';93 %dope = [1, 10, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8] * 5e12;94 %muele = [1500, 1500, 1500, 1500, 1000, 400, 150, 100, 100];95 %muhol = [500, 500, 500, 400, 300, 150, 60, 50, 50]; %these ...lines indicate the maximum electron and hole mobilities ...possible in Si at various doping levels, as given by ...[52], for reference96 %max Na = [>1e20, >1e20, 9e19, 4e19, 2e19, 9e18, ...4e18, 2e18, 8e17, 3e17]97 %vs Ln = [1e - 5, 2e - 5, 5e - 5, 1e - 4, 2e - 4, 5e ...- 4, 1e - 3, 2e - 3, 5e - 3, 1e - 2]98 % = > mun = [95, 95, 95, 100, 110, 130, 160, 195, ...300, 540] %these lines indicate the maximum electron ...diffusion length possible in Si at various doping ...levels, as given by [53], for reference99 N = length(L);100 count = 1:N;101 dE = (E(2:(n + 1)) - E(1:n));102 alpha2 = alpha(2:(n + 1));103 alpha1 = alpha2; % alpha1 and alpha2 are the absorption ...coefficients of then - and p - type materials, ...respectively104 gammaA = flux(2:(n + 1)); % gammaA is the photon flux ...incident on the top ofthe n - type quasineutral region105 %Define constants106 %page 138107108 matrix = [];109110 %%%% Start Modified - SM %%%%111 %Define constants112 q = parameters(2);113 kB = parameters(3);114 T = parameters(4);115 %Silicon data116 %Eg = 1.1117 Nc = parameters(5);118 Nv = parameters(6);119 ni = parameters(7);120 dopants = parameters(8);130APPENDIX A. MATLAB CODE121 Na = dopants;122 mun = parameters(9);123 Nd = dopants;124 mup = parameters(10);125 Dn = kB * T / q * mun;126 Dp = kB * T / q * mup;127 Sn = parameters(11);128 sigman = parameters(12);129 sigmap = sigman;130 vth = parameters(13);131 EpsilonR = parameters(15);132133 %%%% End Modified - SM %%%%134 Sp = Sn;135136 windowwidth = 10?( - 6) * ones(1, N); %minimum emitter ...thickness, in cm137 %windowwidth = 1e - 6 * ones(1, N); %minimum emitter ...thickness, in cm138 %%Lns = [1e - 5, 2e - 5, 5e - 5, 1e - 4, 2e - 4, 5e - 4, 1e ...- 3, 2e - 3, 5e - 3, 1e - 2];139140 for counter_Lns = 1:length(Lns) %different diffusion ...lengths (Lns)141 Ln = Lns(counter_Lns);142 disp(sprintf('Ln = %d', Ln))143 Nr = Dn / (Ln?2 * sigman * vth);144 taun0 = 1 / (sigman * Nr * vth);145 taup0 = 1 / (sigmap * Nr * vth);146147 %page 139148149 taun = taun0;150 taup = taup0;151 %taun0 = 1e - 6;taup0 = taun0; %activate this line to ...set the depletion region minority electron lifetime ...?n0 separately from the quasineutral region ...minority electron lifetime ?n (similarly for holes)152153 Lp = sqrt(taup * Dp);154 epsilon0 = 8.85418e-14;155 epsilons = EpsilonR * epsilon0;156 epsilonp = epsilons;157 epsilonn = epsilons;158159 np0 = ni?2 / Na;160 pn0 = ni?2 / Nd;161162 Vbimax = kB * T / q * log(Na * Nd / ni?2);131APPENDIX A. MATLAB CODE163 V = linspace(0, 99 / 100 * Vbimax);164 NN = length(V);165 Vbi = zeros(NN, N);166167 if isthereadepletionregion == 1168169 x2maxmax = sqrt(2 * epsilonn * epsilonp * Na * ...Vbimax / (q * Nd * (Na * epsilonp + ...170 Nd * epsilonn)));171172 x3maxmax = sqrt(2 * epsilonn * epsilonp * Nd * ...Vbimax / (q * Na * (Na * epsilonp + ...173 Nd * epsilonn)));174 windowwidth(windowwidth<x2maxmax) = x2maxmax;175 for i = 1:N %% different cell thicknesses (L)176 if (x2maxmax + x3maxmax)? L(i)177 x2j(i) = x2maxmax * L(i) / (x2maxmax + ...x3maxmax);178 windowwidth(i) = x2j(i);179 x3j(i) = x3maxmax * L(i) / (x2maxmax + ...x3maxmax);180 Vbi(:, i) = q * Nd / (2 * epsilonn) * ...x2j(i)?2 + q * Na / (2 * epsilonp) * ...x3j(i)?2;181 %%disp('2')182 %page 140183184 elseif (x2maxmax + x3maxmax)<L(i)185 Vbi(:, i) = Vbimax;186 %%disp('1')187 end188189 for ii = 1:NN %% different voltages (Vbi)190 bias(ii, i) = V(ii);191 x2max(ii, i) = sqrt(2 * epsilonn * epsilonp ...* Na * (Vbi(ii, i) - bias(ii, i)) / ...192 (q * Nd * (Na * epsilonp + Nd * ...epsilonn)));193 x3max(ii, i) = sqrt(2 * epsilonn * epsilonp ...* Nd * (Vbi(ii, i) - bias(ii, i)) / ...194 (q * Na * (Na * epsilonp + Nd * ...epsilonn)));195 if (x2max(ii, i) + x3max(ii, i)) ? L(i)196 x4(ii, i) = 0;197 x1(ii, i) = 0;198 x2(ii, i) = x2j(i);199 x3(ii, i) = x3j(i);200 %%Vbi(ii, i) = bias(ii, i) + q * Nd / ...(2 * epsilonn) * x2(ii, i)?2 + q * ...132APPENDIX A. MATLAB CODENa / ...201 %%(2 * epsilonp) * x3(ii, i)?2;202 elseif (x2max(ii, i) + x3max(ii, i))<L(i)203 if x2max(ii, i) ? windowwidth(i)204 x1(ii, i) = 0;205 x2(ii, i) = x2max(ii, i);206 elseif x2max(ii, i)<windowwidth(i)207 x1(ii, i) = windowwidth(i) - ...x2max(ii, i);208 x2(ii, i) = x2max(ii, i);209 else210 disp('error')211 break212 end213 x3(ii, i) = x3max(ii, i);214 x4(ii, i) = L(i) - x1(ii, i) - x2(ii, ...i) - x3(ii, i);215 %%x4(ii, i) = Ln - x1(ii, i) - x2(ii, ...i) - x3(ii, i);216 else217 disp('error')218 break219220 %page 141221 end222 if bias(ii, i) > Vbi(ii, i)223 J0A(ii:NN, i) = zeros(NN - ii + 1, 1);224 J0C(ii:NN, i) = zeros(NN - ii + 1, 1);225 JlA(ii:NN, i) = zeros(NN - ii + 1, 1);226 JlC(ii:NN, i) = zeros(NN - ii + 1, 1);227 Jgdep(ii:NN, i) = zeros(NN - ii + 1, 1);228 Jrdep(ii:NN, i) = zeros(NN - ii + 1, 1);229 bias(ii:NN, i) = zeros(NN - ii + 1, 1);230 break231 end232 d1(ii, i) = x1(ii, i) + x2(ii, i);233 d2(ii, i) = x3(ii, i) + x4(ii, i);234 gammaC(:, i, ii) = flux(2:(n + 1)) .* exp( ...- alpha1 * d1(ii, i) - ...235 alpha2 * x3(ii, i)); % gammaC is the ...photon flux incident on the top of ...the p - type quasineutral region236237 %A5238 JlAperE(:, i, ii) = - (q * gammaA .* dE ./ ...(1 - alpha1.?( - 2) * Lp?( - 2)) .* ...239 ((Sn ./ (alpha1 * Dp) + ones(n, 1) - ...exp( - alpha1 * x1(ii, i)) .* (Sn ..../ ...133APPENDIX A. MATLAB CODE240 (alpha1 * Dp) * cosh(x1(ii, i) / Lp) + ...1 ./ (alpha1 * Lp) * sinh(x1(ii, i) .../ ...241 Lp))) ./ (ones(n, 1) * (Sn * Lp / Dp * ...sinh(x1(ii, i) / Lp) + ...242 cosh(x1(ii, i) / Lp))) - exp( - alpha1 ...* x1(ii, i)))) .* (ones(n, 1));243244 %A4245 JlCperE(:, i, ii) = - (q * dE ./ (1 - ...alpha2.?( - 2) * Ln?( - 2)) .* (1 - 1 ..../ (alpha2 * ...246 Ln) .* (Sp * Ln / Dn * (ones(n, 1) * ...cosh(x4(ii, i) / Ln) - exp( - ...alpha2 * ...247 x4(ii, i))) + ones(n, 1) * sinh(x4(ii, ...i) / Ln) + alpha2 * Ln .* ...248 exp( - alpha2 * x4(ii, i))) ./ (ones(n, ...1) * (Sp * Ln / Dn * sinh(x4(ii, i) .../ ...249 Ln) + cosh(x4(ii, i) / Ln)))) .* ...gammaC(:, i, ii)) .* (ones(n, 1));250251 JdepperE(:, i, ii) = - q * gammaA .* dE .* ...exp( - alpha1 * x1(ii, i)) .* (1 - ...252 exp( - alpha1 * x2(ii, i) - alpha2 * ...x3(ii, i)));253254 %page 142255256 JlA(ii, i) = sum(JlAperE(:, i, ii));257 JlC(ii, i) = sum(JlCperE(:, i, ii));258 Jgdep(ii, i) = sum(JdepperE(:, i, ii));259260 %A3261 J0A(ii, i) = - q * Dp * pn0 / Lp * (Sn * ...Lp / Dp * cosh(x1(ii, i) / Lp) + ...262 sinh(x1(ii, i) / Lp)) ./ (Sn * Lp / Dp ...* sinh(x1(ii, i) / Lp) + ...263 cosh(x1(ii, i) / Lp));264 %A2265 J0C(ii, i) = - q * Dn * np0 / Ln * (Sp * ...Ln / Dn * cosh(x4(ii, i) / Ln) + ...266 sinh(x4(ii, i) / Ln)) ./ (Sp * Ln / Dn ...* sinh(x4(ii, i) / Ln) + ...267 cosh(x4(ii, i) / Ln));268 %A10269 Jrdep(ii, i) = - q * ni * (x2(ii, i) + ...x3(ii, i)) / sqrt(taun0 * taup0) * (2 * ...134APPENDIX A. MATLAB CODE270 sinh(q * bias(ii, i) / (2 * kB * T)) / ...(q * (Vbi(ii, i) - ...271 bias(ii, i)) / (kB * T))) * pi / 2;272 end273 end274 %A1275 Jl = JlA + JlC + Jgdep;276 J0 = J0A + J0C;277 xy = N;278 xx = NN;279 % disp(sprintf('L = %d', L(xy)))280 % disp(sprintf('x1 = %d', x1(xx, xy)))281 % disp(sprintf('x2 = %d', x2(xx, xy)))282 % disp(sprintf('x3 = %d', x3(xx, xy)))283 % disp(sprintf('x4 = %d', x4(xx, xy)))284 % disp(sprintf('x4(1, 1) / Ln = %d', x4(xx, xy) / ...Ln))285 % disp(sprintf('cosh(x4(1, 1) / Ln) = %d', ...cosh(x4(xx, xy) / Ln) ))286 J = (J0 .* (exp(q * bias / (kB * T)) - 1) - Jl + ...Jrdep);287 elseif isthereadepletionregion == 0288 for i = 1:N289 Vbi(:, i) = Vbimax;290 for ii = 1:NN291 bias(ii, i) = V(ii);292 x1(ii, i) = windowwidth(i);293 x4(ii, i) = L(i) - x1(ii, i);294 %%x4(ii, i) = Ln - x1(ii, i);295 if bias(i)>Vbi(ii, i)296 J0A(ii:NN, i) = zeros(NN - ii + 1, 1);297298 %page 143299300 J0C(ii:NN, i) = zeros(NN - ii + 1, 1);301 JlA(ii:NN, i) = zeros(NN - ii + 1, 1);302 JlC(ii:NN, i) = zeros(NN - ii + 1, 1);303 Jgdep(ii:NN, i) = zeros(NN - ii + 1, 1);304 Jrdep(ii:NN, i) = zeros(NN - ii + 1, 1);305 bias(ii:NN, i) = zeros(NN - ii + 1, 1);306 break307 end308 d1(ii, i) = x1(ii, i);309 gammaC(:, i, ii) = flux(2:(n + 1)) .* exp( ...- alpha1 * d1(ii, i)); % gammaC is the ...photon flux incident on the top of the ...p - type quasineutral region310 JlAperE(:, i, ii) = - (q * gammaA .* dE ./ ...(1 - alpha1.?( - 2) * Lp?( - 2)) .* ...135APPENDIX A. MATLAB CODE311 ((Sn ./ (alpha1 * Dp) + ones(n, 1) - ...exp( - alpha1 * x1(ii, i)) .* (Sn ..../ ...312 (alpha1 * Dp) * cosh(x1(ii, i) / Lp) + ...1 ./ (alpha1 * Lp) * sinh(x1(ii, i) .../ ...313 Lp))) ./ (ones(n, 1) * (Sn * Lp / Dp * ...sinh(x1(ii, i) / Lp) + cosh(x1(ii, ...i) / ...314 Lp))) - exp( - alpha1 * x1(ii, i)))) .* ...(ones(n, 1));315316 JlCperE(:, i, ii) = - (q * dE ./ (1 - ...alpha2.?( - 2) * Ln?( - 2)) .* (1 - 1 ..../ ...317 (alpha2 * Ln) .* (Sp * Ln / Dn * ...(ones(n, 1) * cosh(x4(ii, i) / Ln) ...- ...318 exp( - alpha2 * x4(ii, i))) + ones(n, ...1) * sinh(x4(ii, i) / Ln) + alpha2 ...* ...319 Ln .* exp( - alpha2 * x4(ii, i))) ./ ...(ones(n, 1) * (Sp * Ln / Dn * ...320 sinh(x4(ii, i) / Ln) + cosh(x4(ii, i) / ...Ln)))) .* gammaC(:, i, ii)) .* ...321 (ones(n, 1));322323 JlA(ii, i) = sum(JlAperE(:, i, ii));324 JlC(ii, i) = sum(JlCperE(:, i, ii));325 J0A(ii, i) = - q * Dp * pn0 / Lp * (Sn * ...Lp / Dp * cosh(x1(ii, i) / Lp) + ...326 sinh(x1(ii, i) / Lp)) ./ (Sn * Lp / Dp ...* sinh(x1(ii, i) / Lp) + ...327 cosh(x1(ii, i) / Lp));328329 %page 144330331 J0C(ii, i) = - q * Dn * np0 / Ln * (Sp * ...Ln / Dn * cosh(x4(ii, i) / Ln) + ...332 sinh(x4(ii, i) / Ln)) ./ (Sp * Ln / Dn ...* sinh(x4(ii, i) / Ln) + ...333 cosh(x4(ii, i) / Ln));334 %%casdf = cosh(x4(ii, i) / Ln)335 %%px4 = x4(ii, i)336 Jl = JlA + JlC;337 J0 = J0A + J0C;338 J = (J0 .* (exp(q * bias / (kB * T)) - 1) - ...Jl);339 end136APPENDIX A. MATLAB CODE340 end341 end342343 P = J .* bias;344 P(P<0) = 0;345 [maxP, posn] = max(P);346 eff = maxP / (.1) * 100;347 Jsc = J(1, :);348 b = sum(J>0);349 for j = 1:N;350 if b(j) 6= 0351 if b(j) == 100352 Voc(j) = 0;353 elseif bias(b(j) + 1, j) == 0354 Voc(j) = 0;355 else356 Voc(j) = (bias(b(j), j) + bias(b(j) + 1, ...j)) / 2;357 end358 else359 Voc(j) = 0;360 end361 end362363 %%page 145364 ff = maxP ./ (Voc .* Jsc);365366 Lshort = L;367 Lshort(?(ff<0.9999&ff>0)) = 0;368 Vocshort = Voc;369 Vocshort(?(ff<0.9999&ff>0)) = 0;370 Jscshort = Jsc;371 Jscshort(?(ff<0.9999&ff>0)) = 0;372 ffshort = ff;373 ffshort(?(ff<0.9999&ff>0)) = 0;374 effshort = eff;375 effshort(?(ff<0.9999&ff>0)) = 0;376 %matrix = [matrix; Lshort', effshort', Jscshort', ...Vocshort', ffshort'];377 % disp(' L (cm) Jsc (A / cm?2) Voc (V) ff Efficiency (%)')378 % disp([Lshort', Jscshort', Vocshort', ffshort', ...effshort'])379 matrix2 = [matrix2, effshort'];380 matrixJsc = [matrixJsc, Jscshort'];381 matrixVoc = [matrixVoc, Vocshort'];382 matrixff = [matrixff, ffshort'];383 end384 t1 = num2str(Lp, 4);385 t2 = num2str(Ln, 4);137APPENDIX A. MATLAB CODE386 t3 = num2str(Sn, 4);387 t6 = num2str(mup, 4);388 t7 = num2str(mun, 4);389 t8 = num2str(Na, 4);390 t9 = num2str(Nd, 4);391 t10 = num2str(windowwidth, 4);392 t11 = num2str(taun0, 4);393394 %page 146395396 t12 = num2str(taup0, 4);397 maxeff = num2str(max(eff), 3);398 name = sprintf('GaAs Planar matrix Ln = %s, Lp = %s, Sn = ...%s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', t2, t1, t3, t11, ...t12, t8, t9, t10, maxeff);399 name2 = sprintf('GaAs Planar Eff Ln = %s, Lp = %s, Sn = ...%s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', t2, t1, t3, t11, ...t12, t8, t9, t10, maxeff);400 name3 = sprintf('GaAs Planar Jsc Ln = %s, Lp = %s, Sn = ...%s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', t2, t1, t3, t11, ...t12, t8, t9, t10, maxeff);401 name4 = sprintf('GaAs Planar Voc Ln = %s, Lp = %s, Sn = ...%s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', t2, t1, t3, t11, ...t12, t8, t9, t10, maxeff);402 name5 = sprintf('GaAs Planar ff Ln = %s, Lp = %s, Sn = ...%s, taun0 = %s, taup0 = %s, Na = %s, Nd = %s, ...window width = %s, maxeff = %s.txt', t2, t1, t3, t11, ...t12, t8, t9, t10, maxeff);403 nL = length(L);404 nR = length(Lns);405406 %%%% Start Modified - SM %%%%407 h1 = figure(1);408 % surf(log10(Lns), log10(L), matrix2(:, 2:(nR + 1)))409 surf(Lns, L, matrix2(:, 2:(nR + 1)))410 set(gca, 'yscale', 'log')411 set(gca, 'xscale', 'log')412 %page 147413414 xlabel('Ln (cm))')415 ylabel('L (cm))')416 zlabel('Efficiency (%)')417 name2a = sprintf(' %s Planar cell Eff. with Na = %s, Nd = ...%s, Sn = %s, maxEff = %s', materialSymbol, t8, t9, ...t3, maxeff);138APPENDIX A. MATLAB CODE418 title(name2a)419 fileName = sprintf('Eff%sPlanar', materialSymbol);420 saveas(h1, fileName, 'png')421422 h2 = figure(2);423 % surf(log10(Lns), log10(L), matrixJsc(:, 2:(nR + 1)))424 surf(Lns, L, matrixJsc(:, 2:(nR + 1)))425 set(gca, 'yscale', 'log')426 set(gca, 'xscale', 'log')427 %axis 'tight'428 xlabel('Ln (cm))')429 ylabel('L (cm))')430 zlabel('Jsc')431 name2b = sprintf('%s Planar cell Jsc with Na = %s, Nd = ...%s, Sn = %s, maxEff = %s', materialSymbol, t8, t9, ...t3, maxeff);432 title(name2b)433 fileName = sprintf('Jsc%sPlanar', materialSymbol);434 saveas(h2, fileName, 'png')435436437 h3 = figure(3);438 % surf(log10(Lns), log10(L), matrixVoc(:, 2:(nR + 1)))439 surf(Lns, L, matrixVoc(:, 2:(nR + 1)))440 set(gca, 'yscale', 'log')441 set(gca, 'xscale', 'log')442 xlabel('Ln (cm))')443 ylabel('L (cm))')444 zlabel('Voc')445 name2c = sprintf('%s Planar cell Voc with Na = %s, Nd = ...%s, Sn = %s, maxEff = %s', materialSymbol, t8, t9, ...t3, maxeff);446 title(name2c)447 fileName = sprintf('Voc%sPlanar', materialSymbol);448 saveas(h3, fileName, 'png')449450 h4 = figure(4);451 % surf(log10(Lns), log10(L), matrixff(:, 2:(nR + 1)))452 surf(Lns, L, matrixff(:, 2:(nR + 1)))453 set(gca, 'yscale', 'log')454 set(gca, 'xscale', 'log')455 xlabel('Ln (cm))')456 ylabel('L (cm))')457 zlabel('ff')458 name2d = sprintf('%s Planar cell ff with Na = %s, Nd = %s, ...Sn = %s, maxEff = %s', materialSymbol, t8, t9, t3, ...maxeff);459 title(name2d)460 fileName = sprintf('Ff%sPlanar', materialSymbol);139APPENDIX A. MATLAB CODE461 saveas(h4, fileName, 'png')462 disp('Completed')463 %%%% End Modified - SM %%%%464 beepA.5 Silicon Absorption Depth Class1 %Author: Ska - Hiish Manuel (SM)2 %Year: 20133 %Institution: University of British Columbia - Okanagan45 classdef absorption6 %Functions to calculate the extinction coefficient and ...the absorption7 %coefficient for the following ranges8 %used the optical porperties of solids910 %(1) 0 to 3.0 eV, (2) 3.0 to 4.3 eV, (3) 4.3 to 5.3 eV, ...(4) 5.3 to 6.0 eV1112 %variables13 properties14 k;15 alpha;16 omega;17 end18 %constants19 properties20 %properties to calc. extinction coefficient for Si21 %based on Handbook of optical properties Geist Chapter22 epsilon1;23 depsilon1;24 C;25 epsilon2;26 depsilon2;27 epsilon3;28 epsilon4;29 epsilon5;30 epsilonI;31 N;32 dN;33 K4;34 K5;35 C3;36 %p140APPENDIX A. MATLAB CODE37 E1;38 B1;39 B11;40 E2;41 C2;42 gamma;43 F;44 GAMMA;45 Ea;46 Ca;47 gammaA;48 Eg;49 D;50 hbar;51 hv;52 wavelength_nm;53 end5455 methods56 %construct initiates values and calculates57 %(1) 0 to 3.0 eV58 function l = L(obj, y1)59 l = (y1 + abs(y1))./(2);60 end6162 function f = F2(obj, hv)63 f = exp(obj.K4 .* obj.L(hv-obj.epsilon4)+obj.K5 ....* obj.L(hv-obj.epsilon5));64 end6566 function g = G(obj, hv)67 C = obj.C;6869 g = C(1, 1) .* (obj.L(hv-obj.epsilon1-1 .* ...obj.depsilon1)).?2+...70 obj.C3 .* (2 .* ...obj.L(hv-obj.epsilon3)).?(obj.N+obj.dN ....* hv);71 g = g+C(1, 2) .* (obj.L(hv-obj.epsilon1+1 .* ...obj.depsilon2)).?2;72 g = g+C(2, 1) .* (obj.L(hv-obj.epsilon1-1 .* ...obj.depsilon1)).?2;73 g = g+C(2, 2) .* (obj.L(hv-obj.epsilon1+1 .* ...obj.depsilon2)).?2;74 end7576 function k1 = KK1(obj, hv)77 k1 = obj.F2(hv) .* (1.23985 .* 10.?-4./hv) .* ...(obj.G(hv)./hv)./(4 .* pi());141APPENDIX A. MATLAB CODE78 end7980 function kk = KK(obj, epsilonA1, epsilonA2)81 kk = (((epsilonA1.?2 + epsilonA2.?2).?0.5 - ...epsilonA1)./2).?(0.5);82 end8384 function a = ABSORPTION(obj, kk, length)85 a = 4 .* pi() .* kk./length;86 end87 %(2) 3.0 to 4.3 eV8889 %calculates epsilon for equation 6 & 9 in adachi 198990 function [epsilon1, epsilon2] = KK2(obj, omega)91 chi1 = obj.hbar .* omega./obj.E1;92 epsilon1 = - obj.B1 .* chi1.?-2 .* log(1 - ...chi1.?2);9394 for i = 1:length(omega)95 CHI1 = obj.hbar * omega(i)/obj.E1;96 if(obj.hbar * omega(i) < obj.E1)97 epsilon2(i) = pi() * CHI1?-2 * (obj.B1 ...- obj.B11 * (obj.E1 - obj.hbar * ...omega(i))?0.5);98 %epsilon2(i) = 0;99 else100 epsilon2(i) = pi() * obj.B1 * CHI1?-2;101 end102 end103 epsilon2 = transpose(epsilon2);104 end105106 %(3) 4.3 to 5.3 eV107 %calculates epsilon for equation 10 & 11 in adachi 1989108 function [epsilon1, epsilon2] = KK3(obj, omega)109 chi2 = obj.hbar .* omega./obj.E2;110 epsilon1 = obj.C2 .* ...(1-chi2.?2)./((1-chi2.?2).?2+chi2.?2 .* ...obj.gamma.?2);111 epsilon2 = obj.C2 .* chi2 .* ...obj.gamma./((1-chi2.?2).?2+chi2.?2 .* ...obj.gamma.?2);112 end113114 %calculates epsilon for equation 13 & 15 in adachi 1989115 function [epsilon1, epsilon2] = KK4(obj, omega)116 chi2 = obj.hbar .* omega./obj.E2;117 chiC1 = obj.hbar .* omega./obj.E1;118 for i = 1:length(omega)142APPENDIX A. MATLAB CODE119 CHI1 = obj.hbar * omega(i)/obj.E1;120 CHI2 = obj.hbar * omega(i)/obj.E2;121 if (CHI1-1 < 0)122 epsilon2(i) = 0;123 elseif (1-CHI2?2)124 epsilon2(i) = 0;125 else126 epsilon2(i) = pi() * obj.F * CHI2?-2;127 end128 end129 epsilon2 = transpose(epsilon2);130 epsilon1 = -obj.F .* chi2.?(-2) .* ...log((1-chiC1.?2)./(1-chi2.?2));131 end132133 %(4) 5.3 to 6.0 eV134 %function not used135 % function k5 = KK5(obj, omega)136 % %chig = obj.Eg - (obj.hbar .* ...obj.omegaq./obj.hbar./omega);137 % chig = (obj.Eg - obj.hbar .* ...omega)./(obj.hbar./omega);138 % chiA = obj.hbar * omega/obj.Ea;139 % epsilonC1 = obj.Ca .* ...(1-chiA.?2)./((1-obj.chiA.?2).?2+obj.chiA.?2 .* ...obj.gammaA.?2);140 % epsilonC2 = obj.Ca .* obj.gammaA .* ...chiA./((1-obj.chiA.?2).?2+chiA.?2 .* obj.gammaA.?2);141 % %contribution of the indirect bandgap ...semiconductor142 % epsilonC2 = epsilonC2 + obj.D./(obj.hbar .* ...omega.?2) .* (h .* omega-obj.Eg+obj.hbar).?2 .* ...obj.H(1-chig);143 % k4 = epsilonC1;144 % %k4 = KK(epsilonC1, epsilonC2);145 % end146 %(1) alpha and (2) k147 function obj = absorption(s)148 obj.epsilon1 = s.epsilon1;149 obj.depsilon1 = s.depsilon1;150 obj.C = s.C;151 obj.epsilon2 = s.epsilon2;152 obj.depsilon2 = s.depsilon2;153 obj.epsilon3 = s.epsilon3;154 obj.epsilon4 = s.epsilon4;155 obj.epsilon5 = s.epsilon5;156 obj.epsilonI = s.epsilonI;157 obj.N = s.N;158 obj.dN = s.dN;143APPENDIX A. MATLAB CODE159 obj.K4 = s.K4;160 obj.K5 = s.K5;161 obj.C3 = s.C3;162 %p163 obj.E1 = s.E1;164 obj.B1 = s.B1;165 obj.B11 = s.B11;166 obj.E2 = s.E2;167 obj.C2 = s.C2;168 obj.gamma = s.gamma;169 obj.F = s.F;170 obj.GAMMA= s.GAMMA;171 obj.Ea = s.Ea;172 obj.Ca =s.Ca;173 obj.gammaA = s.gammaA;174 obj.Eg = s.Eg;175 obj.D = s.D;176 obj.hbar = s.hbar;177 obj.omega = s.omega;178 obj.wavelength_nm = s.wavelength_nm;179 obj.hv = s.hv;180 %calculate k for the different eV levels181 %(1) 0 to 3.1 eV182 %to reproduce graph from Geist183 figure()184 obj.k = obj.KK1(obj.hv);185 semilogy(obj.wavelength_nm, obj.k)186 axis([300 1200 1 * 10?-6 1])187 [obj.hv obj.wavelength_nm obj.k]188189 % subplot(2, 2, 1);190 % plot(obj.hv, obj.k)191 % axis([0.31 4.133 0 1])192 % subplot(2, 2, 3);193 % plot(obj.wavelength_nm, obj.k)194 % axis([300 1200 1 * 10?-6 1])195 %196 % subplot(2, 2, 2);197 % semilogy(obj.hv, obj.k)198 % axis([0.31 4.133 0 1])199 % subplot(2, 2, 4);200 % semilogy(obj.wavelength_nm, obj.k)201 % axis([1200 4000 0 1])202203 figure()204 [e1A, e2A] = obj.KK2(obj.omega);205 [e1B, e2B] = obj.KK3(obj.omega);206 [e1C, e2C] = obj.KK4(obj.omega);207 e1 = e1A + e1B + e1C + obj.epsilonI;144APPENDIX A. MATLAB CODE208 e2 = e2A + e2B + e2C;209 e1 = real(e1);210 e2 = real(e2);211 subplot(2, 2, 1)212 plot(obj.hv, e1A, obj.hv, e1B, obj.hv, e1C, ...obj.hv, e1)213 axis([1.1 4.5 -20 50])214215 subplot(2, 2, 2)216 plot(obj.hv, e2A, obj.hv, e2B, obj.hv, e2C, ...obj.hv, e2)217 %plot(obj.hv, e2A)218 axis([0 4.5 -20 50])219220 kk = obj.KK(e1, e2);221 subplot(2, 2, 3)222 plot(obj.hv, kk)223224 ALPHA = obj.ABSORPTION(kk, obj.wavelength_nm .* ...10?-9);225 subplot(2, 2, 4)226 plot(obj.hv, ALPHA)227 %(3) 4.3 to 5.3 eV228 %(4) 5.3 to 6.0 eV229230 figure()231232 subplot(2, 2, 1)233 obj.k = obj.KK1(obj.hv);234 asdf = obj.k;235 plot(obj.hv, obj.k)236 axis([0.3 obj.E1 0 1])237238 for i = 1:length(obj.omega)239 %%if(obj.hv(i) ? obj.E1)240 if(obj.hv(i) > 3.0)241 obj.k(i) = kk(i);242 end243 end244245 ASDF = obj.k;246 subplot(2, 2, 2)247 plot(obj.hv, obj.k)248 axis([1.1 4.5 0 6])249 obj.alpha = obj.ABSORPTION(obj.k, ...obj.wavelength_nm .* 10?-7);250 subplot(2, 2, 4)251 semilogy(obj.hv, obj.alpha)252 axis([0.3 4.5 10?3 10?7])145APPENDIX A. MATLAB CODE253 end254 end255 endA.6 Silicon Absorption Depth Implementation1 %Author: Ska - Hiish Manuel (SM)2 %Year: 20133 %Institution: University of British Columbia - Okanagan45 clear; clc;6 load('am.mat');7 c = 299792458;8 hbar = 6.52811928*10?-16;9 %omega = 2.*pi().*c./(wavelength_nm.*(1*10?-9));10 omega = wavelength_ev./hbar;1112 s=struct('epsilon1', 1.09969, ...13 'depsilon1', 0.0583147, ...14 'C', [5030.02 483.916; 1634.30 79.4079],... %'C', [5030.02 ...79.4079; 1634.30 483.916], ... %'C', [5030.02 483.916; ...1634.30 79.4079], ...15 'epsilon2', 0, ...16 'depsilon2', 0.0220161, ...17 'epsilon3', 1.40985, ...18 'epsilon4', 2.15362, ...19 'epsilon5', 2.95893, ...20 'N', 0.394122, ...21 'dN', 1.23084, ...22 'K4', 0.210102, ...23 'K5', 1.03681, ...24 'C3', 1046.08, ... % 'C3', 1046.08, ... %25 'E1',3.38, ...26 'B1',5.22, ...27 'B11',7.47, ...28 'E2',4.27, ...29 'C2',3.01, ...30 'gamma',0.127, ...31 'F',3.51, ...32 'GAMMA',0.05, ...33 'Ea',5.32, ...34 'Ca',0.21, ...35 'gammaA',0.089, ...36 'Eg',1.12, ...37 'D',0.89, ...146APPENDIX A. MATLAB CODE38 'omega',omega, ...39 'hv',wavelength_ev, ...40 'epsilonI', 1.8, ...41 'wavelength_nm', wavelength_nm, ...42 'hbar',hbar );43 si = absorption(s)44 N = length(omega)4546 alpha = si.alpha;47 %flux = am0;48 flux = am15G ./ (6.626e-34 .* c ./ (wavelength_nm .* 1e-9));49 %flux = am15D ./ (6.626e-34 . *c ./ (wavelength_nm .* 1e-9));50 E = wavelength_ev;51 lambda = wavelength_nm;52 figure()53 plot(wavelength_nm, wavelength_ev)54 save('silicon_E_flux_alpha', 'alpha', 'flux', 'E', 'lambda')55 m = [E, alpha, lambda, flux]56 xlswrite('filename.xls', [E, alpha, lambda, flux])147

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