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Modeling solar cells with recombination 2011

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Modeling solar cells with recombination by Prabhjot Kaur Guram A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) November 2011 c© Prabhjot Kaur Guram 2011 Abstract An elementary model for the analysis of a photovoltaic solar cell is pro- posed. This analysis is rooted in the current-voltage device characteristics associated with a p-n junction in conjunction with a model for a solar flux controlled current source; this follows the approach of Prince [M. B. Prince, Journal of Applied Physics, vol. 26, pp. 553-540, 1955], the p-n junction ar- chitecture being that underlying the photovoltaic solar cell. Recombination processes were modeled through two means: (1) an empirical expression for the current-voltage device characteristics with an associated ideality factor, η, whose value determines the importance of recombination processes, and (2) a more advanced expression that includes a recombination current. It is shown that the simplified empirical expression is overly simplified and that its use leads to artifacts, i.e., the suggestion that recombination processes could actually enhance the fill-factor. In contrast, the more realistic current- voltage device characteristic, which includes both ideal and recombination related current densities, suggests that recombination processes actually will reduce the fill-factor. This later observation is in accord with the experi- mental observation. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xx 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The solar spectrum and background material . . . . . . . . 9 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Solar cell technology developments . . . . . . . . . . . . . . . 11 2.3 Blackbody radiation . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 The solar spectrum . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Integrated solar spectrum . . . . . . . . . . . . . . . . . . . . 28 iii Table of Contents 3 General theory of the p-n junction with recombination . . 33 3.1 Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 The geometric layout of the p-n junction . . . . . . . . . . . 34 3.3 Drift and diffusion processes . . . . . . . . . . . . . . . . . . 35 3.4 The p-n junction with no applied bias . . . . . . . . . . . . . 44 3.5 The p-n junction under forward-bias . . . . . . . . . . . . . . 54 3.6 The p-n junction under reverse bias . . . . . . . . . . . . . . 61 3.7 Recombination processes . . . . . . . . . . . . . . . . . . . . 61 4 Solar cell performance analysis . . . . . . . . . . . . . . . . . 68 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Solar cell geometries . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Photovoltaic solar cell behavior . . . . . . . . . . . . . . . . . 73 4.4 Photovoltaic solar cell performance metrics . . . . . . . . . . 73 4.5 Photovoltaic solar cell device modeling . . . . . . . . . . . . 75 4.6 Relationship with the material properties . . . . . . . . . . . 84 4.7 The role of the parasitic resistances on the performance of a photovoltaic solar cell . . . . . . . . . . . . . . . . . . . . . . 85 4.8 The solar spectrum and photovoltaic solar cell performance . 90 4.9 Recombination within solar cells . . . . . . . . . . . . . . . . 95 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 iv List of Tables 2.1 The integrated solar spectra obtained for the upper wave- length in Eq. (2.33) being set to ∞. This corresponds to integrating the solar spectra over the entire range of wave- lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 The electron and hole drift mobilities associated with c-Si and c-Ge at 300 K [27]. . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 The electron and hole diffusion coefficients associated with c-Si and c-Ge at 300 K [28]. . . . . . . . . . . . . . . . . . . . 41 3.3 The electron and hole diffusion lengths associated with c-Si and c-Ge at 300 K [26]. . . . . . . . . . . . . . . . . . . . . . 44 v List of Figures 1.1 Worldwide energy consumption by industrial societies as a function of the year. The data depicted in this plot is from the US Department of Energy [1]. . . . . . . . . . . . . . . . . 2 1.2 The efficiency of c-Si based solar cells as a function of the year. The data is taken from Hezel [17]. . . . . . . . . . . . . 6 1.3 A schematic of a p-n junction. The p-type region is abut- ted next to the n-type region. The neutral and space-charge regions are depicted. . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 A three dimensional electromagnetic cavity of the dimensions L×L×L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Points in n-space. Each point corresponds to an allowed set of nx, ny, and nz values, corresponding to a possible solution of Eq. (2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 The energy per electromagnetic mode as a function of h̄ωkT . The classical result is contrasted with that due to Planck. The results are presented in a dimensionless form through a divison by the classical result, kT . . . . . . . . . . . . . . . . 19 vi List of Figures 2.4 The energy density in an electromagnetic cavity as a function of the wavelength. The temperature is set to 5780 K for the purposes of this analysis. The classical result, i.e., Eq (2.16), is contrasted with result of Planck, i.e., the quantum result, Eq (2.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 The Earth orbiting the Sun at radius rse. The radius of the Sun itself is rs. These distances have been exaggerated on this figure for the purposes of illustration. . . . . . . . . . . . 24 2.6 The solar spectrum, i.e., the power flux spectrum received at the periphery of the Earth’s atmosphere, as a function of wavelength of blackbody radiation for different temperatures. These temperatures correspond to the temperature of the sur- face of the Sun. Eq. (2.31) is used for the purposes of this analysis, where rs is set to 7.0 × 105 km and rse is set to 1.5 × 108 km. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 The angle of incidence, the vertical axis, and the direction of the Sun’s rays. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8 The solar spectra corresponding to the AM0 and AM1.5 cases. The experimental data is taken from NREL [24]. The black- body solar spectrum is also depicted, for the Sun’s surface temperature set to 5780 K [23]. . . . . . . . . . . . . . . . . . 30 2.9 The integrated solar spectra corresponding to the AM0 and AM1.5 solar spectra. The integrated blackbody solar spec- trum is also depicted for the Sun’s surface temperature set to 5780 K [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 vii List of Figures 3.1 The geometric layout of a representative one-dimensional abrupt p-n junction. M denotes the metallurgical junction between the p-type and n-type regions. . . . . . . . . . . . . . . . . . . 36 3.2 The contributions to the overall current flux attributable to the various drift and diffusion processes. It is assumed that the electric field points from right to left [26]. The p-n junc- tion architecture has been assumed. . . . . . . . . . . . . . . 42 3.3 The distribution of charges within a one dimensional abrupt p-n junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 The depletion region formed in a p-n junction under no bias. The net distributed charge, ρnet, is also indicated. E0 denotes the “built-in” electric field. . . . . . . . . . . . . . . . . . . . 47 3.5 The electric field within a p-n junction under no bias. The electric field points from right to left, x = −Wp and x = Wn denoting the space-charge/neutral region boundaries for the p-type and n-type regions, respectively [26]. . . . . . . . . . . 48 3.6 The electric field corresponding to a p-n junction under no bias. Vo denotes the “built-in” potential. . . . . . . . . . . . . 50 3.7 The energy band diagram of an unbiased p-n junction [29]. . 51 3.8 The charge carrier concentrations as a function of the position within the p-n junction. . . . . . . . . . . . . . . . . . . . . . 52 3.9 The energy band diagram of a forward-biased p-n junction [29]. 55 3.10 Depletion region formed in a p-n junction under forward bias. Note that the space-charge region is narrower than for the no bias case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 viii List of Figures 3.11 The excess hole and electron concentrations in a p-n junction Eqs. (3.37) and (3.38) establish the boundary conditions. The x = −Wp represents the location of the space-charge/neutral region boundary in the p-type region of junction. The x = Wn represents the location of the space-charge/neutral region boundary in the n-type region of junction. . . . . . . . . . . . 58 3.12 The contributions to the current density associated with the drift and diffusion processes. . . . . . . . . . . . . . . . . . . . 59 3.13 Depletion region formed in a p-n junction under reverse bias. Note that the space-charge region is wider than for the no bias case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.14 The energy band diagram for a reverse-biased p-n junction [29]. 63 3.15 The injection of charge carriers and the recombination within the space-charge region. . . . . . . . . . . . . . . . . . . . . . 64 3.16 The current density-voltage device characteristics correspond- ing to Eq. (3.48) with Jso set to 1 µA/cm 2 and Jro set to 10 nA/cm2. The temperature is set to 300 K. . . . . . . . . . 66 4.1 Model for the behavior of a photovoltaic solar cell. The p- n junction is in parallel with a solar flux controlled current source. This circuit model follows the suggestion of Prince [30]. 69 4.2 A representative one dimensional p-n junction that may be used for photovoltaic device applications. Note that the p- type layer is thicker than the n-type layer. . . . . . . . . . . . 72 ix List of Figures 4.3 A representative current-voltage device characteristic associ- ated with a photovoltaic solar cell. . . . . . . . . . . . . . . . 74 4.4 Representative current-voltage device characteristics associ- ated with a photovoltaic solar cell for various fill-factor selec- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5 The open-circuit voltage, Voc, scaled by the thermal voltage, kT q , as a function of the ratio, IL Io , for the ideality factor, η, set to unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 The open circuit voltage, Voc, as a function of photocurrent, IL. Io is set to 10 −9 A for the purposes of this analysis. Two selections of η are considered. The temperature is set to 300 K for the purposes of this analysis. . . . . . . . . . . . . . . . 80 4.7 The power delivered to an external load as a function of the applied voltage for two selections of the ideality factor, η. For the purposes of this analysis, Io is set to 10 −9 A and IL is set to 100 mA. The temperature is set to 300 K for all cases. . . 81 4.8 The fill-factor as a function of the photocurrent, IL. Two selections of η are considered. For all cases, Io is set to 10 −9 A and the temperature is set to 300 K. . . . . . . . . . . . . . . 83 4.9 The circuit model for a photovoltaic solar cell with the para- sitic resistances taken into account. . . . . . . . . . . . . . . . 86 x List of Figures 4.10 The current-voltage device characteristic associated with a photovoltaic solar cell, as modeled in Figure 4.9, for a number of selections of the shunt resistance, Rsh. For all cases, Io is set to 10−9 A, IL is set to 100 mA, and the temperature is set to 300 K. This plot is determined using Eq. (4.9). The series resistance, Rs, is set to zero in all cases. . . . . . . . . . . . . 87 4.11 The current-voltage device characteristic associated with a photovoltaic solar cell, as modeled in Figure 4.9, for a number of selections of the series resistance, Rs. For all cases, Io is set to 10−9 A, IL is set to 100 mA, and the temperature is set to 300 K. It is noted that when the current is zero, all the voltages converges on the same point. This plot is determined using Eq. (4.9). The shunt resistance, Rsh, is open-circuited in all cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.12 The power delivered to an external load as a function of the applied voltage for various selections of Rs. For the purposes of this analysis, Io set to 10 −9 A, IL set to 100 mA, and the temperature set to 300 K. . . . . . . . . . . . . . . . . . . . . 91 4.13 The fill-factor, FF , as a function of the series resistance, Rs, for a photovoltaic solar cell, modeled as in Figure 4.9, as a function of the series resistance, Rs. For all cases, Io is set to 10−9 A, IL is set to 100 mA, and the temperature is set to 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 xi List of Figures 4.14 The efficiency as a function of energy gap, Eg, for the AM0 and AM1.5 solar spectra. The efficiency corresponding to the blackbody solar spectra is also depicted. . . . . . . . . . . . . 94 4.15 This current-voltage device characteristic plotted as a func- tion of the voltage across the p-n junction device terminals. For the purposes of this analysis, Io is set to 10 −9 A, Iro is set to 1 µA, IL is set to 100 mA, and the temperature is set to 300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.16 The power delivered to the external load as a function of the applied voltage across the terminals of the p-n junction. For all cases, Io is set to 10 −9 A, Iro is set to 1 µA, IL is set to 100 mA, and the temperature is set to 300 K. . . . . . . . . . 98 4.17 Fill-factor as a function of IL. Both the ideal case and the case with the recombination processes taken into account are considered. For all cases, Io is set to 10 −9 A, Iro is set to 1 µA, IL is set to 100 mA, and the temperature is set to 300 K. 99 xii List of Symbols A .......................... cross sectional area a(subscript) ............... acceptor, e.g., Na is acceptor concentration c .......................... speed of light in vacuum De ...................... diffusion coefficient for electrons Dh ...................... diffusion coefficient for holes d(subscript) ............... donor, e.g., Nd is donor concentration E .......................... energy ~E .................. electric field vector ECN ..................... conduction band edge ECP ..................... conduction band edge EF ...................... Fermi energy level Eg ...................... bandgap energy Eo ...................... built-in electric field e(subscript) ............... electron xiii List of Symbols Ex ...................... electric field in the x-direction EV N ..................... valence band energy edge EV P ..................... valence band energy edge Ef (f), Eλ(λ), Eω(ω) ................. energy per electromagnetic mode f .......................... frequency h .......................... Planck’s constant h̄ ...................... reduced Planck constant h̄ω ............ photon energy h(subscript) ............... hole I .......................... electric current IL ...................... photocurrent Io ...................... scale current Isc ..................... short-circuit current I(v) ....................... current-voltage device characteristic JDx ..................... diffusion current density Jdx ..................... total current density JeDx, JhDx . current density associated with electron diffusion, current den- sity associated with holes diffusion xiv List of Symbols Jedx, Jhdx . current density associated with electron drift, current density associated with holes drift Jo ...................... scale current density Jrecom .................. recombination current density Jro ..................... recombination scale current density Jso ..................... scale current density J(v) ....................... total current density associated with current-voltage characteristics Jx ...................... total current density k .......................... the Boltzmann constant L .......................... length Le, Lh ............... electron diffusion length, hole diffusion length n .......................... electron concentration Nd ...................... donor and ionized donor concentrations ni ...................... intrinsic concentration nM ...................... concentration of electrons at the center of junction nno, ppo .............. equilibrium majority carrier concentrations npo, pno .............. equilibrium minority carrier concentration np ...................... electron concentration xv List of Symbols nx ...................... effect of confinement in x-direction ny ...................... effect of confinement in y-direction nz ...................... effect of confinement in z-direction Ñ(ω) .......... density of modes nf (f), nλ(λ), nω(ω) ..........energy density per electromagnetic mode Pmax .................... maximum power p .......................... hole concentration pM ...................... concentration of holes at the center of the junction pn ...................... hole concentration p(v) ....................... delivered power pf (f), pλ(λ), pω(ω) ... received power flux spectra in frequency, wavelength, angular frequency q .......................... electron charge Rs ...................... series resistance Rsh ..................... shunt resistance rs ...................... radius of the Sun rse ..................... radius of the Earth orbit S(λ) ............... solar spectrum xvi List of Symbols S̃(λ) ......... integrated solar spectrum T .......................... temperature t .......................... time ti ...................... time of the i− th electron uxi ..................... initial velocity in x-direction V, v ....................... voltage Vo ...................... built-in voltage Voc ..................... open-circuit voltage vedx, vhdx . electron drift velocity and hole drift velocity in the x-direction vxi ..................... velocity in the x-direction Wn,Wp ............... width of depletion region on the n-type side and on the p-type side with no applied voltage ∆ ..................... excess charge carrier concentration ∇2 ............... Laplacian operator ∇.∇ ε .............. permittivity εo ............ permittivity of free space η ....................... ideality factor θ ..................... angle xvii List of Symbols λ .................... wavelength µf (f), µλ(λ), µω(ω) ........ energy density associated with an electromag- netic cavity µh, µe ........... hole drift mobility, electron drift mobility ρ ....................... resistivity ρnet ................. net space charge density τe ................... electron lifetime τh ................... hole lifetime Ex .............. electric field in the x-direction Φe ................... electron flux Φh ................... hole flux ω ..................... angular frequency xviii Glossary AM0 air mass zero AM1 air mass one AM1.5 air mass 1.5 ASTM American Society for Testing and Materials c-Ge crystalline germanium c-Si crystalline silicon EJ Exajoules FF fill-factor NREL National Renewable Energy Laboratory WWEA World Wind Energy Association xix Acknowledgements First and foremost, I want to thank my advisor, Dr. Stephen O’Leary. It has been an honor to be his MASc student. I gratefully appreciate his contributions, in terms of time, ideas, and funding. These made my MASc experience productive and stimulating. His advice and comments greatly helped me with my progress. I also appreciate his open mind and confidence, allowing me to propose and develop my own ideas. The joy and enthusiasm he has for his research was contagious and motivational for me, even during the tough times. I am also thankful for the excellent role model he has provided for me. Lastly, I offer my regards and blessings to all those who supported me during the completion of this thesis. xx Chapter 1 Introduction The economic vibrancy of modern industrial economies requires access to cheap and abundant supplies of energy. Today, the uses of energy are numerous. Energy is needed for the transportation of goods and for man- ufacturing. It is also crucial for allowing habitation in regions of the Earth that are either too cold or too hot or both. The demand for energy continues to increase with each passing year; see Figure 1.1. In 2010, industrial soci- eties consumed about 515 Exajoules (EJ) of energy; 1 Exajoule is equivalent to 1018 Joules (J). Projections suggest that the global demand for energy will exceed 700 EJ per year by the year 2030, a 36% increase over that consumed in 2010 [1]. Energy supplies fuel modern human civilization. In- dustrial societies would cease to function if sources of energy were no longer available. There are a number of natural resources that may be drawn upon in order to generate energy. For the past century, fossil fuels have been the pri- mary source of energy for modern industrial societies, such as Canada’s [2]. Unfortunately, in recent years, the finiteness and non-renewable nature of this resource has become apparent [3]. Oil industry insiders suggest that “peak oil” will be achieved shortly. In addition, the demand for energy from emerging economies will rapidly deplete the reservoirs of fossil fuels that 1 Chapter 1. Introduction Figure 1.1: Worldwide energy consumption by industrial societies as a function of the year. The data depicted in this plot is from the US Department of Energy [1]. 2 Chapter 1. Introduction remain. The harmful emissions that arise as a corollary to the use of fossil fuels further underscores the urgent need to develop alternate and renewable sources of energy. In actual fact, renewable sources of energy have been used since the dawn of human civilization. Throughout human history, flowing water has been used in order to supply energy needs. Flour mills of yore often used water in order to provide the mechanical energy needed for the flywheels and pulleys found within such plants [4]. More recently, flowing water has been used in order to supply electrical power. This type of electrical power is often referred to as hydroelectricity. The first use of hydroelectricity dates back as far as the 1870s [5]. Adam Beck’s development of the Niagara Falls power station, the first large scale hydroelectrical generating station on Earth, fur- ther demonstrated the utility of hydroelectricity [6]. Similar projects have now been developed throughout the world. Unfortunately, this resource is now almost fully tapped, and new sources of renewable energy must also be considered [7]. While wind power has been used for many years, it is really only over the past two decades that wind power has been added to the energy mix as a viable candidate for the generation of renewable electrical power. This is particularly true in Europe, where tax incentives have created a large wind power industry. In Denmark, for example, wind power supplies as much as 20% of the electrical energy in the country [8]. In many jurisdictions within North America, windmills now dot the countryside [9], and wind power is now considered a serious potential source of renewable energy; at the present time, however, wind power only supplies 2 % of North Ameri- 3 Chapter 1. Introduction can electrical power demands [10]. Unfortunately, there are some drawbacks associated with wind power that detract from its practicality. The wind is variable in its intensity, and thus, it can not be used in order to supply a steady source of power. The construction of wind turbines is costly and impacts upon the local environment. The noise that is produced as a result of wind power also has taken the sheen off this potential source of renewable energy [11]. The Sun produces a tremendous amount of energy. Arising as a product of thermonuclear fusion processes within it, the solar radiation that is emit- ted from the Sun is mostly in the form of electromagnetic waves [12]. The Earth receives a solar flux of roughly 1400 W/m2 [13]. This radiation fur- nishes the Earth with the energy that the life forms on the Earth require. It is responsible for supplying the Earth with warmth and for photosynthesis, which is the process that allows plants to grow; photosynthesis converts so- lar radiation into organic compounds, which is ultimately the energy source that plants draw upon for sustenance. Estimates suggest that the Earth receives about 5.6 million EJ of solar radiation per year [14]. If completely harnessed, this would supply in excess of 10000 times of all of humanity’s energy needs today. Solar radiation may be harnessed through the use of solar cells. A pho- tovoltaic solar cell is comprised of semiconducting materials prepared in the form of a p-n junction. Photovoltaic solar cells aim to generate elec- trical power in response to exposure to the Sun’s rays. Solar radiation corresponds to a flux of photons, and when these photons are sufficiently energetic, electron-hole pairs can be generated within the semiconductor. 4 Chapter 1. Introduction Through the application of a potential across such a solar cell, electrical power can be drawn from the solar cell. The first modern photovoltaic so- lar cell was that fabricated in 1954 by Pearson, Chapin, and Fuller of Bell Telephone Laboratories [15]. This prototype crystalline silicon (c-Si) based solar cell, which was based on the p-n junction architecture, was reported to have an initial efficiency of 4.5 %. Subsequent improvements, that were introduced in the following months, allowed this efficiency to increase to 6 % [16]. Improvements in the materials used and the device geometries employed, that have been introduced over the past half century, have led to dramatic improvements in the resultant c-Si based solar cell efficiencies, as may be seen from Figure 1.2 [17]. Fundamentally, a modern photovoltaic solar cell is a semiconductor p-n junction. That is, it is comprised of a p-type semiconductor region that is directly abutted against an n-type semiconductor region. Such junctions are found frequently in the microelectronics industry, and the fact that the fabri- cation processes that are employed in their manufacturing are well-developed further adds to the practicality of this particular solar cell device configu- ration. When exposed to sunlight, the absorption of the photons within the sunlight by the semiconductor leads to the generation of electron-hole pairs within the semiconductor. These charge carriers are then separated under the action of the electric field. The potential difference that develops between these separated charge carriers, coupled with the resultant pho- tocurrent, are what is responsible for the generation of electrical power from such a solar cell. A p-n junction is comprised of both neutral and space-charge regions, as 5 Chapter 1. Introduction 1950 1960 1970 1980 1990 2000 2010 5 10 15 20 25 30 Year Ef fic ie nc y (% ) Figure 1.2: The efficiency of c-Si based solar cells as a function of the year. The data is taken from Hezel [17]. 6 Chapter 1. Introduction is shown in Figure 1.3. The space-charge region is formed in the immediate locality of the interface between the p-type and n-type semiconductor re- gions. The neutral regions lie on either side of this space-charge region. The minority charge carriers within both the p-type and n-type semiconductor regions are ultimately what are responsible for the external photocurrent. To zeroth-order, the minority charge carriers within the neutral regions, that are primarily engaging in diffusion processes, may be considered as being supplied by the external current. If the recombination that occurs within the space-charge region is taken into account, however, the external current must also supply the charges that are lost due to recombination within the space-charge region. As photovoltaic solar cells are fabricated using a p-n junction configuration, it is clear that recombination processes are likely to play an important role in determining the performance of such a cell. This thesis is organized in the following manner. In Chapter 2, a brief review of the developments in solar cell technology, that have transpired over the years, is provided. In addition, a detailed description of the solar spectrum is furnished within this chapter, this providing a measure of how the power provided by the Sun is distributed over the electromagnetic spec- trum. Then, in Chapter 3, the operating principles of the p-n junction are discussed, the photovoltaic solar cell being based on the p-n junction archi- tecture. In Chapter 4, the p-n junction operating principles are employed in concert with the solar spectrum in order to analyze the performance of a photovoltaic solar cell. The impact of recombination processes on these results is also examined. Finally, in Chapter 5, conclusions are drawn and recommendations for further research are suggested. 7 Chapter 1. Introduction p−type neutral region n−type neutral region space−charge region Figure 1.3: A schematic of a p-n junction. The p-type region is abutted next to the n-type region. The neutral and space-charge regions are depicted. 8 Chapter 2 The solar spectrum and background material 2.1 Background Modern photovoltaic solar cells are based upon the p-n junction con- figuration. Semiconductors, such as silicon or germanium, are the materi- als from which these junctions are fabricated. When the Sun’s radiation impinges upon such a junction, photons, with energies that exceed the en- ergy gap of the underlying semiconductor, generate electron-hole pairs. The “built-in” electric field, associated with the space-charge region at the p- n junction interface, separates these charge carriers. As a consequence, a potential difference arises across the two electrodes of the junction. This phenomenon is referred to as the “photovoltaic effect,” and, as a result of it, exposure of such a junction to light leads to the generation of current [15]. The resultant current is referred to as a photocurrent, as it is ultimately related to light exposure. The power delivered from the photovoltaic solar cell is essentially proportional to the product of the photocurrent and the potential difference across the junction. In order for a photovoltaic solar cell to work properly, it should per- 9 2.1. Background form four essential functions: (1) light absorption, (2) charge carrier genera- tion, (3) charge carrier separation, and (4) and charge carrier collection [15]. These functions allow a photovoltaic solar cell to supply electrical power to a load. Light absorption arises as a consequence of the interaction of a beam of light with condensed matter, i.e., the semiconductor. As the light beam passes through the material, its intensity diminishes in an exponen- tial fashion. The attenuation of light is directly related to the generation of electron-hole pairs within the material. Photons in the light beam with energies that exceed the energy gap of the semiconductor can lead to the creation of electron-hole pairs. These pairs of charge carriers, which typi- cally recombine when equilibrium is achieved, separate under the action of the “built-in” electric field that exists across the p-n junction; with light exposure generating excess charge carriers, the “built-in” junction is not in equilibrium. In order for these charge carriers to contribute to the external photocurrent, these charge carriers must drift under the action of the elec- tric field within the bulk of the p-n junction, i.e., away from the p-n junction interface towards the external contacts. This chapter is organized in the following manner. In Section 2.2, a brief overview of solar cell technological developments, that have transpired since their introduction, is provided. Then, in Section 2.3, an examination of the energy in the Sun’s electromagnetic rays is furnished. A number of solar spectra are then defined in Section 2.4. Finally the integrated solar spectrum is introduced in Section 2.5. 10 2.2. Solar cell technology developments 2.2 Solar cell technology developments In 1839, the French physicist, A. E. Becquerel, demonstrated the pho- tovoltaic effect. In particular, while studying the properties of electrolytic cells, Becquerel noted that an electrical current is produced when such cells are exposed to light [5]. In contrast, with no light exposure, the electrical current was found to be nil. As this current is directly related to the pres- ence of light, it was dubbed a photocurrent. In 1876, A. William and R. Day observed a photocurrent within the semiconductor selenium [5]. Later on, in 1894, C. Fritts developed what would later be referred to as a photo- voltaic solar cell, comprised of a junction of selenium coated with gold. The efficiency of this cell was about 1 % [15]. While not as efficient as modern photovoltaic solar cells, Fritts’s development was an impressive 19th Century engineering innovation. In the 1950s, c-Si based p-n junctions were being mass produced for a wide variety of electron device applications. As a result, it was just a matter of time before the photovoltaic solar cell was implemented in a c-Si based p-n junction technology. The first successfully implemented c-Si based photo- voltaic solar cell was that fabricated in 1954 at Bell Telephone Laboratories by Pearson, Chapin, and Fuller [15]. This cell, which was based upon the p-n junction architecture, had an initial efficiency which was reported to be about 4.5 %. Later on, however, with improvements in the device design, a 6 % efficiency was achieved. Initial applications for photovoltaic solar cells focused on the emerging space-program and military applications. The first commercial applications for this emerging solar cell technology were found 11 2.2. Solar cell technology developments in commercial satellites, that were starting to be deployed during the late 1960s. Satellites needed sustainable means of supplying their energy needs, and solar cells filled this niche very effectively. The first commercial satel- lite to be powered using solar cells was Telstar from the American Telegraph and Telecommunications Company, the parent company of Bell Telephone Laboratories. This satellite was placed in Earth orbit in 1962 [8]. Owing to their great cost, solar cells were initially not employed in con- sumer electronic products. This has changed over time, however, as solar cell technology, and the materials related to solar cells, have become more widely available. Initial applications of solar cell technology in consumer electronic goods occurred in the mid-1970s (they were initially deployed in watches and calculators) and now solar cells are a staple of the consumer electronic products that are available. Recently, solar cells based on different semiconductor materials, such as cadmium sulphide, gallium arsenide, and cadmium telluride, have been employed in order to achieve higher efficien- cies than that achieved previously through c-Si based solar cells [15]. Gains, in terms of cost-reductions and economies of scale, have also been achieved. In addition to the burgeoning scope of consumer electronic products that employ solar cell technology, since the 1990s photovoltaic solar cells have been fabricated for telecommunication and localized power supply applica- tions [15]. Recently, there has been interest in using photovoltaic solar cells for supplying electricity to the grid [15]. In 2000, the first photovoltaic solar cell production plant, that aimed to produce photovoltaic solar cells for mass power generation purposes, was built in Perrysburg, Ohio [8]. Most recently, research, in terms of photovoltaic solar cells, has focused upon fabricating 12 2.3. Blackbody radiation solar cells of improved geometries and with different materials in order to achieve higher efficiencies and greater cost effectiveness [15]. 2.3 Blackbody radiation The power within the Sun’s rays may be characterized by the solar spec- trum, which provides for the distribution of power over the electromagnetic spectrum. Often times, insights into the spectral dependence of this power spectrum may be achieved by treating the Sun as a blackbody, a blackbody absorbing all of the radiation that it receives. While the Sun is not exactly a blackbody, the blackbody approximation works quite well and provides insight into the spectral dependence of the solar spectrum. It is therefore instructive to consider the power spectrum corresponding to a blackbody. For steady-state sinusoidal excitations, of angular frequency, ω, in a vac- uum, the equation for the electric field, ~E, may be expressed as ∇2~E = −ω 2 c2 ~E, (2.1) where c represents the speed of light in vacuum. For an electromagnetic cavity, of dimensions L×L×L, as shown in Figure 2.1, for zero-field along the boundaries, the possible solutions for Eq. (2.1), for all dimensions, i.e., Ex, Ey, and Ez, where ~E = (Ex, Ey, Ez), are of the form [18], E(x, y, z) = A sin ( nxpix L ) sin ( nypiy L ) sin ( nzpiz L ) , (2.2) where A is a constant and nx, ny, and nz are integers which are greater than or equal to unity. Substituting Eq. (2.2) into Eq. (2.1), it may be seen that ω2 c2 = ( nxpi L )2 + ( nypi L )2 + ( nzpi L )2 . (2.3) 13 2.3. Blackbody radiation L L L x y z Figure 2.1: A three dimensional electromagnetic cavity of the dimensions L×L×L. 14 2.3. Blackbody radiation 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 n z n x ny Figure 2.2: Points in n-space. Each point corresponds to an allowed set of nx, ny, and nz values, corresponding to a possible solution of Eq. (2.1). 15 2.3. Blackbody radiation In n-space, which is depicted in Figure ??, there is a unit density of points in the first octet, i.e, there is one point for every combination of nx, ny, and nz, where these quantities are integers. In the continuum limit, i.e, when nx, ny, and nz become large, the number of points with angular frequency less than a given value of ω, Ñ(ω) = 1 8 4pi 3 n3, (2.4) where n2 = n2x + n 2 y + n 2 z. (2.5) Noting, from Eq. (2.3), that ω2 c2 = pi2 L2 n2, (2.6) it is seen that Ñ(ω) = 1 8 4pi 3 ( ω2L2 c2pi2 )3/2 . (2.7) This suggest that a density of modes may be defined, i.e, where nω(ω)∆ω corresponds to the number of electromagnetic modes in the cavity, between [ω, ω+∆ω], per unit volume [18]. Given that every nx, ny, and nz selection corresponds to two possible electromagnetic modes, i.e, two different po- larizations, differentiation of Eq. (2.7), multiplication by a factor of 2, and division by the volume, L3, results in nω(ω) = ω2 c3pi2 . (2.8) It is often the case that one must instead determine the distribution of 16 2.3. Blackbody radiation modes over frequency or wavelength. Noting that f = ω 2pi , (2.9) and that fλ = c, (2.10) two alternate density of modes distribution functions may also be intro- duced, nf (f) and nλ(λ), where nf (f)∆f and nλ(λ)∆λ denote the number of electromagnetic modes, per unit volume, between [f , f+∆ f ] and [λ, λ+ ∆λ], respectively. Thus, nf (f) = 8pif2 c3 , (2.11) while nλ(λ) = 8pi λ4 . (2.12) These functions, nω(ω), nf (f), and nλ(λ), provide for the density of electro- magnetic modes, per unit volume, with respect to ω, f , and λ, respectively. The density of electromagnetic modes, nω(ω), may now be employed in the determination of the energy density. In particular, it may be shown that the energy density uω(ω) = nω(ω)E(ω), (2.13) where uω(ω)∆ω represents the energy, per unit volume, between angular frequencies [ω, ω+∆ω], and E(ω) denotes the energy per electromagnetic mode of angular frequency ω. Classically, it may be shown that the energy per electromagnetic mode is kT , where T represents the temperature and k denotes the Boltzmann constant, i.e., uω(ω) = 1 pi2 w2 c3 kT. (2.14) 17 2.3. Blackbody radiation In a similar sense, the energy density, with respect to frequency and wave- length, uf (f) and uλ(λ), respectively, may be expressed as uf (f) = 8pif2 c3 kT, (2.15) and uλ(λ) = 8pi λ4 kT , (2.16) where uf (f)∆f and uλ(λ)∆λ represent the energy, per unit volume, between [f , f+∆f ] and [ω, ω+∆ω], respectively. Unfortunately, the classical en- ergy distributions, uω(ω), uf (f), and uλ(λ), depicted in Eqs. (2.14), (2.15), and (2.16), respectively, exhibit a non-integrable singularity at high frequen- cies or low wavelengths. This was referred to as the ultraviolet catastrophe in the 19th Century scientific community [19]. Of course, the integration of the energy spectrum of a blackbody object is in actuality finite, this being experimentally known in the 19th Century. The reasons why the ultraviolet catastrophe does not occur were accounted for in the early 20th Century, and the phenomena underlying these reasons led to a revolution in physics that continues today. In 1900, Planck suggested that light was actually comprised of photons, and that each photon has a discrete energy, which is linearly proportional to its angular frequency, ω [19]. Introducing a constant, now referred to as Planck’s constant, h, the corresponding photon energy may be expressed as h̄ω, where h̄ is referred to as the reduced Planck’s constant, i.e., h̄≡ h 2pi . (2.17) Through a statistical analysis, Planck was able to demonstrate that 18 2.3. Blackbody radiation 10−2 10−1 100 101 102 0 0.5 1 1.5 Classic theory Quantum theory h̄ω/kT E ω (ω )/ k T Figure 2.3: The energy per electromagnetic mode as a function of h̄ωkT . The clas- sical result is contrasted with that due to Planck. The results are presented in a dimensionless form through a divison by the classical result, kT . 19 2.3. Blackbody radiation Eω(ω) = h̄ω exp ( h̄ω kT ) − 1 . (2.18) In Figure 2.3, the energy per mode is plotted as a function of the photon energy, h̄ω. The classical result, kT , is also depicted, it being independent of h̄ω; this spectral dependence is normalized with the respect to the classical result, kT , i.e., the results are presented in a dimensionless form. It is noted that as h̄ω becomes very large, the energy per electromagnetic mode diminishes, eventually becoming negligible. This allows for a reconciliation between theory and experiment, i.e., the ultraviolet catastrophe is eliminated through the exponential decrease in Eω(ω) as ω → ∞, i.e., the resultant function, uω(ω), is now integrable. It is noted that in the classical limit, i.e., when h̄→ 0, Eq. (2.18) reduces to the classical result, i.e., Eω(ω)→ kT. (2.19) This is consistent with the general observation that all quantum results reduce to their corresponding classical results as h̄ → 0, i.e., as one moves towards the classical limit. In the f representation Ef (f) = hf exp ( hf kT ) − 1 , (2.20) while in the λ representation Eλ(λ) = hc λ [ exp ( hc kλT ) − 1 ] , (2.21) where Ef (f) and Eλ(λ) represent the energy per electromagnetic mode. Us- ing the Planck result for Eω(ω), i.e., Eq. (2.18), from Eqs. (2.8) and (2.13), 20 2.3. Blackbody radiation the energy density associated with an electromagnetic cavity may be ex- pressed as uω(ω) = 1 pi2c3  h̄ω3 exp ( h̄ω kT ) − 1 . (2.22) A similar analysis, performed in the f representation, yields uf (f) = 8pif2 c3  hf exp ( hf kT ) − 1 , (2.23) where Eq. (2.23) is determined from the product of Eqs. (2.11) and (2.20). In the wavelength representation, however, uλ(λ) = 1 λ5  8pihc exp ( hc λkT ) − 1 , (2.24) where Eq. (2.24) is determined from the product of Eqs. (2.12) and (2.21). It is noted that all of the energy distribution functions, i.e., uω(ω), uf (f), and uλ(λ), are now integrable. The distribution of energy of an electromagnetic cavity, is plotted as a function of wavelength in Figure 2.4, for both the classical and quantum cases, i.e., Eqs. (2.16) and (2.24), respectively. The temperature is set to the surface temperature of the Sun, i.e., 5780 K, for the purposes of this analysis. The ultraviolet catastrophe is seen at low wavelengths for the classical result [20]. It can be shown that the power flux produced by a blackbody object is related to the energy density within the cavity. In particular, we have pω(ω) = uω(ω) c 4 , (2.25) where pω(ω)∆ω represents the power emitted, between [ω, ω+∆ω], per unit area; the power flux refers to the projected power, per unit area, that is 21 2.3. Blackbody radiation 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10−6 0 5 10 15 x 10 5 classical result quantum result Wavelength (m) En er gy  d en sit y ( J/m 3 ) Figure 2.4: The energy density in an electromagnetic cavity as a function of the wavelength. The temperature is set to 5780 K for the purposes of this analysis. The classical result, i.e., Eq (2.16), is contrasted with result of Planck, i.e., the quantum result, Eq (2.24). 22 2.3. Blackbody radiation radially emitted. Using Eqs. (2.22) and (2.25), it can be shown that pω(ω) = 1 4pi2c2  h̄ω3 exp ( h̄ω kT ) − 1 , (2.26) where integration over this power spectrum leads to the total power radiated from the blackbody object, per unit area. A similar analysis may be used to show that in the f representation, pf (f) = 2pif2 c2  hf exp ( hf kT ) − 1 , (2.27) where pf (f)∆f provides for the power emitted, between frequencies [f, f + ∆f ], per unit area. In the λ representation, however, pλ(λ) = 1 λ5  2pihc2 exp ( hc λkT ) − 1 , (2.28) where pλ(λ)∆λ provides for the power emitted, between wavelengths [λ, λ+ ∆λ], per unit area. All of these power flux spectra are seen to be integrable. The expressions for the power flux spectra that have been determined correspond to the surface of the blackbody. The solar flux received by the Earth is attenuated, of course. Assuming that the solar radiation emitted from the Sun is projected radially from the surface of the Sun, if the Earth orbits the Sun at a radius rse, and if the Sun itself has a radius rs, then the power flux received at the periphery of the Earth’s atmosphere is reduced by an attenuation factor of ( rs rse )2 , as seen in Figure 2.5 [21]. That is, in the ω representation, the power flux spectra, received by the periphery of the Earth’s atmosphere, may be expressed as pω(ω) = 1 4pi2c2 ( rs rse )2 h̄ω3 exp ( h̄ω kT ) − 1 . (2.29) 23 2.3. Blackbody radiation Earth orbit o r s Earth Sun r se Figure 2.5: The Earth orbiting the Sun at radius rse. The radius of the Sun itself is rs. These distances have been exaggerated on this figure for the purposes of illustration. 24 2.4. The solar spectrum In the f representation, however, this received power flux spectra may in- stead be expressed as pf (f) = 2pif2 c2 ( rs rse )2 hf exp ( hf kT ) − 1 . (2.30) In the λ representation, this received power flux spectra may be expressed as pλ(λ) = ( rs rse )2 2pihc2 λ5 [ exp ( hc λkT ) − 1 ] . (2.31) The temperature in these expressions, i.e., Eqs. (2.29), (2.30), and (2.31), corresponds to that at the surface of the blackbody, i.e., the surface of the Sun, as thermal equilibrium is achieved with the surrounding environment through this surface. It is seen that these power spectra are strong functions of this temperature. Treating the Sun as a blackbody, the power spectrum of the solar flux received by the periphery of the Earth’s atmosphere, i.e., the solar spectrum, is a strong function of the temperature of the surface of the Sun. In the λ representation, i.e., Eq. (2.31), the variations exhibited by this power spectrum to changes in the Sun’s surface temperature are depicted in Figure 2.6. For the purposes of this analysis, the radius of the Sun, rs, is set to be 700 × 103 km, while the radius of the Earth orbit, rse, is taken to be 150 × 106 km. 2.4 The solar spectrum The solar spectrum characterizes the distribution of power within the electromagnetic radiation received from the Sun by the Earth. This electro- magnetic radiation is comprised of both electric and magnetic field compo- 25 2.4. The solar spectrum 0 500 1000 1500 2000 2500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 9 7000K 6000K 4000K Wavelength (nm) Po we r f lux  sp ec tra  (W att s/m 3 ) Figure 2.6: The solar spectrum, i.e., the power flux spectrum received at the periphery of the Earth’s atmosphere, as a function of wavelength of blackbody radiation for different temperatures. These temperatures correspond to the temperature of the surface of the Sun. Eq. (2.31) is used for the purposes of this analysis, where rs is set to 7.0 × 105 km and rse is set to 1.5 × 108 km. 26 2.4. The solar spectrum nents, and it reaches the Earth after propagating from the Sun to the Earth through space. It is instructive to define the spectral intensity of the solar radiation received by the Earth in terms of the “air mass” that the Sun’s rays encounter. The air mass takes into account the losses that occur as a consequence of the propagation of the Sun’s electromagnetic flux through the Earth’s atmosphere. Losses arise as a result of the interactions of the electromagnetic radiation from the Sun with aerosol particles, gas molecules, and cloud droplets [22]. At the periphery of the Earth’s atmosphere, the solar spectrum is as determined by the Sun, and the distance of the Earth’s orbit from the Sun. It should be noted, however, that slight deviations from the blackbody spec- trum, arising from absorption processes within the Sun’s atmosphere itself, do occur. As a result, at the periphery of the Earth’s atmosphere, the solar spectrum appears slightly different from that of a blackbody with an appro- priately selected Sun surface temperature [23]. The intensity of the solar radiation received at the periphery of the Earth’s atmosphere is referred to as air mass-zero (AM0), i.e., the sunlight has passed through zero of the Earth’s atmospheric mass. In contrast, at the Earth’s surface, when the Sun is directly overhead, the Sun’s rays have traveled through one atmosphere. The resultant solar spectrum is referred to as air mass-one (AM1), and it will have a lower intensity than the AM0 solar spectrum, i.e., propagating through the Earth’s atmosphere will lead to an attenuation in the received solar spectrum, this attenuation being non-uniformly distributed over the electromagnetic spectrum [22]. If the Sun’s radiation is received at an oblique angle, then the Sun’s 27 2.5. Integrated solar spectrum rays will effectively have passed through more than one atmosphere. Air mass-1.5 (AM1.5), a commonly used reference solar spectrum, refers to the solar radiation received by the surface of the Earth when the angle of in- cidence for the solar radiation is such that the Sun’s rays effectively have passed through 1.5 atmospheres. This corresponds to roughly a 48.2◦ angle of incidence with respect to the vertical axis; see Figure 2.7. In general, the air mass can be defined in terms of the angle between the Sun’s rays and the vertical axis, i.e., the axis normal to the surface of the Earth. That is, AM(number) = 1 cos(θ) , (2.32) where θ refers to the angle of incidence with respect to the vertical axis, i.e., θ = 00 corresponds to the Sun’s rays coming in from directly overhead. The AM0 and AM1.5 solar spectra are depicted in Figure 2.8. The correspond- ing blackbody radiation spectrum is also depicted in Figure 2.8, where the surface temperature of the Sun is set to be 5780 K [23], this being a typical value found in the literature. Integration over these solar spectra, over the entire range of wavelengths, leads to the power flux densities tabulated in Table 2.1. Further details are discussed in the literature. 2.5 Integrated solar spectrum In the analysis of a photovoltaic solar cell, it is instructive to determine the integrated solar spectrum. In the λ representation, the integrated solar spectrum. S̃(λ)≡ ∫ λ 0 S(λ)dλ, (2.33) 28 2.5. Integrated solar spectrum angle of incidence θ vertical axis Sun rays Figure 2.7: The angle of incidence, the vertical axis, and the direction of the Sun’s rays. 29 2.5. Integrated solar spectrum 0 1000 2000 3000 4000 5000 0 0.5 1 1.5 2 2.5 x 10 9 AM0 AM1.5 Blackbody Wavelength (nm) So la r S pe ct ru m  (W att s/m 3 ) Figure 2.8: The solar spectra corresponding to the AM0 and AM1.5 cases. The experimental data is taken from NREL [24]. The blackbody solar spectrum is also depicted, for the Sun’s surface temperature set to 5780 K [23]. 30 2.5. Integrated solar spectrum Table 2.1: The integrated solar spectra obtained for the upper wavelength in Eq. (2.33) being set to ∞. This corresponds to integrating the so- lar spectra over the entire range of wavelengths. Solar Condition Power flux density (W/m2) AM0 1366.1 AM1.5 1000.13 Blackbody 1534.4 where S(λ) denotes the solar spectrum itself. The integrated solar spectra, corresponding to the AM0 and AM1.5 solar spectra depicted in Figure 2.8, are shown in Figure 2.9. The integrated blackbody solar spectrum is also depicted in Figure 2.9 for the case of the surface temperature of the Sun being set to 5780 K [23]. It is noted that as the upper wavelength in the integration of the solar spectrum, i.e., as seen in Eq. (2.33), approaches infinity, the power densities, tabulated in Table 2.1, are achieved. A similar analysis may be performed in the ω and f representations, of course. 31 2.5. Integrated solar spectrum 0 500 1000 1500 2000 2500 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Blackbody AM0 AM1.5 Wavelength (nm) In te gr at ed  S ol ar  P ow er  fl ux  (W /m 2 ) Figure 2.9: The integrated solar spectra corresponding to the AM0 and AM1.5 so- lar spectra. The integrated blackbody solar spectrum is also depicted for the Sun’s surface temperature set to 5780 K [23]. 32 Chapter 3 General theory of the p-n junction with recombination 3.1 Rationale The fundamental building block of most semiconductor devices is the p-n junction. This junction forms the cornerstone upon which modern micro- electronics is built. Accordingly, an understanding of the physics underlying p-n junctions is an important step to developing an appreciation for the op- eration of modern electron devices. Given that the p-n junction lies at the heart of the modern photovoltaic solar cell, understanding the behavior of the p-n junction is also crucial to the understanding of the operation of such a cell. In 1951, Shockley introduced a theory for the operation of the p-n junction, and described the current-voltage device characteristics of the resultant electron device [25]. This theory remains at the core of our under- standing of the operation of such devices today. In this chapter, the general operating principles of the p-n junction are laid out. Rather than going through all of the details provided for in the analysis of Shockley [25], instead only details relevant to photovoltaic solar cell modeling are provided. In particular, the analysis starts with a dis- 33 3.2. The geometric layout of the p-n junction cussion on the geometric layout of the p-n junction. Then, the drift and diffusion charge carrier transport processes, that occur within a p-n junc- tion, are presented. The behavior of the p-n junction under no bias is then discussed. The forward biased p-n junction behavior is then reviewed, an expression for the current-voltage device characteristic being obtained. The behavior of the p-n junction under reverse bias is also presented. Finally, the role that recombination processes play in influencing these device char- acteristics is discussed. This chapter is organized in the following manner. In Section 3.2, the ge- ometric layout of the p-n junction is reviewed. The drift and diffusion charge carrier transport processes, that occur within a p-n junction, are then dis- cussed in Section 3.3. The behavior of the p-n junction under no bias is then presented in Section 3.4. The forward biased p-n junction is reviewed in Section 3.5, a current-voltage device characteristic being determined for the case of no recombination. The reverse biased p-n junction is then con- sidered in Section 3.6. Finally, the role that recombination processes play in influencing these device characteristics is considered in Section 3.7. 3.2 The geometric layout of the p-n junction In general, the p-n junction consists of a p-type semiconductor that is directly abutted against an n-type semiconductor. Most often, a common crystal is used, although heterogeneous p-n junctions have been fabricated. The p-type region is created by doping the underlying semiconductor mate- rial with acceptors, while the n-type region is created by doping the under- 34 3.3. Drift and diffusion processes lying semiconductor material with donors; for the case of c-Si, boron and phosphorous are commonly used acceptors and donors, respectively [26]. The transition between the p-type and n-type regions can be abrupt, or it can be of a gentler grade; the metallurgical junction represents the boundary junction between the p-type and n-type regions. While most p-n junctions are one-dimensional in nature, two-dimensional and three-dimensional p-n junctions have also been used in order to produce photovoltaic solar cells. The geometric layout of a one-dimensional abrupt p-n junction is depicted in Figure 3.1. 3.3 Drift and diffusion processes There are two kinds of charge carriers within semiconductors, electrons and holes. These charge carriers engage in charge transport process that determine the performance of the resultant electron device. There are two fundamental charge carrier transport processes that influence the behavior of a p-n junction: 1) drift, in which the charge carriers transit and under the action of the applied electric field, and 2) diffusion, in which natural equi- librium processes act to redistribute the charge carriers within the semicon- ductor. In this section, these processes are discussed. How these processes influence the behavior of the resultant p-n junction will be considered later. Consider an ensemble of N electrons. Electrons in an ensemble are in a perpetual state of motion and they frequently engage in scattering. Be- tween scattering events, they act as free particles. It will be assumed that the i − th electron begins its transit at t = ti, where ti denotes the time 35 3.3. Drift and diffusion processes p−type M semiconductor n−type semiconductorcontact contact Figure 3.1: The geometric layout of a representative one-dimensional abrupt p-n junction. M denotes the metallurgical junction between the p-type and n-type regions. 36 3.3. Drift and diffusion processes of the last scattering event that the i− th electron has experienced. Under the action of an applied electric field in the x-direction, assuming that the initial electron velocity of the i− th electron in the x-direction is ux(ti), the velocity of this electron in the x-direction at time t may be written as vxi(t) = uxi(ti)− qEx me (t− ti), (3.1) where q denotes the electron charge, me represents the mass of the electron, and Ex is the electric field in the x-direction. The drift velocity of the an electron ensemble is defined to be vedx≡ [vx1 + vx2 + ...+ vxN ] N . (3.2) Noting that the average value of the initial velocity of the electrons over the ensemble is nil, i.e., that ux1(t1) + ux2(t2) + ...+ uxN (tN ) N = 0, (3.3) this drift velocity, defined in Eq. (3.2), may instead be expressed as vedx = −qEx me [ (t− t1) + (t− t2) + ...+ (t− tN ) N ] . (3.4) Noting that the average free time for each electron over the ensemble, i.e., the average time between ti to t when i−th electron is free, may be expressed as t− ti = (t− t1) + (t− t2) + ....+ (t− tN ) N , (3.5) it is seen that Eq. (3.4) may instead be represented as vedx = − qEx me (t− ti), (3.6) 37 3.3. Drift and diffusion processes where (t− ti) corresponds to the average free time for the electrons between scattering events. Defining the electron drift mobility µe = qEx me (t− ti), (3.7) it is thus seen that vedx = −µeEx. (3.8) That is, the electron drift velocity scales linearly with the applied electric field in the x-direction, Ex. A similar analysis may be employed to show that the hole drift mobility in the x-direction, vhdx = µhEx, (3.9) where µh denotes the hole drift mobility, this mobility being defined in a similar manner to the electron drift mobility. The electron and hole drift mobilities, corresponding to c-Si and crystalline germanium (c-Ge) at 300 K, are tabulated in Table 3.1. Table 3.1: The electron and hole drift mobilities associated with c-Si and c-Ge at 300 K [27]. Materials µe (cm 2/V·s) µh (cm2/V·s) c-Si 1400 470 c-Ge 3900 1900 The current density associated with an electric field applied in the x- direction may be expressed as the sum over the contributions due to the 38 3.3. Drift and diffusion processes electrons and holes, respectively. In particular, the current density associ- ated with drifting electrons Jedx = −qnvedx, (3.10) where n denotes the electron concentration (the negative sign is introduced as electrons are negatively charged), while the current density associated with drifting holes Jhdx = qpvhdx, (3.11) where p represents the hole concentration. Using equations Eqs. (3.8) and (3.9), Eqs. (3.10) and (3.11) may instead be written as Jedx = qnµeEx, (3.12) and Jhdx = qpµhEx, (3.13) respectively. Thus, the total current density in the x-direction associated with drifting charge carriers, Jdx = Jedx + Jhdx, (3.14) which may in turn be expressed as Jdx = (qnµe + qpµh)Ex. (3.15) In the absence of an applied electric field, charge carriers, be they elec- trons or holes, diffuse from regions of higher concentration to regions of lower concentration. The flux of electrons in the x-direction φe = −De∂n ∂x , (3.16) 39 3.3. Drift and diffusion processes where De denotes the electron diffusion coefficient. Accordingly, as the electrons have a charge of −q, the current density corresponding to electron diffusion may be expressed as JeDx = qDe ∂n ∂x . (3.17) A similar analysis indicates that the flux of holes in the x-direction may be expressed as φhDx = −Dh ∂p ∂x , (3.18) where Dh denotes the hole diffusion coefficient. Accordingly, as the holes have a charge of q, the hole current density corresponding to hole diffusion may be expressed as JhDx = −qDh ∂p ∂x . (3.19) The total current density in the x-direction associated with diffusive processes may be expressed as the sum over the contributions corresponding to the electrons and holes. Adding Eqs. (3.17) and (3.19), the diffusion current density may be expressed as JDx = qDe ∂n ∂x − qDh ∂p ∂x . (3.20) The diffusion coefficients, corresponding to c-Si and c-Ge at 300 K, are tab- ulated in Table 3.2. It is often the case that one has both an applied electric field and charge carrier concentration gradients. Accordingly, both drift and diffusion pro- cesses must be taken into account when determining the total current flux. These contributions to the overall current flux are schematically depicted 40 3.3. Drift and diffusion processes Table 3.2: The electron and hole diffusion coefficients associated with c-Si and c-Ge at 300 K [28]. Materials De (cm 2/s) Dh (cm 2/s) c-Si 35 12 c-Ge 100 49 in Figure 3.2. To zeroth-order, these charge carrier transport processes are independent of each other, and thus, the total current may be expressed as a sum over that due to drift and that due to diffusion processes, i.e., linear superposition applies. Thus, the total current density Jx = Jdx + JDx. (3.21) While equation Eq. (3.21) will not be explicitly used in the subsequent anal- ysis, in general, it is the equation to use in cases where one has both an applied electric field and variations in the charge carrier concentrations [26]. Recombination processes between the charges play an important role in influencing the performance of devices. When semiconductors are subject to a stimulus, for example, light in the case of a photovoltaic solar cell, excess electron-hole pairs can be generated. These charge carriers also re- combine. For the case of holes in an n-type semiconductor, the excess hole concentration may be expressed as ∆pn ≡ pn− pno, where pn represents the concentration of holes in the n-type region, it being a function of position, and pno denotes the equilibrium concentration in the n-type region; the p script refers to the fact that the charges considered are holes and the n sub- script refers to the fact that they are in the n-type semiconductor region, 41 3.3. Drift and diffusion processes + + + + + + − − − − − + + + + + + − − − − − + + + + + + − − − − − + + + + + + + + + + + + + + + + + − − − − − − − − − − − − − − − − − − − − − − − − − − − p−type n−type Electron drift Electron diffusion x Hole diffusion Hole drift Electric Field − − + + − Figure 3.2: The contributions to the overall current flux attributable to the vari- ous drift and diffusion processes. It is assumed that the electric field points from right to left [26]. The p-n junction architecture has been assumed. 42 3.3. Drift and diffusion processes i.e., they are minority charge carriers. For the case of excess holes combining with a lifetime of τh, the charge continuity equation in the x-direction may be expressed as 1 q ∂Jhx ∂x = −pn − pno τh , (3.22) where this equation accounts for the rate with which charge enters a differ- ential volume in order to compensate for the charge carrier recombination that occurs within. In the absence of an applied electric field, recalling Eq. (3.19), Eq. (3.22) reduces to −Dh∂ 2pn ∂x2 = −pn − pno τh , (3.23) which can be generalized to include a charge generation related term. For just the excess holes in the n-type semiconductor region, neglecting all vari- ations with respect to time, this charge continuity equation reduces to Dh d24pn dx2 = 4pn τh . (3.24) Defining the hole diffusion length, Lh≡ √ Dhτh, Eq. (3.24) may alternatively be expressed as d24pn dx2 = 4pn Lh 2 . (3.25) A similar expression may be determined for the excess electrons in the p-type semiconductor region, i.e., d24np dx2 = 4np Le 2 , (3.26) where Le≡ √ Deτe is referred to as the electron diffusion length and ∆np ≡ np−npo, np representing the concentration of electrons in the p-type region, 43 3.4. The p-n junction with no applied bias it being a function of position, and npo denotes the equilibrium concentra- tion of electrons in the p-type region; the n script refers to the fact that the charges considered are electrons and the p subscript refers to the fact that they are in the p-type semiconductor region, i.e., they are minority charge carriers. The electron and hole diffusion lengths, Le and Lh, respec- tively, corresponding to c-Si and c-Ge at 300 K, are tabulated in Table 3.3. These diffusion lengths will be used in the subsequent analysis. These equa- tions, i.e., Eqs. (3.25) and (3.26), are generally referred to as the ambipolar diffusion equations. Table 3.3: The electron and hole diffusion lengths associated with c-Si and c-Ge at 300 K [26]. Materials Le (µm) Lh (µm) c-Si 41 24 c-Ge 79 48 3.4 The p-n junction with no applied bias It is instructive to consider the distribution of charges within a one- dimensional abrupt p-n junction, as shown in Figure 3.3. The p-type semi- conductor region is doped with acceptors. As a result, at normal operating temperatures, i.e., when the acceptors are fully ionized, these acceptors will be negatively charged and the corresponding bulk semiconductor will pos- sess a surplus of holes, i.e., its polarity will be p-type. Similarly, the n-type semiconductor region is doped with donors. As a result, at normal operat- 44 3.4. The p-n junction with no applied bias M − − − + + + − − − − − − − − − + + + + + + + + + h+ e− p n B− As+ Figure 3.3: The distribution of charges within a one dimensional abrupt p-n junc- tion. 45 3.4. The p-n junction with no applied bias ing temperatures, i.e., when the donors are fully ionized, these donors will be positively charged and the bulk semiconductor will posses a surplus of electrons, i.e., its polarity will be n-type. It should noted that while the acceptors and donors atoms are fixed in their position, i.e., effectively they are immobile, the electrons and holes are free to transit within the mate- rial [26]. As the p-type region of the semiconductor has a high hole concentra- tion, these holes tend to diffuse towards the n-type region. Similarly, as the n-type region of the semiconductor has a high electron concentration, these electrons tend to diffuse towards the p-type region. Holes that diffuse into the n-type region will recombine with the electrons in this region near the junction. Similarly, electrons that diffuse into the p-type region will recombine with the holes in this region near the junction. As a result, the junction region becomes depleted of free charge carriers when contrasted with the bulk p-type and n-type regions far from the junction; see Fig- ure 3.4 in which the distribution of spatially distributed charge is depicted. Thus, a space-charge region exists around the junction, comprised of a pos- itive distribution of charge on the n-type semiconductor side and a negative distribution of charge on the p-type semiconductor side, these charge dis- tributions corresponding to the immobile dopant atoms, i.e, ions, be they acceptors or donors. This space-charge region will create an internal “built-in” electric field that points from the n-type region to the p-type region; see Figures 3.4, and 3.5. This electric field acts to drift holes back into the p-type region and drift electrons back into the n-type region. The greater the amount of 46 3.4. The p-n junction with no applied bias M ρ net W n −Wp − − − + + + − − − − − − − − − + + + + + + + + + p n E o Figure 3.4: The depletion region formed in a p-n junction under no bias. The net distributed charge, ρnet, is also indicated. E0 denotes the “built-in” electric field. 47 3.4. The p-n junction with no applied bias E o E x −Wp Wn0 x Figure 3.5: The electric field within a p-n junction under no bias. The electric field points from right to left, x = −Wp and x = Wn denoting the space-charge/neutral region boundaries for the p-type and n-type re- gions, respectively [26]. 48 3.4. The p-n junction with no applied bias charge that is separated between these regions, the greater the “built-in” electric field. When equilibrium is established, the flux of charge carriers attributed to diffusion is exactly compensated by that due to drift, both for electrons and for holes. Thus, under equilibrium conditions, an electric potential profile, corresponding to the p-n junction, will exist. This poten- tial profile modifies the band diagram associated with this semiconductor device. This potential profile is depicted in Figure 3.6. For a p-n junction under zero bias, the Fermi energy level, EF , remains constant. To determine the band diagram corresponding to such a junction, it is noted that the position of the electron energy levels change according to the relative value of the potential. The resultant band diagram is depicted in Figure 3.7. Note that in the bulk p-type region, i.e., far from the junction, holes are the dominant charge carriers as the Fermi energy level is close to the valence band edge [29]. In contrast, it is noted that in the bulk n-type region, i.e., far from the junction, electrons are the dominant charge carriers as the Fermi energy level is close to the conduction band edge. Across the junction, the charge carrier concentrations change according to the position of the band edges with respect to the Fermi energy level. The dependence of the charge carrier concentrations on the position within the p-n junction is depicted in Figure 3.8. The bulk concentrations of the electrons and holes in the n-type region under no bias may be written as nno and pno, respectively, where the sub- script “o” refers to the case of no bias. The bulk concentrations of the electrons and holes in the p-type region under no bias may be written as npo and ppo, respectively. Assuming that all the donors and acceptors are 49 3.4. The p-n junction with no applied bias Figure 3.6: The electric field corresponding to a p-n junction under no bias. Vo denotes the “built-in” potential. 50 3.4. The p-n junction with no applied bias ECP EF EVP EVN ECN qV o Figure 3.7: The energy band diagram of an unbiased p-n junction [29]. 51 3.4. The p-n junction with no applied bias Figure 3.8: The charge carrier concentrations as a function of the position within the p-n junction. 52 3.4. The p-n junction with no applied bias completely ionized, the “built-in” potential, Vo = kT q ln ( pponno n2i ) , (3.27) where ni denotes the intrinsic charge carrier concentration. Noting that in equilibrium pn = n2i , (3.28) it is seen that in the bulk of the n-type region, pnonno = n 2 i , (3.29) and that in the bulk of the p-type region, pponpo = n 2 i . (3.30) This suggests that Eq. (3.27) may instead be written as Vo = kT q ln ( pponno pnonno ) , (3.31) using Eq. (3.29). Thus, it can be shown that, pno = ppo exp ( −qVo kT ) . (3.32) Alternately, Eq. (3.27) may be rewritten as Vo = kT q ln ( pponno pponpo ) , (3.33) using Eq. (3.30). Thus, npo = nno exp ( −qVo kT ) . (3.34) These expressions relate the minority charge carriers concentrations on one side of the junction with the correspond majority charge carrier concentra- tions on the other side of the junction. 53 3.5. The p-n junction under forward-bias 3.5 The p-n junction under forward-bias When a positive potential is applied across the terminals of a p-n junc- tion, i.e., when the p-n junction is under forward bias, the equilibrium that was established between the drift and diffusion components of the current density for the no bias case is disrupted. As a result, there will now be an injection of excess minority charge carriers across the junction, i.e., ex- cess holes will be injected into the n-type region and excess electrons will be injected into the p-type region. This will lead to a net current across the device. With a positive potential across the p-n junction, the “built-in” potential reduces from Vo to Vo−V , where V denotes the amount by which the p-n junction is forward biased. As a consequence, the band diagram is modified from the no bias case, as is seen in Figure 3.9. Under forward bias, the Fermi energy level is no longer constant. Instead, it changes according to the applied voltage, V , this also being shown in Figure 3.9. As a result, there is a reduction in the height of the barrier [26] and a corresponding reduction in the width of space-charge region [29]; see Figure 3.10. Accordingly, the expressions for the minority charge carrier concentrations are modified from those determined for the zero-bias case, i.e., Eqs. (3.32) and (3.34). An injection of minority charge carriers across the junction under the action of a forward bias will lead to minority charge carrier concentrations that greatly exceed their corresponding steady-state values. These minority charge carriers will recombine as they diffuse into the neutral regions, i.e., away from the space-charge region. Thus, the ambipolar diffusion equations defined in Section 3.3, i.e., Eqs. (3.25) and (3.26), may be used to determine 54 3.5. The p-n junction under forward-bias ECP EF EVP EVN ECN q(V o −V) Figure 3.9: The energy band diagram of a forward-biased p-n junction [29]. 55 3.5. The p-n junction under forward-bias M − − − + + + − − − − − − − − − + + + + + + + + + p n Figure 3.10: Depletion region formed in a p-n junction under forward bias. Note that the space-charge region is narrower than for the no bias case. 56 3.5. The p-n junction under forward-bias the profile of excess minority charge carrier concentrations. Considering Fig- ure 3.11, the excess hole concentration in the n-type region may be shown to be ∆pn = pn(x = Wn) exp ( −(x−Wn) Lh ) , (3.35) where it has been assumed that the n-type neutral region is long and where pn(x = Wn) is the boundary condition at the space-charge/neutral region boundary on the n-type side, this being determined by the charge carrier injection. Similarly, the excess electron concentration in the p-type region may be shown to be ∆np = np(x = −Wp) exp ( (x+Wp) Le ) , (3.36) where it has been assumed that the p-type neutral region is long and where np(x = −Wp) is the boundary condition at the space-charge/neutral region boundary on the p-type side, this being determined by charge carrier injec- tion. The amount by which excess minority charge carriers are injected into a region is related to the reduction in the barrier height, V . Specifically, it may be shown that ∆pn(x = Wn) = pno [ exp ( qV kT ) − 1 ] , (3.37) and that ∆np(x = −Wp) = npo [ exp ( qV kT ) − 1 ] , (3.38) where the total minority charge carrier concentration may be determined by adding the corresponding equilibrium charge carrier concentrations, i.e., pno for the case of holes in the n-type region and npo for the case of electrons in 57 3.5. The p-n junction under forward-bias p n space−charge  region E o −E Excess h+ ppo n no p n (0) np(0) Excess e− x V −Wp Wn npo p no Figure 3.11: The excess hole and electron concentrations in a p-n junction Eqs. (3.37) and (3.38) establish the boundary conditions. The x = −Wp represents the location of the space-charge/neutral region boundary in the p-type region of junction. The x = Wn represents the location of the space-charge/neutral region boundary in the n- type region of junction. 58 3.5. The p-n junction under forward-bias Figure 3.12: The contributions to the current density associated with the drift and diffusion processes. 59 3.5. The p-n junction under forward-bias the p-type region. From Section 3.3, it was demonstrated the total current density is deter- mined by the sum over that due to drift and diffusion processes, their being four possible current components contributing to the overall current den- sity; recall Figure 3.12. It can be shown that the drift and diffusion current densities associated with the majority charge carriers are negligible at the edges space-charge/neutral region boundaries; see Figure 3.12. Accordingly, the current density may be determined by evaluating the sum over the hole and electron diffusion current densities at the different space charge/neutral region boundaries for each minority charge carrier considered. That is J = JhDx|x=+Wn + JeDx|x=−Wp , (3.39) where JhDx|x=+Wn refers to the hole diffusion current on the n-type semi- conductor side, i.e., evaluated at x = +Wn, and JeDx|x=−Wp , i.e., evaluated at x = −Wp, represents the electron diffusion current on the p-type semi- conductor side. Thus, from Eqs. (3.37) and (3.38), and using Eqs. (3.17) and (3.19), it can be shown that J = ( qDh Lhnno + qDe Leppo ) n2i [ exp ( qV kT ) − 1 ] . (3.40) This expression for the current density may alternatively be expressed as J = Jso [ exp ( qV kT ) − 1 ] , (3.41) where all the terms have been defined previously, the scale current density, Jso = ( qDh Lhnno + qDe Leppo ) n2i . (3.42) 60 3.6. The p-n junction under reverse bias Eqs. (3.40) is referred to as the ideal diode current-voltage device characteris- tic associated with a p-n junction, i.e., a p-n junction without recombination processes. The corresponding current may be obtained by multiplying the current density with the area of the junction. 3.6 The p-n junction under reverse bias When a negative potential is applied across the terminals of a p-n junc- tion, i.e., when the p-n junction is under reverse bias, the electric field related to the voltage across the terminals is in the same direction as that related to the “built-in” electric field, as can be seen from Figure 3.13. As a result, the potential barrier is increased and the current across the p-n junc- tion becomes negligible. In addition, the breadth of the space-charge region becomes wider [29]. The resultant band diagram is depicted in Figure 3.14. 3.7 Recombination processes In the event that the charge carriers recombine, these recombining charge carriers must be supplied by the external current. An elementary model for these recombination processes may be obtained by plotting the electron and hole concentrations across the space charge region, as is shown in Figure 3.15. If the mean hole recombination time associated with the n-type region of the space-charge region is denoted τh and the mean electron recombination time associated with the p-type region of the space-charge region is denoted τe, assuming a thick space-charge region, then the recombination current density may be expressed as 61 3.7. Recombination processes M − − − + + + − − − − − − − − − + + + + + + + + + p n Figure 3.13: Depletion region formed in a p-n junction under reverse bias. Note that the space-charge region is wider than for the no bias case. 62 3.7. Recombination processes ECP EF EVP EVN ECN q(V o +V) Figure 3.14: The energy band diagram for a reverse-biased p-n junction [29]. 63 3.7. Recombination processes p n space−charge  region e− ppo nno p n (0)np(0) h+ M x V −Wp Wn OnM pM npo p no Figure 3.15: The injection of charge carriers and the recombination within the space-charge region. 64 3.7. Recombination processes Jrecom = q 12WpnM τe + q 12WnpM τh , (3.43) where nM and pM are the concentrations of electrons and holes at center of the junction, respectively; see Figure 3.15. Assuming that the junction is symmetric in form, it may be shown that pM = nM = niexp ( qV 2kT ) . (3.44) Thus, the current density due to recombination may be expressed as Jrecom = q 1 2 Wp ( 1 τe + 1 τh ) exp ( qV 2kT ) . (3.45) This expression for the current density may alternatively be expressed as Jrecom = Jroexp ( qV 2kT ) , (3.46) where the recombination scale current Jro = q 1 2 Wp ( 1 τe + 1 τh ) . (3.47) With recombination processes taken into account, within the framework of linear superposition, the overall current density may be expressed as J(v) = Jsoexp ( qV kT ) + Jroexp ( qV 2kT ) , (3.48) where the first term represents the current density related to the ideal diode and the second term denotes the current density contribution attributable to recombination processes. These contributions to the current density are plotted as a function of the applied voltage in Figure 3.16, for the case of Jso set to 1 µA/cm2 and Jro set to 10 nA/cm 2, these being representative values 65 3.7. Recombination processes 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10−8 10−6 10−4 10−2 100 102 104 106 V (volts) Cu rre nt  d en sit y ( mA /cm 2 ) Figure 3.16: The current density-voltage device characteristics corresponding to Eq. (3.48) with Jso set to 1 µA/cm 2 and Jro set to 10 nA/cm 2. The temperature is set to 300 K. 66 3.7. Recombination processes found in the literature. The impact that recombination processes play in determining the performance of a photovoltaic solar cell will be investigated further in Chapter 4 using this expression. It is often the case that the current-voltage device characteristic asso- ciated with a p-n junction may be represented using the simple empirical relationship J(v) = Jo [ exp ( qV ηkT ) − 1 ] , (3.49) where η denotes the ideality factor. The ideality factor, η, represents the amount of recombination that occurs. A value of η set to 2 implies that recombination processes dominate, while a value of η set to unity suggests that recombination processes play a negligible role. For the purposes of the subsequent analysis, owing to its simplicity, Eq. (3.49) will initially be employed. Later, as the impact of recombination processes are further ex- amined, Eq. (3.48) will be used instead. 67 Chapter 4 Solar cell performance analysis 4.1 Motivation Fundamentally, a photovoltaic solar cell is created when a p-n junction is exposed to light. The photons in the light generate electron-hole pairs within the junction. These pairs are then separated by the large “built-in” electric field that occurs at the p-n interface within such a junction. The transport of charge carriers, be they electrons or holes, creates a current. As the incident light that the p-n junction receives is directly responsible for this current, within the context of a solar cell device application, such a current is usually referred to as a photocurrent. It is the product of the photocurrent and the voltage developed across the terminals of such a p-n junction that allows for power to be delivered to a load from the solar cell. Following Prince [30], the behavior of a solar cell may be modeled as a p-n junction in parallel with a solar flux controlled current source. The corresponding circuit model is depicted in Figure 4.1. Clearly, a detailed understanding of p-n junction operation is crucial to the understanding of the behavior of this circuit. The functioning of the solar flux controlled 68 4.1. Motivation I − + IL V Figure 4.1: Model for the behavior of a photovoltaic solar cell. The p-n junction is in parallel with a solar flux controlled current source. This circuit model follows the suggestion of Prince [30]. 69 4.1. Motivation current source, which in turn depends upon how the materials within the p-n junction respond to light, is also important. Linking this response to the corresponding solar spectrum is a key focus of this analysis. With the characteristics of the p-n junction established, and the operation of the solar flux controlled current source determined, the performance of such a solar cell may be assessed. This chapter is organized in the following manner. In Section 4.2, the ge- ometries commonly employed for photovoltaic solar cells are reviewed. Then, in Section 4.3, the overall form of the behavior of such a cell is discussed. Performance metrics, corresponding to the photovoltaic solar cell, are then introduced in Section 4.4. Following the approach of Prince [30], using an empirical model for the current-voltage p-n junction device characteristic, i.e., Eq. (3.49), the performance of a photovoltaic solar cell is examined in Section 4.5. This analysis demonstrates a shortcoming of Eq. (3.49), where the ideality factor, η, is set to values beyond unity. The relationship with the underlying material parameters corresponding to these photovoltaic so- lar cells is then discussed in Section 4.6. In Section 4.7, the impact that parasitic resistances have upon the performance of a photovoltaic solar cell are then presented. The relationship between this performance and the solar spectrum are then examined in Section 4.8, the dependence of the efficiency of the solar cell on the energy gap of the material being determined for the different solar spectra considered in this analysis. Finally, using a more rig- orous approach, how recombination processes impact upon the performance of a photovoltaic solar cell is considered in Section 4.9. 70 4.2. Solar cell geometries 4.2 Solar cell geometries There have been a number of different geometries proposed for the pho- tovoltaic solar cell. Fundamentally, all of these configurations are in the form of a p-n junction. A p-n junction is comprised of a p-type semicon- ductor region abutted against an n-type semiconductor region. It is the minority charge carriers in either region that determine the photovoltaic response of such a junction. As the electrons within c-Si have a greater mo- bility than the holes (recall Table 3.1), thicker p-type regions are preferred, i.e, electrons are the minority charge carrier within the p-type c-Si region. Accordingly, most modern c-Si based photovoltaic solar cells are comprised of a thin n-type c-Si layer and a thicker p-type c-Si layer. A representative one-dimensional p-n junction, that may be used for photovoltaic solar cell device applications, is depicted in Figure 4.2. Real photovoltaic solar cell geometries will be more complex, of course. Multidimensional photovoltaic solar cells have been introduced in or- der to achieve greater efficiencies. Lately, for example, Kayes and Atwater have proposed a distribution of radial p-n junctions [31]. Kayes and At- water found that the efficiencies produced from these radial p-n junctions are higher than those corresponding to conventional p-n junctions; this is true for low lifetime materials [31]. Radial p-n junctions offer a greater p-n junction interface surface area. Fundamentally, it is this greater surface area that is responsible for these greater efficiencies. 71 4.2. Solar cell geometries + n−type p−type − V Figure 4.2: A representative one dimensional p-n junction that may be used for photovoltaic device applications. Note that the p-type layer is thicker than the n-type layer. 72 4.3. Photovoltaic solar cell behavior 4.3 Photovoltaic solar cell behavior For the circuit model depicted in Figure 4.1, the photovoltaic solar cell behaves as a battery. Under light illumination, the corresponding photon flux generates electron-hole pairs within the semiconductor. The large elec- tric field that exists at the interface between the p-type and n-type regions separates these charge carriers, and they drift under the action of the ap- plied electric field, leading to a photocurrent. The separation of charges that occurs leads to the development of a potential difference across the termi- nals of the p-n junction, i.e., a photovoltage. This effect is referred to as the photovoltaic effect, and it is found in a variety of natural and engineered structures. It is what is responsible for the behavior of the photovoltaic solar cell. 4.4 Photovoltaic solar cell performance metrics In Figure 4.3, a representative current-voltage device characteristic asso- ciated with a photovoltaic solar cell is depicted. There are three important performance metrics associated with this current-voltage device character- istic: (1) the open-circuit voltage, Voc, (2) the short-circuit current, Isc, and (3) the fill-factor associated with the device, FF . Two of these metrics are depicted in Figure 4.3. The open-circuit voltage, Voc, corresponds to the voltage that exists across the solar cell when no current flows. The short- circuit current, Isc, corresponds to the current that emerges from the solar cell circuit when the voltage across the solar cell is set exactly to zero. The fill-factor, FF , corresponds to the maximum power that may be delivered 73 4.4. Photovoltaic solar cell performance metrics I sc V oc I V Figure 4.3: A representative current-voltage device characteristic associated with a photovoltaic solar cell. 74 4.5. Photovoltaic solar cell device modeling to an external load divided by the product of Voc and Isc, i.e., FF ≡ Pmax VocIsc , (4.1) where Pmax denotes this maximum power. Essentially, the fill-factor pro- vides a measure as to how rounded the current-voltage device characteristic of the photovoltaic solar cell is, a fill-factor of unity corresponding to a completely abrupt current-voltage device characteristic. In Figure 4.4, the current-voltage device characteristics, corresponding to a variety of fill-factor selections, is depicted. Thus far, the focus of the analysis related to the current-voltage device characteristics associated with a photovoltaic solar cell has centered on the fourth quadrant. This is the region of the current-voltage device character- istic over which power may be delivered from the photovoltaic solar cell to an external load. Through a reversal in the polarity of the current, however, this fourth quadrant may be mapped into the first quadrant. This repre- sentation is often used in the characterization of the current-voltage device characteristics associated with a photovoltaic solar cell. For the purposes of this particular analysis, however, the fourth quadrant representation of the current-voltage device characteristics will be employed instead. 4.5 Photovoltaic solar cell device modeling For the purposes of this analysis, the approach of Prince is adopted [30], the current-voltage device characteristics associated with a p-n junction be- ing added to the current-voltage device characteristics of the solar flux con- trolled current source; recall Figure 4.1. The total current that is delivered 75 4.5. Photovoltaic solar cell device modeling I V FF=0.8 FF=0.261 FF=0.308 Figure 4.4: Representative current-voltage device characteristics associated with a photovoltaic solar cell for various fill-factor selections. 76 4.5. Photovoltaic solar cell device modeling across the terminals of the p-n junction is thus the sum over these two contributions. This is the manner in which the current-voltage device char- acteristics associated with a photovoltaic solar cell will be modeled. As was discussed in Section 3.7, the current-voltage device characteristic associated with a p-n junction may be empirically expressed as I(v) = I0 [exp (qv/ηkT )− 1] , (4.2) where I represents the current, v is the voltage applied across the termi- nals, I0 denotes the scale current, q is the charge of the electron, k is the Boltzmann constant, T is the temperature, and η is the ideality factor. The ideality factor, η, lies between 1 and 2 depending on the amount of recom- bination that occurs within the junction. When the solar-flux controlled current source is added to the mix, the resultant current-voltage device characteristic becomes I(v) = I0 [exp (qv/ηkT )− 1]− IL, (4.3) where IL denotes the photocurrent. Note that IL is in the opposite direction of the dark current. This occurs as the “built-in” electric field points in the opposite direction to the forward bias direction; recall Figure 4.1. From Eq. (4.3), it can be shown that the open-circuit voltage, Voc = ηkT q ln ( IL Io + 1 ) ; (4.4) this may be shown by setting the I(v) in Eq. (4.3) to nil. In Figure 4.5, the open-circuit voltage, Voc, is plotted as a function of the ratio, IL Io , for the case of η set to unity, i.e., the ideal case with no recombination. Voc is normalized with respect to the thermal voltage, kTq , for the purposes of 77 4.5. Photovoltaic solar cell device modeling 100 102 104 106 108 0 2 4 6 8 10 12 14 16 18 20 IL/Io V o c/(k T/ q) Figure 4.5: The open-circuit voltage, Voc, scaled by the thermal voltage, kT q , as a function of the ratio, ILIo , for the ideality factor, η, set to unity. 78 4.5. Photovoltaic solar cell device modeling this plot. It is seen that the open-circuit voltage, Voc, logarithmically scales with the current produced by the solar flux controlled current source. For the special case of Io set to 10 −9 A, a value that is representative of the case of c-Si, the dependence of the open-circuit voltage on the photocurrent, IL, is depicted in Figure 4.6, for two selections of η. The temperature is set to 300 K for all cases, 300 K being representative of room temperature. It is seen that for a fixed photocurrent, the open-circuit voltage, Voc, scales linearly (for the temperature being set to 300 K, i.e., room temperature) with the ideality factor, η. For IL set to 100 mA, this being a representative photocurrent value, Voc varies between 0.42 and 0.84 V when η is increased from unity to two. These values are representative of those found in the literature. The short-circuit current may be obtained by setting the voltage across the p-n junction terminals to be zero. Using Eq. (4.3), it may be shown that Isc = IL; (4.5) this is determined by setting v = 0 and taking the absolute magnitude of the resultant current; recall Figure 4.3. That is, the short current corresponds to that provided from the solar flux controlled current source. Within the framework of the elementary model for the p-n junction de- vice characteristics that has been discussed thus far, i.e., Eq. (4.3), the power delivered from the photovoltaic solar cell to an external load, p(v) = −I(v)v. (4.6) This power may be plotted as a function of the voltage applied across the terminals of the p-n junction, v, the negative sign allowing one to determine 79 4.5. Photovoltaic solar cell device modeling 10−6 10−4 10−2 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η=2 η=1 IL(A) V o c(V ) Figure 4.6: The open circuit voltage, Voc, as a function of photocurrent, IL. Io is set to 10−9 A for the purposes of this analysis. Two selections of η are considered. The temperature is set to 300 K for the purposes of this analysis. 80 4.5. Photovoltaic solar cell device modeling −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 120 η=1 η=2 Voltage (V) Po we r (m W) Figure 4.7: The power delivered to an external load as a function of the applied voltage for two selections of the ideality factor, η. For the purposes of this analysis, Io is set to 10 −9 A and IL is set to 100 mA. The temperature is set to 300 K for all cases. 81 4.5. Photovoltaic solar cell device modeling the delivered power as opposed to the absorbed power. For the case of Io set to 10−9 A and IL set to 100 mA, the resultant power delivered to an external load is plotted as a function of the applied voltage, v, in Figure 4.7, for the temperature set to 300 K; two η selections are considered, η = 1 and η = 2. For the case of η set to unity, it is seen that the power delivered to the external load becomes a maximum when the applied voltage is around 0.4 V, the maximum power delivered being around 37 mW. The correspond- ing fill-factor, FF , is seen to be 0.7928, i.e., ∼79.3 %. For the case η set to 2, however, it is seen that the power delivered to the external load becomes a maximum when the applied voltage is around 0.8 V, the maximum power delivered being around 75 mW. The corresponding fill-factor, FF , is seen to be 0.7940 for this case, i.e., ∼79.4 %. It is interesting to note that in this case, a greater ideality factor, i.e., more recombination, results in a slightly higher fill-factor. In order to examine the role that the ideality factor, η, plays in influenc- ing the fill-factor, FF , in Figure 4.8 the fill-factor is plotted as a function of IL for the cases of η set to unity and η set to 2. For all cases, Io is set to 10−9 A and the temperature is set to 300 K. The approach employed in Figure 4.7 is used to determine the corresponding fill-factor, FF . It is noted that the fill-factor increases monotonically with the photocurrent, IL, for all cases. In addition, it is noted that results for the fill-factor corresponding to η set to 2 form an upperbound to that determined for η set to unity. This was confirms the trend that was observed earlier for the specific case of IL set to 100 mA. It is interesting to note that greater recombination seems to produce 82 4.5. Photovoltaic solar cell device modeling 10−6 10−4 10−2 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η=1 η=2 IL(mA) FF Figure 4.8: The fill-factor as a function of the photocurrent, IL. Two selections of η are considered. For all cases, Io is set to 10 −9 A and the temperature is set to 300 K. 83 4.6. Relationship with the material properties higher fill-factors and therefore greater performance. This result seems coun- terintuitive. As will be seen later, it arises as an artifact of over simplistic nature by which the recombination processes within a p-n junction have been treated, i.e., through the empirical expression for current, Eq. (4.3), with η set to a value beyond unity, Io being set at a constant value for all cases. A reanalysis of the role that recombination processes play within p-n junctions, using more a realistic expression for the current-voltage de- vice characteristics, will be presented in Section 4.9. This will show that recombination processes actually reduce the fill-factor. Until that time, the ideality factor, η, is set to unity. 4.6 Relationship with the material properties Thus far, this analysis has relied upon an empirical model for the current- voltage device characteristic associated with the p-n junction, i.e., Eq. (4.3). Ultimately, of course, these current-voltage device characteristics are related to the underlying material parameters. Drawing upon the insights gleaned from the p-n junction device characteristic determined in Chapter 3, i.e., Eq. (3.40), in the absence of recombination, it can be shown that I = Aq [ pno ( Dh τh )1/2 + npo ( De τe )1/2] [ exp ( qv kT ) − 1 ] , (4.7) where A corresponds to the device cross-sectional area, pno is the hole con- centration in the n-type region, Dh is the hole diffusion coefficient, Lh is the hole diffusion length, np is the electron concentration in the p-type region, τe is the electron lifetime, De is the electron diffusion coefficient and Le is the electron diffusion length. Assuming that pno is much greater than npo, 84 4.7. The role of the parasitic resistances on the performance of a photovoltaic solar cell i.e., that pno >> npo, Eq. (4.7) may be instead expressed as I = Aqpno ( Dh Lh )1/2[ exp ( qv kT ) − 1 ] , (4.8) where this expression has been determined using, the relationship between ni, pno, and ppo, i.e., Eq. (3.30). For the material parameters specified in Tables 3.1 and 3.2, it can be shown that for a cross-sectional device area of 1 cm2, the prefactor, Aqpno ( Dh Lh )1/2 , is equal to 10−9 A for the case of pno being set to 1015 cm−3, this pno selection being representative of that found in a modern photovoltaic solar cell. This value is similar to the nominal selection employed for Iso, i.e., 10 −9 A. 4.7 The role of the parasitic resistances on the performance of a photovoltaic solar cell In real photovoltaic solar cells, power losses occur owing to the presence of contact resistances at both the front and back contacts of the device and due to current leakage from the edges of the cell. To determine the role that these effects play in shaping the performance of such a solar cell, the approach of Prince is followed [30], and two parasitic resistances are introduced into the model for the p-n junction based on the photovoltaic solar cell: (1) a series resistance, Rs, which acts in series with the parallel combination of the p-n junction and the solar flux controlled current source, this modeling the contact and body resistances associated with the cell, and (2) a shunt resistance, Rsh, which acts in parallel to the parallel combination of the p-n junction and the solar flux controlled current source, this modeling 85 4.7. The role of the parasitic resistances on the performance of a photovoltaic solar cell I − + solar flux controlled current source IL p−n junction R sh R s V Figure 4.9: The circuit model for a photovoltaic solar cell with the parasitic re- sistances taken into account. 86 4.7. The role of the parasitic resistances on the performance of a photovoltaic solar cell −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −200 −150 −100 −50 0 50 100 Rsh=10 6 Ω Rsh=100 Ω Rsh=70 Ω Rsh=30 Ω Voltage(V) Cu rre nt( mA ) Figure 4.10: The current-voltage device characteristic associated with a photo- voltaic solar cell, as modeled in Figure 4.9, for a number of selections of the shunt resistance, Rsh. For all cases, Io is set to 10 −9 A, IL is set to 100 mA, and the temperature is set to 300 K. This plot is determined using Eq. (4.9). The series resistance, Rs, is set to zero in all cases. 87 4.7. The role of the parasitic resistances on the performance of a photovoltaic solar cell the current leakage. These resistances are depicted in Figure 4.9. With these resistances taken into account, the resultant current across the contacts to the device may be expressed as I(v) = −IL + Ioexp ( q(v + IRs) kT ) + v + IRs Rsh , (4.9) where I0 denotes the scale current, q is the electron charge, k is the Boltz- mann constant, T is the temperature, IL is the photocurrent, Rs is the series resistance, and Rsh is the shunt resistance. The role that the shunt resistance plays in influencing these current- voltage device characteristics is first considered. In Figure 4.10, the current- voltage device characteristics corresponding to a photovoltaic solar cell, as modeled in Figure 4.9, with various selections of the shunt resistance con- sidered, is depicted, the series resistance being set to zero in all cases. For all cases, Io is set to 10 −9 A, IL is set to 100 mA, and the temperature is set to 300 K. It is seen that the shunt resistance, Rsh, plays a relatively mi- nor role in shaping the resultant current-voltage device characteristics. As a result, for the purposes of the subsequent analysis, the presence of shunt resistance will be neglected. These values are realistic values corresponding to a modern photovoltaic solar cell. The role that the series resistance plays in influencing these current- voltage device characteristics is now considered. In Figure 4.11, the current- voltage device characteristics corresponding to a photovoltaic solar cell, as modeled in Figure 4.9, with various selections of the series resistance con- sidered, is depicted, the shunt resistance being set to infinity in all cases. For all cases, Io is set to 10 −9 A, IL is set to 100 mA, and the temperature 88 4.7. The role of the parasitic resistances on the performance of a photovoltaic solar cell −1 −0.5 0 0.5 1 −100 −50 0 50 100 Rs=0 Ω Rs=2 Ω Rs=5 Ω Rs=10 Ω Rs=20 Ω Voltage(V) Cu rre nt(m A) Figure 4.11: The current-voltage device characteristic associated with a photo- voltaic solar cell, as modeled in Figure 4.9, for a number of selec- tions of the series resistance, Rs. For all cases, Io is set to 10 −9 A, IL is set to 100 mA, and the temperature is set to 300 K. It is noted that when the current is zero, all the voltages converges on the same point. This plot is determined using Eq. (4.9). The shunt resistance, Rsh, is open-circuited in all cases. 89 4.8. The solar spectrum and photovoltaic solar cell performance is set to 300 K. It is seen that the series resistance, Rs, plays an important role in shaping the current-voltage device characteristics. In particular, it is seen that the current-voltage device characteristic shifts to the left and becomes more sloped with greater series resistance, Rs. Continuing with the analysis, if the power delivered to an external load, i.e., p(v) = −I(v)v, is plotted as a function of the voltage applied across the terminals, then the optimal power may be determined for a number of series resistance selections; see Figure 4.12. As a result, one can determine the fill-factor, FF , as a function of the series resistance, Rs. In Figure 4.13, the obtained fill-factor is plotted as a function of the series resistance, Rs, for the case of Io set to 10 −9 A, IL set to 100 mA, and the temperature set to 300 K. It is seen that the fill-factor dramatically decreases in response to in- creased Rs. It is interesting to note that at sufficient high Rs, the fill-factor is seen to saturate. Clearly, in photovoltaic solar cell design, minimization of the series resistance is of paramount importance. 4.8 The solar spectrum and photovoltaic solar cell performance The solar spectra that were introduced in Chapter 2 may now be em- ployed in order to determine the performance of a photovoltaic solar cell. Assuming that each photon with energy in excess of the semiconductor bandgap, Eg, corresponds to a collected charge, the photocurrent may be shown to be IL = qA ∫ ∞ Eg S(ω) h̄ω dω, (4.10) 90 4.8. The solar spectrum and photovoltaic solar cell performance −0.4 −0.2 0 0.2 0.4 0.6 0 5 10 15 20 25 30 35 40 Rs=0 Ω Rs=2.5 Ω Rs=5 Ω Rs=12 Ω Voltage(V) Po we r (m W) Figure 4.12: The power delivered to an external load as a function of the applied voltage for various selections of Rs. For the purposes of this analysis, Io set to 10 −9 A, IL set to 100 mA, and the temperature set to 300 K. 91 4.8. The solar spectrum and photovoltaic solar cell performance 0 5 10 15 20 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Rs(Ω) FF Figure 4.13: The fill-factor, FF , as a function of the series resistance, Rs, for a photovoltaic solar cell, modeled as in Figure 4.9, as a function of the series resistance, Rs. For all cases, Io is set to 10 −9 A, IL is set to 100 mA, and the temperature is set to 300 K. 92 4.8. The solar spectrum and photovoltaic solar cell performance where S(ω) denotes the corresponding solar spectrum, h̄ω represents the photon energy, q is the electron charge, and A is the cross sectional area of the device. The efficiency of a device provides for the percentage of the solar radiation that may be delivered to the external load. At optimal conditions, this corresponds to efficiency = Pmax Psolar , (4.11) where Pmax corresponds to the maximum power delivered to the load and Psolar refers to the overall integrated solar flux received by the solar cell, i.e., Psolar = ∫ ∞ 0 S(ω) dω. (4.12) From Eq. (4.1), it is seen that Pmax = FFILVoc, (4.13) where the short-circuit current, IL, is set to Isc, and Voc may be determined using Eq. (4.4). The resultant efficiencies are plotted as a function of the energy gap, Eg, in Figure 4.14. This analysis is performed for the AM0 and AM1.5 solar spectra. Results corresponding to the blackbody solar spectrum are also de- picted. It is interesting to note that in all cases the efficiency is maximized for energy gaps around 1.5 eV. This is very close to the energy gap corre- sponding to c-GaAs. As a result, solar cells produced using this material tend to be more efficient than those produced using their c-Si counterparts. 93 4.8. The solar spectrum and photovoltaic solar cell performance 0.5 1 1.5 2 2.5 3 3.5 4 0 5 10 15 20 25 Blackbody AM0 AM1.5 Bandgap energy (eV) Ef fic ie nc y (% ) Figure 4.14: The efficiency as a function of energy gap, Eg, for the AM0 and AM1.5 solar spectra. The efficiency corresponding to the blackbody solar spectra is also depicted. 94 4.9. Recombination within solar cells 4.9 Recombination within solar cells As was mentioned previously, the elementary empirical current-voltage device characteristic employed for the purposes of the analysis presented in Section 4.5 leads to erroneous results when the ideality factor is increased beyond unity. In order to assess the true role that recombination processes play in influencing these current-voltage device characteristics, the original equation that was developed in Section 3.7, i.e., Eq. (3.48), will be em- ployed instead for reasonable selections of Jso and Jro. For the case of a device with a cross-sectional area, A, set to 1 cm2, for the selections of Jso set to 10 nA/cm 2 and Jro set to 1 µA/cm 2, the current-voltage de- vice characteristic associated with such a p-n junction may be determined. With the solar flux controlled current source added, following the Prince ap- proach [30], the resultant current-voltage device characteristics, neglecting parasitic resistances, may be expressed as I(v) = Io [ exp ( qV kT ) − 1 ] + Iro [ exp ( qV 2kT ) − 1 ] − IL, (4.14) where Io = AJso, Iro = AJro, and IL denotes the photocurrent. This current-voltage device characteristic is plotted as a function of the voltage across the p-n junction device terminals in Figure 4.15, for the case of Io set to 10 −9 A, Iro set to 1 µA, IL set to 100 mA, and the temperature set to 300 K. The power delivered to an external load, p(v) = −I(v)v, can be plotted as a function of the voltage using Eq. (4.14), as is seen in Figure 4.16. It is seen that the maximum power delivered to the load, Pmax, is 30 mW. Thus, the corresponding fill-factor is 0.76, which is noted to be lower than that found for the ideal case; recall, for the ideal case, the fill-factor is 0.7928. 95 4.9. Recombination within solar cells 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 10−4 10−2 100 102 104 106 V (volts) Cu rre nt  (m A) Figure 4.15: This current-voltage device characteristic plotted as a function of the voltage across the p-n junction device terminals. For the purposes of this analysis, Io is set to 10 −9 A, Iro is set to 1 µA, IL is set to 100 mA, and the temperature is set to 300 K. 96 4.9. Recombination within solar cells In Figure 4.17, the fill-factor is plotted as a function of the photocurrent, IL, for the case of Io set to 10 −9 A, Iro set to 1 µA, and the temperature set to 300 K. The result obtained is contrasted with that determined for the ideal case; recall Figure 4.8. It is noted that in all cases, recombination processes reduce the corresponding fill-factor. Thus, the performance is reduced with recombination processes taken into account. This is in accord with experimental observation [15]. 97 4.9. Recombination within solar cells 0 0.1 0.2 0.3 0.4 0.5 0 10 20 30 40 50 60 V (volts) Po we r (m W) Figure 4.16: The power delivered to the external load as a function of the applied voltage across the terminals of the p-n junction. For all cases, Io is set to 10−9 A, Iro is set to 1 µA, IL is set to 100 mA, and the temperature is set to 300 K. 98 4.9. Recombination within solar cells 10−6 10−4 10−2 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 recombination processes ideal case IL(mA) FF Figure 4.17: Fill-factor as a function of IL. Both the ideal case and the case with the recombination processes taken into account are considered. For all cases, Io is set to 10 −9 A, Iro is set to 1 µA, IL is set to 100 mA, and the temperature is set to 300 K. 99 Chapter 5 Conclusions In conclusion, an elementary model for the analysis of a photovoltaic so- lar cell has been introduced. This analysis was rooted in the current-voltage device characteristics associated with a p-n junction, in conjunction with a model for the solar flux controlled current source, following the procedure of Prince [30]. Recombination processes were modeled through two means: (1) an empirical expression for the current-voltage device characteristics with an associated ideality factor, η, whose value determines the role of re- combination processes, and (2) a more advanced expression that includes a recombination current. It is shown that the simplified, empirical expression is overly simplified and that its use leads to artifacts, i.e., the suggestion that recombination processes could actually increase the fill-factor. In con- trast, a more realistic current-voltage device characteristic, i.e., Eq. (4.14), suggests that recombination processes actually reduce the fill-factor. This later observation is in accord with experimental observation. There are a number of issues that could be further explored. The devel- opment of a model, that includes both recombination processes and parasitic resistances, would be useful. The development of a means of extracting the underlying parameter values, from experimental current-voltage device char- acteristics, would also be useful. Finally, the development of a first principles 100 Chapter 5. Conclusions analysis, i.e., based on a combination of the charge continuity equation and the ambipolar diffusion equations, would be helpful. These issues will have to be examined in the future. 101 References [1] B. A. Osif, Using the Engineering Literature, USA: Taylor and Francis, 2006. [2] A. R. McKirdy, The Canadian Renewable Energy Guide, CA: Solar Energy Society of Canada, 1999. [3] OECD Economic Surveys: Canada 2008, Oecd Publishing, vol. 11, 2008. [4] E. Hau, Wind Turbines: Fundamentals, Technologies, Application, Eco- nomics, 2nd Ed., Germany: Springer, 2006. [5] J. E. Welsted, J. C. Everitt, and C. Stadel, The Geography of Manitoba: its Land and its People, CA: University of Manitoba Press, 2006. [6] B. K. Edwards, The Economics of Hydroelectric Power, USA: Edward Elgar Publishing Inc., 2003. [7] Tim Palmer, Lifelines: The Case for River Conservation, 2nd Ed., USA: Rowman and Littlefield Publishers Inc., 1994. [8] P. Gevorkian, Sustainable Energy Systems Engineering: The Complete Green Building Design, USA: McGraw-Hill, 2007. 102 References [9] T. L. Baker, A Field Guide to American Windmills, Norman: University of Oklahama Press, 1985. [10] World wind energy association (WWEA), World Wind Energy Report 2010, 10th World Wind Energy Conference and Renewable Energy Exhibition, Greening Energy: Converting Deserts into Powerhouses, Egypt: Cario, 2011. [11] H. H. Schobert, Energy and Society: An Introduction, USA: Taylor and Francis, 2002. [12] K. N. Liou, An Introduction to Atmospheric Radiation, International Geophyics series, 2nd Ed., USA: Elsevier Science, vol. 84, 2002. [13] C. Laughton, Solar Domestic Water Heating: The Earthscan Expert Handbook for Planning, UK: Earthscan, 2010. [14] K. T. Pickering, and L. A. Owen, An Introduction to Global Environ- mental Issues, 2nd Ed., USA: Routledge, 1997. [15] J. Nelson, The Physics of Solar Cells, UK: Imperial College Press, 2003. [16] A. Luque, and S. Hegedus, Handbook of Photovoltaic Science and En- gineering, 2nd Ed., UK:John Wiley and Sons, 2011. [17] R. Hezel, “Progress in manufacturable high-efficient silicon solar cells,” Advanced in Solid State Physics, vol. 44, pp. 39-49, 2004. [18] J. Mehra, and H. Rechenberg, The Historical Development of Quantum Theory, NY: Springer, 2001. 103 References [19] R. A. Serway, C. J. Moses, and C. A. Moyer, Modern Physics, 3rd Ed., USA: Thomos Learning Inc., 2005. [20] K. Rajagopal, Engineering Physics, India: PHI Learing Private Limited, 2008. [21] J. S. Lewis, Physics and Chemistry of the Solar System, 2nd Ed., USA: Elsevier, 2004. [22] I. N. Mel’nikova, and A. V. Vasilyev, Short-wave Solar Radiation in the Earth’s Atmosphere: Calculation Observation Interpretation, NY: Springer Berlin Heidelberg, 2005. [23] F. T. Mackenzie, and A. Lerman, Carbon in the Geobiosphere: Earth’s Outer Shell, Neitherland: Springer, 2006. [24] American Society for Testing and Materials (ASTM). (1999, June) Reference Solar Spectral Irradiance: ASTM G-173. [Online]. Available: http://www.rredc.nrel.gov/solar/spectra/am1.5/ASTMG173/ASTMG173.html [25] C. Tang, R. N. Noyce, and W. Shockely, “Carrier generation and recom- bination in p-n junction and p-n junction characterstics”, Proceeding of the IRE, pp. 1228-1243, 1997. [26] S. O. Kasap, Principles of Electronic Materials and Devices, Boston: Tata McGraw-Hill, 1997. [27] J. C. Whitaker, The Electronics Handbook, USA: Taylor and Francis, 2005. 104 References [28] S. C. Gupta, Textbook on Optical Fiber Communication and its Appli- cations, India: Prentice Hall, 2004. [29] M. C. Jain, Textbook Of Engineering Physics-Part II, New Delhi: PHI Learning Private Limited, 2010. [30] M. B. Prince, “Silicon solar energy converters,” Journal of Applied Physics, vol. 26, pp. 535-540, 1955. [31] B. M. Kayes, and H. A. Atwater, “Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells,” Journal of Applied Physics, vol. 97, pp. (114302)1-11, 2005. 105

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