UNIVERSITY OF GHANA, LEGON SYNTHESIS AND CHARACTERIZATION OF CUPRIC OXIDE NANOPARTICLES FOR PHOTOVOLTAIC APPLICATIONS BY ALFRED ATO YANKSON This thesis is submitted to the University of Ghana, Legon in partial fulfillment of the requirements for the award of PhD Physics degree MARCH 2015 University of Ghana http://ugspace.ug.edu.gh ii DECLARATION Candidate’s Declaration I hereby declare that except for references to work by others, which have been duly cited, this thesis is the result of my own research and that it is neither in part nor whole been presented for the award of any degree elsewhere. …………………………………………….. CANDIDATE: ALFRED ATO YANKSON DATE: ……………………………………… ………………………………………………… DR. A. KUDITCHER DATE: ……………………………………….. ………………………………………………… PROF. J. K. A. AMUZU DATE: ………………………………………… ………………………………………………… DR. G. G. HAGOSS DATE: ………………………………………… University of Ghana http://ugspace.ug.edu.gh iii ABSTRACT A simple inexpensive chemical route has been identified and used to synthesize cupric oxide nanoparticles suitable for photovoltaic applications. X-ray diffraction analysis showed the synthesized nanoparticles to be a pure cupric oxide phase. The particle size and particle size distribution of the cupric oxide nanoparticles were obtained by transmission electron microscopy (TEM) whereas the crystallite size and crystallite size distribution were obtained by X-ray diffraction peak broadening analysis. The particle size was found to be between 20 nm and 60 nm, an indication that cupric oxide nanoparticles are dominant in the sample produced and analysed. The particle size distribution obtained from cumulative percentage frequency plots features a log-normal function. Absorbance measurements and analysis show that the material has an absorbance peak around 314 nm and an average energy bandgap of 1.48 eV, making it a good candidate for photovoltaic applications. A thin film of average thickness of 1.47 microns made, was used to determine the conductivity of the material, which is 8.541 × 10−2 Ω−1cm−1. Incident radiation of various wavelengths from the highest (600 nm) to the lowest (300 nm), caused decrease in resistance of the thin film sample, an indication that the material responds favourably to visible radiation making it suitable for photovoltaic and photo electronic applications. We emphasize that a low cost approach has been used successfully to synthesize cupric oxide nanoparticles with the suitable optical and electrical properties required for application in the photovoltaic and photo electronic industries. University of Ghana http://ugspace.ug.edu.gh iv ACKNOWLEDGEMENTS I wish to thank my supervisors at the Department of Physics, namely, Dr. A. Kuditcher, Prof. J. K. A. Amuzu, and Dr. G. G. Hagoss for all the guidance, encouragement and the support they gave. I am grateful to the Carnegie Next Generation of Academics in Africa Project for funding this work; I am personally indebted to Prof. Yaa Ntiamoa-Baidu, Director of the Carnegie Next Generation of Academics in Africa Project, and I thank Dr. G. K. Nkrumah- Buandoh for the part he played as Head of Department in making the Carnergie grant possible. I am full of appreciation for the contributions made by Professors; K. G Adanu, R. B. Baeta and K. Oduro-Afriyie. I am grateful to Dr. M. N. Y. H. Egblewogbe, former lab mate, for all the cooperation and to Dr. M. K Addae-Kagyah for offering to support this work in his own style. I appreciate all the support I enjoyed from all senior members at the Department of Physics, University of Ghana. The technicians at the department, led by the Chief Technician, Mr. Harry Ntumi, cooperated and helped in many ways in getting the work done; to all of them I say thank you. The electron microscopy department at the Nuguchi Memorial Institute for Medical Research gave me access to the transmission electron microscope. I express my appreciation to all the technicians there, headed by Mr. Doudu. To God be the Glory, great things He has done. University of Ghana http://ugspace.ug.edu.gh v DEDICATION This thesis is dedicated to Prof. and Mrs. E. K. Agyei and to Dr. A. Kuditcher. To God be the Glory University of Ghana http://ugspace.ug.edu.gh vi TABLE OF CONTENTS DECLARATION ii ABSTRACT iii ACKNOWLEDGEMENT iv DEDICATION v LIST OF FIGURES x LISTS OF SYMBOLS xiii CHAPTER ONE: INTRODUCTION 1.1 OVERVIEW 1 1.2 NANOTECHNOLOGY 7 1.2.1 SURFACE ENERGY 9 1.2.2 ELECTROSTATIC STABILIZATION 19 1.3 ELECTRIC POTENTIAL AT PROXIMITY OF SOLID SURFACE 20 1.3.1 VANDER WAALS ATTRACTION POTENTIAL 24 1.3.2 SYNTHESIS ROUTES 26 1.3.3 PARTICLE SIZE CHARACTERIZATION OF NANOPATICLES 27 1.4 AIMS AND OUTLOOK 28 University of Ghana http://ugspace.ug.edu.gh vii CHAPTER TWO: BAND THEORY, ELECTRICAL CONDUCTION AND THE SOLAR CELL 2.1 ELECTRON IN A CRYSTAL 30 2.1.1 FREE ELECTRON 30 2.1.2 THE PARTICLE IN A BOX 33 2.1.3 VALENCE BAND AND CONDUCTION BAND 36 2.1.4 THE HOLE CONCEPT 42 2.1.5 GENERATION AND RECOMBIATION PHENONMENON 42 2.1.6 DIRECT AND INDIRECT TRANSITIONS 43 2.1.7 GENERATION/ RECOMBINATION CENTERS 45 2.1.8 EXCESS CARRIER LIFE TIME 48 2.1.9 SHOCKLEY READ-HALL RECOMBINATION 49 2.1.10 MINORITY CARREER LIFE TIME 56 2.1.11 SURFACE RECOMBINATION 57 2.2 PHOTOVOLTAIC CONVERSION 59 2.2.1 PN JUNCTION 60 2.2.2 SOLAR CELL 65 2.2.2.1 CHARACTERIZATION OF SOLAR CELL 70 2.3 SCHOTTKY JUNCTION 74 University of Ghana http://ugspace.ug.edu.gh viii 2.3.1 SCHOTTKY DIODE 74 CHAPTER THREE: MATERIALS AND EXPERIMENTAL TECHNIQUES 3.1 MATERIALS 81 3.2 METHODS OF SYNTHESIS 83 3.2.1 FIRST ROUTE 84 3.2.2 SECOND ROUTE 86 3.3 CHARACTERIZATION 87 3.3.1 X-RAY DIFFRACTION 87 3.3.2 ABSORPTION MEASUREMENT 88 3.3.3 TEM 88 3.3.4 FILM DEPOSITION 89 3.3.5 ELECTRICAL CONDUCTIVITY & RESISTIVITY MEASUREMENTS 91 CHAPTER FOUR: RESULTS AND ANALYSIS 4.1 BAND GAP MEASUREMENTS 95 4.2 XRD MEASUREMENTS 107 4.2.1 X-RAY DIFFRACTION PATTERN PEAK BROADENING ANALYSIS 107 4.2.2 WILLIAMSON HALL PLOT 109 4.3 RESULTS FROM TEM MEASUREMENTS 114 4.3.1 MORPHOLOGY 121 University of Ghana http://ugspace.ug.edu.gh ix 4.3.2 STABILITY OF AS-PREPARED CUPRIC OXIDE NANOPARTICLES 121 4.3.3 I-V CHARACTERISTICS AND CONDUCTIVITY OF MATERIAL 122 CHAPTER FIVE: DISCUSSION 5.1 DISCUSSION 128 CHAPTER SIX: CONCLUSIONS AND RECOMMENDATIONS 6.1 CONCLUSION 137 6.2 RECOMMENDATIONS 139 REFERENCES 140 University of Ghana http://ugspace.ug.edu.gh x LIST OF FIGURES Figure 1. A pie chart showing percentage contributions of energy to the world’s energy consumption for 2006. 2 2. Breakdown of incoming solar energy in pet watt. 5 3. Schematic drawing showing two new surfaces created by breaking a rectangle into two pieces. 11 4. Schematic diagram of low index facets of face–centered cubic (FCC) crystal structure. 12 5. Schematic diagram demonstrating sintering and Ostwald ripening processes. 15 6. Schematic diagram illustrating electrical double layer structure and electric potential near the solid surface both Stern and Gouy layers shown. 22 7. Pair of particles used to derive Vander- Waals interaction. 25 8. Showing the energy of the free electron as a parabolic function of its momentum factor k. 32 9. Particle in a box: a) Geometry of potential well; b) Energy levels; c) Wave function; and d) Probability. 33 10. Valence band and conduction band in a metal and in a semiconductor. 38 11. The Periodic table. 39 12. Valence and conduction band in real space. 41 13. Examples of energy band extrema in two crystals. 41 14. Indirect band gap semiconductor and direct bandgap semiconductor. 42 15. Band-to-band recombination in a direct band gap semiconductor (GaAs) and an indirect band gap semiconductor (Si). 44 16. Electron transitions via recombination center at Energy Et. 46 University of Ghana http://ugspace.ug.edu.gh xi 17. Recombination via an acceptor recombination center. 50 18. Schematic picture of the properties of a p-n junction in an equilibrium condition. 62 19. I-V characteristics of a p-n junction, without light and with light. 67 20. Equivalent circuit of a solar cell. 71 21. Formation of Schottky junction between metal and n-type semiconductor. 76 22. I-V characteristics of a Schottky junction exhibiting rectifying properties. 80 23. Set up for chemical bath deposition. 91 24. Set up for measurement of film thickness. 93 25. A UV-VIS absorption spectrum of as synthesized black cupric oxide nanoparticles. 95 26. A UV-VIS absorption spectrum of as synthesized cupric oxide nanoparticles obtained from suspension one hour after suspension. 96 27. A UV-VIS absorption spectrum of as synthesized black cupric oxide nanoparticles obtained after cuvette and its content had been allowed to stand for thirty minutes. 97 28. A UV-VIS absorption spectrum of as synthesized black cupric oxide nanoparticles after cuvette and its content were allowed to stand for fifteen minutes. 98 29. A UV-VIS absorption spectrum of as synthesized black cupric oxide nanoparticles obtained after the sample of as prepared cupric oxide has been used to spin coat a microscope slid on one surface. 99 30. Tauc’s plot (1) for as prepared CuO nanoparticles. 101 31. Tauc’s plot (2) for as prepared CuO nanoparticles. 102 32. Tauc’s plot (3) for as prepared CuO nanoparticles. 103 33. Tauc’s plot (4) for as prepared CuO nanoparticles. 104 34. Tauc’s plot (5) for as prepared CuO nanoparticles. 105 University of Ghana http://ugspace.ug.edu.gh xii 35. XRD spectrum of as synthesized cupric oxide nanoparticles. 109 36. A graph of FWHMCosθ (rad) against Sinθ (rad). 112 37. TEM image of CuO nanoparticles [×50,000]. 113 38. Number frequency histograms showing particle size distribution. 115 39. Number frequency histograms showing particle size distribution. 116 40. Cumulative frequency curve with particle size axes in linear scale. 117 41. Cumulative percentage frequency curve with particle size axes in linear scale with a blue solid line reflecting Gaussian distribution. 118 42. Cumulative percentage frequency curve with particle size axes in logarithmic scale with blue solid line reflecting log-normal distribution. 119 43. I-V characteristics of CuO thin film in (i) dark environment and (ii) when 300 nm – 600 nm radiation is incident on it. 121 44. I-V characteristics of CuO thin film in (i) dark environment and (ii) when 300 nm – 600 nm radiation is incident on it. This shows the linear part of the I-V characteristics used to determine the resistance of the material. 122 45. I-V characteristics of CuO thin film in (i) dark environment (22 oC) and (ii) when it’s exposed to normal day light (28 oC). 123 University of Ghana http://ugspace.ug.edu.gh xiii List of Symbols Symbols 𝛾 Surface energy 1.2 ε The energy equivalent for the atom to return to its original position 1.3 Ω Atomic volume 1.7 Δµ Work done per atom transferred 1.8 𝜇𝑟 The chemical potential of a vapour atom 1.11 𝜇∞ The chemical potential of an atom on a flat surface 1.11 𝑃∞ The equilibrium vapour pressure of the flat solid surface 1.11 K Boltzmann constant 1.11 𝜇𝑐 The chemical potential of an atom on a curved surface 1.12 𝑃𝑐 The equilibrium vapour pressure 1.12 𝑆𝑐 Solubility of a curved surface 1.16 𝑆∞ Solubility of flat surface 1.16 𝐸 Electrical charge density 1.18 𝐸0 Standard electrode potential when concentration of ions is unity 1.18 𝛷𝑅 Electrostatic repulsion between two equally sized spherical particles 1.23 University of Ghana http://ugspace.ug.edu.gh xiv Φ𝐴 Vander Waals attraction 1.25 𝑈𝑛𝑔 Thermal equilibrium rate 2.17 𝜏𝑛 Lifetime of electrons in the steady state regime 2.27 𝜏𝑝 Lifetime of holes in the steady state regime 2.27 𝑛𝑐 𝑖𝑛𝑑 Excess electrons in the conduction band induced by absorption of light 2.51 𝐸+ 𝑠𝑤 Incident energy on a solar cell 2.55 𝑅𝑆𝐻 Shunt resistance 2.60 University of Ghana http://ugspace.ug.edu.gh 1 CHAPTER 1 INTRODUCTION 1.1 Overview Energy is important in all activities of life and the energy consumption rate of a society is correlated with the quality of life members of that society enjoy. Both advanced and emerging economies have depended on conventional sources of energy, which are associated with environmental pollution, global warming with its attendant dangers, and sustainability concerns. Although renewable energy resources are unlimited and renewable energy does not irreparably harm the environment, the present global energy mix incorporates only a small fraction of renewable sources, as can be seen in figure 1, which shows the sources of energy for the year 2006 [1]. In 2011, the International Energy Agency reported that “the development of affordable, inexhaustible and clean solar energy technologies will have huge longer- term benefits. It will increase countries’ energy security through reliance on an indigenous, inexhaustible and mostly import-independent resource, enhance sustainability, reduce pollution, lower the costs of mitigating climate change, and keep fossil fuel prices lower than otherwise.” [2] A clean energy source is one that contributes a negligible amount of greenhouse gases into the atmosphere. It barely contributes other pollutants into the atmosphere, therefore it is more preferred to other sources of energy which heavily pollute the atmosphere. This advantage of environmental friendliness continues to make clean energy sources very attractive to Governments, businesses and researchers. Imperatively, enormous efforts are being put into developing and harnessing clean University of Ghana http://ugspace.ug.edu.gh 2 energy sources all over the globe. Clean energy sources are referred to as renewable or nonconventional. The development of these possibly holds the key to unlocking the energy potential of the earth to meet the ever-growing energy need to sustain civilization. Among renewable energy sources, solar energy stands out with prominence, pointing to a future of clean, affordable, and sustainable electricity production. According to the International Energy Agency (IEA) in its 2012 edition of the World Energy Outlook, it is becoming increasingly difficult and expensive to restrict global warming to 2 °C with each passing year and if action is not taken before 2017, energy infrastructure existing in 2017 will account for all the permissible CO2 emissions [3]. The quest for sustainable energy must go hand in hand with the fight against environmental pollution; it is therefore imperative to pay attention to energy sources Figure 1: A pie chart showing percentage contributions of various sources of energy to the world’s energy consumption for 2006 [1]. The figures are in units of billions of kilowatt hours. University of Ghana http://ugspace.ug.edu.gh 3 that are less polluting in contemporary situations. For many centuries, man has depended on energy from the sun for heating, warming and drying. It is fitting that the sun must again play a central role in efforts to address the energy sustainability and environmental issues that confront contemporary society The sun is a medium yellow star, with hydrogen accounting for almost its entire volume. The hydrogen is at such high temperatures as to cause nuclear fusion. Four hydrogen nuclei combine to produce a helium nucleus, with the release of some radiation. In this process, deuterons are formed when two pairs of protons fuse. The deuterons fuse again with protons resulting in the formation of helium-3. If two helium-3 nuclei combine or fuse, a beryllium-6 nucleus is achieved. This final nuclide is so unstable that it splits up into a helium-4 nucleus and two protons. Equation 1.1 summarizes the nuclear fusion process that occurs in the sun. The net effect of the process is thus the production of two neutrons, two positrons, and gamma rays and the energy released is accounted for by the mass difference between the helium nucleus and the four hydrogen nuclei through the well-known mass-energy equivalence 𝐸 = 𝑚𝑐2 and represents a tremendous amount of energy. Table 1 shows some properties of the sun [4]. Figure 2 shows the incoming radiant solar energy and the processes that account for its removal from the earth [5]. It can be seen that about 174 PW of solar radiation H 1 + H 1 → H 2 + 𝛽+ + 𝜈 H 2 + H 1 → He 3 + 𝛾 He 3 + He 3 → He 4 + H 1 + H 1 1.1 University of Ghana http://ugspace.ug.edu.gh 4 reaches the upper atmosphere. This radiation has a spectrum which is nearly that of a blackbody at a temperature of about 5800 K. The radiation reaching the surface of the earth is modified by scattering in the atmosphere and approximately 30% of the total insolation is returned to space. Solar radiation reaching the surface of the earth can be harnessed for the production of electricity in two ways: either through a solar thermal power plant or by means of photovoltaic devices. In a solar thermal power plant, energy from the sun is used to raise steam, which is utilized by heat engines to generate electricity. Here, Table 1: Some bulk parameters of the sun [4]. Parameter Value Mass 1.9885×1030 kg Volume 1.412×1018 km3 Mean density 1.408×103 kg/m3 Luminosity 3.846×1026 J/s Mean energy production 1.937×10-4 J/kg Surface emission 6.329×107 J/ m2 s Central temperature 1.571×107 K University of Ghana http://ugspace.ug.edu.gh 5 solar concentrators are used to focus radiation from the sun to a point or along a line (usually a metal tube) containing water for the production of steam. The radiation from the sun reaching the earth is converted into thermal energy corresponding to temperatures in the range of between 200 ℃ and 1000 ℃ via concentrating parabolic dish systems at large-scale solar fields. Heat engines with steam turbines coupled to a generator convert the heat energy of the steam to electricity. It has to be mentioned that this system of power generation from a renewable source can be said not to be intermittent because of the fact that the steam generated can be stored in a relatively inexpensive manner and used when necessary to produce need-based electricity. Figure 2: Breakdown of incoming solar energy in petawatt (1015 watts ). Each number is in petawatt and represents a certain percentage of the total incoming solar energy [5]. University of Ghana http://ugspace.ug.edu.gh 6 Photovoltaic (PV) solar power production involves conversion of solar radiation into electricity using photovoltaic devices. Photovoltaic power systems employ solar panels containing semiconductor photovoltaic cells. The world is increasingly turning its attention to photovoltaic power because of its non-depleting source of energy and because it helps bring down the levels of greenhouse gases in the atmosphere. Photovoltaic power is also useful for supporting activities such as health delivery, education and lighting in very remote parts of the world due to its ease of handling after installation, with little or no skill required at all, and its potential for realizing stand-alone systems. Photovoltaic systems hold the potential to increase the world’s energy fraction of clean energy. The International Energy Agency has indicated that by 2012, the world solar PV installed capacity had risen by 50% per year over the last decade. In 2013, the total installed capacity of solar PV capacity increased by 43% or 29.4 GW, representing 15% of total growth in global power generation capacity. Germany alone accounted for more than one quarter of the increase with 7.6 GW of additions. Other countries with major additions include Italy (3.6 GW), China (3.5 GW), USA (3.3 GW), Japan (2.0 GW), and India (1.1 GW) [6]. Energy conversion and charge collection in a PV cell determine its efficiency. These are higher in PV devices based on inorganic materials due to better tolerance of inorganic materials to upward surges in temperature during operation. Solar cells made from inorganic nanomaterials can produce heterojunction cells with an advantage that the photo-generated charge carriers travel short distances that can be smaller than the carrier diffusion length, thereby reducing the incidence of carrier recombination which is a major factor affecting the efficiency of solar cells [7]. University of Ghana http://ugspace.ug.edu.gh 7 Nanomaterials hold a promise for ushering in a new era of high-performance devices that can mitigate the environmental cost associated with the processing of conventional solar PV devices. Consequently, it is of paramount importance to appreciate the technology of nanomaterials, the contributions they can make to solving some of the pressing energy and environmental problems, and their limitations. In the rest of this chapter, a general description of nanotechnology together with synthesis approaches and their challenges is presented. This is followed by the objectives of the work presented in this thesis. 1.2 Nanotechnology Nanotechnology comprises the study and understanding of fundamentals of physics, chemistry, biology and technology at the nanometer scale and the application of nanometer scale structures in various fields to positively affect human life. The manipulation of materials at the nanometer scale brings about different properties as compared to the bulk material. The difference between the behavior of material in bulk and the behavior of the same material at the nanometer scale is the reason for the interest in the study of nanoscale structures. The applications of nanotechnology span a wide spectrum of human endeavour including medicine, electronics, food, fuel cells, underground water purification, chemical sensors, and the fabrication of solar cells. Bulk inorganic materials are heavy and have a small surface to volume ratio, which make the use of bulk inorganic materials in the fabrication of solar cells problematic. At the nanometer scale, the same material will have a very large surface to volume ratio, which means that more electrons will be involved in photoelectric interaction, University of Ghana http://ugspace.ug.edu.gh 8 the basic mechanism underlying the operation of a solar cell. Nanometer size also means short propagation distance compared to carrier diffusion length for the photo ejected charge carriers, leading to low recombination rate and thereby improving the conversion rate of the solar cell. The small size also means that solar cells produced from such materials are lighter, which is good for portable and mobile electronic and electrical gadgets that depend on such cells for electrical power. Nanotechnology brings about the reduction of the size of bulk materials, inducing size- dependent effects arising from:  a significant increase of the surface to volume ratio involving a huge increase of the interfacial area and of the fraction of the species at the surface;  a change of the physico-chemical properties of the species at the surface and in the nanoparticle interior in comparison with that in the bulk or with isolated molecules of the material;  changes of the electronic structure of the species composing the nanoparticle and the nanoparticle as a whole;  changes in the arrangement (lattice structure, interatomic distances) of the species in the nanoparticle and presence of defects;  and confinement and quantum-size effects (due to confinement of charge carriers in a particle having a size comparable with the electron or hole de Broglie wavelength). The fabrication and processing of nanomaterials is the paramount step in nanotechnology. Nanostructured materials are those with at least one dimension falling within the nanometer scale. They include nanoparticles, nanorods, and nanowires. Many technologies exist for the fabrication of nanostructures and nanomaterials. These University of Ghana http://ugspace.ug.edu.gh 9 technologies can be grouped in several ways. One of the fruitful ways is to group them by the phase of the growth media: vapour phase growth, liquid phase growth, solid phase formation, and hybrid growth. The synthesis of nanomaterials can further be classified into bottom-up and top-down approaches. The bottom-up approach has been known and used in materials synthesis for many years. In this approach, the material is built up from the bottom: atom-by-atom, molecule-by-molecule, or cluster –by-cluster. The bottom-up approach provides a better chance to produce nanostructures with fewer defects, improved homogenous chemical composition, and better short- and long-range order. The bottom-up approach is facilitated by the reduction of Gibbs free energy such that the nanomaterials and nanostructures so produced are in a state very close to thermodynamic equilibrium. The top-down approach entails trimming down the size of materials to obtain nanostructures. An exemplary process in this approach is attrition or milling. This approach has a surface imperfection challenge. It is well known that the conventional top-down techniques such as lithography can cause crystallographic damage to the processed patterns and additional defects may be introduced during etching steps. [8, 9]. No single synthesis type exists that is best for all nanomaterials or for all applications. To select the most suitable synthesis route that suits a specific nanomaterial for a given purpose, a good knowledge of the challenges associated with synthesis of nanomaterials in general is needed. The challenges associated with the synthesis of nanomaterials can be traced to the large surface energy due to the enormous surface to volume ratio, Ostwald ripening, and challenges associated with obtaining a desired size, size distribution, morphology, crystallinity, and chemical composition which University of Ghana http://ugspace.ug.edu.gh 10 taken together result in desired physical properties. Issues surrounding surface energy and chemical potential are considered in turn. 1.2.1 Surface energy On a solid surface, atoms or molecules have fewer nearest neighbours or coordination numbers than atoms or molecules within the interior of the solid. This leads to dangling bonds on the surface, and surface atoms or molecules are subject to a force directed inward. The bond length between surface atoms or molecules and subsurface atoms or molecules is smaller than that between interior atoms or molecules. When a solid particle is small, such a decrease in bond length between the surface atoms and interior atoms becomes very significant and the leads to a pronounced reduction in the lattice constant of the entire solid particle. All of these lead to increased excess surface energy for the surface particles of nanoparticles [10]. It is well known that the energy associated with atoms or molecules on the surface of a solid or liquid is different from that associated with atoms or molecules within the bulk. Surface energy is defined by γ = [ ∂G ∂A ] ni,T,P 1.2 and represents the energy required to create an element of surface [11]. In equation 1.2, A is the surface area, G is the Gibbs free energy, T is temperature and P is pressure. Consider a rectangular solid subdivided into two pieces as shown in figure 3. On the newly created surfaces, the surface atoms are subject to forces directed towards the University of Ghana http://ugspace.ug.edu.gh 11 bulk. The energy required to get the surface atoms back to their original positions is given by the surface energy γ = 1 2 Nbερa 1.3 where half the bond strength ε is equivalent to the energy required to get the atom back to its original position, Nb is the number of broken bonds, and ρa is the surface atomic density, the number of atoms per unit area in the new surface. This assumes that the value of ε is the same for surface and bulk atoms, ignoring contributions from higher order neighbours. Equation 1.3 gives a rough estimate of the true surface energy of a solid surface and is only applicable to rigid solids. Surface relaxation will lead to a surface energy less than what has been estimated by the equation above. Consider an elemental crystal with a face-centered cubic (fcc) structure and lattice constant a as an example for illustrating the surface energy on various facets. Figure 3. Schematic drawing showing two new surfaces created by breaking a rectangle into two pieces. University of Ghana http://ugspace.ug.edu.gh 12 Each atom in the FCC structure has a coordination number of 12, and those on the (100) planes have four broken bonds. The surface energy of a (100) plane can be obtained using equation 2: γ{100} = ( 1 2 ) ( 2 a2 ) 4ε = 4ε a2 1.4 In the same manner, each atom on a (110) surface has five broken chemical bonds, each atom on a (111) surface has three and the respective surface energies for these planes are thus γ{110} = 5 √2 ε a2 1.5 and γ{111} = 2 √3 ε a2 1.6 Low index facets have low surface energy according equation 1.3. Thermodynamically, materials are stable only when they are in a state with the lowest Gibbs free energy. University of Ghana http://ugspace.ug.edu.gh 13 Several mechanisms that reduce the overall surface energy, and hence minimize the free energy, can be identified and can be grouped into atomic, individual structure, and the overall system mechanisms. Surface relaxation is an example of an atomic level mechanism. Surface relaxation shifts surface atoms or ions inward into the bulk. This mechanism for surface energy reduction occurs more readily in the liquid phase than in the solid surface. Surface restructuring occurs through combining surface dangling bonds into strained new chemical bonds. Surface adsorption through chemical or physical adsorption of terminal chemical species onto the surface by forming chemical bonds or weak van der Waals forces and composition segregation or impurity enrichment on the surface through solid-state diffusion both lead to surface energy reduction. At the overall system level, mechanisms for reducing the overall surface energy include individual nanostructures combining to form large structures to reduce the overall surface area, if sufficient activation energy exists for such a process to occur and agglomeration of individual nanostructures without altering the individual (100) (110) (111) Figure 4. Schematic diagram of low index facets of face-centered cubic (FCC) crystal structure [11]. University of Ghana http://ugspace.ug.edu.gh 14 nanostructures. Specific combination mechanisms for getting individual nanostructures to form large structures include sintering, in which individual structures merge and Ostwald ripening, in which relatively large structures grow at the expense of small structures. Figure 5 illustrates the two processes. Sintering is pronounced at temperatures above room temperature and it is significant when materials are heated to elevated temperatures of about 70% of the melting point of the given material. Ostwald ripening occurs over a wide range of temperatures, and proceeds at relatively low temperatures when nanostructures are dispersed in a solvent in which they have appreciable solubility. Sintering must be avoided in the synthesis of nanomaterials. Though sintering is significant at high temperatures, the small size of a nanoparticle implies a high surface energy and sintering can be of great concern even when the nanomaterials are brought to moderate temperatures. Sintering involves solid-state diffusion, evaporation- condensation or dissolution-precipitation, viscous flow and dislocation creep. The product of sintering is a polycrystalline material, whereas Ostwald ripening produces a single uniform structure. At the macroscopic level, the reduction in total surface energy is the driving force for both sintering and Ostwald ripening. Microscopically, the differential surface energy of surfaces with different surface curvature is the driving force for the mass transport during sintering and Ostwald ripening. Agglomeration is another way to reduce surface energy. Many nanostructures associate with one another in what is called agglomerates through chemical bonds and physical attraction forces at interfaces. University of Ghana http://ugspace.ug.edu.gh 15 Chemical potential The properties of surface particles differ from those of the interior particles due to the smaller number of bonds linking them to their nearest neighbours compared with the interior particles. Chemical potential is also dependent on the curvature of the particle surface. The transfer of material from an infinite flat surface to a spherical solid surface illustrates this. Figure 5. Schematic diagram demonstrating sintering and Ostwald ripening processes. (a) Sintering operates to combine individual particles to a bulk with solid interfaces to connect each other. (b) Ostwald ripening operates to merge smaller particles into a larger particle [11]. (a) (b) University of Ghana http://ugspace.ug.edu.gh 16 If ⅆ𝑛 atoms are transferred from the flat solid surface to a particle of radius 𝑅, the volume change ⅆ𝑉 of the spherical particle is equal to the atomic volume Ω multiplied by ⅆ𝑛; that is, ⅆ𝑉 = 4𝜋𝑅2ⅆ𝑅 = Ωⅆ𝑛 1.7 The work per atom transferred, ∆𝜇, equals the change in chemical potential, and is given by ∆𝜇 = 𝜇𝑐 − 𝜇∞ = 𝛾 ⅆ𝐴 ⅆ𝑛 = 𝛾8𝜋𝑅Ω. ⅆ𝑅 ⅆ𝑉 1.8 Combining the two equations 1.7 and 1.8 above leads to ∆𝜇 = 2𝛾Ω 𝑅 1.9 Equation 1.9 is known as the Young-Laplace equation. It describes the chemical potential of an atom in a spherical surface with respect to a flat reference surface. The equation can be generalized for all types of curved surfaces. Any curved surface can be described by two principal radii of curvature, 𝑅1 and 𝑅2 [10]. Equation (8) can therefore be written as ∆𝜇 = 𝛾Ω (𝑅1 −1 + 𝑅2 −1) 1.10 The curvature is positive for a convex surface, thus the chemical potential of an atom on a convex surface is higher than that on a flat surface. Mass transfer to a convex surface from a flat surface brings about an increase in surface chemical potential. The surface chemical potential decreases when the transfer is from a flat surface to a concave surface. Thermodynamically, an atom on a convex surface possesses the highest surface chemical potential. This phenomenon is reflected by the difference in vapour pressure and solubility of a solid. University of Ghana http://ugspace.ug.edu.gh 17 Let us assume the vapour of solid phase obeys the ideal gas law. For the flat surface we can write 𝜇𝑣 − 𝜇∞ = 𝑘𝑇 ln 𝑃∞ 1.11 where 𝜇𝑣 is the chemical potential of a vapour atom, 𝜇∞, the chemical potential of an atom on the flat surface, k, the Boltzmann constant, P∞, the equilibrium vapour pressure of the flat solid surface , T, temperature. In a similar manner, for a curved surface we have: 𝜇𝑣 − 𝜇𝑐 = 𝑘𝑇ln𝑃𝑐 1.12 where 𝜇𝑐 , the chemical potential of an atom on the curved surface, and 𝑃𝑐 , the equilibrium vapour pressure of the curved solid surface. From equations (1.11) and (1.12) we obtain: 𝜇𝑐 − 𝜇∞ = 𝑘𝑇ln 𝑃𝑐 𝑃∞ 1.13 Combining with equation (1.10) and rearranging gives us: ln 𝑃𝑐 𝑃∞ = 𝛾Ω (𝑅1 −1 + 𝑅2 −1) 𝑘𝑇 1.14 For a spherical particle, equation (1.14) can be simplified as: ln 𝑃𝑐 𝑃∞ = 2𝛾Ω 𝑘𝑅𝑇 1.15 University of Ghana http://ugspace.ug.edu.gh 18 Equation 1.15 is generally and commonly called the Kevin equation and has been verified experimentally. [9] The same relation can be derived for the dependence of the solubility on surface curvature: ln 𝑆𝑐 𝑆∞ = 𝛾Ω (𝑅1 −1 + 𝑅2 −1) 𝑘𝑇 1.16 where 𝑆𝑐 is the solubility of a curved solid surface, 𝑆∞, the solubility of a flat surface. This equation is also known as Gibbs-Thompson relation. [10] When two particles with different radii, assuming 𝑅1 ≫ 𝑅2 , are put into a solvent, each particle will develop an equilibrium with the surrounding solvent. According to equation (1.16), solubility of the smaller particle will be larger than that of the larger particle. There would be a net diffusion of solute from the region of the smaller particle to the region of the larger particle. For the equilibrium to be maintained, solute will deposit unto the surface of the large particle, whilst the small particle will have to continue dissolving to compensate for the amount of solute diffused away. This leads to the small particle becoming smaller, while the large particle gets larger. This is Ostwald ripening, which occurs also in the forms of solid-state diffusion and evaporation-condensation. Assuming there is no other change between the two particles, then the change of the chemical potential of an atom transferring from a spherical surface of radius 𝑅1 to 𝑅2 is given by: ∆𝜇 = 2𝛾Ω(𝑅2 −1 − 𝑅1 −1) 1.17 University of Ghana http://ugspace.ug.edu.gh 19 Depending on the target parameters, Ostwald ripening can have either positive or negative influence on the material synthesized. It can narrow or widen the size distribution, based on the control of the process conditions. Ostwald ripening has been explored in the preparation of nanoparticles. It is used to narrow size distribution of nanoparticles. Ostwald ripening is promoted by varying processing temperatures. During the preparation of nanoparticles from solution, after the initial nucleation and subsequent growth, when temperature is raised, the solubility of solute in solvent rises to promote Ostwald ripening. 1.2.2 Electrostatic Stabilization Surface charge density A solid emerging from a polar solvent or an electrolyte solution will develop surface charge through one or more of the following mechanisms: 1. Preferential adsorption of ions 2. Dissociation of surface charged species 3. Accumulation or depletion of electrons at the surface 4. Physical adsorption of charged species onto the surface 5. Isomorphic substitution of ions. A fixed surface electrical charge density (electrode potential), E, is established for a given solid surface in some liquid medium. The electrical charge density E is given by the Nernst equation: University of Ghana http://ugspace.ug.edu.gh 20 𝐸 = 𝐸0 + 𝑅𝑇 𝑛𝑖𝐹 ln(𝑎1) 1.18 where E0 is the standard electrode potential when the concentration of ions is unity, ni is the valence state of the ions, R is the gas constant , T is temperature, and F is Faraday’s constant. From equation (1.18), it is obvious that the surface potential of a solid varies with the concentration of the ions in the surrounding solution, and can either be positive or negative. The surface charge in oxides is mainly derived from preferential dissolution or deposition of ions. [11] In the oxide systems typical charge determining ions are protons and hydroxyl groups and their concentrations are determined by pH (pH = - log [H+]). As the concentration of the charge determining ions varies, the surface charge density changes from positive to negative and vice versa. The concentration of charge determining ions corresponding to a neutra l or zero-charged surface is called point of zero charge (p.z.c) or zero-point charge (z.p.c). At pH > p.z.c, the oxide surface is negatively charged, since the surface is covered by hydroxyl groups, OH - , which is the electrical determining ion. At pH< p.z.c, the surface is positively charged and the determining ion is H+.The surface charge density or surface potential, E, in volt, can then be simply related to pH and the Nernst equation can be written as [12]: 𝐸 = 2.303𝑅𝑔𝑇 (𝑝. 𝑧. 𝑐) − pH 𝐹 1.19 At room temperature, equation (1.19) can further be simplified: 𝐸 ≈ 0.06[(𝑝. 𝑧. 𝑐) − pH] 1.20 University of Ghana http://ugspace.ug.edu.gh 21 1.3 Electric potential at proximity of solid surface When the surface charge density of a solid surface is established, there will be a columbic force between the solid surface and the charged species in the proximity to segregate positive and negative charged species. There exist also Brownian motion and entropic force, which homogenize the distribution of various species in solution. In the solution there will always exist both surface determining ions (opposite charge) and counter ions. Although charge neutrality prevails in the system, distributions of charge determining ions and counter ions in the neighbourhood of the solid surface are inhomogeneous and very different. The distributions of both ions are controlled by a combination of the following forces. 1. Coulombic force 2. Entropic force 3. Brownian motion. The concentration of counter ions is highest near the solid surface and decreases as the distance from the surface increases, with the concentration of determining ions changing in the opposite direction, assuming surface charge is positive. This kind of inhomogeneous distributions of ions, in the neighbourhood of the solid surface, leads to the formation of so called double layer structure shown in figure 7 below. The double layer comprises Stern layer and Gouy layer (diffuse double layer); these two layers are separated by the Helmholtz plane. [13] University of Ghana http://ugspace.ug.edu.gh 22 Between the solid surface and the Helmholtz plane is the Stern layer, where the electric potential drops linearly through the tightly bound layer of solvent and counter ions. Beyond the Helmholtz plane until the counter ions reach average concentration in the solvent is the Gouy layer or diffuse double layer. In the Gouy layer, the counter ions diffuse freely and the electric potential does not reduce linearly. The electric potential drops approximately following: h = H Helmholtz plane ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ Diffuse double layer (Gouy Layer) Slip plane ФH Фz Stern Layer Фo h ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ Figure 6. Schematic diagram illustrating electrical double layer structure and the electric potential near the solid surface with both Stern and Gouy layers shown. Surface charge is assumed positive. University of Ghana http://ugspace.ug.edu.gh 23 𝐸 ∝ 𝑒−𝜅(ℎ−𝐻) 1.21 where ℎ ≥ 𝐻, which is the thickness of the Stern layer, 1 𝜅⁄ is known as the Debye- Hückel screening strength and is also used to describe the thickness of the double layer, 𝜅 is given by 𝜅 = { 𝐹2 ∑ 𝐶𝑖𝑍𝑖 2 𝑖 𝜀𝜀0 𝑅𝑔𝑇 } 1 2⁄ 1.22 where F is Faraday’s constant, 𝜀0 is the permittivity of free space, 𝜀 is the dielectric constant of the solvent, and 𝐶𝑖 and 𝑍𝑖 are the concentration and valence of the counter ions of type i. Equation (1.22) clearly shows that the electric potential in the neighbourhood of the solid surface decreases with increased concentration and valence state of counter ions, and increases with an increased dielectric constant of the solvent exponentially. Higher concentration and valence state of counter ions results in a reduced thickness of both Stern layer and Gouy layer. [14] The discussion above also holds for curved surfaces as well, assuming the surface is smooth such that the surface charge density is evenly distributed. For a smooth curved surface the surface charge density is constant, so the electric potential in the surrounding solution can be described by equations (1.21) and (1.22). Interactions between particles are complex, one of these is directly associated with the surface charge and the electric potential adjacent to the interface. Electrostatic repulsion between two particles is as a result of the electric surface charges, which are attenuated to a varied extent by the double layers. When the separation between two particles is large enough, there will be no overlap of two double layers and electrostatic University of Ghana http://ugspace.ug.edu.gh 24 repulsion between them is zero. When the particles approach each other, double layer overlaps and a repulsion develops. An electrostatic repulsion between two equally sized spherical particles is given by: [15] Ф𝑅 = 2𝜋𝜀𝑟𝜀0𝑟𝐸2 exp(−𝜅𝑠). 1.23 1.31 Van der Waals attraction potential When small particles, typically in the micrometer-range or less are dispersed in a solvent, van der Waals attraction force and Brownian motion are significant, whiles the effect of gravity becomes negligible. Van der Waals force is a weak force and it is only significant at very short distance. Brownian motion ensures that the nanoparticles collide with each other all the time. The effect of these two would result in the agglomeration of the nanoparticles. Van der Waals interaction between two nanoparticles is the sum of the molecular interaction for all pair of molecules composed of one molecule in each particle, as well as to all pairs of molecules with one molecule in a particle and one in the surrounding medium such as solvent. Integration of all the van der Waals interactions between two molecules over two spherical particles of radius, r, separated by distance, S, as illustrated in figure 8. gives the total interaction energy or attraction potential. [16] Ф𝐴 = − 𝐴 6 { 2𝑟2 𝑆 2 + 4𝑟𝑆 + 2𝑟2 𝑆 2 + 4𝑟𝑆 + 4𝑟2 + 𝑙𝑛 ( 𝑆 2 + 4𝑟𝑆 𝑆 2 + 4𝑟𝑆 + 4𝑟2 )} 1.24 where the negative sign indicates an attraction force between the two particles, and A is a positive constant termed the Hamaker constant, whose magnitude is of the order University of Ghana http://ugspace.ug.edu.gh 25 of 10−19 to 10−20 J, and depends on the polarization properties of the molecules in the two particles and in the medium which separates them. Figure 7. Pair of particles used to derive van der Waals interaction. Under various boundary conditions, equation (1.24) can be simplified. For example, when the separation between two equal sized spherical particles is significantly smaller than particle radius, i.e., 𝑆 𝑟 << 1⁄ , the simplest expression for the van der Waals attraction could be obtained: Ф𝐴 = −𝐴𝑟 12𝑆 1.25 The combination of van der Waal’s attraction and electrostatic repulsion constitutes the total interaction between two particles which are electrostatically stabilized: Solid 1 r r Solid 2 Solid Liqiud University of Ghana http://ugspace.ug.edu.gh 26 Ф = Ф𝐴 + Ф𝑅 1.26 The interaction between two particles in a suspension is considered to be the combination of van der Waal’s attraction potential and electric repulsion potential. As already pointed out, this thesis focuses on the synthesis of CuO nanoparticles. The rest of this chapter discusses the characterization of various synthesis routes that have been employed by previous researchers Umer et al [82] categorise the approach to synthesis of nano materials into two main types; Chemical and physical approaches. Chemical reduction, micro emulsion (colloidal) technique [22], sonochemical reduction [23], electro chemical [24], microwave assisted [25], and hydrothermal [26] synthesis of nanoparticles are through the chemical approach. Biosynthesis [27] are also considered chemical processes. Physical methods for nanoparticle synthesis include; laser (pulse) ablation [28], vacuum vapour deposition [29], pulsed wire discharge (PWD) [30] and mechanical milling [31 ]. 1.3.2 Synthesis routes Various synthesis routes exist and have been used for the preparation of metal oxide nanoparticles, many methods have been reported for the synthesis of copper based nanomaterials[17-19]. These include the following University of Ghana http://ugspace.ug.edu.gh 27  sol-gel, flame spray  vapour-phase reaction  aqueous precipitation  template method  electrochemical route, sacrificial anode technique,  laser ablation in vacuum .  Laser ablation in liquid medium is the simplest and versatile a technique for the production of metal oxide nanomaterials [20, 21]. 1.3.3 Particle size characterization of nanoparticles The particle size and size distribution of nanoparticles induce the ir unique and often advantageous properties. For example, the melting point of nanoparticles is decreased when the size reaches the nanometer scale [32,33] Hence an all important task in property characterization is particle sizing [34]. In general, there are two basic methods of defining particle size[34]. One method inspects the particles and makes actual measurements of their dimensions. Microscopic techniques, for example, measure many dimensional parameters from the particle images. The second utilizes the relations between particle behaviour and its size. This often implies an assumption of equivalent spherical size developed using a size-dependent property of the particle and relating it to a linear dimension [35]. Equivalent spherical diameters are the diameters of spheres that have the same or equivalent dimensions as the irregular particles themselves. An example of this method is photon correlation spectroscopy (PCS) that the dynamic fluctuation of the scattered light intensity is the basis for the calculation of University of Ghana http://ugspace.ug.edu.gh 28 the average particle size [36]. Particle sizes are not the same throughout the material, it is therefore required that information about the average size and the distribution of sizes about the average is obtained. The equipment used to investigate the part icle size and size distribution in this study are: TEM, X-ray defractometer and reflective/ polarizing microscope. 1.4 Aims and outlook The present work As indicated earlier, the conventional silicon solar cells are very expensive and have rather limited efficiency. Efforts to make use of less expensive inorganic compound for the fabrication of solar cells have become the focus of intense research. These inorganic cells are more durable than the silicon cells because they do not degrade as fast with time. The goal of this research is to contribute to obtaining durable inorganic material obtained from inexpensive sources for the fabrication of photovoltaic cells. We have employed Cu2SO4 .5H2O and 30% Ammonia water for the synthesis of CuO nanoparticles using a simple chemical route in order to produce pure samples which will be suitable for PV applications. Objectives of the work The specific objectives of this work are;  to synthesize CuO nanoparticles using an inexpensive simple chemical route University of Ghana http://ugspace.ug.edu.gh 29  to characterize the products of the synthesis by i. XRD ii. TEM iii. Optical absorbance  To demonstrate through (1) absorption spectra, (2) bandgap determination and (3) voltage and current measurements that our sample can be used for the fabrication of PV cells. The general goal of this thesis is to be able to contribute a great deal of understanding on using cheap and readily available materials to produce CuO nanoparticles, and to show that the potential exists for the use of CuO ( a p-type material with narrow bandgap) nanoparticles to reduce the cost of solar cell applications since the conventional silicon cells are expensive. Inorganic solar cells are durable in that they do not degrade with temperature, it is therefore incumbent on researchers to find inexpensive suitable inorganic materials for fabricating solar cells that are more efficient and less expensive than silicon cells. In that regard inexpensive materials that are easily obtained would be used in the synthesis of CuO nanoparticles under strict synthesis rules in order to produce pure sample which will be suitable for photovoltaic applications. University of Ghana http://ugspace.ug.edu.gh 30 CHAPTER TWO BAND THEORY, ELECTRICAL CONDUCTION AND THE SOLAR CELL 2.1 Electrons in a crystal In order to understand the electrical properties of semiconductors, the behaviour of the electron in a crystal should be looked at together with the concept of energy bands. Let us first understand how an electron behaves in a simpler environment; the classical case of the electron in a vacuum (free electron) and then the electron confined in a box- like potential well. 2.1.1 Free electron This model looks at an electron which is not interacting with its environment: it travels in a medium where the potential is zero. This is a free electron. Let us take a look at a single crystal (one dimensional crystal), the time- independent Schrödinger equation can be written for a zero potential V; − ħ2 2𝑚 ⅆ2 ⅆ𝑥 2 Ѱ(𝑥) = 𝐸 Ѱ(𝑥) 2.1 where E is the electron energy, and m the electronic mass. Since the reference for potential is arbitrary, the potential can be set to be equal to zero (𝑉 = 0) without losing generality [37] The solution to equation (2.1) is Ѱ(𝑥) = 𝐶1𝑒𝑥𝑝(𝑖𝑘𝑥) + 𝐶2𝑒𝑥𝑝(−𝑖𝑘𝑥) 2.2 University of Ghana http://ugspace.ug.edu.gh 31 where: 𝑘 = √ 2𝑚𝐸 ħ2 𝑜𝑟 𝐸 = ħ2𝑘2 2𝑚 2.3 Equation (2.2) is an expression for two waves traveling in the opposite direction. The first part of the right side of the equation represents the motion of electron in the +𝑥 direction, the second part of the same side gives the motion of electron in the −𝑥 direction. k is a vector belonging to the reciprocal space expressed in 𝑚−1 or 𝑐𝑚−1. However, in one- dimensional crystal, k can be considered as a scalar number for all practical purposes. [38] The momentum operator, 𝑝𝑥 , of the electron is given as 𝑝𝑥 = ħ 𝑖 𝜕 𝜕𝑥 For an electron moving in the +𝑥 direction within a one dimensional crystal, the momentum operator is applied to the wave function Ѱ(𝑥) = 𝐶1 exp(𝑖𝑘𝑥) to obtain: 𝑝𝑥 Ѱ(𝑥) = ℏ 𝑖 ⅆѰ(𝑥) ⅆ𝑥 = 𝐶1ℏ𝑘𝑒𝑥𝑝(𝑖𝑘𝑥) = ℏ𝑘 Ѱ(𝑥) The eigenvalues of the operator 𝑝𝑥are given by 𝑝𝑥 = ℏ𝑘 2.4 University of Ghana http://ugspace.ug.edu.gh 32 The conclusion is that the number 𝑘 , called the wave number, is equal to the momentum of the electron, with a factor ℏ. In classical mechanics the electron has a speed v given 𝑣 = 𝑝 𝑚⁄ , which yields 𝑣 = ℏ𝑘/𝑚. The electron energy given by equation (2.3) can thus be related to that derived from classical mechanics: 𝑣 = ℏ𝑘 𝑚⁄ ⇒ 𝐸 = ℏ2 𝑘2 2𝑚 = 1 2 𝑚𝑣2 2.5 The free electron takes any value of energy in a continuous manner in agreement with classical mechanics considerations. Figure 8: Showing the energy of the free electron as a parabolic function of its momentum k . We may now consider a three dimensional crystal, in that case k is a vector of the reciprocal space. It is called the wave vector. The expression exp (𝑖𝒌𝒓) gives the position of the electron, and represents a plane spatial wave moving in the direction of 0 0 k E University of Ghana http://ugspace.ug.edu.gh 33 k. The spatial frequency of the wave is equal to k, and the spatial wavelength is given by 𝜆 = 2𝜋 |𝒌| 2.1.2 The particle in a box We consider an electron that is confined by placing it in an infinitely deep potential well from which it cannot escape. The particle in a box scenario resembles, to a limited extent, an electron in an atom under attraction from the positive nucleus creating a potential well trapping the electron. V x d) c) a) n=3 n=2 n=1 n=1 n=2 n=3 n=2 n=1 n=3 a 0 0 ∞ b) Figure 9: Particle in a box: a) Geometry of potential well; b) Energy levels; c) Wave functions; and d) Probability n = 1,2, and 3 University of Ghana http://ugspace.ug.edu.gh 34 The wave function of the electron confined inside the potential well, vanishes at the edges of the well. Therefore the boundary conditions to the problem are :𝛹(𝑥 ≤ 0) = 𝛹(𝑥 ≥ 𝑎) = 0 . Inside the potential well (0 ≤ 𝑥 ≤ 𝑎) , where 𝑉 = 0 , the time independent Schrödinger equation can be written as: − ℏ2 2𝑚 ⅆ2(𝑥) ⅆ𝑥 2 = 𝐸𝛹(𝑥) 2.6 which can be rewritten in the following form: ⅆ2𝛹(𝑥) ⅆ𝑥 2 + 𝑘2𝛹(𝑥) = 0 𝑤𝑖𝑡ℎ 𝑘 = √2𝑚𝐸 ℏ2⁄ 𝑜𝑟 𝐸 = ℏ2𝑘2 2𝑚 2.7 The solution to this homogenous, second-order differential equation is: 𝛹(𝑥) = 𝐴𝑠𝑖𝑛(𝑘𝑥) + 𝐵𝑐𝑜𝑠(𝑘𝑥) 2.8 From the first boundary condition 𝛹(𝑥 = 0) = 0 we obtain 𝐵 = 0 , the second boundary condition 𝛹(𝑎) = 0 gives 𝐴𝑠𝑖𝑛(𝑘𝑎) = 0 thus: 𝑘 = 𝑛𝜋 𝑎 2.9 where 𝑛 = 1,2,3,…. University of Ghana http://ugspace.ug.edu.gh 35 Hence the wave function is given by: 𝛹𝑛(𝑥) = 𝐴𝑛𝑠𝑖𝑛 ( 𝑛𝜋𝑥 𝑎 ) 2.10 and the energy of the electron is: 𝐸𝑛 = 𝑛2 𝜋 2ℏ2 2𝑚𝑎2 2.11 This result is similar to that obtained for the free electron, the energy is a function of the squared momentum in both situations. It must be noted that the wave number k and energy E in the case of the free electron can take any value, while in the case of the particle- in-the box scenario, the values k and E take are discrete. These values are fixed by the geometry of the well. However, if the width of the potential well becomes very large (𝑎 → ∞) the different values of k become very close to one another, to the extent that they do not remain discrete any more, rather they form a continuum, as in the case for the free electron. Let us consider the values k can take in a finite crystal of macroscopic dimensions. We now look at the example of one-dimensional linear crystal with length L (Figure 11). If we impose 𝛹(𝑥 = 0) = 0 and 𝛹(𝑥 = 𝐿) = 0 as in the case of the particle- in-the-box scenario, equations (2.9) and (2.11) indicate that the permitted values for the momentum and for the energy of the electron will depend on the length of the crystal. This result is clearly unacceptable because the electrical properties of a macroscopic sample do not depend on its dimensions. Employing the Born-von Karman boundary conditions, referred to as cyclic boundary conditions, leads to much better results. To obtain these conditions, we bend the crystal University of Ghana http://ugspace.ug.edu.gh 36 such that 𝑥 = 0 and 𝑥 = 𝐿 become coincident. With the new geometry, for any value of 𝑥 , we have the critical boundary conditions: 𝛹(𝑥 + 𝐿) = 𝛹(𝑥) . Using the free electron wave function (equation 2.2), and taking into account the periodic nature of the problem, we can write: 𝛹(𝑥 + 𝐿) = 𝐴𝑒𝑥𝑝(𝑖𝑘𝑥) 𝑒𝑥𝑝(𝑖𝑘𝐿) = 𝐴𝑒𝑥𝑝(𝑖𝑘𝑥) = 𝛹(𝑥) which imposes: 𝑒𝑥𝑝(𝑖𝑘𝐿) = 1 ⇒ 𝑘 = 2𝜋𝑛 𝐿 2.12 where 𝑛 is an integer. For a three dimensional crystal with dimensions (𝐿𝑥 ,𝐿𝑦 ,𝐿 𝑧), the Born-von Karman boundary conditions can be written as follows: 𝑘𝑥 = 2𝜋𝑛𝑥 𝐿𝑥 , 𝑘𝑦 = 2𝜋𝑛𝑦 𝐿 𝑦 , and 𝑘𝑧 = 2𝜋𝑛𝑧 𝐿𝑧 2.13 where 𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 are integers. 2.1.3. Valence band and conduction band The bonds between atoms in a crystal and the electric transport phenomenon are all due to electrons from the outer most shell. In terms of energy bands the bonding electrons in atoms are found in the last occupied band, where electrons have the highest energy levels for ground-state atoms. However, there exists an infinite number of energy bands. The first contain core electrons such as the 1s electrons which are University of Ghana http://ugspace.ug.edu.gh 37 tightly bound to the atom (nucleus) by strong electrostatic force. The highest bands contain no electrons. The last ground-state which contains electrons is called valence band. The permitted energy band directly above the valence band is the conduction band. In a semiconductor the conduction band has no electrons at low temperature (𝑇 = 0 K). As the temperature increases, some electrons gain enough thermal energy to jump from the valence band into the conduction band, where they are free to move. The energy gap between the bottom of the conduction band and the valence band is called the forbidden gap or bandgap and is represented by 𝐸𝑔 . Generally, the following situations can occur depending on the position of an atom in the periodic table (Figure 12). A: The last (valence) energy band is only partially filled with electrons, even at 𝑇 = 0 𝐾. B: The last (valence) energy band is completely filled with electrons at 𝑇 = 0 𝐾, but the next (empty) energy band overlaps with it (i.e.: an empty energy band shares a range of common energy values; 𝐸𝑔 < 0). C: The last (valence) energy band is completely filled with electrons and no empty band overlaps with it (𝐸𝑔 > 0). University of Ghana http://ugspace.ug.edu.gh 38 From figure 10, in both A and B, electrons with highest energies can easily jump to a slightly higher energy level and move through the crystal after acquiring an infinitesimal amount of energy. That is the electron can leave the atom and move in the crystal without receiving any energy. This is a typical property of metals. In C, a significant amount of energy has to be given to the electron for it to jump from the valence band to the conduction band to enable it leave the atom and move freely or become delocalized in the crystal. A material with such a property is either a semiconductor or an insulator. The difference between the two lies with the quantitative amount of the energy gap Eg. In some semiconductor materials, Eg is such that at room temperature thermal energy or excitation from visible light source can give that energy needed to jump from the valence band to the conduction band. A B C E Eg Figure 10: Valence band (bottom) and conduction band in a metal (A and B) and in a semiconductor or an insulator (C). [J.P McKelvey, University of Ghana http://ugspace.ug.edu.gh 39 Figure 11: The periodic table University of Ghana http://ugspace.ug.edu.gh 40 Apart from the elemental semiconductor elements such as silicon and germanium, compound semiconductors can be synthesized by combining some group IV elements such as silicon carbide (SiC) or combining elements from group III and group V such as aluminium arsenide (AlAs) or combining a transition element like copper (Cu) and a group VI element like oxygen (O). Some non-crystalline materials also exhibit semiconductor properties. Such materials as amorphous silicon, where the distance between atoms varies in a random manner, can behave as semiconductors. The mode of transport of electric charges in these materials are quite different from those in the crystalline semiconductors. [38] Let us represent the energy bands in real space instead of k-space. That will help obtain a diagram such as in figure 12, where the x-axis represents the physical distance in the crystal. The maximum energy of the valence band is denoted 𝐸𝑉 , 𝐸𝐶 the minimum energy of the conduction band and 𝐸𝑔 the width of the energy bandgap. The Fermi level, 𝐸𝐹, represents the maximum energy of an electron in the material at 0 K. At this temperature all the permissible energy levels below the Fermi level are occupied. The Fermi level can also be defined as an energy level that has a 50% probability of of being filled with electrons, even though it might reside in the bandgap. In an insulator or a semiconductor where there exists a band gap between 𝐸𝑉 and 𝐸𝐶 as shown in figure 13, the Fermi level lies somewhere in the bandgap. In a metal the Fermi level lies in an energy band due to the overlap between the valence band and the conduction band. University of Ghana http://ugspace.ug.edu.gh 41 E x EV EC Eg Fermi level. EF Valence band Conduction band Figure 12: Valence and conduction band in real space k[100 ] Band overlap 0 k[100 ] k[111 ] E(k ) k[111 ] 0 Eg E(k ) A B Figure 13: Examples of energy band extrema (minimum of the conduction band and maximum of the valence band in two crystals). In crystal A, Eg is the bandgap energy. There is no bandgap in crystal B because the conduction and the valence bands overlap. University of Ghana http://ugspace.ug.edu.gh 42 2.1.4 The hole concept In electronics, a hole draws its existence from the absence of electron in a given crystal. It is the equivalence of a missing electron in the crystal valence band of a given crystal. Holes can move in a crystal through successive filling of the empty space left by a missing electron. The hole carries an opposite sign to that by an electron but they both have the same magnitude of charge (1.6 × 10−19 C). 2.1.5 Generation/Recombination phenomenon There exist electrons in the conduction band and holes in the valence band in a given crystal of a semiconductor, as long as its temperature is above 0 K. The electron is free to move in the conduction band. It can also occupy a vacant space and in doing the E(k) 0 Eg k[111] k[100] B E(k) 0 Eg k[100] k[111] A Figure 14: A: Indirect band gap semiconductor, B: Direct band gap semiconductor. [H.F. Wolf, Semiconductors, Wiley and sons, p. 51, 1971] University of Ghana http://ugspace.ug.edu.gh 43 electron releases energy. This phenomenon in which a free electron and a hole both disappear is referred to as recombination. An electron can also free itself from a covalent bond if it has enough energy. It jumps from the valence band into the conduction band and is free to move in the crystal. A free hole is created in the process, this leads to the generation of an electron-hole pair. When there is thermodynamic equilibrium, generation and recombination events occur in equal proportions, such that the electron and hole equilibrium concentrations remain constant with respect to time. However, the carrier concentration can increase and reach a state of non-equilibrium when there is an external source of energy, for example; illumination with light. 2.1.6 Direct and indirect transitions In some semiconductor materials such as gallium arsenide (GaAs) the conduction band minimum and the valence band maximum both occur at the same k-value. The k-value which is the wave vector represents the momentum of the carriers. It is shown in figure 17 below that the value of that momentum is zero. This means that when an electron from the conduction band recombines with a hole in the valence band, the law of conservation of momentum is obeyed. When the minimum of the conduction band and the maximum of the valence band occur at the same k-value, as illustrated in figure 14 (B) above, the semiconductor is called a direct band gap semiconductor. In this type of semiconductor, the electron jump from the conduction band into the valence band and this is called “band to band recombination”. In other semiconductor materials such as germanium, the minimum of the conduction band and the maximum of the valence band do not occur at the same k-value as University of Ghana http://ugspace.ug.edu.gh 44 illustrated in figure 14 (A) above. Such a material is called an indirect-band gap semiconductor. In such a material an electron with a momentum 𝑘 = [𝑘𝑚,0,0] recombines with a hole whose momentum 𝑘 = [0,0,0] (Figure 17). This occurs when a certain required momentum is transferred to the electron (or the hole) such that momentum is conserved. This happens through collision with a phonon or with phonons. A precise value of momentum ( −𝑘𝑚 in figure 17) has to be given to the electron , thus band-to-band recombination is an extremely unlikely process in indirect bandgap semiconductors. Instead recombination takes place via trap levels at various k-values within the band gap. E(k) E(k) direct transition Eg Eg 0 K[111] K[111] K[111] K[111] 0 indirect transition GaAs Si Figure 15: Band-to-band recombination in a direct-bandgap semiconductor (GaAs) and an indirect-bandgap semiconductor (Si) University of Ghana http://ugspace.ug.edu.gh 45 Semiconductor materials are transparent to photons with energy ℎ𝑣 smaller than their bandgap energy. Germanium is used in place of glass to make infrared (IR) lenses for wavelengths larger than 2 µm since its bandgap energy is larger than the energy o f 2 µm IR photons. On the other hand, photons with energy equal to or greater than the bandgap energy of the semiconductor, when absorbed can generate electron-hole pairs. Absorption coefficient as a function of wavelength is the measure of the distance travelled by a light wave into a material before it is absorbed. Apart from band-to-band recombination mechanisms, a free electron can recombine with a free hole through recombination centres located within the energy bandgap. These permitted energy levels are introduced by contaminants, impurity atoms or crystal defects. These centres act as catalyst enabling an electron to recombine a t 𝑘 values differing from the 𝑘𝑚 of the conduction band. This is especially true for indirect-bandgap semiconductors, where band-to-band recombination events are unlikely to occur. 2.1.7 Generation/ recombination centres Semiconductor materials may contain some crystal defects such as interstitials (excess semiconductor atoms in the crystal lattice), vacancies (missing semiconductor atoms in the crystal lattice) and dislocations (imperfections in the crystal structure), as well as traces of impurity elements such as atoms or oxygen. These create permitted levels within the energy bandgap. If one of these permitted levels has energy 𝐸𝑡 within the bandgap, it can receive an electron from the conduction band (case A in figure 16), lose an electron to the valence band (case C in figure 18), receive an electron valence band (case D in figure 18 ) or lose an electron to the conduction band (case B in figure 16). A level that is neutral if filled by an electron and positive if empty is called a University of Ghana http://ugspace.ug.edu.gh 46 “donor level”, and a level that is neutral if empty and negative if filled by an electron is called an “acceptor level”. Permitted levels inside the bandgap are called generation- recombination centers, or “recombination centres” . In figure 18 transitions A and C correspond to recombination events, while transitions B and D correspond to generation events. These transitions involve energies that are smaller than the bandgap energy, thus are much more likely to occur than band-to-band transitions, especially in indirect bandgap semiconductors. The terms Gn and Gp in the continuity equations below represent electron-hole pair generation events caused by an external source of energy, such as sunlight penetrating the semiconductor. 𝜕𝑛 𝜕𝑡 = 1 𝑞 ⅆ𝑖𝑣 𝐽𝑛 + (𝐺𝑛 − 𝑈𝑛 ) 2.14 Ec Ei Ev Et B A D C Eg Figure 16: Electron Transitions via a recombination centre at energy 𝐸𝑡. University of Ghana http://ugspace.ug.edu.gh 47 𝜕𝑝 𝜕𝑡 = − 1 𝑞 ⅆ𝑖𝑣𝐽𝑝 + (𝐺𝑝 − 𝑈𝑝) 2.15 Natural, intrinsic generation in a semiconductor material arising from thermal agitation at any temperature above 0 K , is encompassed in the intrinsic recombination- generation rate terms in the continuity equations, 𝑈𝑛 and 𝑈𝑝. The notations from figure 18 establishes that 𝑈𝑛 = 𝐴 − 𝐵 and 𝑈𝑝 = 𝐶 − 𝐷 . If 𝑈𝑛 (or 𝑈𝑝) is positive a net recombination of carriers is taking place. If it is negative, a net generation of carriers occurs. During a recombination event, the energy released can bring about different phenomena: In a band-to-band radiative recombination event, the energy released is in the form of a photon. In an indirect recombination event via an energy level within the bandgap, energy is transferred to the crystal lattice in the form of heat (or phonons). In an Auger recombination event the energy released is transferred to another electron (or hole), which becomes excited to a higher energy level. Recombination of carriers takes place not only within the bulk of the semiconductor crystal, but also at its surface. The surface is the place where the periodicity of the lattce crystal is uninterrupted, and where contac with other substances is made. Within the bulk of the crystal a recombination-generation rate (recombination rate), is defined. The recombination rate for electrons is denoted by 𝑈𝑛and that for holes, 𝑈𝑝 . 𝑈𝑛 and 𝑈𝑝 are accounted for in the continuity equations and University of Ghana http://ugspace.ug.edu.gh 48 represent the number of holes and electrons created or annihilated by intrinsic generation/ recombination processes per cm3 and per second. In a similar fashion, a surface recombination velocity is defined for the surface of the semiconductor crystal. The surface recombination rate for electrons is represented by 𝑆𝑛 and that for holes𝑆𝑝. 𝑆𝑛 and 𝑆𝑝 are boundary conditions for the continuity equations and represent the number of holes and electrons created or annihilated by intrinsic generation/ recombination processes at the surface of a semiconductor crystal per cm2 per second. 2.1.8 Excess carrier lifetime At thermodynamic equilibrium, the generation rate and the recombination rate are equal, such that 𝑈𝑛 = 𝑈𝑝 = 0 and 𝑆𝑛 = 𝑆𝑝 = 0. If for one reason or the other, the carrier concentrations are different form their equilibrium value, generation/ recombination mechanisms will tend to force them back to equilibrium. Actually, 𝑈𝑛, 𝑈𝑝, 𝑆𝑛 and 𝑆𝑝 are directly proportional to how much the carrier concentrations depart from equilibrium: Generation/ Recombination rate: 𝑈𝑛 = 𝑛 − 𝑛0 𝜏𝑛 and 𝑈𝑝 = 𝑝 − 𝑝0 𝜏𝑝 where 𝜏 is measured seconds University of Ghana http://ugspace.ug.edu.gh 49 Surface Recombination Rate: 𝑆𝑛 = 𝑠𝑛(𝑛 − 𝑛0) and 𝑆𝑝 = 𝑠𝑝(𝑝 − 𝑝0) where; 𝑆 = surface recombination rate; measured in cm−2 s−1 𝑠 = surface recombination velocity; measured in cm s−1 𝑛 (or 𝑝) are the electron (or hole) concentration and 𝑛0 (or 𝑝0) , electron (or hole) equilibrium concentration. If the electron concentration is higher than its equilibrium value, recombination events reduce the number of electrons. On the other hand, if the electron concentration is below its equilibrium value generation events will occur. By definition 𝜏𝑛 and 𝜏𝑝 are the lifetime of the excess (or missing) electrons or holes respectively, the equilibrium concentrations being considered as reference. Lifetime is the average time span that excess charge carriers (free electrons or holes) will “survive” before recombining, or the average time that missing electrons will be “missing” before being “regenerated” through a generation event. 2.1.9 Shockley-Read-Hall recombination (SRH) In many instances and in silicon in particular, generation/ recombination events occur through recombination centers located in the energy bandgap. Recombination events of such nature are referred to as SRH recombination events. University of Ghana http://ugspace.ug.edu.gh 50 One can obtain an analytical expression for the recombination rate for electrons and holes, 𝑈𝑛 and 𝑈𝑝 , when there are recombination centers at an energy 𝐸𝑡 within the bandgap. Let us look at a case of electron generation / recombination assuming that the recombination centers are of the acceptor type. Thus the centers are neutral or negative. Let 𝑁𝑡 be the density of the recombination centers and 𝑛𝑡 (𝑛𝑡 ≤ 𝑁𝑡) the concentration of electrons occupying the centers. For easy formulations, the electron generation/ recombination rate 𝑈𝑛, is split into 𝑈𝑛𝑟 and 𝑈𝑛𝑔 representing recombination and thermal generation respectively as indicated in figure 17. The recombination rate due to the centers, 𝑈𝑛𝑟 , is proportional to the concentration of electrons in the conduction band, 𝑛 , and to the concentration of empty (or neutral) recombination centers, 𝑁𝑡 − 𝑛𝑡. Hence one can write: Ec Et Ei Ev Eg Un = Unr - Ung Up = Upr - Upg Figure 17: Recombination via an acceptor recombination center. University of Ghana http://ugspace.ug.edu.gh 51 𝑈𝑛𝑟 = 𝑣𝑡ℎ𝜎𝑛𝑛(𝑁𝑡 − 𝑛𝑡) = 𝑣𝑡ℎ𝜎𝑛𝑁𝑡 (1 − 𝑁𝑡 𝑛𝑡 ) = 𝑣𝑡ℎ𝜎𝑛𝑛𝑁𝑡(1 − 𝑓(𝐸𝑡)) 2.16 where 𝑣𝑡ℎ is the thermal velocity of electrons, defined by the relationship 𝑣𝑡ℎ = √3𝑘𝑇/𝑚 cms−1 , 𝜎𝑛 is called the “ electron capture cross section” and is measured in square centimeters (cm2 ). The capture cross section is a measure of how close an electron must be to a center to be captured by it, while the thermal velocity is the average speed of electrons due to Brownian- like or random motion at a given temperature ( 1 2 𝑚𝑣𝑡ℎ 2 = 3 2 𝑘𝑇, where 𝑘𝑇 is the thermal energy).[39] It is noted that 𝑓(𝐸𝑡) is the probability that a center with energy 𝐸𝑡 is occupied by an electron. The function 𝑓(𝐸𝑡), is the Fermi-Dirac distribution evaluated at the energy of the center , at thermodynamic equilibrium. [40] The thermal equilibrium rate, 𝑈𝑛𝑔 , is the process by which electrons can jump from the recombination centers into the conduction band. It is proportional to the concentration centers occupied by an electron, 𝑛𝑡 = 𝑁𝑡𝑓(𝐸𝑡): 𝑈𝑛𝑔 = 𝑒𝑛 𝑁𝑡𝑓(𝐸𝑡) 2.17 where 𝑒𝑛is a proportionality coefficient which represents the probability of electron emission by the generation/ recombination centers. In a similar fashion, the recombination rate for holes between the recombination center and the valence band is given by: 𝑈𝑝𝑟 = 𝑣𝑡ℎ𝜎𝑝 𝑝𝑛𝑡 = 𝑣𝑡ℎ 𝑁𝑡 𝑓(𝐸𝑡) 2.18 University of Ghana http://ugspace.ug.edu.gh 52 The thermal generation rate, 𝑈𝑝𝑔 , is the process by which holes can jump from neutral recombination centers into the valence band. It is proportional to the concentration of centers not occupied by an electron, 𝑛𝑡 = 𝑁𝑡(1 − 𝑓(𝐸𝑡)): 𝑈𝑝𝑔 = 𝑒𝑝𝑁𝑡(1 − 𝑓(𝐸𝑡)) 2.19 where 𝑒𝑝 is a proportionality coefficient which represents the probability of hole emission by the generation/ recombination centers. We can obtain expressions for calculating the proportionality coefficients 𝑒𝑛 and 𝑒𝑝. For a semiconductor in thermal equilibrium, the regeneration and the recombination rates are equal to zero: 𝑈𝑛 = 𝑈𝑝 = 0 2.20 The number of negatively charged centers, that is filled centers, is given by the relationship 𝑛𝑡 = 𝑁𝑡𝑓(𝐸𝑡), or : 𝑛𝑡 = 𝑁𝑡 1 + 𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝐹 𝑘𝑇 ] At thermodynamic equilibrium, again 𝑈𝑛 = 𝑈𝑛𝑟 − 𝑈𝑛𝑔 = 0. Applying Boltzmann relationship 𝑛(𝑥, 𝑦, 𝑧) = 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝐹 − 𝐸𝑖0 𝑘𝑇 ] 𝑒𝑥𝑝 [ 𝑞Ф0 (𝑥,𝑦, 𝑧) 𝑘𝑇 ] In the absence of internal potential (Ф0 = 0), 𝑈𝑛𝑟 = 𝑈𝑛𝑔can be written as follows: University of Ghana http://ugspace.ug.edu.gh 53 𝑣𝑡ℎ𝜎𝑛𝑛 𝑁𝑡(1 − 𝑓(𝐸𝑡)) = 𝑒𝑛𝑁𝑡𝑓(𝐸𝑡) Using 𝑛 = 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝐹 − 𝐸𝑖 𝑘𝑇 ] the equation above becomes 𝑣𝑡ℎ 𝜎𝑛𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝐹 − 𝐸𝑖 𝑘𝑇 ] (𝑁𝑡 − 𝑁𝑡 1 + 𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝐹 𝑘𝑇 ] ) = 𝑒𝑛 𝑁𝑡 1 + 𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝐹 𝑘𝑇 ] ⇒ 𝑒𝑛 = 𝑣𝑡ℎ 𝜎𝑛 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝑖 𝑘𝑇 ] 2.21 Similarly the hole coefficient can be written as: 𝑒𝑝 = 𝑣𝑡ℎ 𝜎𝑝 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑡 𝑘𝑇 ] 2.22 Next, we obtain an expression for the electrons trapped in the generation/ recombination centers, using the continuity equation, under steady state conditions, the generation/ recombination rate: 𝜕𝑛𝑡 𝜕𝑡 = 𝑈𝑛 − 𝑈𝑝 + 𝐺𝑛 − 𝐺𝑝 = 0 2.23 External generation creates equal amounts of electrons and holes, therefore; 𝐺 = 𝐺𝑛 = 𝐺𝑝, From equation (2.23) we obtain 𝑈𝑛 = 𝑈𝑝. Using equations (2.16) to (2.19), one obtains: University of Ghana http://ugspace.ug.edu.gh 54 𝑈𝑛 = 𝑈𝑝 ⇒ 𝑈𝑛𝑟 − 𝑈𝑛𝑔 = 𝑈𝑝𝑟 − 𝑈𝑝𝑔 ⇒ 𝑣𝑡ℎ 𝜎𝑛 𝑛 𝑁𝑡(1 − 𝑓(𝐸𝑡)) − 𝑒𝑛 𝑁𝑡𝑓(𝐸𝑡) = 𝑣𝑡ℎ 𝜎𝑝 𝑝 𝑁𝑡𝑓(𝐸𝑡) − 𝑒𝑝 𝑁𝑡(1 − 𝑓(𝐸𝑡)) 2.24 Substituting equations 2.21and 2.22 into 2.24 and solving for 𝑓(𝐸𝑡) gives: 𝑓(𝐸𝑡) = 𝜎𝑛 𝑛 + 𝜎𝑝 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑡 𝑘𝑇 ] 𝜎𝑛 (𝑛 + 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝑖 𝑘𝑇 ]) + 𝜎𝑝 (𝑝 + 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑡 𝑘𝑇 ]) 2.25 The generation/ recombination rate can be calculated now: 𝑈 = 𝑈𝑛 = 𝑈𝑝 = 𝑈𝑛𝑟 − 𝑈𝑛𝑔 = 𝑣𝑡ℎ𝜎𝑛𝑛 𝑁𝑡(1 − 𝑓(𝐸𝑡)) − 𝑒𝑛 𝑁𝑡𝑓(𝐸𝑡) Substituting 2.25 into the above expression gives 𝑈 = 𝜎𝑛𝜎𝑝 𝑣𝑡ℎ𝑁𝑡(𝑝𝑛 − 𝑛𝑖 2) 𝜎𝑛 (𝑛 + 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝑖 𝑘𝑇 ]) + 𝜎𝑝 (𝑝 + 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑡 𝑘𝑇 ]) which further reduces into 𝑈 = 𝑝𝑛 − 𝑛𝑖 2 𝜏𝑝 (𝑛 + 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝑖 𝑘𝑇 ]) + 𝜏𝑛 (𝑝 + 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑡 𝑘𝑇 ]) 2.26 where 𝜏𝑛 = 1 𝑁𝑡𝑣𝑡ℎ𝜎𝑛 and 𝜏𝑝 = 1 𝑁𝑡𝑣𝑡ℎ𝜎𝑝 2.27 University of Ghana http://ugspace.ug.edu.gh 55 𝜏𝑛 and 𝜏𝑝 are called “lifetime” of electrons and holes in the steady-state regime, respectively. From equation (2.26) one can see that the recombination rate 𝑈 is directly proportional to 𝑝𝑛 − 𝑛𝑖 2 . The recombination rate represents a force which tends to bring the product 𝑝𝑛 back to its equilibrium value, 𝑛𝑖 2 . We see that: | 𝑈 = 0 𝑖𝑓 𝑝𝑛 = 𝑛𝑖 2 (equilibrium) 𝑈 > 0 𝑖𝑓 𝑝𝑛 > 𝑛𝑖 2 (recombination) 𝑈 < 0 𝑖𝑓 𝑝𝑛 < 𝑛𝑖 2 (generation) The recombination rate is highest when the recombination centers have an energy close to 𝐸𝑖. If we assume that both the hole and the electron cross sections are the same, then equation (2.26) can be written as: 𝑈 = 𝑝𝑛 − 𝑛𝑖 2 𝜏𝑜 (𝑝 + 𝑛 + 2𝑛𝑖𝑐𝑜𝑠ℎ [ 𝐸𝑡 − 𝐸𝑖 𝑘𝑇 ]) 2.28 where 𝜏𝑜 = 1 𝑁𝑡𝑣𝑡ℎ𝜎𝑜 2.29 University of Ghana http://ugspace.ug.edu.gh 56 2.1.10 Minority carrier lifetime The lifetime of the minority carriers is very crucial for the efficiency of certain semiconductor devices whose operation depends on the injection of minority carriers. We consider equation (2.26) in a situation where the excess carrier concentrations, 𝛿𝑛 = 𝑛 − 𝑛𝑜 and 𝛿𝑝 = 𝑝 − 𝑝𝑜 are small compared to equilibrium concentrations: 𝛿𝑛 << 𝑛𝑜 and 𝛿𝑝 << 𝑝𝑜 . This is low-level injection. We can write: 𝑝𝑛 = (𝑝𝑜 + 𝛿𝑝)(𝑛𝑜 + 𝛿𝑛) ⇒𝑝𝑛 − 𝑛𝑖 2 ≅ 𝑝𝑜 𝛿𝑛 + 𝑛𝑜 𝛿𝑝 2.30 Equation (2.26) can be rewritten as: 𝑈 = 𝑝𝑜 𝛿𝑛 + 𝑛𝑜 𝛿𝑝 𝜏𝑝 (𝑛 + 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑡 − 𝐸𝑖 𝑘𝑇 ]) + 𝜏𝑛 (𝑝 + 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑡 𝑘𝑇 ]) 2.31 For centers where the recombination rate is highest (𝐸𝑡 ≅ 𝐸𝑖): 𝑈 = 𝑝𝑜 𝛿𝑛 + 𝑛𝑜 𝛿𝑝 𝜏𝑝(𝑛 + 𝑛𝑖) + 𝜏𝑛 (𝑝 + 𝑛𝑖) 2.32 but 𝑛 = 𝑛𝑜 + 𝛿𝑛 and 𝑝 = 𝑝𝑜 + 𝛿𝑝 and 𝛿𝑛 << 𝑛𝑜 and 𝛿𝑝 << 𝑝𝑜: 𝑈 = 𝑝𝑜 𝛿𝑛 + 𝑛𝑜 𝛿𝑝 𝜏𝑝(𝑛𝑜 + 𝑛𝑖) + 𝜏𝑛(𝑝𝑜 + 𝑛𝑖) 2.33 Recombination rate University of Ghana http://ugspace.ug.edu.gh 57 For a p-type semiconductor, 𝑝𝑜 > 𝑛𝑖 > 𝑛𝑜 , therefore: 𝑈 = 𝛿𝑛 𝜏𝑛 = 𝑛 − 𝑛𝑜 𝜏𝑛 2.34 For an n-type semiconductor, 𝑛𝑜 > 𝑛𝑖 > 𝑝𝑜 , therefore: 𝑈 = 𝛿𝑝 𝜏𝑝 = 𝑝 − 𝑝𝑜 𝜏𝑝 2.35 2:1:11 Surface recombination Recombination of excess carriers also occurs at the surface of the semiconductor crystal. The surface of a semiconductor material acts as the interface between the material and another material. At the surface of the semiconductor material the periodicity of the atoms is disrupted to a very large extent. This and other factors contribute to the usually higher rate of recombination at the surface than in the bulk of the material. Let us define the surface recombination rate for electrons and holes, 𝑆𝑛 and 𝑆𝑝, as the number of carriers disappearing per unit area per second on the semiconductor material surface due to recombination process. The 𝑆𝑛 and 𝑆𝑝 can thus be used as boundary conditions for the continuity equations (2.14) and (2.15). A derivation of the surface recombination will result in the equation (2.36) below: 𝑆 = 𝜎𝑛 𝜎𝑝𝑣𝑡ℎ𝑁𝑠𝑡(𝑝𝑠𝑛𝑠 − 𝑛𝑖 2 ) 𝜎𝑛 (𝑛𝑠 + 𝑛𝑖𝑒𝑥𝑝 [ 𝐸𝑠𝑡 − 𝐸𝑖 𝑘𝑇 ]) + 𝜎𝑝 (𝑝𝑠 + 𝑛𝑖 𝑒𝑥𝑝 [ 𝐸𝑖 − 𝐸𝑠𝑡 𝑘𝑇 ]) 2.36 University of Ghana http://ugspace.ug.edu.gh 58 where 𝑛𝑠 and 𝑝𝑠 are the electron and hole concentrations (measured in cm−3 ) at the surface respectively, 𝑁𝑠𝑡 is the concentration of surface recombination centers (measured in cm−2), and 𝐸𝑠𝑡 is their energy. Similar to bulk recombination, the most efficient recombination centers are those located at midgap energy. If we assume that 𝜎𝑛 ≅ 𝜎𝑝 = 𝜎𝑜 , equation (62) becomes: 𝑆 = 𝜎𝑛 𝑣𝑡ℎ𝑁𝑠𝑡(𝑝𝑠𝑛𝑠 − 𝑛𝑖 2 ) 𝑝𝑠 + 𝑛𝑠 + 2 𝑛𝑖𝑐𝑜𝑠ℎ [ 𝐸𝑠𝑡 − 𝐸𝑖 𝑘𝑇 ] 2.37 We can write the 𝑝𝑛 product at the surface as: 𝑝𝑠 𝑛𝑠 = (𝑝𝑜 + 𝛿𝑝𝑠 )(𝑛𝑜 + 𝛿𝑛𝑠 ) ⇒ 𝑝𝑠 𝑛𝑠 − 𝑛𝑖 2 ≅ 𝑝𝑜 𝛿𝑛𝑠 + 𝑛𝑜 𝛿𝑝𝑠 The recombination rate at the surface of a given semiconductor crystal is larger than the rate inside the crystal. Introducing the surface recombination rate into the continuity equation in the following way: 𝐴 𝐽𝑛 (𝑥𝑜) −𝑞 = 𝐴𝑆𝑛 = 𝐴𝑠𝑛 (𝑛𝑠 − 𝑛𝑜 ) 2.38 for electrons, and : 𝐴 𝐽𝑝(𝑥𝑜) −𝑞 = 𝐴𝑆𝑝 = 𝐴𝑠𝑝 (𝑝𝑠 − 𝑝𝑜 ) 2.39 for holes. The surface recombination rate can be infinite in such cases as at the metal- semiconductor contact. University of Ghana http://ugspace.ug.edu.gh 59 2.2 Photovoltaic Conversion The direct conversion of radiant energy into electrical energy can occur through the use of semiconductor materials, whose conductivity is strongly enhanced by electron excitation, caused by impinging light quanta. Electrical energy is only produced when the excited electrons are not only able to move freely but are made to move in a directed way. A certain force would have to act on these free electrons for them to move. This force will come from an electrical potential gradient, such as is in a p-n junction of doped semiconductor materials. A p-n junction provides an electrical field that will cause the electrons excited by radiation (such as solar) to move in the direction from the p-type to the n-type material and cause the vacancies (holes) left by the excited electrons to move in the opposite direction. [41] An electrical power can be delivered to an external circuit if and only if the electrons and the holes are able to reach the respective edges of the semiconductor material. There is competition from recombination processes which hampers the motion of the charge carriers (electrons and holes) thus factors such as overall dimensions and electron mobility in the material used become paramount in considering materials for conversion of radiant energy directly into electrical energy. There is a large number of materials which exhibit photovoltaic characteristics, however the number of materials that are able to make conversion of solar radiation into electrical energy with efficiency of the order of 20% and with high stability under operation is small. There is substantial amount of research effort aimed at finding University of Ghana http://ugspace.ug.edu.gh 60 materials which make the threshold properties required to generate affordable electricity from solar radiation. 2.2.1 The p-n junction When a p-type and n-type semiconductor materials are joined together such that there exists a common surface between them, they are said to have formed a p-n junction. This will result in electrons flowing at the initial stages. The electron density in the conduction band of the n-type is higher, just as the hole density in the valence band of the p-type is high because there is a surplus of positive charge in the n-type material around the junction, and surplus of electrons in the p-type material. There is the formation of a dipole layer creating an electrostatic potential difference preventing further unidirectional flow of electrons until an equilibrium situation occurs in which the potential difference is such that no net transfer takes place. Let us state the equilibrium condition in terms of Fermi Energy. If 𝜇𝑝and 𝜇𝑛 are the Fermi energies of p-type and n-type materials respectively, 𝜇𝑝 ≠ 𝜇𝑛 at the initial stage, but at equilibrium 𝜇𝑝 = 𝜇𝑛. The change in relative positions of the conduction (valence) bands in the two types of materials with respect a non-doped semiconductor material must be equal to the electron charge −𝑒, times the equilibrium electrostatic potential. [41] The number of electrons in the conduction band may be written as 𝑛𝑐 = ∫ 𝑛′ 𝐸𝑐 ′ 𝐸𝑐 (𝐸′)𝑓(𝐸)ⅆ𝐸 2.40 University of Ghana http://ugspace.ug.edu.gh 61 where 𝐸𝑐 and 𝐸𝑐 ′ are the lower and upper energy limit of the conduction band, 𝑛′(𝐸) is the number of quantum states per unit energy interval, and 𝑓(𝐸) is the Fermi-Dirac distribution. If the electrons in the conduction band are regarded as free, elementary quantum mechanics gives [41] 𝑛′(𝐸) = 4𝜋ℎ−3(2𝑚)3 2⁄ 𝐸1 2⁄ 2.41 where ℎ is the Planck’s constant and 𝑚 is electron mass. The corrections for electrons moving in condensed matter, rather than being free, may to a first approximation, be included by replacing the electron mass by an effective value. If the Fermi energy is not close to the conduction band, 𝐸𝑐 − 𝜇 ≫ 𝑘𝑇, the Fermi-Dirac distribution may be replaced by the Boltzmann distribution, 𝑓𝐵 (𝐸) = exp (−(𝐸 − 𝜇)/𝑘𝑇). 2.42 Evaluating the integral, equation (2.40) then gives an expression of the form 𝑛𝑐 = 𝑁𝑐 exp(−𝐸𝑐 − 𝜇) /𝑘𝑇) 2.43 The number of holes in the valence band is found in an analogous manner, 𝑛𝑣 = 𝑁𝑣exp (−(𝜇 − 𝐸𝑣 )/𝑘𝑇), 2.44 where 𝐸𝑣 is the upper limit energy of the valence band. University of Ghana http://ugspace.ug.edu.gh 62 Figure 18: Schematic picture of the properties of a p-n junction in an equilibrium condition. The x-direction is perpendicular to the junction ( all properties are assumed to be homogenous in the y- and z-directions). The charge (top) is the sum of electron charges in the conduction band and positive hole charges in the valence band, plus C h ar ge El e ct ro n El e ct ro st at ic P o te n ti al φ q EC (p) µ(p) =µ EV (p) φ (p) φ(n) EV (n) EC (n) µ( n)=µ Δφ0= - eΔE0 Distance x xpt xpn xtn 0 Transitional region n - region P - region - ΔEC University of Ghana http://ugspace.ug.edu.gh 63 charge excess or defect associated with the acceptor and donor levels. In the electron energy diagram (middle), the abundance of minority charge carriers (closed circles for electrons and open circles for holes) is schematically illustrated. The equilibrium currents in a p-n junction such as illustrated in figure 18 above can be calculated. Let us firstly consider the electron currents in the conduction band, the thermal electrons in the conduction band in the p-type region can freely flow into the n-type material. The resulting current 𝐼0 ­ (𝑝), may be considered proportional to the number of electrons in the conduction band in the p-region, 𝑛𝑐(𝑝), given by equation (2.43), 𝐼0 ­ (𝑝) = 𝛼𝑁𝑐exp (−(𝐸𝑐(𝑝) − 𝜇(𝑝))/𝑘𝑇) 2.45 where the constant 𝛼 depends on the electron mobility in the material and on the electrostatic potential gradient , grad 𝜙. The electrons excited into the conduction band in the n-type region will have to climb the potential barrier to enable them move into the p-region. The fraction of electrons capable of overcoming this potential barrier is given by a Boltzmann factor of the form (2.43), but with additional energy barrier 𝛥𝐸0 = −∆𝜙0 /𝑒 (−𝑒 being the electron charge), 𝑛𝑐 = 𝑁𝑐exp (−(𝐸𝑐(𝑛) − 𝜇(𝑛) − ∆𝐸0)/𝑘𝑇 . Applying −∆𝐸0 = 𝐸𝑐(𝑝) − 𝐸𝑐(𝑛) and considering the current 𝐼0(𝑛) as being proportional to 𝑛𝑐(𝑛), the corresponding current may be written as 𝐼0 ­ (𝑛) = 𝛼′ 𝑁𝑐exp (−(𝐸𝑐(𝑛) − 𝜇(𝑛))/𝑘𝑇 2.46 University of Ghana http://ugspace.ug.edu.gh 64 where 𝛼′depends on the diffusion parameter and on the relative change in the electron density, 𝑛𝑐 −1grad(𝑛𝑐), looking at the electron motion against the electrostatic potential as a diffusion process. The statistical mechanical condition for thermal equilibr ium demands that 𝛼 = 𝛼′ [42], so equations (2.45) and (2.46) show that the net electron current, 𝐼0 ­ = 𝐼0 ­ (𝑝) + 𝐼0 ­(𝑛), becomes zero when 𝜇(𝑝) = 𝜇(𝑛), which is the condition for thermal equilibrium. The same is true for the hole current, 𝐼0 + = 𝐼0 +(𝑝) + 𝐼0 +(𝑛). Applying an external voltage source to the p-n junction, such that the n-type terminal receives an additional electrostatic potential ∆𝜙𝑒𝑥𝑡 relative to the p-type terminal, the junction is no longer in thermal equilibrium, and the Fermi energy in the p-region is no longer equal to that of the n-region, but satisfies 𝜇(𝑝) − 𝜇(𝑛) = 𝑒−1 ∆𝜙𝑒𝑥𝑡 = ∆𝐸𝑒𝑥𝑡, 2.47 if the Boltzmann distribution of electrons and holes are to maintain their shapes in both p-and n-regions. In a similar fashion, 𝐸𝑐(𝑝) − 𝐸𝑐(𝑛) = −(∆𝐸0 + ∆𝐸𝑒𝑥𝑡) , and assuming the proportionality factors in equations (71) and (72) still bear the relationship 𝛼 = −𝛼′ in the presence of the external potential, the currents are connected by the expression University of Ghana http://ugspace.ug.edu.gh 65 𝐼­(𝑛) = −𝐼­(𝑝)exp (∆𝐸𝑒𝑥𝑡/𝑘𝑇) . The net electron current in the conduction band then becomes 𝐼­ = 𝐼­(𝑛) + 𝐼­(𝑝) = −𝐼­(𝑝) (exp ( ∆𝐸𝑒𝑥𝑡 𝑘𝑇 ) − 1). 2.48 The contributions to the hole current , 𝐼+, behave in a like manner to those of the electron current, and the total current 𝐼 across p-n junction with an external potential ∆𝜙𝑒𝑥𝑡 = −𝑒∆𝐸𝑒𝑥𝑡 may be written as: 𝐼 = 𝐼­ + 𝐼+ = −𝐼(𝑝)(exp (∆𝐸𝑒𝑥𝑡/𝑘𝑇) − 1) 2.49 The relationship between current and potential is called the “characteristic” of the device, and the relation (2.49) for the p-n junction is illustrated in figure 19 by the curve labelled “no light”. The constant saturation current 𝐼(𝑝) is sometimes called “dark current”. 2.3.2 Solar cell A solar cell is constructed when the p-n junction is shaped in such a way that the p- type semiconductor material can be reached by incident solar radiation, eg. by placing a thin layer of p-type materi