Physics of Solar Cells - Buch.de Transport of electrons ... 6.2 Basic mechanisms in solar cells ......

Peter Würfel

Physics of Solar CellsFrom Principles to New Concepts

WILEY-VCH Verlag GmbH & Co. KGaA

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Peter Würfel

Physics of Solar Cells

From Principles to New Concepts

Physics of Solar Cells: From Principles to New Concepts. Peter WurfelCopyright c©2005 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-40428-7

Peter Würfel

Physics of Solar CellsFrom Principles to New Concepts

WILEY-VCH Verlag GmbH & Co. KGaA

Author

Prof., Dr. rer. nat., emerit. Peter WürfelUniversität KarlsruheInstitut für Angewandte Physik [email protected]

Cover Picture

Jörg Trebs, Berlin

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© 2005 WILEY-VCH Verlag GmbH & Co. KGaA,Weinheim

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ISBN 3-527-40428-7

Contents

List of Symbols ix

Preface xi

1 Problems of the Energy Economy 11.1 Energy economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Estimate of the maximum reserves of fossil energy . . . . . . . . . . . . . . 31.3 The greenhouse effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 The temperature of the earth . . . . . . . . . . . . . . . . . . . . . . 6

2 Photons 92.1 Black-body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Photon density nγ in a cavity (Planck’s law of radiation) . . . . . . . 92.1.2 Energy current through an area dA into the solid angle dΩ . . . . . . 142.1.3 Radiation from a spherical surface into the solid angle dΩ . . . . . . 162.1.4 Radiation from a surface element into a hemisphere

(Stefan–Boltzmann radiation law) . . . . . . . . . . . . . . . . . . . 172.2 Kirchhoff’s law of radiation for non-black bodies . . . . . . . . . . . . . . 19

2.2.1 Absorption by semiconductors . . . . . . . . . . . . . . . . . . . . . 212.3 The solar spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Air Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Concentration of the solar radiation . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 The Abbe sine condition . . . . . . . . . . . . . . . . . . . . . . . . 262.4.2 Geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.3 Concentration of radiation using the sine condition . . . . . . . . . . 28

2.5 Maximum efficiency of solar energy conversion . . . . . . . . . . . . . . . 29

3 Semiconductors 373.1 Electrons in semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 Distribution function for electrons . . . . . . . . . . . . . . . . . . . 383.1.2 Density of states De(εe) for electrons . . . . . . . . . . . . . . . . . 393.1.3 Density of electrons . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


vi Contents

3.4 Quasi-Fermi distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.1 Fermi energy and electrochemical potential . . . . . . . . . . . . . . 533.4.2 Work function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Generation of electrons and holes . . . . . . . . . . . . . . . . . . . . . . . 583.5.1 Absorption of photons . . . . . . . . . . . . . . . . . . . . . . . . . 593.5.2 Generation of electron–hole pairs . . . . . . . . . . . . . . . . . . . 62

3.6 Recombination of electrons and holes . . . . . . . . . . . . . . . . . . . . . 653.6.1 Radiative recombination, emission of photons . . . . . . . . . . . . . 653.6.2 Non-radiative recombination . . . . . . . . . . . . . . . . . . . . . . 673.6.3 Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.7 Light emission by semiconductors . . . . . . . . . . . . . . . . . . . . . . 793.7.1 Transition rates and absorption coefficient . . . . . . . . . . . . . . . 79

4 Conversion of Thermal Radiation into Chemical Energy 854.1 Maximum efficiency for the production of chemical energy . . . . . . . . . 88

5 Conversion of Chemical Energy into Electrical Energy 935.1 Transport of electrons and holes . . . . . . . . . . . . . . . . . . . . . . . . 93

5.1.1 Field current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.1.2 Diffusion current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1.3 Total charge current . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Separation of electrons and holes . . . . . . . . . . . . . . . . . . . . . . . 985.3 Diffusion length of minority carriers . . . . . . . . . . . . . . . . . . . . . 1015.4 Dielectric relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.5 Ambipolar diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.6 Dember effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.7 Mathematical description . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Basic Structure of Solar Cells 1096.1 A chemical solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Basic mechanisms in solar cells . . . . . . . . . . . . . . . . . . . . . . . . 1126.3 Dye solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.4 The pn-junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4.1 Electrochemical equilibrium of electrons in a pn-junction in the dark . 1166.4.2 Potential distribution across a pn-junction . . . . . . . . . . . . . . . 1176.4.3 Current–voltage characteristic of the pn-junction . . . . . . . . . . . 120

6.5 pn-junction with impurity recombination,two-diode model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.6 Hetero-junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.7 Semiconductor–metal contact . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.7.1 Schottky contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.7.2 MIS contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.8 The role of the electric field in solar cells . . . . . . . . . . . . . . . . . . . 133

Contents vii

7 Limitations on Energy Conversion in Solar Cells 1377.1 Maximum efficiency of solar cells . . . . . . . . . . . . . . . . . . . . . . . 1377.2 Efficiency of solar cells as a function of their energy gap . . . . . . . . . . . 1407.3 The optimal silicon solar cell . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.3.1 Light trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.4 Thin-film solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4.1 Minimal thickness of a solar cell . . . . . . . . . . . . . . . . . . . . 1477.5 Equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.6 Temperature dependence of the open-circuit voltage . . . . . . . . . . . . . 1497.7 Intensity dependence of the efficiency . . . . . . . . . . . . . . . . . . . . . 1517.8 Efficiencies of the individual energy conversion processes . . . . . . . . . . 151

8 Concepts for Improving the Efficiency of Solar Cells 1558.1 Tandem cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.1.1 The electrical interconnection of tandem cells . . . . . . . . . . . . . 1588.2 Concentrator cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.3 Thermo-photovoltaic energy conversion . . . . . . . . . . . . . . . . . . . 1618.4 Impact ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.4.1 Hot electrons from impact ionization . . . . . . . . . . . . . . . . . 1658.4.2 Energy conversion with hot electrons and holes . . . . . . . . . . . . 165

8.5 Two-step excitation in three-level systems . . . . . . . . . . . . . . . . . . 1688.5.1 Impurity photovoltaic effect . . . . . . . . . . . . . . . . . . . . . . 1688.5.2 Up- and down-conversion of photons . . . . . . . . . . . . . . . . . 172

9 Prospects for the Future 177

Appendix 181

Index 183

List of symbols

h, h = h/(2π) Planck’s constant eVshω photon energy eVa(hω) absorptivityr(hω) reflectivityt(hω) transmissionε(hω) = a(hω) emissivityα(hω) absorption coefficient 1/cmk Boltzmann’s constant eV/Kσ Stefan-Boltzmann constant W/(m2K4)T temperature Kn j concentration of particle type j 1/cm3

e electronh holeγ photonΓ phononne,nh concentration of electrons, holes 1/cm3

ni intrinsic concentration of electrons and holes 1/cm3

NC,NV effective density of states 1/cm3

in conduction band, valence bandεe,εh energy of an electron, hole eVεC energy of an electron eV

at the conduction band minimumεV energy of an electron eV

at the valence band maximumµ j chemical potential of particle type j eVη j electro-chemical potential of particle type j eVχe electron affinity eVϕ electrical potential Ve elementary charge AsV voltage = [ηe(x1)−ηe(x2)]/e VεFC Fermi energy for electron distribution eV

in conduction bandεFV Fermi energy for electron distribution eV

in valence bandm∗

e ,m∗h effective mass of electrons, holes g


x List of Symbols

be,bh mobility of electrons, holes cm2/(Vs)De,Dh diffusion coefficient of electrons, holes cm2/sτe,τh recombination life time of electrons, holes sRe,Rh recombination rate of electrons, holes 1/(cm3s)Ge,Gh generation rate of electrons, holes 1/(cm3s)σe,σh cross-section for the capture cm2

of an electron, hole by an impurityj j current density of particles of type j 1/cm2

jQ charge current density A/cm2

Preface

Mankind needs energy for a living. Besides the energy in our food necessary to sustain ourbody and its functions (100 W), 30 times more energy is used on average to make our lifemore comfortable. Electrical energy is one of the most useful forms of energy, since it canbe used for almost everything. All life on earth is based on solar energy following the inven-tion of photosynthesis by the algae. Producing electrical energy through photovoltaic energyconversion by solar cells is the human counterpart. For the first time in history, mankind isable to produce a high quality energy form from solar energy directly, without the need of theplants. Since any sustainable, i. e. long term energy supply must be based on solar energy,photovoltaic energy conversion will become indispensable in the future.

This book provides a fundamental understanding of the functioning of solar cells. Thediscussion of the principles is as general as possible to provide the basis for present technol-ogy and future developments as well. Energy conversion in solar cells is shown to consist oftwo steps. The first is the absorption of solar radiation and the production of chemical en-ergy. This process takes place in every semiconductor. The second step is the transformationinto electrical energy by generating current and voltage. This requires structures and forcesto drive the electrons and holes, produced by the incident light, through the solar cell as anelectric current. These forces and the structures which enable a directional charge transportare derived in detail. In the process it is shown that the electric field present in a pn junctionin the dark, usually considered a prerequisite for the operation of a solar cell, is in fact morean accompanying phenomenon of a structure required for other reasons and not an essentialproperty of a solar cell. The structure of a solar cell is much better represented by a semicon-ducting absorber in which the conversion of solar heat into chemical energy takes place andby two semi-permeable membranes which at one terminal transmit electrons and block holesand at the second terminal transmit holes and block electrons. The book attempts to developthe physical principles underlying the function of a solar cell as understandably and at thesame time as completely as possible. With very few exceptions, all physical relationships arederived and explained in examples. This will provide the non-physicists particularly with thebackground for a thorough understanding.

Emphasis is placed on a thermodynamic approach which is largely independent of exist-ing solar cell structures. This allows a general determination of the efficiency limits for theconversion of solar heat radiation into electrical energy and also demonstrates the potentialand the limits for improvement for present-day solar cells. We follow a route first taken by W.Shockley and H. J. Queisser.1

1W. Shockley, H. J. Queisser, J. Appl. Phys. 32, (1961), 510.


xii Preface

This book is the result of a series of lectures dealing with the physics of solar cells. I amgrateful to the many students who called my attention to errors and suggested improvements.The material presented here, which differs from the usual treatment of solar cells relying on anelectric field for a driving force, is the result of several years of collaboration with my teacher,W. Ruppel.

In some respects this book is more rigorous than is customary in semiconductor devicephysics and in solar cell physics in particular. The most obvious is that identical physicalquantities will be represented by identical symbols. Current densities will be represented byj and the quantity which is transported by the current is defined by its index, as in jQ forthe density of a charge current or je for the density of a current of electrons. In adheringto this principle, all particle concentrations are given the symbol n, with ne representing theconcentration of electrons, nh the concentration of holes and nγ the concentration of photons.I hope that those who are used to n and p for electron and hole concentrations do not find ittoo difficult to adapt to a more logical notation.

The driving force for a transition from exhausting energy reserves, as we presently do, tousing renewable energies, is not the exhaustion of the reserves themselves, although oil andgas reserves will not last for more than one hundred years. The exhaustion does not bothermost of us, since it will occur well beyond our own lifetime. We would certainly care alot more, if we were to live for 500 years and would have to face the consequences of ourpresent energy use ourselves. The driving force for the transition to renewable energies israther the harmful effect which the byproducts of using fossil and nuclear energy have onour environment. Since this is the most effective incentive for using solar energy, we startby discussing the consequences of our present energy economy and its effect on the climate.The potential of a solar energy economy to eliminate these problems fully justifies the mostintensive efforts to develop and improve the photovoltaic technology for which this book triesto provide the foundation.

Peter Wurfel

1 Problems of the Energy Economy

The energy economy of nearly all and in particular, of the industrialized countries is based onthe use of stored energy, mainly fossil energy in the form of coal, oil and natural gas, as wellas nuclear energy in the form of the uranium isotope U235. Satisfying our energy needs fromreserves, introduces two problems. A source of energy can continue only until it is depleted.Well before this time, that is, at the latest right now, we have to consider how life will continueafter this source of energy is gone and we must begin to develop alternatives. Furthermore,unpleasant side effects accompany the consumption of the energy source. Materials longburied under the surface of the earth are released and find their way into air, water and into ourfood. Up to now, the disadvantages are hardly perceptible, but they will lead to difficulties forfuture generations. In this chapter we estimate the size of the fossil energy resources, which,to be precise, are comprised not only of fossil energy carriers, but also of the oxygen in the airwhich is burned together with them. In addition, we will examine the cause of the greenhouseeffect, which is a practically unavoidable consequence of burning fossil fuels.

1.1 Energy economy

The amount of chemical energy stored in fossil energy carriers is measured in energy units,some more, some less practical. The most fundamental unit is the Joule, abbreviated J, whichis, however, a rather small unit representing the amount of energy needed to heat 1 g of waterby a quarter of a degree, or the amount of energy which a hair drier with a power of 1 kWconsumes in 1 ms. A more practical unit is the kilo Watt hour (kWh), which is 3.6× 106 J.1 kWh is the energy contained in 100 g of chocolate. The only problem with this unit is that itis derived from the Watt, the unit for power, which is energy per time. This makes energy equalto power times time. This awkwardness leads to a lot of mistakes in the non-science press likekW per hour for power, since most people mistake kW for energy which they perceive as themore basic quantity. The energy of fossil fuels is often given in barrels of oil equivalents or in(metric) tons of coal equivalents (t coal equ.).

The following relations apply:1 kWh = 3.6×106 J = 1 kWh1 t coal equ. = 29×109 J = 8200 kWh1 kg oil = 1.4 kg coal equ. = 12.0 kWh1 m3 gas = 1.1 kg coal equ. = 9.0 kWh1 barrel oil = 195 kg coal equ. = 1670 kWh


Peter Würfel

Physics of Solar Cells

© 2005 WILEY-VCH Verlag GmbH & Co.

Physics of Solar Cells - Buch.de Transport of electrons ... 6.2 Basic mechanisms in solar cells ......

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