Solar Energy

Copyright Delft University of Technology, 2014

This copy is provided for free, for personal use only.

Copyright Delft University of Technology, 2014

This copy is provided for free, for personal use only.

Solar Energy

Fundamentals, Technology, and Systems

Klaus Jäger
Olindo Isabella
Arno H.M. Smets
René A.C.M.M. van Swaaij
Miro Zeman

Copyright Delft University of Technology, 2014

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Copyright © 2014, Delft University of Technology
Typeset in DejaVu Sans Condensed and URW Palatino with LATEXAll rights reserved.
No part of this book may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means without the prior written
permission of the copyright owner.

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Preface
Hereby, we present the first version of our book Solar
Energy: Fundamentals, Technology and Systems and hopethat it will be a useful source that helps our readers to
study the different topics of solar energy.

thing is not clear or should be explained more detailed.
Feedback can be given on the edX website.
Preparing such a book is not possible without a lot of
support from others. We want to express our special
Thanks to Arianna Tozzi, who provided us with themodel on estimating the effect of wind speed and irradiance on the module temperature that we present
in Section 18.3. Further, we are very grateful on the information on real-life PV systems provided by Stephanvan Berkel. Ravi Vasudevan and Do Yun Kim are acknowledged for their discussions on PV systems. We
also want to thank Mirjam Theelen for providing feedback on the sections on CIGS and CdTe technologies. Further, we want to thank all the students thatprovided feedback during the 2013/14 PV Systems lecture given at Delft University of Technology.

It covers the topics that are treated in the three lectures on photovoltaics (PV) that are taught at the Delft
University of Technology throughout the Academic
Year: PV Basics, PV Technology, and PV Systems. In
addition the book also covers other forms of solar energy, in particular Solar Thermal applications and Solar
Fuels. Many of the topics that are discussed in thisbook are also covered in the Massive Open Online Course(MOOC) on Solar Energy (DelftX, ET.3034TU) that isgiven by Arno Smets on the edX platform and starts on
1 September 2014.
The students of this MOOC are the first ones that will
use this book for studying solar energy. As this is thevery first version, surely several small errors will bepresent throughout the text. We kindly ask the students
to give us feedback, so that all the errors and typos canbe eliminated quickly. Please let us also know if some-

The AUTHORS
Delft, the Netherlands
1 September 2014v
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vi

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Contents
About this Book

1

I

3

Introduction

1 Energy

5

1.1
1.2
1.3

6
7
11

Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Human and world energy consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Methods of Energy Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Status and prospects of PV technology

15

3 The Working Principle of a Solar Cell

23

II

PV Fundamentals

27

4 Electrodynamic basics

29

4.1
4.2

The electromagnetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
30

vii
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Contents

4.3
4.4
4.5

Optics of flat interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optics in absorptive media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Continuity and Poisson equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Solar Radiation

5.1
5.2
5.3
5.4
5.5

37

The Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radiometric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Blackbody radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solar spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Basic Semiconductor Physics

6.1
6.2
6.3
6.4
6.5

67
69
74
79
81
83

p-n homojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p-n heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Metal-semiconductor junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49
50
52
54
60
67

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bandgap-to-bandgap processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shockley-Read-Hall Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Auger Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Carrier concentration in non-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Semiconductor Junctions

8.1
8.2
8.3

37
38
41
44
45
49

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Atomic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Carrier concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Generation and Recombination of Electron-Hole Pairs

7.1
7.2
7.3
7.4
7.5

31
33
35

83
98
98

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Contents
9 Solar Cell Parameters and Equivalent Circuit

9.1
9.2
9.3

ix
101

External solar cell parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
The external quantum efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
The equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10 Losses and Eﬃciency Limits

113

10.1 The thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.2 The Shockley-Queisser Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.3 Other Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.4 Design Rules for Solar Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

III

PV Technology

135

11 A Little History of Solar Cells

137

12 Crystalline Silicon Solar Cells

141

12.1 Crystalline silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
12.2 Production of silicon wafers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
12.3 Fabricating solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
12.4 High-efficiency concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
13 Thin-film solar cells

163

13.1 Transparent conducting oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
13.2 The III-V PV technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
13.3 Thin-film silicon technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
13.4 Chalcogenide solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
13.5 Organic Photovoltaics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
13.6 Hybrid organic-inorganic solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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14 Third Generation Concepts

207

14.1 Multi-junction solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
14.2 Spectral conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
14.3 Multi-exciton generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
14.4 Intermediate band solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
14.5 Hot carrier solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

IV

PV Systems

15 Introduction to PV systems

217
219

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
15.2 Types of PV systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
15.3 Components of a PV system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
16 Location issues

225

16.1 The position of the sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
16.2 The sun path at different locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.3 The equation of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
16.4 Irradiance on a PV module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
16.5 Direct and diffuse irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
16.6 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
17 Components of PV Systems

251

17.1 PV modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
17.2 Maximum power point tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
17.3 Photovoltaic Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
17.4 Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
17.5 Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

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18 PV System Designxi
303

18.1 A simple approach for designing off-grid systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
18.2 Load profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
18.3 Meteorological effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
18.4 Designing grid-connected PV-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
18.5 Designing off-grid PV systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
19 PV System Economics and Ecology

341

19.1 PV System Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
19.2 PV System Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

V

Alternative Methods of Solar Energy Utilisation

20 Solar Thermal Energy

349
351

20.1 Solar thermal basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
20.2 Solar thermal heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
20.3 Concentrated solar power (CSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
21 Solar fuels369

21.1 Electrolysis of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
21.2 Photoelectrochemical (PEC) water splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Appendix

381

A Derivations in Electrodynamics

383

A.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
A.2 Derivation of the electromagnetic wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
A.3 Properties of electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

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B Derivation of homojunction J-V curves

389

B.1 Derivation of the J-V characteristic in dark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
B.2 Derivation of the J-V characteristic under illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
C Atomic force microscopy and statistical parameters

397

C.1 Atomic force microscopy (AFM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
C.2 Statistical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Bibliography

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About this Book
This book aims to cover all the topics that are relevantfor getting a broad overview on the different aspects ofSolar Energy, with a focus on photovoltaics, which is thetechnology that allows to convert energy transported inlight directly into electrical energy.

and the different technologies in particular. After discussing some basics from electrodynamics in Chapter
4 and solar radiation in Chapter 5, we spend several
chapters on explaining the most important concepts ofsemiconductor physics. After discussing the basics in
Chapter 6, we elaborate on the different generation andrecombination mechanisms in Chapter 7 and introduce
different types of semiconductor junctions in Chapter
8. After introducing the most important parameters forcharacterising solar cells in Chapter 9, we conclude Part
II with a discussion on the efficiency limits of photovoltaic devices in Chapter 10, from which we distill some
general design rules that are very important in Part III.

The Organisation of this book is roughly linked to thethree lectures on photovoltaics (PV), that are given at
the Faculty for Electrical Engineering, Mathematics and Computer Science at the Delft University of Technology throughout the course of the academic year: PV Basics, whichroughly covers the topics covered in Part II on PV Fundamentals; PV Technology which covers the topics
treated in Part III; and PV Systems which is treated inPart IV.

The different PV technologies are discussed in Part III.
After summarising the history of solar cells in Chapter
11, we discuss crystalline silicon technology, which isby far the most important PV technology, in Chapter
12. After that we take a look at the different thin-film
technologies in Chapter 13. Part III is concluded with a
discussion on several third generation concepts that aim
to combine high efficiencies with low cost in Chapter
14.

In total, this book contains five parts. In the introductoryPart I we provide the reader with some general facts onenergy in Chapter 1, summarise the current status of
PV in the world in Chapter 2 and provide a first short
explanation on how solar cells work in Chapter 3.
Part II aims to cover all the physical fundamentals thatare required for understanding solar cells in general
1
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Contents

Part IV is dedicated in the planning of real PV systems.
After a short introduction on PV systems in Chapter
15, we discuss the position of the sun and its implications in great detail in Chapter 16. The different components of a PV system, starting from the modules but
also including all the balance-of-system components are
introduced in Chapter 17. With all this knowledge weelaborate on designing PV systems, for both off-gridand grid-connected situations in Chapter 18. This partis concluded with a discussion of the ecological andeconomical aspects of PV systems in Chapter 19.
Alternative methods of solar energy are discussed inPart V. In Chapter 20 we introduce different concepts
related to solar thermal energy. In Chapter 21, whichis the last chapter of the regular text, we discuss solar
fuels, which allow to store solar energy on the longterm in the form of chemical energy.
The book is concluded with an Appendix, wheresome derivations that are too lengthy for the book areshown.

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Part I

Introduction

3
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1

EnergyAs this book is on Solar Energy, it is good to start thediscussion with some general thoughts on Energy. We
will begin with a quote from The Feynman Lectures on
Physics.

we finish watching nature go through her tricks andcalculate the number again, it is the same.
...
Energy has a large number of different forms, and there
is a formula for each one. These are: gravitational energy, kinetic energy, heat energy, elastic energy, electrical energy, chemical energy, radiant energy, nuclear
energy, mass energy. If we total up the formulas foreach of these contributions, it will not change except
for energy going in and out.

There is a fact, or if you wish, a law, governing all
natural phenomena that are known to date. There is
no known exception to this law—it is exact so far as
we know. The law is called the conservation of energy.
It states that there is a certain quantity, which we callenergy, that does not change in the manifold changes
which nature undergoes. That is a most abstract idea,
because it is a mathematical principle; it says that
there is a numerical quantity which does not change
when something happens. It is not a description of amechanism, or anything concrete; it is just a strange
fact that we can calculate some number and when

It is important to realise that in physics today, we
have no knowledge of what energy is. We do nothave a picture that energy comes in little blobs of adefinite amount. It is not that way. However, there
are formulas for calculating some numerical quantity,

5
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6

1. Energy

where s denotes distance. Energy is usually measured
in the unit of Joule (J), named after the English physicist
James Prescott Joule (1818-1889), which it defined as
the amount of energy required applying the force of 1
Newton through the distance of 1 m, 1 J = 1 Nm.

and when we add it all together it gives . . . always thesame number. It is an abstract thing in that it does nottell us the mechanism or the reasons for the various
formulas [1].

1.1Some definitions

We will now state some basic physical connections
between the three very important physical quantities
of energy, force, and power. These connections are takenfrom classical mechanics but generally valid. We startwith the force F, which is any influence on an object
that changes its motion. According to Newton’s second
law, the force is related to the acceleration a of a bodyvia
F = ma,(1.1)
where m is the mass of the body. The bold characters
denote that F and a are vectors. The unit of force is
Newton (N), named after Isaac Newton (1642-1727). It
is defined as the force required to accelerate the massof 1 kg at an acceleration rate of 1 m/s2 , hence 1 N = 1
kg·m/s2 .
In mechanics, energy E, the central quantity of thisbook, is given as the product of force times distance,E=Z

F (s) ds,

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(1.2)

Another important physical quantity is the power P,
which tells us the rate of doing work, or, which is equivalent, the amount of energy consumed per time unit.
It is related to energy via
E=Z

P(t) dt,(1.3)

where t denotes the time. The power is usually measured in Watt (W), after the Scottish engineer James Watt
(1736-1819). 1 W is defined as one Joule per second, 1
W = 1 J/s and 1 J = 1 Ws.
As we will see later on, 1 J is a very small amount ofenergy compared to the human energy consumption.
Therefore, in the energy markets, such as the electricity
market, often the unit Kilowatt hour (kWh) is used. It is
given as1 kWh = 1000 Wh × 3600s
= 3 600 000 Ws.
h

(1.4)

On the other hand, the amounts of energy in solid statephysics, the branch of physics that we will use to explain how solar cells work, are very small. Therefore,

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7

1.2. Human and world energy consumption

we will use the unit of electron volt, which is the energy a body with a charge of one elementary charge(e = 1.602 × 10−19 C) gains or looses when it is moved
across a electric potential difference of 1 Volt (V),
1 eV = e × 1 V = 1.602 × 10−19 J.

1.2

(1.5)

Human and world energy consumption

After this somewhat abstract definitions we will lookat the human energy consumption. The human body is
at a constant temperature of about 37°C. It hence contains thermal energy. As the body is continuously cooledby the surroundings, thermal energy is lost to the outside. Further, blood is pumped through the blood vessels. As it travels through the vessels, its kinetic energy
is reduced because of internal friction and friction at
the walls of the blood vessels, i.e. the kinetic energy
is converted into heat. To keep the blood moving, theheart consumes energy. Also, if we want our body tomove this consumes energy. Further, the human brain
consumes a lot of energy. All this energy has to besupplied to the body from the outside, in the form offood. A grown up average body requires about 10 000
Kilojoule every day.1 We can easily show that this con1 The energy content of food usually is given in the old-fashioned unit of kiloc-

alories (kcal). The conversion factor is 1 kcal = 4.184 kJ. An average male human

Copyright Delft University of Technology, 2014

sumption corresponds to an average power of the human body of 115.7 W. We will come back to this value
later.
In modern society, humans do not only require energy
to keep their body running, but in fact we consume energy for many different purposes. We use energy forheating the water in our houses and for heating ourhouses. If water is heated, its thermal energy increases,
and this energy must be supplied. Further, we use a lotof energy for transportation of people and products bycars, trains, trucks and planes. We use energy to produce our goods and also to produce food. At the moment, you are consuming energy if you read this bookon a computer or tablet. But also if you read this bookin a printed version, you implicitly consumed the energy that was required to print it and to transport it to
you place.
As we mentioned already above, energy is never produced but always converted from one form to another.
The form of energy may change in time, but the total
amount does not change. If we want to utilise energy towork for us, we we usually convert it from one form toanother more useable form. An example is the electricmotor, in which we convert electrical energy to mechanical energy.
Modern society is very much based on the capability
requires about 2500 kcal a day.

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8

1. Energy

Table 1.1: Total primary energy consumption per capita and$160

average power used per capita of some countries in 2011 [2].

U.S.A.
Netherlands
Germany
China
India

Energyconsumption(kWh/capita)
81 64253 96344 31023 608
6 987

Averagepower use
(W/capita)
9 319
6 160
5 058
2 695
797

$120

Oil price

Country

$140

$100

Values in February 2013 US Dollars
1970s oil crisis

$80

2008
financialcrisis$60$40$20
1861 1876 1891 1906 1921 1936 1951 1966 1981 1996 2011of us humans to covert energy from one form to another form. The most prosperous and technologicallydeveloped nations are also the ones which have accessto and are consuming the most energy per inhabitant.
Table 1.1 shows the primary energy consumption percapita and the average power consumed per capita forseveral countries. We see that the average U.S. citizen
uses an average power of 9 319 W, which is about 80
times what his body needs. In contract, an average citizen from India only uses about 800 W, which is lessthen a tenths of the U.S. consumption.
Many people believe that tackling the energy problem isamongst the biggest challenges for human kind in the
21st century. It is a challenge because of several problems: The first challenge the human kind is facing is a
supply-demand problem. The demand is continuously

Copyright Delft University of Technology, 2014

Figure 1.1: The history of the oil price normalised to the

February 2013 value of the U.S. Dollar [3].

growing. The world population is still rapidly growing, and some studies predict a world population of 9billion around 2040 in contrast to the 7 billion people
living on the planet today. All these people will needenergy, which increases the global energy demand. Further, in many countries the living standard is rapidly
increasing like China and India, where approximately
2.5 billion people are living, which represents morethan a third of the World’s population. Also the increasing living standards lead to an increased energy
demand.
According to the IEA World Energy Outlook 2013, theThis copy is provided for free, for personal use only.

9

1.2. Human and world energy consumption
Other Renewable
11%
Hydro
2%
Nuclear6%

Others
0.2%

Oil
33%

Gas
21%
Coal
27%

Figure 1.2: The primary energy consumption of the world bysource in 2008. The total supply was 143 851 TWh [5].

global energy demand will grow about one third from2011 to 2013 [4]. The increasing demand in energy has
economic impact, as well. If there is more demand fora product, while supply does not change much, theproduct will get more expensive. This basic market
mechanism is also true for Energy. As an example weshow a plot of the annual averaged price for an oil barrel, normalised to the value of the 2008 US Dollar in
Fig. 1.1. We see that prices went up during the oil crisisin the 1970s, when some countries stopped producing
and trading oil for a while. The second era of higher oil
prices started at the beginning of this millennium. Dueto the increasing demand from new growing economies, the oil prices have been significantly increased.
A second challenge that we are facing is related to thefact that our energy infrastructure heavily depends on

Copyright Delft University of Technology, 2014

fossil fuels like oil, coal and gas. Fossil fuels are nothing but millions and millions of years of solar energy
stored in the form of chemical energy. The problem isthat humans deplete these fossil fuels much faster thanthey are generated through the photosynthetic process
in nature. Therefore fossil fuels are not a sustainableenergy source. The more fossil fuels we consume, theless easily available gas and oil resources will be available. Already now we see that more and more oil andgas is produced with unconventional methods, such as
extracting oil from tar sands in Alberta, Canada andproducing gas with fracturing such as in large parts of
the United States. This new methods use much moreenergy to get the fossil fuels out of the ground. Further, off-shore drilling is put regions with ever larger
water depths, which leads to new technological risks as
we have seen in the Deepwater Horizon oil spill in the
Gulf of Mexico in 2010.
A third challenge is that by burning fossil fuels we produce the so-called greenhouse gases like carbon dioxide(CO2 ). The additional carbon dioxide created by humanactivities is stored in our oceans and atmosphere. Figure 1.3 shows the increase in carbon dioxide concentration in the Earth’s atmosphere up to 2000. According to
the International Panel on Climate Change (IPCC) Fifth
Assessment Report (AR5),
The atmospheric concentrations of carbon dioxide,
methane, and nitrous oxide have increased to levels
unprecedented in at least the last 800,000 years. Car-

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10

1. Energy
bon dioxide concentrations have increased by 40%
since pre-industrial times, primarily from fossil fuelemissions and secondarily from net land use change
emissions. The ocean has absorbed about 30% of theemitted anthropogenic carbon dioxide, causing ocean
acidification [7].

Further, in the AR5 it is stated that
Human influence on the climate system is clear. Thisis evident from the increasing greenhouse gas concentrations in the atmosphere, positive radiative forcing,observed warming, and understanding of the climate
system [7].

andFigure 1.3: The atmospheric CO2 content in the last 400 000

years [6].

Human influence has been detected in warmingof the atmosphere and the ocean, in changes in the
global water cycle, in reductions in snow and ice, in
global mean sea level rise, and in changes in someclimate extremes. This evidence for human influence
has grown since AR4. It is extremely likely that human influence has been the dominant cause of theobserved warming since the mid-20th century [7].

Hence, it seems very clear that the increase in carbon
dioxide is responsible for the global warming and climate change, which can have drastic consequences of
the habitats of many people.
Since the beginning of the industrial revolution, mankind is heavily dependent on fossil fuels. Within a few
centuries, we are using solar energy that was incident

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11

1.3. Methods of Energy Conversion

on Earth for hundreds of millions of years, converted in
to chemical energy by the photosynthetic process andstored in the form of gas, coal and oil.
Before the industrial revolution, the main source ofenergy was wood and biomass, which is a secondary
form of solar energy. The energy source was replenished in the same characteristic time as the energy being consumed. In the pre-industrial era, mankind wasbasically living on a secondary form of solar energy.
However, also back then the way we consumed energy
was not fully sustainable. For example, deforestation
due to increasing population density was already playing a role at the end of the first millennium.

1.3

Methods of Energy Conversion

Figure 1.4 shows different energy sources and the ways
we utilise them. We see that usually the chemical energy stored in fossil fuels is converted to usable formsof energy via heat by burning, with an efficiency of
about 90%. Using heat engines, thermal energy can beconverted in to mechanical energy. Heat engines havea conversion efficiency of up to 60%. Their efficiency isultimately limited by the Carnot efficiency limit that wewill discuss in Chapter 10. The far majority of the current cars and trucks works on this principle. Mechanical energy can be converted in to electricity using elec-

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Gravitational

NuclearThermalη=90%Chemical

Coal, oil, gas,biomass, hydrogen

Wind

Hydro-tidal
Heat
enginesη<60%MechanicalElectric
generators
η=90%

Photo
voltaics

Electrolyzers
Fuel Cells

Electrical

Solar thermal

Solar

Figure 1.4: The different energy carriers and how we utilisethem [8].

tric generators with an efficiency of up to 99%. Most of
the World’s electricity is generated with an turbo generator that is connected to a steam turbine, where the coalis the major energy source. This process is explainedin more detail in our discussion on solar thermal electric power in Chapter 20. Along all the process steps of
making electricity out of fossil fuels, at least 50% of theinitial available chemical energy is lost in the various
conversion steps.
Chemical energy can be directly converted into electricity using a fuel cell. The most common fuel used infuel cell technology is hydrogen. Typical conversion efficiencies of fuel cells are 60%. A regenerative fuel cell
can operate in both directions and also convert electrical energy into chemical energy. Such an operation isThis copy is provided for free, for personal use only.

12

1. Energy

called electrolysis; typical conversion efficiencies of hydrogen electrolysers of 50-80% have been reported. We
will discuss electrolysis in more detail in Chapter 21.
In nuclear power plants, energy is released as heat during nuclear fission reactions. With the heat steam is generated that drives a steam turbine and subsequently
an electric generator just as in most fossil fuel power
plants.

1.3.1

Renewable energy carriers

All the energy carriers discussed above are either fossil
or nuclear fuels. They are not renewable because they
are not “refilled” by nature, at least not in a usefulamount of time. In contrast, renewable energy carriers
are energy carriers that are replenished by natural processes at a rate comparable or faster than its rate of
consumption by humans. Consequently, hydro-, windand solar energy are renewable energy sources.
Hydroelectricity is an example of an energy conversion
technology that is not based on heat generated by fossil
or nuclear fuels. The potential energy of rain fallingin mountainous areas or elevated plateaus is converted into electrical energy via a water turbine. With tidal
pools the potential energy stored in the tides can also be
converted to mechanical energy and subsequently electricity. The kinetic energy of wind can be converted into

Copyright Delft University of Technology, 2014mechanical energy using wind mills.
Finally, the energy contained in sunlight, called solar energy, can be converted into electricity as well. If this energy is converted into electricity directly using devices
based on semiconductor materials, we call it photovoltaics (PV). The term photovoltaic consists of the greek
word φως (phos), which means light, and -volt, whichrefers to electricity and is a reverence to the Italianphysicist Alessandro Volta (1745-1827) who invented
the battery. As we will discuss in great detail in thisbook, typical efficiencies of the most commercial solar
modules are in the range of 15-20%.
Solar light can also be converted into heat. This application is called solar thermal energy and is discussed indetail in Chapter 20. Examples are the heating of water flowing through a black absorber material that isheated in the sunlight. This heat can be used for water
heating, heating of buildings or even cooling. If concentrated solar power systems are temperatures of several
hundreds of degrees are achieved, which is sufficient togenerate steam and hence drive a steam turbine and agenerator to produce electricity.
Next to generating heat and electricity, solar energy
can be converted in to chemical energy as well. This iswhat we refer to as solar fuels. For producing solar fuels,
photovoltaics and regenerative fuel cells can be combined. In addition, sunlight can also be directly converted into fuels using photoelectrochemical devices. We

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13

1.3. Methods of Energy Conversion

will discuss solar fuels in Chapter 21.
We thus see that solar energy can be converted into
electricity, heat and chemical energy. The sun has is energy source for almost all the processes that happen
on the surface of our planet. Wind is a result of temperature difference in the atmosphere induced by solar
irradiation, waves are generated by the wind, clouds
and rain are initially formed by the evaporation of water due to sun light. As the sun is the only real energy
source we have, we need to move to an era in which
we start to utilise the energy provided by the sun directly to satisfy our energy needs. The aim of this book
is to teach the reader how solar energy can be utilised
directly.

1.3.2

Electricity

As we see in Fig. 1.5 (a), 17% of all the World’s secondary energy is used as electricity, which is a form of energy that can be easily and cheaply transported withrelative small losses through an electric grid. It is important to realise that without electricity modern society
as we know it would not be possible. Electricity has
been practically used for more than 100 years now. It
provides us the energy to cook food, to wash, to do thelaundry, illuminate the house and streets, and countlessother applications. The access to electricity to electricity
strongly determines the our living standard. Despite

Copyright Delft University of Technology, 2014(a)Nonenergy
use 9%(b)
Electricity
17%
Heat
47%

Other
Renewable
4%

Oil
4%

Other
1%

Hydro
16%
Coal
41%

Nuclear
12%
Gas
22%

Transport
27%

World 2009

World 2011; total: 22 156 TWh

Figure 1.5: (a) The final energy consumption by energy

service [9] and (b) the energy carriers used for electricity generation [10].

this importance of electricity, in 2009 still about 1.3 billion people had no access to electricity.
As we see in 1.5 (b), about 65% of the electricity is generated using fossil fuels, where coal is the dominant
contributor. As coal emits about twice as much CO2 per
generated kWh as natural gas, coal power plants area major contributor to global warming. Nuclear is responsible for 16% of the World’s electricity generation.
With 19%, hydroelectricity is by far the largest contributor among the renewable energy sources.
Of all the generated electricity, about 40% of the electricenergy is used for residential purposes and 47% is usedby industry. 13% is lost in transmission. As you cansee, in 2007, transport did not play a significant role inThis copy is provided for free, for personal use only.

14

1. Energy(a) Other

Renewable
12%
Nuclear4%

Other fossile (b) Import 6%
Other
4%
Renewable 4%
Coal

Others
3%

24%

Coal
1%
Oil
4% GasNuclear
11%
2%

Gas
53%

Hydro
72%

NL 2012; total 103 TWh

Brazil 2007; total: 412 TWh

Figure 1.6: The energy mix used for electricity production in(a) the Netherlands [11] and (b) Brazil [12].

the electricity consumption. However, this is expected
to change as electric cars are becoming more and more
important.
Figure 1.6 shows which energy carriers are mainly used
for electricity generation in the Netherlands and Brazil.
We see that in the Netherlands, electricity generation
heavily depends on the local gas resources, whereas in
Brazil hydroelectricity is the most important resource.

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2

Status and prospects of PV technology

Global PV production volume (MWp)In this Chapter we will give a brief overview on thecurrent status of the PV technology and discuss its prospects.
In Fig. 2.1 the global production of PV modules in recent years is shown. The vertical axis represents the annual production expressed in the total produced power
capacity in MW p . The letter p denotes peak power, thismeans the maximum power a PV module can deliver
if it is illuminated with the standardised AM1.5 solar
spectrum, which we introduce in Section 5.5. On thehorizontal axis the time is shown. We see that the solar
cell production is increasing more than 40% every year,
which is exponential growth.

30000

27213

2009: 87% wafer-type c-Si technology

25000
mono c-Si
poly c-Si
ribbon c-Si
a-Si
ThinCdTe film
CIGS solar
other cells

20000
15000
10000
5000
0

118%

12464
7910
4279
2536

401

40%

560

34%

750

1257 1815
40%

68%

56%

85%

69%

45%

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

Figure 2.1: The global PV production volume in recent years

REFERENCE.

Figure 2.2 shows the worldwide cumulative installed
15
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Global installed peak power (MWp)16

2. Status and prospects of PV technology
110
100
90

110
100
90
80
70
60
50
40
30
20
10
0

2000

Rest of the world (12 554, 12%)60

India (1 205, 1%)
Greece (1 536, 1%)
United Kingdom (1 829, 2%)Czech Republic (2 072, 2%)
Australia (2 412, 2%)
Belgium (2 650, 2%)
France (4 003, 4%)AmericasMiddle East and Africa

5040Japan (6 914, 7%)

European Union

30

80

China
Asia PacificRest of World

2002

2004

70

2006

2008

2010

20
10
0

2012

Figure 2.2: The global installed PV capacity (Data from [13]).

PV power, which is exponentially increasing in time
as well. By far the largest share is installed in Europe.
It is followed by the Asia Pacific Region, where mostof the PV power is installed in Japan. For China weobserve a very strong increase in installed PV power
since 2010. By the end of 2012 the 100 GW p threshold
was passed for the first time [13]. By the end of 2013,
already almost 140 GW p was installed around the globe[14]. Of all the installed PV power at the end of 2013,
almost one third was installed in 2013 alone!
In Fig. the installed PV power in several countries atthe end of 2012 is shown. About 31% of the total PV
capacity is installed in Germany. This is a result of theGerman government’s progressive feed-in-tariff policy

Copyright Delft University of Technology, 2014

Germany (32 411, 31%)

Spain (5 166, 4%)

USA (7 777, 7%)

Italy (16 361, 16%)
China (8 300, 8%)

Figure 2.3: Fraction of PV installations for different countries

by the end of 2012 (Data from [13]).

that was introduced in 2000 [15].1 Considering that Germany lies within an area with a relatively low radiationlevel that is comparable to that of Alaska [16], the large
contribution of solar electricity to Germany’s electricity
production indicates the promising potential of solar
energy for the sunnier parts of the world.
A very strong increase also is observed in Italy, which
accounts for 16% of the world wide PV capacity. China
with a contribution of 8% is the fastest growing market
at the moment, in 2010, China only contributed with 2%
to the global PV capacity. Within the top six, we alsofind the United States, Japan and Spain. Their PV capacity contributes between 4% (Spain) and 7% (U.S.A.).
Also Japan shows a strong growth in PV installations.
1 We will discuss the feed-in tariﬀ scheme in Chapter 19.

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Fraction (%)

17

100
90
80
70
60
50
40
30
20
10
0

MP MP MP

MP MP

MP MP MP MP2000 01

03

05

02

04

EuropeAsia Pacific
M...market
P...production

06

07

Americas

08

MP

MP MP

09

10

China

MP

11 2012
Rest ofWorld

Figure 2.4: Development of the market and production sharesof different PV markets since 2000 (Data from [13]).

After the Fukushima Daiichi nuclear disaster on 11 March2011 the Japanese government introduced some progressive feed-in tariffs to promote and accelerate theintroduction of renewable energy conversion technologies.
It is interesting to see that PV technology is not onlya European affair. The local demand and supply has
been changing rapidly in the last 13 years, which is illustrated in Fig. 2.4. This figure illustrates the evolutionof the world-wide supply and demand of PV modules
in the various regions around the world. We see thatin 2000 the biggest market was Japan with a total shareof 40%. In 2000, Germany introduced the Erneuerbare
Energie Gesetz (Renewable Energy act) which induced a

Copyright Delft University of Technology, 2014

strong growth of the German and hence the European
PV market. By 2008, Europe had a market share ofmore than 80%. Back then, PV was mainly a European
industry. Starting from 2009, the domestic PV markets
in China, the Americas (mainly U.S.) and Asia Pacific(mainly Japan) are increasing very rapidly and catching
up quickly with Europe.
Figure 2.4 also shows the supply side. Up to 2005 we
see that the Asia Pacific and the Europe production
shares were slowly increasing, as their growth was
faster than that of the other regions. Since then the picture changed drastically! Since then the Chinese production share was increasing very strongly. This canbe explained by the fact that the Chinese government
made huge investments in order to scale up PV module
manufacturing in China. In 2012, around 60% of all PV
modules were produced in China.
In 2000, the PV markets was an essentially local, meaning the European companies produced for the European
market etc. The local demands and supplies in Asia, theAmericas and Europe were in balance. In the last years,
the market has become a global market. As a result, in2012 no local balance between supply and demand existed anymore. While the majority of the demand is in
Europe, the majority of the production is in China.
The demand also is strongly stimulated by the decreasing cost price of PV technology. Figure 2.5 shows thelearning curve of PV technology. The learning curveThis copy is provided for free, for personal use only.

Average global sales price(USD/W)18

2. Status and prospects of PV technology

shows in a graphical way, how the cost price develops with increasing experience, where the experienceis expressed by the cumulatively installed PV capacity.
With more PV produced — and hence also with time
— the PV industry gets more experienced. On the one
hand, the industry learns to increase the energy conversion efficiency without increasing the cost via better andbetter understanding the production process and hence
increasing the production yield. On the other hand, industry also learns to produce more efficiently, whichmeans that the man power required per production
unit can be reduced. Also, the materials and energy required for producing the PV modules becomes less andless per production unit. In addition, also up-scaling
reduces the cost. Learning curves usually show an exponentially decreasing cost price, until the technology
or product is fully developed.

100

PV System
10

PV Module

Non-modular cost
1

10-4

10-3

10-2

10-1

100

101

102

103

104

Cumulative installed power (GWp)
Figure 2.5: The learning curve for PV modules and PV sys-

tems (Data from [17]).

Copyright Delft University of Technology, 2014

In Fig. 2.5, the averaged global sales prices of a PV
module versus the cumulative installed power up to
20 GW is shown. Note, that the points up to 20 GW(up to 2009) in the grey area are real data points, while
the points in white area are extrapolation of the general trend. It is important to note that the sales prices,
except for some fluctuations, follow a largely exponential decay. Currently, the average retail price of PV
modules is below 1 US Dollar per Watt-peak. However, cost price of a PV system is not only determinedby the module. The red dots show the decrease in thecost price of complete PV systems. While in the earlyThis copy is provided for free, for personal use only.

19days of the PV technology, the system price was dominated by the module price, currently, the cost of thenon-modular components of PV systems are getting more
and more dominant. With non-modular components,
we refer to components such as the the racking, wiring, inverter, batteries for stand-alone systems, and also
the maintenance costs. All these components are discussed in detail in Chapter 17. The difference between
the red and green line corresponds to the non-modular
costs, which is dropping much slower than that of the
PV modules.
As a consequence, PV technologies with higher energy conversion efficiencies have an advantage, as withhigher efficiency less area is required to install the same
PV power. As the area is directly linked to the nonmodular costs, technologies with higher efficiencies require less modular costs which has an positive effect on
the cost price of the complete PV system. Consequently,
the c-Si PV technology, with module efficiencies ranging
from 14% up to 20% has an advantage with respect tothin-film technologies, that have lower efficiencies.
In Chapter ?? we have seen that hydropower is responsible for 17% of the total worldwide electricity
production while 12% of the electricity is generated innuclear power plants. Howe do these numbers compare to solar electricity? This question is answered inFig. 2.6 (a), where the installed capacity (in GW) of several electricity generation technologies is shown on a

Copyright Delft University of Technology, 2014

logarithmic scale. The figure only considers electricity
generation technologies that are not dependent on fossil
fuels. We see that the installed nuclear power capacity
is hardly growing anymore, while the installed hydropower is still slightly growing in time. Wind is growing
at a much faster rate of 20% per year. Solar has by far
the largest growth rate with an annual increase of installed capacity exceeding 40% since 2008.
However, it is not fair to compare the installed power
between technologies like this, because the numbers
shown in the graph represent the maximum (peak)
power the different technologies can generate insteadof the average power they have delivered in reality.
The relationship between the totally installed power
and the power generated on average is called the capacity factor CF . Of the technologies shown in Fig. 2.6(a) nuclear has by far the highest capacity factor withCF (nuclear) = 90% followed by hydropower withCF (hydro) = 40%. For wind electricity we assume
CF (wind) = 30% and for solar electricity CF (solar) =15%. The low capacity factor for PV systems can be explained by the fact that for most geographical locations,almost half of the solar day is devoid of solar radiation
at night time.
Figure 2.6 (b) shows the effective installed power corrected with the capacity factors. Currently solar energy
generates about an order of magnitude less electricity
than wind energy and more than two orders of mag-This copy is provided for free, for personal use only.

20

100

(b)
Hydro

Effective installed capacity (GW)

1000

Cumulative installed
peak capacity (GWp )(a)

2. Status and prospects of PV technology

Nuclear

10

Wind

1
Solar
0.1
1980

1990

2000

2010

1000

Hydro (CF=0.40)

100

Nuclear (CF=0.90)

2020

10

Wind (CF=0.30)

1
0.1
1980

Solar (CF=0.15)
1990

2000

2010

2020

Figure 2.6: (a) Development of the installed capacity (in GW) of several electricity generation technologies since 1980. (b) The samegraph corrected by the capacity factor CF and extrapolated until 2020.

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21

nitude less than hydro and nuclear electricity. Seen thedevelopment in recent years we however can claim thatthe trend in the growth of solar energy will continue
the coming years. If we therefore extrapolate the trendsof the last decade until 2020 we see that the installed
power of solar energy will exceed nuclear, wind andhydropower by then. It is just a matter of time until
solar electricity will be the most important electricity
generation technology that is not based on combustionof fossil fuels.
Of course, we have to justify why solar electricity cangrow much faster than the other technologies shownin Fig. 2.6. First, solar radiation is available everywhere on Earth and it is available in great abundance.
The amount of solar energy incident on Earth is about
10 000 time larger than the total energy2 consumptionof mankind. As hydroelectricity is powered by waterthat is evaporated by the sun and falls on the groundas rain, it is a secondary form of solar energy. Also
wind arises from temperature and pressure differencesof the atmosphere and hence also is a secondary formof solar energy. As a consequence, solar energy is by far
the largest available form of renewable energy.

hydro and nuclear power plants requires large public or
private investments. While solar electricity can be generated in large PV parks or solarthermal power plants(see Chapter 20) as well, it has an unique advantage:
PV systems can be installed decentralised on every roof.
Electricity consumers can generate a least a part of theirrequired electricity in their own homes, which makes
them partially independent of the electricity market.
In addition, the cost price of PV systems has dropped
below grid parity in large parts of the world [18]. Thismeans, that averaged during the lifetime of the PV system PV generated electricity is cheaper than electricity
from the grid.
We believe that the installation of decentralised PV systems will be the big force behind the solar revolution in
the coming years. It will change the energy landscapemuch faster than most people think, which is justified
in Fig. 2.6 (b). As more and more people become awareof these facts, it is more likely that the growth will befurther enhanced than it will be slowed down.

Secondly, hydro- and nuclear electricity are centralised
electricity generation technologies. For hydro power
plants, big dams are needed. Also nuclear power plants
have large power rates at about 1 GW. Building new
2 We really mean the total human energy consumption and not only electricity!

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22

2. Status and prospects of PV technology

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3

The Working Principle of a Solar CellIn this chapter we present a very simple model of asolar cell. Many notions presented in this chapter will
be new but nonetheless the great lines of how a solar
cell works should be clear. All the aspects presented
in this chapter will be discussed in larger detail in thefollowing chapters.

the light consists of well defined energy quanta, called
photons. The energy of such a photon is given by

E = hν,The working principle of solar cells is based on thephotovoltaic effect, i.e. the generation of a potential difference at the junction of two different materials in response to electromagnetic radiation. The photovoltaic
effect is closely related to the photoelectric effect, where
electrons are emitted from a material that has absorbed
light with a frequency above a material-dependentthreshold frequency. In 1905, Albert Einstein understood that this effect can be explained by assuming that

(3.1)

where h is Planck’s constant and ν is the frequency ofthe light. For his explanation of the photoelectric effect Einstein received the Nobel Prize in Physics in 1921
[19].

The photovoltaic effect can be divided into three basic
processes:
23

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24

3. The Working Principle of a Solar Cell
1. Generation of charge carriers due to the absorption of photons in the materials that form a junction.

(a)

(b)
Ef
EC

EC
Eph

EG

EG
EV

EV
Ei

generation ofan electron-hole pairEphthermalisation, Eph > EG

Figure 3.1: (a) Illustrating the absorption of a photon in asemiconductor with bandgap Eg . The photon with energyEph=hν excites an electron from Ei to E f . At Ei a hole iscreated. (b) If Eph > Eg , a part of the energy is thermalised.

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Absorption of a photon in a material means that its energy is used to excite an electron from an initial energy
level Ei to a higher energy level E f , as shown in Fig.
3.1 (a). Photons can only be absorbed if electron energy levels Ei and E f are present so that their difference
equals to the photon energy, hν = E f − Ei . In an ideal
semiconductor electrons can populate energy levels below the so-called valence band edge, EV , and above theso called conduction band edge, EC . Between those twobands no allowed energy states exist, which could be
populated by electrons. Hence, this energy difference iscalled the bandgap, Eg = EC − EV . If a photon with anenergy smaller than Eg reaches an ideal semiconductor,
it will not be absorbed but will traverse the material
without interaction.
In a real semiconductor, the valence and conduction
bands are not flat, but vary depending on the so-called
k-vector that describes the crystal momentum of thesemiconductor. If the maximum of the valence bandand the minimum of the conduction band occur atthe same k-vector, an electron can be excited from thevalence to the conduction band without a change in the
crystal momentum. Such a semiconductor is called a
direct bandgap material. If the electron cannot be excited
without changing the crystal momentum, we speak ofThis copy is provided for free, for personal use only.

25

an indirect bandgap material. The absorption coefficientin an direct bandgap material is much higher than in
an indirect bandgap material, thus the absorber can bemuch thinner [20].
If an electron is excited from Ei to E f , a void is createdat Ei . This void behaves like a particle with a positiveelementary charge and is called a hole. The absorption of a photon therefore leads to the creation of anelectron-hole pair, as illustrated in Fig. 3.2 ¶. The radiative energy of the photon is converted to the chemicalenergy of the electron-hole pair. The maximal conversion efficiency from radiative energy to chemical energy
is limited by thermodynamics. This thermodynamic limit
lies in between 67% for non-concentrated sunlight and
86% for fully concentrated sunlight [21].
The basic physics required for describing semiconductors is presented in chapter 6.
2. Subsequent separation of the photo-generated
charge carriers in the junction.

Usually, the electron-hole pair will recombine, i.e. theelectron will fall back to the initial energy level Ei , as illustrated in Fig. 3.2 ·. The energy will then be released
either as photon (radiative recombination) or transferred
to other electrons or holes or lattice vibrations (nonradiative recombination). If one wants to use the energy
stored in the electron-hole pair for performing work

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






electronsholesFigure 3.2: A very simple solar cell model. ¶ Absorption ofa photon leads to the generation of an electron-hole pair. ·Usually, the electrons and holes will combine. ¸ With semipermeable membranes the electrons and the holes can beseparated. ¹ The separated electrons can be used to drive anelectric circuit. º After the electrons passed through the circuit,they will recombine with holes.

in an external circuit, semipermeable membranes must bepresent on both sides of the absorber, such that electrons only can flow out through one membrane andholes only can flow out through the other membrane[21], as illustrated in Fig. 3.2 ¸. In most solar cells,
these membranes are formed by n- and p-type materials.
A solar cell has to be designed such that the electrons

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26

3. The Working Principle of a Solar Celland holes can reach the membranes before they recombine, i.e. the time it requires the charge carriers to reachthe membranes must be shorter than their lifetime. Thisrequirement limits the thickness of the absorber.

the maximal energy conversion efficiency of a singlejunction solar cell is considerably below the thermodynamic limit. This single bandgap limit was first calculatedby Shockley and Queisser in 1961 [23].

We will discuss generation and recombination of electrons and holes in detail in chapter 7.

A detailed overview of loss mechanisms and the resulting efficiency limits is discussed in Chapter 10.

3. Collection of the photo-generated charge carriers
at the terminals of the junction.

Finally, the charge carriers are extracted from the solar
cells with electrical contacts so that they can perform
work in an external circuit (Fig. 3.2 ¹). The chemicalenergy of the electron-hole pairs is finally converted
to electric energy. After the electrons passed throughthe circuit, they will recombine with holes at a metalabsorber interface, as illustrated in Fig. 3.2 º.
Loss mechanisms

The two most important loss mechanisms in single
bandgap solar cells are the inability to convert photons
with energies below the bandgap to electricity andthermalisation of photon energies exceeding the
bandgap, as illustrated in Fig. 3.1 (b). These two mechanisms alone amount to the loss of about half the incident solar energy in the conversion process [22]. Thus,

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Part II

PV Fundamentals

27
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4Electrodynamic basics
4.1

In this chapter we introduce the basics of electrodynamics that are required for solar cell physics. First, we introduce the electromagnetic wave equations. The existence of these equations explains the existence of electromagnetic waves, such as light. From there we develop
the equations describing the interaction of an electromagnetic wave with interfaces between two materials;
in this way we naturally derive the basics of optics.

The electromagnetic theory

While electricity and magnetism have been known sinceancient times, it took until the nineteenth century torealise that these two phenomena are two sides on one
medal, namely electromagnetism. We can easily see thisby recalling that electric fields are generated by charges
while magnetic fields are generated by currents, i.e.
moving charges. Let us now assume that we are within
an array of charges. Since charges create an electricfield, we will experience such an field. Now we startmoving with a constant velocity. This is equivalent tosaying that the array of charges moves with respect tous. Since moving charges are a current, we now experience an magnetic field. Thus, when changing from one

Later in the chapter we introduce the Poisson equationand the continuity equations that are very important for
semiconductor physics, which we discuss in chapter 6.
29
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30

frame of reference into another one that moves withrespect to the first one with a constant velocity, electric fields are transformed into magnetic fields and viceversa.
Between 1861 and 1862, the Scottish physicist James
Clerk Maxwell published works in that he managed
to formulate the complete electromagnetic theory by aset of equations, the Maxwell equations. A modern formulation of these equations is given in Appendix A.1.
The transformation of the electric and magnetic fields
between different frames of reference is correctly described by Albert Einstein’s theory of special relativity,
published in 1905.
One of the most important prediction of the Maxwell
equations is the presence of electromagnetic waves.
A derivation is given in Appendix A.2. Maxwell soon
realised that the speed of these waves is (within experimental accuracy) the same as the speed of light thatthen already was known. He brilliantly concluded thatlight is an electromagnetic wave.
In the 1880s the German physicist Heinrich Hertz could
experimentally confirm that electromagnetic waves canbe generated and have the same speed as light. His
work laid the foundation for modern radiocommunication that has shaped the modern world.

4. Electrodynamic basics

light is emitted and absorbed by matter. For this purpose, quantum mechanics is required.

4.2

Electromagnetic waves

As shown in Appendix A.2, electromagnetic waves aredescribed by
 2



∂
∂2
∂2
n2 ∂2 E++E−
=0
(4.1a)∂x2∂y2∂z2
c20 ∂t2
for the electric field E(r, t), where c0 denotes the speedof light in vacuo and n is the refractive index of thematerial. In a similar manner we can derive the wave
equation for the magnetic field,
 2



∂
∂2
∂2
n2 ∂2 H
+
+
H
−
= 0.
(4.1b)
∂x2
∂y2
∂z2
∂t2
c20
The simplest solution to the wave equations (4.1) is
the plane harmonic wave, where light of constantwavelength λ propagates in one direction. Without lossof generality, we assume that the wave travels along thez direction. The electric and magnetic fields in this caseareThe electromagnetic theory can perfectly describe how
light propagates. However, it fails in explaining how

E(r, t) = E0 eikz z−iωt ,(4.2a)

H(r, t) = H0 eikz z−iωt ,(4.2b)

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31

4.3. Optics of flat interfaces

where E0 and H0 are constant vectors (the amplitudes),k z is the wave number and ω is the angular frequency. By
substituting Eq. (4.2a) into Eq. (4.1a) we find that k z
and ω are connected to each other via
n2k2z − 2 ω 2 = 0.
c0

(4.3)

Thus,

nω
.
(4.4)
c
The angular frequency, measured in radians per second
is related to the frequency of the wave ν, measured in
Hertz, via
2πω = 2πν =,(4.5)
Twhere T = 1/ν is the period, measured in seconds. The
wave number k z has the unit of an inverse length. It isrelated to the wavelength λ via
kz =

2πc
c
2π
=
=
.
λ=
kz
nω
nν

k · E0 = k · H0 = H0 ·E0 = 0.

(4.7)

• The electric and magnetic fields are proportional
to the propagation direction, hence electromagnetic
waves are transverse waves.
• The electric and magnetic vectors have a constant,material dependent ratio. If the electric field isalong the x-direction and the magnetic field isalong the y-direction, this ratio is given by
Hy,0 =

nn
Ex,0 =
Ex,0 ,
cµ0
Z0

where

r
Z0 = cµ0 =µ0= 376.7 Ω
e0

(4.8)

(4.9)

is the impedance of free space.

(4.6)

We note that ν and ω are independent of n, while k andλ change when the wave travels to media with differentn.
Electromagnetic waves have some extraordinary properties that are derived in detail in Appendix A.3:
• The electric and magnetic field vectors are perpendicular to each other and also perpendicular to the

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propagation vector,

4.3

Optics of flat interfaces

In this section we repeat the major relations that describe an electromagnetic wave traversing an interface
between a medium 1 a medium 2, as illustrated in Fig.
4.1. We assume the two media to be non-absorptive.
Therefore, only the real parts of the refractive indices
are present. We denote them by n1 and n2 .

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32

4. Electrodynamic basics

A part of the incident light is reflected, where the angleof the scattered light θr is equal to the incident angle θi ,
θr = θi .

incident

reflected

The other part enters medium 2, where the angle of therefracted light θt is related to θi via Snell’s law,n1 sin θi = n2 sin θt ,

n1
n2

θi

θr

interfaceθttransmitted
Figure 4.1: Scheme of light, reflected and refracted by aboundary.

Copyright Delft University of Technology, 2014(4.10)(4.11)

Relations between the magnitudes of the incident, reflected and refracted fields are given by Fresnel equations. We have to distinguish between parallel and perpendicular polarised light. Parallel or perpendicular polarised means that the electric field is parallel or perpendicular to the plane, respectively. For perpendicular polarised light the Fresnel equations are given by
2 n1 cos θi
E0t=,(4.12a)ts =E0i s
n1 cos θi + n2 cos θt
E0rn cos θi − n2 cos θt
rs =
= 1
,
(4.12b)E0i s
n1 cos θi + n2 cos θtwhere the s stands for senkrecht, which is German for
perpendicular. The relations for parallel polarised lightare
2 n1 cos θi
E0t=,(4.13a)tp =E0i pn1 cos θt + n2 cos θi
E0r
n cos θt − n2 cos θi
rp =
= 1
.
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33

4.4. Optics in absorptive media

The intensities are proportional to the square of theelectric field, I ∝ E2 . For unpolarised light, we haveto take the mean values of the two polarisations. For
the reflectivity R we thus obtain
R=


1 2
rs + r2p .
2

For normal incidence this leads to
n1 − n2 2
R ( θ i = 0) =.
n1 + n2(4.14)(4.15)

Because of conservation of energy the sum of R and thetransmittance T must be 1,
R + T = 1.

(4.16)

By combining Eqs. (4.14) with (4.16) and doing some
calculations we findn2 cos θt 1  2(4.17)
ts + t2s ,T = 1−R =n1 cos θi 2
which leads toT ( θ i = 0) =4n1 n2( n1 + n2 )2(4.18)

for normal incidence.
A very important consequence of the Snell’s law is totalreflection. If n2 > n1 , there is a critical angle at which no

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light can leave the layer with n2 any more,sin θcrit =

n1
.
n2

(4.19)

Hence if θ2 ≥ θcrit , no light will be transmitted, buteverything will be reflected back into the layer. For asilicon-air interface (nSi ≈ 4.3), we find θcrit = 13.4◦ .
For the supporting layers used in solar cells, the critical angle is much larger. For a silicon-glass interface(nglass ≈ 1.5), we θcrit = 20.4◦ . And for an interface
between silicon and zinc oxide, which is a transparent
conducting oxide often used in solar cell technology,(nZnO ≈ 2) the critical angle would be θcrit = 30.3◦

4.4

Optics in absorptive media

Let us recap what we have seen that far in this chapter:
Starting from the Maxwell equations we derived thewave equations and looked at their properties for the
special case of plane waves. After that we looked at the
behaviour of electromagnetic waves at the interfaces
between new media. For the whole discussion so far
we implicitly assumed that the media is non-absorbing.
The working principle of solar cells is based on the factthat light is absorbed in an absorber material and thatthe absorbed light is used for exciting charge carriers
that can be used to drive an electric circuit. Therefore

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34

4. Electrodynamic basics

we will use this section to discuss how absorption oflight in a medium can be described mathematically.

Therefore we find for the attenuation of the intensity ofthe electromagnetic fieldIn general, the optical properties of an absorbing medium are described by an complex electric permittivity ẽ,I (z) = I0 exp(−2k00z z) = I0 exp(−αz),ẽ = e0 + ie00 .

where α is the absorption coefficient. Is is related to theother properties via(4.20)

Since the refractive index is given as the square root ofẽ, it also is complex,ñ =√ẽ = n + iκ,(4.21)

where κ denotes the imaginary part of the refractive
index. From Eq. (4.4) it becomes clear that in our casealso the wavenumber becomes complex,k̃ z =

nω
κω
ñω
=
+i
= k0z + ik00z .
ccc(4.22)

00

0(4.23)We thus see that the electric field is attenuated exponentially, exp(−k00z z) when travelling through the absorbing medium. The intensity of the electromagnetic
field is proportional to the square of the electric field,
I (r, t) ∝ |E(r, t)|2 .

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κω
4πκ
=
,
c
λ0

(4.26)

where λ0 = 2πc/ω is the wavelength in vacuo.
Equation (4.25) is known as the Lambert-Beer law. A
magnitude that is often used to judge the absorptivityof a material at a certain wavelength, is the penetration
depth δp ,1
(4.27)
δp = .
αAt this depth, the intensity has decayed to a fraction of
1/e of the initial value.

Let us now substitute Eq. (4.22) into Eq. (4.2a),Ex (z, t) = Ex,0 · eik̃z z−iωt = Ex,0 · e−kz z eikz z−iωt .

α = 2k00z = 2(4.25)(4.24)

In general, the complex refractive index and hence theabsorption coefficient are no material constants but varywith the frequency. Especially α may change across several orders of magnitude across the spectrum, making
the material very absorptive at one wavelength but almost transparent at other wavelength. Absorption spectra will be discussed thoroughly later on when looking
at various photovoltaic materials in Part III.

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35

4.5. Continuity and Poisson equations

4.5

Continuity and Poisson equations

At the end of this chapter we want to mention two
equations that are very important for our treatise of
semiconductor physics in Chapter 6.

4.5.1

By combining Eqs. (4.28) with (4.29) we find the Poisson
equation.

 2
∂2
∂2
ρ
∂
+
+
U=−
.
(4.30)∂x ∂y ∂zee0
In Chapter 6 we only will use the one-dimensional
form given by
∂2 U
ρ
=−
.
(4.31)∂zee0

Poisson equation

The first equation is the Poisson equation that relates thedensity of electric charges ρ(r) to the electrical potential
U (r). For its derivation, we start with the first Maxwell
equation (A.1a). Using Eq. (A.2a) we obtain
∂Ey
∂Ex
∂Ez
ρ
+
+
=
,
∂x
∂y
∂z
ee0

(4.28)

where Ex , Ey and Ez are the components of the electricfield vector, E = ( Ex , Ey , Ez ). Further, we here assume
that we are in an electrostatic situation, i.e. there are
no moving charges . From the second Maxwell equation (A.1b) we know that in that case the electric field
is rotation free. Vector calculus teaches us that then theelectric field is connected to the electric potential via
E=−∂U ∂U ∂U,,∂x ∂y ∂z

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.

(4.29)

4.5.2

Continuity equation

Charge is a conserved quantity. The total amount ofcharge inside a volume V only can be changed via
charges flowing through the boundary surface S of thisvolume. This can be expressed mathematically by theequationdQV {+J dS = 0,(4.32)
dt
S

where J is the current density vector and QV is the total
charge contained within the volume V. It is given by
QV =

y

ρ dV.

(4.33)

V

Equation (4.32) is the integral formulation of the continuity equation. It is equivalent to the differential for-This copy is provided for free, for personal use only.

36

mulation that is given by


∂Jy
∂Jx
∂Jz
∂ρ
+
+
+
= 0,
∂t
∂x
∂y
∂z

4. Electrodynamic basics(4.34)

where Jx , Jy and Jz are the components of the current
density vector, J = ( Jx , Jy , Jz ).

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5

Solar Radiation
5.1

The Sun• 2 positrons (the anti-particles of electrons),• 2 neutrinos,

We begin this chapter on Solar Radiation with a short
introduction about the Sun. The Sun is the central starof our solar system. It consists mainly of hydrogen andhelium. Some basic facts are summarised in table 5.1
and its structure is sketched in Fig. 5.1. The mass of the
Sun is so large that it contributes to 99.68% of the total
mass of the solar system. In the center of the Sun thepressure-temperature conditions are such that nuclear
fusion can take place. In the major nuclear reaction, theproton-proton reaction, via a number of steps four protons
react into

• electromagnetic radiation.
The positrons annihilate with electrons leading to additional radiation. The mass of the helium core is 0.635%
less than that of four protons, the energy difference isconverted into energy according to Einstein’s equation
E = mc2 .

(5.1)

Every second thus, approx, 4 million tons of mass areconverted into energy. However, the power density
at the center of the Sun is estimated by theoretical assumptions only to be about 275 W/m3 .

• a helium core (two protons and two neutrons),
37
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38

5. Solar Radiation

Internal structure:
core
radiative zone
convection zone

Subsurface flows
PhotosphereSun spots

Prominence

sphere. It has a temperature of about 6000 K. It behavesquite perfectly as a blackbody (see section 5.3) and is
the source of the solar radiation that hits the Earth. The
total power density of the solar radiation at the mean
Earth-sun distance on a plane perpendicular to the direction of the Sun, outside the Earth’s atmosphere, is referred to as the solar constant. Its value is approximately
1361 W/m2 .

FlareCoronal hole

Chromosphere
CoronaFigure 5.1: The Sun with its layer structure depicted [24].

The neutrinos hardly interact with matter and thus canleave the solar core without any hinder. Every second,
about 6.5 · 1010 per cm2 pass through the Earth andhence also through our bodies. Neutrinos carry about
2% of the total energy radiated by the Sun.
The remainder of the radiation is released as electromagnetic radiation. The core of the Sun is so densethat radiation cannot travel freely but is continuously
absorbed and re-emitted, such that it takes the radiation between 10 000 and 170 000 years to travel to thesolar surface. The surface of the Sun is called the photo-

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5.2

Radiometric properties

Radiometry is the branch of optics concerned with themeasurement of light. Since photovoltaics deals with sunlight that is converted into electricity it is very important to discuss how the ”amount of energy“ of the light
can be expressed physically and mathematically.
In solar science, not the total amount of the energy isimportant, but the amount of energy per unit time. We
thus will use the power that is given by
P=

dE
.
dt

(5.2)

For our discussion we assume a surface A that is irradiated by light, as illustrated in Fig. 5.2 (a). For obtaining the total power that is incident on the surface, wehave to integrate over the whole surface. Further weThis copy is provided for free, for personal use only.

39

5.2. Radiometric properties
Table 5.1: Some facts on the Sun

Mean distance from the Earth:
Diameter:Volume:Mass:
Density (at its center):Pressure (at its center):Temperature (at its center):Temperature (at the surface):
Energy radiation:
The Earth receives:

149 600 000 km (the astronomic unit, AU)
1 392 000 km (109 × that of the Earth)
1 300 000 × that of the Earth
1.993 × 1027 kg (332 000 times that of the Earth)>105 kg m−3 (over 100 times that of water)over 1 billion atmospheres
about 15 000 000 K
6 000 K
3.8 × 1026 W
1.7 × 1018 W

have to take into account that light is incident from all
the different directions, which we parameterise withthe spherical coordinates (θ, φ). The polar angle θ isdefined with respect to the normal of the surface element dA and φ is the azimuth, as sketched in Fig. 5.2(b). Thus, we have to integrate over the hemisphere,
from that light can be incident on the surface element
dA, as well. We therefore obtain
P=

Z Z
A 2π

Le cos θ dΩ dA.

(5.3)

The quantity Le is called the radiance and it is one of
the most fundamental radiative properties. Its physicaldimension is[ Le ] = W·m−2 ·sr−1 .
The factor cos θ expresses the fact that not the surface

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element dA itself is the relevant property but the projection of dA to the normal of the direction (θ, φ). Thisis also known as the Lambert cosine law.
We can express Eq. (5.3) as integrals of the surface coordinates (ξ, η ) and the direction coordinates (θ, φ),which reads as

P=

Z Z
A 2π

Le (ξ, η; θ, φ) cos θ sin θ dθ dφ dξ dη.

(5.4)

Since sunlight consists of of a spectrum of different frequencies (or wavelengths), it is useful to use spectral

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40

5. Solar Radiation

properties. These are given by
dP
,
dν
dLe
,
Leν =
dν

dP
,
dλ
dLe
Leλ =
,
dλ

Pν =

(a)

(b)

Pλ =

(5.5)
(5.6)

etc. Their physical dimensions are

A
dΩ
θ
dA
η
ξ

Figure 5.2: (a) Illustrating a surface A irradiated by light fromvarious directions and (b) a surface element dA that receives
radiation from a solid angle element dΩ under an angle θ withrespect to the surface normal.

[ Pν ] = W·Hz−1 = W·s,

[ Pλ ] = W·m−1 ,[ Leν ] = W·m−2 ·sr−1 ·s,[ Leλ ] = W·m−2 ·sr−1 ·m−1 ,Since wavelength and frequency are connected to eachother via νλ = c, Pν and Pλ are related via
 c 
dP dλ
dP
=
= Pλ · − 2 ,
(5.7)
Pν =
dν
dλ dν
ν
and similarly for Leν and Leλ . The − sign is because of
the changing direction of integration when switching
between ν and λ and usually is omitted.
The spectral power in wavelength thus can be obtained
via
Z Z
Pλ =
Leλ cos θ dΩ dA,
(5.8)
A 2π

and analogously for Pν . The radiance is given by
Le =

∂4 P
1
,
cos θ ∂A ∂Ω(5.9)

and similarly for Leν and Leλ .

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41

5.3. Blackbody radiation

Another very important radiometric property is the irradiance Ie that tells us the power density at a certain
point (ξ, η ) of the surface. It often also is called the(spectral) intensity of the light. It is given as the integral of the radiance over the solid angle,
Ie =

Z
2π

Le cos θ dΩ =Z
2π

Le (ξ, η; θ, φ) cos θ sin θ dθ dφ.

(5.10)The spectral irradiance Ieν or Ieλ is calculated similarly.
The physical dimensions are[ Ie ] = W·m−2 , [ Ieν ] = W·m−2 ·s, [ Ieλ ] = W·m−2 ·m−1 .
The irradiance also is given as
Ie =

∂2 P
,
∂A

(5.11)and similarly for Ieν and Ieλ . Irradiance refers to radiation, that is received by the surface. For radiation
emitted by the surface, we instead speak of radiant emittance, Me , Meν , and Meλ .
As we discussed earlier, the energy of a photon is proportional to its frequency, E ph = hν = hc/λ. Thus, thespectral power Pλ is proportional to the spectral photon
flow Nph, λ ,hc(5.12)Pλ = Nph, λ ,λand similarly for Pν and Nph, ν . The total photon flow N

Copyright Delft University of Technology, 2014is related to the spectral photon flow via
Nph =

Z ∞
0

Nph, ν dν =Z ∞0Nph, λ dλ.

(5.13)The physical dimensions of the (spectral) photon floware

[ Nph ] = s−1 ,[ Nph, ν ] = 1,

[ Nph, λ ] = s−1 ·m−1 .

The (spectral) photon flux Φph is defined as the photon
flow per area,∂2 NphΦph =,(5.14)∂Aand similarly for Φph, ν and Φph, λ . The physical dimensions are[Φph ] = s−1 m−2 , [Φph, ν ] = m−2 , [Φph, λ ] = s−1 m−2 ·m−1 .
By comparing Eqs. (5.11) and (5.14) and looking at Eq.
(5.12), we findhc
Ieλ = Φph, λ ,(5.15)
λand analogously for Ieν and Φph, ν .

5.3Blackbody radiation

If we take a piece of e. g. metal and start heating it up,
it will start to glow, first in reddish colour getting more

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42

and more yellowish if we increase temperature evenfurther. It thus emits electromagnetic radiation that wecall thermal radiation. Understanding this phenomenon
theoretically and correctly predicting the emitted spectrum was one of the most important topics of physics
in the late nineteenth century.
For discussing thermal radiation, the concept of theblackbody is very useful. A blackbody, which does notexist in nature, absorbs all the radiation that is incident
on it, regardless of wavelength and angle of incidence.
Its reflectivity therefore is 0. Of course, since it also will
emit light according to its equilibrium temperature, itdoes not need to appear black to the eye.
Two approximations for the blackbody spectrum were
presented around the turn of the century: First, in 1896,
Wilhelm Wien empirically derived the following expression for the spectral blackbody radiance:


C1
C2
W
Leλ (λ; T ) = 5 exp −,(5.16)λT
λwhere λ and T are the wavelength and the temperature,
respectively. While this approximation gives good results for short wavelengths, it fails to predict the emitted
spectrum at long wavelengths, thus in the infrared.
Secondly, in 1900 and in a more complete version in
1905, Lord Rayleigh and James Jeans, respectively, derived
2ck B TRJ
Leλ (λ; T ) =,(5.17)
λ4

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5. Solar Radiation

where k B ≈ 1.381 · 10−23 J/K is the Boltzmann constant.
The derivation of this equation was based on electroRJ
dynamic arguments. While Leλ is in good agreement
to measured values at long wavelengths, it diverges
to infinity for short wavelength. Further, the radiantemittance, which is obtained via integration over all
wavelength, diverges towards infinity. This so called
ultraviolet catastrophe demonstrates that Rayleigh andJeans did not succeed in developing a model that canadequately describe thermal radiation.
In 1900, Max Planck found an equation, that interpolates between the Wien approximation and the RayleighJeans law,BB
(λ; T ) =
Leλ

2hc2
1


,
λ5 exp hc − 1(5.18a)λk B T

where c ≈ 2.998 · 108 m/s is the speed of light in vacuo
and h ≈ 6.626 · 10−34 m2 kg/s is the nowadays called
Planck constant. Via Eq. (5.7) we find the Planck law expressed as a function of the frequency ν,BBLeν(ν; T ) =

2hν3
1


,
c2 exp hν − 1(5.18b)kB T

It is remarkable to see that the Planck law contains
three fundamental constants, c, k B , and h, which areamongst the most important constants in physics.

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43

5.3. Blackbody radiation

Figure 5.3 shows the spectrum of a blackbody of 6000
K temperature and the Wien approximation and theRayleigh-Jeans law. We indeed see that the Wien approximation fits well at short wavelengths, while theRayleigh-Jeans law matches well at long wavelengths
but completely fails at short wavelengths.
Both the Wien approximation [Eq. (5.16)] and theRayleigh-Jeans law [Eq. (5.17)] can be directly derived
from the Planck law:

Planck30000

For short wavelength,
20000





Spectral radiance W/ m2 · nm · sr



40000

exp

Rayleigh-Jeans
10000

0

500

1000



 1,

such that the −1 can be ignored and we arrive at the
Wien approximation with C1 = 2hc2 and C2 = hc/k B .

Wien
0

hc
λk B T

1500

2000

2500

Wavelength (nm)Figure 5.3: The blackbody spectrum at 6000 K as calculatedwith the Wien approximation, the Rayleigh-Jeans law and the
Planck law.

For long wavelength we can use the approximation


hc
hc
exp
−1 ≈
,
λk B T
λk B T
which directly results in the Rayleigh-Jeans law.
The total radiant emittance of a black body is given by
MeBB ( T ) =

Z ∞Z
2π

σ=

2π 5 k4B
J≈ 5.670 · 10−8
2
3
15c h
sm2 K4

where

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BBLeλ(λ; T ) cos θ sin θ dθ dφ dλ = σT 4 ,0(5.19)(5.20)

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44

5. Solar Radiation

λmax T = b ≈ 2.898 · 10

−3m · K.

(5.21)

Figure 5.4 shows the spectra for three different temperatures. Note the strong increase in radiance with temperature and also the shift of the maximum to shorter
wavelengths.

6000 K
30000

20000



Another important property of blackbody radiation
is Wien’s displacement law, which states that the
wavelength of maximal radiance is indirectly proportional to the temperature,

40000

Spectral radiance W/ m2 · nm · sr

is the Stefan-Boltzmann constant. Equation (20.5) is
known as the Stefan-Boltzmann law. As a matter of fact,
it already was discovered in 1879 and 1884 by Jožef
Stefan and Ludwig Boltzmann, respectively, i.e. about
twenty years prior to the derivation of Planck’s law.
This law is very important because it tells us that if
the temperature of a body (in K) is doubled, it emits
16 times as much power. Little temperature variations
thus have a large influence on the total emitted power.

5000 K
10000
4000 K
0

0

500

1000

1500

2000

2500

Wavelength (nm)Figure 5.4: The blackbody spectrum at three different temper-

atures5.4

Wave-particle duality

In Planck’s law, as stated in Eqs. (5.18), the constant
h appeared for the first time. Its product with the frequency, hν = hc/λ has the unit of an energy. Planck

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45

5.5. Solar spectra

himself did not see the implications of h. It was Einstein, who understood in 1905 that Planck’s law actually has to be interpreted such that light comes inquanta of energy with the size

5.5

Solar spectra

Nowadays, these quanta are called photons. In termsof classical mechanics we could say that light shows the
behaviour of particles.

As we already mentioned in chapter 3, only photons ofappropriate energy can be absorbed and hence generate
electron-hole pairs in a semiconductor material. Therefore, it is important to know the spectral distribution of
the solar radiation, i.e. the number of photons of a particular energy as a function of the wavelength λ. Twoquantities are used to describe the solar radiation spectrum, namely the spectral irradiance Ieλ and the spectral
photon flux Φph (λ). We defined these quantities alreadyin section 5.2.

On the other hand, we have seen in Chapter 4 that lightalso shows wave character which becomes obvious whenlooking at the propagation of light through space or atreflection and refraction at a flat interface. It also was
discovered that other particles, such as electrons, show
wave-like properties.

The surface temperature of the Sun is about 6000 K. If
it would be a perfect black body, it would emit a spectrum as described by Eqs. (5.18), which give the spectral radiance. For calculating the spectral irradiance a
blackbody with the size and position of the Sun would
have on Earth, we have to multiply the spectral radiance with the solid angle of the Sun as seen from Earth,

Eph = hν.

(5.22)

BBBB( T; λ)ΩSun .
Ieλ( T; λ) = LeλThis behaviour is called wave-particle duality and is avery intriguing property of quantum mechanics that was
discovered and developed in the first quarter of thetwentieth century. Many discussion was held on how
this duality has to be interpreted - but this is out of the
focus of this book. So we just will accept that depending on the situation light might behave as wave or as
particle.

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We can calculate ΩSun with

2
RSun
.
ΩSun = π
AU − REarth(5.23)(5.24)

Using RSun = 696 000 km, an astronomical unit AU =149 600 000 km, and REarth = 6370 km, we findΩSun ≈ 68 µsr.

(5.25)

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46

5. Solar Radiation

The blackbody spectrum is illustrated in Fig. 5.5. The
spectrum outside the atmosphere of the Earth is alreadyvery different. It is called the AM0 spectrum, becauseno (or “zero”) atmosphere is traversed. AM0 also isshown in Fig. 5.5. The irradiance at AM0 is Ie (AM0) =1361 W·m−2 .

AM :=

1
.
cos θ



6000 K blackbody radiation

2.0

AM0 radiation
AM1.5 radiation



1.5
1.0
0.5
0.0

0

500

1000

1500

2000

2500

Wavelength (nm)(5.26)For example, when the Sun is 60° from the zenith, i.e.
30° above the horizon, we receive an AM2 spectrum.
Depending on the position on the Earth and the position of the Sun in the sky, terrestrial solar radiation varies both in intensity and the spectral distribution. The
attenuation of solar radiation is due to scattering andabsorption by air molecules, dust particles and/or aer-

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2.5
Spectral irradiance W/ m2 nm

When solar radiation passes through the atmosphereof the Earth, it is attenuated. The most important parameter that determines the solar irradiance under clear
sky conditions is the distance that the sunlight has totravel through the atmosphere. This distance is theshortest when the Sun is at the zenith, i.e. directly overhead. The ratio of an actual path length of the sunlight to this minimal distance is known as the optical
air mass. When the Sun is at its zenith the optical air
mass is unity and the spectrum is called the air mass
1 (AM1) spectrum. When the Sun is at an angle θ withthe zenith, the air mass is given by

Figure 5.5: Different solar spectra: the blackbody spectrum ofa blackbody at 6000 K, the extraterrestrial AM0 spectrum andthe AM1.5 spectrum.

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5.5. Solar spectra

osols in the atmosphere. Especially, steam (H2 O), oxygen (O2 ) and carbon dioxide (CO2 ) cause absorption.
Since this absorption is wavelength-selective, it resultsin gaps in the spectral distribution of solar radiation
as apparent in Fig. 5.5. Ozone absorbs radiation withwavelengths below 300 nm. Depletion of ozone fromthe atmosphere allows more ultra-violet radiation toreach the Earth, with consequent harmful effects upon
biological systems. CO2 molecules contribute to the absorption of solar radiation at wavelengths above 1 µm.
By changing the CO2 content in the atmosphere the absorption in the infrared, which has consequences forour climate.
Solar cells and photovoltaic modules are produced bymany different companies and laboratories. Further,many different solar cell technologies are investigatedand sold. It is therefore of utmost importance to definea reference solar spectrum that allows a comparison of all
the different solar cells and PV modules. The industrial
standard is the AM1.5 spectrum, which corresponds toan angle of 48.2°. While the “real” AM1.5 spectrum corresponds to a total irradiance of 827 W·m−2 , the industrial standard corresponds to Ie (AM1.5) = 1000 W·m−2
and is close to the maximum received at the surface of
the Earth. The power generated by a PV module under
this conditions is thus expressed in the unit Watt peak,
Wp.

47

ticular place on the Earth is extremely variable. In addition to the regular daily and annual variation due tothe apparent motion of the Sun, irregular variations
have to be taken into account that are caused by local
atmospheric conditions, such as clouds. These conditions particularly influence the direct and diffuse components of solar radiation. The direct component ofsolar radiation is that part of the sunlight that directly
reaches the surface. Scattering of the sunlight in the atmosphere generates the diffuse component. A part ofthe solar radiation that is reflected by the Earth’s surface, which is called albedo, may be also present in the
total solar radiation. We use a term global radiation torefer to the total solar radiation, which is made up ofthese three components.
The design of an optimal photovoltaic system for a particular location depends on the availability of the solar
insolation data at the location. Solar irradiance integrated over a period of time is called solar irradiation.
For example, the average annual solar irradiation in the
Netherlands is 1 000 kWh/m2 , while in Sahara the average value is 2 200 kWh/m2 , thus more than twice as
high. We will disuss these issues in more detail in in
Chapter 15.

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48

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5. Solar Radiation

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6

Basic Semiconductor Physics
6.1

Introduction

are relevant to solar cell operation. This discussion will
give us a basic understanding of how solar cells basedon other semiconductor materials work.

With this chapter we start with the discussion of some
important concepts from semiconductor physics, which
are required to understand the operation of solar cells.
After giving a brief introduction into semiconductor
physics in this chapter, we will discuss the most important generation and recombination mechanisms in
Chapter 7. Finally, we will focus on the physics of semiconductor junctions in Chapter 8.

The central semiconductor parameters that determine thedesign and performance of a solar cell are:
1. Concentrations of doping atoms, which can be of
two different types: donor atoms, which donate freeelectrons or acceptor atoms, which accept electrons.
The concentrations of donor and acceptor atoms
are denoted by ND and NA , respectively, and determine the width of the space-charge region of ajunction, as we will see in Chapter 8.

The first successful solar cell was made from crystalline
silicon (c-Si), which still is by far the most widely used
PV material. Therefore we shall use c-Si as an example
to explain the concepts of semiconductor physics that

2. The mobility µ and the diffusion coefficient D of
charge carriers is used to characterise the transport
49

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50
(a)

6. Basic Semiconductor Physics(b)

Figure 6.1: (a) A diamond lattice unit cell representing a unit

cell of single crystal Si [25], (b) the atomic structure of a part ofsingle crystal Si.

of carriers transport due to drift and diffusion, respectively, which we will discuss in Section 6.5.

3. The lifetime τ and the diffusion length L of theexcess carriers characterise the recombinationgeneration processes, discussed in Chapter 7.

4. The band gap energy Eg , and the complex refractive index n − ik, where k is linked to the absorption
coefficient α, characterise the ability of a semiconductor to absorb electromagnetic radiation.

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6.2

Atomic structure

The atomic number of silicon is 14, which means that are
14 electrons orbiting the nucleus. In ground state configuration, two electrons are in the first shell, both living in the 1s orbital. Further, eight electrons are in thesecond shell, two in the 2s and six in the 2p orbitals.
Hence, four electrons live in the third shell, which is
the outermost shell for a Si atom. Only these four electrons interact with other atoms, for example via forming chemical bonds. They are called the valence electrons.
Two Si atoms are bonded together when they shareeach other’s valence electron. This is the so called covalent bond that is formed by two electrons. Since Si atoms
have four valence electrons, they can be covalently bonded to four other Si atoms. In the crystalline form each
Si atom is covalently bonded to four neighbouring Si
atoms, as illustrated in Fig. 6.1.
In the ground state, two valence electrons live in the
3s orbital and the other two are present in the three
3p orbitals (px , py and pz ). In that state only the twoelectrons in the 3p orbitals can form bond as the 3s orbital is full. In a silicon crystal, where every atom issymmetrically connected to four others, the Si atoms arepresent as so-called sp3 hybrids. The 3p and 3s orbitals
are mixed forming 4 sp3 orbitals. Every of these fourorbitals is occupied by one electron that can form a covalent bond with a valence electron from a neighbour-

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51

6.2. Atomic structure

ing atom.
All bonds have the same length and the angles between
the bonds are equal with 109.5°. The number of bondsthat an atom has with its immediate neighbours in theatomic structure is called the coordination number or coordination. Thus, in single crystal silicon, the coordination number for all Si atoms is four, we can also saythat Si atoms are fourfold coordinated. A unit cell canbe defined, from which the crystal lattice can be reproduced by duplicating the unit cell and stacking the duplicates next to each other. Such a regular atomic arrangement is described as a structure with long rangeorder.
A diamond lattice unit cell represents the real lattice structure of monocrystalline silicon. Figure 6.1 (a)shows the arrangement of the unit cell and Fig. 6.1 (b)the atomic structure of single crystal silicon. One candetermine from Fig. 6.1 (a) that there are eight Si atoms
in the volume of the unit cell. When a lattice constantof c-Si is 5.4 Å one can easily calculate that the densityof atoms is approximately 5 × 1022 cm−3 . Figure 6.1 (a)shows the crystalline Si atomic structure with no foreign atoms. In practice, a semiconductor sample always
contains some impurity atoms. When the concentrationof impurity atoms in a semiconductor is insignificant
we refer to such semiconductor as an intrinsic semiconductor.
At practical operational conditions, e.g. at room tem-

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perature,1 there are always some of the covalent bonds
broken. The breaking of the bonds results in liberating
the valence electrons from the bonds and making themmobile through the crystal lattice. We refer to theseelectrons as free electrons (henceforth simply referred
to as electrons). The position of a missing electron in abond, which can be regarded as positively charged, isreferred to as a hole. This situation can be easily visualised by using the bonding model illustrated in Fig. 6.2.
In the bonding model the atomic cores (atoms withoutvalence electrons) are represented by circles and thevalence or bonding electrons are represented by lines
interconnecting the circles. In case of c-Si one Si atom
has four valence electrons and four nearest neighbours.
Each of the valence electron is equally shared with thenearest neighbour. There are therefore eight lines terminating on each circle. In an ideal Si crystal at 0 K all
valence electrons take part in forming covalent bonds
between Si atoms and therefore no free electrons arepresent in the lattice. This situation is schematicallyshown in Fig. 6.2 (a).
At temperatures higher than 0 K the bonds start tobreak due to absorbing thermal energy. This process
results in creation of mobile electrons and holes. Figure 6.2 (b) shows a situation when a covalent bond isbroken and one electron departs from the bond leav1 In semiconductor physics most of the time a temperature of 300 K is assumed.

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52

6. Basic Semiconductor Physics

ing a so-called hole behind. A single line between theatoms in Fig. 6.2 (b) represents the remaining electronof the broken bond. When a bond is broken and a holecreated, a valence electron from a neighbouring bondcan “jump” into this empty position and restore thebond. The consequence of this transfer is that at thesame time the jumping electron creates an empty position in its original bond. The subsequent “jumps” of avalence electron can be viewed as a motion of the hole,a positive charge representing the empty position, inthe direction opposite to the motion of the valence electron through the bonds.
Since breaking of a covalent bond leads to the formation of an electron-hole pair, in intrinsic semiconductors
the concentration of electrons is equal to the concentration of holes. In intrinsic silicon at 300 K approximately
1.5 × 1010 cm−3 broken bonds are present. This number
then gives also the concentration of holes, p, and electrons, n. Hence, at 300 K, n = p = 1.5 × 1010 cm−3 . This
concentration is called the intrinsic carrier concentrationand is denoted as ni .

(a)

(b)

... nucleus + core electrons... valence electron... hole... “free electron”... electron bonds

Figure 6.2: The bonding model for c-Si. (a) No bonds arebroken. (b) A bond between two Si atoms is broken resulting ina free electron and hole.

6.3

Doping

The concentrations of electrons and holes in c-Si canbe manipulated by doping. Doping of silicon meansthat atoms of other elements substitute Si atoms in the

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53

6.3. Doping
(a)

(b)

P+

... nucleus + core electrons... valence electron
P+ ... ionised P4+−
B− ... ionised B4

B−... hole... “free” electron... electron bonds

Fig. 6.3. When introducing phosphorus atom into the cSi lattice, four of the five phosphorus atom valence electrons will readily form bonds with the four neighbouring Si atoms. The fifth valence electron cannot take partin forming a bond and becomes rather weakly boundto the phosphorus atom. It is easily liberated from thephosphorus atom by absorbing the thermal energy,
which is available in the c-Si lattice at room temperature. Once free, the electron can move throughout thelattice. In this way the phosphorus atom that substitutes a Si atom in the lattice “donates” a free (mobile)electron into the c-Si lattice. The impurity atoms thatenhance the concentration of electrons are called donors.
We denote the concentration of donors by ND .

crystal lattice. The substitution has to be carried out byatoms with three or five valence electrons, respectively.
The most used elements to dope c-Si are boron (B) andphosphorus (P), with atomic numbers of 5 and 15, respectively.

An atom with three valence electrons such as boron
cannot form all bonds with four neighbouring Si atoms
when it substitutes a Si atom in the lattice. However,it can readily “accept” an electron from a nearby Si-Si
bond. A thermal energy that the c-Si lattice contains
at room temperature is sufficient to enable an electronfrom a nearby Si-Si bond to attach itself to the boron
atom and complete the bonding to the four Si neighbours. In this process a hole is created that can movearound the lattice. The impurity atoms that enhance theconcentration of holes are called acceptors. We denotethe concentration of acceptors by NA .

The process of doping action can best be understoodwith the aid of the bonding model and is illustrated in

Note that by substituting Si atoms with only one typeof impurity atoms, the concentration of only one typeFigure 6.3: The doping process illustrated using the bonding

model. (a) A phosphorus (P) atom substitutes a Si atom in thelattice resulting in the positively-ionised P atom and a free electron, (b) A boron (B) atom substitutes a Si atom resulting in thenegatively ionised B atom and a hole.

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54

of mobile charge carriers is increased. Charge neutrality of the material is nevertheless maintained because
the sites of the bonded and thus fixed impurity atoms
become charged. The donor atoms become positively
ionised and the acceptor atoms become negatively ionised.

6. Basic Semiconductor Physics
Low doping

1012

Moderate doping

Heavy doping

1014
1016
1018
Dopant concentration (cm-3)

1020

Figure 6.4: The range of doping levels used in c-Si.

The possibility to control the electrical conductivityof a semiconductor by doping is one of most important semiconductor features. The electrical conductivity in semiconductors depends on the concentration ofelectrons and holes as well as their mobility. The concentration of electrons and holes is influenced by theamount of the impurity atoms that are introduced into
the atomic structure of semiconductor. Figure 6.4 shows
the range of doping that is used in case of c-Si. We denote a semiconductor as p-type or n-type when holes
or electrons, respectively, dominate its electrical conductivity. In case that one type of charge carriers has
a higher concentration than the other type these carriers are called majority carriers (holes in the p-type andelectrons in the n-type), while the other type with lower
concentration are then called minority carriers (electrons
in the p-type and holes in the n-type).

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6.4

Carrier concentrations

6.4.1

Intrinsic semiconductors

Any operation of a semiconductor device depends onthe concentration of carriers that transport charge inside
the semiconductor and hence cause electrical currents.
In order to determine and to understand device operation it is important to know the precise concentration ofthese charge carriers. In this section the concentrationsof charge carriers inside a semiconductor are derived
assuming the semiconductor is under equilibrium. The
term equilibrium is used to describe the unperturbed
state of a system, to which no external voltage, magnetic field, illumination, mechanical stress, or otherperturbing forces are applied. In the equilibrium state
the observable parameters of a semiconductor do notchange with time.

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55

6.4. Carrier concentrations(b)electron energy(a)conduction band

EC

EC

EV

EV

Eg
valence bandx

Figure 6.5: (a) The basic energy band diagram and (b) widelyused simplified version of the energy diagram.

In order to determine the carrier concentration one
has to know the function of density of allowed energy states of electrons and the occupation function of
the allowed energy states. The density of energy statesfunction, g( E), describes the number of allowed states
per unit volume and energy. The occupation functionis the well known Fermi-Dirac distribution function,f ( E), which describes the ratio of states filled with anelectron to total allowed states at given energy E. In
an isolated Si atom, electrons are allowed to have only
discrete energy values. The periodic atomic structureof single crystal silicon results in the ranges of allowed
energy states for electrons that are called energy bands
and the excluded energy ranges, forbidden gaps or bandgaps. Electrons that are liberated from the bonds determine the charge transport in a semiconductor. There-

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fore, we further discuss only those bands of energy
levels, which concern the valence electrons. Valence
electrons, which are involved in the covalent bonds,
have their allowed energies in the valence band (VB) andthe allowed energies of electrons liberated from the covalent bonds form the conduction band (CB). The valenceband is separated from the conduction band by a bandof forbidden energy levels. The maximum attainable
valence-band energy is denoted EV , and the minimum
attainable conduction-band energy is denoted EC . Theenergy difference between the edges of these two bands
is called the band gap energy or band gap, Eg , and it is
an important material parameter.
EG = EC − EV .

(6.1)

At room temperature (300 K), the band gap of crystalline silicon is 1.12 eV.2 A plot of the allowed electronenergy states as a function of position is called the energy band diagram and for intrinsic c-Si is shown in
Fig. 6.5.
The density of energy states at an energy E in the conduction band close to EC and in the valence band close

2 An electron-volt is equal to the energy, which an electron acquires when it
passes through a potential of 1 volt in vacuum (1eV = 1.602× 10-19 J).

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56

6. Basic Semiconductor Physics

to EV are given bygc ( E ) =gv ( E ) =

!3
√
2 p
4 2πm∗n
E − EC ,h3√!3
4 2πm∗p t 2 p
E − EV ,h3(6.2a)(6.2b)

where m∗n and m∗p is the effective mass of electrons andholes, respectively. As the electrons and holes move inthe periodic potential of the c-Si crystal, the mass has
to be replaced by the effective mass, which takes the effect of a periodic force into account. The effective massis also averaged over different directions to take anisotropy into account.
The Fermi-Dirac distribution function is given byf ( E) =1,EF
1 + exp E−
kT

(6.3)

where k is Boltzmann’s constant (k = 1.38 × 10−23 J/K)
and EF is the so-called Fermi energy. kT is the thermalenergy, at 300 K it is 0.0258 eV. The Fermi energy is theelectrochemical potential of the electrons in a material
and in this way it represents the averaged energy ofelectrons in the material. The Fermi-Dirac distributionfunction for different temperatures is shown in Fig. 6.6.
The carriers that contribute to charge transport are electrons in the conduction band and holes in the valence

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band. The total concentration of electrons in the conduction band and the total concentration of holes inthe valence band is obtained by multiplying density ofstates function with the distribution function and integrating across the whole energy band,n=p=

Z EtopEC
Z EV

gc ( E) f ( E)dE,(6.4a)

gv ( E) [1 − f ( E)] dE,(6.4b)

Ebottom

Substituting the density of states and the Fermi-Dirac
distribution function into Eq. (6.4) the resulting expressions for n and p are obtained after solving the equations. The full derivation can be found for example inReference [20].
EF − ECn = NC exp
for EC − EF ≥ 3kT, (6.5a)kTEV − EF
p = NV exp
for EF − EV ≥ 3kT, (6.5b)kT
where NC and NV are the effective densities of the conduction band states and the valence band states, respectively. For crystalline silicon, we have at 300 K
NC = 3.22 × 1019 cm−3 ,(6.6a)

NV = 1.83 × 1019 cm−3 .

(6.6b)

When the requirement that the Fermi level lies in theband gap more than 3kT from either band edge is sat-

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57

6.4. Carrier concentrations

(a)

(b)

(c)

(d)

E

E

E

E

T=0K

T1 > 0 K

T2 > T1

EF

0

0.5

1 f (E )

EC
EF
EV

EF

EF

0

0.5

1 f (E )00.5

1 f (E )00.5

1 f (E )Figure 6.6: The Fermi-Dirac distribution function. (a) For T = 0 K, all allowed states bellow the Fermi level are occupied by twoelectrons. (b, c) At T > 0 K not all states below the Fermi level are occupied and there are some states above the Fermi level that areoccupied. (d) In an energy gap between bands no electrons are present.

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58

6. Basic Semiconductor Physics

isfied the semiconductor is referred to as anondegenerate
semiconductor.

6.4.2

If an intrinsic semiconductor is in equilibrium, we haven = p = ni . By multiplying the corresponding sides ofEqs. (6.5) we obtainEV − EC2np = ni = NC NV exp
kT


(6.7)
Eg
= NC NV exp −,kT

It has been already mentioned in Section 6.3 that theconcentrations of electrons and holes in c-Si can be manipulated by doping. The concentration of electrons
and holes is influenced by the amount of the impurity
atoms that substitute silicon atoms in the lattice. Underassumption that the semiconductor is uniformly doped
and in equilibrium a simple relationship between the
carrier and dopant concentrations can be established.
We assume that at room temperature the dopant atoms
are ionised. Inside a semiconductor the local charge
density is given by+ρ = e p + ND
− n − NA− ,(6.10)

which is independent of the position of the Fermi level
and thus valid for doped semiconductors as well. When
we denote the position of the Fermi level in the intrinsic material EFi we may writeEFi − ECni = NC exp
kT


(6.8)
EV − EFi= NV exp.
kT
From Eq. (6.8) we can easily find the position of EFi tobe
EC + EV
kT
NV
EFi =
+
ln
2
2
NC


(6.9)
Eg
kT
NV
= EC −
+
ln
.
2
2
NC
The Fermi level EFi lies close to the midgap [( EC +EV )/2]; a slight shift is caused by the difference in theeffective densities of the valence and conduction band.

Copyright Delft University of Technology, 2014

Doped semiconductors

where e is the elementary charge (e = 1.602 × 10−19+−
C). NDand NAdenote the density of the ionised donor
and acceptor atoms, respectively. As every ionised atom+−
corresponds to a free atom (hole), ND
and NAtell usthe concentration of electrons and holes due to doping,respectively.
Under equilibrium conditions, the local charge of theuniformly doped semiconductor is zero, which meansthat the semiconductor is charge-neutral everywhere.
We thus can write:+
p + ND− n − NA− = 0.

(6.11)As previously discussed, the thermal energy available
at room temperature is sufficient to ionise almost all theThis copy is provided for free, for personal use only.

59

6.4. Carrier concentrations

assumption that NA = 0 , Eq. (6.13) becomes

dopant atoms. We therefore may assume
+
ND
≈ ND

and

−
NA
≈ NA ,(6.12)

p − n − NA = 0.

(6.17)

Under the assumption thatand hencep + ND − n − NA = 0,

(6.13)

which is the common form of the charge neutrality equation.
Let us now consider an n-type material. At room temperature almost all donor atoms ND are ionised anddonate an electron into the conduction band. Under theassumption that NA = 0 , Eq. (6.13) becomesp + ND − n = 0,

(6.14)(6.15)

we can expect that the concentration of holes is lower
than that of electrons, and becomes very low when ND
becomes very large. We can calculate more accurate theconcentration of holes in the n-type material from Eq.
(6.7).
n2n2
p = i ≈ i  n.
(6.16)n
ND
In case of a p-type material almost all acceptor atoms
NA are ionised at room temperature and accept an electron and leaving a hole in the valence band. Under the

Copyright Delft University of Technology, 2014(6.18)

we can expect that the concentration of electrons islower than that of holes. We can calculate more accurate the concentration of electrons in the p-type material
from Eq. (6.7).
n=Under the assumption that+
ND ≈ ND≈n−NA ≈ NA≈pn2i
n2≈ i  p.
p
NA(6.19)

Inserting donor and acceptor atoms into the lattice of
crystalline silicon introduces allowed energy levels into
the forbidden bandgap. For example, the fifth valence
electron of the P atom does not take part in forming abond, is rather weakly bound to the atom and easily
liberates from the P atom. The energy of the liberated
electron lies in the CB. The energy levels, which we denote ED , of the weakly-bound valence electrons of thedonor atoms have to be positioned close the CB. Noticethat a dashed line represents the ED . This means that
an electron that occupies the ED level, is localised to thevicinity of the donor atom.
Similarly, the acceptor atoms introduce allowed energy
levels E A close to the VB. Doping also influences theThis copy is provided for free, for personal use only.

60

6. Basic Semiconductor Physics

(a)

(b)

(c)
EC

EC

ED

EF

EC

EF = EFi
EF
EV

EV
intrinsic

EA

EV
n-typep-typeFigure 6.7: A shift of the position of the Fermi energy in the band diagram and the introduction of the allowed energy level into thebandgap due to the doping.

position of the Fermi energy. When increasing the concentration of electrons by increasing the concentrationof donors the Fermi energy will increase, which is represented by bringing the Fermi energy close to the CB
in the band diagram. In the p-type material the Fermi
energy is moving closer the VB. A change in the Fermienergy position and the introduction of the allowed energy level into the bandgap due to the doping is illustrated Fig. 6.7.

The position of the Fermi level in an n-type semiconductor can be calculated using Eqs. (6.5a) and (6.15)
and in a p-type semiconductor using Eqs. (6.5b) and

Copyright Delft University of Technology, 2014(6.18), respectively,



NC
ND


NV
EF − EV = kT ln
NA
EC − EF = kT ln

6.5

for n-type,(6.20a)for p-type.

(6.20b)Transport properties

In contrast to the equilibrium conditions, under operational conditions a net electrical current flows through asemiconductor device. The electrical currents are generated in a semiconductor due to the transport of charge

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61

6.5. Transport properties

ExampleThis example demonstrates how much the concentration of electrons and holes can be manipulated by doping. A c-Si wafer is uniformly
doped with 1 × 1017 cm−3 P atoms. P atoms act as donors and therefore at room temperature the concentration of electrons is almost equal
to the concentration of donor atoms:+
n = ND≈ ND = 1017 cm−3 .
The concentration of holes in the n-type material is calculated from Eq. (6.15),
n2i
1.5 × 1010p=
= 2.22 × 103 cm−3 .
=n1017
We notice that there is a difference of 14 orders between n (1017 cm−3 ) and p (2.25 × 103 cm−3 ). It is now obvious why electrons in
n-type materials are called the majority carriers and holes the minority carriers. We can calculate the change in the Fermi energy due to thedoping. Let us assume that the reference energy level is the bottom of the conduction band, EC = 0 eV. Using Eq. (6.9) we calculate theFermi energy in the intrinsic c-Si.
!


Eg
kT
NV
1.12 0.0258
1.83 × 1019
EFi = EC −
+
ln
=−
+
ln
= −0.57 eV.
2
2
NC
2
2
3.22 × 1019
The Fermi energy in the n-type doped c-Si wafer is calculated from Eq. (6.5a)
EF = EC + kT ln

n
NC



= 0.0258 × ln

1017
3.22 × 1019!

= −0.15 eV.

We notice that the doping with P atoms has resulted in the shift of the Fermi energy towards the CB. Note that when n > NC , EF > EC
and the Fermi energy lies in the CB.

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62

6. Basic Semiconductor Physics

by electrons and holes. The two basic transport mechanisms in a semiconductor are drift and diffusion.

6.5.1

drift current is in the same direction as the electric field.
This is illustrated in Fig. 6.8.
The electron and hole drift-current densities are thengiven as

Drift

Drift is charged-particle motion in response to anelectric field. In an electric field the force acts on the
charged particles in a semiconductor, which accelerates
the positively charged holes in the direction of the electric field and the negatively charged electrons in the
direction opposite to the electric field. Because of collisions with the thermally vibrating lattice atoms andionised impurity atoms, the carrier acceleration is frequently disturbed. The resulting motion of electrons
and holes can be described by average drift velocities
vdn and vdp for electrons and holes, respectively. In caseof low electric fields, the average drift velocities are directly proportional to the electric field as expressed by
vdn = −µn ξ,
vdp = µ p ξ.

(6.21a)(6.21b)

The proportionality factor is called mobility µ. It is acentral parameter that characterises electron and holetransport due to drift. Although the electrons move inthe opposite direction to the electric field, because thecharge of an electron is negative the resulting electron

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Jn, drift = −envdn = enµn ξ,J p, drift = epvdp = epµ p ξ.

(6.22a)(6.22b)

Combining Eqs. (6.22a) and (6.22b) leads to the total
drift current,
Jdrift = e( pµ p + nµn )ξ.
(6.23)
Mobility is a measure of how easily the charge particles
can move through a semiconductor material. For example, for c-Si with a doping concentration ND or NA ,respectively, at 300 K, the mobilities areµn ≈ 1 360 cm2 V−1 s−1 ,µ p ≈ 450 cm2 V−1 s−1 .
As mentioned earlier, the motion of charged carriers isfrequently disturbed by collisions. When the number ofcollisions increases, the mobility decreases. Increasing
the temperature increases the collision rate of charged
carriers with the vibrating lattice atoms, which results
in a lower mobility. Increasing the doping concentration of donors or acceptors leads to more frequent collisions with the ionised dopant atoms, which results in alower mobility as well. The dependence of mobility on

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63

6.5. Transport properties

doping and temperature is in more detailed discussed
in standard textbooks for semiconductor physics anddevices, such as Reference [20].
(a)

(b)

6.5.2

ElectronElectronECelectric field ξelectric field ξ
EV

Hole

Hole

Diﬀusion

Diffusion is a process whereby particles tend to spreadout from regions of high particle concentration into regions of low particle concentration as a result of random thermal motion. The driving force of diffusion isa gradient in the particle concentration. In contrast to
the drift transport mechanism, the particles need not becharged to be involved in the diffusion process. Currents resulting from diffusion are proportional to thegradient in particle concentration. For electrons andholes, they are given byJn, diff = eDn ∇n ,J p, diff = −eD p ∇ p ,(6.24a)(6.24b)

Combining Eqs. (6.24a) and (6.24b) leads to the total
diffusion current,
Figure 6.8: Visualisation of (a) the direction of carrier fluxes

due to an electric field and (b) the corresponding band diagram.

Jdiff = e( Dn ∇n − D p ∇ p).

(6.25)The proportionality constants, Dn and D p are called theelectron and hole diffusion coefficients, respectively. The
diffusion coefficients of electrons and holes are linked
with the mobilities of the corresponding charge carriers

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64

6. Basic Semiconductor Physics

ExampleTo obtain some idea about values of diffusion coefficients, let us
assume an c-Si wafer at room temperature, doped with donors,
ND = 1014 cm−3 . According to Eq. (6.26),kTµn = 0.0258 V × 1360 cm2 V−1 s−1 = 35 cm2 s−1 .
DN =econtinuity equations for the electrons and holes can beformulated. We have to take drift, diffusion, and recombination as well as generation processes into account;the latter are discussed in Chapter 7.
For electrons, the rate with that the concentration
changes is given by
∂n
∂n
∂n
∂n
∂n
=
+
+
+
,
∂t
∂t drift∂t diff∂t R∂t G

where R and G denotes recombination and generation,
respectively. For holes, we obtainby the Einstein relationship that is given by
Dp
Dn
kT
=
=
.
µn
µp
e

(6.26)

Figure 6.9 visualises the diffusion process as well as theresulting directions of particle fluxes and current.
Combining Eqs. (6.23) and (6.25) leads to the total current,
J = Jdrift + Jdiff= e( pµ p + nµn )ξ + e( Dn ∇n − D p ∇ p).

6.5.3(6.27)

Continuity equations

As we have discussed in Section 4.5, charge is a conserved quantity. Now we will have a look at how the

Copyright Delft University of Technology, 2014(6.28a)

∂p
∂p
∂p
∂p
∂p
=
+
+
+
.
∂t
∂t drift∂t diff∂t R∂t G(6.28b)In these equations ∂n/∂t (∂p/∂t) is the time ratechange in the electron (hole) concentration
Without any generation or recombination, the numberof electrons and holes is conserved, i.e. Eq. 4.34 can beapplied. We thus obtain∂n
1∂n∇ × Jn ,+=∂t drift∂t diffe∂p∂p1+= − ∇ × Jp,∂t drift∂t diffe(6.29a)(6.29b)where Jn (J p ) is the electron (hole) current density.

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65

6.5. Transport properties

Jn = eDn dn/dx

Jn = eDn dn/dx

densitydensitydensitydn/dx = 0  Jn = 0

Figure 6.9: Visualisation of electron diffusion.

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66

6. Basic Semiconductor Physics

The equations can be written in a more compact form
when introducing substitutions,∂n= − Rn ,∂t R∂p= −R p ,
∂t R
∂n
=
∂t G
∂p
=
∂t G

(6.30a)(6.30b)Gn ,(6.31a)Gp .

(6.31b)Rn (R p ) denotes the net thermal recombinationgeneration rate of electrons (holes), Gn (G p ) is the generation rate of electrons (holes) due to other processes,
such as photogeneration. We will discuss these processes in detail in Chapter 7. Substituting Eqs. (6.29),(6.30) and (6.31) into Eq. (6.28) finally leads to1∂n=∇ × Jn − Rn + Gn ,
∂t
e
∂p
1
= − ∇ × J p − R p + Gp .
∂teCopyright Delft University of Technology, 2014(6.32a)(6.32b)This copy is provided for free, for personal use only.

7

Generation and Recombination of Electron-Hole Pairs
7.1

Introduction

more. Instead, now an inequality is valid,
np > n2i .

(7.1)

After the pulse stops, the excess electrons will recombinewith holes, until the equilibrium state is reached again.
Depending on the properties of the semiconductor different types of recombination can and will occur. In
this chapter we will discuss the most important of thesemechanisms.

Assume that a piece of a semiconductor is illuminatedby a light pulse, which leads to excitations of electrons
from the valence band to the conduction band and consequently to the creation of holes in the valence band.
This illumination will disturb the semiconductor fromthe state of thermal equilibrium. In the valence band an
excess concentration of holes p > p0 is present, wherep0 denotes the equilibrium concentration. Similarly, inthe conduction band the electron concentration is larger than the equilibrium concentration, n > n0 , whichmeans that excess carriers are present. It is clear that inthis non-equilibrium state Eq. (6.7) is not fulfilled any-

The recombination rate strongly determines the performance of the solar cells. On the one hand, it will reduce the current that can be collected and hence utilised from the solar cell. However, the photo-generation
rate often is several magnitudes higher than the recombination rate, such that the effect of recombination on
67

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68

7. Generation and Recombination of Electron-Hole Pairs

(a)

(b)

photonEnergyEnergyphononMomentum

photonMomentum

Figure 7.1: Illustrating the dispersion diagram of (a) an direct

bandgap semiconductor and (b) a indirect bandgap semiconductor.

the solar cell current is negligible. On the other hand,
the recombination rate strongly determines the saturation current density; the more recombination, thehigher the saturation current density. As we will see in
Chapter 8, a high saturation current density has a detrimental effect on the solar cell voltage, and hence on theenergy conversion efficiency.
Before we can actually start with the treatment on the
different generation and recombination mechanisms, we
mention that we have to distinguish between direct andindirect semiconductors. Figure 7.1 shows the energymomentum space of the electrons, which also is called
the electronic dispersion diagram. On the vertical axis the

Copyright Delft University of Technology, 2014

energy state in the electronic bands is plotted. On thehorizontal axis the momentum of the charge carrier isshown. This momentum is also called the crystal momentum, it is also related to the wave vector k of the electron. It is important to realise that the position of thevalence and conduction band may differ in different
directions of the lattice coordination. We can understand this by realising that the crystal can look very different if we look at it from different directions. Hence,
also the energy levels in that electrons and holes canpropagate across the crystal depend on the direction.
The dispersion diagram of silicon is discussed in moredetail in Chapter 12.
For a direct band gap material the highest point of thevalence band is vertically aligned with the lowest pointof the conduction band, as shown in Fig. 7.1 (a). Thismeans that exciting an electron from the valence to the
conduction band requires only the energy provided bya photon without any additional momentum transfer.
In contrast, for an indirect band gap the highest pointof the valence band is not aligned with the lowest pointof the conduction band, as shown in Fig. 7.1 (b). Therefore, exciting an electron from the valence to conduction
band requires energy provided by a photon and momentum, provided by a phonon.
Therefore, the excitation of an electron induced byphoton absorption is more likely to happen for direct
band gap materials than for indirect band gap materials

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69

7.2. Bandgap-to-bandgap processes

and hence the absorption coefficient for direct band gap
materials is significantly higher than for indirect band
gap materials. The same principle makes the reverse
process of radiative recombination more likely to happen in a direct band gap material. In an indirect band
gap material additional momentum is required to makethe electron and hole recombine.
Crystalline silicon is an indirect band gap material. In
such a material, the radiative recombination is inefficient and recombination will be dominated by theAuger recombination mechanism discussed in Section
7.4. For direct band gap materials such as gallium arsenide under moderate illumination conditions, radiative recombination will be the dominant loss mechanism of charge carriers. For very high illumination conditions, Auger recombination starts to play a role aswell. Gallium arsenide based solar cells are discussed in
Chapter 13.

7.2

Bandgap-to-bandgap processes

Generation and recombination processes that happen
from bandgap to bandgap are also called direct generation and recombination. They are much more likely tohappen in direct bandgap materials, as there no change
in momentum is required for an electron that is excited

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into the conduction band. These processes most usuallyare radiative, which means that a photon is absorbed
when an electron-hole pair is created, and a photon isemitted if electron-hole pairs recombine directly.
We will use this section also to introduce general concepts important for generation and recombination, suchas the minority carrier lifetime.

7.2.1

Radiative Generation

When light penetrates into a material it will be (partially) absorbed as it propagates through the material. If
the photon energy is higher than the bandgap energy ofthe semiconductor, it is sufficient to break bonds and toexcite a valence electron into the conduction band andleaving a hole behind in the valance band. This process
is illustrated in Fig. 7.2 (a).
The absorption profile in the material depends on
the absorption coefficient of the material, which iswavelength dependent. The most frequent approachto calculate the absorption profile of photons in semiconductor devices is by using Lambert-Beer’s law that wealready introduced in Eq. (4.25), where it was formulated for the decay of intensity. Here we formulate itwith the photon flux density Φ( x, λ), which decreases exponentially with the distance x that it travelled throughThis copy is provided for free, for personal use only.

70

7. Generation and Recombination of Electron-Hole Pairs

the absorber,
Φ( x, λ) = Φ0 (λ) exp [−α(λ) x ] ,
(a)

EC

Eph
EV
(b)

EC
Eph

EV

where Φ0 is the incident photon flux density and α(λ)
is the absorption coefficient. The photon flux density
is defined as the number of photons per unit area, unit
time and unit wavelength. It is related to the spectral
power density P(λ) associated with the solar radiationviaλΦ0 ( λ ) = P ( λ ) .
(7.3)
hc
The spectral generation rate gsp ( x, λ) , which is thenumber of electron-hole pairs generated at a depth x in
the film per second unit volume and unit wavelength,by photons of wavelength λ, is calculated according to
gsp ( x, λ) = ηg Φ0 (λ)α(λ) exp[−α(λ) x ],... nucleus + core electrons... free electron... valence electron... electron bonds... hole
Figure 7.2: Visualisation of bandgap-to-bandgap (a) genera-tion and (b) recombination processes using the bonding modeland the energy band diagram.

(7.2)

(7.4)

where we assumed zero reflection. ηg is the generation
quantum efficiency, usually assumed equal unity. This
assumption means that every photon generates one andonly one electron-hole pair. The optical generation rate
GL ( x ) is calculated from the spectral generation rate byintegrating over a desired wavelength spectrum,
GL ( x ) =Z λ2

gsp ( x, λ) dλ.

(7.5)

λ1

It has the unit [ GL ] = cm−3 s−1 . The optical generation
rate is related to the absorption profile A( x ) in the film

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71

7.2. Bandgap-to-bandgap processesviaequilibrium, the expression
G L ( x ) = η g A ( x ).

(7.6)

Hence,A( x ) =Z λ2

Φ0 (λ)α(λ) exp [−α(λ) x ] dλ.

(7.7)

np = n2i(7.9)

must be valid. Hence, also recombination must takeplace at the same rate as the generation,

λ1

Rth = Gth .
Because of the photogeneration excess electrons andholes will be created. The rates of the generated concentration of generated excess electrons and holes per
second are the same, therefore we can write∂p∂n== GL .
∂t light∂t light

7.2.2

(7.8)

Direct recombination

We now will discuss direct recombination, that mainly
occurs in direct bandgap semiconductors, such as gallium arsenide. It is illustrated in Fig. 7.2 (b). In this section we roughly follow the derivation by Sze [26].
Let us first look at the situation at thermal equilibrium. If
the temperature is higher than 0 K, the crystal lattice isvibrating. This vibrational energy will be sufficient tobreak bonds from time to time, which leads to the generation of electron-hole pairs at a generation rate Gth ,where the th stands for thermal. As we are in thermal

Copyright Delft University of Technology, 2014(7.10)

We may assume that the direct recombination rate is
proportional to the concentration of electrons in the
conduction band and to the concentration of the available holes in the valence band,R∗ = βnp,(7.11)

where β is a proportionality factor. For the thermal recombination we have
Rth = βn0 p0 .

(7.12)

We now look at a situation where the semiconductor isilluminated such that a constant generation rate GL ispresent throughout the volume of the semiconductor.
In this situation excess electrons and holes are created.
As the electron and hole concentrations are increasing,also the recombination rate will increase according to
Eq. (7.11). At some point, the generation and recombination rates will be the same, such that n and p do notchange any more. This situation is called the steady stateThis copy is provided for free, for personal use only.

72

7. Generation and Recombination of Electron-Hole Pairs

ExampleLet us calculate the total absorption in a d = 300 µm thick c-Si wafer for light with a wavelength of 500 nm. The optical constants of c-Si
this wavelength are: refractive index is n = 4.293, extinction coefficient k = 0.045 and absorption coefficient α = 1.11 × 104 cm−1 . The
incident irradiance is 1000 Wm−2 .
First we calculate the photon flux density at 500 nm corresponding to the irradiance of 1000 Wm−2 . Using Eq. (7.3) we obtain
Φ0 ( λ ) = P ( λ )λ1000 Wm−2 × 500 × 10−9 m== 2.5 × 1021 m−2 s−1 .
hc
6.625 × 10−34 Js × 2.998 × 108 ms−1

Using Eq. (4.15) we calculate how many incident photons are reflected from the surface,R=ñ0 − ñ1 2
1.0 − 0.0i − (4.293 − 0.045i ) 2= 0.38.
=ñ0 + ñ1
1.0 − 0.0i + 4.293 − 0.045i

Using Lambert-Beer’s law [Eq. (7.2)] we calculate the photon flux density at the backside of the wafer, i.e. at 300 ¯m distance from thesurface. We take the reflected light into account by adapting Eq. (7.2),

Φ(d) = Φ0 (1 − R) exp[−α(λ)d] = 2.5 × 1021 m−2 s−1 × (1 − 0.38) exp −1.11 × 106 m−1 × 300 × 10−6 m ≈ 0.
The total absorption in the wafer is the difference between the photon flux density at the surface after reflection and the photon flux density
at the back of the wafer,
A = ∆Φ = Φ0 (1 − R) − Φ(d) = 2.5 × 1021 m−2 s−1 × (1 − 0.38) − 0 = 1.55 × 1021 m−2 s−1 .
When we assume that all the absorbed photons generate one electron-hole pair (ηg = 1), we can calculate the photocurrent density corresponding to the absorbed photon flux,
J ph = eA = 1.602 × 10−19 C × 1.55 × 1021 m−2 s−1 = 248.31 Cm−2 s−1 = 248.31 Am−2 .

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73

7.2. Bandgap-to-bandgap processes

situation. The total recombination and generation rates
are given byR∗ = βnp = β(n0 + ∆n)( p0 + ∆p),
G = Gth + GL ,(7.13)(7.14)

where n0 and p0 are the equilibrium concentrations. ∆nand ∆p are the excess carrier concentrations that are
given by∆n = n − n0 ,∆p = p − p0 .

(7.15a)(7.15b)In steady state R∗ and G are equal, hence
GL = R∗ − G = Rd ,(7.16)

where Rd denotes the net radiative recombination rate.
By substituting Eqs. (7.11) and (7.12) into Eq. (7.16), we
obtainGL = Rd = β(np − n0 p0 ).
(7.17)
We now assume the semiconductor to be n-type andlow level injection, which means that ∆n  n and p  n.
Under these assumptions the recombination rate becomesp − p0,(7.18)
Rd ≈ βn0 ( p − p0 ) =
τpd
where
τpd =

1
βn0

Copyright Delft University of Technology, 2014(7.19)

is the lifetime of the minority holes in the n-type semiconductor. Clearly, if no excess carriers are present, Rd = 0.
The excess carrier concentration is given as the sum of
the generation rate and the lifetime,p − p0 = GL τpd .

(7.20)To understand the meaning of the lifetime, we consider
a situation where the light and hence generation at therate GL is suddenly shut off. Without loss of generality
we may assume that the light is shut of at the instantt = 0. As no generation is present any more, the excess carrier concentration will change according to thedifferential equationp ( t ) − p0
dp
=−
.
dt
τpd

(7.21)

If we solve this equation with the boundary conditionp(t = 0) = p0 + GL τpd , we findtp(t) = p0 + GL τp exp −
τpd

!
.

(7.22)We therefore see that the minority carrier lifetime is thetime constant at which an excess carrier concentration
decays exponentially, if no external generation is taking
place any more.
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74

7. Generation and Recombination of Electron-Hole Pairs

p and n  p) we find similar expressions,
Rd ≈ βp0 (n − n0 ) =n − n0,τnd(7.23)

where the lifetime of the electrons is given by
τnd =

1
.
βp0

(7.24)Let us assume that in a semiconductor several recombination mechanisms are present, with recombination
rates R1 , R2 , . . . . The total recombination rate then is
given byRtot = R1 + R2 + . . .
(7.25)
If we have an n-type semiconductor under low injection
we may assume
Rtot =

p − p0p − p0p − p0++... =.
τp1
τp2
τp,tot(7.26)We therefore have an overall lifetime that is related to
the lifetimes of the different processes via
1

1
1
=
+
+...
τp,totτp1
τp2(7.27)The more recombination mechanisms are present, theshorter the overall lifetime of the excess minority carriers.

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The last aspect that we want to discuss in this section isa situation where the excess of generated carriers is not
uniform throughout the semiconductor. In this case diffusion of the excess carriers takes place. The excess carriers will diffuse in the semiconductor till they recombine. We define the average distance that the minority
carriers can diffuse among majority carries before being
annihilated as minority-carrier diffusion length. They aredefined as:
Ln =
Lp =

√

Dn τn

q

D p τpfor electrons in ap-type material,for holes in an
n-type material,(7.28a)(7.28b)where Dn and D p are the diffusion coefficients as introduced in Section 6.5.

7.3

Shockley-Read-Hall Recombination

In the Shockley-Read-Hall (SRH) recombination process, which is illustrated in Fig. 7.3, the recombination of electrons and holes does not occur directlyfrom bandgap to bandgap. It is facilitated by a impurity atom or lattice defects. Their concentration is usuallysmall compared to the acceptor or donor concentrations. These recombination centres introduce allowed

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75

7.3. Shockley-Read-Hall Recombination

ExampleEC

To get an idea about the diffusion lengths, let us assume room
temperature, mobility of electrons in a p-type c-Si wafer tobe µn ≈ 1250 cm2 V−1 s−1 , which corresponds to doping of
NA = 1014 cm−3 , and τn = 10−6 s. For the given conditions, the electron diffusion length in the p-type c-Si can becalculated from Eq. (7.28a).
r
p
kT
µn τn
Ln = D N τn =ep
= 0.0258 V × 1250 cm2 V−1 s−1 × 10−6 s = 57 ¯m.

ETEV... nucleus + core electrons... impurity atom... valence electron... free electron... hole... electron bonds

Figure 7.3: Visualisation of Shockley-Read-Hall recombina-

tion using the bonding model and the energy band diagram.

energy levels (ET ) within the forbidden gap, so-called
trap states. An electron can be trapped at such a defectand consequently recombines with a hole that is attracted by the trapped electron. Though this process seems
to be less likely than the direct thermal recombination,
it is the dominant recombination-generation process insemiconductors at most operational conditions . The
process is typically non-radiative and the excess energy
is dissipated into the lattice in form of heat. The nameis a reverence to William Shockley, William T. Read andRobert N. Hall, who published the theory of this recombination mechanism in 1952 [27, 28].
We distinguish between two kinds of traps: first, donortype traps that are neutral when they contain an elec-

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tron and positively charged when they do not containan electron. Secondly, acceptor-type traps that are negatively charged when they contain and electron andneutral when they do not contain an electron.
The SRH statistics is based on four processes that are
involved in recombination in a single-electron trap:• r1 : capture of an electron from the conduction
band,• r2 : emission of an electron to the conduction band,• r3 : capture of a hole from the valence band, andThis copy is provided for free, for personal use only.

76

7. Generation and Recombination of Electron-Hole Pairs
Table 7.1: Processes associated with single-electron trapping• r4 : emission of a hole to the valence band.

and their rates.

These processes are illustrated in Fig. 7.4 for both
donor- and acceptor-type traps. The electron and holecapture rates are proportional to the free carrier concentration, n or p, respectively, the thermal velocity vth , thetrap density NT , the trap occupancy by electron, f , orholes, 1 − f , and the electron and hole capture crosssection of the traps, σn and σp . The emission rates areproportional to the trap density and the electron or hole
occupancy of the traps, as well as the emission coefficient for electron or holes, en or e p , respectively. Allprocesses and their rates are listed in Table 7.1.
The thermal velocity is the average velocity of the electrons and holes due to thermal movement. It can beobtained by setting the thermal and the kinetic energy
equal. Since electrons and holes have three degrees of
freedom, we obtain
1 ∗ 2
3
2 mn vth,n = 2 kT

1 ∗ 2
3
2 m p vth,p = 2 kT,(7.29)

where m∗n and m∗p are the effective masses of the electrons and holes, respectively. For electrons in silicon
and gallium arsenide, the thermal velocity is about 107
cm/s. For the following derivation we assume that vth
is the same for electrons and holes.
The electron capture cross section σn describes the effectiveness of the trap state to capture an electron. It isa measure of how close an electron has to come to the

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Donor-like traps
ProcessRate
r1 electron capturenvth σn+ NT (1 − f )r2 electron emission e0n NT fr3 hole capture
pvth σp0 NT fr4 hole emissione+p NT (1 − f )Acceptor-like traps
ProcessRate
r1 electron capturenvth σn0 NT (1 − f )r2 electron emission en− NT fr3 hole capture
pvth σp− NT fr4 hole emission
e0p NT (1 − f )trap to be captured. It has the unit of area, cm2 . Similarly, σp describes the effectiveness of a trap state tocapture a hole.
The derivation of the recombination efficacy, below, isvalid for both donor- and acceptor-like traps. Therefore,
the capture cross-section and emission coefficients inTable 7.1 are generalised by omitting their charge state.
Depending on the type of trap considered the appropriate cross-sections and emission coefficients need to be
substituted.
According to the Fermi-Dirac statistics the carrier dis-This copy is provided for free, for personal use only.

77

7.3. Shockley-Read-Hall Recombination

(a)
r1

EC
r2
ET
r3

r4
EV

(b)

r1

EC
r2
ET
r3

r4
EV

Figure 7.4: Schematic illustration of the processes involved

with SRH recombination in a single-electron trap state for (a) adonor-type and (b) an acceptor-type trap.

tribution in a semiconductor in thermal equilibrium depends on the chemical potential of the carriers, whichis referred to as the Fermi level EF . When the device isilluminated or a bias voltage is applied, the carriers oneither side of the mobility gap are no longer in equilibrium. Yet, they do relax to a state of quasi-equilibriumwith their respective bands. This leads to the definitionof the quasi-Fermi levels for electrons and holes, EFn
and EFp , which determine the carrier concentrations under non-equilibrium conditions. Note that in thermal
equilibrium EFn = EFp = EF . General expressions forthe free electron and hole concentrations n and p, respectively, both under equilibrium and non-equilibrium
conditions, readEFn − ECn = NC exp,(7.30a)kTEV − EFp
p = NV exp,(7.30b)kT
where EC ( EV ) is the conduction (valence) band edge
and NC ( NV ) the effective density of states in the conduction (valence) band, respectively. According to the
Fermi-Dirac statistics the occupation function in thermal
equilibrium is given byf ( ET ) =1 + exp1
ET − E F
kT,(7.31)

where ET is the trap energy.

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78

7. Generation and Recombination of Electron-Hole Pairs

In thermal equilibrium no net recombination occurs,such that r1 = r2 and r3 = r4 . Substituting the rateequations from Table 7.1 and Eqs. (7.30–7.31) yields thefollowing expressions for the emission coefficients:ET − EC,(7.32a)en = vth σn NC expkTEV − ETe p = vth σp NV exp.
(7.32b)kT
By substituting NC and NV by the intrinsic carrier concentration ni times an exponential according to Eq.
(6.8), we obtainET − EFi,(7.33a)en = vth σn ni exp
kT
EFi − ETe p = vth σp ni exp.
(7.33b)kT
Consider now a non-equilibrium steady-state situation.
Under equilibrium, it is assumed that the emission coefficients are approximately equal to the emission coefficients. As recombination involves exactly one electron and one hole, at steady state the rate at which theelectrons leave the conduction band equals the rate at
which the holes leave the valence band. The recombination rate is therefore equal to
RSRH =

dpdn== r1 − r2 = r3 − r4 .
dt
dt

Copyright Delft University of Technology, 2014(7.34)

By substituting the rates from Table 7.1, the expression
for the steady-state occupation function can be determined to bef ( ET ) =vth σn n + e p.
Vth σn n + vth σp p + en + e p(7.35)

Finally, the recombination rate is obtained by substituting Eq. (7.35) in the rate equations as in Eq. (7.34),
yielding
RSRH = v2th σp σn NT

np − n2i,
vth σn n + vth σp p + en + e p(7.36)

where ni is the intrinsic carrier concentration as in Eq.
(6.7).
We can simplify the general expression of Eq. (7.36)
when we assume the same capture cross sections forelectrons and holes, σn = σp ≡ σ0 , which yieldsET − EFi,(7.37)
en + e p = 2vth σ0 ni coshkT
and hence
RSRH = vth σNT

np − n2i
.
− EFi
n + p + 2ni cosh ET kT(7.38)

We now look at an n-type semiconductor at low injection rate, meaning that the amount concentration of excess electrons is small compared to the total electronThis copy is provided for free, for personal use only.

79

7.4. Auger Recombination

concentration, n ≈ n0 , where n0 is the electron concentration under thermal equilibrium. Further, we may
assume n  p. By applying these assumptions to Eq.
7.38 we obtainp − p0
RSRH = vth σNT− EFini
1 + 2 n0 cosh ET kT(7.39)
p − p0= c p NT ( p − p0 ) =,τp,SRH
where c p is called the hole capture coefficient. τp,SRH is the
lifetime of holes in an n-type semiconductor.
In a similar manner, we can derive for a p-type semiconductor at a low injection rate
RSRH = vth σNTn − n0− EFi
1 + 2 pn0i cosh ET kTn − n0= cn NT (n − n0 ) =,τn,SRH(7.40)

with the electron capture coefficient cn and the electron
lifetime τn,SRH .
We see that the lifetime is related to the capture coefficients via
τp,SRH =1
c p NTand

τn,SRH =

1
.
cn NT

(7.41)The lifetime of the minority carriers due to ShockleyRead-Hall recombination therefore is indirectly propor-

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(a)

EC

(b)

EV

EC
EV

Figure 7.5: Schematic illustration of Auger recombination

with (a) two electrons and (b) two holes involved.

tional to the trap density NT . Hence, for a good semiconductor device it is crucial to keep NT low.
The values of the minority-carrier lifetimes can varya lot. When the trap concentration in c-Si is very low,τn (τp ) can achieve values around 1 ms. On the other
hand, the intentional introduction of gold atoms into
Si, which introduce efficient traps into Si, can decreaseτn (τp ) to values around 1 ns. Typical minority-carrier
lifetimes in most c-Si devices are usually around 1 µs.
For an efficient collection of photo-generated carriers in
c-Si solar cells the minority carrier lifetimes should be
in range of tens of milliseconds.

7.4

Auger Recombination

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80

7. Generation and Recombination of Electron-Hole Pairs

both transfer in energy and momentum must occur for
an electron in the conduction band to recombine witha hole in the valence band. In indirect semiconductors,
Auger recombination becomes important. In difference
to direct and SRH recombination, which involve twoparticles, i.e. an electron and a hole, Auger recombination is a three particle process, as illustrated in Fig. 7.5.
In Auger recombination, momentum and energy of therecombining hole and electron is conserved by transferring energy and momentum to an another electron(or hole). If the third particle is an electron, it is excited
into higher levels in the electronic band. This excitedelectron relaxes again, such that the energy is transferred vibrational energy of the lattice, or phonon modes,
and finally heat. Similarly, if the third particle is a hole,
it is excited into deeper levels of the valence band, from
where it rises back to the valence band edge via transferring its energy to phonon modes.
As Auger recombination is a three-particle process, theAuger recombination rate RAug strongly depends on the
charge carrier densities for the electrons n and holes p.
The recombination rates for electron-electron-hole (eeh)and electron-hole-hole (ehh) processes are given byReeh = Cn n2 p,(7.42a)Rehh = C p np2 ,(7.42b)respectively, where Cn and C p are the proportionality
constants that are strongly dependent on the temper-

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ature [26]. Reeh is dominant when the electrons are themajority charge carriers, while Rehh is dominant when
the holes are the majority charge carries. Adding themleads to the total Auger recombination rate,
RAug = Reeh + Rehh = Cn n2 p + C p np2 ,(7.43)In strongly doped n-type silicon with a donor concentration ND under low injection we can assume thatn ≈ ND and hence that the eeh process is dominant.
We then can write
2
Reeh = Cn ND
p.

(7.44)

Hence, the lifetime can be approximated with
τeeh =

1
.
2
Cn ND

(7.45)

Similarly, for strongly doped p-type silicon with acceptor concentration NA we may assume p ≈ NA andhence the ehh process being dominant,2n.
Rehh = C p NA

(7.46)

1.
2
C p NA(7.47)The lifetime then isτehh =As the Auger recombination under these conditions is
proportional to the square of the doping levels, it becomes the more important the higher the doping is.

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81

7.5. Carrier concentration in non-equilibrium

For an n-type semiconductor under high-injection conditions, as it occurs for example for concentrator solar
cells with very high irradiance values the lifetime canbe approximated with
τAuger, hi =

1
1
=
,
2
(Cn + C p )∆n
Ca ∆n2(7.48)

where ∆n = n − n0 = p − p0 is the excess carrier density[29]. Under these circumstances Auger recombination
also might become important for direct bandgap materials such as gallium arsenide.

7.5

Carrier concentration in nonequilibrium

When a semiconductor is illuminated additional electrons and holes are generated in the material by the absorption of photons. The photo-generated carriers interact
with the semiconductor lattice. The extra energy thatthe electron-hole pairs receive from the photons withenergies larger than the band gap of the semiconductor
is released into the lattice in form of heat. After thisso called thermalisation process, which is very fast andtakes approximately 10−12 s, the carrier concentrations
achieve a steady-state. In this non-equilibrium state
the electron and hole concentrations are different thanthose in the equilibrium state. In non-equilibrium states

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two Fermi distributions are used to describe the electron and hole concentrations. One Fermi distribution
with the quasi-Fermi energy for electrons, EFC , describes
the occupation of states in the conduction band withelectrons. Another Fermi distribution with the quasiFermi energy for holes, EFV , describes the occupation ofstates in the valence band with electrons, and therefore
determines also the concentration of holes. Using theband diagram with the quasi-Fermi levels the processof creation of electron-hole pairs and their subsequent
thermalisation that describe the carrier concentration
under illumination is illustrated in Fig. 7.6. The difference between the quasi-Fermi levels is the electrochemical energy, µeh , of the generated electron-hole pairs
which represents the measure for the conversion efficiency of solar radiation.
The density of electrons and holes under nonequilibrium conditions is described by
EFC − ECn = NC exp,kTEV − EFV
p = NV exp,
kT(7.49a)(7.49b)It then follows that under non-equilibrium conditionsEV − EC
EFC − EFV
np = NC NV exp
exp
kT
kT


(7.50)EFC − EFV
= n2i exp.
kT

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82

7. Generation and Recombination of Electron-Hole Pairs

(a)

(b)

levels in an energy band diagram, one can easily determine whether current flows inside the semiconductor.

EC

EC
EG

μeh
EV

EV

Figure 7.6: Thermalisation of photo-generated electron-holepairs resulting in non-equilibrium charge-carrier concentrations
described by the quasi-Fermi levels.

By using the quasi-Fermi level formalism for describing
the concentration of charge carriers in non-equilibrium
conditions the electron and hole current densities inside
a semiconductor, J N and J P , can be expressed with
J N = nµn ∇ EFC ,J P = pµ p ∇ EFV .

(7.51a)(7.51b)

One can notice from Eqs. (7.51) that when a quasiFermi level varies with position,
dEFC
6= 0
dx

or

dEFV
6= 0,
dx

the current is flowing inside the semiconductor. By
checking the position dependence of the quasi-Fermi

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8

Semiconductor Junctions
Almost all solar cells contain junctions between (different) materials of different doping. Since these junctions
are crucial to the operation of the solar cell, we will discuss their physics in this chapter.

MS junction. The Schottky barrier formed at the metalsemiconductor interface is a typical example of the MS
junction.

A p-n junction fabricated in the same semiconductormaterial such as c-Si is an example of an p-n homojunction. There are also other types of junctions: A p-n junction that is formed by two chemically different semiconductors is called a p-n heterojunction. In a p-i-n junctions, the region of the internal electric field is extended by inserting an intrinsic,i, layer between the p-type
and the n-type layers. The i-layer behaves like a capacitor and it stretches the electric field formed by the p-njunction across itself. Another type of the junction is ajunction between a metal and a semiconductor, a so-called

8.1 p-n homojunctions
8.1.1

Formation of a space-charge region
in the p-n junction

Figure 8.1 shows schematically isolated pieces of a ptype and an n-type semiconductor and their corresponding band diagrams. In both isolated pieces the
charge neutrality is maintained. In the n-type semicon83

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84

8. Semiconductor Junctions

ductor the large concentration of negatively-charged
free electrons is compensated by positively-charged ionised donor atoms. In the p-type semiconductor holes
are the majority carriers and the positive charge ofholes is compensated by negatively-charged ionised acceptor atoms. For the isolated n-type semiconductor we
can writen = nn0 ≈ ND ,.
p = pn0 ≈ n2i ND .

(8.1a)(8.1b)For the isolated p-type semiconductor we havep = p p0 ≈ NA ,.
n = n p0 ≈ n2i NA .

(8.2a)(8.2b)

When a p-type and an n-type semiconductor are
brought together, a very large difference in electron
concentration between n- and p-type regions causes a
diffusion current of electrons from the n-type material
across the metallurgical junction into the p-type material.
The term “metallurgical junction” denotes the interface
between the n- and p-type regions. Similarly, the difference in hole concentration causes a diffusion currentof holes from the p- to the n-type material. Due to this
diffusion process the region close to the metallurgicaljunction becomes almost completely depleted of mobile
charge carriers. The gradual depletion of the charge carriers gives rise to a space charge created by the charge

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n-type region

p-type region

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

Band diagrams:

electrons

holes

Figure 8.1: Schematic representation of an isolated n-type andp-type semiconductor and corresponding band diagrams.

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85

8.1. p-n homojunctions

space charge
regionn-type region

p-type region

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

electrons+

E-field

–

8.1.2
holes

Figure 8.2: Formation of a space-charge region, when n-type
and p-type semiconductors are brought together to form ajunction. The coloured part represents the space-charge region.

of the ionised donor and acceptor atoms that is notcompensated by the mobile charges any more. This region of the space charge is called the space-charge region
or depleted region and is schematically illustrated in Fig.
8.2. Regions outside the depletion region, in which the
charge neutrality is conserved, are denoted as the quasineutral regions.
The space charge around the metallurgical junction results in the formation of an internal electric field which
forces the charge carriers to move in the opposite dir-

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ection than the concentration gradient. The diffusion
currents continue to flow until the forces acting on the
charge carriers, namely the concentration gradient andthe internal electrical field, compensate each other. The
driving force for the charge transport does not exist any
more and no net current flows through the p-n junction.

The p-n junction under equilibrium

The p-n junction represents a system of charged
particles in diffusive equilibrium in which the electrochemical potential is constant and independent of position. The electro-chemical potential describes an average energy of electrons and is represented by the Fermi
energy. It means that under equilibrium conditions the
Fermi level has constant position in the band diagramof the p-n junction. Figure 8.3 shows the energy-band
diagram of a p-n junction under equilibrium. The distance between the Fermi level and the valence and/or
conduction bands does not change in the quasi-neutral
regions and is the same as in the isolated n- and p-typesemiconductors. Inside the space-charge region, theconduction and valence bands are not represented bystraight horizontal lines any more but they are curved.
This indicates the presence of an electric field in thisregion. Due to the electric field a difference in the electrostatic potential is created between the boundariesof the space-charge region. Across the depletion re-This copy is provided for free, for personal use only.

86

8. Semiconductor Junctions

gion the changes in the carriers concentration are compensated by changes in the electrostatic potential. The
electrostatic-potential profile is also drawn in Fig. 8.3
The concentration profile of charge carriers in a p-njunction is schematically presented in Fig. 8.4. In thequasi-neutral regions the concentration of electrons andholes is the same as in the isolated doped semiconductors. In the space-charge region the concentrations ofmajority charge carriers decrease very rapidly. This fact
allows us to use the assumption that the space-charge
region is depleted of mobile charge carriers. This assumption means that the charge of the mobile carriers
represents a negligible contribution to the total spacecharge in the depletion region. The space charge in thisregion is fully determined by the ionised dopant atoms
fixed in the lattice.
The presence of the internal electric field inside the pn junction means that there is an electrostatic potential difference, ψ0 , across the space-charge region. We
shall determine a profile of the internal electric fieldand electrostatic potential in the p-n junction. First we
introduce an approximation, which simplifies the calculation of the electric field and electrostatic-potential.
This approximation (the depletion approximation) assumes
that the space-charge density, ρ, is zero in the quasineutral regions and it is fully determined by the concentration of ionised dopants in the depletion region.
In the depletion region of the n-type semiconductor it

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n-type Siψp-type Si

ψ0
0
-ℓn

-eψ0

x

ℓp
χe

EC
EF

E1
E2
EG

-eψ0

EV
E
Figure 8.3: Energy-band diagram of the p-n junction. The

electrostatic potential profile (red curve) is also presented in the
figure.

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87

8.1. p-n homojunctions

n-type silicon

p-type silicon

n=nn0=ND

p=pp0=NAis the concentration of positively charged donor atoms,
ND , which determines the space charge in this region.
In the p-type semiconductor, the concentration of negatively charged acceptor atoms, NA , determines the spacecharge in the depletion region. This is illustrated inFig. 8.5. Further, we assume that the p-n junction is a
step junction; it means that there is an abrupt change
in doping at the metallurgical junction and the doping
concentration is uniform both in the p-type and the ntype semiconductors.
According to Fig. 8.5 the position of the metallurgicaljunction is placed at zero, the width of the space-charge
region in the n-type material is denoted as `n and thewidth of the space-charge region in the p-type material
is denoted as ` p . The space-charge density is described
by

ln n
ln p

n=np0=pi2/NA

p=pn0=ni2/ND
Position

Figure 8.4: Concentrations profile of mobile charge carriers ina p-n junction under equilibrium.

ρ( x ) = eND
ρ( x ) = −eNA

for
for

−`n ≤ x ≤ 0,
0 ≤ x ≤ `p,(8.3a)(8.3b)

where ND and NA is the concentration of donor and acceptor atoms, respectively. Outside the space-charge region the space-charge density is zero. The electric fieldis can be calculated from the Poisson’s equation, in one
dimension can be written as
dξ
ρ
d2 ψ
=−
=−
,
dx
εr ε0
dx2

(8.4)

where ψ is the electrostatic potential, ξ is the electricfield, ρ is the space-charge density, ε r is the semicon-

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88

8. Semiconductor Junctions

n-typeductor dielectric constant and ε 0 is the vacuum permittivity. The vacuum permittivity is e0 = 8.854 · 10−14
F/cm and for crystalline silicon ε r = 11.7.. The electric field profile can be found by integrating the spacecharge density across the space-charge region,

p-type

ρ

eND

ξ=

x

1
εr ε0

Z

ρ dx.

(8.5)

eNA

Substituting the space-charge density with Eqs. (8.3)
and using the boundary conditions

ξmaxξ (−`n ) = ξ ` p = 0,

(8.6)

ξ

we obtain as solution for the electric field
x
ψ
ψ0
ℓn

0

ℓp

x

Figure 8.5: The space-charge density ρ( x ), the electric fieldξ ( x ), and the electrostatic potential ψ( x ) across the depletion
region of a p-n junction under equilibrium.

e
ND (`n + x )
εr ε0

e
NA ` p − xξ (x) =εr ε0
ξ (x) =

for−`n ≤ x ≤ 0,

(8.7a)

for0 ≤ x ≤ `p.

(8.7b)At the metallurgical junction, x = 0, the electric field is
continuous, which requires that the following condition
has to be fulfilled
NA ` p = ND `n .

(8.8)

Outside the space-charge region the material is electrically neutral and therefore the electric field is zero there.

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89

8.1. p-n homojunctions

The profile of the electrostatic potential is calculated
by integrating the electric field throughout the spacecharge region and applying the boundary conditions,
ψ=−

Z

ξdx.

(8.9)

We define the zero electrostatic potential level at theoutside edge of the p-type semiconductor. Since we assume no potential drop across the quasi-neutral region
the electrostatic potential at the boundary of the spacecharge region in the p-type material is also zero,ψ(` p ) = 0.

(8.10)

Using Eqs. (8.7) for describing the electric field in the
n-type and p-type parts of the space-charge region, respectively, and taking into account that at the metallurgical junction the electrostatic potential is continuous,
the solution for the electrostatic potential can be writtenaseψ (x) = −ND ( x + `n )2
2 εr ε0
for − `n ≤ x ≤ 0,e
ND `2n + NA `2p +
2 εr ε0(8.11a)2e
ψ (x) =
N x − `pfor 0 ≤ x ≤ ` p .
2 εr ε0 A(8.11b)Under equilibrium a difference in electrostatic potential, ψ0 , develops across the space-charge region. The

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electrostatic potential difference across the p-n junction is an important characteristic of the junction andis denoted as the built-in voltage or diffusion potential of
the p-n junction. We can calculate ψ0 as the difference
between the electrostatic potential at the edges of thespace-charge region,ψ0 = ψ (−`n ) − ψ ` p = ψ (−`n ) .
(8.12)
Using Eq. (8.11a) we obtain for the built-in voltage

e 
ψ0 =
ND `2n + NA `2p(8.13)
2 εr ε0
Another way to determine the built-in potential ψ0 is to
use the energy-band diagram presented in Fig. 8.3.
eψ0 = EG − E1 − E2 .

(8.14)

Using Eqs. (6.1) and (6.20), which determine the bandgap, and the positions of the Fermi energy in the n- andp-type semiconductor, respectively,
EG = EC − EV ,E1 = EC − EF = kT ln ( NC /ND ) ,E2 = EF − EV = kT ln ( NV /NA ) .
We can write





NV
NC
− kT ln
NA
ND


NV NC
.
= EG − kT ln
NA NDeψ0 = EG − kT ln(8.15)

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90

8. Semiconductor Junctions

Using the relationship between the intrinsic concentration, ni and the band gap, EG [see Eq. (6.7)],E
n2i = NC NV exp − G ,kTductors and can be calculated with
s


1
1
2ε r ε 0+
W = `n + ` p =
ψ0
.
e
NA
ND

we can rewrite Eq. (8.15) and obtain

The space-charge region is not uniformly distributed in
the n- and p- regions. The widths of the space-charge
region in the n- and p-type semiconductor are determined by the doping concentrations as illustrated by Eqs.
(8.17). Knowing the expressions for `n and ` p we candetermine the maximum value of the internal electricfield, which is at the metallurgical junction. By substituting ` p from expressed by Eq. (8.17b) into Eq. (8.7b)
we obtain the expression for the maximum value of theinternal electric field,
s


NA ND
2e
ξ max =
ψ0.
(8.19)εr ε0
NA + ND

kT
ψ0 =
ln
e

NA ND
n2i

!
(8.16)

This equation allows us to determine the built-in potential of a p-n junction from the standard semiconductor
parameters, such as doping concentrations and the intrinsic carrier concentration. With the built-in potential
we can calculate the width of the space charge regionof the p-n junction in the thermal equilibrium. Substituting ψ0 using Eq. (8.16) into Eq. (8.12) and taking theboundary condition [Eq. (8.6) into account, the resulting
expressions for `n and ` p are obtained. The full derivation can be found for example Ref. [20].
s


NA
1
2ε r ε 0ψ0,(8.17a)`n =q
ND NA + NDs

2ε r ε 0
N
1
`p =
ψ0 D
.
(8.17b)q
NA NA + ND

(8.18)The total space-charge width, W, is the sum of the partial space-charge widths in the n- and p-type semicon-

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91

8.1. p-n homojunctions

ExampleA crystalline silicon wafer is doped with 1016 acceptor atoms per cubic centimetre. A 1 micrometer thick emitter layer is formed at the surface of the wafer with a uniform concentration of 1018 donors per cubic centimetre. Assume a step p-n junction and that all doping atoms
are ionised. The intrinsic carrier concentration in silicon at 300 K is 1.5 · 1010 cm−3 .
Let us calculate the electron and hole concentrations in the p- and n-type quasi-neutral regions at thermal equilibrium. We shall use Eqs.
(8.1) and (8.2) to calculate the charge carrier concentrations.
P-type region:p = p p0 ≈ NA = 1016 cm−3 .
.
2 n = n p0 = n2i p p0 = 1.5 · 1010
1016 = 2.25 · 104 cm−3

N-type region:n = nn0 ≈ NA = 1018 cm−3 .
.
2 p = pn0 = n2i nn0 = 1.5 · 1010
1018 = 2.25 · 102 cm−3

We can calculate the position of the Fermi energy in the quasi-neutral n-type and p-type regions, respectively, using Eq. (6.20a). We
assume that the reference energy level is the bottom of the conduction band, EC = 0 eV.
.
N-type region:EF − EC = −kT ln ( NC /n) = −0.0258 ln 3.32 · 1019 1018 = −0.09 eV.
.
P-type region:EF − EC = −kT ln ( NC /n) = −0.0258 ln 3.32 · 1019 2.24 · 104 = −0.90 eV.
The minus sign tells us that the Fermi energy is positioned below the conduction band.
The built-in voltage across the p-n junction is calculated using Eq. (8.16),
#
!
"
NA ND
1016 1018
kT
ln= 0.0258 V
ψ0 =2 = 0.81 V.
en2i
1.5 · 1010

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92

8. Semiconductor Junctions
The width of the depletion region is calculated from Eq. (8.18),
s

 s


1
1
2ε r ε 0
2 · 11.7 · 8.854 · 10−14
1
1
ψ0
+
=
W=
· 0.81+ 18 = 3.25 · 10−5 cm = 0.325 µm.
eNAND
1.602 · 10−191016
10
A typical thickness of c-Si wafers is 300 µm. The depletion region is 0.3 µm which represents 0.1% of the wafer thickness. It is importantto realise that almost the whole bulk of the wafer is a quasi-neutral region without an internal electrical field.
The maximum electric field is at the metallurgical junction and is calculated from Eq. (8.19).
s
!


NA ND
2 · 1.602 · 10−192e1016 1018
ξ max =
ψ0=
= 5 · 104 V cm−1 .
·
0.81
εr ε0
NA + ND
11.7 · 8.854 · 10−141016 + 1018

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93

8.1. p-n homojunctions

8.1.3

The p-n junction under applied
voltage

When an external voltage, Va , is applied to a p-n junction the potential difference between the n- and p-type
regions will change and the electrostatic potential acrossthe space-charge region will become (ψ0 − Va ). Remember that under equilibrium the built-in potential is negative in the p-type region with respect to the n-type
region. When the applied external voltage is negativewith respect to the potential of the p-type region, theapplied voltage will increase the potential difference
across the p-n junction. We refer to this situation as p-njunction under reverse-bias voltage. The potential barrier across the junction is increased under reverse-bias
voltage, which results in a wider space-charge region.
The band diagram of the p-n junction under reversebiased voltage is presented in Fig. 8.6 (a). Under external voltage the p-n junction is not under equilibrium
any more and the concentrations of electrons and holes
are described by the quasi-Fermi energy for electrons,
EFC , and the quasi-Fermi energy for holes, EFV , respectively. When the applied external voltage is positive
with respect to the potential of the p-type region, theapplied voltage will decrease the potential difference
across the p-n junction. We refer to this situation as p-njunction under forward-bias voltage. The band diagramof the p-n junction under forward-biased voltage ispresented in Fig. 8.6 (b). The potential barrier across the

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junction is decreased under forward-bias voltage andthe space charge region becomes narrower. The balance
between the forces responsible for diffusion (concentration gradient) and drift (electric field) is disturbed.
Lowering the electrostatic potential barrier leads to ahigher concentration of minority carriers at the edgesof the space-charge region compared to the situationin equilibrium. This process is referred to as minoritycarrier injection. This gradient in concentration causes
the diffusion of the minority carriers from the edge into
the bulk of the quasi-neutral region.
The diffusion of minority carriers into the quasi-neutral
region causes a so-called recombination current, Jrec ,since the diffusing minority carriers recombine with the
majority carriers in the bulk. The recombination current is compensated by the so-called thermal generation
current, Jgen , which is caused by the drift of minority
carriers, which are present in the corresponding doped
regions (electrons in the p-type region and holes in the
n-type region), across the junction. Both, the recombination and generation currents have contributions fromelectrons and holes. When no voltage is applied to the
p-n junction, the situation inside the junction can beviewed as the balance between the recombination andgeneration currents,
J = Jrec − Jgen = 0

for Va = 0 V.

(8.20)

It is assumed that when a moderate forward-bias
voltage is applied to the junction the recombination cur-This copy is provided for free, for personal use only.

94

8. Semiconductor Junctions

n-type Si(a)p-type Siψ(b)n-type Si

p-type Si

ψ

+Va
-Va
0

0

x
-eψ

x

-eψ

χe

χe

EC
EFC

EC

EFC

EG

EV

EG

EFV
-e(ψ0-Va)EFV-e(ψ0+Va)

E

EV

E

Figure 8.6: Energy band diagram and potential profile (in red colour) of a p-n junction under (a) reverse bias, and (b) forward bias.

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95

8.1. p-n homojunctions

rent increases with the Boltzmann factor exp(eVa /kT ),


e Va
.
(8.21)
Jrec (Va ) = Jrec (Va = 0) exp
kTThis assumption is called the Boltzmann approximation.
The generation current, on the other hand, is almost
independent of the potential barrier across the junction
and is determined by the availability of the thermallygenerated minority carriers in the doped regions,
Jgen (Va ) ≈ Jgen (Va = 0) .

(8.22)The external net-current density can be expressed as
J (Va ) = Jrec (Va ) − Jgen (Va )




eVa
−1 ,
= J0 exp
kT(8.23)

where J0 is the saturation-current density of the p-njunction, given by
J0 = Jgen (Va = 0) .

(8.24)

Equation (8.23) is known as the Shockley equation that
describes the current-voltage behaviour of an ideal p-n
diode. It is a fundamental equation for microelectronics device physics. The detailed derivation of the darkcurrent density of the p-n junction is carried out in Appendix B.1. The saturation-current density is given by


DP
DN
+
.
(8.25)
J0 = e n2i
L N NAL P ND

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The saturation-current density depends in a complex
way on the fundamental semiconductor parameters.
Ideally the saturation-current density should be as low
as possible and this requires an optimal and balanced
design of the p-type and n-type semiconductor properties. For example, an increase in the doping concentration decreases the diffusion length of the minority
carriers, which means that the optimal product of thesetwo quantities requires a delicate balance between thesetwo properties.
The recombination of the majority carriers due to the
diffusion of the injected minority carriers into the bulkof the quasi-neutral regions results in a lowering of theconcentration of the majority carriers compared to theone under equilibrium. The drop in the concentrationof the majority carriers is balanced by the flow of the
majority carriers from the electrodes into the bulk. In
this way the net current flows through the p-n junction under forward-bias voltage. For high reverse-bias
voltage, the Boltzmann factor in Eq. (8.23) becomes verysmall and can be neglected. The net current density isgiven by
J (Va ) = − J0 ,(8.26)

and represents the flux of thermally generated minority
carriers across the junction. The current density-voltage(J-V) characteristic of an ideal p-n junction is schematically shown in Fig. 8.7.

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96

8. Semiconductor Junctions

ln J

J

Slope e/kT
ln J 0-J 0V
Reversebias

V

Forwardbias

Figure 8.7: J-V characteristic of a p-n junction; (a) linear plotand (b) semi-logarithmic plot.

8.1.4

The p-n junction under illumination

When a p-n junction is illuminated the additional
electron-hole pairs are generated in the semiconductor.
The concentration of minority carriers (electrons in thep-type region and holes in the n-type region) strongly
increases. This increase in the concentration of minority carriers leads to the flow of the minority carriers
across the depletion region into the quasi-neutral regions. Electrons flow from the p-type into the n-type
region and holes from the n-type into the p-type region. The flow of the photo-generated carriers causes
the so-called photo-generation current, Jph , which adds
to the thermal-generation current, Jgen . When no external contact between the n-type and the p-type re-

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gions is established, which means that the junction isin the open-circuit condition, no net current can flow
inside the p-n junction. It means that the current resulting from the flux of photo-generated and thermallygenerated carriers has to be balanced by the opposite
recombination current. The recombination current will
increase through lowering of the electrostatic potential barrier across the depletion region. This situationof the illuminated p-n junction under open-circuit condition using the band diagram is presented in Fig. 8.8(a). The electrostatic-potential barrier across the junction
is lowered by an amount of Voc . We refer to Voc as the
open-circuit voltage. Under non-equilibrium conditions
the concentrations of electrons and holes are describedby the quasi-Fermi energy levels. It is illustrated in Fig.
8.8 (a) that the electrochemical potential of electrons,
denoted by EFC , is higher in the n-type region than in
the p-type region by an amount of eVoc . This meansthat a voltmeter will measure a voltage difference of
Voc between the contacts of the p-n junction. Under
illumination, when the n-type and p-type regions areshort circuited, the photo-generated current will also
flow through the external circuit. This situation is illustrated in Fig. 8.8 (b). Under the short-circuit condition
the electrostatic-potential barrier is not changed, butfrom a strong variation of the quasi-Fermi levels insidethe depletion region one can determine that the current
is flowing inside the semiconductor.
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97

8.1. p-n homojunctions

n-type Si(a)p-type Siψ(b)n-type Siψp-type Si

-Voc
0

-eψ

0

x

x

-eψ

χe

χe

EC
EC

EFV

EFC

-eVoc
EFC

EFV

-e(ψ0-Voc)

EV

-eψ0

EV
E

E

Figure 8.8: Energy band diagram and electrostatic-potential (in red colour) of an illuminated p-n junction under the (a) open-circuitand (b) short-circuit conditions.

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98

8. Semiconductor Junctions

illuminated p-n junction, only a fraction of the photogenerated current will flow through the external circuit.
The electro-chemical potential difference between the
n-type and p-type regions will be lowered by a voltage
drop over the load. This in turn lowers the electrostatic
potential over the depletion region which results in
an increase of the recombination current. In the superposition approximation, the net current flowing throughthe load is determined as the sum of the photo- andthermal generation currents and the recombination current. The voltage drop at the load can be simulated byapplying a forward-bias voltage to the junction, therefore Eq. (8.23), which describes the behaviour of thejunction under applied voltage, is included to describe
the net current of the illuminated p-n junction,
J (Va ) = Jrec (Va ) − Jgen (Va ) − Jph




eVa
− 1 − Jph .
= J0 exp
kT

B.2. and its value under uniform generation rate, G, is
Jph = eG ( L N + W + L P ) ,(8.28)

where L N and L P is the minority-carrier-diffusion
length for electrons and holes, respectively, and W is
the width of the depletion region. It means only carriers generated in the depletion region and in the regions
up to the minority-carrier-diffusion length from the depletion region contribute to the photo-generated current. When designing the thickness of a solar cell, Eq.
(8.28) must be considered. The thickness of the absorbershould not be thicker than the region from which thecarriers contribute to the photo-generated current.

8.2 p-n heterojunctions(8.27)

8.3

Metal-semiconductor junctions

Both the dark and illuminated J-V characteristics ofthe p-n junction are represented in Fig. 8.9. Note, thatin the figure the superposition principle is reflected.
The illuminated J-V characteristic of the p-n junctionis the same as the dark J-V characteristic, but it is shifted down by the photo-generated current density Jph .
The detailed derivation of the photo-generated current
density of the p-n junction is carried out in Appendix

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99

8.3. Metal-semiconductor junctions

Pmax

J
P

P=J×V

Dark
Voc
V

Vmp
Peak power
J mp

Illuminated
J ph

J sc
Figure 8.9: J-V characteristics of a p-n junction in the dark

and under illumination.

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100

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8. Semiconductor Junctions

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9

Solar Cell Parameters and Equivalent Circuit
9.1

External solar cell parameters

Short-circuit current density

solar cell are short circuited. The short-circuit current ofa solar cell depends on the photon flux density incident
on the solar cell, which is determined by the spectrumof the incident light. For a standard solar cell measurements, the spectrum is standardised to the AM1.5 spectrum. The Isc depends on the area of the solar cell. In
order to remove the dependence of the solar cell area
on Isc , often the short-circuit current density is used to describe the maximum current delivered by a solar cell.
The maximum current that the solar cell can deliver
strongly depends on the optical properties of the solarcell, such as absorption in the absorber layer and reflection.

The short-circuit current Isc is the current that flows
through the external circuit when the electrodes of the

In the ideal case, Jsc is equal to Jph as can be easily derived from Eq. (8.27). Jph can be approximated by Eq.

The main parameters that are used to characterise theperformance of solar cells are the peak power Pmax , theshort-circuit current density Jsc , the open-circuit voltage Voc ,and the fill factor FF. These parameters are determinedfrom the illuminated J-V characteristic as illustrated inFig. 8.9. The conversion efficiency η can be determinedfrom these parameters.

9.1.1

101
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102

9. Solar Cell Parameters and Equivalent Circuit(8.28), which shows that in case of an ideal diode (forexample no surface recombination) and uniform generation, the critical material parameters that determine
Jph are the diffusion lengths of minority carriers. Crystalline silicon solar cells can deliver under an AM1.5
spectrum a maximum possible current density of 46
mA/cm2 . In laboratory c-Si solar cells the measured Jsc
is above 42 mA/cm2 , while commercial solar cell havean Jsc exceeding 35 mA/cm2 .

9.1.2

Open-circuit voltage

The open-circuit voltage is the voltage at which no current flows through the external circuit. It is the maximum voltage that a solar cell can deliver. Voc corresponds to the forward bias voltage, at which the darkcurrent compensates the photocurrent. Voc depends on
the photo-generated current density and can be calculated from Eq. (8.27) assuming that the net current is
zero,


J ph
kT
ln
+1 .
(9.1)
Voc =
e
J0
This equation shows that Voc depends on the saturation
current of the solar cell and the photo-generated current. While Jph typically has a small variation, the key
effect is the saturation current, since this may vary byorders of magnitude. The saturation current density, J0 ,Copyright Delft University of Technology, 2014

depends on the recombination in the solar cell. Therefore, Voc is a measure of the amount of recombinationin the device. Laboratory crystalline silicon solar cells
have a Voc of up to 720 mV under the standard AM1.5
conditions, while commercial solar cells typically have
Voc exceeding 600 mV.

9.1.3

Fill factor

The fill factor is the ratio between the maximum power( Pmax = Jmp Vmp ) generated by a solar cell and the
product of Voc with Jsc (see Fig. 8.9),
FF =Jmp Vmp.
Jsc Voc(9.2)

Assuming that the solar cell behaves as an ideal diodethe fill factor can be expressed as a function of opencircuit voltage Voc 1 ,FF =voc − ln (voc + 0.72),
voc + 1

(9.3)

where voc = Voc · e/(kT ) is a normalised voltage. Eq.
(9.3) is a good approximation of the ideal value of FF
for voc > 10. The FF as a function of Voc is illustrated
in Fig. 9.1. This figure shows that FF does not change
1 M.A. Green, Solar Cells; Operating Principles, Technology and System Applications, Prentice-Hall, 1982.

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103

9.1. External solar cell parameters
1

However, in practical solar cells the dark diode current
Eq. (8.23) does not obey the Boltzmann approximation.
The non-ideal diode is approximated by introducing an
ideality factor n, into the Boltzmann factor,

0.9

Fill Factor (−)

0.8
0.7
0.6

n=1
n=1.5
n=2

0.5
0.4

exp

0.3
0.2
0.1
0

0.4

0.6

0.8

1

1.2

1.4

1.6

Voc (V)
Figure 9.1: The FF as a function of Voc for a solar cell with

ideal diode behaviour.

drastically with a change in Voc . For a solar cell witha particular absorber, large variations in Voc are notcommon. For example, at standard illumination conditions, the difference between the maximum open-circuit
voltage measured for a silicon laboratory device and atypical commercial solar cell is about 120 mV, givinga maximal FF of 0.85 and 0.83, respectively. However,the variation in maximum FF can be significant forsolar cells made from different materials. For example,
a GaAs solar cell may have a FF approaching 0.89.

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eVa
.
nkT

Fig. 9.1 also demonstrates the importance of the diodeideality factor when introduced into the normalised
voltage in Eq. (9.3). The ideality factor is a measure of
the junction quality and the type of recombination in asolar cell. For the ideal junction where the recombination is represented by the recombination of the minority carriers in the quasi-neutral regions the n is equal
to 1. However, when other recombination mechanisms
occur, the n can have a value of 2. A high n value not
only lowers the FF, but since it signals a high recombination, it leads to a low Voc . Eq. 9.3) describes a maximum achievable FF. In practice the FF is often lower
due to the presence of parasitic resistive losses.

9.1.4

Conversion eﬃciency

The conversion efficiency is calculated as the ratio
between the maximal generated power and the incident
power. The irradiance value Pin of 1000 W/m2 for theAM1.5 spectrum has become a standard for measuring

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104

9. Solar Cell Parameters and Equivalent Circuit

the conversion efficiency of solar cells,η=

Jmp VmpPmax
Jsc Voc FF
=
=
.
Pin
Pin
Pin

(9.4)

Typical external parameters of a crystalline silicon solar
cell as shown are; Jsc ≈ 35 mA/cm2 , Voc up to 0.65 V
and FF in the range 0.75 to 0.80. The conversion efficiency lies in the range of 17 to 18%.

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105

9.1. External solar cell parameters

ExampleA crystalline silicon solar cell generates a photo-current density of Jph = 35 mA/cm2 . The wafer is doped with 1017 acceptor atoms per
cubic centimetre and the emitter layer is formed with a uniform concentration of 1019 donors per cubic centimetre. The minority-carrier
diffusion length in the p-type region and n-type region is 500 · 10−6 m and 10 · 10−6 m , respectively. Further, the intrinsic carrier
concentration in silicon at 300 K is 1.5 · 1010 cm−3 , the mobility of electrons in the p-type region is µn = 1000 cm2 V−1 s−1 and holes
in the n-type region is µ p= 100 cm2 V−1 s−1 . Assume that the solar cell behaves as an ideal diode. Calculate the built-in voltage, theopen-circuit voltage and the conversion efficiency of the cell.
Jph = 350 Am−2 .
NA = 1017 cm−3 = 1023 m−3 .
ND = 1019 cm−3 = 1025 m−3 .
L N = 500 · 10−6 m.
L P = 10 · 10−6 m.
D N = (kT/e)µn = 0.0258 V · 1000 · 10−4 cm2 V−1 s−1 = 2.58 · 10−3 m2 s−1 .
DP = (kT/e)µ p = 0.0258 V · 100 · 10−4 cm2 V−1 s−1 = 2.58 · 10−4 m2 s−1 .
Using Eq. (8.16) we calculate the built-in voltage of the cell,!
1023 1025
kT
NA ND
ψ0 =
ln
=
0.0258V·ln= 0.93 V.
e(1.5 · 1016 )2n2i

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106

9. Solar Cell Parameters and Equivalent Circuit

According to the assumption that the solar cell behaves as an ideal diode, the Shockley equation describing the J-V characteristic is applicable. Using Eq. (8.25) we determine the saturation-current density,!
DP
2.58 · 10−3 m2 s−1
2.58 · 10−4 m2 s−1
DN−19
16 2 −62+= 1.602 · 10
C (1.5 · 10 ) m+
J0 = e ni
L N NAL P ND500 · 10−6 m 1023 m−3100 · 10−6 m 1025 m−3= 1.95 · 10−9

CA
= 1.95 · 10−9 2 .
m2 sm

Using Eq. (9.1) we determine the open-circuit voltage,
kT
ln
Voc =
e



Jph
J0



+1

= 0.0258 V ln

350 Am−2+11.95 · 10−9 Am−2!

= 0.67 V.

The fill factor of the cell can be calculated from Eq. (9.3). First, we normalise Voc ,
0.67 VkT== 26.8.
voc = Voce0.0258 V
Hence,

voc − ln (voc + 0.72)= 0.84.
voc + 1
Finally, the conversion efficiency is determined using Eq. (9.4),
FF =

η=

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Jsc Voc FF
350 Am−2 0.67 V 0.84=
= 0.197 = 19.7%.
Pin1000 Wm−2

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107

Ext. Quantum Efficiency (–)

9.2. The external quantum eﬃciency
1

ciency is then determined as

0.8

EQE(λ0 ) =

0.6

0.2

400

600

800

1000

1200

Wavelength (nm)Figure 9.2: The external quantum efficiency of a high-quality

crystalline silicon based solar cell.

9.2

(9.5)

where e is the elementary charge and Φph is the photon
flux incident on the solar cell. Since Iph is dependent
on the bias voltage, the bias voltage must be fixed. The
photon flux is usually determined by measuring theEQE of a calibrated photodiode under the same lightsource.

0.40Iph (λ0 ),e Φph (λ0 )The external quantum eﬃciency

The external quantum efficiency EQE(λ0 ) is the fractionof photons incident on the solar cell that create electronhole pairs in the absorber, which are successfully collected. It is wavelength dependent and is usually measured by illuminating the solar cell with monochromatic
light of wavelength λ0 and measuring the photocurrent
Iph through the solar cell. The external quantum effi-

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Figure 9.2 illustrates a typical EQE for a high quality
crystalline silicon based solar cell. We can identify the
major optical loss mechanisms for such a solar cell: For
short wavelengths only a small fraction of the light isconverted into electron-hole pairs. Most photons are
already absorbed in the layers that the light traverses
prior to the absorber layer. For long wavelengths, thepenetration depth2 of the light exceeds the optical thickness of the absorber. Then the absorber itself becomestransparent so that most of the light leaves the solar
cell before it can be absorbed. We can see that for thistype of solar cells the EQE is close to 1 for a broad
wavelength band. Hence, in this band almost all absorbed photons are converted into electron-hole pairsthat can leave the solar cell.
2 According to Lambert-Beer’s law, the intensity of light in an absorbing layer
decays exponentially, I (z)∝ exp (−αz), where α is the absorption coeﬃcient.
The penetration depth is then defined as dpen (λ0 )=1/α(λ0 ). The absorption coefficient α is related to the imaginary part k̃ of the complex refractive index viaα(λ0 )=4π k̃/λ0 .

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108

When a bias voltage of 0 V is applied, the measured
photocurrent density equals the short circuit current
density. When applying a sufficiently large reversed
bias voltage, it can be assured that nearly all photogenerated charge carriers in the intrinsic layer are collected. Thus, this measurement can be used to study
the optical effectiveness of the design, i.e. light trapping
and light absorption in inactive layers, such as the TCO
layer, doped layers and the back reflector.
Measuring the EQE

EQE spectra are measured using an EQE-setup that also
called spectral response setup. For this measurement, usually a wavelength selective light source, a calibrated
light detector and a current meter are required. Usually,
the used light sours is a xenon gas discharge lamp that asa very broad spectrum covering all the wavelengths important for the solar cell performance. With the help offilters and monochromators a very narrow wavelengthband of photon energies can be selected that then canbe incident on the solar cell.
As already seen in Eq. (9.5), EQE(λ) is proportional
to the the current divided by the photon flux. Whilethe current can be easily determined using an Ampere
meter, the photon flux must be determined indirectly.
This is done by performing a measurement with acalibrated photodetector (or solar cell), whose EQE is

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9. Solar Cell Parameters and Equivalent Circuit

known. Via this measurement we findΦph (λ0 ) =ref ( λ )Iph
0e EQEref (λ0 )

,

(9.6)

By combining Eqs. (9.5) and (9.6) we therefore findIph (λ0 )EQE(λ0 ) = EQEref (λ0 ) ref.
Iph (λ0 )(9.7)

The EQE therefore can be determined by performing
two current measurements. Of course it is very important that the light source is sufficiently stable during thewhole measurement as we assume that the photon fluxin the reference measurement and the actual measurement is unchanged.
If we perform the EQE measurement under short circuit
conditions, the measurement can be used to determine the short circuit current density Jsc . Determining Jsc
via the EQE has the advantage that it is independentof the spectral shape of the used light source, in contrast to determining the Jsc via an JV measurement.
Secondly, on lab scale the real contact area of solar cells
is not accurately determined during JV measurements.
When using shading masks, the EQE measurement is
independent on the contact area. Hence, for accurately
measuring the sort circuit current density, it is not sufficient to rely on JV measurements only, but EQE setups
have to be used.

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109

9.3. The equivalent circuit

For determining Jsc we combine that the photon flux at
a certain wavelength with the EQE at this wavelength,
leading to the flow of electrons leaving the solar cellat this wavelength. Jsc then is obtained by integrating
across all the relevant wavelength,
Jsc = −e

Z λ2

(a)

EQE(λ)ΦAM1.5 (λ) dλ.

I
+

(9.8)

λ1

For crystalline silicon, the important range would befrom 300 to 1200 nm.

9.3

The equivalent circuit

The J-V characteristic of an illuminated solar cell thatbehaves as the ideal diode is given by Eq. (8.27),
J (Va ) = Jrec (Va ) − Jgen (Va ) − Jph




eVa
− 1 − Jph .
= J0 exp
kT
This behaviour can be described by a simple equivalent circuit, illustrated in Fig. 9.3 (a), in which a diodeand a current source are connected in parallel. The diode is formed by a p-n junction. The first term in Eq.
(8.27) describes the dark diode current density while
the second term describes the photo-generated current
density. In practice the FF is influenced by a series resistance Rs , and a shunt resistance R p . The influence of

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Iph

Id

V

−
(b)

I
Rs

Iph

Id

Rp

+
V

−
Figure 9.3: The equivalent circuit of an (a) ideal solar cell anda (b) solar cell with a series resistance Rs and a shunt resistance
Rp.

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110

100

Current Density (A/m2)(b)

Rs = 0.0 Ω
Rs = 2.5 Ω
Rs = 5.0 Ω
Rs = 7.5 Ω
Rs = 10. Ω

200

0

100

Rp = 104 Ω

-100
-200
-300
-400
-500

Rs
0

0.1

0.2

Rp = 0.001 Ω
Rp = 0.005 Ω
Rp = 0.010 Ω
Rp = 0.030 Ω
Rp = 10400 Ω

200

Current Density (A/m2)(a)

9. Solar Cell Parameters and Equivalent Circuit

0.3

0.4

0.5

Voltage (V)0.60Rs = 0 Ω

-100

Rp

-200
-300
-400

Voc
0.7

0.8

-500

Voc
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Voltage (V)Figure 9.4: Effect of the (a) series resistance and (b) parallel resistance on the J-V characteristic of a solar cell.

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111

9.3. The equivalent circuit

these parameters on the J-V characteristic of the solar
cell can be studied using the equivalent circuit presented in Fig. 9.3 (b). The J-V characteristic of the onediode equivalent circuit with the series resistance andthe shunt resistance is given bye (V − AJRs )V − AJRs
J = J0 exp−1 +− Jph ,
kT
Rp
(9.9)
where A is the area of the solar cell. The effect of Rs
and R p on the J-V characteristic is illustrated in Fig. 9.4.
In real solar cells the FF is influenced by additional recombination occurring in the p-n junction. This nonideal diode is often represented in the equivalent circuit by two diodes, an ideal one with an ideality factor
equal to unity and a non-ideal diode with an ideality
factor larger than one. The equivalent circuit of a real
solar cell is presented in Fig. 9.5. The J-V characteristicof the two-diode equivalent circuit is given bye (V − AJRs )−1
J = J01 exp
n1 kT




e (V − AJRs )−1+ J02 exp
(9.10)n2 kT
V − AJRs+− Jph ,
Rp

I
Rs
Iph

Id1

1

n1=1

2

Id2
n2>1

Rs

+
V
−

Figure 9.5: The equivalent circuit of a solar cell based on thetwo-diode model.

where J01 and J02 are the saturation currents of the twodiodes, respectively. n1 and n2 are the ideality factors of
the two diodes.

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112

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9. Solar Cell Parameters and Equivalent Circuit

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10

Losses and Efficiency Limits
10.1

In the previous chapters we have learned the basic
physical principles of solar cells. In this chapter we willbring the different building blocks together and analyse,
how efficient a solar cell theoretically can be.

The thermodynamic limit

The most general efficiency limit is the thermodynamic
efficiency limit. In this limit, the photovoltaic device isseen as a thermodynamic heat engine, as illustrated inFig. 10.1. Such a heat engine operates between two heat
reservoirs; a hot one with temperature TH and a cold
one temperature TC . For the heat engine, three energy
flows are relevant. First, the heat flow Q̇ H from the
hot reservoir to the engine. Secondly, the work Ẇ thatis performed by the engine and thirdly, heat flowing
from the engine to the cold reservoir that serves as aheat sink, Q̇C . Clearly, the third energy flow is a loss andconsequently, the efficiency of the heat engine is given

After discussing different efficiency limits and the major
loss mechanisms, we will finalise this chapter with theformulation of three design rules that should always be
kept in mind when designing solar cells.
It is very important to understand, why a solar cell cannot convert 100% of the incident light into electricity.
Different efficiency limits can be formulated, each taking different effects into account.
113
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114

hot reservoir

10. Losses and Eﬃciency LimitsQ H

engineQ C

cold reservoir(heat sink)W
Figure 10.1: Illustrating the major heat flows in a generic heatengine.

by
η=

Ẇ
.
Q̇h

(10.1)The second law of thermodynamics teaches us that the entropy of an independent system never decreases. It only
increases or stays the same. While the heat flows Q̇ H
and Q̇C carry entropy, the performed work W is anentropy-free form of energy. Thermodynamics teaches
us that there is an efficiency limit for the transformationof heat into entropy-free energy. An (ideal) engine thathas this maximal efficiency is called a Carnot engine andits efficiency is given by
ηCarnot = 1 −

TC
.
TH

(10.2)For a Carnot engine, the entropy does not increase.
Note that all the temperatures must be given in a temperature scale where the absolute zero takes the value

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0. For example, the Kelvin scale is such a scale. From
Eq. (10.2) we can already see two important trends thatare basically true for every heat engine, i.e. also steam
engines of combustion engines. The efficiency increases,
if the higher temperature TH is increased and/or the
lower temperature TC is decreased.
Let us now look at a solar cell that we imagine as aheat engine operating between an absorber of temperature TA (this is our hot reservoir) and a cold reservoir,
which is given by the surroundings and that we assume
to be of temperature TC = 300 K. What this heat engine
actually does is that it converts the energy stored in theheat of the absorber into entropy-less chemical energythat is stored in the electron-hole pairs. We may here
assume that the transformation of chemical energy into
electrical energy happens lossless, i.e. with an efficiencyof 1. Clearly, the efficiency of this thermodynamic heatengine is given byηTD = 1 −TC.
TA

(10.3)

The absorber will be heated by absorbing sunlight. As
we look at the ideal situation, we assume the absorber
to be a black body that absorbs all incident radiation.
Further, we assume the sun to be a blackbody of temperature TS = 6000 K. As we have seen in Chapter 5,
the solar irradiance incident onto the absorber is given

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115

10.2. The Shockley-Queisser Limitby
IeS = σTS4 Ωinc ,(10.4)

where Ωinc is the solid angle covered by the incident
sunlight. As the absorber is a black body of temperature TA it also will emit radiation. The emittance of theabsorber is given by
EeA = σTA4 Ωemit .

Note that η A is the larger the lower TA while the efficiency of the heat engine ηTD is the larger the higherTA .
For the total efficiency of the ideal solar cell we combine Eq. (10.3) with Eq. (10.7) and obtain(10.5)

ηSC =Ωemit is the solid angle into that the absorber can emit.
We can easily see that the efficiency of the absorption
process is given by
ηA =T4
IeS − EeA
EAΩ
= 1 − eS = 1 − emit A4 .
S
Ωinc TS
Ie
Ie(10.6)

The absorber efficiency can be increased by increasingΩinc , which can be achieved by concentrating the incident sunlight. Under maximal concentration sunlightwill be incident onto the absorber from all angles of thehemisphere, i.e. Ωmax
inc = 2π. We assume the absorber
to be open towards the surroundings and hence the sunon the top side. Its bottom side is connected to the heatengine such that radiative loss only can happen via the
top side. Therefore, also Ωemit = 2π. The maximal absorber efficiency is therefore achieved under maximal
concentration and it is given by
TA4
.
η max
=
1
−
A
TS4

Copyright Delft University of Technology, 2014(10.7)T4
1 − A4
TS

!
1−

TC
TA


.

(10.8)

Figure 10.2 shows the absorber efficiency, the thermodynamic efficiency and the solar cell efficiency. We
see that the solar cell efficiency reaches its maximumof about 85% for an absorber temperature of 2480 K.
Please note that the solar cell model presented in thissection does not resemble a real solar cell but that it is
only intended to discuss the physical limit of converting solar radiation into electricity. Several much moredetailed studies on the thermodynamic limit have been
performed. We want to refer the interested reader toworks by Würfel [21] and Markvart et al. [30–32].

10.2

The Shockley-Queisser Limit

We now will take a look at the the theoretical limit for
single-junction solar cells. This limit is usually referred
to as the Shockley-Queisser (SQ) limit, as they where theThis copy is provided for free, for personal use only.

116

10. Losses and Eﬃciency Limits

1.0

ηA

ηTD

0.8
Efficiency (–)

first ones to formulate this limit based purely on physical assumptions and without using empirically determined constants [23]. We will derive the SQ limit in a
two-step approach. First, we will discuss the losses due
to spectral mismatch. Secondly, we also will take into account that the solar cell will have a temperature different from 0 K which means that it emits electromagnetic
radiation according to Planck’s law. Just like Shockley
and Queisser, we will do this with the detailed balance
approach.

ηSC

0.6
0.4

10.2.1

0.2
0.0

0

1000

2000

3000

4000

5000

6000

Absorber temperature TA (K)

Figure 10.2: The absorber efficiency η A , the thermodynamic

efficiency ηTD and the combined solar cell efficiency ηSC underfull concentration for a solar temperature of 5800 K and an
ambient temperature of 300 K.

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Spectral mismatch

There are two principal losses that strongly reduce theenergy conversion efficiency of single-junction solar
cells. As discussed in Chapter 8, an important part ofa solar cell is the absorber layer, in which the photonsof the incident radiation are efficiently absorbed resulting in a creation of electron-hole pairs. In most cases,
the absorber layer is formed by a semiconductor material, which we characterise by its bandgap energy Eg .
In principle, only photons with energy higher than theband gap energy of the absorber can generate electronhole pairs. Since the electrons and holes tend to occupy energy levels at the bottom of the conduction
band and the top of the valence band, respectively,
the extra energy that the electron-hole pairs receivefrom the photons is released as heat into the semicon-

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117

10.2. The Shockley-Queisser Limit

ductor lattice in the thermalisation process. Photons withenergy lower than the band gap energy of the semiconductor absorber are in principle not absorbed andcannot generate electron-hole pairs. Therefore thesephotons are not involved in the energy conversion process. The non-absorption of photons carrying less energy than the semiconductor band gap and the excessenergy of photons, larger than the band gap, are thetwo main losses in the energy conversion process using solar cells. Both of these losses are thus related tothe spectral mismatch between the energy distributionof photons in the solar spectrum and the band gap of asemiconductor material.
Shockley and Queisser call the efficiency that is obtained when taking the spectral mismatch losses into
account the ultimate efficiency, that is given according
to the hypothesis that ‘each photon with energy greaterthan hνg produces one electronic charge e at a voltage of
Vg = hνg /e’ [23].
Let us now determine the fraction of energy of theincident radiation spectrum that is absorbed by asingle-junction solar cell. When we denote λ g as the
wavelength of photons that corresponds to the band
gap energy of the absorber of the solar cell, only thephotons with λ ≤ λ g are absorbed. The fraction pabs of
the incident power that is absorbed by a solar cell and

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used for energy conversion can be expressed as
R λg hcΦph, λ dλ,pabs = R0∞ hcλ
0 λ Φph, λ dλ(10.9)

where Φph, λ is the spectral photon flux of the incident
light as defined in Chapter 5. The fraction of the absorbed photon energy exceeding the bandgap energy islost because of thermalisation. The fraction of the absorbed energy that the solar can deliver as useful energy is then given by
R λg
Eg 0 Φph, λ dλpuse = R λ.
(10.10)g hc
0λ Φph, λ dλBy combining Eqs. (10.9) and (10.10), we can determine
the ultimate conversion efficiency,
R λg
Eg 0 Φph, λ dλ.
(10.11)
ηult = pabs puse = R ∞ hc
0 λ Φph, λ dλFigure 10.3 illustrates the fraction of the AM1.5 spectrum that can be converted into a usable energy by acrystalline silicon solar cell. Figure 10.4 shows the ultimate conversion efficiency of a solar cells band gapof a semiconductor absorber for three different radiation spectra, black-body radiation at 6000 K, AM0
and AM1.5 solar radiation spectra. The figure demonstrates that in the case of a crystalline silicon solar cellThis copy is provided for free, for personal use only.

118

10. Losses and Eﬃciency Limits(Eg = 1.12 eV) the losses due to the spectral mismatchaccount for almost 50%. It also shows that an absorbermaterial for a single junction solar cell has an optimalband gap of 1.1 eV and 1.0 eV for the AM0 and AM1.5
spectra, respectively. Note that the maximum conversion efficiency for the AM1.5 spectrum is higher thanthat for AM0, while the AM0 spectrum has a higher
overall power density. This is because of the fact that
the AM1.5 spectrum has a lower power density in partsof the spectrum that are not contributing to the energy
conversion process as can be seen in Fig. 10.3. The dipsin the AM1.5 spectrum also result in the irregular shapeof the conversion efficiency as function of the band gap.

1.4
1.2
1.0



Spectral irradiance W/ m2 nm



1.6

0.8
0.6
0.4

10.2.2

0.2
0.0

0

500

1000

1500

2000

2500

Wavelength (nm)Figure 10.3: The fraction of the AM1.5 spectrum that can beconverted into a usable energy by a crystalline silicon solar cellwith Eg = 1.12 eV.

Detail balance limit of the eﬃciency

Similar to Shockley and Queisser we now will formulate
the detail balance limit of the efficiency. But before we start
we briefly will discuss the reason that the ultimate efficiency formulated above is not physical for solar cellswith temperatures higher than 0 K.
Let us estimated that the solar cell is embedded in anenvironment of ambient temperature of 300 K and thatthe solar cell temperature also is 300 K. As the solar cell
will be in thermal equilibrium with its surroundings,
it will absorb thermal radiation according to the ambient temperature and it will also emit the same amount

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119

10.2. The Shockley-Queisser Limit

0.6

AM 0
AM 1.56000 K

0.5
Efficiency (–)

of radiation. Therefore recombination of electron-holepairs will be present in the semiconductor leading toa recombination current different from zero. As wehave seen in Eq. (9.1), the open-circuit voltage will be
reduced with increasing recombination current, which
is an efficiency loss.
We start the derivation of the detailed balance limit
with recalling the definition of the efficiency from Eq.
(9.4),Jph Voc FFη=.
(10.12)Pin
For calculating ηult we made the assumption that ‘each
photon with energy greater than hνg produces one electronic
charge e at a voltage of Vg = hνg /e’. Under the same assumption, we obtain for the short circuit current density

0.4
0.3
0.2
0.1
0

0.5

1

1.5

2

2.5

3

Bandgap energy (eV)Figure 10.4: The ultimate conversion efficiency for the black

body spectrum at 6000 K, the AM0 and AM1.5 solar radiation
spectra, limited only by the spectral mismatch as a function of
the band gap of a semiconductor absorber in single junction
solar cells.

Jph ( Eg ) = −eZ λg0Φph, λ dλ(10.13)

with λ g = hc/Eg . Note that we here implicitly assumed
that the photo-generated current Jph is equivalent to theshort circuit current. This assumption is valid as the recombination current originating from thermal emission
is orders of magnitude lower than the photo-generated
current. By combining Eqs. (10.11) and (10.13) we findJph = −P ηePin ηult = − in ult .
Eg
Vg(10.14)

Let us now define the bandgap utilisation efficiency ηV

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120

10. Losses and Eﬃciency Limits

that is given by
Voc
ηV =
Vg

(10.15)and tells us the fraction of the bandgap that can beused as open-circuit voltage (Shockley and Queisser use
the letter v for this efficiency). We now combine Eqs.
(10.12), (10.12) and (10.15) and find
η = ηult ηV FF.

(10.16)For determining the efficiency in the detailed balance
limit, we therefore must determine the bandgap utilisation efficiency and the fill factor. Let us start with ηV .
According to Eq. (9.1), the open circuit voltage is given
as


Jph
kT
Voc =
ln
+1 .
(10.17)eJ0
The only unknown in this equation is the dark current J0 . We assume the solar cell to be in thermal equilibrium with its surroundings at an ambient temperature of Ta = 300 K. Further, we assume that the solar
cell absorbs and emits as a black body for wavelengths
shorter than the bandgap wavelength of the solar cellabsorber. For wavelengths longer than the bandgap weassume the solar cell to be completely transparent thus
to neither absorb nor emit. This is the same assumptionthat we already used for the absorption of sunlight.
BB as
Using the equation for the blackbody radiance Leλ

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given in Eq. (5.18a) we find for the radiative recombination current
Z λg ZBB
Leλ(λ; Ta ) cos θ dΩ dλ

 −1


Z λg
2hc2
hc
= −2eπdλ,−1
exp
λk B Ta
λ5
0(10.18)where the factor 2 arises from the fact that we assume
the solar cell to emit thermal radiation both at its front
and back sides.

J0 ( Eg ) = −2e02πCombining Eqs. (10.15) with (10.17) we find"#Jph ( Eg )ηV ( Eg ) = kT/Eg ln+1 .
J0 ( Eg )(10.19)Figure 10.5 shows the bandgap utilisation efficiency forthree different spectra of the incident sunlight. For abandgap of 1.12 eV this efficiency is about ηV ≈ 77%.
For the fill factor we take the empirical but very accurate approximationFF =voc − ln (voc + 0.72)
voc + 1

(10.20)with voc = eVoc /kT. We already discussed this approximation in Eq. (9.3).
Figure 10.6 finally shows the Shockley-Queisser efficiency limit for three different spectra of the incident

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121

10.2. The Shockley-Queisser Limit
0.4

AM 0
AM 1.56000 K

Efficiency (–)

0.3

0.9

Efficiency (–)

0.8

0.2

0.1

0.7

AM 0
AM 1.56000 K

0.6

0

0.5

1

1.5

2

2.5

3

Bandgap energy (eV)0.5

Figure 10.6: The Shockley-Queisser efficiency limit for the

0.4

black body spectrum at 6000 K, and the AM0 and AM1.5 solar
spectra.
0.5

1

1.5

2

2.5

3

Bandgap energy (eV)Figure 10.5: The bandgap utilisation efficiency ηV for theblack body spectrum at 6000 K, and the AM0 and AM1.5 solar
spectra.

Copyright Delft University of Technology, 2014

light. For the AM1.5 spectrum the limit is about 33.1%
at 1.34 eV. For AM0 it is 30.1% at 1.26 eV.
The major loss mechanisms that are taken into accountin the Shockley-Queisser limit are illustrated in Fig.
10.7. The major losses are non-absorbed photons below
the bandgap and thermalised energy of photons above
the bandgap. The other losses are due to the voltage
loss because of thermal radiation and the fill factor being different from 100%.

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122

10. Losses and Eﬃciency Limits

10.2.3

It is very important to note that the Shockley-Queisser(SQ) limit is not directly applicable to solar cells madefrom crystalline silicon. The reason for this is that silicon is a so-called indirect bandgap semiconductor as we
will discuss in detail in Chapter 12. This means that
Auger recombination, which is a non-radiative recombination mechanism, is dominant. For the derivation of
the SQ limit we assumed that only radiative recombination is present. Clearly, this assumption cannot be valid
for crystalline silicon solar cells. Several attempts to calculate the efficiency limit while taking radiative recombination mechanisms into account were performed in
the past. A study from 2013 by Richter et al. derives anefficiency limits of 29.43% for silicon solar cells .

10.9

other losses

Efficiency (–)

0.8
0.7
0.6
0.5

thermalisation
lossesbelow bandgap losses

0.4
0.3
0.2

SQ limit

0.1
0

0.5

1

1.5

2

2.5

Eﬃciency limit for silicon solar
cells

3

Bandgap energy (eV)Figure 10.7: The major loss mechanisms in the Shockley

Queisser limit. For this calculation the AM1.5 spectrum wasused as incident light.

As the Shockley-Queisser limit only considers radiative recombination, it is most valid for direct band gap
materials such as GaAs. Because of its direct band gap,
radiative recombination is the limiting recombination
mechanism for GaAs.

10.3

Other Losses

The Shockley-Queisser limit is a very idealised model.
For example all optical losses are neglected. Now we

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10.3. Other Losses

will discuss several other loss mechanisms that were
not taken account above. At the end of this section wewill derive at an equation for the efficiency were allthese important losses are taken into account.

10.3.1

Optical losses

For the Shockley-Queisser limit, we only took the
bandgap energy Eg into account for deriving the efficiency limit. However, the real performance is alsostrongly influenced by the optical properties given as
the complex refractive index ñ = n − ik, which is afunction of the wavelength.
As we already discussed in Section 4.3, a part of thelight is reflected from and the other part is transmitted when light arrives on an interface between two
media. The interface is therefore characterised by the
wavelength dependent reflectance R(λ) and transmittance T (λ). All the reflections and transmissions at thedifferent interfaces in the solar cell result in a total reflectance between the solar cell and the surrounding air.
Hence, a part of the incident energy that can be converted into a usable energy by the solar cell is lost byreflection. We shall denote the total effective reflectancein the wavelength range of interest as R∗ .
As we will discuss in more detail in Chapter 12, in
most c-Si solar cells thin metal strips are placed on the

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123

front side of the solar cell that serve as front electrode.
The metal-covered area does not allow the light to enter
the solar cell because it reflects or slightly absorbs theincident light. The area that is covered by the electrodeeffectively decreases the active area of the solar cell.
When we denote the total area of the cell as Atot andthe cell area that is not covered by the electrode as A f ,the fraction of the active area of the cell is determinedby the ratio
Af
Cf =
,
(10.21)
Atot
which is called the coverage factor C f . These loss iscalled the shading loss. The design of the front electrodeis of great importance since it should one the one handminimise losses due to the series resistance of the front
electrode, i.e. should be designed with sufficient crosssection. The optimal design of the front electrode istherefore a trade-off between a high coverage factor anda sufficiently low series resistance of the front electrode.
When light penetrates into a material, it will be partially absorbed as it propagates through the material.
The absorption of light in the material depends on its
absorption coefficient and the layer thickness, as wehave seen in Section 4.4. In general, light is absorbed
in all layers of the solar cell. All the absorption in layers different from the absorber layer is loss. It is called
the parasitic absorption. Further, due to the limited thickness of the absorber layer, not all the light entering theabsorber layer is absorbed. Incomplete absorption inThis copy is provided for free, for personal use only.

124

10. Losses and Eﬃciency Limits

the absorber due to its limited thickness is an additional loss that lowers the energy conversion efficiency.
The incomplete absorption loss can be described by
the internal optical quantum efficiency IQEop , which
is defined as the probability of a photon being absorbed
in the absorber material. Since there is a chance thata highly energetic photon can generate more than one
electron-hole pair, we also define the quantum efficiency
for carrier generation ηg which represents the number ofelectron-hole pairs generated by one absorbed photon.
Usually ηg is assumed to be unity.

10.3.2

Solar cell collection losses

Not all charge carries that are generated in a solar cell
are collected at the electrodes. The photo-generated carries are the excess carriers with respect to the thermal
equilibrium and are subjected to the recombination. The
carriers recombine in the bulk, at the interfaces, and/or
at the surfaces of the junction. The recombination is determined by the electronic properties of materials thatform the junction, such as density of states introduced
into the band gap by the R-G centers. The concentration of R-G centers strongly influences the minoritycarrier lifetimes as discussed in Chapter 7.
The contributions of both the electronic and opticalproperties of the solar cell materials to the photovoltaic performance are taken into account in the absolute

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external quantum efficiency that we already defined in
Chapter 9. The EQE can be approximated by
EQE(λ) = (1 − R∗ )IQEop (λ)ηg (λ)IQEel (λ),(10.22)where the IQEel denotes the electrical quantum efficiency and is defined as the probability that a photogenerated carrier is collected.
When we take the shading losses and the EQE into account, we find the short-circuit current density to be∗ ∗∗Jsc = Jph (1 − R∗ )IQEop
ηg IQEelCf ,(10.23)where the ∗ denote averages across the relevant
wavelength range and Jph is as defined in Eq. (10.13).

10.3.3

Additional limiting factors

We have seen in Section 10.2 that the Voc of a solar celldepends on the saturation current J0 and the photogenerated current Jsc of the solar cell. The saturation
current density depends on the recombination in thesolar cell. Recombination cannot be avoided and depends on the doping of the different regions (n-type
and p-type regions) of a junction and the electronic
quality of materials forming the junction. The doping
levels and the recombination determine the bandgap
utilisation efficiency ηV that we already defined in Section 10.2.

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125

10.3. Other Losses

For determining the Shockley-Queisser limit we assumed for the FF that the solar cell behaves as an ideal
diode. In a real solar cell, however, the FF is lower than
the ideal value because of the following reasons:• The voltage drop due to the series resistance Rs of
a solar cell, which is introduced by the resistanceof the main current path through which the photogenerated carriers arrive to the external circuit. The
contributions to the series resistance come from thebulk resistance of the junction, the contact resistance between the junction and electrodes, and theresistance of the electrodes themselves.
• The voltage drop due to leakage currents, which ischaracterised by the shunt resistance R p of a solarcell. The leakage current is caused by the current
through local defects in the junction or due to theshunts at the edges of solar cells.

R λg hc• The recombination in a non-ideal solar cell results ina decrease of the FF.

10.3.4

Conversion eﬃciency

Again, we start with the expression for the efficiency
Jsc Voc FFη=.
Pin
Using Eqs. (10.23) and (10.14) and we findη=ηult Voc FF∗ ∗∗(1 − R∗ )IQEop
ηg IQEel
Cf
Vg

{z
1

where we used Eqs. (10.9) and (10.10). By filling in thedefinitions for pabs , puse , ηV and C f , we obtain [33].

R λg
Eg 0 Φph, λ dλA f eVoc∗ ∗∗(1 − R∗ ) IQEop
ηg IQEelFF .
R λg hc
Atot EgΦdλph, λ} | 0 λ {z} | {z } | {z } | {z } |{z} | {z } |{z}2This describes the conversion efficiency of a solar cellin terms of components that represent particular losses
in energy conversion.

Copyright Delft University of Technology, 2014(10.25)∗ ∗∗= pabs puse (1 − R∗ )IQEop
ηg IQEelC f ηV FF,

Φph, λ dλη = R0∞ hcλ
0 λ Φph, λ dλ|(10.24)

3

4

5

6

7

(10.26)8

1. Loss due to non-absorption of long wavelengths,
2. Loss due to thermalisation of the excess energy ofThis copy is provided for free, for personal use only.

126

10. Losses and Eﬃciency Limitspphotons,n

3. Loss due to the total reflection,
4. Loss by incomplete absorption due to the finite
thickness,

E Fn
eVoc

5. Loss due to recombination,

E Fp

6. Loss by metal electrode coverage, shading losses,
7. Loss due to voltage factor,
8. Loss due to fill factor.

10.4

Design Rules for Solar Cells

Now, as we extensively have discussed all the factorsthat limit the efficiency of a solar cell we are able todistill three design rules. We will use this design rules inPart III when we discuss different PV technologies. The
three design rules are
1. Utilisation of the band gap energy,
2. Spectral utilisation,
3. Light trapping.
We now will take a closer look at each of these design
rules.

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Figure 10.8: The open circuit voltage of a solar cell is determ-

ined by the splitting of the quasi-Fermi levels.

10.4.1

Bandgap utilisation

As we have seen earlier in this chapter, the open circuit voltage Voc is always below the voltage Vg corresponding to the bandgap, which we characterised withthe bandgap utilisation efficiency ηV . The open circuit
voltage is determined by the extent to which the quasiFermi levels are able to split, which is limited by the
charge carrier recombination mechanisms. We will discuss that various PV materials have different recombination mechanism that limit the utilisation of the band
gap energy.
Let us now discuss the first design rule, bandgap utilisation. Figure 10.8 shows a p-n-junction, with the p-doped
region on the left and the n-doped region right. Fur-This copy is provided for free, for personal use only.

10.4. Design Rules for Solar Cells

ther the quasi-Fermi levels are depicted. The extent ofsplitting between the quasi-Fermi levels determines theopen circuit voltage.
The open circuit voltage, for example expressed in Eq.
(10.17) also can be expressed in terms of the generation
rate GL , the life time τ0 of the minority charge carriers
and the intrinsic density of the charge carriers ni in thesemiconductor material,


2kT
GL τ0
Voc =
ln
.
(10.27)eni
The derivation of this equation is out of the scope of
this book.
Let us now take a closer look on Eq. (10.27). If we increase the irradiance, or in other words, the generation
rate of charge carriers, the open circuit voltage is increased. This is a welcome effect which is utilised inconcentrator photovoltaics, which we will briefly discuss in Section 13.2. Secondly, we see that the lifetime
τ0 plays an important role. The larger the lifetime of
the minority charge carrier, the larger the open circuit
voltage can be. Or in other words, the longer the lifetime, the larger the possible splitting between the quasiFermi levels and the larger the fraction of the bandgap
energy that can be utilised.
The lifetime of the minority charge carrier is determined by the recombination rate. As discussed in

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127

Chapter 7, we have to consider three different recombination mechanisms: radiative, Shockley-Read-Hall,
and Auger recombination. While radiative and Auger
recombination depend on the semiconductor itself, SRH
recombination is proportional to the density of traps
or impurities in the semiconductor. In the three recombination mechanisms energy and momentum are transferred from charge carriers to phonons or photons.
The efficiency of the different recombination processes
depends on the nature of the band gap of the used
semiconductor material used. We distinguish between
direct and indirect bandgap semiconductors. Crystalline
silicon is an indirect band gap material. The radiative
recombination in an indirect band gap material is inefficient and recombination will be dominated by theAuger mechanism. For direct band gap materials such
as GaAs under moderate illumination conditions, radiative recombination will be the dominant loss mechanism of charge carriers. For very high illumination
conditions, Auger recombination starts to play a role
as well.
To summarise, we find that in the defect rich solar cells,
the open circuit voltage is limited by the SRH recombination. In low-defect solar cells based on indirect
band gap materials, the open circuit voltage is limited
by Auger recombination. In low-defect solar cells basedon direct band gap materials, the open circuit voltage islimited by radiative recombination.

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128

10. Losses and Eﬃciency Limits

Besides band gap utilisation, it also is important to discuss the relationship between the maximum thickness
for the absorber layer of a solar cell and the dominant
recombination mechanism. As we have seen in Chapter
7, the recombination mechanism also affects the diffusion length of the minority charge carrier. The diffusion
length Ln of minority electrons is given byLn =√Dn τn ,

(10.28)where Dn is the diffusion coefficient and τ is the lifetime of the minority charge carrier. Similarly, we canformulate the diffusion length for minority holes, L p .
It is important to realise that the thickness of the absorber layer should not exceed the diffusion length. To
understand this requirement, we consider photons thatwould penetrate far into the absorber layer being absorbed. We want these charge carriers to be separated
at the p-n junction or at the back contact. If the distance of these charge carriers from the p-n junction or
the back contact is exceeding the diffusion length, theexcited charge carriers will recombine within the typicaldiffusion length before arriving at the p-n junction orback contact. In other words, their lifetime is too short.
This means that all charge carriers generated at a distance larger than the diffusion length from the p-n junction or the back contact cannot be collected and hence
are lost. If the charge carriers are generated with diffusion length, they can be collected. This means that the

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diffusion length of the minority charge carrier, limits
the maximum thickness of the solar cell.
To summarise this section, the open circuit voltage islimited by the dominant recombination mechanism. The
dominance of radiative, Auger or Shockley-Reed-Hall
recombination depends on the type of semiconductormaterials used in the solar cell and the illumination
conditions. We will discuss several different cases on
Part III on PV technology.

10.4.2

Spectral utilisation

The spectral utilisation is mainly determined by the
choice of materials from which the solar cell is made of.
As we have seen in Section 10.2, and mainly Eq. (10.13),
the photocurrent is determined by the bandgap of thematerial. For a bandgap of 0.62 eV corresponding to awavelength of 2000 nm, we could theoretically generatea short circuit current density of 62 mA/cm2 . If we consider c-Si, having a band gap of 1.12 eV (1107 nm), we
arrive at a theoretical current density of 44 mA/cm2 .
The optimal bandgap for single-junction solar cells isdetermined by the Shockley-Queisser limit, as illustrated in Fig. 10.6. For single junction solar cells, semiconductor material as such silicon, gallium arsenide
and cadmium telluride have a band gap close to the
optimum.

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129

10.4. Design Rules for Solar Cells

In Part III we will discuss various concepts that allow
to surpass the Shockley-Queisser limit. Here, we will
briefly discuss the concept of multijunction solar cells.
In this devices, solar cells with different bandgaps arestacked on top of each other. As illustrated in 10.9,
the excess energy can be reduced significantly, and thespectral utilisation will improve.

10.4.3(a)Excess energy

Light trapping

The third and last design rule that we discuss is lighttrapping. In an ideal solar cell, all light that is incident
on the solar cell should be absorbed in the absorberlayer. As we have discussed in Section 4.4, the intensityof light decreases exponentially as it travels through anabsorptive medium. This is described by the LambertBeer law that we formulated in Eq. (4.25),
I (d) = I0 exp(−αd).

Excess energy(b)(10.29)From the Lambert-Beer law it follows that at the side,at which the light is entering the film, more light is absorbed in reference to the back side. The total fractionof the incident light absorbed in the material is equal to
the light intensity entering the absorber layer minus theintensity transmitted through the absorber layer,
I abs (d) = I0 [1 − exp(−αd)].

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Figure 10.9: Illustrating the lost excess energy in (a) a single-(10.30)juntion and (b) a multi-junction solar cells.

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130

10. Losses and Eﬃciency Limits

Absorption coefficient α (cm-1)

107

GaAs
InP
Ge
Si

106
105
104
103
102
101
100

200 400

600

800 1000 1200 1400 1600 1800 2000

Wavelength λ (nm)Figure 10.10: Absorption coefficients of different semicon-

ductors.

Ideally, we would like to absorb a solar cell 100% of theincident light. Such an absorber is called optically thickand has a transmissivity very close to 0. As we can see
from the Lambert-Beer law, this can be achieved byeither absorbers with a large thickness d or with verylarge absorption coefficients α.
Figure 10.10 shows the absorption coefficients for four
different semiconductor materials: germanium (Ge), silicon (Si), gallium arsenide (GaAs) and indium phosphide (InP). We notice that germanium has the lowest band gap. It starts to absorb at long wavelengths,

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which corresponds to a low photon energy. GaAs has
the highest band gap, as it starts to absorb light at
the smallest wavelength, or highest photon energy.
Secondly, if we focus on the visible spectral part from
300 nm up 700 nm, we see that the absorption coefficients of InP and GaAs are significantly higher than forsilicon. This is related to the fact that InP and GaAs are
direct band gap materials as discussed earlier. Materials
with an indirect band gap have smaller absorption coefficients. Only in the very blue and ultraviolet part below 400 nm, Si has a direct band gap transition. Silicon
is a relatively poor absorber. Therefore for the same
fraction of light thicker absorber layers are required incomparison to GaAs.
In general, for all semiconductor materials the absorption coefficient in the blue is orders of magnitude
higher than in the red. Therefore the penetration depthof blue light into the absorber layer is rather small. In
crystalline silicon, the blue light is already fully absorbed within a few nanometers. The red light requires
an absorption path length of 60 µm to be fully absorbed. The infrared light is hardly absorbed, and afteran optical path length of 100 µm only about 10% of thelight intensity is absorbed.
As the absorption of photons generates excited charge
carriers, the wavelength dependence of the absorption
coefficient determines the local generation profile of thecharge carriers. At the front side where the light enters

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131

10.4. Design Rules for Solar Cells

the absorbing film, the generation of charge carriers
is significantly higher than at the back side. It follows
that the EQE values measured in the blue correspond to
charge carriers generated close to the front of the solarcell, whereas the EQE in the red part represents charge
carriers generated throughout the entire absorber layer.
Further, it is important to reduce the optical loss mechanisms such as shading losses, reflection, and parasitic absorption that we already discussed in Section 10.3.1.

The first method is based on a clever utilisation of theFresnel equations that we discussed in Section 4.3. For
understanding how this can work, we first will take alook at interfaces with silicon, the most common usedmaterial for solar cells. Let us consider light of 500 nm
wavelengths falling onto an air-silicon interface perpendicularly. At 500 nm, the refractive index of air isn0 = 1 and that of silicon is ns = 4.3. With the Fresnel
equations we hence find that the optical losses due toreflection are significant with 38.8%.

Total reflectivity

For reducing the reflection, anti-reflective coatings(ARC) can be used. Light that is impinging onto a surface between two media with different refractive indiceswill always be partly reflected and partly transmitted.
In order to reduce losses, it is important to minimise
these reflective losses.

0.8

n0 = 1.0
ns = 4.3

λ = 500 nm

n1opt = n0 ns » 2.1

0.6
0.4
0.2
0.0
1.0

1.5

2.0
2.5
3.0
Refractive index n1

3.5

4.0

Figure 10.11: Illustrating the effect of an interlayer withrefractive index n1 in between n0 and ns on the reflectivity.

The reflection can be significantly reduced by introducing an interlayer with a refractive index n1 with a

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132

10. Losses and Eﬃciency Limits

value in between that of n0 and ns . If no multiple reflection or interference is taken into account, it can easily be shown that the reflectivity becomes minimal if
n1 =√waves by simply adding themC ( x, t) = A( x, t) + B( x, t)= A0 eikx−iωt + B0 eikx−iωt+iφ
n0 n s .

(10.31)This is also seen in Fig. (10.11), where n1 takes all thevalues in between n0 and ns . In this example, including a single interlayer can reduce the reflection at theinterface from 38.8% down to 22.9%. If more than one
interlayers are used, the reflection can be reduced evenfurther. This technique is called refractive index grading.
In another approach constructive and destructive interference of light is utilised. In Chapter 4 we alreadydiscussed that light can be considered as an electromagnetic wave. Waves have the interesting properties thatthey can interfere with each-other, they can be superimposed. For understanding this we look at two waves
A and B that have the same wavelength and cover thesame portion of space,A( x, t) = A0 eikx−iωt ,(10.32a)B( x, t) = B0 eikx−iωt+iφ .

(10.32b)

The letter φ denotes the phase shift between A and B
and is very important. We can superimpose the two

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= A0 eikx−iωt + B0 eikx−iωt eiφ= A0 + B0 eiφ eikx−iωt .

(10.33)

The amplitude of the superimposed wave is thus
C0 = A0 + B0 eiφ .

(10.34)

Depending of the phase shift, the superimposed wave
will be stronger or weaker than A and B. If φ is inphase, i.e. a multiple of 2π, i.e. 0, 2π, 4π, . . ., we will
have maximal amplification of waves,
C0 = A0 + B0 · 1 = A0 + B0 .

(10.35)This situation is called constructive interference.
But if φ is in antiphase, i.e. from the set π, 3π, 5π, . . . wehave maximal attenuation of the waves, or destructive
interference,
C0 = A0 + B0 · (−1) = A0 − B0 .

(10.36)

Based on this principle, we can design an anti-reflection
coating, as illustrated in Fig. 10.12. The green wave
shows the reflection back from the first interface andthe red wave shows the wave which is reflected backThis copy is provided for free, for personal use only.

133

10.4. Design Rules for Solar Cells(a)from the second interface. If we look at the two waves
coupled out of this system, they appear to be in antiphase. As a result the total amplitude of the electricfield of the outgoing wave is smaller and hence thetotal irradiance coupled out of the system is smaller
as well.

(b)
g
rin

It can be shown easily that the have antiphase, when
the product of the refractive index and thickness of theinterlayer is equal to the wavelength divided by four,

Destructive
interference

Constructive
interference

n1 d =

Reflected
waves 180°
out of phase

Reflected
waves
in phase

Using an antireflection coating based on interference
demands that the typical length scale of the interlayer
thickness must be in the order of the wavelength.

e

e

rk

da

}

ng
fri

Path length
difference =odd multipleof λ/2

f
ht

ig
br

}

Path length
difference =even multipleof λ/2

Figure 10.12: Illustrating the working principle of an anti-

reflective coating based on interference [34].

λ0.
4(10.37)The last approach that we discuss for realising antireflective coating is using textured interfaces. Here, we
consider the case where the typical length scales of thesurface features are larger than the typical wavelengthof light. In this case, which is also called the geometrical limit, the reflection and transmission of the lightrays are fully determined by the Fresnel equations andSnell’s law.
The texturing helps to enhance the coupling of lightinto the layer. For example, for light that is perpendicular incident, light that is reflected at one part of the
textured surface can be reflected into angles in which
the trajectory of the light ray is incident a second time

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134

10. Losses and Eﬃciency Limits
Incident
light
36°

reflected

refl.

Figure 10.13: Illustrating the effect of texturing.

somewhere else on the interfaces, as illustrated in Fig.
10.13. Here, another fraction of the light will be transmitted into the layer and effectively less light will bereflected, when compared to a flat interface in-between
the same materials.

it should be optically thick in order to absorb as muchlight as possible. In principle, if the light can be reflected back and force inside the absorber until everything
is absorbed. However, at every internal reflection partof the light is transmitted out of the film.
But if the light would travel through the layer at anangle larger than the critical angles of the front andback interfaces of the absorber, it could stay there until everything is absorbed without any loss. Unfortunately, it is very difficult if not impossible to design
such a solar cell.
Note that textured interfaces do not help in this case.
For the same reason more light can be coupled into theabsorber using such a texturisation, also more light iscoupled out of the absorber!

In summary, we have discussed three types of antireflective coatings: Rayleigh films with intermediate
refractive indexes, anti-reflection coatings based on destructive interference and enhanced in-coupling of light
due to scattering at textured interfaces.
At the end of this chapter, we want to discuss something that is very important for thin-film solar cells. Because of reducing production cost but also because of
reducing bulk recombination it is desirable to have theabsorber layer as thin as possible. On the other hand,

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Part IIIPV Technology

135
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11

A Little History of Solar Cells
We will start our discussion on PV technology with abrief summary of the history of solar energy in generaland of photovoltaics in particular. Already in the seventh century BCE, humans used magnifying glasses toconcentrate sunlight and hence to make fire. Later, theancient Greeks and Romans used concentrating mirrors
for the same purpose.

be considered as the World’s first solar collectors.
In 1839, the French phycisist Alexandre-EdmondBecquerel, discovered the photovoltaic effect at an ageof only 19 years. He observed this effect in an electrolytic cell, which was made out of two platinum electrodes, placed in an electrolyte. An electrolyte is anelectrically conducting solution; Becquerel used silver
chloride dissolved in an acidic solution. Becquerel observed that the current of the cell was enhanced when
his setup was irradiated with sunlight.

In the 18th century the Swiss physicist Horace-Bénédict
de Saussure build heat traps, which are a kind of miniature green houses. He constructed hot boxes, consisting of a glass box, within another bigger glass box,with a total number of up to five boxes. When exposed
to direct solar irradiation, the temperature in the innermost box could rise up to values of 108°C; warm
enough to boil water and cook food. These boxes canIn the 1860s and 1870s, the French inventor Augustin
Mouchot developed solar powered steam engines using
the World’s first parabolic trough solar collector, that wewill discuss in Chapter 20. Mouchot’s motivation was
137

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138

his believe that the coal resources were limited. At thattime, coal was the energy source for driving steam enginges. However, as coal became cheaper, the Frenchgovernment decided that solar energy was too expensive and stopped funding Mouchet’s research.
In 1876, the British natural philosopher William Grylls
Adams together with his student Richard Evans Day
demonstrated the photovoltaic effect in a junction basedon platinum and the semiconductor selenium, howeverwith a very poor performance. Seven years later, theAmerican inventor Charles Fritts managed to make a
PV-device based on a gold-selenium junction. The energy conversion efficiency of that device was 1%.
In 1887, the German physicist Heinrich Hertz discovered the photo-electric effect, already briefly mentioned in Chapter 3. In this effect, electrons are emitted from a material that has absorbed light with awavelength shorter than a material-dependent thresholdfrequency. In 1905 Albert Einstein published a paper
in which he explained the photoelectric effect with assuming that light energy is being carried with quantised packages of energy [19], which we nowadays callphotons.
In 1918 the Polish chemist Jan Czochralski invented
a method to grow high-quality crystalline materials.
This technique nowadays is very important for growing monocrystalline silicon used for high-quality siliconsolar cells that we will study in detail in Chapter 12.

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11. A Little History of Solar Cells

The development of the c-Si technology started in thesecond half of the 20th century.
In 1953, the American chemist Dan Trivich was the first
one to perform theoretical calculations on the solar cell
performance for materials with different bandgaps.
The real development of solar cells as we know them
today, started at the Bell Laboratories in the United states.
In 1954, their scientists Daryl M. Chapin, Calvin S.
Fuller, and Gerald L. Pearson, made a silicon-basedsolar cell with an efficiency of about 6% [35]. In thesame year, Reynolds et al. reported on the photovoltaic
effect for cadmium sulfide (CdS), a II-VI semiconductor[36].
In the mid and late 1950s several companies and laboratories started to develop silicon-based solar cells in order to power satellites orbiting the Earth. Among thesewere RCA Corporation, Hoffman Electronics Corporation but also the Unites States Army Signal Corps.
In these days, research on PV technology was mainly
driven by supplying space applications with energy.
For example, the American satellite Vanguard 1, which
was launched by the U.S. Navy in 1958, was poweredby solar cells from Hoffman Electronics. It was the
fourth artificial Earth satellite and the first one to be
powered with solar cells. It was operating until 1964
and still is orbiting Earth. In 1962 Bell Telephone Laboratories launched the first solar powered telecommunications satellite and in 1966 NASA launched the first Or-

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139

biting Astronomical Observatory, which was poweredby a 1 kW photovoltaic solar array.
In 1968, the Italien scientist Giovanni Francia built thefirst concentrated-solar power plant near Genoa, Italy.
The plant was able to produce 1 MW with superheatedsteam at 100 bar and 500°C.
In 1970, the Soviet physicist Zhores Alferov developed
solar cells based on a gallium arsenide heterojunction.
This was the first demonstrator of a solar cell basedon III-V semiconductor materials that we will discussin Section 13.2. In 1976, Dave E. Carlson and Chris
R. Wronski developed the first thin-film photovoltaic
devices based on amorphous silicon at RCA Laboratories. We will discuss this technology in Section 13.3.
In 1978, the Japanese companies SHARP and Tokyo
Electronic Application Laboratory bring the first solar
powered calculators on the market.
Because of the oil crisis, induced by the OPEC oil embargo in 1973 and as a consequence a sharply rising oil
price, the public interest in photovoltaic technology for
terrestrial application was increasing in the 1970s. In
that time, PV technology moved from a niche technology for space application to a technology applicablefor terrestrial applications. In the late 1970s and 1980s
many companies started to develop PV modules andsystem for terrestrial applications.
In 1980 the first thin film solar cells based on a copper-

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sulfide/cadmium-sulfide junction was demonstrated
with a conversion efficiency above 10% at the University of Delaware. In 1985, crystalline silicon solar cellswith efficiencies above 20% were demonstrated at the
University of New South Wales in Australia.
From 1984 through 1991 the World’s largest solar
thermal energy generating facility in the world was
built in the Mojave Desert in California. It consits of
9 plants with a combined capacity of 354 Megawatts.
In 1991 the first high efficiency Dye-sensitized solar
cell was published by the École polytechnique fédérale
de Lausanne in Switzerland by Michael Grätzel andcoworkers. The Dye-sensitized solar cell is a kind ofphoto-electrochemical system, in which a semiconductor material based on molecular sensitizers, isplaced between a photoanode and an electrolyte. We
will introduce this technology in Section 13.6.
In 1994, the U.S. National Renewable Energy Laboratory placed in Golden, Colorado, demonstrated a concentrator solar cell based on III-V semiconductormaterials. Their cell based on a indium-galliumphosphide/gallium-arsenide tandem junction exceeded
the 30% conversion limit.
In 1999, the total global installed photovoltaic power
passed 1 GW p . Starting from about 2000, environmental
issues and economic issues started to become more
and more important in the public discussion, which re-This copy is provided for free, for personal use only.

140

11. A Little History of Solar Cells

newed the public interest in solar energy. Since 2000,
the PV market therefore transformed from a regional
market to a global market, as discussed in Chapter 2.
Germany took the lead with a progressive feed-in tariff policy, leading to a large national solar market andindustry [15].
Since about 2008, the Chinese government has beenheavily investing in their PV industry. As a result,
China has been the dominant PV module manufacturer
for several years now. In 2012 the world-wide solar energy capacity surpassed the magic barrier of 100 GW p[13]. Between 1999 and 2012, the installed PV capacity
hence has grown with a factor 100. In other words, inthe last 13 years, the average annual growth of the installed PV capacity was about 40%.

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12

Crystalline Silicon Solar Cells
As we already discussed in Chapter 6, most semiconductor materials have a crystalline lattice structure. As astarting point for our discussion on crystalline silicon
PV technology, we will take a closer look at some properties of the crystal lattice.

12.1

Crystalline silicon

In such a lattice, the atoms are arranged in a certain
pattern that repeats itself. The lattice thus has long-rangeorder and symmetry. However, the pattern is not thesame in every direction. In other words, if we make alarge cut through the lattice, the various planes you canmake do not look the same.

Figure 12.1: A unit cell of a diamond cubic crystal [25].

141
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142
(a)

12. Crystalline Silicon Solar Cells
(100)

(b)

(111)

[111]

4
2

Figure 12.2: The (a) 100 and the (b) 111 surfaces of a silicon

crystal.

Crystalline silicon has a density of 2.3290 g/cm3 and adiamond cubic crystal structure with a lattice constant of
543.07 pm, as illustrated in Fig. 12.1. Figure 12.2 shows
two different sections through a crystalline silicon lattice, which originally consisted out of three by three by
three unit cells. The first surface shown in Fig. 12.2 (a)is the 100 surface, whose surface normal is the 100 direction. At a 100 surface, each Si atom has two back bonds
and two valence electrons pointing to the front. The
second surface, shown in Fig. 12.2 (b), is the 111 surface. Here, every Si atom has three back bonds and one
valence electron pointing towards the plane normal.

Energy E (eV)[100]

3.4 eVEC

1.12 eV

0

EV

–2
–4
–6
–8
–10

L

Λ

Γ

∆

X

U,K

Σ

Γ

wavevector k
Figure 12.3: The band diagram of crystalline silicon.

To understand the importance of these directions, welook at the electronic band dispersion diagram for silicon,

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12.1. Crystalline silicon

Because of the required change in momentum, for for
an indirect band gap material it is less likely that aphoton with an energy exceeding the bandgap can excite the electron, with respect to a direct bandgap material like gallium arsenide (GaAs) or indium phosphite (InP). Consequently, the absorption coefficientof crystalline silicon is significantly lower than that of
direct band gap materials, as we can see in Fig. 12.4.
While in the visible part of the spectrum c-Si absorbs
less than the GaAs and InP, below 364 nm, it absorbs

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107

Absorption coefficient α (cm-1)

shown in Fig. 12.3. On the vertical axis, the energy position of the valence and conduction bands is shown.
The horizontal axis shows the crystal momentum, i.e.
the momentum of the charge carriers. The white area
represents the energy levels in the forbidden band gap.
The band gap of silicon is determined by the lowest
energy point of the conduction band at X, which corresponds to the 100 direction, and the highest energy
value of the valence band, at Γ. The band gap energy isthe difference between those two levels and is equal
1.12 eV, or 1107 nm, when expressed in wavelengths.
1107 nm is in the infrared part of the spectrum of light.
This bandgap is an indirect bandgap, because the charge
carriers must change in energy and momentum to be
excited from the valence to the conduction band. As
we can see, crystalline silicon has a direct transition as
well. This transition has an energy of 3.4 eV, which isequivalent to a wavelength of 364 nm, which is in theblue spectral part.

GaAs
InP
Ge
Si

106
105
104
103
102
101
100

200 400

600

800 1000 1200 1400 1600 1800 2000

Wavelength λ (nm)Figure 12.4: Absorption coefficients of different semiconduct-ors.

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144

just as much as GaAs and InP, because there silicon has
a direct band to band transition as well. Germanium(Ge), also indicated in the figure, is an indirect bandgap
material, just like silicon. The bandgap of Ge is 0.67
eV, which means it already starts to absorb light atwavelengths shorter than 1850 nm. In the visible partof the spectrum, germanium has some direct transitions
as well.
Let us now take another look at the design rules forsolar cells that we introduced in Section 10.4. First, welook at spectral utilisation. The c-Si band gap of 1.12 eVmeans that in theory we can generate a maximum shortcircuit current density of 45 mA per square centimeter.
Let us now consider the second design rule, i.e. lighttrapping. First, we look at a wavelength around 800
nm, where c-Si has an absorption coefficient of 1000
cm−1 . Using the Lambert-Beer law [Eq. (4.25)], we easily can calculate that for realising an an absorption of
90% of the incident light at 800 nm, the required absorption path length is 23 µm. Secondly, we look at 970
nm wavelength, where c-Si has an absorption coefficientof 100 cm−1 . Hence, an absorption path length of 230µm is required to absorb 90% of the light. 230 µm isa typical thickness for silicon wafers. This calculation
demonstrates that the light trapping techniques become
important for crystalline silicon absorber layers above awavelength of about 900 nm.
Let us now consider the design rule of bandgap utilisa-

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12. Crystalline Silicon Solar Cells
(a)

(b)

Monocrystalline

Multicrystalline

Figure 12.5: Illustrating a (a) monocrystalline and a (b) mul-

ticrystalline silicon wafer.

tion, which is determined by the recombination losses.
As silicon is an indirect band gap material, only Auger
recombination and Shockley-Read-Hall (SRH) recombination will determine the open circuit voltage, while
radiative recombination can be neglected. Considering SRH recombination, the recombination rate of thecharge carriers is related to the electrons trapped at defect states. When looking at the defect density in thebulk of silicon, we can differentiate between two major types of silicon wafers: monocrystalline silicon andmulticrystalline silicon, which is also called polycrystalline
silicon.
Monocrystalline silicon, also known as called singlecrystalline silicon, is a crystalline solid, in which the
crystal lattice is continuous and unbroken without any

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145

grain boundaries over the entire bulk, up to the edges.
In contrast, polycrystalline silicon, often simply abbreviated with polysilicon, is a material that consistsof many small crystalline grains, with random orientations. Between these grains grain boundaries arepresent. Figure 12.5 shows two pictures of monocrystalline and multicrystalline wafers. While a monocrystalline silicon wafer has one uniform color, in multicrystalline silicon, the various grains are clearly visible forthe human eye. At the grain boundaries we find latticemismatches, resulting in many defects at these boundaries. As a consequence, the charge carrier lifetime forpolycrystalline silicon is shorter than for monocrystalline silicon, because of the SRH recombination. The
more grain boundaries in the material, the shorter the
lifetime of the charge carriers. Hence, the grain size
plays an important role in the recombination rate.
Figure 12.6 shows the relationship between the open
circuit voltage and the average grain size for various
solar cells developed around the world, based on themulticrystalline wafers [37]. The larger the grain size,
the longer the charge carrier lifetimes and the larger theband gap utilisation and hence the open circuit voltage
will be. On the right hand side of the graph the open
circuit voltages of various solar cells, based on monocrystalline wafers, is shown. As monocrystalline silicon has no grain boundary, much larger open circuit
voltages can be obtained.

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Open circuit voltage Voc (mV)

12.2. Production of silicon wafers
750
700
650
600
550
500

IMEC

450

IMEC
BP

400
350
10-2

10-1

100

Mitsubishi
Astropower
ASE/ISFH
ISE
Daido
ETL

UNSW
MPI
ISE
Ipe
SONY
ZAE
Canon

Tonen
PHASE

101

102

103

104

Average grain size (μm)
Figure 12.6: The relationship between the open circuit voltage

and the average grain size [37]12.2

Production of silicon wafers

After the initial considerations on designing c-Si solar
cells, we now will discuss how monocrystalline andmulticrystalline silicon wafers can be produced. In Fig.
12.7 we illustrate the production process of monocrystalline silicon wafers.
The lowest quality of silicon is the so-called metallurgical silicon, which is made from quartzite. Quartzite
is a rock consisting of almost pure silicon dioxide(SiO2 ). For producing silicon the quartzite is molten
in a submerged-electrode arc furnace by heating it up

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146

12. Crystalline Silicon Solar Cells

carbonSiemens process

CO

HCl
Si+3HCl
H2+HSiCl3

SiO2+2C
Si+2COquartzite

arc furnace

H2, ClH2

metallurgicalsilicon powder

chemical reactor

HSiCl3

HSiCl3

distillation

chemical vapour
deposition

sawingsiliconingot

float zone process

Chozralski process

polysiliconsilicon wafers
Figure 12.7: Illustrating the production process of monocrystalline silicon wafers.

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147

12.2. Production of silicon wafers

to around 1900°C, as illustrated in Fig. 12.7. Then, themolten quartzite is mixed with carbon. As a carbon
source, a mixture of coal, coke and wood chips is used.
The carbon then starts reacting with the SiO2 . Since
the reactions are rather complex, we will not discussthem in detail here. The overall reaction how ever canbe written as
SiO2 + 2 C → Si + 2 CO

(12.1)As a result, carbon monoxide (CO) is formed, which
will leave the furnace in the gas phase. In this way, thequartzite is purified from the silicon. After the reactions
are finished, the molten silicon that was created during
the process is drawn off the furnace and solidified. The
purity of metallurgic silicon, shown as a powder in Fig.
12.7, is around 98% to 99%.
About 70% of the worldwide produced metallurgicalsilicon is used in the aluminium casting industry formake aluminium silicon alloys, which are used in automotive engine blocks. Around 30% are being used formake a variety of chemical products like silicones. Onlyaround 1% of metallurgical silicon is used as a raw
product for making electronic grade silicon.
The silicon material with the next higher level of purity
is called polysilicon. It is made from a powder of metallurgical silicon in the Siemens process. In the process,the metallurgical silicon is brought into a reactor and

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exposed hydrogen chloride (HCl) at elevated temperatures in presence of a catalyst. The silicon reacts withthe hydrogen chloride,
Si + 3 HCl → H2 + HSiCl3 ,(12.2)

leading to the creation of trichlorosilane (HSiCl3 ). Thisis a molecule that contains one silicon atom, threechlorine atoms and one hydrogen atoms. Then, the trichlorosilane gas is cooled and liquified. Using distillation, impurities with boiling points higher or lower
than HSiCl3 are removed. The purified trichlorosilane
is evaporated again in another reactor and mixed withhydrogen gas. There, the trichlorosilane is decomposedat hot rods of highly purified Si, which are at a hightemperature in between around 850°C and 1050°C. The
Si atoms are deposited on the rod whereas the chlorineand hydrogen atoms are desorbed from the rod surface back in to the gas phase. As a result a pure silicon material is grown. This method of depositing silicon on the rod is one example of chemical vapour deposition (CVD). As the exhaust gas still contains chlorosilanes and hydrogen, these gasses are recycled and usedagain: Chlorosilane is liquified, distilled and reused.
The hydrogen is cleaned and thereafter recycled back into the reactor. The Siemens process consumes a lot ofenergy.
Another method for producing poly silicon granules isusing Fluid Bed Reactors (not shown in Fig. 12.7). This
process is operated at lower temperatures and con-This copy is provided for free, for personal use only.

148

sumes much less energy. Polycrystalline silicon canhave an purity as high as 99.9999%, or in other words
only one out of million atoms is an atom different fromSi.
The last approach we briefly mention is that of upgradedmetallurgical silicon (not shown in Fig. 12.7). In this process metallurgical silicon is chemically refined by blowing gasses through the silicon melt removing the impurities. Although processing is cheap, the silicon purity islower than that achieved with the Siemens or the Fluid
Bed reactor approaches.
Now we introduce two methods that are industriallyused for making monocrystalline silicon ingots, i.e. large
cylinders of silicon that consist of one crystal only. Thismeans that inside the ingot no grain boundaries arepresent. Such a monocrystalline ingot and both themethods are sketched in Fig. 12.7.
The first method we discuss is based on the Czochralski process that was discovered by the Polish scientist
Jan Czochralski in 1916. In this method, highly purified
silicon is melted in a crucible at typical temperature of
1500°c. Intentionally boron or phosphorus can be added for making p-doped or n-doped silicon, respectively. A seed crystal that is mounted on rotating shaft isdipped in to the molten silicon. The orientation of thisseed crystal is well defined; it has either is either 100
or 111 oriented. The melt solidifies at the seed crystal
and adopts the orientation of the crystal. The crystal is

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12. Crystalline Silicon Solar Cells

rotating and pulled upwards slowly, allowing the formation of a large, single-crystal cylindrical column fromthe melt — the ingot. For successfully conducting theprocess, temperature gradients, the rate of pulling theshaft upwards and the rotational speed must be wellcontrolled. Due to improved process control throughout the years, nowadays ingots of diameters of 200 mm
or even 300 mm with lengths of up to two meters canbe fabricated. To prevent the incorporation of impurities, the Czochralski this process takes place in an inert
atmosphere, like argon gas. The crucible is made fromquartz, which partly dissolves in the melt as well. Consequently, monocrystalline silicon made with the Czochralski method has a relatively high oxygen level.
The second method to make monocrystalline silicon is
the float zone process, which allows fabricating ingots
with extremely low densities of impurities like oxygenand carbon. As a source material, a polycrystalline rod
made with the Siemens process is used. The end of therod is heated up and melted using an induction coil
operating at radio frequency (RF). The molten part isthen brought in contact with the seed crystals, where it
solidifies again and adopts the orientation of the seed
crystal. Again 100 or 111 orientations are being used.
As the molten zone is moved along the polysilicon rod,the single-crystal ingot is growing as well. Many impurities remain in and move along with the molten
zone. Nowadays, during the process nitrogen is intentionally added in order to improve the control over mi-

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12.2. Production of silicon wafers

crodefects and the mechanical strength of the wafers.
One advantage of the float—zone technique is that themolten silicon is not in contact with other materials likequartz, as this is the case when using the Czochralski
method. In the float-zone process the molten silicon is
only in contact with the inert gas like argon. The silicon
can be doped by adding doping gasses like diborane(B2 H6 ) and phosphine (PH3 ) to the inert gas to get pdoped and n-doped silicon, respectively. The diameterof float-zone processed ingots generally is not larger
than 150 mm, as the size is limited by surface tensions
during the growth.
Not only monocrystalline silicon ingots, but also multicrystalline silicon ingots, which consist of many small
crystalline grains, can be fabricated (not shown in Fig.
12.7). This can be made by melting highly purified silicon in a dedicated crucible and pouring the molten
silicon in a cubic shaped growth crucible. There, the molten silicon solidifies in to multi-crystalline ingot in aprocess called silicon casting. If both melting and solidification is done in the same crucible it is referred to as
directional solidification. Such multicrystalline ingots canhave a front surface area of up to 70 × 70 cm2 and aheight of up tp 25 cm.
Now, as we know how to produce monocrystalline andmulticrystalline ingots we will discuss how to makewafers out of them. The process that is used to make thewafers is sawing, as illustrated in Fig. 12.7. Logically,

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sawing will damage the surface of the wafers. Therefore, this processing step is followed by an polishingstep. The biggest disadvantage of sawing is that a significant fraction of the silicon is lost as kerf loss, whichusually is determined by the thickness of the wire or
saw used for sawing. Usually, it is in the order of 100µm. As typical wafers used in modern solar cells havethicknesses in the order of 150 µm up to 200 µm, thekerf loss is very significant.
A completely different process for making siliconwafers is the silicon ribbon technique (not shown in Fig.
Fig. 12.7). As this technique does not include any sawing step, no kerf loss occurs. In the silicon ribbon technique a string is used that is resistant agains high temperatures. This string is pulled up from a silicon melt.
The silicon solidifies on the string and hence a sheet of
crystalline silicon is pulled out of the melt. Then, theribbon is cut into wafers. Before the wafers can be processed further in order to make solar cells, some surface
treatments are required. The electronic quality of ribbon
silicon is not as high as that of monocrystalline.
To summarise, we discussed how to make metallurgicalsilicon out of quartzite and how to fabricate polysilicon.
We have seen that monocrystalline ingots are madeusing either the Czochralski or the float-zone process,while multicrystalline ingots are made using a casting
method. Wafers are fabricated by sawing these ingots.
A method that does not have any kerf losses in the rib-

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150

12. Crystalline Silicon Solar Cells

front contactp-type wafer

antireflective
coatingn+-type emitter
p+-type layer1022(metal grid)back contact

Figure 12.8: Scheme of a modern crystalline silicon cell.

Generation rate (m-3)(to the back contactof the next cell)≈ 200 μm

serial connections

1021
1020
1019
1018
1017

0

50

100

150

200

250

300

Cell depth x (µm)
Figure 12.9: The generation profile in crystalline silicon.

bon silicon approach.

Before we start with the actual discussion on how c-Si
solar cells are fabricated, we briefly discuss the operating principles of c-Si solar cells. Especially, we discuss
several technical aspects that play an important role inthe collection of the light, excitation of charge carriers
and the reduction of optical losses.

used there, both the p-doped and n-doped regions
have the same thickness. This is not the case in realc-Si devices. For example, the most conventional typeof c-Si solar cells is built from a p-type silicon wafer,as sketched in Fig. 12.8. However, the n-type layer on
the top of the p-wafer is much thinner than the wafer;it typically has a thickness of around 1 µm. Often, thislayer is called the emitter layer. As mentioned before,
the whole wafer has typically thicknesses in between
100 and 300 µm.

In Chapter 8 we discussed how an illuminated p-njunction can operate as a solar cell. In the illustrations

Because of the Lambert-Beer law [Eq. (4.25)], the intensity if light inside the silicon bulk decays exponentially

12.3

Fabricating solar cells

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151

12.3. Fabricating solar cellswith the depth. Hence also the generation profile willshow an exponential decay as discussed in Section 7.2.1
and shown in Fig. 12.9. Therefore, the largest fractionof the light is absorbed close to the front surface of thesolar cell. In the first 10 µm by far the most charge carriers are generated. By making the front emitter layer
very thin, a large fraction of the light excited charge
carriers generated by the incoming light are createdwithin the diffusion length of the p-n junction.

First, the emitter layer: At the p-n junction the minority charge carriers, which are excited by the light, areseparated at the p-n junction: the minority electrons in
the p-layer drift to the n-layer, where they have to becollected. Since the silicon n-emitter is not sufficiently
conductive we have to use the much more conductive
metal contacts, which are placed on top of the emitter
layer. Very often, the metal contacts are made of thecheap metal aluminium.

The emitter layer can be created with a solid state diffusion process. In this process, the wafers are placed in a
furnace at around 850°C, where the dopant atoms arepresent in atmosphere in the form of phosphine (PH3 ).
These atoms react at these high temperatures with thesurface. Further, at these temperatures the phosphine
atoms are mobile in the silicon crystal. Based on Fick’slaw, the therefore can diffuse into the wafer,

This means that the electrons have to diffuse laterally
through the emitter layer to the electric front contact
to be collected. Which factors are important for a goodtransport of the electrons to the contact? First, the lifetime of the charge carriers needs to be high. A highlifetime guarantees large open circuit voltages, or inother words the optimal utilisation of the band gap energy. For increasing the lifetime, recombination losses
must be reduced as much as possible.

J (x) = −D

dn
.
dx

(12.3)

This law states that the particle flux J is proportional
to the negative gradient of the particle density n. D isthe diffusion constant. The diffusion process has to becontrolled such that the dopants penetrate into the solid
to establish the desired emitter thickness.
Now we are discuss the collection of charger carriers
in a crystalline silicon solar cell. The crucial components that play a role in charge collection are the emitter layer, the metal front contacts and the back contact.

Copyright Delft University of Technology, 2014

Recombination not only reduces the Voc , it also limits
the collected current. As mentioned earlier, in silicon
two recombination mechanisms are present: ShockleyRead-Hall recombination and Auger recombination.
First, let us take a look at Shockley-Read-Hall recombination. A bare c-Si surface contains many defects,
because the surface silicon atoms have some valence
electrons that cannot make molecular orbitals due tothe absence of neighbouring atoms. These valence orbitals containing only one electron at the surface act

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152

like defects. They are also called dangling bonds. At
the dangling bonds, the charge carriers can recombine
through the SRH process. The probability and speed
at which charge carriers can recombine is usually expressed in terms of the surface recombination velocity.
Since a large fraction of the charge carriers are generated close to the front surface, a high surface recombination velocity at the emitter front surface will leadto significant charge carriers losses and consequently
lower short-circuit current densities. In high-quality
monocrystalline silicon wafers, for example, no defectrich boundaries are present in the bulk. Thus, the lifetime of charge carriers is limited by the recombination
processes at the wafer surface.
In order to reduce the surface recombination two approaches are used. First, the defect concentration on
the surface is reduced by depositing a thin layer of adifferent material on top of the surface. This material
partially restores the bonding environment of the silicon atoms. In addition, the material must be an insulator, it must force the electrons to remain in and movethrough the emitter layer. Typical materials used forthis chemical passivation layers are silicon oxides (SiOx )and silicon nitride (Six Ny ).1 A silicon oxide layer can beformed by heating up the silicon surface in an oxygenrich atmosphere, leading to the oxidation of the surface
1 Stoichiometric silicon oxide and silicon nitride have the chemical formula SiO
2and Si3 N4 , respectively. As the layers used for passivation are not stoichiometric, we
indicate the elementary fractions with x and y.

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12. Crystalline Silicon Solar Cells

silicon atoms. Si3 N4 can be deposited using plasmaenhanced chemical vapour deposition (PE-CVD) that
we will discuss in more detail in Chapter 13.
A second approach for reducing the surface recombination velocity is to reduce the minority charge carrier
density near the surface. As the surface recombination
velocity is limited by the minority charge carriers density, it is beneficial to have the minority charge carrier
density at the surface as low as possible. By increasing the doping of the emitter layer, the density of theminority charge carriers can be reduced. This results inlower surface recombination velocities. However, this is
in competition with the diffusion length of the minority charge carriers. The blue part of the solar spectrum
leads to generation of many charge carrier very close
to the surface, i.e. in the emitter layer. For utilisingthese light excited minority charge carriers, the diffusion length of the holes has to be large enough to reachthe depletion zone at the p-n junction as indicated by
the blue arrow. However, increasing the doping levels
leads to a decreasing diffusion length of the minority
holes in the emitter. Therefore, too high doping levels
or too thick emitter layers would result in a poor blue
response or – in other words – low External Quantum
Efficiency (see Chapter 9) values in the blue part of thespectrum. Such an emitter layer could be called “deadlayer” as the light excited minority charge carriers can
not be collected.

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12.3. Fabricating solar cells

As a next step, we take a closer look at the metalemitter interface. Because electrons must be easily conducted from the emitter to the metal, insulating passivation layers such as SiOx or Six Ny cannot be used.
Therefore, the metal-semiconductor interface has more
defects and hence an undesirably high interface recombination velocity. Additionally, a metal-semiconductor
junction induces a barrier for the majority charge carriers. It is, however, out of the scope of this book to
discuss the physics of this barrier in more detail. We
just must keep in mind that this high barrier will giverise to a higher contact resistance. Again high doping levels can reduce the recombination velocity at themetal-semiconductor interface and reduce the contact
resistance. In order to minimise the recombination at
the interface defects as much as possible, the area of themetal-semiconductor interface must be minimised andthe emitter directly below the interface should be heavily doped, which is indicated with n++ . The sides ofthe metal contacts are buried in the insulting passivation layer. The area below the contact has been heavily
doped. These two approaches reduce the recombinationand collection losses at the metal contact.
The solar cell shown in Fig. 12.8 has a classic metal grid
pattern on top. We see two high ways for the electrons
in the middle of top surface of the solar cell. They arecalled busbars. The small stripes going from the busbars
to the edges of the solar cell are called the fingers. Let
now R be the resistance of such a finger. If L, W, and H

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are the length, width and height of the fingers, respectively, and ρ is the resistivity of the metal, R is given by
R=ρ

L
.
WH

(12.4)

This equation shows that the longer the path length for
an electron, the larger the resistance the electrons experience. Further, he smaller the cross-section (W · H ) of
the finger, the larger the resistance will be. Note, thatthe resistance of the contacts will act as a series resistance in the equivalent electric circuit. Larger series resistance will result in lower fill factors of the solar cell as
discussed in Chapter 9. Hence, electrically large crosssections of the fingers are desirable.
To arrive at a metallic contact, electrons in the emitter
have to travel laterally through the emitter. Because of
the resistivity of the n-type silicon, the charge carriersin the emitter layer also experience a resistance. It canbe shown that the power loss due to the resistivity ofthe emitter layer scales with the spacing between two
fingers to the power 3. EXERCISE???
As the metal contacts are at the front surface, they act
as unwelcome shading objects, or in other words lightincident on the metallic front contact area cannot be
absorbed in the PV-active layers. Therefore, the contact area should be kept as small as possible, whichis in competition with the fact that the finger crosssection should be maximised. So basically, the finger

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154

12. Crystalline Silicon Solar Cells

(a)

(b)
Total

Resistive
emitter Resistive
contact

Power loss

Power loss

Total

Shadowshown in Fig. 12.10 (b). The larger W the larger shading losses will be. But with increasing W the resistancedecreases. Again, here an optimum exists at which thepower losses are minimal. We see that optimising thefront contact pattern is a complex interplay between thefinger width and spacing.

Shadow

Finger spacing

Resistive

Finger width

Figure 12.10: Power loss due to (a) spacing and (b) widths ofthe metallic fingers on top of a c-Si solar cell.

height should be as high as possible while the finger
width should you would like to have a small as possible width and a high as possible height to comply
with these requirements.
We can see that several effects compete with other.
Fig. 12.10 (a) shows the relationship between power
losses and finger spacing. With increasing finger spacing the power losses of the solar cell decreases becauseof less shading. On the other hand, the losses due tothe increased resistivity in the emitter layer increase.
Hence, there is an optimal spacing distance at that thepower loss is minimal. A similar plot can be made forthe power loss in dependence of the finger width; it is

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For designing the back contact we find similar issues.
Just as the electrons are to be collected in the front ntype layer, the holes are to be collected at the back contact. Electrons are the only charge carriers that exist in
metal. Therefore the holes have to recombine with theelectrons at the back semiconductor-metal interface. If
the distance between the p-n interface and the backcontact is smaller than the typical diffusion length ofthe minority electrons, the minority electrons can belost at the defects of the back contact interface becauseof SRH recombination
Several methods can be used to reduce this loss. First,
the area between the metal contact and the semiconductor can be reduced, just as for the front contact. To
do this, point contacts can be used, while the rest of therear surface is passivated by an insulating passivation
layer, similar as we already discussed for the emitter
front surface. The recombination loss of electrons at theback contact can be further reduced by a introducing
a back surface field. Above the point contacts a highly
doped p-doped region is placed, which is indicated byp+ .

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155

12.3. Fabricating solar cells

Al

p+

Al

p+-typeTo understand how the back surface field works, wetake a look at the band diagram shown in Fig. 12.11.
The interface between the normally doped p-region
and the highly doped p+ -region acts like an n-p junction. Here, this junctions acts as a barrier that prevents
minority electrons in the p-region from diffusing to theback surface. Further, the space charged field behaves
like a passivation layer for the defects at the back contact interface and allows to have higher minority carrier
densities in the p-doped bulk.
After this thorough discussion on managing the charge
carriers, we now take a closer look on managing thephotons in a crystalline silicon solar cell. Several opticalloss mechanisms must be addressed. These are shading,reflection losses, parasitic absorption losses in the nonPV active layers, and transmission through the back of
the solar cell. As already mentioned, shading losses arecaused by the metallic front contact grid.

p-typen-typeFigure 12.11: Effect of the back surface field illustrated in aband diagram.

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Secondly, reflection from the front surface is an important loss mechanisms. We briefly mention two approaches to design anti-reflective coatings (ARC) for reducing these losses. First, reflection can be minimised based on the Rayleigh film principle: by putting afilm with a refractive index smaller than that of siliconwafer between the cell and the wafer losses can be reduced. The optimal value for the refractive index of theintermediate layer equals the square root of the productThis copy is provided for free, for personal use only.

156

12. Crystalline Silicon Solar Cellsof refractive indexes of the two other media,√
nopt = nair nSi .

(a)(12.5)

At a wavelength of 500 nm, the optimal refractive index
for a layer in-between air and silicon is 2.1. Note thatin practice a solar cell in encapsulated under a glass or
polymer plate, which will have a beneficial effect on therefractive index grading as well, reducing the reflection
losses further. Secondly, using the concept of destructive interference, the thickness and refractive index ofan antireflection coating can be chosen such that in acertain wavelength range the reflection is minimised.
This happens when the light reflected from the air-ARC
interface is in anti-phase with the light reflected fromthe ARC-Si interface, as discussed in Chapter 4. We findthat the thickness of such a layer should be a quarter ofthe wavelength in the layer,dARC =

λ0,4n(12.6)

where λ0 denotes the wavelength in vacuo and n is therefractive index of the antireflective layer. For a refractive index of 2.1, a layer thickness of 60 nm would leadto destructive interference at 500 nm. As mentioned
earlier, a typical material for passivation is silicon nitride. we have discussed earlier, one of the typical passivation layers of standard crystalline silicon is silicon
nitride. Fig. 12.12 (a) shows a multicrystalline wafer
without any ARC. It appears silverish, which means

Copyright Delft University of Technology, 2014(b)

Figure 12.12: A multicrystalline silicon wafer (a) without and(b) with an antireflective coating from silicon nitride.

that it is highly reflective. In Fig. 12.12 (b) a similar
wafer is shown after it was passivated with Six Ny . We
see that it has a dark-blue appearance, hence its reflection is much lower. Interestingly enough, the refractive index of Six Ny at 500 nm is in the range of 2 to 2.2,
close to the optimum mentioned earlier. The blue appearance indicates that reflection in the blue is stronger
than at other wavelengths.
The reflection losses also can be minimised by texturing the wafer surface. Light that is reflected at the textured surface can be reflected at angles in which the
trajectory of the light ray is incident somewhere else on
the surface, where it still can be coupled into the silicon. Additionally, the scattering at textured surfaces
will couple the light under angles different from theThis copy is provided for free, for personal use only.

157

12.4. High-eﬃciency concepts
(a)

(b)

finger

p+

n+

n
oxide
p-type wafer +pp+
Figure 12.13: (a) A textured multicrystalline Si wafer and (b) awafer with a black silicon surface.

“inverted ” pyramids

rear contact

p+

oxide

Figure 12.14: Illustrating the structure of a PERL solar cell.

interface normal into to the wafer. Therefore, the average path length of the light in the absorber will be
increased, which leads to stronger absorption of thelight. For realising the textures, usually wet etching approaches are used. Interestingly enough, 100 surfaces
are etched much faster than the 111 surfaces. Such a behaviour is called unisotropic etching. Thus, when a 100
wafer is etched, the 100 surface will be etched away,
while the (slanted) 111 surface facets will remain. Therefore, we obtain a wafer with a pyramid structure consisting of 111 facets only, as illustrated in Fig. 12.13(a) that shows such a wafer that was processed in the
DIMES Research Center in Delft. With a different etching approach, it is even possible to make silicon to appear completely black, as shown in Fig. 12.13 (b). Thisis called black silicon.

Copyright Delft University of Technology, 201412.4

High-eﬃciency concepts

In the last section of this chapter we discuss three examples of high-efficiency concepts based on crystalline
silicon technology. As already discussed earlier, different types of silicon wafers with different qualities canbe used. Naturally, for achieving the highest efficiencies, the bulk recombination must be as small as possible. Therefore the high efficiency concepts are basedon monocrystalline wafers.

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158

12.4.1

12. Crystalline Silicon Solar Cells

The PERL concept

The first high efficiency concept was developed in thelate 1980s and the early 1990s at the University of New
South Wales in the group of Martin Green. in the lateeighties and early nineties. Figure 12.14 shows an illustration of the PERL concept, which uses a p-type floatzone silicon wafer. With this concept, conversion efficiencies of 25% were achieved [38]. The abbreviation
PERL stands for Passivated Emitter Rear Locally. This abbreviation indicates two important concepts that havebeen integrated in this technology: First, the opticallosses of the PERL solar cell at the front side are minimised using three techniques:
1. The top surface of the solar cell is textured withinverted-pyramid structures. This microscopic texture allows a fraction of the reflected light to be incident on the front surface for a second time, which
enhances the total amount of light coupled in to
the solar cell.
2. The inverted pyramid structure is covered with adouble-layer anti-reflection coating (ARC), which
results in an extremely low top surface reflection.
Often a double layer coating of magnesium fluoride (MgF2 ) and zinc-sulfide (ZnS) is used as anantireflection coating.
3. The contact area at the front side has to be as small
as possible, to reduce the shading losses. In the

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PERL concept the very thin and fine metal fingers
are processed using photolithography technology.
Secondly, the emitter layer is smartly designed. As discussed in the previous sections, the emitter should behighly doped underneath the contacts, which in the
PERL concept is achieved by heavily phosphorus diffused regions. The rest of the emitter is moderately
doped, or in other words lightly diffused, to preserve
an excellent blue response. The emitter is passivatedwith a silicon oxide layer on top of the emitter to suppress surface recombination as much as possible. With
the PERL concept, the surface recombination velocity
could be reduced so far that open circuit voltages withvalues of above 700 mV could be obtained.
At the rear surface of the solar cell, point contacts havebeen used in combination with thermal oxide passivation layer, which passivates the non-contacted area, andhence reduces the undesirable surface recombination.
A highly doped boron region, created by local Boron
diffusion, operates as a local back surface field, to limit
the recombination of the minority electrons at the metalback contact.
The PERL concept includes some expensive processing
steps. Therefore, the Chinese company Suntech developed in collaboration with University of New South
Wales a more commercially viable crystalline siliconwafer technology, which is inspired on the PERL cellconfiguration.

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159

12.4. High-eﬃciency concepts

12.4.2The interdigitated back contact(IBC) solar cell

A second successful cell concept is the interdigitatedback contact (IBC) solar cell. The main idea of the IBC
concept is to have no shading losses at the front metal
contact grid at all. All the contacts responsible for collecting of charge carriers at the n- and p-side are positioned at the back of the crystalline wafer solar cell. A
sketch of such a solar cell is shown in Fig. 12.15 (a).
An advantage of the IBC concept is that monocrystalline float-zone n-type wafers can be used. This is interesting because n-type wafers have some interesting
advantages with respect to p-type wafers. First, then-type wafers do not suffer from light induced degradation. In p-type wafers both boron and oxygen
are present, which under light exposure start to makecomplexes that act like defects. This light induced degradation causes a reduction of the power output with
2-3% after the first weeks of installation. No such effect is present in n-type wafers. The second advantage
is that n-type silicon is not that sensitive for impurities
like for example iron impurities. As a result, less efforts have to be made to fabricate high quality n-typesilicon, and thus high quality n-type silicon can be processed cheaper than p-type silicon. On the other hand,
p-doped wafers have the advantage that the boron doping is more homogeneously distributed across the wafer
as this is possible for n-type wafers. This means that

Copyright Delft University of Technology, 2014(a)n+-doped
front diffusion

Passivating SiO2 layer

ARCn-type silicon (240 μm)p+Backsidemirrorn+p+

Passivating
SiO2 layern+p+Localized
contactsn+

Backside
gridlines(b)

Figure 12.15: (a) Illustrating the structure of an IBC solar celland (b) the contacts on its back.

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160

within one n-type wafer the electronic properties canvary, which lowers the yield of solar cell production
based on n-type monocrystalline wafers.
Besides that IBC cells are made from n-type wafers,
they lack one large p-n junction. Instead, IBC cells have
many localised junctions. The holes are separated at ajunction between the p+ silicon and the n-type silicon,
whereas the electrons are collected using n+ -type silicon. The semiconductor-metal interfaces are kept as
small as possible in order to reduce the undesired recombination at this defect-rich interface. Another advantage is that the cross section of the metal fingers canbe made much larger, because they are at the back andtherefore do not cause any shading losses. Thus, resistive losses at the metallic contacts can be reduced. Sinceboth electric contacts are on the back side, it contains
two metal grids, as illustrated in Fig. 12.15 (b). The passivation layer should made from a low-refractive-index
material such that it operates like a backside mirror.
It will reflect the light above 900 nm, which is not absorbed during the first pass back in to the absorberlayer. Thus, this layer enhances the absorption path
length.
At the front side of the IBC cell losses of light-excitedcharge carriers due to surface recombination is suppressed by a front surface field similar to the back surface field discussed earlier. This field is created witha highly doped n+ region at the front of the surface.

Copyright Delft University of Technology, 2014

12. Crystalline Silicon Solar Cells

Thus, a n+ -n junction is created that acts like a n-pjunction. It will act as a barrier that prevents the lightexcited minority holes in the n-region from diffusing towards the front surface. The front surface field behaves
like a passivation for the defects at the front interface
and allows to have higher levels for the hole minority
density in the n-doped bulk.
Reflective losses at the front side are reduced in a similar way as for PERL solar cells: Deposition of double
layered antireflection coatings and texturing of the front
surface.
The IBC concept is commercialised by the U.S. company SunPower Corp. They have achieved high solar cell
efficiencies of 24.2%.

12.4.3Hetero junction (HIT) solar cells

The third high-efficiency concept are heterojunction withintrinsic thin layer (HIT) solar cells. Before we discuss
the technological details, we briefly recall the principles
of heterojunctions, that already were discussed in Section 8.2.
Homojunctions, which are present in all the c-Si solar
cell types discussed so far in this chapter, are fabricatedby different doping types within the same semiconductor material. Hence, the band gap in the p- and nregions is the same. A junction consisting of a p-doped

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161

12.4. High-eﬃciency concepts
n-type FZ c-Si

p-type a-Si:H

fingers

ECP
eVbin
EFn

ECn
EVn

Egn

ΔEC

ΔEC

EgP
EVP

EFP

Figure 12.16: Illustrating the band diagram of a heterojunc-

p/i a-Si:H
i/n a-Si:H

textured n-type c-Si wafer

Figure 12.17: Illustrating the structure of a HIT solar cell.

tion.

semiconductor material and an n-doped semiconductor
made from another material is called a heterojunction.
In HIT cells, the heterojunction is formed in between
two different silicon-based semiconductor materials: On
the one hand, we use a n-type float zone monocrystalline silicon wafer. The other material is hydrogenatedamorphous silicon (a-Si:H), that we will discussed in
more detail in Section 13.3. a-Si:H is a silicon material in which the atoms are not ordered in a crystalline
lattice but in a disordered lattice. For the moment weonly must keep in mind that a-Si:H has a band gap ofaround 1.7 eV which is considerably higher than that ofc-Si (1.12 eV).
In Fig. 12.16 a band diagram of a heterojunction in

Copyright Delft University of Technology, 2014

between n-doped crystalline silicon and p-doped
amorphous silicon in the dark and thermal equilibrium is sketched. We see that next to the induced field
because of the space charge region, some local energy
steps are introduced. These steps are caused by the twodifferent bandgaps for the p and n regions. The valenceband is higher positioned in the p-type amorphous silicon than in the n-type crystalline silicon. This will allow the minority charge carriers in the n-type c-Si, theholes, to drift to the p-type silicon. However, the holes
experience a small barrier. While they could not travel
across such a barrier in classical mechanics, quantum
mechanics allows them to tunnel across this barrier.
Let us now take a closer look to HIT solar cells. The
HIT concept was developed by the Japanese company

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162

Sanyo, that is currently a part of the Japanese Panasonic
Corp. As we can see in Fig. 12.17, the HIT cell configuration has two junctions: The junction at the front side
is formed using a thin layer of only 5 nanometers of
intrinsic amorphous silicon which is indicated by thecolor red. A thin layer of p-doped amorphous silicon
is deposited on top and here indicated with the color
blue. The heterojunction forces the holes to drift to thep-layer. At the rear surface a similar junction is made:First, a thin layer of intrinsic amorphous silicon is deposited on the wafer surface, indicated by red. On topof the intrinsic layer an n-doped amorphous silicon isdeposited indicated, which is indicated by the yellow
color.
As discussed earlier, for high quality wafers, like thisn-type float zone monocrystalline silicon wafer, the recombination of charge carriers at the surface determines the charge carrier lifetime. The advantage of the
HIT concept is that the amorphous silicon acts as a very
good passivation layer. With this approach the highest
possible charge carrier lifetimes are accomplished.
Thus, c-Si wafer based heterojunction solar cells havethe highest achieved open circuit voltage among thedifferent crystalline silicon technologies. The current record cell2 has an open circuit voltage of 0.74 V and en
efficiency of 25.6%.

12. Crystalline Silicon Solar Cells

conductive properties of the p-doped amorphous silicon
are relatively poor. While in the homojunction solar
cells the lateral diffusion to the contacts takes place inthe emitter layer, in a HIT solar cell this occurs througha transparent conducting oxide (TCO) material, like indium tinoxide (ITO), which is deposited on top of the
p-doped layer. TCOs are discussed in more detail in
Section 13.1.
The same contacting scheme is applied at the n-typeback side. This means that this solar cell can be used
in a bifacial configuration: it can collect light from thefront, and scattered and diffuse light falling on the
back-side of the solar cell. Another important advantage of the HIT technology is that the amorphous silicon layers are deposited using cheapplasma enhanced
chemical vapour deposition (PE-CVD) technology at
low temperatures, not higher than 200°C. Therefore,making the front surface and back surface fields in HIT
solar cells is very cheap.
In this chapter we discussed many aspects of crystalline
silicon solar cell technology. Before the solar cells canbe installed, they must be packed as a PV module. How
such modules are made is discussed in Part IV on PV
systems in Section 17.1.

How do the charge carriers travel to the contacts? The
2 As of May 2014.

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13

Thin-film solar cells
In Chapter 12 we have discussed the PV technology
based on c-Si wafers, which currently is by far the dominant PV technology. It is very likely that it will staydominant for a long time to go.

parameters and may also be thickness dependent.’
Thin-film solar cells were expected to become cheaper
than first generation solar cells. However, due to thecurrent price decline in wafer based solar cells thin-film
solar cells have not yet become interesting from an economic point of view yet.1 In general thin-film cells havea lower efficiency than c-Si solar cells. GaAs is an exception to this rule of thumb [38]. In contrast to wafer
based silicon solar cells that are self-supporting, thinfilm solar cells require a carrier that gives them mechanical stability. Usual carrier materials are glass, stainless steel or polymer foils. It is thus possible to produce
flexible thin-film solar cells.

In this chapter we will look at an alternative family oftechnologies, namely thin film technologies, also referred
to as the second generation PV technology. These solar
cells are made from films that are much thinner than
the wafers that form the base for first generation PV.
According to Chopra et al. [40], ‘a thin film is a film thatis created ab initio by the random nucleation process of
individually condensing/reacting atomic/ionic/ molecular species on a substrate. The structural, chemical,
metallurgical and physical properties of such a material
are strongly dependent on a large number of deposition

1 According to the PHOTON module price index, the price for wafer-based
modules has decreased around 40% within one year as of 25 May 2012 [41].

163
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Abundance (atoms of element per 106 atoms of Si)

164

13. Thin-film solar cells

109
Major industrial metals in red
Precious metals in purple
Rare earth elements in blue

Rock-forming elements

106
103

O

Si
Al
Na

H
C
Li
B
Be

Mg
F
N

100

K

Ca

P
S
Cl

Fe

Ti
Mn

Sr Zr
Cu Zn
Rb
VCr
Nb
Ga
Y
Sc Co Ni
As
Ge Br
Mo
Se

10-6
0

Ce
Nd
Pb
Gd
La
Sm DyEr Hf
Yb Ta
Pr
Cs
W
Eu
Tl
Cd SbITb Ho Lu
Tm
Ag
Hg Bi
In

Ru

10-3

Ba

Rh

Sn

Pd

Te

Rarest "metals"

10

20

30
40
50
60
Atomic number Z

Re
Os

Th

U

Au
Pt
Ir

70

80

90

Figure 13.1: The abundance of elements (as atomic fraction)

In thin-film solar cells the active semiconductor layers
are sandwiched between a transparent conductive oxide (TCO) layer and the electric back contact. Often aback reflector is introduced at the back of the cell in order to minimise transmissive solar cell losses. As wewill see in this chapter, many different semiconductors
are used for thin-film solar cells. Figure 13.1 shows theabundance of metals in the Earth’s crust. Some semiconductors require very rare elements such as indium(In), selenium (Se), or tellurium (Te). For terawatt scale
photovoltaics, solar cells should be based on abundantelements only.
Thin-film PV technologies easily can fill a book on its
own, see for example the book edited by Poortmans
and Arkhipov [42]. In this chapter we therefore only
can give a general introduction into the different thinfilm technologies. We will focus on the working principles of the various devices, the current status and thefuture challenges of the various technologies. But before
we start with this discussion, we will begin with a short
introduction on transparent conducting oxides (TCOs).

in the Earth’s upper crust. Data provided by the USGS [39].

13.1 Transparent conducting
oxidesDue to the paramount importance of the TCO layer forthe solar cell performance we briefly discuss its main

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165

13.1. Transparent conducting oxides

properties. The TCO layer acts as electric front contact of the solar cell. Furthermore, it guides the incident light to the active layers. It therefore should be both
highly conductive and highly transparent in the active
wavelength range. The first resistance measurements on
thin-films of what we nowadays call TCOs were published by Bädeker in 1907 [43].

Figure 13.2 shows the transmission, reflection and absorption spectra of a flat ZnO:Al layer. Following Kluth,
we divide this spectrum into three parts [45]: For shortwavelength, the transmission is very low due to thehigh absorption of light with energies higher than the
bandgap. For longer wavelength, with photon-energies
below the bandgap, the transmission is very high. We
here see interference fringes that can be used to determine the film thickness. After a broad highly transmissive
wavelength band, the absorption increases again. This

Copyright Delft University of Technology, 2014

0.4
0.2

transmission

500

1000

plasma wavelength λ p

T,R,A (–)

0.6

interference
fringesparasitic absorption
due to TCO-bandgap

0.8

e
fre tionsed orprea bsincrier acar

Typical TCO layers are made from fluor-doped tin oxide (SnO2: F), aluminium-doped zinc oxide (ZnO: Al),boron-doped zinc oxide (ZnO: B), hydrogen-doped (hydrogenated) indium oxide (In2 O3: H) [44], and indium
tin oxide, which is a mixture of about 90% indium
oxide (In2 O3 ) and 10% tin oxide (SnO2 ). These films
are processed using either sputtering (ZnO:Al), lowpressure-chemical vapour deposition (LP-CVD, used e.g.
for ZnO:B), metal-organic chemical vapour deposition(MO-CVD), or atmospheric-pressure chemical vapour
deposition (AP-CVD, used e.g. for SnO2: F).

1absorption

reflection

0
1500

2000

2500

Wavelength (nm)Figure 13.2: Transmission, reflection and absorption of a

ZnO:Al layer (d = 880 nm).

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166

13. Thin-film solar cells

absorption is called free carrier absorption an can be
explained with the Drude theory of metals that was developed in 1900 [46, 47]. In this model, the frequencydependent electric permittivity is given byω 2p
ω 2 + i ωt

(13.1),

where χ(ω ) is the dielectric susceptibility, t is the relaxation time,2 n − i k̃ is the complex refractive index andω p denotes the plasma frequency that is given by
ωp =

Ne2
.
e0 m∗e 2(13.2)

Here, N is the density of free charge carriers, e is theelementary charge, e0 is the permittivity of vacuum andm∗e is the effective electron mass in the TCO layer.
The real and imaginary part of the susceptibility are
given byt2,
ω 2 t2 + 1t/ωω 2p 2 2.
ω t +1(<χ)(ω ) = −ω 2p(13.3a)(=χ)(ω ) =(13.3b)2 The relaxation time denotes the average time between two collisions, i.e. two
abrupt changes of velocity, of the electrons.

Copyright Delft University of Technology, 2014e(ω ) ≈ 1 −

ω 2p
ω2

,

(13.4)

while the imaginary part is negligible.

2e(ω ) = n(ω ) − i k̃(ω )= 1 + χ(ω ) = 1 −If ωt  1, e can be simplified to

If this approximation is valid around ω p , the material istransparent for ω > ω p (e > 0). For ω < ω p , e becomes
negative, i.e. the refractive index is purely imaginary
and the material therefore has a reflectivity of 1. In thiscase, the material changes dramatically from transparent to reflective, as ω p is crossed.
If the approximation ωt  1 is not valid, =χ cannot
be neglected. The imaginary part will increase withdecreasing frequency, i.e. with increasing wavelengths
the absorption increases. For wavelengths longer thanthe plasma wavelength the material becomes more reflective, what we also see in Fig. 13.2. For application
in solar cells, the TCO should be highly transparent in
the active region of the absorber. Therefore the plasmawavelength should at least be longer than the bandgap
wavelength of the absorber. On the other hand theplasma frequency is proportional to the free carrier
density N. A longer plasma wavelength therefore corresponds to a lower N. Finding an optimum between
high transparency and high carrier densities is an important issue in designing TCOs for solar cell applications.
Even though the Drude model gives a good approxima-

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13.1. Transparent conducting oxides

tion of the free carrier related phenomena in TCOs, thismodel often is too simple. Therefore several authors
used extended Drude models with more parameters[48–50].
Of all TCO materials currently available, the trade off
between transparency and conductivity is best for indium tin oxide [51]. However, indium is a rare earth
element with a very low abundance of 0.05 ppm in the
Earth’s crust, similar to the abundance of silver (0.07
ppm) and mercury (0.04 ppm) [52], which makes it lesspreferable for cheap large-scale PV applications. Therefore other TCO materials are thoroughly investigatedand used in industry. Amongst them are aluminium
doped zinc oxide, boron doped zinc oxide and fluorine
doped tin oxide. The abundances of the used elements
are: aluminium: 7.96%, zinc: 65 ppm, boron: 11 ppm,
fluorine: 525 ppm, and tin: 2.3 ppm [52].
The morphology of selected TCO samples

As we have briefly mentioned in Chapter 10 and will
discuss in more detail in Chapter ??, light management
is crucial for designing solar cells. Very often, the TCO
layers in thin-film solar cells have nano-textured surfaces. The incident light is scattered at these interfaces
such that the average photon path length in the absorber layer is enhanced. Thus, more light can be absorbed and the photocurrent density can be increased.

Copyright Delft University of Technology, 2014

167

However, the texture may also influence the growth oflayers that are deposited onto the TCO and thus alter
the electrical properties, which may lead to a reduction
in Voc and the fill factor. Further, the increased interface
area due to the texture may lead to more surface recombination and also reduce the voltage.
Figure 13.3 shows the surface morphology, rms roughness σr and correlation length `c of some nano-textured
TCO samples made from three different materials.
Fluorine-doped AP-CVD tin oxide (SnO2 :F) of Asahi Utype [54] and boron-doped LP-CVD zinc-oxide (ZnO:B)of so-called ‘B-type’ from PV-LAB of the École polytechnique fédérale de Lausanne (EPFL), Switzerland [55],obtain their nano-structure already during the deposition process. The nano-structure is due to the crystal
growth of the TCO layers and has a pyramid-like shape.
In contrast, RF-sputtered, aluminium-doped zinc oxide(ZnO:Al) is flat after deposition (σr ≈ 3 nm, depending on the deposition condition als higher). To obtain anano-texture, ZnO:Al is etched in a 0.5% HCl solution[56, 57]. This etching process leads to crater-like features.
The lateral size of these craters is influenced by thegrain size of the zinc oxide crystals. Fig. 13.3 (c) and(d) shows etched ZnO:Al with broad craters. Fig. 13.3(e) and (f) shows etched ZnO:Al with narrow craters.
The surface morphologies shown in Fig. 13.3 were obtained with atomic force microscopy (AFM). Also the twostatistical parameters rms roughness σr and correlation

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168

13. Thin-film solar cells

290 (e)2 μm

1.3 (d)510 (f)0.0

ZnO:B
σ r ≈ 220 nm, ℓc ≈ 470 nm

2 μm

ZnO:Al (60 s, broad craters)
σ r ≈ 90 nm, ℓc ≈ 625 nm

0760

z (nm)z (μm)
2 μm

2 μm

ZnO:Al (20 s, narrow craters)
σ r ≈ 50 nm, ℓc ≈ 215 nm

ZnO:Al (40 s, broad craters)
σ r ≈ 60 nm, ℓc ≈ 520 nm

SnO2 :F (Asahi-U)
σ r ≈ 40 nm, ℓc ≈ 175 nm
(b)

0

0

z (nm)0z (nm)z (nm)2 μm

390

z (nm)

290 (c)

(a)

2 μm

0

ZnO:Al (40 s, narrow craters)
σ r ≈ 100 nm, ℓc ≈ 375 nm

Figure 13.3: The morphology, rms roughness, and correlation length for some selected TCO samples [53].

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169

13.2. The III-V PV technology

length `c were obtained with AFM scans of 256×256
points over an area of 20×20 µm2 . AFM and the determination of the statistical parameters are explainedin Appendix C.

13.2

The III-V PV technology

Of all the thin-film technologies, the III-V (speak threefive) PV technology results in the highest conversion
efficiencies under both one sun standard test conditions
and concentrated sun conditions. Therefore, it is mainly
used in space technology and in solar concentrator technology.
As some concepts use a crystalline germanium or an
GaAs wafer as substrate, it might not be considered
as a real thin film PV technology, in contrast to thinfilm silicon, CdTe, CIGS or organic PV, which we will
discuss later in this chapter. However, the III-V based
absorber layers themselves can be considered as thincompared to the thickness of crystalline silicon wafers.
The III-V materials are based on the elements with threevalence electrons like aluminium (Al), gallium (Ga) or indium (In) and elements with five valence electrons like
phosporus (P) or arsenic (As). Various different semiconductor materials such as gallium arsenide (GaAs),
gallium phosphide (GaP), indium phosphide (InP), indium

Copyright Delft University of Technology, 2014

Figure 13.4: A unit cell of a gallium arsenide crystal in thezinc blend structure [58].

arsenide (InAs), and more complex alloys like GaInAs,
GaInP, AlGaInAs and AlGaInP have been explored.
We now will take a closer look on GaAs, which is the
III-V semiconductor most used for solar cells. The GaAs
has a zinc blend crystal structure, illustrated in Fig. 13.4.
It is very similar to the diamond cubic crystal structurewith the difference that it has atoms of alternating elements at its lattice sites. In contrast to c-Si, where every
Si atom has four neighbours of the same kind, in GaAs,
every Ga has four As neighbours, while every As atom
has four Ga neighbours. When compared to silicon,
GaAs has a slightly larger lattice constant of 565.35 pm,

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170

13. Thin-film solar cells
X-valley[100]
Γ-valleyEL

1.71 eVEg

1.42 eVEX

1.90eV

EnergyL-valley[111]

Heavy holes
T = 300K

Wavevector
Figure 13.5: The electronic band dispersion diagram of gal-

lium arsenide.

and significant denser than silicon with a density of
5.3176 g/cm3 (Si: 543.07 pm, 2.3290 g/cm3 ). Note thatboth Ga and As are are roughly twice as dense as silicon. In contrast to silicon, which is highly abundant,
the abundance of gallium in the Earth’s crust is only
about 14 ppm [59]. GaAs therefore is a very expensive
material. Arsenic is highly toxic; it is strongly suggested
that GaAs is carcinogenic for humans [60].
Figure 13.5 shows the electronic band dispersion diagram of gallium arsenide. We can easily see that GaAs

Copyright Delft University of Technology, 2014is a direct band gap material, i.e. , the highest energy
level in the valence band is vertically aligned with thelowest energy level in the conduction band. Hence,
only transfer of energy is required to excite an electronfrom the valence to the conduction band, but no transfer of momentum is required. The band gap of GaAs
is 1.424 eV [20]. Here we look again at the absorption
coefficient versus the wavelength. Consequently, as you
can see in Fig. 12.4, the absorption coefficient of GaAs
is significantly larger than that of silicon. The same istrue for InP, another III-V material that also is shownin Fig. 12.4. Because of the high absorption coefficient.
Because of the high absorption coefficient, the sameamount of light can be absorbed in a film more than
one order of magnitude as thin when compared to silicon. Another advantage of the direct III-V semiconductor materials their sharp band gap. Above Eg , theabsorption coefficient increases quickly.

Let us now take a look at the utilisation of the band
gap energy. Since GaAs is a direct band gap material,radiative recombination processes become important. On
the other hand, Shockley-Read-Hall recombination canbe kept at a low level because III-V films can be deposited tih epitaxy processes that result in high purity films.

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171

13.2. The III-V PV technology

13.2.1

Multi-junction cells

III-V PV devices can reach very high efficiencies because they are often based on the multi-junction
concept, which means that more than one band gap isused. As discussed in Chapter 10, the maximum theoretical efficiency of single-junction cells is describedby the Shockley-Queisser limit. A large fraction of theenergy of the energetic photons are lost as heat, while
photons with energies below the band gap are lost as
they are not absorbed.
For example, if we use a low bandgap material, a large
fraction of the energy carried by the photons is not
used. However, if we use more band gaps, the sameamount of photons can be used but less energy iswasted as heat. Thus, large parts of the solar spectrum
and larges part of the energy in the solar spectrum canbe utilised at the same time, if more than one p-n junctions are used.
In Fig. 13.6 a typical III-V triple junction cell is illustrated. As substrate, a germanium (Ge) wafer ie used.
From this wafer, the bottom cell is created. Germanium
has a band gap of 0.67 eV. The middle cell is based on
GaAs and has a band gap of about 1.4 eV. The top cellis based on GaInP with a band gap in the order of 1.86
eV.

Contact=
AR

Top cell window/emitter

n+-GaAsn-AllnP
n-GalnP
n-GalnP

Top cell base/BSF
Wide-Eg tunnel junction
Middle cell window/emitter

p-AlGalnPp++n++Middle Cell Window

el l
pC

To

n-GaAsp-GaAsMiddle cell base/BSF

TC & MC crystal quality:
Nucleation, buffer,
Interface control,
Lattice-matching

p-GalnPp++n++buffernucleation
n-Ge

l

ne
tun
Eg
e
l
d l
Wi e Ce
l
dd w
i
M do
n
i
W
dd
Mi

el l
le C
on

ti
nc
l ju
e
nn
Tu
om
ott

ll
Ce

B

p-Ge substratecontact

Figure 13.6: Illustrating a typical III-V triple junction solarcell.

Let now take a closer look on how a multi-junction
solar cell works. Light will enter the device from

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172

13. Thin-film solar cells
30

Iph

I

ID

Voc,1

25

J (mA/cm2)

20
15
10

3
Iph

2
1

I

ID

Voc,2

1+2+3

5
0

Iph

I

ID

Voc,3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

Voltage (V)Figure 13.7: The J-V curves of the three junctions used in

the III-V triple junction cell and J-V curves of the III-V triple
junction cell (in black).

the top. As the spectral part with the most energetic
photons like blue light has the smallest penetration
depth in materials, the junction with the highest band
gap always acts as the top cell. On the other hand, as
the near infrared light outside the visible spectrum has
the longest penetration depth, the bottom cell is the cellwith the lowest band gap.
Figure 13.7 shows the J-V curve of the three single pn junctions. We observe p-n junction 1 has the highest

Copyright Delft University of Technology, 2014

Figure 13.8: The equivalent circuit of the three junctions

connected in series.

open circuit voltage and the lowest short circuit current density, which means that this p-n junction has thehighest band gap. In contrast, p-n junction 3 has a low
open circuit voltage and a high current density, consequently it has the lowest band gap. p-n junction 2
has a band gap in between. Hence, if we are designinga triple-junction cell out of these three junctions, junction 1 will act as the top cell, junction 2 will act as the
middle cell and junction 3 will act as the bottom cell.

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13.2. The III-V PV technology

For understanding how the J-V curve of the triple junction looks like, we take a look at the equivalent circuit.
Every p-n junction in the multijunction cell can be represented by the circuit of a single-junction cell, as discussed in chapter 9. As the three junctions are stacked
onto each other, they are connected to each other inseries, as illustrated in Fig. 13.8. In a series connection,
the voltages of the individual cell add up in the triple
junction cell. Further, the current density in a seriesconnection is equal over the entire solar cell, hence thecurrent density is determined by the p-n junction generating the lowest current. The resulting J-V curve is also
illustrated in Fig. 13.7. We see that the voltages add up
and the current is determined by the cell delivering thelowest current.
Figure 13.9 (a) shows a typical band diagram of such
triple junction. The top cell with high band gap isshown at the left hand side and the bottom cell is at theright hand side. However, this band diagram does not
represent reality. If we would place three p-n junctions
in series, for example the p-layer of the top cell and then-layer of the middle cell would form a p-n junctionin the reverse direction than the p-n junctions of thethree single junction solar cells. These reverse junctions
would significantly lower the voltage of the total triple
junction.
For preventing the creation of such reverse junctions,so-called tunnel junctions are included. These tunnel

Copyright Delft University of Technology, 2014

(a)

n

p

n

p

n

p

(b)

n

p

n

p

n

p

Figure 13.9: The band diagram of the III-V triple junction cell(a) without and (b) with tunnel junctions.

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174

junctions align the valence band at one side with theconduction band at the other side of the tunnel junction, as illustrated in Fig. 13.9 (b). They have a highband gap to prevent any parasitic absorption losses.
Further, tunnel junctions are relatively thin and havean extremely narrow depletion zone. As a result, theslopes of the valence and conduction band are so steepthat the electrons from n-layer can tunnel through thesmall barrier to the p-layer, where they recombine withthe holes. It is important to have the tunnel junctions
with a low resistance such that the voltage loss acrossthem is low.
In the triple junction cell, two tunnel junctions arepresent. First, a tunnel junction via that the holes in
the p-layer of the top cell have to recombine with theelectrons of the n-layer of the middle cell. Secondly, atunnel junction where the holes in the p-layer of the
middle cell recombine with the electrons of the n-layerof the bottom cell. The electrons in the top cell n-layer
are collected at the front contact and the hole in the player of the bottom cell are collected at the back contact.
It is important to realise that the recombination current
at the tunnel junctions represents the current density of
the complete triple junction.
Let us take a look at the external parameters of a typical lattice matched triple junction solar cell from SpectroLab, which is a subsidiary of The Boeing Company.
Because this cell was developed for space applications,

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13. Thin-film solar cells

it is not tested under AM1.5 conditions, but under AM0
conditions with an irradiance of 135 mW/cm2 . The
total open circuit voltage is 2.6 Volts. The short circuitcurrent density is 17.8 mA/cm2 , which corresponds to aspectral utilisation up to the lowest band gap of 0.67 eVofJtot = 3 · 17.8 mA/cm2 = 53.4 mA/cm2 .

(13.5)The maximal theoretical current that could be generated
when absorbing AM0 for all wavelength shorter than1850 nm, which corresponds to 0.67 eV, is 62 mA/cm2 .
The average EQE of the solar cell thus is 86%, which isan very impressive value. Further, this cell has a very
high fill factor of 0.85, which means that it has a conversion efficiency of 29.5% under AM0 illumination.
Figure 13.10 shows the EQE of the subcells of a typical III-V triple junction cell. We note that the shapeof the spectral utilisation of all the curves approaches
the shape of block functions, which would be the mostideal shape. These block shapes are possible because
the III-V materials have very sharp band gap edges andhigh absorption coefficients. We see that the bottom cell
generates much more current than the middle and top
cells. Hence, a lot of current is lost in the bottom cell.
This ineffective use of the near infrared part can be
reduced using quadruple junctions instead of triple
junctions. In these cell, an additional cell is placed inbetween the middle and bottom cells of the triple junc-

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13.2. The III-V PV technology

tion. The spectral utilisation can be even increased further by moving to multi-junction solar cells consisting of 5 or even 6 junctions. The major challenge forthese cells is that lattice matching can not be guaranteed anymore. Lattice-mismatched multi-junctions arecalled metamorphic multijunctions. They require buffer
layers that have a profiling in the lattice constant, going from the lattice constant of one p-n junction to thelattice constant of the next p-n junction. Using this technology, Spectrolab could demonstrated an impressive
38.8% conversion efficiency of a 5-junction solar cell under 1-sun illumination [38].

External Quantum Efficiency (%)

100
90
80
70
60
50
40
30
GaInP

20

GaInAs

Ge

10
0

300

500

700

900 1100 1300 1500 1700 1900

Wavelength (nm)Figure 13.10: The external quantum efficiency of a three-junction III-V solar cell (Data from SPECTROLAB).

Copyright Delft University of Technology, 2014

The III-V PV technology is very expensive. Hence, suchcells are mainly used for space-applications and in concentrator technology, where high performance is more
important that the cost. For reducing the cost per Wattpeak, a solar concentrator can be used. In such a concentrator, the irradiance of a large area is concentrated onto
the small solar device. One advantage of concentrated
sunlight is that the open circuit voltage is increasing
with the increasing irradiance as well, as long as Auger
recombination of light excited charge carriers does not
become dominant. It is, however, very important thatthe small solar cell devices are actively cooled, as theirperformance decreases with increasing temperatures.
This temperature dependence is caused by stronger recombination and hence higher dark currents and is discussed in more detail in Section 18.3. Additionally, concentrator systems require sun-tracking systems, as onlyThis copy is provided for free, for personal use only.

176

13. Thin-film solar cells

direct sunlight can be concentrated. The tracking systems ensure that the optical concentrator system tracks
the sun and guarantees the optimal light concentrationon the small PV device during the entire day. Hence,
the non-modular costs of concentrator systems are larger than that of conventional PV system based on c-Si
or other thin-film technologies.

As the layer-by-layer growth process is very slow, it allows the deposition of compact materials without any
vacancy defects. Furthermore, processing at high vacuum conditions prevents the incorporations of impurities. Hence, III-V semiconductors can be deposited up
to a very high degree of purity. Dopants can be added
to make it n- or p-type.

The current world record conversion efficiency for all
solar cell technologies on lab scale is 44.4% for a threejunction III-V solar cell under 302-fold concentrated sun
light conditions. This result was achieved by Sharp [38].

Typically, III-V semiconductor layers are deposited by using metal-organic chemical vapour deposition(MOCVD). Typical precursor gasses are trimethylgallium [Ga(CH3 )3 ], trimethylindium [In(CH3 )3 ], trimehtylaluminium [Al2 (CH3 )6 ] , arsine gas (AsH3 ) andphosphine gas (PH3 ). Surface reactions of the metalorganic compounds and hydrides, which contain therequired metallic chemical elements, create the rightconditions for the epitaxial crystalline growth. Epitaxial
growth is a very expensive process. Similar techniques
are also used in the microchip production process.

13.2.2

Processing of III-V semiconductormaterials

As already mentioned above, high quality III-V semiconductor materials can be deposited using epitaxy deposition methods. In this method, crystalline overlayers
are deposited on a crystalline substrate, such that they
adopt the crystal lattice structure of the substrate. The
precursor atoms from that the layers are grown areprovided by various elemental sources. For example,
if GaAs is deposited with epitaxy, Ga and As atoms are
directed to a growth surface under ultra-high vacuum
conditions. For growing III-V semiconductors, germanium substrates usually are used. On this substrate
the GaAs crystalline lattice is grown layer-by-layer andadopts the structure of the crystalline substrate.

Copyright Delft University of Technology, 2014

A big challenge of depositing III-V semiconductor materials is that the lattice constants of the various materials are different, as seen in Fig. 13.11. We see that every
III-V semiconductor has a unique bandgap-lattice constant combination. Hence, interfaces between different
III-V materials show a lattice mismatch, as illustrated inFig. 13.12. Because of this mismatch, not every valence
electron is able to make a bond with a neighbour. This
problem can be solved by lattice matching, as this also
is done in the triple junction cell discussed in Section

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177

13.2. The III-V PV technology

latticeconstant a0
dangling
bond
4.0
3.6

Bandgap (eV)

3.2

ZnS

MgSe

GaN

2.8
AlP

2.4
2.0
1.6

CdS

ZnTe

Figure 13.12: Illustrating the lattice mismatch at an interface
CdSe

AlSb

InP

between two different III-V materials.
CdTe

Si

0.8

InN

0.4
0.0

AlAs

GaAs

BAs

1.2

ZnSc

GaP

BP

latticeconstant a1

4.5

5.0

Ge
5.5

GaSb
InAs

6.0

InSb

6.5

Lattice Constant (Å)
Figure 13.11: The bandgap and lattice constant of various

III-V semiconductor materials.

Copyright Delft University of Technology, 2014

13.2.1. For understanding what this means, we take another look at the phase diagram shown in Fig. 13.12.
The triple junction is processed on a p-type germanium
substrate, the bottom cell is a Ge cell. One top it is best
to place a junction that has a higher band gap but thesame lattice constant. As we can see in the plot, GaAs
has exactly the same lattice constant as Germanium.
Therefore, between GaAs and Germanium interfaces
can be made without any coordination defects related
to mismatched lattices. For reasonable current matching
the desired band gap of the top cell should be around
1.8 eV. However, we see that no alloys, based on solely
two elements, exist. But if we take a mixture of gallium,
indium and phosphorus, we can make a III-V alloys

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178

13. Thin-film solar cellswith a band gap of 1.8 eV material and with matchinglattice constant. Consequently, triple junction cells basedon GaInP, GaAs and Germanium can be fully lattice
matched.

13.3

Thin-film silicon technology

In this section we are going to discuss the thin-film silicon PV. An advantage of thin-film silicon solar cells isthat they can be deposited on glass substrates and evenon flexible substrates.

13.3.1Thin-film silicon alloys

Thin-film silicon materials usually are deposited withchemical vapour deposition (CVD) processes that we will
discuss in more detail in Section 13.3.3. In chemical vapour deposition different precursor gasses are brought
into the reaction chamber. Due to chemical reactions,layer is formed on the substrate. Depending on theused precursors and other deposition parameters suchas the gas flow rate, pressure, and temperature, various different alloys with different electrical and opticalparameters can be deposited. We will discuss the mostimportant alloys in the following paragraphs.

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We start with two alloys consisting of silicon and hydrogen: hydrogenated amorphous silicon (a-Si:H) and hydrogenated nanocrystalline silicon (nc-Si:H), which is also
known as microcrystalline silicon. Usually in discussions the term ‘hydrogenated’ is left out for simplicity.
Hydrogenated means that some of the valence electrons
in the silicon lattice are passivated by hydrogen, whichis indicated by the ‘:H’ in the abbreviation. The typicalatomic hydrogen content of these alloys is from 5% up
to about 15%. The hydrogen passivates most defects inthe material, resulting in a defect density around 1016
cm−3 [61], which is suitable for PV applications. Pureamorphous silicon (a-Si) would have an extremely highdefect density (> 1019 cm−3 ) [42], which would resultin fast recombination of photo-excited excess carriers.
Similarly, we can make alloys from germanium and hydrogen: hydrogenated amorphous germanium (a-Ge:H) andhydrogenated nanocrystalline germanium (nc-Ge:H).
Let us now take a look at alloys of silicon with fourvalence electrons with other elements with four valence
electrons, carbon and germanium. In thin-film silicon solar
cells, both hydrogenated amorphous and nanocrystalline silicon-germanium alloys (a-SiGe:H and nc-SiGe:H)
are being used. Silicon is also mixed using the fourvalence electron material, carbon, leading to hydrogenated amorphous silicon carbide (a-SiC:H).
Another interesting alloy is obtained when oxygen with
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179

13.3. Thin-film silicon technology

1
2

Si

4
H

3

Figure 13.13: Illustrating the atomic structure of amorphoussilicon with four typical defects: (1) monovacancies, (2) divacancies, (3) nanosized voids with mono- and (4) dihydrides.
With kind permission of M. A. Wank [62].

hydrogenated nanocrystalline silicon oxide is often used inthin-film silicon solar cells.
All these alloys can be doped, usually boron is used asa p dopant while phosphorus is the most common ndopant.
Many of the alloys mentioned above are present as
amorphous materials. It is hence important to discuss the
structure of amorphous lattices. In this discussion wewill limit ourselves to amorphous silicon, since it is thebest studied amorphous semiconductor and the general properties of the other amorphpous alloys are similar. In Chapter 12 we thoroughly discussed crystalline

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silicon, which has an ordered lattice in which the orientation and structure is repeated in all directions. For
amorphous materials this is not the case, but the lattice
is disordered as it is illustrated in Fig. 13.13. This figure shows a so-called continuous random network (CRN).
On atomic length scales, or also called short-range order,
the atoms still have a tetrahedral coordination structure,
just like crystalline silicon. But the silicon bond angles
and silicon-silicon bond lengths are slightly distortedwith respect to a crystalline silicon network. However,at larger length scales, or also referred to as long range,
the lattice does not look crystalline anymore. Not all
silicon atoms have four silicon neighbours, but somevalence electrons form dangling bonds, similar to theunpassivated surface electrons already discussed in
Chapter 12. Recent studies show that these dangling
bonds are not distributed homogeneously throughout
the amorphous lattice, but that they group in divacancies, multivacancies or even nanosized voids smets:2007.
The surfaces of these volume deficiencies are passivatedwith hydrogen.
Another phase of hydrogenated silicon alloys is thenanocrystalline lattice. It is a heterogeneous material
with a structure that even is more complex than thatof amorphous materials. Nanocrystalline silicon consists of small grains that have a crystalline lattice andare a few tens of nanometres big. These grains are embedded in a tissue of hydrogenated amorphous silicon.
Figure 13.14 shows the various phases of thin-film sil-This copy is provided for free, for personal use only.

180

13. Thin-film solar cells

Wavelength (μm)

Figure 13.14: The various phases of thin-film silicon with ahighly crystalline phase on the left and amorphous silicon on
the right[63].

icon [63]. On the left-hand side, a fully crystalline phaseis shown, which is close to that of poly-crystalline silicon, except that is has much more cracks and poresin it. On the right hand side, the phase represents theamorphous lattice. Please note again, that the terms
microcrystalline and nanocrystalline refer to the samematerial. Going from right to left, the amorphous phaseis changing in to a mixed phase with a few small crystalline grains to a phase which is dominated by large
crystalline grains and a small fraction of amorphous tissue. Research has shown that the best nanocrystalline
bulk materials used in solar cells has a network close to
the amorphous-nanocrystalline silicon transition region andits crystalline volume fraction is in the order of 60%.

Copyright Delft University of Technology, 2014

Absorption coefficient (cm-1)

2.48

1.24

0.83

0.62

0.50

0.41

105
104
103
a-Si Ge:H

102

a-Si:H

101

c-Si

100
0.5

1.0

1.5

2.0

2.5

3.0

Photon energy (eV)Figure 13.15: The absorption coefficients of different thin-film

silicon materials.

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13.3. Thin-film silicon technology

The band gap of nanocrystalline silicon is close to thatof crystalline silicon (≈ 1.12 eV) due to the crystalline
network in the grains. The band gap of amorphous silicon is in the order of 1.6 up to 1.8 eV, which can betuned by the amount of hydrogen incorporated in to
the silicon network. It is larger than that of crystalline silicon because of the distortions in bond angles
and bond lengths. It is out of the scope of this bookto discuss the reasons for this increase in band gap in
more detail. An important consequence of a disordered
amorphous lattice is that the electron momentum ispoorly defined in contrast to crystalline silicon. As wediscussed in Chapter 12, both energy and momentum
transfer are needed to excite and electron from thevalence band to the conduction band. Hence, crystalline silicon is an indirect band gap material. This is nottrue for amorphous silicon, which is a direct band gap
material. Therefore the absorptivity of a-Si:H is muchhigher than that of c-Si, as we can see in Fig. 13.15. We
see that the absorption coefficient for amorphous silicon in the visible spectrum is much larger that thatof crystalline silicon. In some wavelength regions it isabout two orders of magnitude larger, which meansthat much thinner silicon films can be used in reference to the typical wafers in crystalline silicon solar
cells. In the figure, also data for amorphous silicongermanium are shown. a-SiGe:H has lower band gap
and even higher absorption coefficient in the visible. Its
band gaps are in the range of 1.4 up to 1.6 eV. Amorph-

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ous silicon carbide alloys have band gaps of 1.9 eV andlarger. Finally, nanocrystalline silicon oxides have bandgaps exceeding 2 eV.

13.3.2The design of thin-film silicon solar
cells

The first successful a-Si:H solar cell with an efficiencyof 2.4% was reported by Carlson and Wronski in 1976[64]. We now will discuss how such solar cells can be
designed and what are the issues in designing them.
When compared to c-Si, hydrogenated amorphous andnanocrystalline silicon films have a relatively high defect density of around 10−16 cm−3 . As mentioned before, because of the disordered structure not all valence
electrons are able to make bonds with the neighbouring atoms. The dangling bonds can act as defects that
limit the lifetime of the light excited charge carriers.
This is described with Shockley-Read-Hall recombination, which controls the diffusion length. Because of thehigh defect density the diffusion length of charge carriers in hydrogenated amorphous silicon is only 100 nm
up to 300 nm. Hence, the transport of charge carriers ina thick absorber can not rely on diffusion.
Therefore, amorphous silicon solar cells are not basedon a p-n junction like wafer based c-Si solar cells.
Instead, they are based on a p-i-n junction, which

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13. Thin-film solar cells

(a)

(c)

glass

glass

transparent conducting
oxide (TCO)

transparent conducting
oxide (TCO)

silicon p-i-n junction

silicon p-i-n junction

ZnO or SiOx back reflector

back contact

back contact(b)
p-a-SiC:H

intrinsic a-Si:H

n-a-Si:H

conduction band
EFermivalence band

Figure 13.16: Illustrating the (a) layer structure and the (b)band diagram of an amorphous silicon solar cell. (c) Thin-film
silicon solar cells hava nanotextured interfaces.

Copyright Delft University of Technology, 2014means that an intrinsic (undoped) layer is sandwiched
between thin p-doped and n-doped layers, as illustrated
in Fig. 13.16 (a). While the i-layer is several hundredsof nanometres thick, the doped layers are only about 10
nm thick. Between the p- and n-doped layers a built-in
electric field across the intrinsic absorber layer is created. This is illustrated in the electronic band diagram
shown in Fig. 13.16 (b). If the layers are not connectedto each other, the Fermi level in the p- n layers is closer
to the valence and the conduction bands, respectively.
For the intrinsic layer, it is in the middle of the bandgap. When the p, i, and n layers are connected to eachother, the Fermi level has to be the same throughout thejunction, if it is in the dark and under thermal equilibrium. This creates a slope over the electronic band in
the intrinsic film as we can see in the illustration. This
slope reflects the built-in electric field.
Another way to look at the field is to consider the ilayer as an dielectric in between to charged plates (thedoped layers). Neglecting any edge effects, the electricfield in the i layer is given by
E=

σ
,
ee0

(13.6)

where σ is the charge density on the doped layers ande is the electric permittivity of the i-layer material. Notethat the electric field in the i layer is constant. Follow-

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183

13.3. Thin-film silicon technology

ing Eq. (4.29), we find the voltage in the i layer to be
U (x) = −Z

E( x ) dx = U0 −

σ
x.
ee0

(13.7)We see that the voltage and hence the energy in the ilayer is linear, just as shown in the band diagram ofFig. 13.16 (b).
Because of the electric field, the light excited charge
carrier will move through the intrinsic layer. As discussed in Section 6.5.1, the holes move up the slope
in the valence band towards the p layer and the electrons move down the slope in the conduction band
towards the n layer. Such a device, where electronic
drift because of an electric field is the dominant transport mechanism is called a drift device. In contrast, awafer based crystalline silicon solar cell, as discussed
in Chapter 12, can be considered as a diffusion device.
Note, that because of the intrinsic nature of the absorber layer the hole and electron density is in the sameorder of magnitude. On the other hand, in the p layer
the holes are the majority charge carriers and the dominant transport mechanism is diffusion. Similarly, in the
n layer the electrons are the majority charge carriers
and again diffusion in the dominant transport mechanism. Because of the low diffusion length, both p and nlayers must be very thin thin.
Now we take another look at Fig. 13.16 (a). The cell
sketched there is deposited in superstrate configuration.

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This means that the layer that is passed first by the incident light in the solar cell is also deposited first in the
production process. Also the term p-i-n layer refers tothis superstrate configuration, as it indicates the orderof the depositions: because in thin-film silicon holes
have a significantly lower mobility than electrons, thep layer is in front of the n layer and hence the p layer
is deposited first. As then the generation rate is higher
close to the p layer, more holes can reach it.
In superstrate thin-film silicon solar cells, usually glass
is used as a superstrate because it is highly transparent and can easily handle all the chemical and physicalconditions in that all the depositions are carried out.
Before the p-i-n layers can be deposited, a transparent
front contact has to be deposited. Usually transparent conducting oxides are used, that are discussed in more detailin Section 13.1.
Another possible configuration is the substrate configuration. There, either the substrate acts like a back contact or the back contact is deposited on the substrate.
Consequently, no light will pass through the substrate.
Thin-film silicon solar cells deposited in the substrateconfiguration are also called n-i-p cells as the n layer isdeposited before the i layer and the p layer.
Usually, thin-film silicon solar cells have no flat interfaces, as shown in Fig. 13.16 (a), but nano-textured interfaces, as illustrated in Fig. 13.16 (c). These textured
interfaces scatter the incident light and hence prolong

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184

the average path length of the light through the absorber layer. Therefore, more light can be absorbed andthe photocurrent can be increased. This is an exampleof light management that is discussed in more detail in
Chapter ??.
For the p layer, often not amorphous silicon but higher
band gap materials such as silicon carbide or silicon oxides are used in order to minimise parasitic absorption
mainly in the blue part of the spectrum. Usually, boron
is used as a dopant. Also for the n layer nowadays often silicon oxides are used, but sometimes still a-Si:H
is used. Phosphorus is the main dopant used here. The
n-SiOx :H is very transparent. Therefore it also can beused as a back reflector structure when its thickness ischosen such that destructive interference occurs at thei-n interface minimising the electric field strength andhence parasitic absorption in the n layer. Additionally,
also several tenth of nm thick TCO can be used withthe same purpose. Further, a metallic back reflector that
also acts as the electric back contact is used. Because of
its attractive cost, mainly Al is used for this purpose.
However, also silver that has a higher reflectivity can beused, yet is it more expensive.
The band gap of hydrogenated amorphous silicon isin the order of 1.75 eV, hence it only is absorptive forwavelength shorter than 700 nm. The highest current
densities achieved in single junction amorphous silicon solar cell are 17 up to 18 mA·cm−2 , whereas the

Copyright Delft University of Technology, 2014

13. Thin-film solar cells

maximum theoretic current that could be achieved up
to 700 nm is in the order of 23 mA·cm−2 . Thus, theEQE averaged over the spectrum is in the order of 74
to 77%.
The highest open circuit voltages that have been experimentally achieved are in the order of 1.0 eV. With respect to a band gap of 1.75 eV the band gap utilisationis quite low because of the high levels of SRH recombination and the relative broad valence and conductionband tails. The highest stabilised efficiencies of a single
junction solar cells is 10.1% [38]. It was obtained by the
research Oerlikon Solar Lab in Switzerland, which currently is a subsidiary of the Japanese Tokyo Electron
Ltd.
Besides a-Si:H, also nanocrystalline silicon films areused for the intrinsic absorber layers in p-i-n solar cell
as well. The spectral utilisation of the nc-Si:H is better than that of amorphous silicon because of the lower
band gap of nc-Si:H. However, to utilise the spectral
part from 700 up to 950 nm, thicker films are required,because of the indirect band gap of the silicn crystallites. Typical intrinsic film thicknesses are in between
1 µm and 3 µm. The current nc-Si:H record cell has ashort circuit current density of 28.8 mA·cm−2 , an open
circuit voltage of 523 mV and an efficiency of 10.8%[38]. This result was achieved by the Japanese Instituteof Advanced Industrial Science and Technology (AIST).
Neither a-Si:H nor nc-Si:H has an optimal spectral util-

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185

13.3. Thin-film silicon technology

(a)

glass
transparent conducting
oxide (TCO)

ZnO or SiOx intermediate reflectora-Si:H p-i-n junction
TCO or SiOx back reflector

nc-Si:H p-i-n junction

metal back contact(b)
p-a-SiC:H n-a-SiC:H
i-a-SiC:H p-nc-Si:H

i-nc-Si:H n-nc-Si:H
conduction
band
EFermivalenceband

150-300 nm

1000-2000 nm

Figure 13.17: Illustrating the (a) layer structure and the (b)band diagram of an micromorph silicon solar cell.

Copyright Delft University of Technology, 2014

isation. Therefore also in thin-film technology the multijunction approach is used, just like for III-V solar cells.
Probably the most studied concept is the micromorph
concept illustrated in Fig. 13.17 (a), which is a double
junction concept consisting of an a-Si:H and a nc-Si:H
junction. Like before, the solar cell with the highest
bandgap is used as a top cell that converts the most energetic photons into electricity, while the lower bandgap
material is used for the bottom cell and converts thelower energetic photons.
Figure 13.17 (b) shows the typical band diagram ofsuch a micromorph solar cell, that is also called a tandem solar cell. On the left hand side the electronic band
diagram of the amorphous silicon top cell is shown;on the right hand side the electronic band diagram ofnanocrystalline silicon bottom cell is shown. The blueand green short-wavelength light is absorbed in the top
cell, where electron-hole pairs are generated. Similarly,
the red and infrared long-wavelength light is absorbed
in the bottom cell, where also electron-hole pairs are
generated. Let us take a closer look on the two electronhole pairs excited in the top and bottom cells, respectively. The hole generated in the amorphous top cellmoves to the top p layer and the electron excited in thebottom cell drifts to the bottom n layer. Both can becollected at the front and back contacts. However, theelectron excited in the top cell drifts to the top n layer
and the hole generated in the bottom cell drifts to thep layer. Just as for the III-V multijunction devices, theThis copy is provided for free, for personal use only.

186

13. Thin-film solar cells
3025

nc-Si:HJ (mA/cm2)20

a-Si:H

15

a-Si:H/nc-Si:H
tandem

10
5
0

0.0

0.3

0.6

0.9

1.2

1.5

Voltage (V)Figure 13.18: The J-V curve of a micromorph solar cell andits isolated subcells.

electrons and holes have to recombine at a tunnel recombination junction between the n layer of the top cell andthe p layer of the bottom cell. Often a very thin and defect rich layer is used for this purpose. Just as for the
III-V multijunction solar cells, the total current density
is equal to that of the junction with the lowest current
density. Therefore, for an optimised multijunction cell
all current densities in the various sub cells have to bematched in order to achieve the best spectral utilisation.
Figure 13.18 shows the J-V curves of a single junction

Copyright Delft University of Technology, 2014

a-Si:H solar cell and of a single junction nc-Si:H solarcell. Let us assume that the Voc of the high band gap
a-Si:H top cell has an open circuit voltage of 0.9 V anda relatively low short circuit density of 15 mA·cm−2 ,whereas the low band gap material of nc-Si:H has alower open circuit voltage of 0.5 Volts and a higher
short circuit current density of 25 mA·cm−2 . If wemake a tandem of both junctions, the resulting current density of the double junction is lower than thecurrents in both bottom cells. Because the open circuit
voltage is approximately proportional to ln( Jph /J0 ),the open circuit voltages of the junctions in a tandem
cell will be slightly lower than for similar single junction cells. The total current utilisation of the tandem
cell is determined by the bottom cell, i.e. 25 mA·cm−2 .
Given the examples of the single junctions here, thebest current matching of both cells would deliver 12.5
mA·cm−2 . The current tandem record cell has an efficiency of 12.3% and was manufactured by the Japanesecompany Kaneka [38].
Just as for the III-V technology, also with thin-film technology multijunction cells with more than two junctions can be made. For example the former US company United Solar Ovonic LLC made a thin-film silicon
based triple junction device with an a-Si:H top cell, ana-SiGe:H middle cell and a nc-Si:H bottom cell, illustrated in Fig. 13.19 (a). As illustrated in this figure, also
various other combinations for triple junctions can bemade, for example a-Si:H/nc-Si:H/nc-Si:H. Figure 13.19

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187

13.3. Thin-film silicon technology

(a)

(b)

glass

glass

transparent conducting
oxide (TCO)

transparent conducting
oxide (TCO)

a-Si:H p-i-n junction

1a-Si:H p-i-n junction

nc-Si:H p-i-n junction

2

a-SiGe:H p-i-n junction

nc-Si:H p-i-n junction

3nc-Si:H p-i-n junction

spectral irradiance[mW/(cm2μm)]

180
160
140

metal back contact

1

100

2

3

80
60
40
20
0

metal back contact

AM 1.5120

400

500 600 700

800 900 1000 1100 1200

Wavelength (nm)Figure 13.19: (a) Two possible structures and (b) the typical spectral utilisation (EQE·AM1.5) of a triple-junction thin-film silicon
cell.

Copyright Delft University of Technology, 2014

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188(b) shows the spectral utilisation3 of the three junctions
and the total cell of the record device by United Solar.
In difference to the EQE of the multijunction III-V cell(Fig. 13.10), here the individual cells have various overlaps instead of being block functions as for the III-V
device. Here, light with wavelengths below 450 nm is
utilised by the top cell only, around 550 nm light is utilised by the top cell and middle cell, wavelengths at 650
nm are utilised by all three junctions, and wavelengths
above 900 nm are utilised by the bottom cell only. Consequently, optimising thin film silicon multijunction
solar cells is a complex interplay between the various
thicknesses and light trapping concepts used in thesolar cell. The initial efficiency of the current record triple
junction cell by Uni Solar Ovonic is 16.3%.
The term initial refers to a big issue for amorphousthin-film silicon alloys: they suffer from light induced
degradation which leads to a reduction of the efficiency.
For example, for the initial 16.3% record cell the efficiency drops below the 13.4% of the current stabilised
record cell. This record cell consists of an a-Si:H/ncSi:H/nc-Si:H stack and was produced by the South
Korean LG Corp. [38].
Light induced degradation, which is also referred to as
the Staebler-Wronski effect (SWE), is one of the biggest
challenges for the thin film solar cells. It was alreadydiscovered one year after the first a-Si:H solar cells
3 The spectral utilisation is given as EQE·AM1.5.

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13. Thin-film solar cells

were made in 1977 [65]. Because of the recombinationof light excited charge carriers, meta-stable defects are
created in the absorber layers. The increased defectdensity leads to increased bulk charge carrier recombination, which mainly affects the performance of theamorphous solar cells. After about 1000 hours of illumination, the efficiency of amorphous thin-film solar
cells stabilises at around 85% - 90% of the initial efficiency and stays stable for the rest of its lifetime. If theSWE could be tackled, thin-film silicon devices easily
could achieve stable efficiencies well above 16%.
Just as for III-V cells, current matching is very important for thin-film silicon multijunction cells. First, nanotextured surfaces scatter the incident light in order toenhance the average photon path length and hence toincrease the absorption in the various absorber layers ofthe multijunction cell. Scattering becomes more important for the bottom cell because its nc-Si:H film is the
thickest layer in the device has to absorb most of the
red and infrared, where nc-Si:H is not that absorptive.
Secondly, intermediate reflector layers are used as a tool
to redistribute the light between the junctions aboveand below them. More specifically, the top and bottom
junctions are separated with a low reflective index material, such as nc-SiOx :H. Because of the refractive index
difference of the top absorber and the nc-SiOx :H, more
light is reflected back into the top cell. Thus, the a-Si:H
top cell can be made thinner, which makes it less sensitive to light-induced degradation.

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13.3. Thin-film silicon technology

13.3.3Making thin-film silicon solar cells

Let us now discuss how lab-scale thin-film silicon solar
cells are made in the DIMES Laboratory in Delft, theNetherlands. Before the deposition can start, the sample
has to be cleaned in a ultrasonic cleaning bath. During
cleaning, dirt and dust particles are removed, which
could lead to a shunt between the front and back contact in the final solar cell. Now, the TCO layer can bedeposited. In Delft, we can deposit ZnO:Al or ITO
with sputtering processes. After sputtering, ZnO:Al canbe etched with acids in order to achieve a crater-likestructure for light scattering. Alternatively, the sample
can already be covered with a TCO layer, for example
SnO2: F from the Japanese Asahi company, which alreadyhas a pyramid-like structure for light scattering because of its deposition process. Also on SnO2: F a 5-10
nm thick ZnO:Al layer is deposited that protects the
SnO2: F from being reduced by hydrogen-rich plasmapresent during the deposition of the thin-film siliconlayers. During sputtering, the zinc oxide target is bombarded using an ionised noble gas like Argon. The generated aluminium zinc oxide species are sputtered into
the chamber and deposited on to the substrate.
Before the thin-film silicon layers are deposited, a thin
Al strip is deposited on the side of the sample, that will
act as electric front contact when the cells are measured.
Processing of the different thin-film silicon layers often

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189

happens in multi-chamber setups that allow to processeach layer in a separate chamber and therefore can prevent cross-contamination, e.g. from p and n dopants,
which would reduce the quality of the layers. Afterthe sample is mounted on a suitable substrate holder, it
enters the setup via a load lock, in which the substrate isbrought under low pressure before it’s moved into theprocessing chambers. This avoids the processing chambers to be contaminated with various unwelcome atoms
and molecules present in ambient air. Then the sample
is brought into a central chamber, from where all process chambers can be accessed.
A typical precursor gas for depositing thin-film Si layers is silane (SiH4 ), which often is diluted with hydrogen (H2 ). The precursor gasses are brought into theprocess chamber at low pressure. A radio-frequency(RF) or very-high-frequency (VHF) bias voltage between
two electrodes is used to generate a plasma. This
plasma leads to dissociation of the SiH4 atoms into radicals such as SiH3 , SiH2 or SiH. This radicals react withthe substrate, where a layer is deposited. With increasing the H2 content, the material becomes more nanocrystalline, as sketched in Fig. 13.14.
As dopants, mainly diborane (B2 H6 ) is used for p layers and phosphine (PH3 ) is used for the n layers. For
silicon carbide layers, additionally to the silane, methane (CH4 ) is used. Silicon oxide layers can be made bycombining silane with carbon dioxide (CO2 ).

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190

13. Thin-film solar cells

After the p-i-n junction is deposited, the sample iscovered with a mask, which defines the areas on thatthe metallic back contacts are deposited. In Delft, we
deposit silver with evaporation. Little pieces of silver are
put in a boat that is heated by a very high current thatis flowing through it. The silver melts and evaporated
silver particles can move freely through a vacuum until
they hit the sample, where a silver layer is deposited.
Now the solar cells are ready, every metallic square
defines a little solar cell. When they are measured, thesmall metal strip is used as front contact and the backof the metallic square is the electric back contact.
Of course, such a configuration is not suitable for largescale thin-film PV modules. How such a module ismade, is discussed in Section 17.1 during our general
discussion on PV modules.

13.3.4

Crystalline silicon thin-film solarcells

At the end of our discussion on thin-film silicon technology we briefly discuss crystalline silicon thin-film solar
cells, where the advantages of crystalline silicon technology and thin-film silicon technology should be combined [66]. This means, that high-quality crystalline
films of only several tenths of micrometers that are positioned on a substrate or a superstrate are used as ab-

Copyright Delft University of Technology, 2014

sorbers.
Different approaches are investigated for reaching thisgoal: Kerf-less wafering techniques are investigated
where thin wafers can be prepared without any kerf
loss. The wafers then are transferred on a glass substrate. With this technique, efficiencies of 10.0% wereobtained with a 50 µm thick absorber [66].
Alternatively, films of large-grain nanocrystalline siliconor amorphous silicon can be deposited. The amorphousfilms can be crystallised after deposition, for example ina thermal annealing step. Because of the low electricalquality of the defect-rich silicon films, the maximallyachieved efficiency is 10.5% and was achieved by the
Chinese company CSG Solar [66].
The highest efficiencies are reached when the thin crystalline layers are prepared in an epitaxy process, just as
for the high-performance III-V solar cells. The epitaxy
films then are transferred on glass. The current record
cell made with this method has an efficiency of 20.1%
on a 43 µm thick substrate and was made by the American company SOLEXEL [38].

13.4

Chalcogenide solar cells

The third class of thin-film solar cells that we discuss
are the large class of chalcogenide solar cells, where ourThis copy is provided for free, for personal use only.

13.4. Chalcogenide solar cells

191

focus mainly will be on CIGS and cadmium telluride(CdTe) solar cells. The term chalcogenides refers to all
chemical compounds consisting of at least one chalcogen anion from the group 16 (also known as group VI)
with at least one or more electropositive elements. Fiveelements belong to group 16: oxygen (O), sulphur (S),selenium (Se), tellurium (Te), and the radioactive polonium (Po). Typically, oxides are not included in discussions of chalcogenides. Because of its radioactivity,
compounds with Po are of only very limited relevancefor semiconductor physics.

13.4.1

Chalcopyrite solar cells

The first group of chalcogenide solar cells that we discuss are chalcopyrite solar cells. The name of this class ofmaterials is based on chalcopyrite (copper iron disulfide,
CuFeS2 ). Like all the chalcopyrites, its forms tetragonal
crystals, as illustrated in Fig. 13.20.
Many chalcopyrites are semiconductors. As they consist of elements from groups I, III, and VI, they are also
called I-III-VI semiconductors or ternary semiconductors. In principle, all these combinations can be used:

Figure 13.20: The crystal structure of copper indium diselenide,a typical chalcopyrite. The colors indicate copper (red), selenium (yellow) and indium (blue). For copper gallium diselenide,
the In atoms are replaced by Ga atoms [67].

  Cu
AlS Ag  Ga Se
Au
In
Te 2

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192

13. Thin-film solar cells
3.0

Bandgap (eV)2.5

y is a number in between 0 and 1.
CuAlSe2
CuGaS22.0
CuGaSe2
1.5

CuInS2AgInSe2
CuInTe2

1.0

CuInSe2
5.4

5.6

5.8

6.0

6.2

Lattice Constant (Å)
Figure 13.21: The bandgap vs. the lattice constant for several

chalcopyrite materials. Data taken from [42].

Figure 13.21 shows the bandgap vs. the lattice structure
for several chalcopyrite materials.
The most common chalcopyrite used for solar cells is a
mixture copper indium diselenide (CuInSe2 , CIS) and copper gallium diselenide (CuGaSe2 , CGS). This mixture iscalled copper indium gallium diselenide [Cu(Inx Ga1-x )Se2 ,CIGS], where the x can vary between 0 and 1. Several
research groups and companies use a compound that
also contains sulfur; it is called copper indium gallium
diselenide/disulfide [Cu(Inx Ga1-x )(Sey S1-y )2 , CIGSS], where

Copyright Delft University of Technology, 2014

The physical properties of CIGS(S) are rather complex
and many different views exist on these properties
among scientists. CuInSe2 has a band gap of 1.0 eV, thebandgap of CuInS2 is 1.5 eV and CuGaSe2 has a band
gap of 1.7 eV. By tuning the In:Ga ratio x and the Se:S
ratio y, the band gap of CIGS can be tuned from 1.0
eV to 1.7 eV. As CIGS(S) is a a direct band gap semiconductor material, it has a large absorption coefficient,hence an absorber thickness of 1-2 µm is sufficient for
absorbing a large fraction of the fraction of the lightwith energies above the band gap energy. Also thetypical electron diffusion length is the order of a few
micrometers. CIGS(S) is a p-type semiconductor, thep-type character resulting from intrinsic defects in thematerial that amongst others are related to Cu deficiencies. The many different types of defects in CIGS(S) andtheir properties are a topic of ongoing research.
Figure 13.22 (a) illustrates a typical CIGS solar cellstructure deposited on glass, which acts as a substrate.
On top of the glass a molybdenum layer (Mo) of typically 500 nm thick is deposited, which acts as the electric back contact. Then, the p-type CIGS absorber layer
is deposited with a thickness up to 2 µm. Onto the pCIGS layer, a thin n-CIGS layer is deposited, for example an indium/gallium rich Cu(Inx Ga1-x )3 Se5 alloy.
The pn-junction is formed by stacking a thin cadmium
sulfide (CdS) buffer layer of around 50 nm thickness onto

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13.4. Chalcogenide solar cells

n-ZnO:Al

i-ZnO

i-ZnO
p-CdS buffer
n-CIGS
p-CIGSn-CdS(b)n-ZnO:Al(a)

1.1 eV
2.5 eVburiedjunction

p-CIGSEC
EF
EV

3.3 eVMo back contact

glass substrate

Figure 13.22: The (a) layer structure and (b) band diagram ofa typical CIGS solar cell.

the CIGS layers. The n-type region is extended with the
TCO layer, that also is of n type: First an intrinsic zincoxide (ZnO) layer deposited followed by a layer of Al
doped ZnO. The Al is used as an n dopant for the ZnO.
Similar to thin film silicon technology, the n-type TCO
acts as the transparent front contact for the solar cell.
Figure 13.22 (b) shows the electronic band diagram ofa CIGS solar cell. The light enters the cell from the left,
via the ZnO. The p-type CIGS absorber layers used inindustrial modules typically has a band gap of 1.1-1.2
eV, which is achieved using Cu(Inx Ga1-x )Se2 with x ≈0.3 [68]. The n-type CdS buffer layer has a band gapof 2.5 eV. The band gaps of the n- and p-type materials
are different, which means that such CIGS cells can be

Copyright Delft University of Technology, 2014

can be considered as heterojunctions. The band gap of
ZnO is very large with values of 3.2 eV or even higher,
which minimises the parasitic absorption losses in thisdevice [69].
The defect density at the surface is higher than in thebulk, which could be a loss mechanism for the minority charge carriers. This recombination can be reducedby placing an n-type CIGS layer between the p-type
CIGS and the n-type CdS layers, as already mentioned
above. The pn-junction within the CIGS is called a buried junction. There the electron-hole pairs are separated.
In p-CIGS, which is Cu-deficient, the dominant recombination mechanism is Shockley-Read-Hall recombination in the bulk. In contrast, in Cu rich CIGS films the
SRH recombination at the CIGS/CdS interface becomesdominant.
A very important issue in the development of CIGS
solar cells is the role of sodium (Na). A low contamination with sodium appears to reduce the recombinationin p-type CIGS materials because of better recombination of the grain boundaries. The reduced recombination rate results in a higher band gap utilisation andthus a higher open circuit voltages. Typically, the optimal concentration of sodium in the CIGS layers isabout 0.1%. Often, as a Na source soda-lime glass isused, that is also the substrate for the solar cell. If no
soda-lime glass is used, the Na has to be intentionallyadded during the deposition process. The scientificThis copy is provided for free, for personal use only.

194

question why Na significantly improves several properties of the CIGS films is still not completely solved.
Currently, also the influence of potassium (K) is heavily
investigated.
Fabrication of CIGS solar cells

CIGS films can be deposited with various different
technologies. Because many of these activities are developed within companies, not much detailed information is available on many of these processing techniques. One example of a processing technique is coevaporation under vacuum conditions. Using various
crucibles of copper, gallium, selenium and indium, theprecursors in are co-evaporated onto a substrate invarious steps. Two approaches can be used: First, coevaporation on a heated substrate. During the process
an additional selenium source is used, such that a CIGS
film can be formed. The second approach is sputtering onto a substrate at room temperature. After that,the substrate is thermally annealed under presence ofselenium vapour, such that the CIGS structure can beformed. Alternatively, a selenium rich layer can be deposited on top of the initially deposited alloy, followedby an annealing step. Because of the variety and complexity of the reactions taking place during the ‘selenisation’ process, the properties of CIGS are difficult to control. Companies that use or have used co-evaporation
process are Würth Solar, Global Solar and Ascent Solar

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13. Thin-film solar cells

Technologies. Among CIGS companies using sputter
approaches are Solar Frontier, Avancis, MiaSolé and
Honda Soltec.
An alternative approach to produce CIGS layers isbased on a wafer bonding technique. Two different
films are deposited onto a substrate and a superstrate,
repspectively. Then, the films are pressed together onhigh pressure. During annealing, the film is released
from the superstrate and a CIGS film remains on a substrate.
Non-vacuum techniques can be based on depositing
nanoparticles of the precursor materials on a substrate
after which the film is sintered. During the sintering process films are made out of powder. For achieving this,the powder is heated up to a temperature below themelting point. At such high temperatures, atoms in theparticles can diffuse across the particle boundaries. The
particles thus fuse together forming one solid. This process has to be followed by a selenisation step.
An important advantage of the CIGS PV technology
is that on it has achieved the highest conversion efficiencies among the different thin-film techologies except the III-V technology. The current record for labscale CIGS solar cells processed on glass is 20.8% andwas achieved by the German research institute ZSW[38]. This record cell has a Voc of 0.757 V, a Jsc of 34.77
mA/cm2 , and a FF of 79.2%. The world record on flexible substrates has been obtained at the Swiss Fed-

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13.4. Chalcogenide solar cells

eral Laboratories for Materials Science and Technology(EMPA). On flexible polymer foil they achieved a conversion efficiency of 20.4% [70].
For making CIGS modules, the interconnection is done
similar as for other thin-film technologies and is discussed in detail in Section 17.1. As generally true forthe different PV technologies, the record efficiencies ofmodules are significantly lower than that of lab-scale
cells. The record efficiencies of 1 m2 modules are in theorder of 13%, whereas the aperture-area efficiencies are
just above 14% as confirmed by NREL. The German
manufacturer Manz AG has presented an 15.9% aperture area efficiency and a total area efficiency of 14.6%.
The Japanese company Solar Frontier claims a 17.8%
aperture area efficiency on a small module of 900 cm2
size.
Despite the very high conversion efficiencies, the CIGS
technology faces several challenges. As CIGS is a deposited in a complex deposition processes, it is challenging to perform large area deposition with a high production yield. The production yield is the percentage of
all the modules coming out of the production line that
fulfil the product specifications.
Kesterites

Figure 13.1 shows the abundance in the Earth’s crustfor several elements. Is we can see, indium is a very

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195

rare element. As we have seen above, In is a crucial
element of CIGS solar cells. Because of its scarcity, Inmight be the limiting step in the upscaling of the CIGSPV technology to future Terawatt scales. In addition,
the current thin-film display industry depends on In as
well, as ITO is integrated in many display screens.

As a consequence, other chalcogenic semiconductors
are investigated that do not contain rare elements. A
interesting class of materials are the kesterites which
are quarternary or pentary semiconductors consistingof four or five elements, respectively. While the namegiving mineral kesterite [Cu2 (ZnFe)SnS4 ], where zincand iron atoms can substitute each other, is not used asa semiconductor, kesterite without iron (Cu2 ZnSnS4 ) isused. It also is known as copper zinc tin sulfide (CZTS)
and is a I2 -II-IV-VI4 semiconductor. Other kesterites
are for example copper zinc tin selenide (Cu2 ZnSnSe4 ,CZTSe), or ones using a mixture of sulphur and selenium, Cu2 ZnSn(SSe)4 (CZTSS).

In difference to CIGS, CZTS is based on non-toxic andabundantly available elements. The current record efficiency is 12%. It is achieved with an CZTSS solar cellson lab-scale by IBM [38].

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196

13.4.2

13. Thin-film solar cells

Cadmium telluride solar cells

In this section we will discuss the cadmium telluride(CdTe) technology, which currently is the thin-film technology with the lowest demonstrated cost per W p . We
start with discussing the physical properties of CdTe,
which is a II-VI semiconductor because it consists of theII valence electron element cadmium (Cd) and the VI
valence electron element tellurium (Te). Like the III-V
semiconductors discussed in Section 13.2, CdTe forms azincblende lattice structure where every Cd atom is bonded to four Te atoms and vice versa.
The band gap of CdTe is 1.44 eV, a value which is close
to the optimal band gap for single junction solar cell.
CdTe is a direct band gap material, consequently only
a few micrometres of CdTe are required to absorb all
the photons with an energy higher than the band gap
energy. If the light-excited charge carriers should be
efficiently collected at the contacts, their diffusion has to
be in the order of the thickness.
CdTe can be n-doped by replacing the II-valence electron atom Cd with a III-valence electron atom like aluminium, gallium or indium. n-doping can be achieved
as well by replacing a VI-valence tellurium atom witha VII-valence electron element like fluorine, chlorine,
bromine and iodine atoms. The III- and VII-valence
atoms act as shallow donors. Also a tellurium vacancy
acts like a donor.

Copyright Delft University of Technology, 2014

p-doping of CdTe can be achieved by replacing Cd witha I-valence electron atom like copper, silver or gold. It
can be achieved as well by replacing Te atoms with Vvalence electron elements such as nitrogen, phosphorus
or arsenicum. These elements act as shallow acceptors.
But also a Cd vacancy acts as an acceptor. In solar cells,
p-doped CdTe is used. However, it is difficult to obtain
CdTe with a high doping level.
Figure 13.23 (a) shows the structure of a typical CdTe
solar cell. First, transparent front contact is deposited
onto the glass superstrate. This can be tin oxide or cadmium stannate, which is a Cd-Sn-oxide alloy. On top ofthat the n-layer is deposited, which is a cadmium sulfide layer, similar to the n-uffer layer in CIGS solar cells(Section 13.4.1). Then, the p-type CdTe absorber layer
is deposited with a typical thickness of a few micrometres. Making a good back contact on CdTe is rather
challenging because the material properties of CdTe do
not allow a large choice of acceptable metals. Heavily
doping the contact area with a semiconductor material
improves the contact qualities, however, achieving highdoping levels in CdTe is problematic. Copper containing contacts have been used as back contacts, however,on the long term they may face instability problems
due to diffusion of Cu through the CdTe layer up to the
CdS buffer layer. Nowadays, stable antimony telluride
layers in combination with molybdenum are used.
The band diagram of a CdTe solar cell is shown on Fig.

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197

13.4. Chalcogenide solar cells

13.23 (b). The p-type semiconductor CdTe has a band
gap of 1.44 eV whereas the n-type CdS has a band gapof 2.4 eV. Consequently, the junction is a heterojunction,
similar to what we have seen for CIGS PV devices. The
light excited minority electrons in the p-layer are separated at the heterojunction and collected at the TCO
based front contact. The holes are collected at the backcontact.

(a)front glass
n-doped SnO2
n-CdS buffer
p-CdTeManufacturing CdTe solar cells

metal back contact

(b)
ITO

SnO2 CdS

CdTe

TeSb2Te3 NiVEFEC

2.4 eV
3.7 eV
3.6 eV

0.33 eV

1.45 eV

ΔEV = 0.95 ± 0.1 eV

ΔEV= 0.6 ±0.1 eV

ΔEV = 1.2 ± 0.2 eVEV

Figure 13.23: The (a) layer structure and (b) band diagram ofa typical CIGS solar cell. Band diagram data taken from [71].

Copyright Delft University of Technology, 2014

Usually, the CdS/CdTe layers are processed using
the closed space sublimation method. In this method, thesource and the substrate are placed at a short distancefrom each other, ranging from a few millimetres up
to several cm, under vacuum conditions.The source
can either consist of granulates or powders of CdTe.
Both, the source and the substrate are heated, where thesource is kept at a higher temperature than the substrate. This temperature difference induces a driven
force on the precursors, which are deposited on thesubstrate. Such, the bulk p-type CdTe is formed. In theprocess chamber, an inert carrier gas like argon or nitrogen can be used.
The US company First Solar Inc. is the leading company in the CdTe technology [72]. Like the German
company Antec Solar, they use the closed space sublimation method. Another company producing CdTe

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198

solar cells is the German Calyxo. First Solar is by far
the largest CdTe manufacturer. In 2008, First Solar had
an annual production rate of 500 MW. In 2006 and 2007
it already was one of the biggest solar module manufacturers in the world.
The record conversion efficiency of lab-scale solar CdTe
solar cells is 19.6% [38] and was obtained by GE Global
Research in 2013. The open circuit voltage of this cellis 857.3 mV, the short circuit current density is 28.59
mA/cm2 , and the FF is 80.0%. The current record module based on CdTe technology has a conversion efficiency of 16.1% at an area of 7200 cm2 .
The current cost of modules from First Solar product isin the order of 0.68 to 0.70 USD/W p and is expected todrop to 0.40 USD/W p , hence keeping the cost per W p
lower than that of modules based on crystalline siliconwafers.
An important aspect that needs to be addressed is
the toxicity of cadmium. However, insoluble Cd compounds like CdTe and CdS are much less toxic than theelementary Cd. Nonetheless, it is very important to prevent cadmium entering into the ecosystem. It is an important question whether, CdTe modules could becomea major source of Cd pollution. AS OF»», First Solar
has an installed production capacity of 2 GW/year.
With this capacity, about 2% of the total industrial Cd
consumption would be taken by First Solar. Nevertheless, recycling schemes have been set up for installed

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13. Thin-film solar cells

CdTe solar modules. For example, First Solar has a recycling scheme in which a deposit of 5 dollar cents per
W p is included to cover the cost for the recycling at theend of the module lifetime.
Maybe an even bigger challenge for the CdTe technology is the supply with tellurium. Let us take anotherlook at Fig. 13.1 that shows the abundance of the various elements in the Earth’s crust. Tellurium is one of
the rarest stable solid elements in the Earth’s crust with
an abundance of about 1 µg/kg, which is comparable
to that of platinum [73]. Because of its scarcity, the supply with Te might be the limiting step to upscale the
CdTe PV technology to future Terawatt scales. On theother hand, Tellurium as source material has only had
a few users so far. Thus, for Te no dedicated mining
has been explored so far. In addition, new supplies oftellurium-rich ores were found in Xinju, China. Therefore at this moment it remains unclear to which extent
the CdTe PV technology might be limited by the supply
with tellurium.

13.5

Organic Photovoltaics

So far we discussed inorganic thin-film semiconductormaterials such as III-V semiconductors, amorphous andnanocrystalline silicon, CIGS and CdTe solar cells. In
this section we will take a look at organic photovoltaics.

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13.5. Organic Photovoltaics

The used absorber materials are either conductive organic
polymers or organic molecules that are based on carbon,
which may form a cyclic, a-cyclic, linear or mixed compound structure.
Figure 13.24 shows some example of organic materials
that can be used for PV applications: P3HT, Phtalocyanine, PCBM and Ruthenium Dye N3. All these materials can be considered as large conjugated systems,
which means that carbon atoms in the chain have analternating single or a double bond and every atomin the chain has a p-orbital available. In such conjugated compounds, the p-orbitals are delocalised, whichmeans that they can form one big mixed orbital. Hence,the valence electron of the original p-orbital is sharedover all the orbitals. A classical example would be thebenzene molecule, which is a cyclic conjugated compound. As we can see in Fig. 13.25 (a), This moleculehas 6 carbon atoms and six p-orbitals, which mix andfrom two circular orbitals that contain three electrons
each. These electrons do not belong to one single atom
but to a group of atoms.
In contrast, a methane molecule (CH4 ) is tretrahedrally
coordinated, which means that it has 4 equivalent sp3
hydride bonds with a bond angle of 109.5°. An ethene(C2 H4 ) molecule has three equivalent sp2 hybrid bonds
with a bond angle of 120° plus an electron in a pz orbital. Two neighbouring pz orbitals form a so-called πorbital, as illustrated in 13.25 (b).

Copyright Delft University of Technology, 2014199

In Chapter 6 we have discussed that two individual
sp3 hybrid orbitals can make an anti-bonding and abonding state. The same is valid for the two pz orbitals
forming a molecular π orbital. They can make bonding and anti-bonding π states. Therefore, conjugatedmolecules can have similar properties as semiconductormaterials.
At room temperature, most electrons are in the bonding
state, which is also called the highest occupied molecular orbital (HUMO). The anti-bonding state can be considered as the lowest unoccupied molecular orbital (LUMO).
As the conjugated molecules are getting longer, the
HOMO and LUMO will broaden and act similar tovalence and conduction band in conventional semiconductors. The energy difference between the HOMO andLUMO levels can be considered as the band gap of thepolymer material.
To discuss whether an organic material is p-type or ntype, we first have to introduce the concept of the vacuum level, shown in Fig. 13.26 (a). The vacuum level isdefined as the energy of a free stationary electron thatis outside of any material, or in other words, in vacuo.
This level often is used as the level of alignment for
the energy levels of two different materials. The ionisation energy is the energy required to excite an electronfrom the valence band or HOMO to the vacuum state.
The electron affinity is the energy obtained by moving
an electron form the vacuum just outside the semicon-

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200

13. Thin-film solar cells

PCBM-C61

Rhutenium dye N3

COOH

O
O

S

P3HT

COO-

N
C

N

N

Phtalocyanine

Ru

S

C

N

N
N

COO-

COOH
Figure 13.24: Some examples of organic molecules used for organic photovoltaics.

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201

13.5. Organic Photovoltaics
(a)
H

H

H

H

H
H
Benzene6 pz orbitals

(b)

H

H

H

H

H

H
H

H

H
H

H

H
H

H
Benzenedelocalised
π orbital

Ethenedelocalised
π orbital

Figure 13.25: The chemical structure and π orbitals of (a) benzene and (b) ethene.

ductor or conjugated polymer to the bottom of the conduction band or LUMO. When a material has a low
ionisation potential, it can release an electron out of
the material relatively easy, i.e. it can act as an electron
donor. On the other hand, when a material has a highelectron affinity, it can easily accept an additional electron in the LUMO or conduction band, it thus acts as
an electron acceptor.
As we have discussed earlier, in inorganic semiconductors an electron can be excited from the valence bandto the conduction band leaving a hole in the valenceband. Such an electron-hole pair is only weakly boundand both entities are easily separated and can diffuse
away from each other. In organic materials this is not
the case. Absorption of a photon with sufficient en-

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ergy results in the creation of an exciton, illustrated inFig. 13.26 (b) ¶. An exciton is an excited electron-hole
pair that is still in bound state because of the mutual
Coulomb forces between the particles. Such excitons
can diffuse through the material, but they have a low
life time in organic materials, recombining back to theground state within a few nanoseconds. Hence, the diffusion length of such excitons is in the order of only 10
nm.
If an electron donor and an electron acceptor material
are brought together, an interface is formed between
those two. The HOMO and LUMO of both polymers
can be aligned considering their energy levels with reference to the vacuum level, as illustrated in Fig. 13.26(b). At the interface we see a difference in the HOMO

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202

13. Thin-film solar cells
(a)

(a)

vacuum level
electron affinity

ionisation energy
LUMO

(b)
diffusion
lengthElectrode 1 (e.g. ITO)Electrode 1 (e.g. ITO)

electron donor

LUMO
electron acceptor

HOMO
HOMO
electron donor

electron acceptor

Electrode 2 (Al, Mg, Ca)bulkheterojunction Electrode 2 (Al, Mg, Ca)

Figure 13.27: Illustrating (a) the layer structure of organic(b)vacuum level
LUMO




LUMO




HOMO

HOMO

excitonelectron donor

electron acceptor

Figure 13.26: Illustrating (a) the energy levels in organic PV

materials and (b) the separation of electrons and holes in an
exciton at the acceptor-donor interface..

Copyright Delft University of Technology, 2014

solar cells and (b) an organic solar cell with a bulk heterojunction.

and LUMO levels. Because of this difference, an electrostatic force exists between the two materials. If thematerials are chosen such that the difference is large
enough, these local electric fields can break up the exciton ·. The electron then can be injected into the electron acceptor and a hole remains in the electron donor
material ¸.
Figure 13.27 (a) shows the structure of a typical organic
solar cell. Here, we consider a solar cell consisting ofboth organic acceptor-type and donor-type materials.
Similar as for semiconductor materials, a heterojunction
based on two different materials or conjugated compounds can be constructed. As mentioned before, theThis copy is provided for free, for personal use only.

203

13.6. Hybrid organic-inorganic solar cells

typical diffusion length in the organic materials is only
about 10 nm. Hence, the thickness of the solar cell in
principle is strongly limited by the diffusion length,
while it has to be at least 100 nm to absorb a sufficient fraction of the light. Therefore, organic solar cells
are based on bulk heterojunction photovoltaic devices
that is illustrated in Fig. 13.27 (b), where the electrondonor and the electron-acceptor materials are mixed
together and form a blend. In this way, typical lengthscales in the order of the exciton diffusion lengths canbe achieved. Hence, a large fraction of the excitons
can reach an interface, where they are separated into
an electron and a hole. The electrons move though theacceptor material to the electrode and the holes movethrough the donor material to be collected at the otherelectrode. Usually, the holes are usually collected at a
TCO electrode, for example indium tin oxide (ITO). The
electrons are collected at a metal back electrode.
The record organic solar cells are based on double junctions. Heliatek achieved a 12.0% solar cell efficiency onlab-scale, with unknown stability.
An advantage of organic solar cells is that they have
low production costs. With chemical engineering theband gap can be tuned. Organic solar cell can be integrated into flexible substrates. Important disadvantages
are a low efficiency, low stability and low strength compared to inorganic PV cells.
Although organic solar cells themselves can be cheaply,

Copyright Delft University of Technology, 2014

the need of highly expensive encapsulation materials
to protec the organic materials from humidity, moistureand air limits the industrial application. To my knowledge, no company is producing organic PV modules at
the moment. Konarka Technologies has been active inthe past. Their small PV modules had efficiencies in therange of 3 up to 5% and worked for a couple of years
only.

13.6

Hybrid organic-inorganic
solar cells

In this section we discuss two hybrid concepts, where
the junction consists of both inorganic and organic compounds. These concepts are dye-sensitised solar cells andperovskite solar cells.

13.6.1

Dye-sensitised solar cells

An alternative solar cell concept based on organic materials is the dye-sensitised solar cell (DSSC), which isa photo-electrochemical system. It contains titanium dioxide (TiO2 ) nanoparticles, organic dye particles, an electrolyte and platinum contact. Figure 13.28 (a) shows anillustration of a DSSC. The dye sensitiser, which acts
like an electron donor, is the photo-active component

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204

13. Thin-film solar cells(a)Transparent Electrode
Electrolyte

TiO2
l3

3l

Light
Counter Electrode(b)

2Dye + + 3I- → 2Dye + I 3LUMO

Conduction band

I-

S*
-

I3
Valence band

TiO 2

HOMO

S

Dye +

ElectrolytePt

I 3 - + 2e - → 3I -Figure 13.28: (a) A sketch of a dye sensitised solar cell and (b)the relevant energy levels in its operation.

Copyright Delft University of Technology, 2014of the solar cell. The second major component are the
TiO2 nanoparticles that act as the electron–acceptor.
Similar to organic bulk heterojunction cells, the dye ismixed with TiO2 .
As a photoactive dye sensitiser, ruthenium polypyridine
is used. The operation is illustrated in 13.28 (b). If aphoton is absorbed by the ruthenium polypyridine, itcan excite an electron from its ground state, the S state,
to an excited state, referred to as S*. In this case the S
can be considered as a HOMO and the S* state can beconsider as LUMO. The S* energy level is above theenergy level of the conduction band of the TiO2 . As aresult, the light excited electrons are injected into theTiO2 nanoparticles, while the dye-sensitiser molecule remains positively charged. The electrons move throughthe TiO2 to the TCO based back contact in a diffusion
based transport mechanism. Via the electric circuit, theelectrons move to the front contact. The front contact iselectrically connected to the dye via an electrolyte. Electrolytes are solutions or compounds that contain ionised
entities that can conduct electricity. The typical electrolyte used for DSSC contains iodine. The positively
charged oxidised dye molecule is neutralised by a negatively charge iodide. Three negatively charged iodides,
neutralise two dye molecules and create one negatively
charged tri-iodide. This negatively charged tri-iodide
moves to the counter electrode where it is reduced using two electrons into three negatively charged iodine.
To facilitate the chemical reactions, these photoelectro-

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13.6. Hybrid organic-inorganic solar cells

205

chemical cells require a platinum back contact.
The performance of a DSSC depends on the HOMO
and LUMO level of the dye material, the Fermi level ofTiO2 nanoparticles and the redox potential of the iodide
and triodide reactions. The current record efficiency ofdye-sensitised PV devices on lab-scale is 11.9% and wasachieved by Sharp [38].
The major advantage of DSSC are the low production
costs. A disadvantage is the stability of the electrolyte
under various weather conditions: At low temperatures
the electrolyte can freeze, which stops the device fromgenerating power and might even result in physicaldamage. High temperatures result in thermal expansionof the electrolyte, which make encapsulating modules
more complicated. Another challenge is the high cost of
the platinum electrodes, hence replacing the platinum
with cheaper materials is a topic of ongoing research.
Secondly, more stable and resistive electrolyte materials must be developed. Thirdly, research is done on
improved dyes that enhance the spectral and band gap
utilisation of the solar cells.
Yet, no dye-sensitised PV products are available commercially.

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206

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13. Thin-film solar cells

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14

Third Generation Concepts
The term third generation photovoltaics refers to all novel
approaches that aim to overcome the Shockley-Queisser(SQ) single bandgap limit, preferably at a low cost. The
SQ limit that we discussed extensively in Chapter 10 isa thermodynamic approach to estimate the maximum
efficiency of a single junction solar cell in dependenceof the bandgap of its semiconductor material. For its
derivation we assumed that the AM1.5 spectrum is incident on a solar cell. Further, we do not allow the solar
cell to increase in temperature but we force it to keep
the ambient temperature of 300 K. This means that allenergy absorbed by the solar cell can escape the solar
cell by either the generated current density or by radiative recombination of charge carriers. Under theseassumptions the efficiency limit is around 33% in theband gap range from 1.0 eV up to 1.8 eV.
Now we are going to look at some very fundamental
limitations of classical single-junction solar cells.
First, in single-junction solar cells only on band gap material is used. Hence, a large fraction of the energy ofthe most energetic photons is lost as heat as illustrated
in Fig. 10.7.
Secondly, most solar cells concepts are based on an incident irradiance level of 1 sun. However, higher irradiance means more current generation and also higher
voltage levels, resulting in a higher overall efficiency.
Thirdly, every photon only excites one electron in the
conduction band creating only one electron-hole pair.
207

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208

14. Third Generation Concepts

The energy of highly energetic photons could be utilised better if they would create more than one excitedelectron in the conduction band.

cells when looking at the III-V technology (Section 13.2)
and thin-film silicon technology (Section 13.3). For completeness, we here will summarise this technology.

Fourthly, the photons with energies below the bandgap
are not used. Hence, they do not result in charge carrier
excitation, as illustrated in Fig. 10.7.

In multi-junction cells, several cell materials with different bandgaps are combined in order to maximise theamount of the sun light that can be converted into electricity, as we already illustrated in Fig. 10.9. To realise
this, two or more cells are stacked onto each other. The
top cell has the highest bandgap, in order to absorb andconvert the short wavelength (blue) light. Light withwavelengths longer than the bandgap-wavelength can
traverse the top cell and be absorbed in the cells below
with lower bandgaps. The bottom cell has the lowest
bandgap to absorb the long wavelength (red and nearinfrared) light. In order to optimise the performance ofmulti-junction solar cells with two electrical terminals,
matching the currents of all the subcells (current matching) is crucial. Multi-junction cells with more terminals do not have this restriction, but their production ismore complicated.

If these fundamental limitations could be solved, PVconcepts with conversion efficiencies exceeding the
Shockley-Queisser limit could be developed. In thischapter, we will discuss several third generation concepts: Multi-junction solar cells, concentrator photovoltaics, spectral up and down conversion, multi-exciton
generation, intermediate band-gap solar cells and hot
carrier solar cells. Note, that besides multi-junction
and the concentrator approach, none of these concepts
have resulted in high efficiency solar cells or even been
demonstrated yet. These other concepts are still in fundamental research phase and it is not clear whether
they will ever become a large scale PV technology [74].

14.1

Multi-junction solar cells

The first limitation discussed in the list above can be
attacked by using multi-junction solar cells. Dependingon the author, these solar cells are seen as part of thesecond or the third generation, and indeed we alreadybriefly discussed them in Chapter 13 on thin-film solar

Copyright Delft University of Technology, 2014

In thin-film silicon tandem cells, an a-Si:H top cell isstacked onto a nc-Si:H bottom cell. In order to achieve
current matching, the top cell is much thinner than thebottom cell. The cell can be further optimised by using
an intermediate reflector between the top and the bottom cell in order to reflect the blue light back into the
top cell while letting the red light pass to the bottom
cell. The reported record efficiency of a-Si:H/nc-Si:H

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209

14.2. Spectral conversion

tandem cells is 12.3% and for a a-Si:H/nc-Si:H/nc-Si:H
it is 13.4% [38].
Multi-junction cells containing III-V semiconductors areat present the most efficient solar cells. The current
world record efficiency is 44.4% for a triple-junction
GaInP/GaAs/GaInNAs cell that is used in a concentrated PV-system [38]. Because of the high concentration of 302 the overall efficiency increases and hence
we also tackle the second limitation mentioned in thetable above. As a result the SQ limit can be exceededby more than 10%.

C

Spectral conversion

Single-junction solar cells have the limitation that every
photon can only generate one collected electron at a
maximum. In theory, this limitation can be tackled byspectral conversion. The main idea of spectral conversion
to use an additional layer consisting of a “magical” material such that the incident spectrum can be altered.
Materials that are investigated for spectral conversion
are organic dyes, quantum dots, lanthanide ions, andtransition metal ion systems [75].

Copyright Delft University of Technology, 2014

e-

V

h+

vs.
V
upconversion
Figure 14.1: Illustrating the principle of spectral up-

conversion.

14.2.1

14.2

CSpectral upconversion

In spectral upconversion, two or more low-energy
photons excite electron-hole pairs in several steps, as
illustrated in Fig. 14.1. If these electron-hole pairs recombine radiatively, they emit a photon with a higher
energy. As a result, two or more low-energy photons
are converted in one high-energy photon, which can beabsorbed in the PV-active material.
The up-converting layers should be placed at theback of the solar cell as low-energy photons can passthrough the solar cell absorber to the up-converting
layer and can be converted there to high energy
photons that are absorbed by the solar cell in a second
pass [75]. In this way, also up-converting layers withThis copy is provided for free, for personal use only.

210

C

14. Third Generation Concepts

e-

C

-

e

e

V

h+

h+

-

vs.
V

h+
downconversion

Figure 14.2: Illustrating the principle of spectral down-

conversion.

low efficiencies become interesting. Due to parasitic absorption it is not advised to put these layers at the frontof the solar cell.

14.2.2

Spectral downconversion

The idea of spectral down-conversion is to split one highenergy photon in multiple lower energetic photons [76],as illustrated in Fig 14.2. A high-energy photon is absorbed at the front of the solar cell and converted into
at least two photons with lower energies. If the energyof the initial photon is Eph > 2Eg and the energy the
resulting two photons is still larger than that of theband gap of the absorber material, both photons canbe absorbed and used for exciting charge carriers. As aresult, a high energetic photon, for example in the blue,

Copyright Delft University of Technology, 2014

can result in two excited electrons in the blue part. In
other words, the maximum theoretical EQE of 100% at
the wavelength of the blue photon can be increased to200%. If the photon had sufficient energy to be split
into three photons with an energy higher than the bandgap, a theoretical EQE of 300% could be obtained. In
difference to upconversion, a down-converting layer
has to be at the front of the solar cell, as highly energetic photons are always absorbed in the absorber layer.
Hence, parasitic absorption might be a problem in thistechnology.
One possibility that is investigated for realising spectral down-conversion is to use so-called quantum dots
(QDs). These are small spherical nanoparticles made ofsemiconductor materials with typical diameters of a few
nanometers, as illustrated in Fig. 14.3 (a). These semiconductor particles still behave like a semiconductormaterial; however, due to so-called quantum confinement
the band gap of the semiconductor quantum dots canbe larger than that of the same semiconductor in a bulk
configuration. The band gap of the QDs can be tuned
by varying their size. The smaller the particles, the larger the band gap. This enables interesting opportunities
for bandgap engineering, such as a multijunction solar
cells based on junctions with different QDs of different
sizes in every junction.
If QDs should be used for down-conversion, an ensemble of nanoparticles is embedded in a host mater-This copy is provided for free, for personal use only.

211

14.2. Spectral conversion

(a)

CB

CB

1Pe
1Se

Eg
1Sh
1Ph

VB

VB

6 nm

2 nm

(b)
Si NC

Si NC

CB

Process:

2
VB

1

CB

1

3

3
3

2

VB
3

Figure 14.3: Illustrating (a) the effect of the quantum dot(QD) size on the bandgap and (b) spectral down-conversion
with QDs.

Copyright Delft University of Technology, 2014

ial, where the particles are in very close proximity ofeach other. Figure 14.3 (b) shows the electronic band
gap diagram of two nanoparticles. Now, a high-energy
photon is absorbed by one QD and hence one electron
is excited into the conduction band of the particle. In
difference to a bulk semiconductor, the excess energyof the photon is not necessarily lost as heat, but it canbe transferred as a quantised energy package to a neighbouring quantum dot. Here a second electron is excited
into conduction band of the second quantum dot. As aresult, two electron-hole pairs have been created out of
one photon. If non-radiative recombination mechanisms
like Auger recombination and SRH recombination are
sufficiently suppressed, the electron-hole pairs in bothquantum dots can radiatively recombine such that eachof the two QDs emits one reddish photon. In summary,
one incident bluish photon is converted in to two reddish photons, which can be absorbed by a PV material.
Figure 14.4 shows some experimental results by
Jurbergs et al. on down conversion based on siliconquantum dots in a narrow spectral range [77]. The horizontal axis represents the photon emission wavelength.
At around 790 nm a down-conversion efficiency of 60%
achieved. The EQE exceeds 100% in the blue region
from 3.1 up to 3.4 eV. The major challenge is have QD
layers with a spectral response exceeding 100% at lower
photon energies, because the solar spectrum contains
far more photons in that spectral range.

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212

14. Third Generation Concepts
Si NC

Si NC

CB

Process:
CB

2

2
VB

1

VB

1

Figure 14.5: Illustrating multi exciton generation withquantum dots.

In practice, only small enhancements in efficiency due
to up/down-converters have been reported [74].

14.3

Figure 14.4: Down conversion at Si quantum dots [77].

Multi-exciton generation

Another approach to enhance the charge carrier excitation by a single energetic photon is called multiple exciton generation (MEG). Here, more than one electronhole pairs is generated from high energy photons. In
difference to spectral down-conversion here all thefundamental energy conversion steps occur in the PVactive layers.
Like down conversion, in principle MEG can be realisedwith quantum dots, as illustrated in Fig. 14.5. Again, in

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213

E x te rna l Q ua ntum E ffic ie nc y ( % )14.4. Intermediate band solar cells

120
110
100

(a)

Bandgap:
0.73 eV
0.72 eV
0.71 eV
AR coated

̶

Au

̶

p

IB materialn̶

PbSe QDs
ZnO

90

ITO

80

Glass

eV

70
60
2.6

(b)

+
2.8

3.0

3.2

Photon Energy (eV)

3.4

+

3.6

Figure 14.6: Multi-exciton generation using PbSe quantum

dots with EQEs exceeding 100% [78].

one particle an electron is excited into the conduction
band and the excess energy is transferred to a neighbouring QD, where a second electron is excited in to
conduction band of the second quantum dot. However,here the charge carriers in the two electron-hole pairs
are separated before they can recombine such that one
incident photons results in more then one generated
electrons [79]. Just as for spectral down-conversion, also
here quantum efficiencies exceeding 100% are theoretically possible when one incident photon creates statistically more than charge carriers. Figure 14.6 shows that
EQEs exceeding 100% can be achieved as demonstrated
by Semonin et al. [78]. Here, the absorber layer consists
PbSe quantum dots.

Copyright Delft University of Technology, 2014+

Figure 14.7: Band diagram (a) with the intermediate band de-picted and (b) with the quasi-Fermi levels exceeding the energyof the low-energy photons.

14.4

Intermediate band solar cells

The concept of intermediate band solar cells (IB) tries totackle the problem photons with energies below thebandgap cannot be utilised for current generation. As
shown in Fig. 14.7 (a), in intermediate band cells energy
levels are created artificially in the bandgap of the absorber material [80]. As in conventional single-junction
solar cells, photons with a sufficient energy can excite
an electron from the valence band in to the conduction
band. However, in difference to conventional semiconductors photons with energies below the bandgap can
excite an electron from the valence band in to the inter-

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214

14. Third Generation Concepts

mediate band. A second low-energy photon is required
to excite the electron from the intermediate band into
the conduction band. As illustrated in 14.7 (a), therefore
two photons with small energies can result in quasiFermi level splitting larger than the energy of each ofthese photons.
Various studies have been performed on how intermediate band cells can be realised, as for example summarised in Reference [80]. In such solar cells, a layer
with the intermediate states is placed in between pand n-layers. For example quantum dots can be used
to realise the intermediate states. Further, various bulk
materials are studied for realising the intermediate
states. One major problem of experimental IB cells isthat the voltages are lower than the voltages of reference cells without IB structures. Yet, the efficiency loss
due to the lower voltage cannot be overcome by subbandgap current density originating from the IB transitions.

Selective energy
contacts
V > EgEgelectroncontact

hot electron and holeenergy distributions

eV

hole
contact

Figure 14.8: Illustrating the working principle of a hot carrier

14.5

Hot carrier solar cells

solar cell.

The idea of hot carrier solar cells is to reduce the energy
losses due to relaxation and hence thermalisation. As
illustrated in Fig. 14.8, this should be achieved by collect electron-hole pairs of high energy photons just after
light excitation before they have a chance to relax backCopyright Delft University of Technology, 2014

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14.5. Hot carrier solar cells

215

to the edges of the electronic bands. In the figure, thepopulation of the charge carrier levels reflects the situation just after the excitation by the absorption of aphoton. This distribution is not in thermal equilibrium
as many electrons are excited into position further up in
the conduction band and the holes are excited down tolower levels in the valence band. These charge carriers
are called hot electrons and holes [22, 81].
It takes only a few picoseconds (10−12 s) for the hot
charge carriers to relax back to the edges of the electronic bands. The idea of hot carrier cells it to collectthe charge carriers as long as they are still hot. Hence,an energy larger than the band gap energy can be utilised per excited charge carrier and the average bandgap
utilisation would exceed the bandgap.
The fundamental challenge is to collect the hot carriers before they relax back to the edges of the electronic
bands. Such a concept would require selective contacts,
which only select electrons above a particular energy
level in the conduction band and holes below a certain
energy level in the valence band, respectively. At the
moment the main challenge is to increase the lifetimeof the hot charge carriers, such that they have time tomove from the absorber material to the selective contacts.

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216

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14. Third Generation Concepts

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Part IV

PV Systems

217
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15

Introduction to PV systems
15.1

Introduction

15.2

Types of PV systems

PV systems can be very simple, consisting of just a PV
module and load, as in the direct powering of a water pump motor, which only needs to operate when thesun shines. However, when for example a whole houseshould be powered, the system must be operational
day and night. It also may have to feed both AC andDC loads, have reserve power and may even include aback-up generator. Depending on the system configuration, we can distinguish three main types of PV systems: stand-alone, grid-connected, and hybrid. The basic PV system principles and elements remain the same.
Systems are adapted to meet particular requirements byvarying the type and quantity of the basic elements. A

After we have discussed the fundamental scientific theories required for solar cells in Part II and have taken alook at modern PV technology in Part III, we now will
use the gained knowledge to discuss complete PV systems. A PV system contains many different components
besides the PV modules. For successfully planning a
PV system it is crucial to understand the function of thedifferent components and to know their major specifications. Further, it is important to know the effect on thelocation of the (expected) performance of a PV system.
219
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220

15. Introduction to PV systems

(a)

(b)

PV Modules
PV Modules
Charge controller

=
~
DC

DC
AC

Inverter

Loads

Water reservoir
Water pump

Batteries

Figure 15.1: Schematic representation of (a) a simple DC PV system to power a water pump with no energy storage and (b) acomplex PV system including batteries, power conditioners, and both DC and AC loads.

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221

15.2. Types of PV systems

modular system design allows easy expansion, whenpower demands change.

15.2.1

Stand-alone systems

Stand-alone systems rely on solar power only. Thesesystems can consist of the PV modules and a load onlyor they can include batteries for energy storage. Whenusing batteries charge regulators are included, whichswitch off the PV modules when batteries are fully
charged, and may switch off the load to prevent thebatteries from being discharged below a certain limit.
The batteries must have enough capacity to store theenergy produced during the day to be used at nightand during periods of poor weather. Figure 15.1 shows
schematically examples of stand-alone systems; (a) asimple DC PV system without a battery and (b) a large
PV system with both DC and AC loads.

PV Modules=~

Inverter
Distribution board

AC
Loads

15.2.2

Grid-connected systems

Grid-connected PV systems have become increasingly
popular for building integrated applications. As illustrated in Fig. 15.2, they are connected to the grid via
inverters, which convert the DC power into AC electricity. In small systems as they are installed in residential homes, the inverter is connected to the distribution

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Electricity grid
Figure 15.2: Schematic representation of a grid-connected PV

system.

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222

15. Introduction to PV systems

board, from where the PV-generated power is transferred into the electricity grid or to AC appliances in
the house. These systems do not require batteries, since
they are connected to the grid, which acts as a buffer
into that an oversupply of PV electricity is transported
while the grid also supplies the house with electricity intimes of insufficient PV power generation.
Large PV fields act as power stations from that all thegenerated PV electricity is directly transported to theelectricity grid. They can reach peak powers of up to
several hundreds of MW p . Figure 15.3 shows a 25.7
MW p system installed in Germany.

15.2.3Hybrid systems

Hybrid systems consist of combination of PV modules
and a complementary method of electricity generationsuch as a diesel, gas or wind generator. A schematic ofan hybrid system shown in Fig. 15.4. In order to optimise the different methods of electricity generation, hybrid systems typically require more sophisticated controls than stand-alone or grid-connected PV systems.
For example, in the case of an PV/diesel system, thediesel engine must be started when the battery reaches
a given discharge level and stopped again when battery
reaches an adequate state of charge. The back-up generator can be used to recharge batteries only or to supply
the load as well.

Copyright Delft University of Technology, 2014

Figure 15.3: The 25.7 MW p Lauingen Energy Park in Bav-

arian Swabia, Germany [82].

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223

15.3. Components of a PV system

15.3

PV Modules
Charge controller=~

DC
Back-upgenerator

InverterAC

Batteries
Figure 15.4: Schematic representation of a hybrid PV systemthat has a diesel generator as alternative electricity source..

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Components of a PV system

As we have seen earlier in this book, a solar cell canconvert the energy contained in the solar radiation into
electrical energy. Due to the limited size of the solar
cell it only delivers a limited amount of power under
fixed current-voltage conditions that are not practicalfor most applications. In order to use solar electricity
for practical devices, which require a particular voltage
and/or current for their operation, a number of solar
cells have to be connected together to form a solar panel,also called a PV module. For large-scale generation ofsolar electricity solar panels are connected together intoa solar array.
Although, the solar panels are the heart of a PV system, many other components are required for a working
system, that we already discussed very briefly above.
Together, these components are called the Balance of System (BOS). Which components are required depends onwhether the system is connected to the electricity gridor whether it is designed as a stand-alone system. The
most important components belonging to the BOS are:
• A mounting structure is used to fix the modules andto direct them towards the sun.
• Energy storage is a vital part of stand-alone systems because it assures that the system can deliver electricity during the night and in periods ofThis copy is provided for free, for personal use only.

224

15. Introduction to PV systems

bad wheather. Usually, batteries are used as energystorage units.
• DC-DC converters are used to convert the module
output, which will have a variable voltage depending on the time of the day and the weather conditions, to a fixed voltage output that e. g. can beused to charge a battery or that is used as input for
an inverter in a grid-connected system.
• Inverters or DC-AC converters are used in gridconnected systems to convert the DC electricity
originating from the PV modules into AC electricity that can be fed into the electricity grid.
• Cables are used to connect the different components of the PV system with each other and to theelectrical load. It is important to choose cables ofsufficient thickness in order to minimise resistive
losses.
Even though not a part of the PV system itself, the electric load, i.e. all the electric appliances that are connected to it have to be taken into account during the planning phase. Further, it has to be considered whether theloads are AC or DC loads.

AC
Chargecontroller

PV Modules

Battery

Back-upgenerator

AC load=~DC

Inverter

DC loadFigure 15.5: A schematic of the different components of a PV

system.

The different compontents of a PV system are schematically presented in Fig. 15.5 and will be discussed indetail in Chapter 17.

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16

Location issues
In Chapter 5 we discussed solar radiation on earth andintroduced the AM1.5 spectrum, normalised to a total
irradiance of 1000 W/m2 . This spectrum is used toevaluate the performance of solar cells and modules
in laboratories and industry. The AM1.5 spectrum represents the solar irradiance if the centre of the solar discis at an angle of 48.2° off the zenith (or 41.2° above thehorizon).

mosphere and how this influences the direct and diffuse spectrum. We also will discuss the influence of themounting angle and position of a PV module on theirradiance at the module.

16.1

The position of the sun

For planning a PV system it is crucial to know the position of the sun in the sky at the location of the solar
system at a given time. In this section we explain how
this position can be calculated.

Of course, the sun not always is at this position, but
the position is dependent on the time of the day andthe year and of course it is dependent on the location
on earth. In this section we will discuss how to calculate the position of the sun at every location on earth
at an arbitrary time and date. Furthermore we will
discuss scattering of sunlight when it traverses the at-Since celestial objects like the sun, the moon and thestars are very far away from the earth it is convenient
to describe their motion projected on a sphere with ar225

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226

16. Location issues

Zenith
Ce
les
tia
ere
ph
ls

Merid

Altitude a

E
90°

S
180°

ian

Objectζ
Observer

Azimuth A

υ
N
0°

ξ

on
Horiz

W
270°

Figure 16.1: Illustrating the definition of the altitude a and theazimuth A in the horizontal coordinate system. Note that Northis at the bottom of the figure.

bitrary radius and concentric to the earth. This sphereis called the celestial sphere. The position of every celestial object thus can be parameterised by two angles.
For photovoltaic applications it is most convenient to
use the horizontal coordinate system, where the horizonof the observer constitutes the fundamental plane. In thiscoordinate system, the position of the sun is expressedby two angles that are illustrated in Fig. 16.1: The altitude a that is the angular elevation of the centre of thesolar disc above the horizontal plane. Its angular range
is a ∈ [−90◦ , 90◦ ], where negative angles correspondto the object being below the horizon and thus not visible. The azimuth A that is the angle between the line ofsight projected on the horizontal plane and due North.
It is usually counted eastward, such that A = 0◦ , 90◦ ,180◦ , 270◦ correspond to due North, East, South andWest, respectively. Its angular range is A ∈ [0◦ , 360◦ ]. Ina different convention also used by the PV community,
due South corresponds to 0◦ and is counted westward,
the angles then are in between −180◦ and 180◦ . Figure
16.1 also shows the meridian, which is great circle on thecelestial sphere passing through the celestial North andSouth poles as well as the zenith.
Instead of using the spherical coordinates a and A, wealso could use Cartesian coordinates, that we here call
ξ (xi), υ (upsilon) and ζ (zeta) and that are also depicted in Fig. 16.1. The principal direction is parallel to ξ.
The Cartesian coordinates are connected to the spherical

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227

16.1. The position of the suncoordinates via(16.1)

North
ecliptic
pole

Note that ξ 2 + υ2 + ζ 2 = 1 for all points on the celestial
sphere.

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Axial
tilt
ε

stial equator
Cele

ec
21 D

rc

a

EA

23 Septic c

Ecliptic

irc l e

un
21 J

RTH

21 Mar

ox

e

uin
eq
l
a

rn
Ve

er

Earth orbits the sun in an elliptic orbit at an average
distance of about 150 million kilometres. Due to the elliptic orbit the speed of Earth is not constant. This isbecause of Kepler’s second law that states that “A line
joining a planet and the Sun sweeps out equal areas
during equal intervals of time.” On the celestial spherethe sun seems to move on a circular path with one revolution per year. This path is called the ecliptic andillustrated in Fig. 16.2. For describing the apparent
movement of the sun on the celestial sphere is is convenient to use coordinates in that the ecliptic lies in thefundamental plane. These coordinates are called the ecliptic coordinates. As principal direction, the position of
the sun at the spring (vernal) equinox (thus around 21
March) is used, which is indicated by the sign of Aries,
à. As obvious from Fig. 16.2, à lies both in the ecliptic
plane as well as in the equatorial plane. The ecliptic coordinate system is sketched in Fig. 16.3 (a). The two angular coordinates are called the ecliptic longitude λ andthe ecliptic latitude β. Note, that in this coordinate system the rotation of the earth around its axis is not taken
into account.

Northcelestial
pole

ph

  

ξ
cos a cos Aυ =  cos a sin A  .
ζ
sin a

Ce

ti
les

al

s

Figure 16.2: Illustrating the ecliptic, i.e. the apparent move-

ment of the sun around earth. Further, the celestial equator andthe direction of the vernal equinox are indicated. The sizes of Sun
and Earth are not in scale.

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228

16. Location issues

In ecliptic coordinates approximate position the position of the sun can be expressed easily. The approximation presented here has an accuracy of about 1 arcminute within two centuries of 2000 and is publishedby the the Astronomical Applications Departement of the
U.S. Naval Observatory.
To express the position of the sun we first have to express the time D elapsed since Greenwich noon, Terrestrial Time, on 1 January 2000, in days. For astronomic purposes it may be convenient to relate D to theJulian date JD via
D = JD − 2451545.0.

(16.2)The Julian Date1 that is defined as the number of days
since 1 January 4713 BC in a proleptic2 Julian calendar
or since 24 November 4717 BC in a proleptic Gregoriancalendar.
Now, the mean longitude of the sun corrected to the aberration of the light is given by
q = 280.459◦ + 0.98564736◦ D(16.3)

Because of the elliptic orbit of earth and hence a varying speed throughout the year, we have to correct withthe so-called mean anomaly of the Sun,g = 357.529◦ + 0.98560028◦ D.
1 Calculating the Julian Date is implemented in MatLab.

(16.4)

2 Proleptic means that a calendar is applied to dates before its introduction.

Copyright Delft University of Technology, 2014

It may be convenient to normalise q and g to the range[0◦ , 360◦ ) by adding or subtracting multiples of 360◦ .
Now the ecliptic longitude of the sun is given by
λS = q + 1.915◦ sin g + 0.020◦ sin 2g.

(16.5)The ecliptic latitude can be approximated by
β S = 0.

(16.6)For estimating the radiation it might also be convenient
to approximate the distance of the Sun from the Earth.
In astronomical units (AU) this is given by
R = 1.00014 − 0.01671 cos g − 0.00014 cos 2g.

(16.7)As stated above, for PV applications it is convenient touse horizontal coordinates. We therefore have to transform from ecliptic coordinates into the horizontal coordinates. This is done via three rotations that are to be
performed after each other consecutively:First, we have to transform from ecliptic coordinates
into equatorial coordinates. As illustrated in Fig. 16.2,
the fundamental plane of these coordinates is tilted tothe ecliptic with an angle e,e = 23.429◦ − 0.00000036◦ D(16.8)The principal direction is again given by the vernal
equinox à. In Fig. 16.3 (b) the equatorial coordinate

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229

16.1. The position of the sun

(a)

(b)

e
er
ph

Object

Ce
les
tia
l

S
Centre of
the Eartheerphtion δ
clina
De

ptic
Ecliude β
it
alt

Eclip
tic

celestial pole
North

Object

Ce
les
tia
l

S

cliptic pole
North eEcliplongitic
tude
λ



Cel

estia
l equator

Centre of
the Earth

Righascentsion
α

Figure 16.3: Illustrating (a) the ecliptic coordinate system and (b) the equatorial coordinate system.

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230

16. Location issues

system is sketched. The two coordinates are called theright ascension α and the declination δ. The transformation from ecliptic coordinates to equatorial coordinates
is a rotation by the angle e about the vernal equinox asrotational axis. Mathematically this is expressed by

 


cos δ cos α
100
cos β cos λ cos δ sin α  = 0 cos e − sin e   cos β sin λ  .
sin δ
0 sin ecos e
sin β
(16.9)(16.10)



EARTH

N

θL

h = θ L − α,

GREEN W

M

G

where θ L is the local mean sidereal time, i.e. the angle
between the vernal equinox and the meridian. All theseangles are illustrated in Fig. 16.4. A sidereal day isthe duration between two passes of the vernal equinox through the meridian and it is slightly shorter
than a solar day. We can understand this by realisingthat the earth has to rotate by 360◦ and approximately
360◦ /365.25 between two passes through the meridian.
The duration of mean sidereal day is approximately 23h, 56 m and 4 s.

Sunour angle hlocal hL MSTSecondly, we have to take the rotation of the eartharound its axis into account. We do this by using theso-called hour angle h instead of the right ascension α.
Those two angles are connected to each other via

VernalequinoxST

IC H

OBSER VERl o n git u d e

λ0

r i g ht a s c e n si o nαFigure 16.4: Illustrating the right ascension α, the local hourangle h, the Greenwich Mean Sidereal Time GMST and the local
mean sidereal time θ L .

For calculating θ L we first have to determine the Greenwich Mean Sidereal Time (GMST), which is (approxim-

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231

16.1. The position of the sun

ately) given by
GMST =18.697374558 h+ 24.06570982441908 h · D(16.11)

2+ 0.000026 h · T ,where D is as defined above and T is the number ofcenturies past since Greenwich noon, Terrestrial Time,on 1 January 2000,
D.
(16.12)36525
For many applications, the quadratic term may be omitted. GMST is given in hours and has to be normalised
to the range [0 h, 24 h). We then can obtain the local
mean sidereal time in degrees withT=

15◦
+ λ0 ,
hour
where λ0 is the longitude of the observer.
θ L = GMST

(16.13)We thus have to the following transform about the rotational axis of the earth,

 


cos δ cos h
cos θ Lsin θ L 0
cos δ cos α cos δ sin h  =  sin θ L − cos θ L 0  cos δ sin α  .
sin δ
0
0
1
sin δ
(16.14)Note that this transform is no rotation but a reflection
at an angle of θ L /2. We thus now transformed the principal direction of the coordinate system from vernal
equinox to the local mean sidereal time.

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Thirdly we transform to the horizontal coordinate system by rotating with the latitude angle φ0 of the observer. While the ecliptic and equatorial coordinate systems use the centre of the earth as origin, the horizontal
coordinate system uses the actual position on the surface of Earth as origin. However, because of the distance of celestial objects in general and the Sun in particular being much larger than the radius of the Earth,
we may neglect this translational shift of the origin of
the coordinate systems.
The rotational axis of the third transform is the axis thatis normal to both the principal direction θ L and the rotational axis of the earth (δ = 90◦ ).
 0 ξ
sin φ0 0 − cos φ0
cos δ cos hυ0  =  0  cos δ sin h  . (16.15)
10ζ
cos φ0 0
sin φ0sin δHowever, here the directions ξ 0 and υ0 point to due
South and West, respectively. We thus must apply
ξ 0 → −ξ and υ0 → −υ in order to get the directions
such as they are defined in Fig. 16.1 (ξ and υ pointing
to due North and East, respectively). We do this withthe matrix transform    0
ξ−10 0ξ
 υ  =  0 −1 0  υ 0  .
(16.16)ζ00 1
ζBy combining Eqs. (16.9), (16.14)–(16.16) we directly cantransform from ecliptic to horizontal coordinates,

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232

  
ξ
−1
υ =  0
ζ0

16. Location issues

0 0
sin φ0−1 0  0
0 1
cos φ0010− cos φ0cos θ L  sin θ L
0sin φ00sin θ L− cos θ L
0


0
1
0 0
1
0

0
cos e
sin e



0
cos β cos λ− sin e   cos β sin λ  . (16.17)
cos e
sin β

Please note that matrix multiplications do not commute, i.e. the order in which the rotations are applied must not bealtered. Now, we apply that the ecliptic latitude of the sun β S = 0. By calculating Eq. (16.32) we find
ξ S = cos aS cos AS = − sin φ0 cos θ L cos λS − (sin φ0 sin θ L cos e − cos φ0 sin e) sin λS ,
υS = cos aS sin AS = − sin θ L cos λS + cos θ L cos e sin λS ,
ζ S = sin aS= cos φ0 cos θ L cos λS + (cos φ0 sin θ L cos e + sin φ0 sin e) sin λS ,(16.18a)(16.18b)(16.18c)where we also used the the relationship between Cartesian and spherical horizontal coordinates from Eq. (16.1). Dividing Eq. (16.18b) by Eq. (16.18a) and leaving Eq. (16.18c) unchanged leads to the final expressions for the solar position,
tan AS =− sin θ L cos λS + cos θ L cos e sin λS
υS
=
,
ξS
− sin φ0 cos θ L cos λS − (sin φ0 sin θ L cos e − cos φ0 sin e) sin λS

sin aS = ζ S = cos φ0 cos θ L cos λS + (cos φ0 sin θ L cos e + sin φ0 sin e) sin λS .

(16.19a)(16.19b)

AS and aS now can be derived by applying inverse trigonometric functions. While arcsin uniquely delivers an altitude in between −90◦ and 90◦ , applying arctan leads to ambiguities. For deriving an azimuth in between 0◦ and360◦ , we have to look in which quadrant is lying. Therefore we use ξ and υ from Eqs. (16.18a) and (16.18b), respectively. We findξ > 0 ∧ υ > 0 ⇒ AS = arctan f (. . .),ξ<0⇒ AS = arctan f (. . .) + 180◦ ,ξ > 0 ∧ υ < 0 ⇒ AS = arctan f (. . .) + 360◦ .

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233

16.1. The position of the sunf (. . .) denotes the function at the right hand side of Eq. (16.19a). Note that an altitude aS < 0◦ corresponds to thesun being below the horizon. This means that the Sun is not visible and no solar energy can be harvested.
The approximations presented on the previous pages are accurate within arcminutes for 200 centuries of 2000. Several years ago, NREL presented a much more complicated model, the so-called Solar Position Algorithm (ASP), withuncertainties of only ±0.0003◦ in the period from 2000 BC to 6000 AD [83].
ExampleAs an example we will calculate the position of the Sun in Delft on 14 April 2014 at 11:00 local time.
For determining the solar position we need next to date and time (in UTC) the latitude and longitude. Since the time zone in Delft on 14
April is the CEST, the Central European Summer Time, the time difference with UTC is +2 hours, such that 11:00 CEST corresponds to9:00 UTC. According to Google Maps, the latitude and longitude of the Markt in the centre of Delft are given by
φ0 = 52.01◦ N = +52.01◦ ,λ0 = 4.36◦ E = + 4.36◦ .
For the calculation we first have to express date and time as the time elapsed since 1 January 2000 noon UTC.
9
D = 4 · 366 + 10 · 365 + 2 · 31 + 28 + 13 − 0.5 + 24
= 5216.875.

Now we can calculate the mean longitude q and the mean anomaly g of the sun according to Eqs. (16.3) and (16.4),
q = 280.459◦ + 0.98564736◦ D = 22.4580712◦ ,g = 357.529◦ + 0.98560028◦ D = 99.28246073◦ ,where the values were normalised to [0◦ , 360◦ ). From Eq. (16.5) we thus obtain for the latitude of the Sun in ecliptic coordinates
λS = q + 1.915◦ sin g + 0.020◦ sin 2g = 24.34162696◦ .

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234

16. Location issues

For the axial tilt e of the Earth we obtain from Eq. (16.8)
e = 23.429◦ − 0.00000036◦ D = 23.42712193◦ .
The Greewich Mean Sidereal Time (GMST) is given by Eq. (16.11),
GMST = 18.697374558 h + 24.06570982441908 h · D + 0.000026 h · T 2 = 22.49731535 h,
where we used T = D/36525 and normalised to [0 h, 24 h). We then find for the local mean sidereal time θ L
θ L = GMST

15◦
+ λ0 = 341.8197303◦ ,hour

Now we have all variables required to calculate the solar position. From Eqs. (16.19) and (16.20) we thus findtan AS =− sin θ L cos λS + cos θ L cos e sin λS= −1.318180633.
− sin φ0 cos θ L cos λS − (sin φ0 sin θ L cos e − cos φ0 sin e) sin λS

sin aS = cos φ0 cos θ L cos λS + (cos φ0 sin θ L cos e + sin φ0 sin e) sin λS = 0.589415473,
which leads to the solar altitude aS =36.1◦ and the solar azimuth AS =127.2◦ .

Copyright Delft University of Technology, 2014

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235

16.3. The equation of time

16.2

The sun path at diﬀerent locations

In this section we discuss the solar paths throughout
the year at several locations around the earth. Figures
16.5-16.8 shows four examples: Delft, the Netherlands(φ0 = 52.01◦ N), the North Cape, Norway (φ0 = 71.17◦
N), Cali, Colombia (φ0 = 3.42◦ N), and Sydney, Australia (φ0 = 33.86◦ S). Note that all the times are givenin the apparent solar time (AST). While in Delft and on
the North Cape, the Sun at noon always is South of thezenith, in Sydney it is always North. In Cali, close to
the equator, the Sun is either South or North, depending on the time of the year. Since the North Cape isnorth of the arctic circle, the Sun does not set around 21
June. This phenomenon is called the midnight sun. On
the other hand, the sun always stays below the horizon
around 21 December - this is called the polar night.

year, such that it seems to run along the shape of an
Eight. This closed curve is called the analemma.
The difference between the apparent solar time (AST),
i.e. the timescale where the sun really is highest at noon
every day, and mean solar time is described by the socalled equation of time, which is defined as
EoT = AST − MST.

The equation of time is given as the difference between
the mean longitude q, as defined in Eq. (16.3), and theright ascendent αS of the sun in the equatorial coordinate system,
EoT( D ) = [q( D ) − αS ( D )]1 hour.
15 deg

The equation of time

Figure 16.9 shows the position of the sun throughout
the year in Delft at 8:30, 12:00 and 15:30 mean solar
time (MST), i.e. UTC + λ0 , where the longitude λ0 is expressed in hours. We see that the Sun not only changes
from altitude but also from azimuth in the course of a

Copyright Delft University of Technology, 2014(16.22)

The right ascendent is connected to the ecliptic longitude of the sun λS , as given in Eq. (16.5) via
tan αs = cos e tan λS ,16.3

(16.21)(16.23)and e is the axial tilt as given in Eq. (16.8).
The equation of time has two major contributors: theanomaly due to the elliptic orbit of the Earth aroundthe Sun and the Axial tilt of the rotational axis of
the Earth with respect to the ecliptic. Both effects areshown in Fig. 16.10. In this figure, also the total EoT isshown, which is nearly the sum of the two contributors(The maximal deviation is less than a minute).

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236

16. Location issues= 52.01◦ N). The sun path was calculated with theSun path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].
Figure 16.5: The sun path in apparent solar time in Delft, the Netherlands (φ0Copyright Delft University of Technology, 2014

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16.3. The equation of time237

= 71.17◦ N). The sun path was calculated with
the Sun path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].

Figure 16.6: The sun path in apparent solar time on the North Cape, Norway (φ0Copyright Delft University of Technology, 2014

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238

16. Location issues

Figure 16.7: The sun path in apparent solar time in Cali, Colombia (φ0 = 3.42◦ N). The sun path was calculated with the Sun pathchart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].

Copyright Delft University of Technology, 2014

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16.3. The equation of time239

= 33.86◦ S). The sun path was calculated with the Sunpath chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [84].

Figure 16.8: The sun path in apparent solar time in Sydney, Australia (φ0Copyright Delft University of Technology, 2014

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240

12:00
13 Jun60◦

Solar Altitude

50◦

15 Apr 1 Sep

08:30

40◦
30◦

11 Feb

20◦

15:30

E

SE

S

15
10

Equation of Timeeffect of axial tilt

5
0
-5
-10

effect of anomaly-15
01 Feb 01 Apr 01 Jun 01 Aug 01 Oct 01 Dec

3 Nov

Figure 16.10: The effect of the anomaly of the terrestrial orbit

25 Dec10◦0◦Apparent−Mean Solar Time (min)16. Location issues

SWW

and the axial tilt of the Earth rotation axis on the difference
between apparent and mean solar time. The equation of timenearly is the sum of these two effects.

Solar Azimuth

Figure 16.9: The analemma, i.e. the apparent curve of thesun throughout the year when observed at the same meansolar time every day. The analemma is shown for Delft (52◦ N
latitude) at three points in time during the day.

Copyright Delft University of Technology, 2014

We see that the largest negative shift is on 11 February,
where, the apparent noon is about 14 min 12 s prior
to the mean solar noon. The largest positive shift is on
3 November, when the apparent solar noon is about
16 min 25 s past the mean solar noon. These points
are also marked in Fig. 16.9. There, also the zeros of
the EoT are shown, which are on 15 April, 13 June, 1
September, and 25 December.

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241

16.4. Irradiance on a PV module
Zenitheul n Malm

Horizonthen can describe the position of the module by the direction of the module normal in horizontal coordinates( A M , a M ), where the altitude is given by a M = 90◦ − θ.
Let now the sun be at the position ( AS , aS ). Then the
direct irradiance on the module G M is given by theequation

M

ere
ph
ls

θM

Ce
les
tia

od

Altitud
ea

M

r
no

M

od

ule

dir
Gdir
M = Ie cos γ,

N
0°

θM

th A M
Azimu(16.24)

where γ = ^( A M , a M )( AS , aS ) is the angle between
the surface normal and the incident direction of thesunlight. Now we know that the scalar product of twounit vectors is equal to the cosine of the enclosed angle.
Thus, we can write,

Figure 16.11: Illustrating the angles used to describe theorientation of a PV module installed on a horizontal plane.

16.4

Irradiance on a PV modulecos γ = n M · nS .

(16.25)The normal vectors are given by 
cos a M cos A MξM
n M = υ M  =  cos a M sin A M  ,sin a M
ζM
  

ξS
cos a S cos A S
nS = υ S  =  cos a S sin A S  .
ζS
sin a S ,After having discussed how to calculate the positionof the Sun everywhere on the Earth and having looked
at several examples, it now is time to discuss the implications for the irradiance present on solar modules.
For this discussion we assume that the solar module ismounted on a horizontal plane and that it is tilted under an angle θ M , as illustrated in Fig. 16.11. The angle
between the projection of the normal of the module
onto the horizontal plane and due north is A M . We

Copyright Delft University of Technology, 2014(16.26)(16.27)

where we used the relationship between Cartesian andspherical horizontal coordinates given in Eq. (16.1).

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242

16. Location issues

Hence, we find

Modules mounted on a tilted roofcos γ =n M · nS
= cos a M cos A M cos aS cos AS+ cos a M sin A M cos aS sin AS + sin a M sin aS= cos a M cos aS (cos A M cos AS + sin A M sin AS )+ sin a M sin aS= cos a M cos aS cos ( A M − AS ) + sin a M sin aS .
(16.28)Thus we obtain for the irradiance

When a module is mounted on a horizontal plane, itis easy to determine its normal n M . However, when amodule is to be mounted on an arbitrarily tilted roof
things become more complicated. We thus will derive
how to calculate the normal of the module in horizontal
coordinates when the coordinates with respect to theroof are given. In fact, we thus have to transform themodule normal from the roof coordinate system to the horizontal coordinate system.

dir
Gdir
M = Ie [cos a M cos aS cos ( A M − AS ) + sin a M sin aS ]= Iedir [sin θ cos aS cos ( A M − AS ) + cos θ sin aS ] .
(16.29)Note that this equation only holds when the sun isabove the horizon ( aS > 0) and the azimuth of the sun
is within ±90◦ of A M , AS ∈ [ AS − 90◦ , AS + 90◦ ]. Otherwise, Gdir
M = 0.
If the azimuth of the solar position is the same as theazimuth of the module normal A M = AS , Eq. (16.29)
becomesdir
Gdir
M = Ie [cos a M cos aS + sin a M sin aS ]= Iedir cos ( a M − aS ) .

(16.30)

When using the tilt angle θ = 90◦ − a M we finddirGdir
M = Ie sin ( θ + aS ) .

Copyright Delft University of Technology, 2014

As illustrated in Fig. 16.12 (a), the orientation of theroof in horizontal coordinates is characterised by theazimuth A R and the altitude a R of its normal n R . The
module is installed on the roof, and its orientation withrespect to the roof is best described in the roof coordinate system, where the fundamental plane is parallel tothe roof and the principal direction is along the gradient of the roof, as illustrated in Fig. 16.12 (b). In thissystem, the module normal is given by the azimuth φ M
and the altitude is given by δM . The coordinate transform itself is transformed by combining two rotations:First, we rotate with the angle 90◦ − a R around the axis
that is perpendicular to both n R and the gradient direction of the roof. Secondly, we rotate with the angle
A R + 180◦ along the zenith. We thus obtain(16.31)This copy is provided for free, for personal use only.

243

16.4. Irradiance on a PV module



 
 
ξM
cos a M cos A M− cos A Rυ M  =  cos a M sin A M  =  − sin A R
ζM
sin a M0sin A R− cos A R00
sin a R0  01
cos a R

0
1
0



− cos a Rcos δM cos φ M  cos δM sin φ M  .
0
sin a Rsin δM(16.32)The coordinates of the module in the horizontal coordinate system then are given byξ M = cos a M cos A M = − cos A R sin a R cos δM cos φ M + sin A R cos δM sin φ M + cos A R cos a R sin δM ,
υ M = cos a M sin A M = − sin A R sin a R cos δM cos φ M − cos A R cos δM sin φ M + sin A R cos a R sin δM ,
ζ M = sin a M= cos a R cos δM cos φM + sin a R sin δM .

(16.33a)(16.33b)(16.33c)

Dividing Eq. (16.33b) by Eq. (16.33a) and leaving Eq. (16.33c) unchanged leads to the final expressions for the module orientation in horizontal coordinates,− sin A R sin a R cos δM cos φM − cos A R cos δM sin φM + sin A R cos a R sin δM,− cos A R sin a R cos δM cos φM + sin A R cos δM sin φM + cos A R cos a R sin δM
sin a M = cos a R cos δM cos φ M + sin a R sin δM .

tan A M =(16.34a)(16.34b)

Finally, the cosine of the angle between the module orientation and the solar position is given bycos γ = n M · nS =cos aS cos( A R − AS ) (cos a R sin δM − sin a R cos δM cos φ M )+ cos aS sin( A R − AS ) cos δM sin φM + sin aS (cos a R cos δM cos φM + sin a R sin δM ) .

Copyright Delft University of Technology, 2014(16.35)This copy is provided for free, for personal use only.

244

16. Location issues(a)(16.36b)

θR

th A R
Azimu

R

Altitudea

N
0°

Roof normal nR

(b)

od

M

e
ul n Malmr
no(16.37a)−0 − 0 + 0= 0.
− sin a R cos δM + 0 + cos A R sin δM(16.37b)

Copyright Delft University of Technology, 2014

t
ien
d
a
Gr

M

tan A M =

Ro
of

θM

In the second example, the roof is facing southwards
and tilted under an angle θ R . We thus have a R =90◦ − θ R and A R = 180◦. Now the module tilted under an angle θ M with respect to the roof and mounted perpendicular to the gradient of the roof. Hence,δM = 90◦ − θ M and φ M = 180◦ . Using Eqs. (16.34) wefind
sin a M = cos a R cos δM · (−1) + sin a R sin δM .
= cos( a R + eta M ) = sin( a R − θ M ).

Horizon(16.36a)Altitudeδ−0 − 0 + 1 · cos a R sin δM
0 + sin A R cos δM sin ·1 + 0= cos a R tan δM .

tan A M =

Ce
les
tia
ere
ph
ls

sin a M = cos a R cos δM · 0 + sin a R sin δM .
= sin a R sin δM

θR

RoFirst, we look at a roof that faces eastward and has atilt angle θ R . Then, a R = 90◦ − θ R and A R = 90◦ .
On this roof a solar module is installed under a tilting angle θ M with respect to the roof. The modules aremounted parallel to the gradient of the roof. We thus
have δM = 90◦ − θ M and φ M = 270◦ . From Eqs. (16.34)
we thus obtain

Zenithof al n Rrmno

We will try to understand these results by discussing
easy examples:

Roof
M

Roof gradient

od

ule

θM

th φ M
Azimu

Figure 16.12: Illustrating the angles used to describe (a) theorientation of a roof on a horizontal plane and (b) the orientation of a module mounted on a roof.

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245

16.5. Direct and diﬀuse irradiance

16.5

Direct and diﬀuse irradianceAs sunlight traverses the atmosphere, it is partiallyscattered, leading to an attenuation of the direct beamcomponent. On the other hand,the scattered light also
partially will arrive at on the earths surface as diffuse
light. For PV applications it is important to be able toestimate the strength of the direct and diffuse components.
First, we discuss a simple model that allows to estimate
the irradiance on a cloudless sky in dependence of the
air mass and hence the altitude of the sun. As we haveseen in section 5.5, the air mass is defined as
1
1
AM =
=
,
cos θ
sin aS(16.38)where we used that the angle between the sun and thezenith θ is connected to the solar altitude via θ = 90◦ −aS . This equation, however, does not take the curvatureof the earth into account. If the curvature is taken into
account, we find [85]AM( aS ) =1. (16.39)
sin aS + 0.50572(6.07995 + aS )−1.6364

To estimate the direct irradiance at a certain solar altitude aS and altitude of the observer h, we can use thefollowing empirical equation [86]
h
i
0.678
Iedir = Ie0 (1 − ch) · 0.7(AM ) + ch ,(16.40)

Copyright Delft University of Technology, 2014

with the constant c = 0.14. The solar constant is given
as Ie0 = 1361 Wm-2 . In a first approximation, the diffuse
irradiance is about 10% of the direct irradiance. For the
global irradiance we hence obtain [87]globalIe≈ 1.1 · Iedir .

(16.41)

A more accurate model was developed in the framework of the European Solar Radiation Atlas [88]. In thatmodel the direct irradiance for clear sky is given by
Iedir = Ie0 ε exp [−0.8662 TL (AM2) m δR (m)] .

(16.42)I0 is the solar constant that takes a value of 1361 W/m2 .
The factor ε allows to correct for deviations of the sunearth distance from its mean value. aS is the solar altitude angle. TL (AM2) is the Linke turbidity factor withthat the haziness of the atmosphere is taken into account. In this equation its value at an Air Mass 2 isused. m is the relative optical air mass, and finallyδR (m) is the integral Rayleigh optical thickness. The
different components can be evaluated as follows:
The correction factor ε is given byε=Ie ( R)
R2
=
.
0
Ie
AU2

(16.43)The distance between earth and sun as a multiple ofastronomic units (AU) is given in Eq. (16.7). We thus

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246

16. Location issues

obtainε = (1.00014 − 0.01671 cos g − 0.00014 cos 2g)2 , (16.44)

They also present and expression for the diffuse irradiance of the light, which is given bywhich leads to annual variations of about ±3.3%.
The Linke Turbidity factor approximates absorption andscattering in the atmosphere and takes both absorption
by water vapour and scattering by aerosol particles into
account. It is a unit-less number and typically takes values between 2 for very clear skies and 7 for heavily polluted skies.
The relative optical air mass m expresses the ratio of
the optical path length of the solar beam through theatmosphere to the optical path through a standard atmosphere at sea level with the Sun at zenith. In can beapproximated as a function of the solar altitude aS byexp(−z/zh )m( aS ) =.
sin aS + 0.50572( aS + 6.07995)−1.6364(16.45)Here, z is the site elevation and zh is the scale height of
the Rayleigh atmosphere near the Earth surface, given
by 8434.5 m.
Finally, the Rayleigh optical thickness δR (m) is given by1=6.62960 + 1.75130 m − 0.12020 m2
δR ( m )3

4+ 0.00650 m − 0.00013 m .

Copyright Delft University of Technology, 2014

In their paper, Rigollier et al. also take the effect of ofrefraction at very low altitudes into account [88]. This,
however is not relevant for our application.

(16.46)

Iedif = Ie0 εTrd [ TL (AM2)] Fd [ aS , TL (AM2)],(16.47)

where Trd is the diffuse transmission function at zenith,
which is a second-order polynomial of TL (AM2) Trd
has typical values in between 0.05 for very clear skies
and 0.22 for a very turbid atmosphere. Fd is a diffuse
angular function, given as a second-order polynomial ofsin aS . For more details we refer to the paper [88].

16.6

Shadowing

Shadowing had to be kept in mind when planning a PV
system that consists of several rows of PV modules,
which are placed behind each other. In this section wewill determine how far behind the module the shadow
reaches in dependence of the solar position, the module orientation and the length l of the module. Figure
16.13 shows the important notions that we use in thisderivation.
For the determination we look at a module that is tilted
at an angle θ M . Its normal angle has an azimuth A M .

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247

16.6. Shadowing

Zenith
Ce
les
tia

θM

ere
ph
ls

nM
M

P
d
E

P’

nS

h
θM

ℓ
F

a
Altitude S

Altitudea

This module touches the ground at two corner points
that we call E and F. Without loss of generality, we
may assume that E is at the origin of our horizontal
coordinate system, E = (0, 0, 0). Further, P is the corner
point of top the module lying above E. The lengthof the module, i.e. the distance between E and P is l,
EP = l. The shadow of this point on the horizontal
plane, we denote by P0 . Then, the length of the shadow d
is defined as the shortest distance between P0 and theline g, which connects E with F.

uAzim

For the determination of d we first must derive P, andthen P0 . The normal of the module n M is given by
t

N
S 0°
A
h

Azimuth AM 
sin θ M cos A M
cos a M cos A M
n M =  cos a M sin A M  =  sin θ M sin A M  . (16.48)
sin a Mcos θ M

Figure 16.13: Derivation of the length d of a shadow behind a

PV module with ground length l and height h. The module has
the normal n M while the position of the sun is given by the direction nS . The length of the shadow d is given as the distance
between the line connecting the lower module corners E and F
and the projection of the upper corner P on the ground ( P0 ).

The direction vector r of the line g connecting E with F
is then given as
sin A Mr = − cos A M  .
0(16.49)Then, we can calculate the direction vector h of the line

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248

16. Location issues

We find the shadow P0 of the point P as the intersectionof the line s with the horizontal plane, z = 0,

that connects E with P with the vector producth = −n M × r sin θ M cos A M
sin A M= −  sin θ M sin A M  × − cos A M 
cos θ M0
− cos θ M cos A M=  − cos θ M sin A M  ,sin θ Ml sin θ M + t sin aS = 0.
(16.50)sin θ M.
sin aS(16.55)The coordinates of P0 = ( Px0 , Py0 , 0) then are given as

For calculating the position of P0 we define the line s,
which goes through P and points towards the sun,

Px0 = −l (cos θ M cos A M + sin θ M cot aS cos AS ) , (16.56a)
Py0 = −l (cos θ M sin A M + sin θ M cot aS sin AS ) . (16.56b)
As stated already earlier, the length of the shadow d isgiven as the shortest distance between P0 and the line gconnecting E and F. Let g0 be the line through P0 that isperpendicular to g, g ⊥ g0 . Since E = (0, 0, 0) we findfor g and g0
g(u) =
0(16.52)

When the position of the sun is described by its altitudeaS and azimuth AS , we find− cos θ M cos A M
cos aS cos AS
s(t) = l  − cos θ M sin A M  + t  cos aS sin AS 
sin θ Msin aS ,(16.53)Copyright Delft University of Technology, 2014

Hence,
t = −lwhere the − sign is because of the fact that the horizontal coordinate system is left-hand. Since h has
length 1, i.e. it is a unit vector, we easily can derive theposition of the point P with
− cos θ M cos A M(16.51)
P = E + l · h = l  − cos θ M sin A M  .
sin θ Ms(t) = P + t · nS .

(16.54)0

u · r,(16.57a)0(16.57b)g (v) = P +v · r ,where the direction vector r0 is given as
cos A Mr0 =  sin A M  .
0(16.58)d is the distance between P0 and the intersection of gThis copy is provided for free, for personal use only.

249

16.6. Shadowing

with g0 . At this intersection we have

operations

g ( u ) = g 0 ( v ),

d = l [cos θ M + sin θ M cot aS cos( A M − AS )] .

sin A M
cos A Mu − cos A M  = P0 + v  sin A M 00(16.62)

As a rule of thumb, d should be at least three times l,d > 3l.

u sin A M = Px0 + v cos A M

Example−u cos A M = Py0 + v sin A M

A PV system should be installed on a flat roof in Naples (Italy).
The area of the roof that can be utilized for installing the PV
system is 10×10 m2 . The roof is oriented such that the sides
are parallel to the East-West and North-South directions,
respectively.

u sin A M cos A M = Px0 cos A M + v cos2 A M−u cos A M sin A M = Py0 sin A M + v sin2 A M
By adding the last two equations we findPx0 cos A M + v cos2 A M + Py0 sin A M + v sin2 A M = 0,
(16.59)and hence

The owner of the roof decides to use Yingli PANDA 60 modules with dimensions of 1650×990×40 mm3 . The modules are
installed facing south with a tilt of 30◦ .

v = − Px0 cos A M − Py0 sin A M .

He wants to install as many modules as possible under
the condition that on the shortest day of the year no mutual
shading must occur for the duration of 6 hours.

(16.60)Using Eqs. (16.56), we derivev =l [(cos θ M cos A M + sin θ M cot aS cos AS ) cos A M+ (cos θ M sin A M + sin θ M cot aS sin AS ) sin A M ]=l (cos θ M cos2 A M + sin θ M cot aS cos AS cos A M+ cos θ M sin2 A M + sin θ M cot aS sin AS sin A M ).
(16.61)
Because the direction vector r0 of the line g0 is a unit
vector, the length of the shadow d is equal to v. Therefore we obtain from Eq. (16.61) with some trigonometric

Copyright Delft University of Technology, 2014

Should the modules be mounted with the long or short side
touching the ground? How many modules can be mounted inthis case?
Answer: The shortest day of course is 21 December. The solar
position on this day at 9:00 and 15:00 is
Time
9:00
15:00

Altitude (◦ )
13.59
13.13

Azimuth (◦ )138.55222.17This copy is provided for free, for personal use only.

250

16. Location issues

Because of the Equation of Time, the sun is not at its highest
point at exactly 12:00 noon. We see, that the solar altitude at
15:00 is just slightly lower than at 9:00. Thus, when using
9:00 for calculating the length of the shadow, the duration
without mutual shading will be slightly shorter than 6 hours.
Thus, we use the position at 15:00 for the calculation.

Thus, we can mount four rows behind each other, because
3d + d0 = 3 · 2430 + 857 = 8147 mm,
which is lower than 10 m. In one row fit 6 modules because
6 · 1650 = 9900 mm. Thus, we can place 24 modules.

The length of the shadow can be calculated with equation(16.62)d = l [cos θ M + sin θ M cot aS cos( A M − AS )] .
We have θ M =30◦ , A M =180◦ , aS =13.13◦ , and AS =222.17◦ .
If the module is mounted to the ground on the short side, wehave l =1650 mm. Hence, we find d=4050 mm. The area
directly beneath the module at the last row is
d0 = l cos θ M = 1429 mm.
Thus, we can mount three rows behind each other, because
2d + d0 = 2 · 4050 + 1429 = 9529 mm,(16.63)

which is less than 10 m. In one row fit 10 modules because 10 ·990 = 9900 mm. Thus, we can place 30 modules.
If the module is mounted to the ground on the long side, wehave l =990 mm. Hence, we find d=2430 mm. The area
directly beneath the module at the last row is
d0 = l cos θ M = 857 mm.

Copyright Delft University of Technology, 2014

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17

Components of PV Systems
17.1

PV modules(c), consists of several PV modules that are electricallyconnected and mounted on a supporting structure. Finally, a PV array consists of several solar panels. An example of such an array is shown in Fig. 17.1 (d). This
array consists of two strings of two solar panels each,where string means that these panels are connected inseries.

In this section we will discuss PV modules (or solar
modules), their fabrication and how to to determine
their performance.
Before we start with the actual treatment of PV modules, we briefly want to introduce different terms. Figure 17.1 (a) shows a crystalline solar cell, which we discussed in Chapter 12. For the moment we will consider
only modules that are made from this type of solar
cells. A PV module, is a larger device in which manysolar cells are connected, as illustrated in Fig. 17.1 (b).
The names PV module and solar module are often usedinterchangeably. A solar panel, as illustrated in Fig. 17.1

17.1.1

Series and parallel connections in
PV modules

If we make a solar module out of an ensemble ofsolar cells, we can connect the solar cells in different
ways: first, we can connect them in a series connec251

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252

17. Components of PV Systems

(a)

(b)

(d)

solar cell

PV module(c)seriesconnection

seriesconnection

string of panels
PV panelstring of panels
PV arrayFigure 17.1: Illustrating (a) a solar cell, (b) a PV module, (c) a solar panel, and (d) a PV array.

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253

17.1. PV modules

1.8 V

(c)

(d)

3I sc

9
8
Current I (A)

(a)

0.6 V

(b)

7
2I sc
6
5
4

parallelseries

I sc

3
2
1

Voc
0

0.6 V

0.3

0.6

3Voc

2Voc
0.9
1.2
Voltage V (V)

1.5

1.8

Figure 17.2: Illustrating (a) a series connection of three solar cells and (b) realisation of such a series connection for cells with aclassical front metal grid. (c) Illustrating a parallel connection of three solar cells. (d) I-V curves of solar cells connected in series andparallel.

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254

tion as shown in Fig. 17.2 (a). In a series connection
the voltages add up. For example, if the open circuit
voltage of one cell is equal to 0.6 V, a string of threecells will deliver an open circuit voltage of 1.8 V. For
solar cells with a classical front metal grid, a series connection can be established by connecting the bus bars at
the front side with the back contact of the neighbouring
cell, as illustrated in Fig. 17.2 (b). For series connected
cells, the current does not add up but is determinedby the photocurrent in each solar cell. Hence, the total
current in a string of solar cells is equal to the current
generated by one single solar cell.
Figure Fig. 17.2 (d) shows the I-V curve of solar cells
connected in series. If we connect two solar cells inseries, the voltages add up while the current stays thesame. The resulting open circuit voltage is two times
that of the single cell. If we connect three solar cells inseries, the open circuit voltage becomes three times aslarge, whereas the current still is that of one single solar
cell.
Secondly, we can connect solar cells in parallel as illustrated in Fig. 17.2 (c), which shows three solar cells
connected in parallel. If cells are connected in parallel, the voltage is the same over all solar cells, while
the currents of the solar cells add up. If we connect e.g.
three cells in parallel, the current becomes three times
as large, while the voltage is the same as for a single
cell, as illustrated in Fig. 17.2 (d).

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17. Components of PV Systems

The reader may have noticed that we used I-V curves,
i.e. the current-voltage characteristics, in the previous
paragraphs. This is different to Parts II and III, where
we used I-V curves instead, i.e. the current density voltage characteristics. The reason for this switch fromJ to I is that on module level, the total current that themodule can generate is of higher interest than the current density. As the area of a module is a constant, theshapes of the I-V and J-V curves of a module are similar.
For a total module, therefore the voltage and current
output can be partially tuned via the arrangements ofthe solar cell connections. Figure 17.3 (a) shows a typical PV module that contains 36 solar cells connected
in series. If a single junction solar cell would have ashort circuit current of 5 A, and an open circuit voltage
of 0.6 V, the total module would have an output of
Voc = 36 · 0.6 V = 21.6 V and Isc = 5 A. However,
if two strings of 18 series-connected cells are connected in parallel, as illustrated in Fig. 17.3 (b), the outputof the module will be Voc = 18 · 0.6 V = 10.8 V and
Isc = 2 × 5 A = 10 A. In general, for the I-V characteristics of a module consisting of m identical cells in series
and n identical cells in parallel the voltage multiplies
by a factor m while the current multiplies by a factor n.
Modern PV modules often contain 60 (10 × 6), 72 (9 × 8)
or 96 (12 × 8) solar cells that are usually all connected
in series in order to minimise resistive losses.

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17.1. PV modules

17.1.2(a)(b)

Figure 17.3: Illustrating a PV module consisting (a) of a stringof 36 solar cells connected in series and (b) of two strings of 18
solar cells each that are connected in parallel.

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PV module parameters

In a nutshell, for a PV module a set of parameters canbe defined, similar than for solar cells. The most common parameters are the open circuit voltage Voc , the shortcircuit current Isc and the module fill factor FF M . On module level, we have to distinguish between the aperturearea efficiency and the module efficiency. The aperture area
is defined as the area of the PV-active parts only. The
total module area is given as the aperture area plus
the dead area consisting of the interconnections andthe edges of the module. Clearly, the aperture area efficiency is larger than the module efficiency.
Determining the the efficiency and the fill factor of a
PV module is less straight-forward than determining
voltage and current. In an ideal world with perfectly
matched solar cells and no losses, one would expectthat the efficiency and fill factor at both the moduleand cell levels to be the same. This is not the case inreal life. As mentioned above. The cells are connectedwith each other using interconnects that induce resistive losses. Further, there might be small mismatches
in the interconnected cells. For example, if m × n cells
are interconnected, the cell with the lowest current in astring of m cells in series determines the module current. Similarly, the string with the lowest voltage in
the n strings that are connected in parallel dictates themodule voltage. The reason for mismatch between individual cells are inhomogeneities that occur during

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17. Components of PV Systems

the production process. Hence, in practice PV module
perform a little less than what one would expect from
ideally matched and interconnected solar cells. This loss
in performance translates to a lower fill factor and efficiency at module level. If the illumination across themodule is not constant or if the module is heats up
non-uniformly, the module performance reduces evenfurther.
Often, differences between cell and module performance are mentioned in datasheets that are provide by
the module manufacturers. For example, the datasheetof a Sanyo HIT-N240SE10 module gives a cell level efficiency of 21.6%, but a module level efficiency of only
19%. Despite all the technological advancements being
made at solar cell level for improving the efficiency, still
a lot must be done at the PV systems level to ensure ahealthy PV yield. For the performance of a PV system,
not only the module performance is important, but also
the yield of the PV system.

17.1.3

Partial shading and bypass diodes

PV modules have so-called bypass diodes integrated. To
understand the reason for using such diodes, we haveto consider modules in real-life conditions, where they
can be partially shaded, as illustrated in Fig. 17.4 (a). The
shade can be from an object nearby, like a tree, a chimney or a neighbouring building. It also can be caused

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by a leaf that has fallen from a tree. Partial shading
can have significant consequences for the output of thesolar module. To understand this, we consider the situation in which one solar cell in the module shaded fora large part shaded. For simplicity, we assume that all
six cells are connected in series. This means that thecurrent generated in the shaded cell is significantly
reduced. In a series connection the current is limitedby the cell that generates the lowest current, this cellthus dictates the maximum current flowing through themodule.
In Fig. 17.4 (b) the theoretical I-V curve of the five unshaded solar cells and the shaded solar cell is shown. If
the cells are connected to a constant load R, the voltage
across the module is dropping due to the lower current
generated. However, since the five unshaded solar cells
are forced to produce high voltages, they act like a reverse bias source on the shaded solar cell. The dashedline in Fig. 17.4 (b) represents the reverse bias loadput on the shaded cell, which is the I-V curve of thefive cells, reflected across the vertical axis equal to 0 V.
Hence, the shaded solar cell does not generate energy,
but starts to dissipate energy and heats up. The temperature can increase to such a critical level, that the encapsulation material cracks, or other materials wear out.
Further, high temperatures generally lead to a decreaseof the PV output as well.
These problems occurring from partial shading can be

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257

17.1. PV modules

(a)

(b)
1
R

0
Current (A)current(c)current-1

Dissipatedenergy-2

J < Jsc-3
R-5

Current
fromshaded6th cell

5Voc

Current
from 5 cells

Jsc
-4

-3

-2

-1
1
0
Voltage (V)

2

3

4

5

Figure 17.4: Illustrating (a) string of six solar cells of which one is partially shaded, which (b) has dramatic effects on the I-V curveof this string. (c) Bypass diodes can solve the problem of partial shading.

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258

17. Components of PV Systems

prevented by including bypass diodes in the module,
as illustrated in 17.4 (c). As discussed in Chapter 8,
a diode blocks the current when it is under negative
voltage, but conducts a current when it is under positive voltage. If no cell is shaded, no current is flowing
through the bypass diodes. However, if one cell is (partially) shaded, the bypass diode starts to pass current
through because of the biasing from the other cells. As
a result current can flow around the shaded cell and themodule can still produce the current equal to that of aunshaded single solar cell.
For cells that are connected in parallel, partial shading is less of a problem, because the currents generated in the others cells do not need to travel throughthe shaded cell. However, a module consisting of 36
cells in parallel have very high currents (above 100 A)
combined with a very low voltage (approx. 0.6 V). This
combination would lead to very high resistive losses in
the cables; further an inverter that has only 0.6 V as input will not be very efficient, as we will see in Section
17.3. Therefore, combining the cells in series and using
bypass diodes is much better an option to do.

17.1.4Fabrication of PV modules

As discussed in the subsection 17.1.5, a PV module
must withstand various influences in order to survivea lifetime of 25 years or even longer. In order to ensure

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front glass
front encapsulant
solar cells
Al frameback
encapsulant
back layers

Al framejunction box
Figure 17.5: The components of a typical c-Si PV module.

a long lifetime, the components of that a PV module isbuilt must be well chosen. Fig. 17.5 shows the typicalcomponents of a usual crystalline silicon PV module.
Of course, the layer stack may consist of different materials dependent on the manufacturer. The major components are [89]:• Soda-lime glass with a thickness of several millimetres, which provides mechanical stability
while being transparent for the incident light. It
is important the glass has a low iron content because iron leads to absorption of light in the glass
which can lead to losses. Further, the glass must
be tempered in order to increase its resistance to impacts.

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259

17.1. PV modules• The solar cells are sandwiched in between two layers of encapsulants. The most common material
is ethylene-vinyl-acetate (EVA), which is a thermoplastic polymer (plastic). This means that it goes
into shape when it is heated but that these changes
are reversible.
• The back layer acts as a barrier against humidity
and other stresses. Depending on the manufacturer, it can be another glass plate or a composite
polymer sheet. A material combination that is often
used is PVF-polyester-PVF, where PVF stands forpolyvinyl fluoride, which is often known by its brandname Tedlar® . PVF has a low permeability for vapours and is very resistive against weathering. A
typical polyester is polyethylene terephthalate (PET)• A frame usually made from aluminium is put
around the whole module in order to enhance themechanical stability.
• A junction box usually is placed at the back of themodule. In it the electrical connections to the solar
cell are connected with the wires that are used toconnect the module to the other components of the
PV system.
One of the most important steps during module production is laminating, which we briefly will explain forthe case that EVA is used as encapsulant [89]. For lamination, the whole stack consisting of front glass, theCopyright Delft University of Technology, 2014

encapsulants, the interconnected solar cells, and theback layer are brought together in a laminator, which
is heated above the melting point of EVA, which isaround 120°C. This process is performed in vacuo inorder to ensure that air, moisture and other gasses are
removed from within the module stack. After someminutes, when the EVA is molten, pressure is applied
and the temperature is increased to about 150°C. Now
the curing process starts, i.e. a curing agent, which ispresent in the EVA layer, starts to cross-link the EVA
chains, which means that transverse bonds between the
EVA molecules are formed. As a result, EVA has elastomeric, rubberlike properties.
The choice of the layers that light traverses before entering the solar cell is also very important from an optical point of view. If this layers have an increasing
refractive index, they act as antireflective coating andthus can enhance the amount of light that is in-coupledin the solar cell and finally absorbed, which increases
the current produced by the solar cell.

17.1.5

Lifetime testing of PV Modules

The typical lifetime of PV systems is about 25 years.
In these as little maintenance as possible should be required on the system components, especially the PV
modules are required to be maintenance free. Furthermore, degradation in the different components of that

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260

the module is made should be little: manufacturers typically guarantee a power between 80% and 90% of theinitial power after 25 years. During the lifetime of 25
years or more, PV modules are exposed to various external stress from various sources [90]:• temperature changes between night and day as well
as between winter and summer,• mechanical stress for example from wind, snow andhail,• stress by agents transported via the atmosphere suchas dust, sand, salty mist and other agents,• moisture originating from rain, dew and frost,• humidity originating from the atmosphere,• irradiance consisting of direct and indirect irradiance from the sun; mainly the highly-energetic UV
radiation is challenging for many materials.
Before PV modules are brought to the market, they areusually tested extensively in order to assure their stability against these various stresses. The required tests areextensively defined in the standards IEC 61215 for modules based on crystalline silicon solar cells and in IEC
61646 for thin-film modules. Since the modules cannot
be tested during a period of 25 years, accelerated stresstesting must be performed. The required tests are [91]:• Thermal cycles for studying whether thermal stress

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17. Components of PV Systems

leads to broken interconnects, broken cells, electrical bond failure, adhesion of the junction box,. . .
• Damp heat testing to see whether the modules suffer from corrosion, delamination, loss of adhesion
and elasticity of the encapsulant, adhesion of thejunction box,. . .
• Humidity freeze testing in order to test delamination,adhesion of the junction box, . . .
• UV testing, because UV light can lead to delamination, loss of adhesion and elasticity of the encapsulant, ground fault due to backsheet degradation.
Mainly, UV light can lead to a discoloration of theencapsulant and back sheet, which means that they
get yellow. This can lead to losses in the amount oflight that reaches the solar cells.
• Static mechanical loads in order to test whether
strong winds or heavy snow loads lead to structural failures, broken glass, broken interconnect ribbons or broken cells.
• Dynamic mechanical load, which can lead to roken
glass, broken interconnect ribbons or broken cells.
• Hot spot testing in order to see whether hot spots
due to shunts in cells or inadequate bypass diodeprotection are present.
• Hail testing to see whether the module can handle
the mechanical stress induced by hail.

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17.1. PV modules• Bypass diode thermal testing to study whether overheating of these diodes causes degradation of theencapsulant, backsheet or the junction box.
• Salt spray testing to see whether salt that is present
in salty mist or that is used in salty water for snow
and ice removal leads to corrosion of PV module
components.
How these tests are to be performed is defined in otherstandards, for example IEC 61345 for UV testing andIV 61701 for salt-mist corrosion testing Usually thesetests are carried out by organisations like TÜV Rheinland. Refining the test requirements and understanding which accelerated tests are required to guarantee alifetime of 25 years and more is subject to ongoing research and development.

17.1.6Thin-film modules

Making thin-film modules is very different from making modules from c-Si solar cells. While for c-Si technology producing solar cells and producing PV modules are two distinct steps, in thin-film technology producing cells and modules cannot be separated fromeach other. To illustrate this we look at a PV module
where the thin-films are deposited in superstrate configuration on glass, as illustrated in Fig. 17.6. For making
such a module, a transparent front contact, a stack of

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(photo)active layers that also contain one or more semiconductor junctions, and a metallic back contact are deposited onto each other. In industrial production, theglass plates on that these layers are deposited can bevery large, with sizes significantly exceeding 1 × 1 m2 .
Such a stack of layers deposited onto a large glass
plate in principle forms one very large solar cell thatwill produce a very high current. Since all the current
would have to be transported across the front and backcontacts, which are very thin, resistive losses in themodule is even a bigger problem than for c-Si modules. Therefore, the module is produced such that it
consists of many very narrow cells of about 1 cm width
and the length being equal to the module length. Thesecells then are connected in series across the width ofthe module. On the very left and right of the module
metallic busbars collect the current and conduct it tothe bottom of the module where they are connectedwith external cables.
The series connection is established with laser scribing.
In total, three laser scribes are required for separating
two cells from each other and establishing a series connection between them. The first laser scribe, called P1,
is performed after the transparent front contact is deposited, as shown in Fig. 17.6 (a). The wavelength ofthe laser is such that the laser light is absorbed in thefront contact and the material is evaporated, leaving
a “gap” in the front contact, as shown in Fig. 17.6 (b).

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262

Then the photoactive layers are deposited onto the front
contact and also fill the gaps. Then, the second laser
scribe, called P2, is performed, as illustrated in Fig.
17.6 (c). The laser wavelength has to be chosen suchthat it is not absorbed in the transparent front contact
but in the absorber layer. For example, if the absorberconsists of amorphous silicon, green laser light can beused. The P2 scribe leaves a gap in the absorber layer,
as illustrated in Fig. 17.6 (d). The next step is the deposition of the metallic back contact that also fills theP2 gap. Finally, the third laser scribe (P3) is performed
as illustrated in Fig. 17.6 (e). The wavelength for thisscribe has to be chosen such that it is neither absorbed
in the front contact nor in the absorber stack, so it is,for example, infrared. The P3 scribe shoots a gap into
the back contact, as shown in Fig. 17.6 (f).
To understand the action of the laser scribes, we takea look at Fig. 17.6 (f): the P1 scribe filled with absorbermaterial forms an barrier, since the absorber is orders ofmagnitudes less conductive than the transparent front
contact. Similar, the P3 scribes forms an insulating gap
in the metallic back contact. However, the P2 scribe that
also is filled with metal forms a highly conducting connection between the front and back contacts – here theactual series connection is performed.

17. Components of PV Systems

(a)

(b)
front contact
glass

P1

(c)

(d)
absorberfront contact
glass

P2

(e)

(f)

current

back metal

P3
interconnect width
Figure 17.6: Schematic of creating an interconnect in thin-film

module. (Explanation given in the text).

For example, for making CIGS solar cells, first the molybdenum back contact is deposited on top of the glass
substrate and the cell areas are defined by P1 laser

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17.1. PV modules

scribes. Then the CIGS p layer and the CdS n layer aredeposited including a P2 laser scribe step. Finally the
intrinsic and n-doped zinc oxide is deposited, followed
by a final P3 laser scribe step. Now the front TCO electrode is connected with the Molybdenum back contactof the next solar cell.
The performance of such an interconnect established via
laser scribes and hence the total module performanceis determined by several things. First, the P2 scribe has
to be highly conductive, This means that it has to be
wide enough and that there must be no barrier at theinterface between the front contact and the metal of theP2 scribe. Further, the P1 and P3 scribes must perform
good barriers to effectively separate the cells from eachother. Thirdly, the region between the P1 and P3 scribes
does not contribute to the the current generated by themodule. Therefore, the ratio between this width andthe total cell width (including the scribes) has to be
as small as possible. Another issue is the fact that thethree laser scribes are performed in different steps of
production and thus often in different machines. Further, the distance between the scribes might be different
at the different processes when they are performed at
different temperatures. This, aligning the glass plates
in all the production steps is extremely important for
manufacturing high-quality thin-film modules.
The production steps and also the exact processing of
the laser scribes is of course dependent on which thin-

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film technology is used and even on the manufacturer
itself. However, the basic principles and the action behind these processes is valid in general.
One advantage of thin-film PV technology is that they
can be deposited onto flexible substrates. For example,
the Dutch company HyET Solar developed a technology, where thin-film silicon layers are deposited onto
a temporary aluminium substrate [92]. After the solar
cell layers are encapsulated on the back side, the temporary substrate is etched away, and the front side isencapsulated. This results in a very low weight flexible substrate, which can be integrated for example incurved roof top elements. A very big advantage is thatsuch very light modules can be installed on simple roof
top constructions that only can handle little ballast. On
such roofs, heave PV panels with glass cannot be installed. Further, if such flexible modules are directly
integrated into roofing elements, installation costs canbe reduces significantly. Often, installation costs are thelargest contributor to the non-modular costs of a PV
system. Currently, only thin-film silicon technologies
have demonstrated flexible modules with reasonableefficiencies.

17.1.7

Some examples

Table 17.1 shows some parameters of PV modules using
different PV technologies:

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17. Components of PV Systems

Table 17.1: Specifications of different PV modules.

Technology
Rated power
Rated current
Rated voltage
Short circuit current
Open circuit voltage
Dimensions
Max warranty on Pmpp

Pmpp
Impp
V mpp
Isc
V oc

(Wp )
(A)
(V)
(A)
(V)
(m×m)
(years)

SunPower
X21-345
c-Si
345
6.02
57.3
6.39
68.2
1.56×1.05
25

AvancisPowerMax140
CIS
140
2.98
47
3.31
61.5
1.6×0.67
25• A SunPower module made based on monocrystalline silicon solar cells,

17.2

KanekaU-EA120a-Si/nc-Si
114
2.18
55
2.6
71
1.2×1.00
25

First Solar
FS-392
CdTe
92.5
1.94
47.7
2.11
60.5
1.2×0.60
25

Maximum power point tracking• a Kaneka amorphous silicon /nanocrystalline silicon tandem (a-Si:H/nc-Si:H) module, andIn this section we discuss the concept of Maximum
power point tracking (MPPT). This concept is very uniqueto the field of PV Systems, and hence brings a very
special application of power electronics to the field ofphotovoltaics. The concepts discussed in this section are
equally valid for cells, modules, and arrays, although
MPPT usually is employed at PV module/array level.

• a module of First Solar based on cadmium telluride(CdTe) technology.

As discussed earlier, the behaviour of an illuminatedsolar cell can be characterised by an I-V curve. Interconnecting several solar cells in series or in parallel
merely increases the overall voltage and/or current, but• an Avancis module based on copper indium diselenide (CIS) technology,

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17.2. Maximum power point tracking

Current I

higher irradiance GM
highertemperature T

current, at that the PV module operates at any given
point in time. For a given irradiance and temperature,
the operating point corresponds to a unique ( I, V ) pairwhich lies onto the I-V curve. The power output at thisoperating point is given by
P = I · V.

Voltage V
Figure 17.7: Effect of increased temperature T or irradianceG M on the I-V curve.

does not change the shape of the I-V curve. Therefore,
for understanding the concept of MPPT, it is sufficient
to consider the I-V curve of a solar cell. The I-V curveis dependent on the module temperature on the irradiance, as we will discuss in detail in Section 18.3. For
example, an increasing irradiance leads to an increased
current and slightly increased voltage, as illustrated inFig. 17.7. The same figure shows that an increasing temperature has a detrimental effect on the voltage.
Now we take a look at the concept of the operating
point, which is the defined as the particular voltage andCopyright Delft University of Technology, 2014(17.1)The operating point ( I, V ) corresponds to a point on
the power-voltage (P-V) curve, shown in Fig. 17.8. For
generating the highest power output at a given irradiance and temperature, the operating point should suchcorrespond to the maximum of the (P-V) curve, whichis called the maximum power point (MPP).
If a PV module (or array) is directly connected to anelectrical load, the operating point is dictated by that
load. For getting the maximal power out of the module,
it thus is imperative to force the module to operate at
the maximum power point. The simplest way of forcing
the module to operate at the MPP, is either to force thevoltage of the PV module to be that at the MPP (called
Vmpp ) or to regulate the current to be that of the MPP(called Impp ).
However, the MPP is dependent on the ambient conditions. If the irradiance or temperature change, the IV and the P-V characteristics will change as well andhence the position of the MPP will shift. Therefore,changes in the I-V curve have to be tracked continuously such that the operating point can be adjusted toThis copy is provided for free, for personal use only.

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17. Components of PV Systems

be at the MPP after changes of the ambient conditions.
This process is called Maximum Power Point Tracking or
MPPT. The devices that perform this process are called
MPP trackers. We can distinguish between two categories of MPP tracking:
Pmpp

Isc
Impp

Power P

Current I

MPP

Voltage V

Vmpp

Voc

Figure 17.8: A generic I-V curve and the associated P-V

curve. The maximum power point (MPP) is indicated.

• Indirect MPP tracking, for example performed with
the Fractional Open Circuit Voltage method.
• Direct MPP tracking, for example performed with
the Perturb and Observe method or the the Incremental Conductance method.
All the MPPT algorithms that we discuss in this section
are based on finding the and tuning the voltage until
VMPP is found. Other algorithms, which are not discussed in this section, work with the power instead andaim to find IMPP .

17.2.1

Indirect MPPTFirst, we discuss indirect MPP Tracking, where simple
assumptions are made for estimating the MPP based ona few measurements.
Fixed voltage method

For example, in the fixed voltage method (also called
constant voltage method), the operating voltage of the

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17.2. Maximum power point tracking

solar module is adjusted only on a seasonal basis. Thismodel is based on the assumption that for the samelevel of irradiance higher MPP voltages are expectedduring winter than during summer. It is obvious that
this method is not very accurate. It works best at locations with minimal irradiance fluctuations between
different days.
Fractional open circuit voltage method

One of the most common indirect MPPT techniques isthe fractional open circuit voltage method. This method
exploits the fact that – in a very good approximation –the Vmpp is given by
Vmpp = k · Voc ,(17.2)

where k is a constant. For crystalline silicon, k usually takes values in between 0.7 and 0.8. In general,k of course is dependent on the type of solar cells.
As changes in the open circuit voltage can be easily
tracked, changes in the Vmpp can be easily estimatedjust by multiplying with k. This method thus can be implemented easily. However, there are also certain drawbacks.
First, using a constant factor k only allows to roughly
estimate the position of the MPP. Therefore, the operating point usually will not be exactly on the MPP but
in its proximity, with is called the MPP region. Secondly,

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every time the system needs to respond to a change inillumination conditions, the Voc must be measured. For
this measurement, the PV module needs to be disconnected from the load for a short while, which will leadto a reduced total output of the PV system. The more
often the Voc is determined, the larger the loss in output
will be. This drawback can be overcome my slightly
modifying the method. For this modification a pilot
PV cell is required, which is highly matched with therest of the cells in the module. The pilot cell receives
the same irradiance as the rest of the PV module, anda measurement of the pilot PV cell’s Voc also gives anaccurate representation of that of the PV module, hence
it can be used for estimating Vmpp . Therefore, the operating point of the module can be adjusted without
needing to disconnect the PV module.

17.2.2

Direct MPPT

Now we discuss direct MPP tracking, which is more involved than indirect MPPT, because current, voltage
or power measurements are required. Further, the system must response more accurately and faster than inindirect MPPT. We shall look at a couple of the mostpopular kind of algorithms.

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17. Components of PV Systems

Table 17.2: A summary of the possible options in the P&O

algorithm.

Change inPower
Positive
Negative
Positive
Negative

Next
Perturbation
Positive
Negative
Negative
Positive

Perturb and observe (P&O) algorithm

The first algorithm that we discuss is the Perturb andObserve (P&O) algorithm, which also is known as "hillclimbing" algorithm. In this algorithm, a perturbation
is provided to the voltage at that the module is currently driven. This perturbation in voltage will lead to achange in the power output. If an increasing voltage
leads to an increasing in power, the operating point
is at a lower voltage than the MPP, and hence further
voltage perturbation towards higher voltages is required to reach the MPP. In contrast, if an increasing
voltage leads to a decreasing power, further perturbation towards lower voltages is required in order toreach the MPP. Hence, the algorithm will converge towards the MPP over several perturbations. This principle is summarised in Table 17.2.

PMPP
C
B
Power P

Prior
Perturbation
Positive
Positive
Negative
Negative

A

PMPP’

C’

Voltage V VMPP’ VMPP
Figure 17.9: The perturb&observe algorithm struggles fromrapidly changing illumination conditions.

A problem with this algorithm is that the operat-

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269

17.2. Maximum power point tracking

ing point is never steady at the MPP but meandering around the MPP. If very small perturbation steps
are used around the MPP, this meandering, however,can be minimised. Additionally, the P&O algorithm
struggles from rapidly changing illuminations. For example, if the illumination (and hence the irradiance)changes in between two sampling instants in the process of convergence, then the algorithm essentially fails
in its convergence efforts, as illustrated in Fig. 17.9 In
the latest perturbation, the algorithm has determinedthat the MPP lies to the at a higher voltage than ofpoint B, and hence the next step is a perturbation toconverge towards the MPP accordingly. If the illumination was constant, it would end up at C and the algorithm would conclude that the MPP is at still higher
voltages, which is correct. However, as the illumination
changes rapidly before the next perturbation, the next
perturbation shifts the operating point to C’ instead toC, such thatPC’ < PB

(17.3)

While the MPP still lies to the right of C’, the P&O algorithm thinks that it is on the left of C’. This wrong
assumption is detrimental to the speed of convergence of the P&O algorithm, which is one of the critical figures of merit for MPPT techniques. Thus, drasticchanges in weather conditions severely affect the efficacy of the P&O algorithm’s.

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Incremental conductance method

Next, we look at the Incremental Conductance Method.
The conductance G1 of an electrical component isdefined asIG=(17.4)V
At the MPP, the slope of the P-V curve is zero, hence
dP
= 0.
dV

(17.5)dPd( IV )
dI
=
= I+V
.
dV
dV
dV

(17.6)

We can write

If the sampling steps are small enough, the approximationdI∆I≈(17.7)dV∆V
can be used. We call ∆I/∆V the incremental conductanceand I/V the constanteneous conductance. Hence, we have
∆I
I
=−
∆V
V
∆I
I
>−
∆V
V
∆I
I
<−
∆V
V

if

V = Vmpp ,(17.8a)if

V < Vmpp ,(17.8b)if

V > Vmpp .

(17.8c)1 The electrical conductance G must not be confused with the irradiance incident on the module G M .

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17. Components of PV Systems
V (t ), I (t ), Vref (t )DI = I (t ) - I (t -Dt )DV = V (t ) - V (t -Dt )Decrease
Vref

<

DI
I
?V
DV

>

Increase
Vref=
Maintain
Vref
RETURN
Figure 17.10: A conceptual flowchart of the incremental con-ductance algorithm.

These relationships are exploited by the incremental
conductance algorithm.
Figure 17.10 shows a conceptual flowchart. Note that
this flowchart is not exhaustive. While both, the instantaneous voltage and current are the observable parameters, the instantaneous voltage is also the controllable
parameter. Vref is the voltage value forced on the PV
module by the MPPT device. It is the latest approximation of the Vmpp . For any change of the operating
point, the algorithm compares the instantaneous with
the incremental conductance values. If the incremental
conductance is larger than the negative of the instantan-

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eous conductance, the current operating point is to theleft of the MPP; consequently, Vref must be incremented.
In contrast, if the incremental conductance is lower than
the negative of the instantaneous conductance, the current operating point is to the left of the MPP and the isconsequently decremented. This process is iterated until
the incremental conductance is the same as the negative
instantaneous conductance, in which case Vref = Vmpp .
The incremental conductance algorithm can be more efficient than the P&O algorithm as it does not meanderaround the MPP under steady state conditions. Further, small sampling intervals make it less susceptible
to changing illumination conditions. However, underconditions that are strongly varying and under partial
shading, the incremental conductance method mightalso become less efficient. The major drawback of thisalgorithm is the complexity of its hardware implementation. Not only currents and voltages must be measured, but also the instantaneous and incremental conductances must be calculated and compared. How sucha hardware design can look like however is beyond thescope of this book.

17.2.3

Some remarks

While a MPPT is used to find the MPP by changing thevoltage, it does not perform changes of the operating
voltage. This is usually done by a DC-DC converter

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17.3. Photovoltaic Converters
global MPP
local MPP
Power P

Current I

local MPP

curve is similar to that of a single cell. Let us now consider a system that is partially shaded, as illustrated
in Fig. 17.11. Then, the P-V curve will have different
local maxima. Depending on the used MPPT algorithm,
it is not sure at all that the algorithm finds the global
maximum. Different companies use proprietary solutions to tackle this issue. Alternatively, each string canbe connected to a separate MPPT. There are invertersthat have connections for several strings (usually two).

Voltage V
Figure 17.11: The P-V curve of partially shaded system thatexhibits several local maxima.

17.3

Photovoltaic Convertersthat will be discussed in section 17.3.2.
In modern PV systems, the MPPT is often implemented
within other system components like the inverters or
charge controllers. The list of techniques presented inthis section not exhaustive, we just discussed the mostcommon ones. The development of more advanced
MPPT techniques is going on rapidly and many scientific papers as well as patents are published in thisarea. Furthermore, manufacturers usually use proprietary techniques.
Up to now we only looked at situations the total I-V

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A core technology associated with PV systems is thepower electronic converter. An ideal PV converter
should draw the maximum power from the PV panel
and supply it to the load side. In case of grid connected
systems, this should be done with the minimum harmonic content in the current and at a power factor close
to unity. For stand-alone systems the output voltage
should also be regulated to the desired value. In thissection a short review of different topologies often associated with PV systems is given. The semiconductor
switches in the following are assumed to be ideal.

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17.3.1

17. Components of PV Systems

System configurations

Before digging into details about different converter topologies used for power conversion in PV systems, ageneral overview of different system architecture will
be presented. The system architecture determines how
PV modules are interconnected and how the interface
with the grid is established. Which of these system architectures will be employed in a particular PV plantdepends on many factors such as environment of theplant (whether the plant is situated in an urban environment or at an open area), scalability, costs etc. Figure17.12 an overview of different system architectures isgiven. The main advantages and disadvantages of thedifferent architectures are discussed below.
In general solar inverters should have the following
characteristics [93]:• Inverters should be highly efficient because theowner of the solar system requires the absolute
maximum possible generated energy to be delivered to the grid/load.
• Special demands regarding the potential between
solar generator and earth (depending on the solar
module type).
• Special safety features like active islanding detection capability.
• Low limits for harmonics of the line currents. This

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requirement is enforced by law in most countries
since the harmonic limits of both sources and loads
connected to the grid are regulated.
• Special demands on electromagnetic interference(EMI), which are regulated by law in most countries. The goal of these minimise the unwanted
influence of EMI on other equipment in the vicinity or connected to the same supply. Think for example of the influence of a mobile phone on an old
radio.
• In many instances the solar system is to be installed outdoors and inverters should adhere tocertain specification regarding temperature and humidity conditions, e.g. IP 54.
• Design for high ambient temperatures.
• Designed for 20 years operation under harsh environmental conditions.
• Silent operation (no audible noise).
We have to distinguish between single-phase and threephase inverters. For low powers, as they are common
in small residential PV systems, single-phase inverters
are used. They are connected to one phase of the grid.
For higher powers, three-phase inverters are used thatare connected to all phases of the grid. If a high power
would be delivered to one phase, the currents flowing
across the three phases would become very asymmetricThis copy is provided for free, for personal use only.

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17.3. Photovoltaic Converters(a)Central inverter

(b)

(c)

Module inverter(d)String inverter

Multi-string inverter

Figure 17.12: Different system architectures employed in PV systems.

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274

leading to several problems in the electricity grid.
Note that the term inverter is often used for two different things: First, it is used for the actual inverter, whichis the electronic building block that performs the DC-toAC conversion, as described in section 17.3.3. Secondly,
the term inverter also is used for the total unit produced by manufacturers, that nowadays usually contains, an MPP tracker, a DC-DC converter, an DC-AC
converter and possibly also a charge controller of also abattery is connected.
Central inverters

This is the simplest architecture employed in PV systems. Here, PV modules are connected in strings leading to an increased system voltage. Many strings arethen connected in parallel forming a PV array, which isconnected to one central inverter. The inverter performs
maximum power point tracking and power conversion
as shown in Fig. 17.12 (a), where a three-phase inverter
is depicted. This configuration is mostly employed invery-large scale PV production, with central inverter
usually being DC to three phase [94].
Many different inverter topologies are utilised as athree phase inverter. Sometimes they are organised asa single DC to three phase unit but sometimes as athree separate DC-to-AC single phase units working
with a phase displacement of 120 degrees each. Having

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17. Components of PV Systems

all the PV modes connected in a single array in such acentralised configuration offers the lowest specific cost
(cost per kWp p of installed power). Since central inverters only use a few components, they are very reliable
what makes them the preferred option in large scale PV
power plants.
In spite of their simplicity and low specific cost, central
inverters suffer from the following disadvantages:
1. Due to the layout of the system, a large amount of
power is carried over considerable distances using
DC wiring. This can cause safety issues because
fault DC currents are difficult to interrupt. Special precaution measures must to be taken suchas ticker insulation on the DC cabling and special
circuit breakers, which can increase the costs.
2. All strings operate at the same maximum power
point, which leads to mismatch losses in the modules. This is significant disadvantage. Mismatch
losses increase even more with ageing and withpartial shading of sections of the array. Mismatch
between the different strings may significantly reduce the overall system output.
3. Low flexibility and expandability of the system.
Due to the high ratings a system is normally designed as a unit and hence difficult to extend. In
other words the system design is not very flexible.
4. Power losses in the string diodes, which are put inThis copy is provided for free, for personal use only.

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17.3. Photovoltaic Converters

series with each string to prevent current circulation inside strings.
Module Integrated or module oriented inverters

A very different architecture is that of the module integrated inverters, as shown in 17.12 (b). These inverters
operate directly at one or several PV modules and havepower ratings of several hundreds of watts. Becauseof the low voltage rating of the PV module, these inverters require often require a two stage power conversion. In a first stage boosts the DC voltage is boosted
to the required value while it is inverted to AC in thesecond stage. Often, a high frequency transformer isincorporated providing full galvanic isolation, which
enhances the system flexibility even further. As theirname suggests, these inverters are usually integrated
with PV panel (so called ‘AC PV panels’). In this way
the highest flexibility and the expandability of system
is obtained. One of the most distinguishing features ofthis system is the “plug and play” characteristic, whichallows to build a complete (and readily expandable) PV
system at a low investment cost. Another advantage ofthese inverters, is minimisation of the mismatch losses
that can occur because of non-optimal MPPT.
All these advantages come at certain expenses. Because
these inverters are be mounted on a PV module, theymust operate in harsh environment such as high tem-

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perature and large daily and seasonal temperature variations. Also, the specific are the highest of all the inverter topologies. Many topologies for module integrated inverters have been proposed, with some of them
being already implemented in commercially available
inverters.
String Inverters

String inverter, as illustrated in 17.12 (c), combine theadvantages of central and module integrated inverter
concepts with little tradeoffs. A number of PV modules that are connected in series form a PV string witha power rating of up to 5 kWp and with the open circuit voltage of up to 1 kV. Now, a number of smaller
inverters can be used to connect the PV system to thegrid.
One disadvantage of the topology is the fact that thehigh DC voltage requires special consideration, similarly as this already was the case for the central inverterarchitecture. Here, this issue is even more importantbecause string inverters are usually being installed inhouseholds or on office buildings, without designatedsupport structure or increased safety requirements. In
general a qualified electrician is needed to perform theinterconnections between the modules and the inverter.
The protection of the system also requires special consideration, with emphasis on proper DC cabling.

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17. Components of PV Systems

Although partial shading of the string will influence
the overall efficiency of the system, each string can independently be operated at its MPP. Also, because no
strings are connected in parallel, there is no need forseries diodes as in the case of PV arrays with many
parallel strings. This reduces losses associated withthese diodes. However, it still is a risk that within astring hot-spot occurs because of unequal current andpower sharing inside the string.
Multi String Inverters

The Multi String inverter concept, illustrated in Fig.
17.12 (d), has been developed to combine the advantageof the higher energy yield of a string inverter with thelower costs of a central inverter [95]. Lower power DCDC converters are connected to individual PV strings
each having its own MPP tracking, which independently optimises the energy output from each PV string.
To expand the system within a certain power range
only a new string with a DC-DC converter has to be
included. All DC-DC converters are connected via a DC
bus through a central inverter to the grid.

Team
connection

Figure 17.13: The team concept of inverters.

Team Concept

Aside of the four system architecture already described,
many other concepts are also present in literature.

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277

17.3. Photovoltaic Converters

However, these concepts are less widely utilised thanthe ones already presented. One of the alternative concepts is so called team concept, which combines thestring technology with the master-slave concept. A
combination of several string inverters working withthe team concept is shown in Figure 17.13. At very
low irradiation the complete PV array is connected
to a single inverter. This reduces the overall losses as
any power electronic converter is designed such that it
has maximum efficiency near full load. With increasing
solar radiation more inverters are being connected dividing the PV array into smaller units until every string
inverter operates close to its rated power. In this mode
every string operates independently with its own MPP
tracker. At low solar radiation the inverters are controlled in a master-slave fashion.
In literature many other concepts are discussed. For
further reading on this subject, we refer for example toRef. [96].

17.3.2

DC-DC converters

DC-DC converters fulfil multiple purposes. In an inverter, DC power is transformed into AC power. The
DC input voltage of the inverter often is constant while
the output voltage of the modules at MPP is not. Therefore a DC-DC converter is used to transform the variable voltage from the panels into stable voltage used

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by the DC-AC inverter. Additionally, as already statedin section 17.2, the MPP Tracker controls the operating
point of the modules, but it cannot set it. This also isdone by the DC-DC converter. Further, in a stand-alone
system the MPP voltage of the modules might differ
from that required by the batteries and the load. Also
here, a DC-DC converter is useful. Three topologies areused for DC-DC converters: buck, boost, and buck-boost
converters. They are described below.

Step-Down (Buck) Converter

Figure 17.14 (a) illustrates the simplest version of abuck DC-DC converter. The unfiltered output voltage
waveform of such a converter operated with pulsewidth-modulation (PWM) is shown in Fig. 17.14 (b).
If the switch is on, the input voltage Vd is applied to theload. When the switch is off, the voltage across the load
is zero. From the figure we see that the average DC
output voltage is denoted as Vo . From the unfilteredvoltage, the average output voltage is given as
Z Ts

1
ton
V.
(ton Vd + toff · 0) =TsTs d(17.9)The different variables are defined in Fig. 17.14 (b). To
simplify the discussion we define a new term, the duty
Vo =

1
Ts

0

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278

17. Components of PV Systems

cycle D, as
(a)

D :=

+
SwitchcontrolsignalVo = D · Vd .

+
Vo

Load

−

−

(b)
Ts

vo
t on

t off

Vd

Vo

0

t

Figure 17.14: (a) A basic buck converter without any filtersand (b) the unfiltered switched waveform generated by thisconverter.

(17.11)

In general the output voltage with such a high harmonic content is undesirable, and some filtering is required. Figure 17.15 (a) shows a more complex modelof a step-down converter that has output filters included and supplies a purely resistive load. As filter
elements an inductor L and a capacitor C are used. The
relation between the input and the output voltages,
as given in Equ. 17.11, is valid in continuous conduction mode, i.e. when the current through the inductor
never reaches zero value but flows continuously. We
can change the ratio between the voltages on the input
and output side by changing the duty cycle D. A detailed discussion about different modes of operation of
a buck converter can be found in Reference [97].
In steady-state operation the time integral of the voltage
across the inductor v L taken during one switching cycle
is equal to zero. If this is not the case, the circuit is notin steady state. Thus, in steady state, we obtain the following inductor volt-second balance:
Z Ts
0Copyright Delft University of Technology, 2014(17.10)and hence

Vd
RL

ton
Ts

v L dt =Z ton
0v L dt +Z Ts
tonv L dt = 0.

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17.3. Photovoltaic Converters

Solving this equation leads to(a)id

Vo = DVd ,+which is the same result as in Eq. 17.11.
iL

Vd

+

vL

Step-Up (Boost) Converter

io

L

+

−
Vo
Load

C

−
(b)

−

io

iL
vL

L

−

Vd

−

C

In a boost converter, illustrated in Fig. 17.15 (b), an input DC voltage Vd is boosted to a higher DC voltage
Vo . By applying the inductor volt-second balance acrossthe inductor as explained in Eq. 17.12, we find
Vd ton + (Vd − V0 )toff = 0.

(17.14)Using the definition for the duty cycle we find++(17.13)

Vo
Load

−

Figure 17.15: (a) A buck converter with filters and (b) a boost

1
V0
=
.
Vd
1−D

(17.15)The above relation is valid in the continuous conduction mode. The principle of operation is that energy
stored in the inductor (during the switch is on) is later
released against higher voltage Vo . In this way theenergy is transferred from lower voltage (solar cellvoltage) to the higher voltage (load voltage).
Buck-Boost Converter

converter.

In a buck-boost converter the output voltage can be both
higher or lower than the input voltage. The simpli-

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17. Components of PV Systems
id+

MPP Tracker

I1
Vd

+
vL

−

I2

−
iL

Vo
Load

C

− L

Rmeas
V1

V2

Figure 17.17: A combination of a unit performing an MPPT

Figure 17.16: A buck-boost converter.

algorithm and a DC-DC converter (adapted from [98]).

fied schematic of a buck-boost converter is depicted inFig. 17.16. Using inductor volt second balance as in Eq.
17.12, we find

MPP Tracking

and hence

Vd ton + (−Vo )toff = 0(17.16)

Vo
D
=
Vd
1−D

(17.17)in the continuous conduction mode.
The above described topologies are only the most basic
DC-DC converter topologies. The interested reader canfind more in-depth information in Reference [97].

Copyright Delft University of Technology, 2014

R

+

+

io

−

In section 17.2 we extensively discussed Maximum
Power Point Tracking. Or – more specific we discussed
different algorithms that are used for performing MPPT.
In these algorithms, usually the operating point of themodule is set such that its power output becomes maximal. However, the MPPT algorithm itself cannot actually adjust the voltage or current of the operating point.
For this purpose a DC-DC converter is needed. Figure17.17 shows such a combination of the unit performing
the MPPT and DC-DC converter. As illustrated in the
figure, this MPPT unit measures the voltage or the cur-This copy is provided for free, for personal use only.

281

17.3. Photovoltaic Converters

rent on the load side and can very those by adapting
the duty cycle of the DC-DC converter. In this illustration, current and voltage on the load side are measured,
but they also can be measured on the PV side.

ExampleAssume a PV module has its MPP at VPV=
17 V and
IPV = 6 A at a given level of solar irradiance. The module has
to power a load with a resistance R L = 10 Ω. Calculate theduty cycle of the DC-DC converter, if a buck-boost converter isused.
The maximum power from the module is PMPP = VMPP ·IMPP = 102 W. If this power should be dissipated at the resistor, we have to use the relation
PR = UR2 /R
and hence find for the voltage at the resistorp
VR = PMPP R L = 31.94 V
Using Eq. (17.17),
Vo
D
=
Vd
1−D
with Vo = VR and Vd = VPV we find D = 0.65.

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17.3.3

DC-AC converters (inverters)

Earlier in this section we have discussed different architectures that are used for the power conversion in PV
systems. Further, we looked at DC-DC converters thatare mainly used in combination with MPPTs in orderto push the variable output from the PV modules to alevel of constant voltage.
As nowadays most appliances are designed for the
standard AC grids, for most PV systems a DC-AC converter is required. As already stated earlier, the term
inverter is used both for the DC-AC converter and the
combination of all the components that form the actual
power converter.

The H-bridge inverter

Figure 17.18 shows a very simple example of a so called
H-bridge or full-bridge inverter. On the left, the DC input is situated. The load (or in our case the AC output)
is situated in between four switches. During usual operation we can distinguish between three situations:
A All four switches open: No current flows across theload.
B S1 and S4 closed, S2 and S3 open: Now a current isflowing to through the load, where + is connected

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17. Components of PV Systems
id

+
S1

S2

+
Vd

−
Load

−

+

S3

S4

−

Figure 17.18: A simple representation of an H-bridge.

to the left- and − is connected to the right-handside of the load.

positive and negative voltages is exactly what is happening in AC. In the easiest operation mode, the Hbridge switches between situations B and C continuously, which will provide a square wave. Note that
there will always be a dead time in the order of µs inorder to prevent short-circuiting. While such a squarewave might be useful for some applications, it is notsuited at all for grid-connected installations. The reason
for this are harmonic distortions. To understand this welook at a Fourier transform of a square wave, which is
given as
41
Vsquare (t) =sin(2πνt) + sin(6πνt)
π3
(17.18)1+ sin(10πνt) + · · · .
5

WARNING It must be assured that S1 and S2 never areopen at the same time because this would lead toshort-circuiting. The same is true for S3 and S4 .

Thus, such a square wave contains not only the principal sine function with frequency ν, but also all thehigher harmonics with frequencies 3ν, 5ν, and so on.
These higher harmonics can lead to distortions of theelectricity grid and thus must be reduced as much as
possible. One principle how to do this is pulse-widthmodulation (PWM) that we already discussed in the section on DC-DC conversion (17.3.2). In this configuration, each leg (the one via S1 and S4 and the one via S2and S3 ) in fact acts as a buck converter.

From this list we see that the H-bridge configuration
allows to put the load on three different levels, whichare +Vd , 0, and −Vd . Continuously switching between

If a low-pass filter consisting of capacitors and inductors is used, as in Fig. 17.19, the high frequency components are filtered out and hence a very smooth sine

C S1 and S4 open, S2 and S3 closed: Now a current isflowing to through the load, where − is connectedto the left- and + is connected to the right-handside of the load.

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17.3. Photovoltaic Converters

+

+VDC
S1

0

S2

VDC

VPWM

C

VAC

Figure 17.19: Illustration of an H-bridge containing a low-

pass filter for removing the high-frequency components of thesignal.

curve can be obtained that complies to the regulations
for grid-connected systems. Figure 17.20 shows the unfiltered PWM output and the filtered sine output.
Note that in Fig. 17.19 diodes are connected in parallel to the switches. The reason for placing these diodes is the following: If the switch goes from closed
to open very fast, no current can flow through the inductor anymore, meaning that the change of current
flowing through the inductor is very high. This induces
a voltage given bydI,
dt

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Mean voltage output

that is obtained with a low-pass filter.

−

Vinduced = − L-VDC

Figure 17.20: The unfiltered PWM signal and the sine signal

S4

S3

PMW output of bridge(17.19)which is the higher the faster the current changes. Ashigh induced voltages will damage the electric circuitry,
they must be prevented. The diode ensures that current
also can flow after the switch opens. Hence current isflowing always and no high induced voltages appear.
Since this configuration is grid-connected, we easily candetermine the required DC input voltage, which must
be minimal the peak voltage of the AC voltage. For aneffective AC voltage of 240 V, as used in large√ parts of
the world, the peak voltage is Vpeak = 1.1 2 · 240 V =373 V, where we also took a 10% tolerance into account.
If the PV array delivers a lower voltage, thus a boost
converter would be required prior in addition to theinverter. Alternatively, a transformer can be used thattransforms a lower-voltage AC signal to the 240 V AC
signal. Such a system is sketched in Fig. 17.23. This system has the advantage of galvanic separation with all
the advantages discussed below, but on the other handThis copy is provided for free, for personal use only.

284

17. Components of PV Systems

+

+
S2

VDC

L1
L2
L3

VDC

VAC

Load
S4

−

−

Figure 17.21: Illustrating a three-phase inverter.

Figure 17.22: Illustrating a half-bridge inverter.

the transformer reduces the overall efficiency.
Figure 17.21 shows a three-phase inverter. As it can beseen, it is very similar to the single-phase inverter discussed above, just it contains three legs of that eachcreates a sine-wave output with a phase shift of 120°
between them.

Half-bridge inverters

A simpler inverter topology, the so-called half-bridge
inverter, is shown in Fig. 17.22. In contrast to the Hbridge configuration, here, two switches are replacedby capacitors and the midpoint in-between the two ca-

Copyright Delft University of Technology, 2014

pacitors is directly connected to the ground. The halfbridge configuration is much simpler than the H-bridge
configuration, but it has some drawbacks.

The main drawback is requirement for a high DC link
voltage, which needs to be two times higher than thatof a full bridge inverter. For a effective AC voltageof 240 V this would be 746 V. Because the topology
provides two-levels in the output voltage (in difference
to three levels for the full bridge configuration, a higher
current ripple is present in the output filter inductor.
Hence, larger value of the output filters are required.

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17.3. Photovoltaic Converters

17.3.4

Some remarks

Switches

All the DC-DC transformers and DC-AC transformers
discussed above contain switches. Traditionally, in linecommutated inverters, thyristors are used as switches. A
thyristor is a electronic component consistent of pnpn
layers. It thus contains 3 p-n junctions. One disadvantage is that thyristors cannot be turned off, but only
turned on. Thus, one has to wait for the next zero passof the grid signal [98]. The current flow thus is rectangular which leads to a very high harmonic content
requiring additional filters in order to make the output
compatible with the electricity grid. Nowadays, thyristors only are used for inverters with a power of 100 kW
and higher.
All other inverters are self-commutated inverters thatgenerate an output with very little harmonic content
as described above and in Fig. 17.20. The switches there
are fully-controllable such that pulse-width modulation becomes possible. As switches, GTOs (gate turn-off
thyristors), IGBTs (insulated-gate bipolar transistors) or
MOSFETs (metal-oxide-semiconductor field-effect transistors) are used. More information can be found forexample in Reference [20].

Copyright Delft University of Technology, 2014

Overall configuration

Figure 17.23 shows an example of a transformer-lessabsorber as it could be sold for household systems. Besides the actual DC-AC converter, which is realised asan H-bridge, it also contains a DC-DC boost converter
and an MPPT that uses the voltage and current measured on the PV side as input. The system sketched also
contains several electronic components for increased
safety: a residual current measurement to detect leakage
currents above a certain threshold and to shut-down
the inverter if these currents appear. Further, a gridmonitoring unit to prevent islanding (see below) [98].

Potential induced degradation (PID)

As mentioned earlier, in PV systems that have atransformer-less inverter, no galvanic separation
between the DC- and the AC-parts of the system isgiven. Because of this lack of galvanic isolation, a potential of -500 V or more between the PV modules andthe ground can occur, which can lead to potential induced degradation (PID). Many thin-film contain a TCO
front contact that is deposited in superstrate configuration on the glass top plate. Positively charged sodium
ions than can travel into the TCO because of this potential. This leads to corrosion and consequently to performance loss of the module. Also, for crystalline mod-This copy is provided for free, for personal use only.

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17. Components of PV Systems

PV Array

EMC
FilterBoost
Converter

DC
Bus

PWM
Bridge

EMC
FilterL
230 V ~N

Earth-fault
Monitoring

MPP
Regulation

Central Control

Residual
Current
Measurement

Grid
Monitoring

Data
Interface

Figure 17.23: Illustrating an example of a transformer-less inverter unit as it is sold for residential PV systems. As switches, MOS-

FETs are used. (Adapted from [98]).

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17.3. Photovoltaic Converters

ules, PID can be a problem [98]. Therefore, for systems
containing thin-film modules the inverter must have atransformer.

IslandingA potential danger of grid-connected systems is islanding. Imagine that a potential PV system is installed ina street where the electricity grid is shut-down in orderto do maintenance work on the electricity cables. If it
is a sunny day, the PV system will produce power and– without protection – would deliver the power to thegrid. The electricity worker thus can be in danger. Thisphenomenon is called islanding and due to its danger it
must be prevented.

i.e. the fraction of the output power Po to the sum of
Po with the dissipated power Pd . For estimating theefficiency we thus must estimate the dissipated (lost)
power, which can be seen as a sum of several components,
Pd = PL + Pswitch + other losses,(17.21)
i.e. the power lost in the inductor PL and the power lostin the switch. PL is given as2
PL = IL,
rms R L ,(17.22)where IL, rms is the mean current flowing through theinductor. The losses in the switch strongly depend on
the type of switch that is used. Other losses are for example resistive losses in the circuitry in-between theswitches.

The inverter therefore must be able to detect, when theelectricity grid is shut-down. If this is the case, also theinverter must stop delivering power to the grid.

For a complete inverter unit it is convenient to define
the efficiency asP(17.23)
ηinv = AC ,PDC

Eﬃciency of power converters

which is the ratio of the output AC power to the DC input power. Figure 17.24 shows the efficiency of a commercially available inverter for different input voltages.
As we can see, in general the lower the output power,
the less efficient the inverter. This efficiency characteristic must be taken into account when planning a
PV system. Further, the efficiency is lower if the input
voltage deviates from the nominal value.

For planning a PV system it is very important to know
the efficiency of the power converters. This efficiency of
DC-DC and DC-AC converters is defined asη=Po,
Po + Pd

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17. Components of PV Systems

17.4

Inverter Efficiency (%)

100

95
120 V DC
260 V DC335 V DC

90

85

0

10

20

30

40

50

60

70

80

90 100Fraction of Rated Output Power (%)
Figure 17.24: The power-dependent efficiency for several

input voltages of a Fronius Galvo 1.5-1 208-240 inverter at 240 V
AC. The nominal input voltage is 260 V DC. (Data taken from
Ref. [99]).

Copyright Delft University of Technology, 2014

Batteries

In this section we discuss a vital component not only of
PV systems but of renewable energy systems in general.
Energy storage is very important at both small and large
scales in order to tackle the intermittency of renewableenergy sources. In the case of PV systems, the intermittency of the electricity generation is of two kinds:
first, diurnal fluctuations, i.e. the difference of irradianceduring the 24 hour period. Secondly, the seasonal fluctuations, i.e. the difference of irradiance between thesummer and winter months. There are several technological options for realising storage of energy. Therefore it
is important to make an optimal choice.
Fig. 17.25 shows a Ragone chart, where the specific energy
is plotted against the peak power density. Because it uses
a double-logarithmic chart storage technologies withvery different storage properties can be compared in
one plot. Solar energy application require a high energy
density and – depending on the application – also a
reasonably high power density. For example, we cannot
use capacitors because of their very poor energy density. For short term to medium term storage, the mostcommon storage technology of course is the battery.
Batteries have both the right energy density and power
density to meet the daily storage demand in small andmedium-size PV systems. However, the seasonal storage problem at large scales is yet to be solved.

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289

17.4. Batteries
ultracap.

Power density (W/kg)

104

10

3

10

Li-ionNiCd NiMH bat.
H2,
combustion
gasolineelektrolyte
cap.
doublelayer cap.

102

10

-1

Li-ion cap.

CH4

Lead
acid

H2, fuel cell

10
10
10
Energy density (Wh/kg)
02Secondary batteries or rechargeable batteries, as they are
more commonly called, convert chemical energy to electrical energy reversibly. This means that they can be recharged when an over-potential is used. In other words,
excess electrical energy is stored in these secondary batteries in the form of chemical energy. Typical examples
for rechargeable batteries are lead acid or lithium ion batteries. For PV systems only secondary batteries are ofinterest.

103

17.4.1

Types of batteries

Figure 17.25: A Ragone chart of different energy storage

methods. Capacitors are indicated with “cap”.

Lead-acid batteries are the oldest and most mature technology available. They will be discussed in detail laterin this section.

The ease of implementation and efficiency of the batteries is still superior to that of other technologies, likepumping water to higher levels, compressed air energy
storage, conversion to hydrogen, flywheels, and others.
Therefore we will focus on battery technology as a viable storage option for PV systems.

Other examples are nickel-metal hydride (NiMH) andnickel cadmium (NiCd) batteries. NiMH batteries havea high energy density, which is comparable to thatof lithium-ion batteries (discussed below). However,
NiMH batteries suffer from a high rate of self discharge. On the other hand, NiCd batteries have much
lower energy density than lithium ion batteries. Furthermore, because of the toxicity of cadmium, NiCd
batteries are widely banned in the European Union for
consumer use. Additionally, NiCd batteries suffer fromwhat is called the memory effect: the batteries loose theirusable energy capacity if they are repeatedly charged
after only being partially discharged. These disadvant-Batteries are electrochemical devices that convert chemical energy into electrical energy. We can distinguishbetween primary or secondary batteries. Primary batteries convert chemical energy to electrical energy irreversibly. For example, zinc carbon and alkaline batteries areprimary batteries.

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290

17. Components of PV Systems+−

Positive
electrolyte
tank

Negative
electrolyte
tank
VO+2 « VO2+

Pump

V 2+ « V 3+

protonexchangemembrane

Pump

Figure 17.26: Schematic on a vanadium redox-flow battery

that employs vanadium ions.

ages make NiMH and NiCD batteries unsuitable candidates for PV storage systems.
Lithium ion batteries (LIB) and lithium-ion polymer batteries, which are often referred to as lithium polymer(LiPo) batteries, have been heavily investigated in recent years. Their high energy density already has madethem the favourite technology for light-weight storage
applications for example in mobile telephones. However, these technologies still suffer from high costs andlow maturity.
The last and the most recent category of batteries that
we will discuss in this treatise are redox-flow batteries.

Copyright Delft University of Technology, 2014

Lead-acid batteries and LIB, the two main storage options for PV systems, are similar in the sense that theirelectrodes undergo chemical conversion during charging and discharging, which makes their electrodes todegenerate with time, leading to inevitable “ageing” of
the battery. In contrast, redox flow batteries combine
properties of both batteries and fuel cells, as illustrated
in Fig. 17.26. Two liquids, a positive electrolyte and a negative electrolyte are brought together only separated bya membrane, which only is permeable for protons. The
cell thus can be charged and discharged without the reactants being mixed, which in principle prevents the liquids from ageing. The chemical energy in a redox flow
battery is stored in its 2 electrolytes, which are stored
in two separate tanks. Since it is easy to make the tanks
larger, the maximal energy that can be stored in sucha battery thus is not restricted. Further, the maximal
output power can easily be increased by increasing thearea of the membrane, for example by using more cells
at the same time. The major disadvantages is that sucha battery system requires additional components suchas pumps, which makes them more complicated thanother types of batteries.
Figure 17.27 shows a Ragone plot as it is used to compare different battery technologies. In contrast to Fig.
17.25, here the the gravimetric energy density and volumetric energy density are plotted against each other.
The gravimetric energy density is the amount of energy
stored per mass of the battery; it typically is meas-

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291

1000
800
600

logies still very expensive. Thus, because of to the unequalled maturity and hence low cost of the lead-acidbatteries, they are still the storage technology of choicefor PV systems.

Thin-film Li

smaller

Volumetric energy density (Wh/l)

17.4. Batteries

ured in Wh/kg. The volumetric energy density is theamount of energy stored per volume of battery; it isgiven in Wh/l. The higher the gravimetric energy density, the lighter the battery can be. The higher the volumetric energy density, the smaller the battery can be.

Figure 17.28 shows a sketch of a lead-acid battery. A
typical battery is composed of several individual cells,of which each has a nominal cell voltage around 2 V.
Different methods of assembly are used. In the block assembly, the individual cells share the housing and areinterconnected internally. For example, to get the typical battery voltage of 12 V, six cells are connected inseries. As the name suggests, lead acid batteries usean acidic electrolyte. More specifically they use diluted
sulphuric acid H2 SO4 . Two plates of opposite polarity are inserted in the electrolyte solution, which act as
the electrodes. The electrodes contain grid shaped lead
carrier and porous active material. This porous active
material has a sponge-like structure, which provides
sufficient surface area for the electrochemical reaction.
The active mass in the negative electrode is lead, whilein the positive electrode lead dioxide (PbO2 ) is used.

Figure 17.27 shows that lead acid batteries have boththe lowest volumetric and gravimetric energy densities
among the different battery technologies. Lithium ion
batteries show ideal material properties for using themas storage devices. Redox-flow batteries are very promising. However, both LIB and redox-flow batteries arestill in a development phase which makes these techno-

When the battery is discharged, electrons flow from thenegative to the positive electrode through the external
circuit, causing a chemical reaction between the plates
and the electrolyte. This forward reaction also depletes
the electrolyte, affecting its state of charge (SoC). Whenthe battery is recharged, the flow of electrons is reversed, as the external circuit does not have a load, but

400

Li-ionNiMH

Li-polymer

200
Pb-acid0

lighter

200
100
300
Gravimetric energy density (Wh/kg)

400

Figure 17.27: A Ragone chart for comparing different second-ary battery technologies with each other.

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292

17. Components of PV Systems

Through partition
conectors+

Tapered terminal post
Coverexternal
circuit−Post straps
Plate lugs

Vent plugs

Positive plate

Envelope
separators

Container

Negative plate
Sediment space

Element rests
Positive plate

H2SO4

Negative plate

Figure 17.28: Schematic of a lead-acid battery.

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17.4. Batteries

a source with a voltage higher than that of the battery
enables the reverse reaction. In the PV systems, thissource is nothing but the PV module or array providing clean solar power.

17.4.2

Battery parameters

Let us now discuss some parameters that are used tocharacterise batteries.
VoltageFirst, we will discuss the voltage rating of the battery.
The voltage at that the battery is rated is the nominal
voltage at which the battery is supposed to operate. The
so called solar batteries or lead acid grid plate batteries
are usually rated at 12 V, 24 V or 48 V.
Capacity

When talking about batteries, the term capacity refers
to the amount of charge that the battery can deliverat the rated voltage. The capacity is directly proportional to the amount of electrode material in the battery. This explains why a small cell has a lower capacity
than a large cell that is based on the same chemistry,even though the open circuit voltage across the cell will

Copyright Delft University of Technology, 2014

be the same for both the cells. Thus, the voltage of thecell is more chemistry based, while the capacity is more
based on the quantity of the active materials used.
The capacity Cbat is measured in ampere-hours (Ah).
Note that charge usually is measured in coulomb (C).
As the electric current is defined as the rate of flow ofelectric charge, Ah is another unit of charge. Since 1 C= 1 As, 1 Ah = 3600 C. For batteries, Ah is the moreconvenient unit, because in the field of electricity theamount of energy usually is measured in watt-hours(Wh). The energy capacity of a battery is simply given
by multiplying the rated battery voltage measured in
Volt by the battery capacity measured in Amp-hours,
Ebat = Cbat V,(17.24)

which results in the battery energy capacity in Watthours.
C-rateA brand new battery with 10 Ah capacity theoretically
can deliver 1 A current for 10 hours at room temperature. Of course, in practice this is seldom the case due toseveral factors. Therefore, the C-rate is used, which is ameasure of the rate of discharge of the battery relative
to its capacity. It is defined as the multiple of the current over the discharge current that the battery can sustain over one hour. For example, a C-rate of 1 for a 10

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294

17. Components of PV Systems

Ah battery corresponds to a discharge current of 10 A
over 1 hour. A C-rate of 2 for the same battery would
correspond to a discharge current of 20 A over half anhour. Similarly, a C-rate of 0.5 implies a discharge current of 5 A over 2 hours. In general, it can be said thata C-rate of n corresponds to the battery getting fullydischarged in 1/n hours, irrespective of the battery capacity.

This efficiency covers the fact that the charging voltage
is always a little above the rated voltage in order todrive the reverse chemical (charging) reaction in thebattery.
Secondly, we have the coulombic efficiency (or Faraday
efficiency), which is defined as the ratio of the total
charge got out of the battery to the total charge put into
the battery over a full charge cycle,
ηC =Battery eﬃciency

For designing PV systems it is very importent to know
the efficiency of the storage system. For storage systems,
usually the round-trip efficiency is used, which is given
as the ratio of total storage system input to the total
storage system output,
Eoutηbat =
100%.
Ein(17.25)For example, if 10 kWh is pumped into the storage system during charging, but only 8 kWh can be retrieved
during discharging, the round trip efficiency of the storage system is 80%. The round-trip efficiency of batteries
can be broken down into two efficiencies: first, the voltaic efficiency, which is the ratio of the average discharging voltage to the average charging voltage,
ηV =Vdischarge
100%.
Vcharge

Copyright Delft University of Technology, 2014(17.26)

Qdischarge
100%.
Qcharge(17.27)The battery efficiency then is defined as the product ofthese two efficiencies,
ηbat = ηV · ηC =Vdischarge Qdischarge
100%.
Vcharge Qcharge(17.28)When comparing different storage devices, usually
this round-trip efficiency is considered. It includes all
the effects of the different chemical and electrical nonidealities occurring in the battery.
State of charge and depth of discharge

At another important battery parameter is the State ofCharge (SoC), which is defined as the percentage of thebattery capacity available for discharge,
SoC =

Ebat
.
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295

17.4. Batteries

Thus, a 10 Ah rated battery that has been drained by
2 Ah is said to have a SoC of 80%. Also the Depth ofDischarge (DoD) is an important parameter. It is definedas the percentage of the battery capacity that has beendischarged,
C V − Ebat
DoD = bat.
(17.30)Cbat VFor example, a 10 Ah battery that has been drainedby 2 Ah has a DoD of 20%. The SoC and the DoD arecomplimentary to each other.

Cycle lifetime

(a)

(b)

Figure 17.29: (a) Qualitative illustration of the cycle lifetimeof a lead-acid battery in dependence of the DoD and temperature [100]. (b) Effect of the temperature on the battery capacity[100].

The cycle lifetime is a very important parameter. It isdefined as the number of charging and discharging
cycles after that the battery capacity drops below 80%of the nominal value. Usually, the cycle lifetime is specified by the battery manufacturer as an absolute number. However, stating the battery lifetime as a single
number is a oversimplification because the different
battery parameters discussed so far are not only related
to each other but are also dependent on the temperature.

DoD. The smaller the DoD, the higher the cycle lifetime. Thus, that the battery will last longer if the average DoD can be reduced during the lifetime of thebattery. Also, battery overheating should be strictly controlled. Overheating can because of overcharging andsubsequent over-voltage of the lead acid battery. To prevent this, charge controllers are used that we address inthe next section.

Figure 17.29 (a) shows the cycle lifetime as a function of
the DoD for different temperatures. Clearly, colder operating temperatures mean longer cycle lifetimes. Furthermore, the cycle lifetime depends strongly on theTemperature eﬀects

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While the battery lifetime is increased at lower temperatures, one more effect must to be considered. The

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296

temperature affects the battery capacity during regular
use too. As seen from Fig. 17.29 (b), the lower the temperature, the lower the battery capacity. This is because,the chemicals in the battery are more active at higher
temperatures, and the increased chemical activity leads
to increased battery capacity. It is even possible to reachan above rated battery capacity at high temperatures.
However, such high temperatures are severely detrimental to the battery health.

Ageing

The major cause for ageing of the battery is sulphation.
If the battery is insufficiently recharged after being discharged, sulphate crystals start to grow, which cannot
be completely transformed back into lead or lead oxide.
Thus the battery slowly loses its active material massand hence its discharge capacity. Corrosion of the lead
grid at the electrode is another common ageing mechanism. Corrosion leads to increased grid resistance due
to high positive potentials. Further, the electrolyte candry out. At high charging voltages, gassing can occur,
which results in the loss of water. Thus, demineralised
water should be used to refill the battery from time totime.

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17. Components of PV Systems

17.4.3Charge controllers

After having discussed different types of batteries anddifferent battery characteristics, it is now time to discuss charge controllers. Charge controllers are used in
PV systems that use batteries, which are stand-alone
systems in most cases. As we have seen before, it isvery important to charge and discharge batteries at
the right voltage and current levels in order to ensure
a long battery lifetime. A battery is an electrochemical device that requires a small over-potential to be
charged. However, batteries have strict voltage limits,which are necessary for their optimal functioning. Further, the amount of current sent to the battery by the
PV array and the current flowing through the battery
while being discharged have to be within well-defined
limits for proper functioning of the battery. We haveseen before that lead acid batteries suffer from overcharge and over-discharge. On the other hand, the PV
array responds dynamically to ambient conditions like
irradiance, temperature and other factors like shading.
Thus, directly coupling the battery to the PV array andthe loads is detrimental to the battery lifetime.
Therefore a device is needed that controls the currents
flowing between the battery, the PV array and the load
and that ensures that the electrical parameters present
at the battery are kept within the specifications givenby the battery manufacturer. These tasks are done bya charge controller, that nowadays has several different

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17.4. Batteries

DC

PV Modules

during severe winter days at low irradiance, the load
exceeds the power generated by the PV array, such thatthe battery is heavily discharged. Over-discharging thebattery has a detrimental effect on the cycle lifetime, as
discussed above. The charge controller prevents the battery from being over-discharged by disconnecting thebattery from the load.

Load

Chargecontroller

Batteries

Figure 17.30: Illustrating the position of the charge controller

in a generic PV system with batteries.

functionalities, which also depend on the manufacturer.
We will discuss the most important functionalities. A
schematic of its location in a PV system is shown in
Fig. 17.30.
When the sun is shining at peak hours during summer,
the generated PV power excesses the load. The excessenergy is sent to the battery. When the battery is fully
charged, and the PV array is still connected to the battery, the battery might overcharge, which can cause several problems like gas formation, capacity loss or overheating. Here, the charge controller plays a vital roleby de-coupling the PV array from the battery. Similarly,

Copyright Delft University of Technology, 2014

For optimal performance, the battery voltage has to bewithin specified limits. The charge controller can help
in maintaining an allowed voltage range in order toensure a healthy operation. Further, the PV array will
have its Vmpp at different levels, based on the temperature and irradiance conditions. Hence, the charge controller needs to perform appropriate voltage regulation
to ensure the battery operates in the specified voltage
range, while the PV array is operating at the MPP.
Modern charge controllers often have an MPP tracker
integrated.
As we have seen above, certain C-rates are used as battery specifications. The higher the charge/discharge
rates, the lower the coulombic efficiency of the battery.
The optimal charge rates, as specified by the manufacturer, can be reached by manipulating the current flowing into the battery. A charge controller that contains
a proper current regulation is also able to control the
C-rates. Finally, the charge controller can impose thelimits on the maximal currents flowing into and fromthe battery.

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298

If no blocking diodes are used, it is even possible thatthe battery can “load” the PV array, when the PV array is operating at a very low voltage. This means thatthe battery will impose a forward bias on the PV modules and make them consume the battery power, whichleads to heating up the solar cells. Traditionally, blocking diodes are used at the PV panel or string level toprevent this back discharge of the battery through the PV
array. However, this function is also easily integrated in
the charge controller.
We distinguish between series and shunt controllers,
as illustrated in 17.31. In a series controller, overcharging is prevented by disconnecting the PV array until aparticular voltage drop is detected, at which point thearray is connected to the battery again. On the other
hand, in a parallel or shunt controller, overcharging isprevented by short-circuiting the PV array. This meansthat the PV modules work under short circuit mode,and that no current flows into the battery. These topologies also ensure over-discharge protection using powerswitches for the load connection, which are appropriately controlled by the algorithms implemented into thecharge controller algorithm. As charge controllers area necessary component of of stand-alone PV systems,
they should have a very high efficiency.
As we have seen above, temperature plays a crucialrole in the functioning of the battery. Not only does
temperature affect the lifespan of the battery, but it also

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17. Components of PV Systems

changes its electrical parameters significantly. Thus,
modern charge controllers have a temperature sensor
included, which allows the charge controller to adjust
the electrical parameters of the battery, like the operating voltage, to the temperature. The charge controller
thus keeps the operating range of the battery within
the optimal range of voltages. The charge controller isusually kept in close proximity to the battery, such thatthe operating temperature of the battery is close to thatof the charge controller. However, when the battery isheavily loaded, the battery might heat up, leading todifferences between the temperature expected by the
charge controller and the actual temperature of the battery. Therefore, high end charge controllers also taketemperature effects due to high currents into account.

17.5

CablesThe overall performance of PV systems also is strongly
dependent on the correct choice of the cables. We therefore will discuss how to choose suitable cables. But westart our discussion with color conventions.
PV systems usually contain DC and AC parts. For correctly installing a PV system, it is important to know
the color conventions. For DC cables,• red is used for connecting the +-contacts of thedifferent system components with each other while

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299

17.5. Cables

(a)

ILoad

IPV

(b)

ILoad

IPV

Ibat

Ibat

PV
Array

Chargecontroller
core

Vbat

Load

PV
Array

Chargecontroller
core

Vbat

Load

Figure 17.31: Basic wiring scheme of (a) a series and (b) a shunt charge controller.

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300

17. Components of PV Systems• black is used for interconnecting the −-contacts.
RL

For AC wiring, different colour conventions are usedaround the world.

Rcable• For example, in the European Union,
blue is used for neutral,
green-yellow is used for the protective earth andbrown (or another color) is used for the phase.
• In the United States and Canada,
silver is used for neutral,
green-yellow, green or a bare conductor is used
for the protective earthand black (or another color) is used for the phase.

Figure 17.32: Illustrating a circuit with a load R L and cable
resistance Rcable . The connections drawn in between the components are lossless.

as• In India and Pakistan,
black is used for neutral,
green is used for the protective earth andblue, red, or yellow is used for the phase.
Therefore it is very important to check the standardsof the country in that the PV system is going to be installed.
The cables have to be chosen such that resistive losses areminimal. For estimating these losses, we look at a very
simple system that is illustrated in Fig. 17.32 The system consists of a power source and a load with resistance R L . Also the cables have a resistance Rcable , which
also is sketched. The power loss at the cables is given

Copyright Delft University of Technology, 2014

Pcable = I · ∆Vcable ,(17.31)

where ∆Vcable is the voltage drop across the cable,
which is given as∆Vcable = V

Rcable.
R L + Rcable(17.32)Using
we find

V = I ( R L + Rcable )(17.33)

Pcable = I 2 Rcable .

(17.34)Hence, as the current doubles, four times as much heat
will be dissipated at the cables. It now is obvious why
modern modules have connected all cells in series.

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301

17.5. Cables

Let us now calculate the resistance of a cable withlength ` and cross section A. It is clear, that if ` isdoubled, also Rcable doubles. In contrast, if A doubles,
Rcable decreases to half. The resistance thus is given by

ductivities areρCu = 1.68 · 10−8 Ω · m = 1.68 · 10−2 Ω
σCu = 5.96 · 107

Rcable = ρ

`
1 `
=
,
A
σA

(17.35)where ρ is the specific resistance or resistivity and σ is the
specific conductance or conductivity. If both, ` and A are
given in metres, their units are [ρ] = Ω · m and [σ] =S/m, where S denotes the unit for conductivity, whichis Siemens.
For electrical cables it is convenient to have ` in metresand A in mm2 . When using this convention, we findthe following units for ρ and σ:Sm= 59.6 S,
mmm2

ρAl = 2.82 · 10−8 Ω · m = 2.82 · 10−2 Ω
σAl = 3.55 · 107

mm2
,
m

mm2
,
m

S
m
= 35.5 S.
mmm2

Usual thicknesses for cables are 0.75 mm2 , 1.5 mm2 ,2.5 mm2 , 4 mm2 , 6 mm2 , 10 mm2 , 16 mm2 , 25 mm2 ,35 mm2 , et cetera. Since DC circuits are driven at lower
voltages than AC currents, the currents are higher, requiring thicker cables.

m2
106 mm2mm2= 1Ω= 106 Ω
,
m
m
m
S
m
m
m
1
= 1S 2 = 1Ω 6
= 10−6 Ω
.
2
m
m
10 mm
mm2

1Ω·m = 1Ω

The most widely used metals used for electrical cables
are copper and aluminium. Their resistances and con-

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302

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17. Components of PV Systems

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18

PV System Design
In the previous chapters we discussed how to estimate
the irradiance on a PV module in dependence of theposition on Earth, date and time (Chapter 16). Further
we introduced and discussed all the different components of PV Systems in Chapter 17.

the other components due to changing irradiance andweather conditions are taken into account. Since these
performance changes can be quite high, they can alter
the optimal system design considerably.
There are two main paradigms for designing PV systems. First, the system can be designed such that thegenerated power and the loads, i.e. the consumed
power, match. A second way to design a PV system
is to base the design on economics. We must distinguish between grid-connected and off-grid systems. As
we will see, grid-connected systems have very different
demands than off-grid systems.

In this chapter, we will put the things together and use
them such that we can design complete PV systems. Of
course, PV system designs can be made with different
levels of complexity. For a first approximation the STC
performance of the PV modules and the performanceof the other components (like the inverter) at STC conditions and the number of Equivalent Sun Hours (ESH)at the location of the PV system are sufficient. The notion of ESH will be discussed below. In a more detailed
approach, performance changes of the modules andThis chapter is organised as follows: First, as an example, we will discuss a design of a simple off-grid
303

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304

18. PV System Design

system in Section 18.1. After that we will take a moredetailed look on load profiles in Section 18.2. In Section
18.3 and on how weather and irradiance conditions
affect the performance of PV modules and BOS components, mainly inverters. Finally, in Sections 18.4 and18.5 we learn how to design grid-connected and offgrid systems, respectively.

18.1

A simple approach for designing oﬀ-grid systems

PV Modules
Charge controller

In this section, we will design a simple off-grid system,as depicted in Fig. 18.1. The design presented here isbased on very simple assumptions and does not takeany weather-dependent performance changes into account. Nonetheless, we will see the major steps that arenecessary for designing a system. Such a simple design
can be performed in a six step plan:

=
~

DC
AC

InverterLoads

1. Determine the total load current and operationaltime
2. Add system losses
3. Determine the solar irradiation in daily equivalent
sun hours (ESH)
4. Determine total solar array current requirements

Copyright Delft University of Technology, 2014

Batteries
Figure 18.1: Illustrating a simple off-grid PV system with AC

and DC loads (see also Fig. 15.1 b).

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305

18.1. A simple approach for designing oﬀ-grid systems

5. Determine optimum module arrangement for solararrayergy requirements of the devices expressed in Ah are calculated
as follows:

6. Determine battery size for recommended reserve
time

Device A: 15 W · 6 h = 90 Wh
Device B: 20 W · 3 h = 60 Wh

1. Determine the total load current and operational
time

Total: 90 Wh + 60 Wh = 150 Wh
150 Wh/12 V = 12.5 Ah

Before starting determining the current requirementsof loads of a PV system one has to decide the nominal
operational voltage of the PV system. Usual nominal
voltages are 12 V or 24 V. When knowing the voltage,
the next step is to express the daily energy requirements of loads in terms of current and average operational time expressed in Ampere-hours [Ah]. In caseof DC loads the daily energy [Wh] requirement is calculated by multiplying the power rating [W] of an individual appliance with the average daily operationaltime [h]. Dividing the Wh by the nominal PV system
operational voltage, the required Ah of the appliance isobtained.
ExampleA 12 V PV system has two DC appliances A and B requiring
15 and 20 W respectively. The average operational time per day
is 6 hours for device A and 3 hours for device B. The daily en-

Copyright Delft University of Technology, 2014

In case of AC loads the energy use has to be expressed
as a DC energy requirement since PV modules generate DC electricity. The DC equivalent of the energy useof an AC load is determined by dividing the AC loadenergy use by the efficiency of the inverter, which typically can be assumed to be 85%. By dividing the DC
energy requirement by the nominal PV system voltage
the Ah is determined.
ExampleAn AC computer (device C) and TV set (device D) are connected to the PV system. The computer, which has rated power
40 W, runs 2 hours per day and the TV set with rated power
60 W is 3 hours per day in operation. The daily energy requirements of the devices expressed in DC Ah are calculated asfollows:

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306
Device C: 40 W · 2 h = 80 Wh
Device D: 60 W · 3 h = 180 Wh
Total: 80 Wh + 180 Wh = 260 Wh
DC requirement: 260 Wh/0.85 = 306 Wh
306 Wh/12 V = 25.5 Ah

2. Add system losses

Some components of the PV system, such as charge regulators and batteries require energy to perform theirfunctions. We denote the use of energy by the systemcomponents as system energy losses. Therefore, thetotal energy requirements of loads, which were determined in step 1, are increase with 20 to 30% in order tocompensate for the system losses.

ExampleThe total DC requirements of loads plus the system losses(20%) are determined as follows:(12.5 Ah + 25.5 Ah) · 1.2 = 45.6 Ah

Copyright Delft University of Technology, 2014

18. PV System Design
3. Determine the solar irradiation in daily equivalent sun hours (EHS)

How much energy a PV module delivers depends on
several factors, such as local weather conditions, seasonal changes, and installation of modules. PV modules
should be installed under the optimal tilt-angle in orderto achieve best year-round performance. It is also important to know whether a PV system is expected to be
used all-year round or only during a certain period ofa year. The power output during winter is much lessthan the yearly average and in the summer months thepower output will be above the average.
In the PV community, 1 equivalent sun means a solar
irradiance of 1000 W/m2 . This value corresponds tothe standard, at which the performance of solar cells
and modules is determined. The rated parameters ofmodules (see for example Table 17.1) are determined atsolar irradiance of 1 sun.
When solar irradiation data are available for a particular location than the equivalent sun hours can be determined. For example, in the Netherlands the averageannual solar irradiation is 1000 kWh/m2 . According
to the AM1.5 spectrum, 1 sun delivers 1000 W/m2 = 1
kW/m2 . Hence, the Dutch average annual solar irradiation can be expressed as1000 kWh · m−2= 1000 h,
1 kW · m−2

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18.1. A simple approach for designing oﬀ-grid systems

307

Figure 18.2: The average global horizontal insolation of the World given in kWh/year and PSH/day [101]. Note that this map onlyshows latitudes in between 60° N and 60° S.

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308

18. PV System Designi.e. 1000 equivalent sun hours, which corresponds to
2.8 h per day. Figures 18.2 and 18.3 show the global
horizontal irradiation given in units of daily equivalent
sun hours (ESH) and in kWh/year. Note that these figures show horizontal values, meaning that a PV module would get this irradiance if it would be flat. For atilted module, the annual irradiance can be considerably higher, as we have seen in Chapter 16.

4. Determine total solar array current requirements

The current that has to be generated by the solar array is determined by dividing the total DC energy requirement of the PV system including loads and systemlosses (calculated in step 2 and expressed in Ah) by the
daily equivalent sun hours (determined in step 3).

ExampleThe total DC requirements of loads plus the system losses
are 45.6 Ah. The daily EHS for the Netherlands is about 3
hours. The required total current generated by the solar arrayis 45.6 Ah / 3 h = 15.2 A.

Copyright Delft University of Technology, 2014

Figure 18.3: The average global horizontal insolation of theNetherlands given in kWh/year and PSH/day [101].

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18.1. A simple approach for designing oﬀ-grid systems
5. Determine optimum module arrangement forsolar array

Usually, PV manufacturers produce modules in a whole
series of different output powers. In the optimum arrangement of modules the required total solar arraycurrent (as determined in step 4) is obtained with theminimum number of modules. Modules can be either
connected in series or in parallel to form an array.
When modules are connected in series, the nominal
voltage of the PV system is increased, while the parallel
connection of modules results in a higher current.
The required number of modules in parallel is calculated by dividing the total current required from thesolar array (determined in step 4) by the current generated by module at peak power (rated current in thespecification sheet). The number of modules in series isdetermined by dividing the nominal PV system voltage
with the nominal module voltage (in the specification
sheet under configuration). The total number of modules is the product of the number of modules requiredin parallel and the number required in series.
ExampleThe required total current generated by the solar array is
15.2 A. We have Shell SM50-H modules available. The specification of these modules is given in Table 17.1. The rated

Copyright Delft University of Technology, 2014309
current of a module is 3.15 A. The number of modules in parallel is 15.2 A / 3.15 A = 4.8 < 5 modules. The nominal
voltage of the PV system is 12V and the nominal module
voltage is 12 V. The required number of modules in series thus
is 12 V / 12 V = 1 module. Therefore, the total number ofmodules in the array is 5 · 1 = 5 modules.

6. Determine battery size for recommended reserve
time

Batteries are a major component of stand-alone PV systems. The batteries provide load operation at night orin combination with the PV modules during periodsof limited sunlight. For a safe operation of the PV system one has to anticipate periods with cloudy weather
and plan a reserve energy capacity stored in the batteries. This reserve capacity is referred to as PV systemautonomy, which means the period of time that the system is not dependent on energy generated by PV modules. It is given in days. The system autonomy depends
on the type of loads. For critical loads such as telecommunications components the autonomy can be 10 days
and more, for residential use it is usually 5 days or less.
The capacity [Ah] of the batteries is calculated by multiplying the daily total DC energy requirement of the
PV system including loads and system losses (calculated in step 2 and expressed in Ah) by the number ofdays of recommended reserve time. In order to prolong

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310

18. PV System Design

Table 18.1: Worksheet for designing a simple off-grid PV system based on rough assumptions.
Daily DC loads requirements
DC load

W×

h =

Wh

1

Daily DC energy use (DC loads)

1

Daily DC energy use (AC loads)+

1

Daily DC energy use (all loads)=

PV system nominal voltage/

2

Total DC loads energy use:h =Daily Ah requirements (system)=

Design EHS/

4Total solar array current

=

5

W×

=
×

3

Daily AC loads requirements
DC loadDaily Ah requirements (all loads)
Add PV system losses

Wh

Select module type
Module rated current/

Number of modules in parallel=

PV system nominal voltage

6

Modules nominal voltage/

Number of modules in series=

Number of modules in parallel×

Total number of modules=

Determine battery capacity

Total AC loads energy use:

Daily Ah requirements (system)/0.85 = DC energy requirement

Recommended reserve time×

Usable battery capacity/

Minimum battery capacity=

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18.2. Load profiles

664

(a)

(b)

Luis Castañer et al.

Luis Castañer et al.

(c)

311

(d)

4 (a)load
Individual
having a momentary
andrunningand
currents
and a numFigure 18.4: Illustrating different load profiles.
Figure 4 (a) Figure
Individual
havingloada momentary
and runningcurrents
a number n of occurrences,(b)a load
usedofa numberof day,hours(c)perday, (c) coincident
loads,
ber n of occurrences,(b) load used
numberhours percoincident
loads,and (d) noncoincident
loads. from
Elaborated
from IEEE1144-1996.
standard 1144-1996.
and (d) noncoincident
loads. Elaborated
IEEE standard

the life of the battery it is recommended to operate the18.2 Load profiles
battery using only 80% of its capacity. Therefore, theminimal capacity of the batteries is determined by diNow we take a look at the load profile. Figure 18.4 il4(c)respectively).
and (d), respectively).
Therealso be load profiles
viding the required capacity by a factor of(see0.8.Figure(see4(c)Figure
andlustrates(d),There thatmay
also may
be loaddifferent shapes
loads
can
have.profiles(a) Aa longerof periodof time24thathours
thatto needrequiring arequiring
longer simple
period24 thanhours
bea to beload time
drawsthana constantamountofneedpower
forFigure 4 (a) Individual load havingdescribed;
a momentary
andrunning
currents
and aandnum-a maximum daily load should be
oad having a momentary and running currents and a numin
this
case,
an
average
described; ina number
this case,an averageandHowever,a maximum
daily load powershould doesbecertaintime. (b)the consumed
Exampleload used a number of hours
per day, (c) coincident loads,
ber n of occurrences, (b) load usedgiven.
of hours per day, (c) coincident loads,
not
need
to
be
constant
but
can
also
show
peaks
that
given.
ads. Elaborated from IEEE standard 1144-1996.
and (d) noncoincident loads. Elaborated from IEEE standard 1144-1996.
The total DC requirements of loads plus the system losses
This information allows
the derivationof appliances
the load profile,at least
to switching
onleastororoff.
This information correspondallows the derivationof electricalthe load profile,or at
are 45.6 Ah. The recommended reserve time capacity forthe typicalAload
profile fora 24-hour-periodload. This loads
is performedbyhouseholdof course
hasseveralthat
theBattery
typical load profile
for a 24-hour-periodload.
This different
is performed by
the installation side in the Netherlands is 5 days.
multiplying
load
current on
by atthethedurationand (coincident)
then adding orall the(c)eachcan beswitchedsame time
capacityrequiredtheprofiles
system is 45.6 Ah · 5 multiplying
Ah.each load current by the duration and then adding all the(d), respectively). There
may also
be byload(see Figure 4(c) and=(d), 228
respectively).
beloadprofiles load. If the duration of the(d) findat maydifferent
times(non-coincident).
products There
tothealsodailyampere-hourTheminimalbattery
capacityforasafe
operation
thereforeisproducts
finddailyampere-hourload.
If the duration of theperiod of time than 24 hours that need to requiring
bea longer
period toofmomentary
time thethan load
24 is
hours
that need
to 1benotknown,
thenminuteshould be
taken
228 Ah/0.8 = 285 Ah.
Analysingload
profilescanbeperformed
within- as recommomentary
load
is
not
known,
then1
minuteshouldbetaken
asrecome, an average and a maximum daily load shoulddescribed;
bein this case, an averageandamaximum
daily
load
shouldbemended increasingreference[3]. The andloadhence
profileaccuracy.
also provides
the necessary
complexity
The
simplest
mended ininformationreference [3].
profiledischarge
also provides
theneednecessary
given.
aboutThe
thetoload
maximumrates
thattobasis.
be sustained
methodisdetermine
the
loads
on
a
24-hourTo
allows the derivation of the load profile, or at leastThis information
allows thederivationof the load discharge
profile, or rates
at least
informationabout
the maximumthat need to be sustainedby
thebattery.
DesigningasimplePVsystemas
described
in
this
secdo
this,
an
arbitrary
day
can
be
taken
and
the
electrie for a 24-hour-period load. This is performedtheby typical load profile
for a 24-hour-period load. This is performed bybyasthetion cancarriedoutallusing
a worksheet
in battery.
Tablecity consumption can be monitored. However, several
d current by the duration
andbethenaddingthemultiplying each load current2.1
by theduration Consumption
and then adding all theElectricity
LightingSeveral
and Electrical
18.1,
where
the
PV
systemdesignrules
aresummarised.
loads do not fit in suchbya scheme.
examples fordaily ampere-hour load. If the duration of productstheto find 2.1
the dailyampere-hourload.
If theby
durationof theElectricity
Consumption
Lighting
andElectrical
Appliancest known, then 1 minute should be taken as recommomentary load isAppliances
not known, then 1 minute should be taken as recomgives aalso
summary
the necessary
load values for the main domestic appli[3]. The load profile also provides the necessary
mended in reference [3]. TheTable
load 1profile
providesofthe
Table
1
gives
a
summaryoftheload
valuesfor theconsumption
main domestic
appli- in the
maximum discharge rates that need to be sustained
ances.
The
principal
results
onenergyfor lighting
information
be copysustained
Copyright Delft University of Technology,
2014about the maximum discharge rates that need toThis
is provided for free, for personal use only.
ances.
The
principal
results
onenergyconsumptionforlighting
intheby the battery.

sumption by Lighting and Electrical

312

18. PV System Design

this already were treated in Section 18.1. For example,
washing machines and dishwashers do not fit in a 24hour scheme because typically they are only used several times in a week. Additionally, several loads areseasonal in nature, for example, air conditioning or
heating, in case this is performed with a heat pump.
Therefore it is advisable to look at load profiles for awhole year.
The total energy consumed in a year is given by
EYL =

Z
year

PL (t) dt.

(18.1)

It is expressed in kWh/year.

18.3

Meteorological eﬀects

Standard Test Conditions (STC) of Photovoltaic (PV)modules are generally not representative of the realworking conditions of a solar module. For example,
high levels of incident irradiation, may cause the temperature of a module to rise many degrees above the
STC temperature of 25°C, therefore lowering the module performances. For example, in the in a climate suchas the one of the Netherlands, real operating conditions
for PV systems correspond to relatively low levels of irradiance combined with a cold and windy weather. In
order to effectively estimate real time energy production

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from a given set of ambient parameters, it is essential todevelop an accurate thermal model of the solar module,
which evaluates the effects of the major meteorologicalparameters on the final temperature of a solar module.
In this section, we will develop an accurate thermal
model for estimating the PV module working temperature as a function of meteorological parameters will
be developed. The model is based on a detailed energybalance between the module itself and the surrounding environment. Both the installed configuration of thearray together with external parameters such as direct
incident solar irradiance on the panels, wind speed andcloud cover will be taken into account. From the calculated equilibrium temperature, the overall efficiency of
the PV array will be calculated by separately assessing
the temperature and irradiance effect on the efficiency
at STC.

18.3.1

Simplified thermal model for a
photovoltaic array

The temperature strongly influences the performancesof a PV module [102]. While the level of incident irradiation can be easily determined by measuring it witha pyranometer, the temperature reached inside the cellis much harder evaluate. In order to give an estimateof the average operating temperature of the module,
olar manufacturers provide, together with rated per-This copy is provided for free, for personal use only.

313

18.3. Meteorological eﬀects

ence between the module temperature and the ambient
temperature ( TM − Ta ), where the NOCT is used as areference point, [103]TM = Ta +TNOCT − 20◦
GM .
800

(18.2)

Equation (18.2) is however too simplistic and thus leads
to significant errors in the modules temperature evaluations. The reason is that it does not take environmental conditions such as wind speed and mounting
configuration of the array into account [103].

Figure 18.5: Rise in temperature above the ambient level withrespect of increasing solar irradiance [103].

formances at STC, also the so called Nominal Operating
Cell Temperature (NOCT). This value corresponds to theperformances of a cell under an irradiance level of 800
W/m2 , ambient temperature of 20°C and an external
wind speed of 1 m/s [102].
Simplified steady state models use a linear relationship between the solar irradiance G M and the differ-

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In order to take into mounting configuration of themodule into account, the Installed Nominal Operating Conditions Temperature (INOCT) has been defined[104]. This value is described as the cell temperatureof an installed array at NOCT conditions. Its value cantherefore be obtained from the NOCT and the mounting configuration. The evaluation of how the INOCT
varies with the module mounting configuration has
been experimentally determined by measuring the
NOCT at various mounting heights. The results canbe found in literature and are here summarised in the
Table 18.2.
Evaluating the influence of external meteorologicalparameters on the the module temperature is more
complex and has been approached by developing a detailed thermal energy balance between the module andthe surroundings. Here, the module is assumed to be aThis copy is provided for free, for personal use only.

314

18. PV System Design

Table 18.2: Derivation of the INOCT from the NOCT forvarious mounting configurations [104].

Rack Mount
Direct Mount
Standoﬀ

INOCT = NOCT − 3◦ C
INOCT = NOCT + 18◦ C
INOCT = NOCT + X

where X is given by
W (inch)1
36

X (°C)
112− 1

ule surface and the sky44etop σ TM− Tsky,where etop = 0.84 is the emissivity of the module
front glass and σ is the Stefan-Boltzmann constantas defined in Eq. (5.20). and between the rear surface and the ground
4
4
eback σ TM
− Tgr.
,
where the emissivitiy of the back is assumed to beeback = 0.89.

single uniform mass at temperature TM . The three typesof heat transfer between the module itself with the surroundings are conduction, convection and radiation. The
contributions considered in the model are:• Conductive heat transfer between the module andthe mounting structure. We neglect this contribution due to the small area of the contact points[105].

• Heat received from the Sun in the form of insolation ϕG M , where ϕ is the absorptivity of the module.

By separately considering each of the above mentioned
contributions, we write down the balance for the heattransfer, [104]• Convective heat exchange with surrounding air
from the front and rear side of the modulehc ( TM − Ta ),where hc denotes the overall convective heat transfer coefficient of the module.
• Radiative heat exchange between the upper mod-

Copyright Delft University of Technology, 2014

mc



dTM
4
4
= ϕGM − hc ( TM − Ta ) − eback σ TM
− Tgr.
dt44− etop σ TM
− Tsky.

(18.3)
Before providing a solution for this differential equation, it is appropriate to remark the fact that we areconsidering the entire module as a uniform piece at aThis copy is provided for free, for personal use only.

315

18.3. Meteorological eﬀects

qrad, sky

the cell to heat flow and therefore justifies the uniform
temperature approximation.

Tskyqconvection

Ta

qsun

qrad, grounda+θ

qconvection
a

Tgr.

TM
θ

Figure 18.6: Representation of heat exchange between a tilted

module surface and the surroundings.

temperature TM . However, this is not entirely realistic
since modules are made of various layers of different
materials surrounding the actual solar cells. It is the
purpose of this section to evaluate the temperature of
the inner cell, which is the place where absorption ofsolar radiation effectively place. This temperature will
be to some extent higher than the surface module temperature TM due to the heat produced in the cell due
to light absorption. The approximation of considering
the temperature uniform throughout the module layers
however is justified because of the relatively low thickness of the active cell together with the low heat capacity of the cell material compared to the other layers[105]. This results into a very low thermal resistance of

Copyright Delft University of Technology, 2014

In addition to this, a steady state approach will here beconsidered, meaning that the module temperature will
not change over time for each of the 10-minute time
steps. In reality, the temperature follows an exponential
decay lagging behind variations in irradiation level. It
is defined as Time Constant of a module, which is ‘thetime it takes for the module to reach 63% of the totalchange in temperature resulting from a step change in
irradiance’ [105]. Time constants for PV modules are
generally of the order of approximately 7 minutes. Fortime steps greater that the Time Constant, as it is in ourcase, the module can be approximated as being in a
steady state condition. In light of this assumption, theterm on the left hand side of Eq. (18.3) vanishes.
It is now possible to proceed with the solution of thethermal energy balance equation. The formula can belinearised by noticing that 
a4 − b4 = a2 + b2 ( a + b)( a − b).

(18.4)

Since



2
2
TM
+ Tsky( TM + Tsky )changes less than 5% for a 10°C variation in TM , wecan consider this term to be constant when T M varies[104]. Therefore the energy balance can be simplified

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316

18. PV System Design

18.3.2

becoming linear with respect to T M . By defining22
hr, sky = etop σ TM
+ Tsky( TM + Tsky ),22+ Tgr.
hr, gr. = eback σ TM
( TM + Tgr. ),(18.5a)(18.5b)we can rewrite Eq. (18.3) and findϕG M −hc ( TM − Ta ) − hr, sky ( TM − Tsky )−hr, gr. ( TM − Tgr. ) = 0.

(18.6)

By rearranging the terms, the formula can be explicitly
expressed as a function of T M ,TM =ϕG + hc Ta + hr, sky Tsky + hr, gr. Tgr.
hc + hr, sky + hr, gr.

(18.7)

However, since hr, gr. and hr, sky are also function of themodule temperature, the equation needs to be solved
iteratively: an initial module temperature is assigned
and by hr, gr. and hr, sky are updated each iterations. A
nearly exact solution can be obtained after 5 iterations.
Before solving iteratively Eq. (18.7) there are still many
unknown variables that need to be determined, which
we will do in the following sections.

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Calculating the convective transfer
coeﬃcients

Convection is a form of energy transfer from one place
to another caused by the movement of a fluid. Convective heat transfer can be either free or forced dependingon the cause of the fluid motion.
In free convection, heat transfer is caused by temperature differences which affect the density of the fluid
itself. Air starts circulating due to difference in buoyancy between hot (less dense) fluid and cold (denser)
fluid. A circular motion is therefore initiated with rising
hot fluid and sinking cold fluid. Free convection only
takes place in a gravitational field [106].
Forced convection, on the other hand, is caused by afluid flow due to external forces which therefore enhance the convective heat exchange. The heat transfer
depends very much on whether the induced flow overa solid surface is laminar or turbulent. In the case of
turbulent flow, an increased heat transfer is expectedwith respect to the laminar situation. This fact is due
to an increased heat transport across the main directionof the flow. On the contrary, in laminar flow regime,
only conduction is responsible for transport in the crossdirection. For this reason, forced convection is always
studied separately in the laminar and turbulent regime[106]. The overall convection transfer is made of thetwo relative contribution for free and forced compon-

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317

18.3. Meteorological eﬀects

ents. Mixed convective coefficient can be obtained bytaking the cube root of the cubes of the forced and convective coefficients according to the equation [102]h3mixed = h3forced + h3free .

(18.8)

Since the convective heat transfer coefficients will bedifferent on the top and rear surface of the module, determining the total heat transfer coefficient has to be
decoupled between the top hcT and rear hcB surfaces. The
overall heat transfer will eventually be determined by
the sum of the two components.

Convective heat transfer on the top surface

0.86 Re−0.5ρcair w,
Pr0.670.028 Re−0.2
hturb.
ρcair w.
forced =Pr0.4

Copyright Delft University of Technology, 2014

Re =(18.9a)(18.9b)wDh,ν

(18.10)where w is the wind speed at the height of the PV array,
Dh is the hydraulic diameter of the module, which is used
as relevant length scale, and ν is the kinematic viscosityof air. Pr is the Prandtl number which is the ratio of the
momentum diffusivity to the thermal diffusivity. It isconsidered to be 0.71 for air. Finally, ρ and cair are thedensity and heat capacity of air, respectively. The hydraulic diameter of a rectangle of length L and widthW, and thus of the PV module, is given as
Dh =Convective heat transfer has to be distinguished in free
and forced components. For the forced component wefurther have to distinguish between laminar and turbulent flow. We obtain for the laminar and turbulent
convective heat transfer coefficients

hlam.
forced =Re is the Reynolds number that expresses the ratio of of
the inertial forces to viscous forces,

2LW
.
L+W

(18.11)Figure 18.7 shows the variation of the forced heat transfer coefficient at various wind speeds. We notice anoverall increase in the heat transfer with increasing
wind speeds due to the increased force transfer component. There are two different regions in the graphwhich represent the laminar and turbulent regimes,
respectively. The laminar flow extends till around 3
m/s and is characterised by a lower convective heatexchange compared to the turbulent regime.
In a good approximation, hforced and w are proportional

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318

18. PV System Design

to each other and we may write

12
hforced (Wm-2 K-1 )(18.12a)0.8
hturb.
forced ' w .

(18.12b)

To determine the free heat transfer coefficient we only
need to utilise the dimensionless Nusselt number Nu,
which expresses the ratio between the convective andconductive heat transfer [104]

14

10
8

Nu =

6

hfree Dh
= 0.21(Gr · Pr)0.32 ,k(18.13)where k is the heat conductivity of air and Gr is theGrasshoff number that is the ratio between the buoyancy
and viscous forces,

4
2
0

0.5
hlam.
forced ' w ,

0

1

2

3

4

5

6

7

8

Wind speed @ 10 m (m/s)

Figure 18.7: Forced convective heat transfer coefficient withincreasing wind speeds

Gr =gβ( T − Ta ) Dh3.
ν2(18.14)Here, g is the acceleration due to gravity on Earth and β
is the volumetric thermal expansion coefficient of air,which can be approximated to be β = 1/T.
With the value of both free and forced coefficient we
can calculate the total mixed heat convective masstransfer coefficient using Eq. (18.8).
qhmixed = hcT = 3 h3forced + h3free .

Copyright Delft University of Technology, 2014(18.15)This copy is provided for free, for personal use only.

319

18.3. Meteorological eﬀects
Convective heat transfer coeﬃcient on the rear surface of the module.

Convection on the back side of the module will belower than on the top because of the mounting structure and the relative vicinity to the ground. For example, a rack mount configuration, which is approximately installed at 1 m height, will achieve a larger
heat exchange rate than a standoff mounted array thatis mounted 20 cm above the ground. We model the effect of the different mounting configurations by scaling
the convection coefficient obtained for the top of themodule. We determine the scaling factor by performing
an energy balance at the INOCT conditions [104],

The back side convection is therefore given by
hcB = R · hcT .

(18.19)We therefore find the overal convective heat transfer
coefficient to behc = hcT + hcB .
(18.20)18.3.3

Other parameters

Sky temperature evaluation(18.16)
We define R as the ratio of the actual to the ideal heatloss from the back side,

The sky temperature can be expressed as a function of
the measured ambient temperature, humidity, cloud
cover and cloud elevation [104]. On a cloudy day, usually when the cloud cover is above 6 okta1 , the sky
temperature will approach the ambient temperature,
Tsky = Ta [105]. However, on a clear day the sky temperature can drop below Ta and can be estimated by44 )hcB ( TINOCT − Ta ) + eback σ( TINOCT− Tgr.

Tsky = 0.0552 · Ta3/2 .

ϕG M − hcT ( TINOCT − Ta ) − hr, sky ( TINOCT − Tsky ) =hcB ( TINOCT − Ta ) + hr, gr. ( TINOCT − Tgr. )

R=

4
4 )
hcT ( TINOCT − Ta ) + etop σ( TINOCT− Tsky.

(18.17)Wind speed at module height evaluation

Substituting this into Eq. (18.16) at INOCT conditions
yieldsR=

44 )ϕG M − hcT ( TINOCT − Ta ) − etop σ( TINOCT− Tsky44 )hcT ( TINOCT − Ta ) + etop σ( TINOCT− Tsky.

(18.18)

Copyright Delft University of Technology, 2014(18.21)Since the anemometer used for the evaluation of thewind speed is at a higher height than the module array,
1 Okta is a measure for the cloud cover, where 0 is clear sky and 8 is a completely cloudy sky.

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320

18. PV System Designw = wr

yM
yr

1

5

,

(18.22)where y M and yr denote the module and anemometerheights, respectively [104].
The elevation factor of 1/5 is determined by the landscape surrounding the installation, which is supposed
to be open country.

56
Module temperature (°C)

the real wind speed experienced by the module will bescaled down with

52
48
44
40
0

1

2

3

4

5

6

7

8

Wind speed @ 10 m (m/s)

Absorptivity and emissivity of the module

Figure 18.8: Influence of the wind speed on the temperat-

In the thermal model developed, absorptivity is definedas the fraction of the incident radiation that is converted into thermal energy in the module. This value islinked to the reflectivity R and efficiency of the moduleby the equation [104]ϕ = (1 − R)(1 − η ).

(18.23)

Typical value for reflectivity of solar modules are 0.1[104]. As far as the emissivity is concerned, a value of0.84 has been used for the front glass surface and 0.89
for the back surface [107].
Now that all the unknown variables have been determined it is possible to the make iterations to calculate the

Copyright Delft University of Technology, 2014

ure of a solar module at a fixed radiation of 1000 W/m2 andambient temperature of 25°C on a clear day.

final temperature of the module as a function of thelevel of irradiance, wind speed and ambient temperature.

18.3.4

Evaluation of the thermal model

In the previous section, all the variables that are required to solve the thermal model have been derived.
Therefore, the thermal model now can be solved via it-This copy is provided for free, for personal use only.

321

18.3. Meteorological eﬀects

eration.

In Fig. 18.9 the thermal model and the NOCT model
presented in Eq. (18.2) are compared to each other in
dependence of the incident radiation for 1 m/s windspeed. We see that for 1000 W/m2 the NOCT model
predicts a module temperature of 54.3°C regardless of
the wind speed, which is much above the values obtained with the thermal model. The contribution offree convective and radiative heat exchange lower themodule temperature with respect to the NOCT model.
The difference between the two models is more pronounced at low levels of radiation, but for irradianceshigher than 1000 W/m2 the two curves start diverging
again. This can be understood when we consider that
the NOCT model has been developed as a linearisation of the temperature/irradiation dependency aroundthe NOCT conditions [103]. The accuracy of the model
therefore decreases with the increasing distance fromthe NOCT.

Copyright Delft University of Technology, 2014

56
52
Module temperature (°C)

Figure 18.8 below shows the effect of the wind speed
on the module temperature at a fixed level of irradiance(1000 W/m2 ) and an ambient temperature of 25 °C on
a clear day. It is obvious that the module temperature
is significantly above the value of 25 °C for all windspeeds. Thus, STC are not representative for real operating cell conditions. Two different regions are visible in
the graph corresponding to laminar and turbulent flow,
respectively.

48
44
40
36
32
28

Thermal model @ 1 m/s
NOCT model

24
20

0

200

400

600

800

1000

Incident irradiance (Wm-2 )Figure 18.9: The module temperature calculated with the

NOCT and the thermal model in dependence of the incident
irradiance for 1 m/s wind speed.

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322

18. PV System Design

18.3.5Eﬀect of temperature on the solar
cell performance

The effect of a module temperature deviating from the
25°C of STC is expressed by the temperature coefficients that given on the data sheet provided by solar
manufacturers. When knowing the temperature coefficient of a certain parameter, its value at a certain temperature TM can be estimated with

η ( TM , GSTC ) =Pmpp ( TM , GSTC ).
GSTC(18.27)

If the efficiency temperature coefficient ∂η/∂T is not
given in the datasheet, it can be obtained by rearranging
η ( TM , GSTC ) = η (STC) +Current I∂Voc(STC)( TM − TSTC ),∂T(18.24)∂Isc
Isc ( TM , GSTC ) = Isc +(STC)( TM − TSTC ), (18.25)∂T∂Pmpp
Pmpp ( TM , GSTC ) = Pmpp +(STC)( TM − TSTC ),∂T(18.26)
Voc ( TM , GSTC ) = Voc +higher irradiance GM
highertemperature T

Voltage V
Figure 18.10: Effect of a temperature increase on the I-V solar
cell characteristic.

∂η(STC)( TM − 25◦ C). (18.28)∂T

An increase in the solar cell temperature will shift theI-V curve as shown in Fig. 18.10. The slight increase

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323

18.3. Meteorological eﬀects

in the short circuit current at higher temperatures iscompletely outweighed by the decrease in open circuit
voltage. The overall effect is of a general linear decreasein the maximum achievable power and therefore a decrease in the system efficiency and fill factor. This effectis due to an increase of the intrinsic carrier concentration at higher temperatures which in turns leads to anincrease of the reverse saturation current I0 , which represents a measure of the leakage of carriers across thesolar cell junctions, as we have seen in Chapter 8. The
exponential dependence of I0 from the temperature is
the main cause of the linear reduction of Voc with thetemperature,

18.3.6

Eﬀect of light intensity on the solar
cell performance

Intuitively, performances of a solar cell decrease considerably with decreasing light intensity incident on themodule with respect to STC. The evaluation of the extent of this reduction is however less straightforward
than for the case of the temperature since solar manufacturer often do not explicitly provide a reduction
factor of the efficiency at every light intensity level.
By definition the efficiency is given by
η=

Voc =

kT
ln
e



Isc
I0


.

(18.29)Isc Voc FF
GM(18.30)The maximum variation of the FF for light intensity
between 1 and 1000 W/m2 is about 2% for CdTe, 5%
for a-Si:H, 22% for poly-cristalline silicon, and 23% formono-crystalline silicon [108].
The short circuit current of a solar cell is directly proportional to the incoming radiation,

On the other hand, the slight increase in the generated
current is due to a moderate increase in the photogenerated current resulting from an increased numberof thermally generated carriers. The overall reduction of
power at high temperature shows that cold and sunnyclimates are the best environment where to place a solar
system.

Copyright Delft University of Technology, 2014

Isc ' λG M ,(18.31)

where λ is simply a constant of proportionality. By expressing Voc as in Eq. (18.29), the efficiency can be written as
η ' FF λkT(ln GM + ln λ − ln I0 ).
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324

18. PV System Design

18.3.7

By defining
kT
,
e
kT
b = FF λ (ln λ − ln I0 ),e
a = FF λ(18.33a)(18.33b)the efficiency can be finally written asη (25◦ C, G M ) = a ln G M + b

Overall module performance

By combining the two effects of temperature and lightintensity, the final efficiency of the module at every
level of irradiance and temperature can be determinedas [109]
η ( TM , G M ) = η (25°C, G M ) [1 + κ ( TM − 25°C)] , (18.39)(18.34)whereκ=

The values of the coefficients a and b are device specificparameters and are rarely given by the manufacturer.
The overall trend of the efficiency is represented by astraight line on a logarithmic scale [108].
From this model the values of Isc , Voc and the efficiency
at a irradiance level G M can be determined from the
STC as follows
ln G M,(18.35)ln GSTC
G
Isc (25°C, G M ) = Isc (STC) M ,(18.36)GSTC
PMPP (25°C, G M ) = FF Voc (25°C, G M ) Isc (25°C, G M ),(18.37)
Voc (25°C, G M ) = Voc (STC)η (25°C, G M ) =PMPP (25°C, G M ),A M GM

where A M is the module area.

Copyright Delft University of Technology, 2014(18.38)∂η
1.
∂T η (SCT)(18.40)

Typical values for κ are −0.0025/◦ C for CdTe,−0.0030/◦ C for CIS, and −0.0035/◦ C for c-Si [109].
All the other parameters such as Isc , Voc , and Pmpp canalso be evaluated at every level of irradiance and module temperature by simply adapting Eq. (18.39) to thecorresponding coefficients and parameters.
In Fig. 18.11 the difference between η (25°C, G M ) andη ( TM , G M ), which takes both irradiance G M and module temperature TM into account, is shown. The graphhas been derived by keeping the wind speed constantat a value of 1 m/s and by determining TM iterating
the heat thermal model at each level of irradiance fora constant ambient temperature of 25°C. At low levelsof irradiance no effects of the module temperature areobserved. At higher level of incident light intensity thedifference between the two curves becomes more pronounced. since only ηmpp ( G M , TM ) takes into accountThis copy is provided for free, for personal use only.

325

18.3. Meteorological eﬀects
14.0

Module efficiency (%)

13.8

16

Module efficiency (%)

14
12

13.4
13.2
13.0
12.8
12.6

GM = 200 Wm-2
400 Wm-2
1000 Wm-2

12.4
12.2

10

12.0

8
6

0

1

2

3

4

5

6

7

8

Wind speed @ 10 m (m/s)

4

Figure 18.12: Overall efficiency of the module with increasing

DC efficiency @ Tmod ≡ 25◦ C
DC efficiency @ Tmod

2
0

13.6

0

200

400

600

800

wind speed at various irradiance levels.
1000

Incident irradiance (Wm-2 )Figure 18.11: Comparison of η (25°C, G M ) and η ( TM , G M )in dependence of the light irradiance at a wind speed of 1 m/s
and ambient temperature of Ta = 25◦ C for a c-Si solar cell.

the marked reduction in efficiency resulting from therising TM .
Figure 18.12 shows the overall efficiency at various
light intensities in dependence of the wind speed. Once
more, the beneficial effect of turbulent motion in cooling the module is reflected by an increase of efficiency
with the wind speed.
Finally, the power output of a module before the BOS isgiven by
PDC = η ( G M , TM ) · G M · A M ,(18.41)

Copyright Delft University of Technology, 2014

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326

18. PV System Design

where Atot is the total module area. It is related to thearea of one module A M via

and the power output at STC is given by
PSTC = η ( GSTC , 25°C) · GSTC · A M ,(18.42)
Atot = NT · A M ,(18.45)The energy yield of the DC side then is defined asP
η ( G M , TM ) · G M
YDC = DC · 100% =· 100%.
PSTC
η ( GSTC , 25°C) · GSTC
(18.43)

18.4

Designing grid-connected PVsystem

Now we can use all the things learned in this and theother chapters for actually designing a PV system. For
designing the PV system we use the energy balance approach, meaning that we design the system such thatthe generated energy and the consumed energy during
one year match. Of course, there are also other waysof designing systems for example based on economic
arguments.
For the energy balance we first need to calculate theannual load, which already happened in section 18.2.
The energy yield at the DC side is given byY
EDC= AtotZ
year

G M (t)η (t) dt,

Copyright Delft University of Technology, 2014(18.44)

where NT is the number of modules. The energy balance now writes asY
EDC= EYL · SF,(18.46)

where SF is a sizing factor that usually is assumed to be
1.1. We therefore can calculate the required number ofmodules,&'
EYL · SFR
NT =,(18.47)A M · year G M (t)η (t) dt
where d x e denotes the ceiling function, i.e. the lowest
integer that is greater or equal than x.
Now it is important to decide how many modules areto be connected in series ( NS ) and in parallel ( NP ). Of
course,NT = NS · NP .
(18.48)Such a PV array hence consists of P strings of S modules each. The NT determined in Eq. (18.86) not necessarily needs to be an even number. For example, if
NT = 11, you might want to choose for NT = 12 panels,
because you can install them as S × P= 12 × 1, 6 × 2,
4 × 3, 3 × 4, 2 × 6 or 1 × 12 strings. In principle, it isThis copy is provided for free, for personal use only.

327

18.4. Designing grid-connected PV-systempreferable to connect as many modules as possible inseries since then the currents on the DC side and hence
the cable losses stay low. Further, thinner cables can bechosen. However, many modern inverters can connect
two or even more individual strings, which can be important if the installation contains two or more areas
with different shading, for example on two different
sides of a roof. Further, it can be chosen to connect different strings to different inverters as well, such that thesystem uses several string inverters. NS and hence thevoltage at the inverter is also restricted by the chosen
inverter typ — or, if seen the other way round, the inverter has to be chosen such that its operating voltage
fits well to the string voltage.
In a conservative assumption, the power on the DCside at STC now is given as
STC
STC
.
PDC
= NT × PMPP

(18.49)The inverter must be chosen such that its maximal
inv
power PDC,
max is above the maximal PV output,
inv
STC
PDC,
max > PDC .

(18.50)Further, the nominal DC power of the inverter should
be approximately equal to the PV power,
STC
PDC0 ≈ PDC.

(18.51)

Usually, for PDC0 < 5 kWp, single-phase inverters areused while for PDC0 > 5 three-phase inverters are advised.

Copyright Delft University of Technology, 2014

Also the inverter efficiency is dependent on the input
power and voltage. A model discussing the inverter
efficiency is presented below.

18.4.1Inverter eﬃciency

As we already discussed in Chapter 17, modern inverters fulfil two major functions: First, maximum power point
tracking (MPPT), and secondly, the actual inverter function, ie converting the incoming direct current (DC) toalternating current (AC) that can be fed into the electricity grid.
In order not to waste electricity produced by the PV array, an inverter should always work as close as possible
at its maximum achievable efficiency. However, the inverter efficiency of an inverter strongly depends on the
DC input voltage as well as the total DC input power
as well as on the DC input voltage of the system. The
inverter efficiency ηinv with respect to the input DC
power at various DC voltage level is usually given at
the data sheet, at least for some values. Often, only thepeak efficiency is given as a single value.
Weighted Eﬃciencies

A more reliable way of expressing the inverter efficiency in a single number is to use weighted efficiencies,

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328

18. PV System Design

which combine the inverter efficiencies over a wide
range of solar resource regimes [110]. Two different
weighted efficiencies are commonly used. First, theEuropean Efficiency, which represents a low-insolation
climate such as in central Europe, and the California Energy Commission (CEC) efficiency, which represents thePV system performance in high-insolation regions suchas in the southwest of the United States [110]. They are
given byηEuro = 0.03 η 5% + 0.06 η10% + 0.13 η20% +0.10 η30% + 0.48 η50% + 0.20 η100% ,(18.52a)

ηCEC = 0.04 η10% + 0.05 η20% + 0.12 η30% +0.21 η50% + 0.53 η75% + 0.05 η100% ,(18.52b)where ηx% denotes the efficiency at x% of nominal
power of the inverter. Note that the CEC efficiency contains a 75% value that is not present in the European
efficiency.
Even though the weighted efficiencies represents a
more accurate approximation of the effective yearlyworking performance of the inverter compared to themere peak efficiency, it still only is an approximation of
the average performance of a system in the European
climate.
If a better estimate of the real time energy yield of an extended PV system is needed, a more accurate representation of the instantaneous inverter performance at every

Copyright Delft University of Technology, 2014

Figure 18.13: Field test results for a 2.5 kW Solectria PVI2500

inverter recorded during a period of 13 days for system operation at Sandia Laboratories [111].

level of input power and voltage must be developed,
for example the model discussed below.
Sandia National Laboratories (SNL) model

Due to a lack of detailed data from inverters manufacturers, many research institutes around the world havepublished extended data, which are publicly available
online. These data present efficiency curves for a large

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329

18.4. Designing grid-connected PV-system

range of inverters as a function of a several characteristic parameters. One database that can be used is theone provided by the Sandia National Laboratories [112].
Figure 18.13 shows an example of an inverter efficiency
curve CITE. For the graph, field test measurements
were taken during a period of 13 days with changing
weather condition. While the relationship between the
AC and DC power appears to be linear at first, a closer
look to the graph reveals that this is not entirely thecase, as shown in Fig. 18.14. The power consumptionof the inverter itself together with the electrical characteristics of the switching modes and circuits at different power levels results in a degree of non linearity between AC and DC power at a given DC voltage
level. Assuming that the inverter efficiency is a constantvalue throughout the whole DC power range is equivalent to assuming a linear relationship between DC andAC power, which has been shown not to be the case.
The dependency of the inverter efficiency on the DC
input voltage is very complex phenomenon that stilllacks a full physical explanation [113]. The differences
between the different inverter types can partially be explained with different types of switches used. Figure18.15 shows the voltage dependent inverter efficiency
for different inverter types. These curves were determined with the Sandia model that is described in the next
paragraph.

Figure 18.14: A closer look on to the relationship between AC

output and DC input power for an inverter [111].

In the following, we will introduce a mathematical

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330

18. PV System Design

Inverter efficiency (%)

100

95
VDC < Vnom
VDC = Vnom
VDC > Vnom

90

85

80

Solaron 500 HE [3159502-XXXX] (480 V)[CEC 2011]
0

50

100

200

250

300

350

400

DC power (kW)100

Inverter efficiency (%)

150

Several of the parameters are depicted in Fig. 18.14. The
parameters are [111]:
PAC : AC-power output from inverter based on input
power and voltage, (W).

95

PDC : DC-power input to inverter, typically assumed tobe equal to the PV array maximum power, (W).

VDC < Vnom
VDC = Vnom
VDC > Vnom90

VDC : DC-voltage input, typically assumed to be equal
to the PV array maximum power voltage, (V).
PAC0 : Maximum AC-power “rating” for inverter at reference or nominal operating condition, assumed to
be an upper limit value, (W).

8580model developed at Sandia Research Institute that has
been chosen due to its accuracy of both the effect of
DC input power and voltage on the AC power output.
In this model the relationship between PAC and PDC isgiven by [111]


PAC0
PAC =
− C ( A − B) · ( PDC − B) − C ( PDC − B)2 ,A−B(18.53)where the coefficients A, B, and C are given by
A = PDC0 1 + C1 VDC − VDC0 ,(18.54a)
B = PS0 1 + C2 VDC − VDC0 ,(18.54b)
1 + C3 VDC − VDC0 .
(18.54c)
C = C0Samil Power SolarRiver9000TL-US (208 V)[CEC 2013]
0

3

6

9

12

15

PDC0 : DC-power level at which the ac-power rating isachieved at the reference operating condition, (W).

DC power (kW)Figure 18.15: Variation in the inverter efficiency with DC

Copyright
Delft Universityof Technology,
2014
input voltage
for two different
inverters.

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18.4. Designing grid-connected PV-system

VDC0 : DC-voltage level at which the ac-power rating isachieved at the reference operating condition, (V).
PS0 : DC-power required to start the inversion process,or self-consumption by inverter, strongly influences
inverter efficiency at low power levels, (W).
C0 : Parameter defining the curvature (parabolic) of therelationship between AC-power and DC-power at
the reference operating condition, default value ofzero gives a linear relationship, (1/W).
Ci : Empirical coefficient allowing PDC0 to vary linearlywith DC-voltage input, default value is zero, (i =1, 2, 3, 1/V).
This model takes the following losses into account[113]:• Self consumption of the inverter. This value corresponds to the DC power required to start the inversion process.
• Losses proportional to the output power due tofixed voltage drops in semiconductors and switching losses.
• Ohmic losses
The accuracy of the model depends on the data available for determining the performance parameters. An
initial estimate can be performed using the little information provided by the manufacturer. Using all the re-

Copyright Delft University of Technology, 2014

331

quired parameters will provide a model with an errorof approximately 0.1% between the modeled and measured inverter efficiency [113].
All the parameters required for handling Eq. (18.53) are
given in the Sandia laboratory database where a fulllist of a wide range of inverters with nominal powers
from 200 W up to 1 MW are covered [112]. This model
therefore uses the instantaneous value of PDC and VDC
produced by the entire PV array to evaluate the AC
power output and inverter efficiency.
Maximum Power Point Tracker and additional losses

The efficiency of the MPPT has not been included explicitely in the Sandia performance model. This is because the efficiency of most inverters ranges between
98% and nearly 100% at every level of input power andthe voltage provided is within the accepted minimum
and maximum window for the MPPT to function correctly. A decrease of 1% in the system performance has
therefore been used to take the MPPT losses into account.
An an additional decrease of 3% in the system performance can be considered to cover losses caused by mismatch between modules (−1.5%), ohmic cable losses(−0.5%) and soiling (−1%) [109], if the cable losses
are not determined via the method described in section
17.5.

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332

18.4.2

18. PV System Design

Performance Analysis

where Ei,Y sys is the solar energy incident on the PV system throughout the year. It can be calculated with

Now we will put all things together that we discussed
earlier in this section. The instantaneous power output on
the AC side can be described with
PAC (t) = A M G M (t) η M (t) ηinverter (t) ηMPPT (t) ηother .
(18.55)The system efficiency then is given asηsystem (t) =PAC (t)· 100%,
Atot G M (t)(18.56)

which leads us to the instantaneous AC-side yield (alsoknown as performance ratio),
PAC (t)· 100%.
PSTCYAC (t) =(18.57)

Then, the yearly energy yield at the AC side can be calculated with
Y
EAC
=

Z
year

PAC (t) dt;(18.58)

it is given in Wh/year. Another important parameter isthe annual efficiency of the systemYηsystem=Y
EAC

Ei,Y sys· 100%,

Copyright Delft University of Technology, 2014(18.59)Ei,Y sys = AtotZ
year

G M (t) dt.

(18.60)The last parameter we look at is the yearly electricity
yield
Y
EAC
YE =
,
(18.61)NS NP · PSTC
which is given by Wh/(year kWp).
At the end of the design phase it is important to check
whether the system really fulfils the requirements. If
the yearly energy yield exceeds the annual load, thesystem is well designed. Otherwise, another iteration
has to be done in order to scale up the system. However, as stated earlier, for a grid-connected system it
also can be a choice not the require the whole load toby covered by PV electricity.

18.5

Designing oﬀ-grid PV systems

In this last section of this chapter on PV system design
we take a closer look on the design of off-grid PV systems (also called stand-alone systems). Choosing agood design is more critical for off-grid systems thanfor grid-connected systems. The reason for this is that

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333

18.5. Designing oﬀ-grid PV systems

off-grid systems cannot fall back on the electricity grid,
which increases the requirements on the reliability ofthe off-grid system.
A major component of off-grid systems is the storage
component, which can store energy in times when the
PV modules generate more electricity than required andit can deliver energy to the electric appliances when theelectricity generated by the PV modules in not sufficient. A major design parameter for off-grid systems is
the required number of autonomous days, i.e. the number of days a fully charged storage must be able to deliver energy to the system until discharged.
The sizing of both the PV array and the storage component, usually a battery bank, is interconnected. Here,two parameters arise: first, Efail , which is the energy required by the electric load that cannot be delivered by
the PV system, for example if the batteries are emptied
after several cloudy days. Secondly, Edump , which is theenergy produced by the PV array that neither is used
for driving a load nor is stored in the battery, for example, if the batteries already are full after a number ofsunny days. Now, we can define the Loss of Load Probability,
Efail
LLP = R.
(18.62)
year PL ( t ) dt
Of course, the lower LLP, the more stable and reliable
the PV system. Table 18.3 shows recommended LLPs
for different applications.

Copyright Delft University of Technology, 2014

Table 18.3: Recommended Loss of Load Probability (LLP) forsome Applications

Application
Domestic illumination
Appliances
Telecommunications

Recommended LLP

10−2
10−1
10−4

Figure 18.16 (a) shows a schematic of an off-grid system with all the required components, i.e. the PV array,a maximum power point tracker (MPPT), a charge controller (CC), a discharge controller (DC), a battery bank(BB), an inverter, and the load. The CC prevents theBB from being overcharged by the PV system, while
the DC prevents the battery to be discharged below
the minimal allowed SoC. In the simplest case, they(dis)connect the PV system and load from the battery
by switches. For optimal charging, however, nowadays
often CC with pulse-width modulation (PWM) areused. By far not all off-grid systems contain an MPPT,
thus it is represented with a dashed line. However, if
an MPPT is present it usually is delivered in one unit
together with the CC. Usually, the DC and inverter are
combined in one battery inverter unit. Especially at larger
systems the inverter currents may become very large.
For example, if a 2400 W load is present for a shorttime, this means 100 A on the DC side in a 24 V system. Therefore the battery inverter usually is directly
connected to the BB with thick cables. In very smallThis copy is provided for free, for personal use only.

334

18. PV System Design

(a)

(b)

VMPP

Charge
M
P
P
T

I CC

I inv

VBB

VBB

VBB

I BB
MPPT + CC
DC + Inverter

Inverter
Controller

Controller

Battery
Bank (BB)MPPT + CC + DC + Inverter

IBB-chI AC

Discharge

VAC

L
O
A
D

Current I

I MPP

charging
regimeVBB-chVOC-BB

VBB-disV

IBB-dis+VOC-BB–Ri
I BB

discharging
regime+VBB–

Figure 18.16: (a) PV topology of an off-grid system. (b) Simplified I-V curve and schematic of the battery bank.

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335

18.5. Designing oﬀ-grid PV systems

systems with maximal powers of several hundreds ofWatt, the CC and DC may be combined in one unit.

the battery, as seen by the equation
IBB =18.5.1

A closer look at the battery

The battery bank is the workhorse of any off-grid system, because it is its stable power source. It is thus very
important to understand how the battery bank will act
in the PV system. Therefore, we will discuss it in moredetail in this section.
When we want to understand how a battery works wehave to consider the net effect of all the forces that try to
charge or discharge the battery. In Fig. 18.16 (a) the mostimportant currents and voltages that we need and thefollowing discussion are depicted. In Fig. 18.16 (b) aschematic and an I-V curve of an idealised batterybank is shown. As we can see, this battery bank is
considered to consist of an voltage source with a total
voltage VOC-BB and an internal resistance Ri . In reality,
VOC-BB will not be constant, but a function of the stateof charge, the ambient temperature, and others,
VOC-BB = f (SoC, T, . . .).

(18.63)In the following discussion, however, we will neglect
this dependence and assume VOC-BB to be constant. Because of Ri , the voltage VBB will differ from VOC-BB andalso is dependent on the current IBB flowing throughCopyright Delft University of Technology, 2014

1
(VBB − VOC-BB ) .
Ri(18.64)Note that we use the convention, where IBB is positive
when the battery is charged and negative when it isdischarged. Let us now derive an expression for thevoltage of the battery bank (VBB ) as a function of theother PV system parameters. We start with the poweron the left and right-hand sides of the MPPT,ηMPP IMPP VMPP = ICC VBB =: β,(18.65)βηMPP IMPP VMPP
=
.
VBB
VBB

(18.66)and hence
ICC =Consequently, β = 0 means that the PV system is not
active, for example during night.
In a similar manner, we look at the power on the left
and right hand side of the inverter:
PL = IAC VAC = ηinv Iinv VBB =: ηInv α,(18.67)and consequently
Iinv =

αPL=.
ηinv VBB
VBB(18.68)Combining Eqs. (18.66) and (18.68), we find
IBB = ICC − Iinv =

β−α
.
VBB

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336

18. PV System Design

Clearly, α = 0 indicates that no load is present.
Combining Eqs. (18.64) and (18.69), multiplying withRi VBB and rearranging leads to a quadratic equation,
2
VBB− VOC-BB VBB − Ri ( β − α) = 0.

(18.70)with the solutions
V±
VBB= OC-BB ±2s

VOC-BB2

2+ R i ( β − α ).

(18.71)

The correct solution is the "+" solution as we can check
with
+
VBB
( β − α = 0) = VOC-BB .
(18.72)Hence, the final solution iss
VOC-BB
VOC-BB 2
VBB =++ R i ( β − α ).
22(18.73)Let us now take a short look on this solution. If thebattery is charged, ICC is higher than Iinv and hence( β − α) and IBB are positive. From Eq. (18.73) it follows
that in this case VBB is higher than VOC-BB . On the other
hand, if the battery is discharged, ICC is lower than Iinv
and hence ( β − α) and IBB are negative. Therefore, VBB
will be lower than VOC-BB .
Note that only the net current IBB = ICC flows in or outof the battery. Therefore, only this current determines

Copyright Delft University of Technology, 2014

Table 18.4: Recommended number of autonomous days d A atseveral latitudes.

Latitude (◦ )
0-30
20-50
50-60

Recommended d A
5-6
10-12
15

the power loss in the battery,2
PBB (loss) = IBB
Ri .

(18.74)This power always is lost, irrespective of the sign of
IBB .

18.5.2

Designing a system with energybalanceNow we discuss how to design a PV system based on
the principle of energy balance. For determining the adequate components, the analysis for the load side and
the PV side is performed separately. Let us begin withthe load side. We can determine the annual load EYL as
discussed in section 18.2 and Eq. (18.1),
EYL =

Z
year

PL (t) dt.

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337

18.5. Designing oﬀ-grid PV systems

From that we can estimate the average daily load with
ELD =

1 Y
E .
365 L

(18.76)

Next, we have to choose an adequate number of daysof autonomy d A . Some values given in table 18.4. Now,
we can calculate the required energy of the batterybank,
E D · SFbat
EBB = d A L,(18.77)
DoDmaxwhere SFbat is the sizing factor of the battery, which
is similar to the sizing factor for the PV array alreadydefined in section 18.4. DoDmax is the maximally allowed depth of discharge of the batteries. The rated
energy of the chosen batteries is
Ebat = VOC-bat Cbat ,(18.78)where Cbat is the battery capacity (unit Ampere-hours).
Hence, the required number of batteries is


EBB
.
(18.79)Nbat =
Ebat
Similarly as for grid-connected systems we now need tochoose a suitable inverter. Therefore we must consider
the maximal load power PLmax . This we can do, for example, by looking at the appliance with the maximal
power consumption, or by adding up the power of all

Copyright Delft University of Technology, 2014

appliances. It may be more beneficial to choose a system design, such that not all appliances can be used at
the same time. The final decision of course is up to thedesigner of the system. Similar as for grid-connected
systems, the inverter must fulfil several requirements:
First, its maximally allowed power output must exceed
the maximal power required by the appliances,
inv
max
PDC,
max > PL .

(18.80)

Secondly, the nominal power of the inverter should beapproximately equal to the maximal load power,
PDC, 0 ≈ PLmax .

(18.81)

Thirdly, the nominal inverter input voltage should beapproximately equal to the nominal voltage of the battery back,
VDC, inv ≈ VOC-BB .

(18.82)For a more detailed analysis of the inverter performance the Sandia model can be used that was discussed
in section 18.4.
Typical voltages for the battery bank are 12 V, 24 V, 48
V or 96 V. It can be adjusted by the number of batteries
that are connected in series,
S
Nbat
=

VOC-BB.
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338

18. PV System Design

From that we also can determine the number of batteries that must be connected in parallel,
&
'
Nbat
P
.
(18.84)
Nbat =S
Nbat
Now, after designing the load side is completed withchoosing inverter and batteries, we look at the PV sideof the system. Sizing the PV array is very similar to theprocedure used for grid-connected systems in section
18.4. The energy balance can be written down as The
energy balance now writes asY
EDC= EYL · SF,(18.85)where V MPP-mod denotes the annual average of the
MPP voltage of the PV modules. Of course, the maximally allowed input voltage of the MPPT-CC unit mustnot be exceeded by the PV array,
max
VMPP ≥ NS · Vmod-MPP.

(18.88)The number of required parallel PV strings is given by


NT
NP =
.
(18.89)NS

18.5.3

Performance Analysis

where SF is a sizing factor that usually is assumed to be
1.1. We therefore can calculate the required number ofmodules,&'
EYL · SFR
NT =.
(18.86)A M · year G M (t)η (t) dt

Like for grid-connected systems, also for off-grid systems, a performance analysis should be done to evaluate the chosen design. For this analysis, it is useful touse an algorithm that can simulate the performance of
the PV system throughout the year. Conceptually, thisalgorithm can look as follows

For minimising losses, the MPP voltage of the PV arrayand the nominal voltages of the inverter and the battery
pack should be approximately equal, since otherwise
the losses of the DC-DC converter that is included in
the MPPT-CC unit will be higher. The number of PV
modules that are connected in series in the PV array isgiven by
VOC-BBNS =,(18.87)
V mod-MPP

1. Set a starting state of charge (SoC) of the batterybank.

Copyright Delft University of Technology, 2014

2. Then calculate the SoC of the battery throughout
the year for time steps ∆t,• Depending on the actual load and PV array
output, determine the battery current IBB (t).
• Determine the actualised SoC.

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339

18.5. Designing oﬀ-grid PV systems

Of course, the function of the charge controller, i.e.
its switching behaviour, must be accurately mimicked by the algorithm.
3. Determine when the system cannot deliver the required load and hence Efail .
4. Now, calculate the Loss of Load Probability of thesystem (LLP).
5. Finally, determine the yearly energy yield on theY .
AC-side of the system, EAC
Mathematically, the annual energy yield on the AC side
can be expressed by
Y
EAC
=

Z
year

PL (t) dt − Efail + ∆EBB .

(18.90)In contrast to the expression used for determining theAC yield for grid-connected systems in Eq. (18.58), thisequation contains two additional components: Efail , theenergy that is required by the load but cannot be delivered and ∆EBB is the difference in energy stored in
the battery bank between the beginning and end of the
year,


∆EBB = SoCend − SoCbeginning Cbat VBB .
(18.91)

Because of Efail > 0,
Y
<
EAC

Z
year

PL (t) dt.

(18.92)As already stated earlier, the Loss of Load Probability isgiven by
Efail
LLP = R.
(18.93)
year PL ( t ) dt
For evaluating the design, it is very important to lookat LLP:• LLP acceptable:– Edump low: The system design is OK.
– Edump high: Resize the PV array.
• LLP not acceptable:– Increase the size of the PV array.
– Increase the capacity of the battery bank.

If ∆EBB < 0, the system might not be sustainable. On
the other hand, if it is > 0, Edump will increase in thefollowing, if the average meteorological conditions areunchanged.

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340

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18. PV System Design

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19

PV System Economics and Ecology
19.1

PV System Economy

can be calculated withpayback time =initial investment.
annual return(19.1)

We conclude our discussions on PV systems with looking on several important topics on the economics of
PV systems. Note that the economics of PV can be discussed at several levels, such as the consumer level, themanufacturing level, the level of PV installers, and thetechnology level where PV is compared to other electricity generation technologies on the scale of the electricity grid.

Translated to the consumer level, the payback time isthe time it takes to recover the initial investment of the
PV system as the system continuously reduces the electricity bill. Please note that the financial payback time isdifferent from the energy payback time that we will discuss in Section 19.2 Let us look at an example:

We will start this discussion with the definition of thepayback time, which in finance is defined as the amountof time required to recover the cost of an investment. It

Let us assume that family Smith have installed a PV systemwith a power of 1 kW p on their rooftop. The initial investment

Example341
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342
was e 8000. Family Smith has an annual electricity bill ofe 2000. The installation of the PV system leads to an averageannual reduction of the electricity bill of e 800. As a part oftheir consumed electricity is provided by their PV system,
the electricity bill is therefore constantly reduced. Hence, theaverage annual return on their PV system is e 800. As a consequence, the Smith’s have earned the final investment backafter 5 years, the payback time hence is 5 years.

The payback time is strongly influenced by the annual
solar radiation on the PV system. As we have seen in
Chapter 16, this is dependent on the orientation of the
PV modules and on the location of the PV systems.
In general we can say that the sunnier the location,
the greater the PV yield and the shorter the payback
time. Another factor that influences the payback timeis the grid electricity costs: the higher these costs, theshorter the payback time. Finally, the payback time also
is strongly dependent on the initial costs of the PV system.
In practice, often more factors must be taken into account than in the simple example above. This will increase the complexity of calculating the payback time.
For instance, if we are considering a significant periodof time, also the change of the value of money has tobe taken into account, which is due to inflation. For
example today e 1000 will have a different purchasing
power than in ten years time. Another factor that should
be considered are policies regarding renewable energy.

Copyright Delft University of Technology, 201419. PV System Economics and Ecology

For example, subsidies and feed-in tariffs can affect theinitial investments and savings.

Let us briefly discuss the concept of feed-in tariffs. At theconsumer level, the feed-in tariff is the price at which a
consumer can sell renewable electricity to the electricity
provider. We distinguish between two kinds of feedin tariffs, gross and net. Gross feed-in tariffs are paid
for all the electricity the panels produce, irrespectiveof the consumer’s electricity consumption. In contrast,
net feed-in tariffs promise a higher rate for the surplus
electricity fed into the grid after domestic use of theconsumers is subtracted. For implementing feed-in tariff schemes at consumer level, the facility of net metering
is pivotal.

Another very important concept is the Levelized Cost of
Electricity (LCoE), which is defined as the cost per kWhof electricity produced by a power generation facility. It
is usually used to compare the lifetime costs of different
electricity generation technologies. To be able to estimate the effective price per kWH, the concept of LCoE
allocates the costs of an energy plant across its full lifecycle. It is somehow similar to averaging the upfront
costs of production over a long period of time. Depending on the number of variables that are to be taken into
account, calculating the LCoE can become very com-

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343

19.1. PV System Economy

plex. In a simple case the LCoE can be determined with
I + M +F

(19.2)

Variable O&M
Levelized capital cost

200

Fixed O&M

150
100
50

al
rm

la
r
So

in

d

(o

ffs

th
e

ho

PV

)
re

d
in
W

Nu
cle

ar

0

W

Depending on the location of a PC system and the initial investment required for the PV system, the LCoE
for PV can vary a lot between different projects. Additionally, the discount rate r used for the calculation willstrongly influence the LCoE.

Transmission Investment

250

ls

Figure 19.1 shows the LCoE for different methods ofelectricity generation. We see that the LCoE of wind energy is lowest, it is even below the LCoE of fossil fueland nuclear based electricity. The LCoE of PV generated electricity is still slightly above that of the nonrenewable technologies.

300

ue

The sums expand across the whole lifetime of the system n, every year accounts for one summand. It are theinvestment expenditures in the year t, Mt are the operational and maintenance expenditures in the year t, andFt are the fuel expenditures in the year t. Of course, for
PV Ft ≡ 0. Further, Et is the electricity yield in the year
t. Finally, r is the discount rate which is a factor used
to discount future costs and translating them into thepresent value.

il f

.

ss

E

∑nt=1 (1+tr)tFo∑nt=1 t (1+tr)t t

Estimated LCoE (US$/MWh)

LCoE =

Figure 19.1: The levelised cost of electricity for different

electricity generation technologies.

For the electricity supplier, the LCoE is a valuable indicator of the cost competitiveness of a certain energy
technology. It is also a good indicator for determining

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344

19. PV System Economics and Ecology

The final concept that we discuss is that of grid parity.
For solar electricity this is the situation at which it canbe generated at an LCoE that is equal to the electricity
price from the conventional electric grid. Of course, if
the PV electricity price is below the grid price, the situation becomes even better,
LCOEPV ≤ LCoEconventional .

(19.3)

Grid parity whenlines crossTypical sites

Solar electricity pricethe electricity price: for making profit this price must
be above the LCoE. Surely, the supplier cannot determine the electricity price independently, as it is strongly
influenced by policy factors such as feed-in tariffs, subsidies and other incentives.

Best sites
Earliest

LatestGrid power
pricesWholesale
Volume of PV systems installed

Figure 19.2: Grid parity for PV systems [114].

In principle, the concept of grid parity can be generalised to the other renewable technologies as well. However, there is one significant difference between PV
and other renewable technologies such as wind andhydro electricity. Wind and hydro electricity installations usually only can be financed by companies but
are no option for a single consumer. In contrast, PV canbe scaled down to the level of a single module, suchthat a house owner can become an electricity producer with his small scalable PV installation on his roof.
As a consequence, the consumer offering PV electricity to the grid now is effectively competing with theretail grid price, which conventionally includes othercosts like transmission, distribution, and so on. As forthe consumer these costs are no part of the PV electri-

Copyright Delft University of Technology, 2014

city price, PV grid parity for retail grid prices can bereached faster, as shown in Fig. 19.2.
The graph is showing the volume of installed PV systems versus the price of PV electricity. The installedvolume can be directly correlated with time, as the past
decade has seen the implemented PV volume rise tremendously. As capital costs decline with increasing
volumes, the price of PV generated electricity is expected to decrease in the future. On the other hand,the price of fossil fuels is expected to rice because ofincreasing scarcity and cost linked to the right to emit
CO2 emissions into the atmosphere costs. These trends

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19.2. PV System Ecology

will lead to increasing prices for electricity generated
with combusting fossil fuels. Grid parity is reachedwhen the prices of PV electricity and grid electricity
are crossing. Naturally, with incentives and subsidies
grid parity can be reached even faster, however grid
parity reached with public financial support often is
disregarded. Therefore it is said that true grid parity isreached when the PV electricity prices fall below thegrid prices without any subsidies. Real grid parity forexample already is reached in the sunny country of
Spain.
To conclude, grid parity is a very useful concept to indicate the feasibility of a renewable energy technology.
The closer a technology is to grid parity, the easier it
can be integrated in the electricity mix. With the advancements in technology and the maturity of manufacturing processes, grid parity for solar is expected to bereached at many locations around the World in the next
years.

19.2

PV System Ecology

Besides discussing the economics of PV systems, it also
is very important to consider their ecological and environmental aspects. The main reason for that is that theaim of photovoltaics is to generate electricity without
any considerable effect on the environment. It is there-

Copyright Delft University of Technology, 2014345

fore very important to check the ecological aspects ofthe different PV technologies. In this section we will
discuss different concepts to quantify the environmental
impact of PV systems.
The concept of the carbon footprint estimates the emissions of CO2 caused by manufacturing the PV modules
and compares them with the reduction of CO2 emissions due to the electricity generated with PV insteadof combusting fossile fuels. A more analytical approachis to look at the total energy required to produce either
the PV modules or all the components of a PV system.
As the different PV technologies vary considerably inthe required production processes, the energy consumption for producing 1 kW p varies considerably between
the different technologies. If a complete life cycle assessment (LCA)is performed, it is tried to trace the energy
and carbon footprints of the PV panels throughout their
lifetime. Therefore LCA also is known as cradle-to-grave
analysis.
We now are going to introduce several indicators thatare used to judge the different ecological aspects. The
Energy Yield Ratio is defined as the ratio of the total energy yield of a PV module or system throughout its
lifetime with all the energy that has to be invested in
the PV system in that time. This invested energy not
only contains the energy for producing the components,
transporting them to the location and installing them
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346

19. PV System Economics and Ecology

ent components at the end of their lifecycle.
As the energy required for producing a PV system depends strongly on the PV technology and also on thequality of the panels, the energy yield ratio for the different technologies varies a lot. While the energy yieldratio for PV modules can be as large as 10 to 15, PV
systems usually have a lower ratio because of the energy invested in the components other than the modules.

CIGS modules
With frame, min
CdTe modules

A very important concept is the Energy Payback Time,
which is defined as the total required energy investment over the lifetime divided by the average annual
energy yield of the system,
totally invested energy.
average annual energy yield(19.4)
Note that the energy payback time is different from theeconomic payback time introduced in Section 19.1.

Without frame, min
Amorphous modules

Without frame, max

Polycrystalline modules

Energy Payback Time =The energy payback time of typical PV systems is
between 1 and 7 years and it also depends on location
issues such as the orientation of the PV array as well as
the solar irradiance throughout the year.

With frame, max

Monocrystalline modules

0

4 000

8 000

12 000

16 000

20 000Primary energy use for manufacturing PV systems (kWh/kWp)
Figure 19.3: Specific energy used to produce PV modules ofdifferent technologies (Data from [115]).

Figure 19.3 shows the specific energy required for producing PV modules with different technologies. As we
can see, the differences between the technologies arelarge. The specific energy required for producing thinfilm modules from materials such as amorphous silicon,

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347

Energy Payback time (years)

19.2. PV System Ecology

7

High irradiation(2200 kWh/m2/year)

Medium irradiation(1700 kWh/m2/year)

Low irradiation(1100 kWh/m2/year)

c-Si a-Si

c-Si a-Si

c-Si a-Si

6
5
4
3
2
1
0

Roof

c-Si a-Si

GroundRoofc-Si a-Si

GroundRoofc-Si a-Si

GroundFigure 19.4: The energy payback time for the different com-ponents constituting PV systems (Data from [114]).

cadmium telluride and CIGS is significantly below thatof modules made from polycrystalline and monocrystalline silicon, where the specific energy can reach values up to 12000-18000 kWh/kW p . Because of further
improvements in the module efficiency and the manufacturing process we however may expect that that the
specific energy follows a decreasing trend.
For a PV system it is more difficult to allocate the energy that was used for its production, as all the components constituting the balance of system have to betaken into account as well. For example, for components like batteries and inverters the technologies andmanufacturing processes may vary a lot between the

Copyright Delft University of Technology, 2014

different products available on the market. Nonetheless, studies were carried out that estimated the energy
required by whole PV systems. Generally, as we cansee in Fig. 19.4 it is found that the energy required for
the BOS is significantly below that used for manufacturing the modules. In the figure, amorphous and crystalline silicon modules are compared. As expected, we
see that the energy payback time in regions with highsolar irradiance have significantly shorter energy payback times than regions with low irradiance. While aSi:H based modules have a shorter energy payback timethan c-Si based modules, the energy payback time forthe module frame and the BOS of a-Si:H based systems
can be significantly higher than that of c-Si based systems. This can be explained with the lower efficiency ofa-Si:H that increases the required framing material per
Wp.
The irradiance strongly influences the energy payback
time and varies between two years (high irradiance)and six years (low irradiance). Roof-mounted systems
always have a shorter energy payback time than systems mounted on the ground, mainly because of the
BOS that is more energy extensive for ground-mounted
systems.
No matter which PV technology is chosen, the energy
payback time always is far below the expected system
lifetime, which usually is between 25 and 30 years. For
the PV systems discussed in Fig. 19.4, the energy yield

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19. PV System Economics and Ecology

ratio is between 4 and 10. Hence, the energy invested
in the PV system is paid back several times throughout the life cycle of the PV system. The urban legendthat PV modules require more energy to be produced
than they will ever produced thus is not backed by any
data. In contrast, the net energy produced is much larger than the energy required for PV production.
However, a lot of work still needs to be done and canbe done. Some studies indicate that the energy requiredfor producing PV modules can be reduced by up to
80%. Further, as the amount of installed PV systems
becomes larger and larger, recycling of the components
at the end of the lifecycle becomes very important. For
example, the European Union introduced already several directives that induced recycling schemes for c-Si
based PV modules.
The last environmental issue that we want to mention
is pollution caused by the production of PV modules.
As many sometimes toxic chemicals are required for
producing PV modules, this can be a serious threat tothe environment. Therefore it is very important to havestrong legislation in order to prevent pollution of thesurroundings of PV factories. Especially in countries
with weak environmental legislation pollution can be asevere problem that affects the environment and people
living in the surroundings negatively.

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Part V

Alternative Methods of Solar Energy
Utilisation

349
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20

Solar Thermal Energy
20.1

Solar thermal energy is an application of solar energythat is very different from photovoltaics. In difference
to photovoltaics, where we used electrodynamics andsolid state physics for explaining the underlying principles, solar thermal energy is mainly based on the lawsof thermodynamics. In this chapter we give a brief introduction to that field. After introducing some basics
in Section 20.1, we will discuss Solar Thermal Heatingin Section 20.2 and Concentrated Solar (electric) Power(CSP) in Section 20.3.

Solar thermal basics

We start this section with the definition of heat, whichsometimes also is called thermal energy. The moleculesof a body with a temperature different from 0 K exhibit a disordered movement. The kinetic energy of thismovement is called heat. The average of this kinetic energy is related linearly to the temperature of the body.1
Usually, we denote heat with the symbol Q. As it is aform of energy, its unit is Joule (J).
If two bodies with different temperatures are brought
together, heat will flow from the hotter to the cooler
1 The interested reader will find more details in every textbook on physics orthermodynamics. This definition is taken from Ref. [116].

351
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20. Solar Thermal Energy

Sensible heat is that form of heat that results in changes
in temperature. It is given as
Q = mC p ( T2 − T1 ),(20.1)

where Q is the amount of heat that is absorbed by thebody, m is its mass, C p is its heat capacity and ( T2 − T1 )is the temperature difference. On the other hand, if abody absorbs or releases latent heat, the temperature
stays constant but the phase changes. This happens forexample when ice is melting: When its temperature isequivalent to its melting point, heat that is absorbed by
the ice will not result in increasing temperature but intransformation into from the solid to the liquid phase,
which is water. Mathematically, this is expressed as

T2
Temperature (T)body and as a result the cooler body will be heated.
Dependent on its physical properties and temperature,
this heat can absorbed in the cooler body in two forms,
sensible heat and latent heat.

Solidsensibleheat

Phase-change
latent heat

T*

Liquidsensibleheat

T1
Storage heat (Qs)
Figure 20.1: Illustrating the difference of sensible and latentheat.

the liquid will heat up further, the heat now is stored asliquid sensible heat.

(20.2)

Now we will take a look at the three basic mechanismsof heat transfer: conduction, convection and radiation.

The two forms of heat are illustrated in Fig. 20.1, whichshows what happens when a body absorbs heat. In the
beginning, the body is solid and has temperature T1 . It
then heats up and the heat is stored as solid sensible
heat. When its melting point T ∗ is achieved, its temperature will not increase any more but the phase willchange from solid to liquid. After everything is molten,

Conduction is the transfer of heat in a medium due toa temperature gradient. For a better understanding welook at the heating of a house during winter, as illustrated in Fig. 20.2. The inside of the house is warmat 20°C, but the outside is cold with a temperature of
0°C. Because of this heat gradient, heat will be transferred from the inside through the wall to the outside.
Mathematically, heat conduction can be described by

Q = mL,
where L is the specific latent heat.

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353

20.1. Solar thermal basics
(a)

20°C

0°C

(b)

Figure 20.3: Illustrating (a) natural and (b) forced conductive

outsidewall

heat between a solid and a surrounding fluid.

insidex
Figure 20.2: Illustrating conductive heat transfer through awall.

the Fourier law,dQconddT= −kA ,
dt
dx

(20.3)

where dQ/dt is the heat flow in W, k is the thermal
conductivity of the wall [given in W/(Km)], A is thecontact area given in m2 and dT/dx is the temperature
gradient in the x direction given in K/m. If we assume
the wall to be made from a uniform material, we canassume that dT/dx is constant throughout the wall andwe can replace it by ∆T/∆x, where ∆x is the thicknessof the wall.
Convection is the second possible mechanism for heattransfer. It is the transfer of heat by the movement of a

Copyright Delft University of Technology, 2014

fluid. We distinguish between two forms of convection,
as sketched in Fig. 20.3: In forced convection the movement of the fluid is caused by external variables, whilein natural convection the movement of the fluid is caused
by density differences due to temperature gradients. In
both cases, the heat transfer from a medium of temperature T1 into a fluid of temperature T2 can be describedby Newton’s law
dQconv= −hA( T1 − T2 ),dt(20.4)

where h is the heat transfer coefficient which is given inW/(m2 K). We will not discuss the calculation of h indetail but we want to mention that it depends on various factors such as the velocity of the fluid, the shapeof the surface or the kind of flow that is present, i.e.
whether it is laminar or turbulent flow.
The third heat transfer mechanism is radiative heat trans-

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20. Solar Thermal Energy

fer, which is the most important mechanism of heattransfer for solar thermal systems. As we already discussed in Chapter 5, thermal radiation is electromagnetic radiation propagated through space at the speedof light. Thermal radiation is emitted by bodies depending on their temperature. This is caused when excited electrons fall back on their ground level and emita photon and hence electromagnetic radiation.
As discussed in Chapter 5, a black body is an idealised
concept of a body, which is a perfect absorber of radiation, independent on the wavelength or directionof the incident light. Further, it is a perfect emitterof thermal radiation. The Stefan-Boltzmann law [Eq.
(20.5)] describes the total irradiant emittance of a black
body of temperature T,
MeBB ( T ) = σT 4 ,(20.5)

where M is given in W/m2 and σ ≈ 5.670 ·
10−8 W/(m2 K4 ) is the Stefan-Boltzmann constant. In
nature, no black bodies exist. However we can describe
them as so-called grey bodies. The energy emitted by agrey body still can be described by Plank’s law [Eq.
(5.18a)] when it is multiplied with a wavelength dependent emission coefficient e(λ). For a black body wewould have e(λ) ≡ 1.

Copyright Delft University of Technology, 2014

warm watersecondaryheat
supplycollector arraycollector circuit

pump

heat storage

fresh water

Figure 20.5: Illustrating the main components of a solar water

heating system.

20.2

Solar thermal heating

About half of the world’s energy consumption is in theform of heat. About two thirds of the heat demand iscovered by coal, oil, and natural gas, as we can see in
Fig. 20.4 (a). As shown in Fig. 20.4 (b), heat is mainly
used in the industrial sector for facilitating for example
chemical processes and in the residential sector forheating and warm water supply.
Figure 20.4 (c) shows the total energy demand of a typical household in the United States. We see that spaceheating and water heating represent 59% of the totalenergy consumption. If also the demand for cooling istaken into account, about two third of the energy consumption are related to the use of heat.

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355

20.2. Solar thermal heating(a)(b) Agriculture
Other 4.6%
1.3%

Waste Other
Biofuels 2.3% 2.9%
2.8%Services
12.3%
Coal 37.9%
Gas 47.7%

Residential
38.3%
Oil 6.2%

Industry
43.5%

(c)

Appliances,electronics,
and lighting Space heating
41%
35%
Water
heating18%

Air conditioning
6%

Figure 20.4: (a) The used primary energy suppliers for the generation of heat and (a) the demand of heat by sector [9]. (c) Energy

consumption of U.S. homes [117]. All data is for 2009.

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356

The residential demand for heat can at least partiallycovered with a solar water heater, which is a combinationof a solar collector array, an energy transfer system anda storage tank, as sketched in Fig. 20.5. The main partof a solar water heater is the collector array, which absorbs solar radiation and converts it into heat. This heatis absorbed by a heat transfer fluid that passes throughthe collector. The heat can be stored or used directly.
The amount of the hot water produced by a solar water
heater throughout a year depends on the type and sizeof the solar collector array, the size of the water storage, the amount of sunshine available at the site andthe seasonal hot water demand pattern.
There are several ways to classify solar water heating
systems. The first way is by the fluid heated in the collector. When the fluid used in the application is thesame that is heated in the collector it is called a direct
or open loop. In contrast, when the fluid heated in thecollector goes to a heat exchanger to heat up the utility
fluid, it is called an indirect or closed loop.
The second way to classify the systems is by the way
the heat transfer fluid is transported. This can either bepassive, where no pumps are required, or by forced circulation, using a pump. The passive solar water heating
systems, uses natural convection to transport the fluid
from the collector to the storage. This happens because
the density of the fluid drops when the temperature increases, such that the fluid rises from the bottom to the

Copyright Delft University of Technology, 201420. Solar Thermal Energy

top of the collector – this is the same as natural convection that we discussed in Section 20.1. The advantage of passive systems is that they do not require any
pumps or controllers, which makes them very reliableand durable. However, depending on the quality of theused water, pipes can be clogged, which considerably
reduced the flow rate.
On the other hand, active systems like the one sketchedin Fig. 20.5 require pumps that force the fluid to circulate from the collector to the storage tank and the restof the circuit. These systems usually are more expensive
than passive systems. However, they have the advantage that the flow rates can be tuned more easily.

20.2.1

Solar Thermal Collectors

Now we will take a closer look at the solar collector,
in which the working fluid is heated by the solar radiation. The collector determines how efficiently thethe incident light is used. It usually consists of a blacksurface, called the absorber, and a transparent cover.
The absorber is able to absorb most of the incident energy from the sun, Qsun , raising its temperature andtransferring that heat to a working fluid. Hence, the absorber can be cooled and the heat can be transferredelsewhere.
Not all of Qsun can be used, as there are some losses,

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20.2. Solar thermal heating

Transparent
cover

Qcol

Qrad
Qrefl
Qsun

Qconv

Figure 20.6: Illustrating the major energy fluxes in a covered

solar collector.

Copyright Delft University of Technology, 2014as illustrated in Fig. 20.6. A part Qrefl is lost as reflection either in the encapsulation or in the absorber itself.
Other losses are related to the heat exchanged with thesurrounding air by the convection mechanism, Qconv
and radiation from the hot absorber, Qrad . When weput all these energies in an energy balance, we find forthe heat Qcol that can be collected by the collector
Qcol = Qsun − Qrefl − Qconv − Qrad .

(20.6)The efficiency of the collector depends mainly on two
factors: the extent to which the sunlight is converted
into heat by the absorber and the heat losses to thesurroundings. It will therefore depend on the weather
conditions and the characteristics of the collector itself.
To reduce losses, insulation from the surroundings isimportant, especially when the temperature difference
between the absorber and the ambient is high.
Usually, collectors are classified in three categories: uncovered, covered and vacuum, as shown in Fig. 20.7.
Uncovered collectors do not have a transparent cover, so
the sun strikes directly the absorber surface, hence thereflection losses are minimised. This collector type onlyis used for applications where the temperature differences between the absorber and the surroundings are
small, for example for swimming pools. Covered collectors are covered by a transparent material, providing extra insulation but also increasing the reflection losses.
These collectors are used for absorber temperatures of
up to 100°C. Finally, in vacuum collectors the absorber isThis copy is provided for free, for personal use only.

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20. Solar Thermal Energy

(a)

(b)
Transparent cover(c) Vacuum chamber
AbsorberInsolation
Absorber

Absorber Heat pipe

Figure 20.7: Illustrating (a) an uncovered, (b) a covered and (c) a vacuum collector.

encapsulated in vacuum tubes. In that case, little heat islost to the surroundings. The manufacturing process ofthese collectors in more complicated and expensive, but
the collector can be used for high temperature applications since the convection losses to the surroundings aremuch lower than for the other types.
Modern solar collectors often use light trapping techniques just as solar cells. For example, transparent
conducting oxide layers at the front window, that we
discussed in Section 13.1, can be used. If they havethe plasma frequency in the infrared, almost all thesolar radiation can enter the window, but the longwavelength infrared radiation originating from the hotparts in the collector cannot leave the collector as it isreflected back at the window.
Collectors also can be classified by their shape. We candistinguish between flat-plate collectors and concentrating collectors. Flat-plate collectors consist of flat absorbers that are oriented towards the sun. They can

Copyright Delft University of Technology, 2014deliver moderate temperatures, up to around 100°C.
They use both direct and diffuse solar radiation andno tracking systems are required. Their main applications are solar water heating, heating of building, airconditioning and industrial processes heat. In contrast,
concentrating collectors are suited for systems that requirea higher temperature than achievable with flat collectors. The performance of concentrating collectors can be
optimised by decreasing the area of heat loss. This isdone by placing an optical device in between the sourceof radiation and the energy absorbing surface. becauseof this optical device the absorber will be smaller andhence will have a lower heat loss compared to a flatplate collector at the same absorber temperature. One
disadvantage of concentrator systems is that they require a tracking system to maximise the incident radiation at all times. This increases the cost and leads toadditional maintenance requirements.

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20.2. Solar thermal heating

Like for PV systems,2 also for solar heat collector arrays it is important to decide whether the collectors
should be connected in series or in parallel. Connecting collectors in parallel means that all collectors havethe same input temperature, while for collectors connected in series the outlet temperature of one collector is
the input temperature of the next collector. Most commercial and industrial systems require a large numberof collectors to satisfy the heating demand. Therefore
most of the time, a combination of collectors in series
and in parallel is used. Parallel flow is used more frequently because it is inherently balanced and minimises the pressure drop. In the end, the choice of series
or parallel arrangement will ultimately depend on thetemperature required by the application the system.

flowing through the collectors from quickly reachinghigh temperatures.
Several storage technologies are available; some of themcan even be combined to cover daily and seasonal fluctuations. In general, heat can be stored in liquids, solids
or phase?change materials, abbreviated as PCM. Water
is the most frequently used storage medium for liquid
systems, because it is inexpensive, non-toxic, and it has
a high specific heat capacity. In addition, the energy canbe transported by the storage water itself, without theneed for additional heat exchangers.
The usable energy stored in a water tank can be calculated with
Qstored = VρC p ∆T,

20.2.2Heat storage

Now we will discuss the heat storage, which is an extremely important component as it has an enormous
influence on the overall system cost, performance andreliability. Its design affects the other basic elements
such as the collector or the thermal distribution system.
The task of the storage is twofold. First, it improves
the utilisation of the collected solar energy. Secondly, it
improves the system efficiency by preventing the fluid
2 See Section 17.1.

Copyright Delft University of Technology, 2014(20.7)

where V is the volume of the tank, ρ is the density ofwater, C p is the specific heat capacity of water and ∆Tis the temperature range of operation. The lower temperature limit is often set by external boundaries, suchas the temperature of the cold water, or by specific process requirements. The upper limit may be determinedby the process, the vapour pressure of the liquid or theheat loss of the water storage. For example, for residential water heating systems the maximally allowed
temperature is set to 80°C because at higher temperatures calcium carbonate will be released from the water,
clogging the warm water tubes [118].

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20. Solar Thermal Energy

The heat loss of the tank, Q̇loss , can be determined withQ̇loss = U A∆T,

(20.8)

where A is the area of the heat storage tank. U is theglobal heat exchange coefficient and is a measure forthe quality of the insulation. Usually it varies between
2 and 10 W/K. Further, U is also a function of the different media between which the heat exchange takes
place.
The same principles can be applied to small and big
storage systems. Small water energy storage can coverdaily fluctuations and is usually in the form of watertanks with volumes from several hundreds of litres up
to several thousands of litres. Large storage systems canbe used as a seasonal storage. Often they are realised asunderground reservoirs.
Another type of energy storage are so-called packed beds,
which are based on heat storage in solids. They use theheat capacity of a bed of loosely packed particulate material to store the heat. A fluid, usually air, is circulated
through the bed to add or remove energy. A variety
of solids can be used, rock being the most common. In
operation, flow is maintained through the bed in one
direction during addition of heat and in the opposite
direction during removal. The bed is in general heatedduring the day with hot air from the collector. In theevening and during night energy is removed with airtemperatures around 20°C flowing upward.

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Another way to store heat in buildings is to use thesolids of that the walls and roofs are built as a storage
material. A case of particular interest is the collectorstorage wall, which is arranged such that the solar radiation is transmitted through a glazing and absorbed in
one side of the wall. As a result, the temperature of thewall increases and the energy can be transferred fromthe wall to the room by radiation and convection. Someof these walls are vented to transfer more energy to theroom via forced convection.
The last method to store heat that we will discuss is
to use phase change materials (PCM). While in all thestorage methods discussed so far energy is stored assensible heat, in PCM storage takes place as latent heat,
which is used for a phase change, without any change
in the material temperature. PCM used for energy storage must have a high specific latent heat L, such that alarge amount of energy can be stored. In addition, thephase change must be reversible, and survive cycles
without significant degradation. The stored amount ofheat can be calculated with
Qstored = m [Cs ( T ∗ − T1 ) + L + Cl ( T2 − T ∗ )] ,(20.9)

where the different temperatures are also sketched inFig. 20.1. We consider the specific heat capacity in thesolid phase, Cs , to be constant from the initial temperature T1 up to the phase change temperature T ∗ , the latent heat of the material L and the specific heat capacity
in the liquid phase Cl from T ∗ up to the final temper-

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20.2. Solar thermal heating

ature T2 . Materials that are commonly used s PCM aremolten salts, such as Na2 SO4 , CaCl2 or MgCl2 . PCM
storage in general is used for high temperature applications.

20.2.3

System design

Often, at least at larger latitudes with large differences
between summer and winter, solar thermal systems
include a boiler as a backup. Its main function is toprovide the necessary energy when the solar power is
not sufficient. It is basically a normal heater that adds
the remaining heat needed to achieve the desired temperature in the storage tank. As energy source, usually
natural gas, oil, or biomass is used. Sometimes, the additional heat also is provided with electricity or a heatpump.
Up to now we discussed how to collect and store theenergy, but we did not consider yet how to transport
it from the collector to the storage system. The transport of heat is done with a collector circuit, which usually transports heat using either a liquid or gas. If a liquid is used, it is important that it neither should freeze
nor boil, even at the most extreme operating conditions.
Further, the medium should have a large specific heat
capacity, a low viscosity, and it should be non-toxic,
cheap and abundant. The most common fluids are water, oils or air.

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As mentioned above, the flux can be caused either naturally by the temperature gradients, forced by a pump,or by a heat pipe, in which the fluid is allowed to boil
and condense again. The optimal choice will depend on
the specific system design. Further, heat losses in thecollector circuit must be taken into account, especiallywhen the pipes are very long. During the planningphase it therefore is important to minimise the circuit
length.
In systems with a pump often a controller is present that
regulates the fluxes of fluid through the collector, thestorage and the boiler in order to assure the desired
temperature of the. This can for example be done bycalculating the optimal flux in dependence of the fluid
temperature at the collectors and in the storage suchthat the heat transfer is maximised.

20.2.4

Solar air conditioning

Another interesting application is solar air conditioning(also called solar cooling), which seems a bit contradictory on first sight. Before we start with the actual discussion on solar air conditioning, we briefly recap theprinciple of a compressor-driven air conditioning system, which is sketched in Fig. 20.8. Similar to every heatpump, the task of an air conditioning system is to transport heat from a cool reservoir B to a warmer reservoir A. Such a system has typically four components:

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20. Solar Thermal Energy

thermal expansion
valve

evaporator

condenser

released during phase changes.

compressor

warm side B

cold side A

Figure 20.8: A compressor-driven cooling circuit.

a compressor, a condenser, a thermal expansion valve(throttle) and an evaporator.
In the compressor (1), vaporised coolant is compressed
to a higher pressure. Under release of its latent heat tothe surroundings, it is condensed to its liquid state in
the condenser (2). Then the coolant passes a thermal
expansion valve (3), such that its pressure on the otherside of this valve is so low that it can evaporated again.
For the evaporation, it must absorb latent heat fromthe cool reservoir A. Then, the coolant containing thelatent heat from A is compressed again. We see that
this cooling circuit utilises the latent heat stored and

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Naturally, heat only flows from warm to cold reservoires. For reversing the heat flow, as it happens in aheat engine, additional energy must be added to thesystem. This happens in the compressor and traditionally is supplied as electric energy. Probably the simplest
option to realise solar cooling is to generate this electricity by a conventional PV system or a solar thermal
power system as discussed in Section 20.3. Further, it isalso possible to drive the compressor with mechanic energy obtained from solar energy directly with a Rankine
cycle, which is discussed in 20.3.
The choice of the coolant is based on the temperature
regime in which the air conditioner is supposed to operate. However, also their effect on the environment
has to be taken into account. For example the use, chlorofluorocarbons that were widely used as coolants wasstrongly banned in the 1980s because of their destructive effect on the ozone layer [119].
Another concept that that directly can be driven bysolar heat is that of an absorption cooling machine. In difference to compressor-driven cooling, no mechanicalenergy is required for absorption cooling, but the cooling process is directly driven by heat that can be supplied by solar collectors. Instead of one coolant, here
two substancesare used: a refrigerant and an absorbant. Often, ammonia (NH3 ) is used as refrigerant, while water
is used as an absorbant. Under atmospheric pressure,This copy is provided for free, for personal use only.

20.2. Solar thermal heating363

ammonia has a boiling point of -33°C. Further, it is very
soluble in water, a property that is utilised in the cooling process. Figure 20.9 sketches the cooling circuit ofan absorption cooler. In the boiler (1), a mixture of bothsubstances is heated. Because of its low boiling point
the refrigerant will leave the mixture in the gas phases.
Absorbant that is still present in this gas is separated(2) and brought back into the boiler. Then, the refrigerant is then is condensed in a condenser (3). Then, thecondensed refrigerant is brought to a thermal expansion
valve (4). Similar to the compressor circuit, latent heatis used to evaporate the expanded refrigerant, such thatcooling takes place. In the last step, the refrigerant isabsorbed by the absorbant and the circuit starts again.
Instead of using a throttle valve to expand the refrigerant, it also can be mixed with a third gas, for examplehydrogen (H2 ), such that the partial pressure of the refrigerant is reduced allowing it to evaporate. In thatversion, the whole system operates at one pressure.

Figure 20.9: Illustrating an absorption cooling circuit asinvented by Einstein and Szilard [120]. (Annotations by P.
Brandon Malloy.)

Copyright Delft University of Technology, 2014

The last option of cooling that we discuss is that ofsolar desiccant cooling, illustrated in Fig. 20.10. In sucha system, air air is dried when passing through a desiccant, like silica. Then water is sprayed into the air. As it
evaporates, the air is cooled and humidified, such thatit guarantees optimal interior conditions. Air that leaves
the interior is heated up using a solar collector. The hot
air streams through the desiccant. Hence, the desiccantis tried and can be reused for adsorbing humidity.

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20. Solar Thermal Energy

20.3

In this section we take a closer look at concentrated solar
power (CSP), where high temperature fluids are used insteam turbines to produce electricity. Much of the earlyattention on CSP systems was on small-scale applications, mainly for water pumping. However, since about
1985 several large-scale power systems with a power
output up to 80 MW have been built.

solarcollector
air heater for
desiccantregeneration

exterior

interior

desiccant(rotating)heat
exchanger

humidifier

Figure 20.10: Desiccant cooling.

Concentrated solar power(CSP)

The CSP technology is especially interesting for desert
regions, where almost all the solar radiation is incident as direct radiation. As illustrated in Fig. 20.11, thesesystems consist basically a collector, where the solar
energy is absorbed, a storage system, usually water orphase-change storage, a boiler that acts as a heat exchanger between the operational fluids of the collector
and the heat engine, and the heat engine itself, whichconverts the thermal energy into mechanical energy.
In large CSP plants, the heat engine is a steam turbine,where the energy stored in the hot steam is partiallyconverted to rotational (mechanic) energy as the steam
expands along the turbine. This thermodynamic process
is called Rankine cycle, called after the Scottish engineer
William John Macquorn Rankine (1820-1872).
The mechanical energy can be further used to drive anelectrical generator. The collectors are built as concen-

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20.3. Concentrated solar power (CSP)

trator systems, in order to reach the high temperaturesof several hundreds of degree Celsius, which are required to operate the heat engine.
One problem of CSP systems is that the efficiency of thecollector diminishes as its operating temperature rises,
while the efficiency of the engine increases with temperature as we already have seen when discussing thethermodynamic efficiency limit in Section 10.1. Hence,
a compromise between the two has to be found whenchoosing the operating temperature. Current systems
have efficiencies up to about 30% [121],

heat storage

collector arraypump

evaporator

feedwaterpumpcooling towersteam
turbinegeneratorcondenser

electricity
grid

Figure 20.11: Sketching a solar thermal electric power sys-tem.

Copyright Delft University of Technology, 2014

Now we will take a closer look at the solar concentrator
system, illustrated in Fig. 20.12. First concepts alreadywere developed by the ancient scientist Archimedes(287 BC - 212 BC); Leonardo da Vinci (1452 - 1519) designed some concentrators. Different types of concentrators produce different peak temperatures and hence
thermodynamic efficiencies, due to the different ways of
tracking the sun and focusing light. Innovations in thisfield are leading to more and more energy efficient andcost effective systems.
The most common concentrator type with about 96%
of all installed CSP installations is the parabolic trough[126], which consists of a linear parabolic reflector thatconcentrates the light onto an absorber tube located in
the middle of the parabolic mirror, in which the working fluid is located. The fluid is heated to 150 to 350
degrees Celsius, and then used in a heat engine. This

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20. Solar Thermal Energy

technology is far developed, well-known examples are
the Nevada Solar One power plant and the power plantthat the Plataforma Solar de Almeria (PSA) in Spain.

(a) Parabolic troughAnother concentrator concept is that of Fresnel concentrators, where flat mirrors are used, as illustrated in
20.12. Flat mirrors allow more reflection in the sameamount of space as parabolic mirrors, reflect more sunlight, and are much cheaper.

(d) Solar power tower

Other important concentrator systems is the dish Stirling or dish engine system. It consists of a parabolic reflector that concentrates light to the focal point, where
the working fluid absorbs the energy and is heated up
to about 500°C. The heat is then converted to mechanical energy using a Stirling engine. With about 31.25%
efficiency demonstrated at Sandia National Laboratories, this design has currently the highest demonstrated
efficiency [121].

Figure 20.12: Different types of concentrators used in CSP

The last concentrator type is that of solar power tower
plants. Here, an array of dual-axis tracking reflectors,commonly named heliostats, are arranged around a
tower. There the concentrated sunlight hits a central receiver, which contains the working fluid, which can beheated to 500 or even 1000°C. Examples for solar power
towers are Solar One and Solar Two in the Unites Stars
and the Eureka project in Spain [126].

(b) Fresnel concentrator(c) Dish Stirlingsystems [122–125].

Copyright Delft University of Technology, 2014

CSP systems can be combined with heat storage, suchas phase-change materials discussed in Section 20.2.

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20.3. Concentrated solar power (CSP)

367

This allows CSP systems to operate during day andnight.

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368

Copyright Delft University of Technology, 201420. Solar Thermal Energy

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21

Solar fuels

In this chapter we discuss how to store solar energy
in the form of chemical energy in so-called solar fuels.
As we have seen in previous chapters, we are able toconvert solar energy to electricity with efficiencies of up
to 44%. Also, we are able to convert solar energy easily
in heat, which we can use for preparing warm water,
heating and even cooling.

summer until winter would require very large battery systems that usually are not feasible. This problem becomes more severe in regions that are far awayfrom the equator such that the differences in solar irradiance between summer and winter are significant.
The same also is true for solar heat that was discussed
in Chapter 20: while a hot water storage tank with avolume of several hundreds of litres is sufficient as daynight storage, seasonal storage that would allow to use
solar-heated water throughout the year would require
storage tanks with a volume of tens of thousands litres.
This tank also would need to be heavily isolated. If wemanage to store solar energy as chemical energy, we do
not have this problem as chemicals can be easily stored.

The biggest obstacle towards a economy that is driven
by renewable energy is not the generation of sustainable energy sources but the storage the renewable energy. As we have seen in Chapter 17, storing electricity with batteries is very difficult. While day-night
storage that makes solar electricity generated during
daytime available in the night is easily done, seasonal
storage that allows to store electricity generated in the

Let us take another look at the Ragone chart in Fig.
369

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370

21. Solar fuels
ultracap.

Power density (W/kg)

104

10

3

10

Li-ionNiCd NiMH bat.
H2,
combustion
gasolineelektrolyte
cap.
doublelayer cap.

102

10

-1

Li-ion cap.

CH4

Lead
acid

H2, fuel cell

10
10
10
Energy density (Wh/kg)
0

2

103

Figure 21.1: A Ragone chart of different energy storage meth-ods. Capacitors are indicated with “cap”.

21.1 that we already analysed in Section 17.4. This
chart shows the amount of energy per mass stored ina certain storage technology versus the specific power
provided by the technology. As we know, by far themost used form of stored energy are fossil fuels. As
we can see from the graph, fossil fuels have high energy density properties, are stable and reliable. But wealso know that fossil fuels are depleting and their use
is detrimental to our environment. Hydrogen (H2 ) has
an even better gravimetric energy density compared tofossil fuels, but it is a very light gas, so the volumetricenergy density, i.e. the energy per unit volume, is much
lower. This is one of the reasons it has not been widely

Copyright Delft University of Technology, 2014

used until now. When we compare batteries and hydrogen as energy storage for solar electricity, we see thatbatteries have a lower energy density. Further, they require a much higher initial investment than hydrogen.
Hence, it seems an interesting option to use hydrogenas a storage material, i.e. as a solar fuel.
As we see in this chapter, we can produce hydrogen using solar energy by utilising electrochemistry. In nature,
photosynthesis takes place, where sunlight is used toconvert carbon dioxide and water into oxygen and sugars, which can be seen as nature has been converting
the energy from the sun into chemical energy for a longtime by converting carbon dioxide and water into oxygen and sugars. Here, we try to mimic nature with inorganic semiconductor materials that re able to split awater molecule into oxygen and hydrogen using the energy of sunlight. This process sometimes is referred toas artificial photosynthesis.
For storing solar energy as chemical energy in the formof hydrogen, water splitting can be used,
2H2 O → 2H2 + O2 .

(21.1)The energy required for this reaction is given by the
Gibbs free energy and it has a value of G = 237.2 kJ/mol.
In solar water splitting this energy is provided by thesun.
Another product that seems very promising for energy
storage is methane (CH4 ), as it is easier to store andThis copy is provided for free, for personal use only.

371

21.1. Electrolysis of water

has less hazardous problems than H2 . It can be produced from hydrogen and carbon dioxide (CO2 ) with
the Fischer-Tropsch process. In this process, first the hydrogen is combined with carbon dioxide in a water gas
shift reaction to obtain carbon monoxide and water inthe form of steam,
H2 + CO2 → 2H2 O + CO.

(21.2)

O2

H2O

2H2

This gas mixture, known as synthesis gas, can be thenrefined to finally obtain methane,
CO + 3H2 → CH4 + H2 O.

(21.3)

When the stored energy is required, the methane can beburned in a combustion reaction,
CH4 + 2O2 → CO2 + 2H2 O(21.4)

that will give water and carbon dioxide as byproducts.
The water can be reused for the electrolysis and thecarbon dioxide can be used for the water gas shift reaction, hence, the cycle is closed. Each reaction in thecycle has a certain efficiency; the overall efficiency canbe obtained by combining all those efficiencies.

PV array

–
–

=

+

Control= electronics
+

Anode

CathodeFigure 21.2: Illustrating a lab-scale Hofmann voltameter thatis used to split water. In the sketch the voltameter is connected
to a PV system via some control electronics.

Many different methods can be used for water splitting.
In this chapter we will discuss two methods: electrolysis
of water and photoelectrochemical water splitting (PEC).

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372

21.1

21. Solar fuelsElectrolysis of water

Figure 21.2 shows a typical lab-scale setup for the hydrolysis of water. It is called a Hofmann voltameter after
the German chemist August Wilhelm von Hofmann(1818-1892). It contains of three upright cylinders thatare joined at the bottom. While electrodes are placedin the left and the right cylinder, the central cylinder
is used to refill the device with water. For starting theelectrolysis, a voltage has to be applied between thetwo electrodes.
Water splitting is a reduction-oxidation reaction, which iscommonly abbreviated as redox reaction. In such redox
reactions, the reaction happens due to the exchangeof electrons between atoms or molecules. They can bedivided in two half reactions, the oxidation and the reduction. For water splitting these half reactions look as
follows: During the oxidation, which happens at theanode, water is split giving oxygen and protons to thesolutions and electrons to the anode,
H2 O → 12 O2 + 2H+ + 2e− .

(21.5)In the reduction, electrons originating from the cathodereact with the protons to hydrogen,
2H+ + 2e− → H2 .

(21.6)The total reaction therefore can be written as
H2 O → 12 O2 + H2 .

Copyright Delft University of Technology, 2014(21.7)

Each half reaction has an associated potential and thesum of the potentials of each half reaction is the potential for the whole redox reaction. Potentials always aredefined with respect to a reference, i.e. a zero point. For
redox reactions, the zero is defined as the hydrogen half
reaction that happens at the cathode. For calculating
the potential at the anode, ie of the oxygen production
reaction, we can apply the equation
V0 =

G
,
nF

(21.8)

where G is the Gibbs free energy already mentioned
above, n is the number of electrons transferred between
the anode and the cathode for the production of one H2
molecule, and F = 96 485 C/mol is the Faraday constantthat tells us the amount of charge per mole of electrons.
Using G = 237.2 kJ/mol and n = 2, as two electrons aretransferred per H2 molecule, we find V 0 = 1.23 V.
In reality, an applied voltage of 1.23 V is not enough todrive the redox reaction because of some voltage losses.
These voltage losses are mainly because of the activation energy at the electrodes as well as Ohmic losses in
the cables, electrodes and also in the solution. For increasing the conductivity of the water, a base or an acidcan be added. For the classic Hofmann voltameter, asmall amount of sulphuric acid (H2 SO4 ) is used.
The potential difference between V0 and the potential
used in the real device is called overpotential ∆V. The

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373

21.2. Photoelectrochemical (PEC) water splitting

overpotential due to the activation energy required at
the electrodes is strongly related to the electrode material and the gas that is produced at that electrode. For
hydrogen production at the cathode, platinized (black)platinum is one of the best materials with an overpotential of −0.07 V. For the oxygen production at the anode, nickel with an overpotential of +0.56 V is very well
suited. Combining these two voltages and the overpotentials we can understand that the typical overpotential is usually around 0.8V.
It is important to realise that all the energy consumed
by the hydrolysis process that is related to the overpotential is a loss that will lead to heating up of the hydrolysis device. Only a fraction
ηWS =

V0
V 0 + ∆V

(21.9)of the supplied energy is actually stored in hydrogen. It
is clear that it is a very important task for science andindustry to reduce the required overpotential. Further,
the current requirement for platinum electrodes makes
hydrolysis not feasible from an economic point of view.
For an overpotential of 0.8 V, we would get an efficiency of around 60%. If we would use a good PV system with an overall efficiency of ηPV = 18% we would
have a total maximal solar-to-hydrogen efficiency ofηSTH = ηPV ηWS = 10.9%.

Copyright Delft University of Technology, 2014(21.10)21.2

Photoelectrochemical (PEC)
water splitting

Hydrogen can also be produced utilising solar energy
by using a photoelectrode, which uses light to produce anelectrochemical reaction, in which water is split into
oxygen and hydrogen. In this process, the photons
reach the surface of the photoelectrode, which is madeof a photoactive semiconductor. As in any other semiconductor, photons with an energy equal or larger
than the semiconductor band gap energy will create anelectron-hole pair. The electrons and holes will be separated by an electric field, and both will be used in thetwo half reactions involved in the overall water splitting process. To generate the required electrical field,
we need a voltage source, for example a solar cell. The
photoelectrode can be either an anode or a cathode.
If the solar-splitting device is made of a solar cell thatis placed behind the photoanode, the solar cell will
receive the light transmitted through the photoanode.
This light creates another electron-hole pair and electricfield that brings the electrons to the photoanode andthe holes to the photocathode with enough potential todrive the redox reaction in the electrolyte. As a result,
the water molecule is split into oxygen and hydrogen.
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374

21. Solar fuels

acts as electron acceptor. As an acceptor, the material
will attract more holes to the interface, which will enhance the reduction half reaction. Hence it acts as thephotocathode and produces hydrogen. If the semiconductor is n-type, it acts as the photoanode. Electrons
are moved to the interface by the internal electric field,
and those electrons are involved in the oxidation half
reaction, such that oxygen is produced.
The semiconductor material has to fulfil several requirements. First, it has to absorb the light that is incident
on its surface. Secondly, charge carrier transport inside the material and separation into the two electrodes
must be efficient. Thirdly, a bandgap of 1.23 eV is notenough to drive the reaction because of the overpotential already discussed above. It has been estimated thatmaterials with an energy band gap close to 2.1 eV havethe potential to split water. Fourthly, the energy levelsof the material have to be adequate to couple with theenergy needed for the reaction. In particular, the energy
levels of the reactions have to be located somewherein the energy band gap of the semiconductor, which iscalled the favourable position. To further enhance thereaction, a catalyst may be added to the semiconductor
surface. Fifthly, on the practical side, it is important thatthe used materials are photochemically stable and relatively cheap. From these criteria we can conclude thatthe main technical challenges to be addressed are lightabsorption, the separation of charges and the catalysisof the reaction.

Copyright Delft University of Technology, 2014

Figure 21.3: The solar-to-hydrogen efficiency and bandgapof different potential photocathode materials. Also the photon
flux is shown [127].

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375

21.2. Photoelectrochemical (PEC) water splitting

The overall efficiency of water splitting depends on the
catalytic efficiency and the separation efficiency of thephotoelectrode. The catalytic efficiency can be improvedby placing a catalyst on the semiconductor surface. For
example, for a ceBiVO4 photoanode the inclusion of acobalt phosphate [Co3 (PO4 )2 ] catalyst on the surface
will ease the oxidation reaction and hence increase thewater splitting efficiency. The separation efficiency canbe improved by introducing an electric field inside thematerial. One way to do this is to introduce gradient
doping, starting from no doping at the surface to 1%

Copyright Delft University of Technology, 20144
Current density (mA/cm2)

The absorption and catalysis problems can be tackled
by carefully choosing the semiconductor material andits corresponding catalyst. Several materials can beconsidered for solar water splitting. Some of the mostpopular materials studied for this application are titanium dioxide (TiO2 ), tungsten oxide (WO3 ), bismuth
vanadate (BiVO4 ), hematite (Fe2 O3 ) or amorphous silicon carbide (a-SiC). As shown in Fig. 21.3, BiVO4 ,Fe2 O3 , and a-SiC have the most promising potential
solar-to-hydrogen efficiency. , as shown in the graph.
Hematite and silicon carbide are the best choice forthe optimal absorption of light, since they have band
gap energies closest to the optimal bandgap of 2.1 eV.
However, if also other factors such as the band position or stability are considered, materials like bismuth
vanadate may also be a viable option. Because of theirlarge bandgap energies, the other materials are not considered.

BiVO4+Gradient doping+Catalyst

BiVO4+ Catalyst

2
BiVO4
dark

0
0.5

1.0
1.5
Potential vs. RHE (V)

2.0

Figure 21.4: The effect on using a catalyst and gradient dop-

ing on the performance of a BiVO4 photoanode [127].

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376

doping close to the back of the electrode. With gradient doping a depletion region is created in between thesemiconductor and the electrolyte. As a result, electrons will move more easily from the electrolyte into
the semiconductor. Combining both effects can highly
improve the efficiency of the overall device. Both the effects of the catalyst and the gradient doping are shownin Fig. 21.4, where the current density of the illuminated photoanode is shown in dependence of the applied
voltage with respect to the reversible hydrogen electrode(RHE). The RHE is a special electrode with the property
that the measured potential does not changed when the
pH value of the solution is changed.
As we already mentioned above, for the photoelectrochemical (PEC) water splitting process, a potential
difference of at least 1.23 V plus the overpotential ∆V
must be present. The value of ∆V will depend on thematerials and electrolyte used, but it is usually around
0.8V. Both potentials added will result in the total potential difference required to drive the redox reaction.
This voltage will be partially covered by the potential
difference created within the photoelectrode when lightshines on it. However, depending on the material, thePEC only creates 0.6 V of the required voltage. For the
extra potential that is required to allow water splitting,the photoelectrode can be combined with solar cells
that provide the extra potential needed for the reaction
to happen. The combination of a photoelectrode and asolar cell forms a photoelectrochemical device.

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21. Solar fuels

In Fig. 21.5 (a) such a PEC device is sketched. The photoelectrode (in this case a photoanode) is connected to asolar cell in series. The photoanode is connected to thepositive contact of the solar cell. The negative contactof the solar cell is connected to another electrode via
an external circuit. This counter electrode may or may
not be photoactive. The electric circuit is closed by theelectrolyte. The band diagram of this device is shownin Fig. 21.5 (b).
Since the photoelectrode and the solar cell are connected in series, the same current will go through both
the devices, similar as for multi-junction solar cells. In
this device, the light is utilised better then it would
be in the solar cell alone. In the photoelectrode, thephotons with energies exceeding that of the photoelectrode bandgap will be absorbed. The fraction of thelight that has not been absorbed or reflected, called thetransmitted spectrum, reaches the solar cell where it canbe absorbed to generate the extra potential difference
required for water splitting. The solar cell must be optimised for the transmitted spectrum, which is different
from the standard AM1.5 spectrum that is usually used
for solar cell optimisation.
As any semiconductor device, the photoelectrode has
its own characteristic J-V curve. When a solar cell anda photoelectrode are combined, the conditions at which
they will work can be estimated by studying their J-V
curve characteristics. Figure 21.6 shows the J-V curves

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377

21.2. Photoelectrochemical (PEC) water splitting

(a)

A

(b)

–

–
O2

H2

GraphiteElectrolyte–

Gradient-doped
BiVO 4

p–i–n

p–i–n
–

–
–

H 2O/H 2
Transmitted

AM 1.5+–+–

H 2O/O 2

+ –

–
+

Co-Pi

–

+
+

Metal
cathodeElectrolyte

Pt coil Co-Pi Gradient-doped FTO Glass ITO
BiVO 42-jn a-Si Ag/Cr/Al
contact

Semiconductor
electrolyte
interface

TCO

TunneljunctionB a ckcontact

Figure 21.5: (a) Illustrating a photoelectrochemical (PEC) device consisting of a BiVO4 photoanode and a tandem a-Si:H/a-Si:H
solar cell. (b) The band diagram of this PEC device [127].

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378

21. Solar fuelsof the solar cell and of the photoanode in this case.
Since both elements are connected in series, the current of both the solar cell and the photoelectrode must
be the same. Hence, the operational point will be where
both J-V curves cross.

Current density (mA/cm2)

Jsc

Jphotosolar cell
photoanodeNow we will estimate the solar-to-hydrogen efficiency
ηSTH that is related to the amount of produced hydrogen. This amount is directly calculated by the current
density measured at the operating point, which we canunderstand by realising that current is the flow of electric charge per time.
When we assume that all generated charges produced
are involved in the production of hydrogen, the overallsolar-to-hydrogen conversion efficiency, is described by
ηSTH =

Potential

Voc
Voperational

Figure 21.6: The J-V characteristics of the photoelectrode andthe solar cell. The operating point is at the crossing of the two.

Copyright Delft University of Technology, 2014

Jph × 1.23 V,
Pin(21.11)

where Jph is the operational current density, Pin is theirradiance arriving at the PEC device, usually it will be1000 W/m2 from the AM 1.5 spectrum.
The only free variable in Eq. (21.11) is the current density: the higher the operational current, the more hydrogen is produced and the higher the device efficiency.
The major focus of current academic research therefore is to improve the current density. Besides utilising
gradient doping water-oxidation catalysts also other improvements can be done to increase the performanceof PEC devices. These are optimising the materials andThis copy is provided for free, for personal use only.

21.2. Photoelectrochemical (PEC) water splitting

379

layer thicknesses in the used solar cell and texturing of
the photoanode for better light trapping.
All these concepts were applied at the Delft University of Technology, where a bismuth vanadate photoanode was combined with a double junction amorphoussilicon solar cell and a platinum cathode. This device
achieved a solar-to-hydrogen efficiency of ηSTH = 4.9%,
which is actually the highest efficiency reported forsuch devices based on a metal oxide photoanode [127].

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380

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21. Solar fuelsThis copy is provided for free, for personal use only.

Appendix

381
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Copyright Delft University of Technology, 2014

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A

Derivations in Electrodynamics
A.1

The Maxwell equations

where r and t denote location and time, respectively.
D is the electric displacement, E the electric field, B is themagnetic induction and H is the magnetic field. ρ F is thefree charge density and J F is the free current density.

The four Maxwell equations couple the electric andmagnetic fields to their sources, i.e. electric charges andcurrent densities, and to each other. They are given by∇ · D(r, t) = ρ F (r),∂B(r, t)∇ × E(r, t) = −,∂t∇ · B(r, t) = 0,
∂D(r, t)∇ × H(r, t) = ++ J F ( r ),∂t

The electric displacement and field are related to eachother via
D = ee0 E,(A.2a)
where e is the relative permittivity of the medium inthat the fields are observed and e0 = 8.854 × 10−12
As/(Vm) is the permittivity in vacuo. Similarly, the magnetic field and induction are related to each other via(A.1a)(A.1b)(A.1c)

B = µµ0 H,(A.1d)(A.2b)

where µ is the relative permeability of the medium in383

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384

A. Derivations in Electrodynamics

that the fields are observed and µ0 = 4π · 10−7 Vs/(Am)is the permeability of vacuum. Eqs. (A.2) are only valid
if the medium is isotropic, i.e. e and µ are independentof the direction. We may assume all the materials important for solar cells to be nonmagnetic, i.e. µ ≡ 1.

A.2

Derivation of the electromagnetic wave equation

We now derive de electromagnetic wave equations insource-free space, ρ F ≡ 0 and j F ≡ 0. For the derivationof the electromagnetic wave equations we start with applying the rotation operator ∇× to the second Maxwell
equation, Eq. (A.1b),
 
∂B
.
(A.3)
∇ × (∇ × E) = −∇ ×∂t
Now we take the fourth Maxwell equation, Eq. A.1d,with j F = 0 and Eqs. (A.2),

and substitute it into Eq. (A.3),

Copyright Delft University of Technology, 2014

 2 
∂ E
.
∂t2

∇ × (∇ × E) = ∇(∇ · E) − ∆E = −∆E,
we find∆E = ee0 µµ0

 2 
∂ E
.
∂t2

(A.5)

(A.6)

In Eq. (A.5) we used that we are in source free space,i.e. ∇E = 0. Equation (A.6) is the wave equation for theelectric field. Note that the factor1
ee0 µµ0
has the unit of (m/s)2 , i.e. a speed to the square. In
easy terms, it is the squared propagation speed of the
wave.1 We now set
1e0 µ 0(A.7)

n2 = e,(A.8)

c20 :=
andwhere c0 is the speed of light vacuo and n is the refractive index of the medium. Since we also assume µ ≡ 1,
we finally obtain for the wave equation for the electric
field


n2 ∂2 E
= 0.
(A.9a)∆E − 2c0 ∂t2

1∂E∇ × B = ee0 ,µµ0∂t∇ × (∇ × E) = −ee0 µµ0

By using the relation(A.4)

1 In reality, if the medium is absorbing, things are getting much more complex.

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385

A.3. Properties of electromagnetic waves
λSubstituting Eq. (A.10a) into the first Maxwell equation,
Eq. (A.1a), yields

E
B

k

k

∇D = ee0 ∇E = 0,
∇E = 0,∇ E0 eikz z−iωt = 0,

Figure A.1: Illustrating the E and B field and the k vector of aplane electromagnetic wave.

In a similar manner we can derive the wave equation
for the magnetic field,n2∆H − 2
c0

A.3

 2 
∂ H
= 0.
∂t2

(A.9b)

Properties of electromagnetic
waves

In section A.2, we found that plane waves can be described byE(x, t) = E0 · eikz z−iωt ,(A.10a)ik z z−iωt(A.10b)H(x, t) = H0 · e.

In this section we study some general properties ofplane electromagnetic waves, illustrated in Fig. A.1.

Copyright Delft University of Technology, 2014

∂ ik z z−iωt∂ ik z z−iωtee++ Ey,0 ∂yEx,0 ∂x| {z }| {z }=0(A.11a)=0∂ ik z z−iωt+ Ez,0 ∂ze= 0,iEz,0 k z eikz z−iωt = 0,
Ez,0 = 0,where we used the notation E0 = Ex,0 , Ey,0 , Ez,0 . In asimilar manner, by substituting Eq. (A.10b) into the first
Maxwell equation, Eq. (A.1c), we obtain
Hz,0 = 0.

(A.11b)Thus, neither the electric nor the magnetic fields havecomponents in the propagation direction but only components perpendicular to the propagation direction (thex- and y-directions in our case).
Without loss of generality we now assume that theelectric field only has an x-component, E0 = ( Ex,0 , 0 0).
Substituting this electric field into the left hand sideof the second Maxwell equation, Eq. (A.1b), yields

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386

A. Derivations in Electrodynamics    ∂

∂
∂
 
 


Ex
∂y Ez − ∂z Ey∂x000 ∂    ∂
∂∂ ik z z−iωt  =  ∂  ∇×E = iEx,0 k z eikz z−iωt  .
 ∂x  ×  Ey  =  ∂z Ex − ∂x Ez  = ∂z Ex = Ex,0 ∂z e
0
0
0
∂
∂
∂
Ez
∂x
∂x Ey − ∂y Ex
(A.12)The right hand side of Eq. (A.1b) yields





∂
∂ ik z z−iωtHx,0 ∂te−iωHx,0 eikz z−iωt∂t Hx∂B(r, t)∂H(r, t)−= − µ0= −µ0  ∂t∂ Hy  = −µ0  Hy,0 ∂t∂ eikz z−iωt  = −µ0  −iωHy,0 eikz z−iωt  .
∂t
∂t
∂
0
0
∂t Hz

(A.13)

is the impedance of free space.
Thus, we obtain for Eq. (A.1b)




0
−iωHx,0 eikz z−iωtiEx,0 k z eikz z−iωt  = −µ0  −iωHy,0 eikz z−iωt  , (A.14)
00i.e. only the y-component of the magnetic field survives.
The electric and magnetic components are related toeach other via
Ex,0 k z = µ0 ωHy,0(A.15)
By substituting Eq. (4.4) into Eq. (A.15), we findnnEx,0 =
Ex,0 ,
Hy,0 =
cµ0
Z0
where

r
Z0 = cµ0 =µ0= 376.7 Ωe0Copyright Delft University of Technology, 2014

In summary, we found the following properties of theelectromagnetic field.
• The electric and magnetic field vectors are perpendicular to each other and also perpendicular to the
propagation vector,
k · E0 = k · H0 = H0 ·E0 = 0.

(A.18)(A.16)• The electric and magnetic fields are proportional
to the propagation direction, hence electromagnetic
waves are transverse waves.

(A.17)• The electric and magnetic vectors have a constant,material dependent ratio. If the electric field isalong the x-direction and the magnetic field isThis copy is provided for free, for personal use only.

387

A.3. Properties of electromagnetic waves

along the y-direction, this ratio is given by
Hy,0 =

nn
Ex,0 =
Ex,0 ,
cµ0
Z0

where

r
Z0 = cµ0 =µ0= 376.7 Ωe0(A.19)(A.20)

is the impedance of free space.
The derivations in this section were done for planewaves. However, it can be shown that the properties
of electromagnetic waves summarised in the itemisation
above are valid for electromagnetic waves in general.

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388

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A. Derivations in Electrodynamics

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B

Derivation of homojunction J-V curves
In this appendix we present the derivations for the J-V
curves in dark and under illumination for p-n homojunctions, as they were discussed in Section

B.1

across the p-n junction. The concentration of the minority carriers at the edge of the space-charge region increases exponentially with the applied forward-bias
voltage but it is still much lower that the concentration of the majority carriers (low-injection conditions).
The concentration of the majority carriers in the quasineutral regions do not change significantly under forward bias. The concentration of charge carriers in a p-njunction under forward bias is schematically presented
in Fig. B.1

Derivation of the J-V characteristic in dark

When an external voltage, Va , is applied to a p-n junction the potential difference between the n-type and ptype regions will change and the electrostatic potential
across the space-charge region will become (ψ0 − Va ).
Under the forward-bias condition an applied external
voltage decreases the electrostatic-potential difference

The concentrations of the minority carriers at the edgesof the space-charge region, electrons in the p-type semiconductor and holes in the n-type semiconductor after389

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390

B. Derivation of homojunction J-V curves

applying forward-bias voltage are described by
n2i eVae kT ,NAeVa
n2 eVapn (b) = pn0 e kT = i e kT .
NDeVan p ( a) = n p0 e kT =n-type silicon

p-type silicon

n=nn0=ND

p=pp0=NAln nnp(a)=np0eeV/(kT)ln ppn(b)=pn0eeV/(kT)n=np0=pi2/NA

p=pn0=ni2/NDb

PositionaFigure B.1: Concentration profiles of mobile charge carriers

in a p-n junction under forward bias (red lines). Concentration
profiles of carriers under thermal equilibrium are shown forcomparison (black lines).

(B.1a)(B.1b)

Since we assume that there is no electric field in thequasi-neutral region the current-density equations ofcarriers reduce to only diffusion terms and are notcoupled by the electric field. The current is based on
the diffusive flows of carriers in the quasi-neutral regions and is determined by the diffusion of the minority carriers. The minority-carriers concentration can becalculated separately for both quasi-neutral regions. The
electron-current density in the quasi-neutral region of
the p-type semiconductor and the hole-current density
in the quasi-neutral region of the n-type semiconductor
are described by
dn
,
dx
dp
J p = −eDP .
dx

Jn =

eD N

(B.2a)(B.2b)

The continuity equations (4.34) for electrons and holes
in steady-state (∂n/∂t = 0 and ∂p/∂t = 0) can be writ-

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391

B.1. Derivation of the J-V characteristic in dark

ten as

By combining Eqs. (B.2a), (B.3a) and (B.4a) we obtain

1 dJN− R N + GN = 0,
(B.3a)e dx
1 dJP−− R P + GP = 0.
(B.3b)
e dx
Under low-injection conditions, a change in the concentration of the majority carriers due to generation
and recombination can be neglected. However, therecombination-generation rate of minority carriers depends strongly on the injection and is proportional tothe excess of minority carriers at the edges of the depletion region. The recombination-generation rate ofelectrons, Rn , in the p-type semiconductor and holes,
R p , in the n-type semiconductor is described by Eqs. (),
∆n
Rn =
,
τn
∆p
,
Rp =
τp

(B.4a)(B.4b)

where ∆n is the excess concentration of electrons in thep-type semiconductor with respect to the equilibrium
concentration n p0 , and τn is the electrons (minority carriers) lifetime. ∆p is the excess concentration of holes
in the n-type semiconductor with respect to the equilibrium concentration pn0 and τp is the holes (minority
carriers) lifetime. ∆n and ∆p are given by∆ n = n p ( x ) − n p0 ,(B.5a)∆ p = pn ( x ) − pn0 .

(B.5b)

Copyright Delft University of Technology, 2014

DN

d2 n p ( x )∆n=− GN .
2τndx(B.6a)

This equation describes the diffusion of electrons in
the p-type semiconductor. Similarly, by combining Eqs.
(B.2b), (B.3b) and (B.4b) we obtainDP

d2 pn ( x )∆p− GP ,=
τpdx2(B.6b)

that describes the diffusion of holes in the n-type semiconductor.
Now we substitute n p ( x ) from Eq. (B.5a) and pn ( x )from Eq. (B.5b) into Eqs. (B.6a) and (B.6b), respectively.
By using that d2 n p0 /dx2 = d2 pn0 /dx2 = 0, and in dark
GN = GP = 0 we obtain
d2 ∆n
∆n
=
,
2
D N τn
dx
d2 ∆p
∆p
.
=
DP τp
dx2

(B.7a)(B.7b)The electron-concentration profile in the quasi-neutral
region of the p-type semiconductor is given by the general solution to Eq. (B.7a),




x
x
∆ n ( x ) = A exp+ B exp −,LN
LN

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392

B. Derivation of homojunction J-V curves√
where L N = D N τn [see Eq. (7.28a)] is the electron
minority-carrier diffusion length. The origin of the xaxis is set to the edge of the depletion region in the ptype semiconductor and denoted as a in Fig. B.1. For
the p-type semiconductor infinite thickness is assumed(approximation of the infinite thickness). The constants A
and B can be determined from the boundary conditions
1. At x = 0, n p ( a) = n p0 exp(qVa /kT ),2. n p is finite at x → ∞, therefore A = 0.
With these boundary conditions the solution for theconcentration profile of electrons in the p-type quasineutral region is found to be

 eVa
− x
n p ( x ) = n p0 + n p0 e kT − 1 e L N .

(B.8a)The hole concentration profile in the quasi-neutral region of the n-type semiconductor is given by the general solution to Eq. (B.7a),∆ p x 0 = A0 exp



x0
LP





x0
+ B0 exp −,LPpwhere L P =DP τp [see Eq. (7.28b)] is the holeminority-carrier diffusion length. The origin of the x 0 axis ( x 0 := − x ) is set at the edge of the depletion region
in the n-type semiconductor and denoted as b in Fig.
B.1. Here, we use the approximation of infinite thickness for the n-type semiconductor. The constants A0

Copyright Delft University of Technology, 2014

and B0 can be determined from the boundary conditions
1. At x 0 = 0, pn (b) = pn0 exp(qVa /kT ),2. p − n is finite at x 0 → ∞, therefore A0 = 0.
We thus find the concentration profile of holes in thequasi-neutral region of the n-type semiconductor to be
 eVa


− x
pn x 0 = pn0 + pn0 e kT − 1 e LP .
(B.8b)
When substituting the corresponding concentration profiles of minority carriers (Eqs.B.8) into Eqs. (B.2) we obtain for the current densities
e D N n p0  eVa− x
e kT − 1 e L N ,JN ( x ) =(B.9a)LNe DP pn0  eVa− x
JP x 0 =e kT − 1 e LP .
(B.9b)
LP
Under the assumptions that the effect of recombination and thermal generation of carriers in the depletion region can be neglected, which means that the electron and hole current densities are essentially constantacross the depletion region, one can write for the current densities at the edges of the depletion region
e D N n p0  eVa(B.10a)J N | x =0 = J N | x 0 =0 =e kT − 1 ,LNe DP pn0  eVa
J P | x 0 =0 = J P | x =0 =e kT − 1 .
(B.10b)LP

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393

B.2. Derivation of the J-V characteristic under illumination

The total current density flowing through the p-n junction at the steady state is constant across the device
therefore we can determine the total current density as
the sum of the electron and hole current densities at theedges of the depletion region,
J (Va ) = JN | x=0 + JP | x=0(B.11)
e D N n p0
eVa
e DP pn0
e kT − 1 .
+
=
LN
LP

n-type silicon

p-type silicon

n=nn0=ND

p=pp0=NAln nUsing Eqs. (8.1b) and (8.2b), Eq. (B.11) can be rewritten
as




eVa
J (Va ) = J0 exp
−1 ,
kT

ln p
Gτn

Gτp

(B.12)

n=np0=pi2/NA

p=pn0=ni /ND
2

where J0 is the saturation-current density of the p-njunction which is given byb

PositionaFigure B.2: Concentration profiles of mobile charge carriersJ0 =e D N n2i
e DP n2i+
L N NAL P ND!
.

(B.13)

in an illuminated p-n junction with uniform generation rate
G (red line). Concentration profiles of charge carriers underequilibrium conditions are shown for comparison (black line).

Equation (B.12) is known as the Shockley equation that
describes the current-voltage behaviour of an ideal p-n
diode. It is a fundamental equation for microelectronics
device physics.

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394

B.2

B. Derivation of homojunction J-V curves

Derivation of the J-V characteristic under illumination

When a p-n junction is illuminated the additional
electron-hole pairs are generated through the junction. In case of moderate illumination the concentration of majority carriers does not change significantly
while the concentration of minority carriers (electrons
in the p-type region and holes in the n-type region) willstrongly increase. In the following section it is assumed
that the photo-generation rate, G, is uniform throughout the p-n junction, this is called the uniform generationrate approximation. This assumption reflects the situation
where the device is illuminated with a long-wavelength
light, which is weakly absorbed by the semiconductor.
The concentration of charge carriers in a p-n junction
with uniform photo-generation rate is schematicallypresented in Fig. B.2.

Eqs. (B.6a) described have seen the steady-state situation for minority carriers. In section B.1 we then proceeded with assuming that the generation rate is zero.
Now we analyse the case where the junction is illuminated, i.e. the generation rate is not zero. The equations

Copyright Delft University of Technology, 2014

thus can be rewritten to
∆n
d2 ∆n
G
=
−
,
2
D N τn
DN
dx
∆p
d2 ∆p
G
=
−
.
2
DP τp
DP
dx

(B.14a)(B.14b)Under the assumption that G / D N and G/DP are constant, the general solution to Eq. (B.14) isx− x∆ n ( x ) = G τn + C e L N + D e L N ,
0
x0

−x
∆ p x 0 = G τp + C 0 e LP + D 0 e LP .

(B.15a)(B.15b)

The constants in Eqs. (B.15) can be determined fromthe same boundary conditions as were used in the analysis of the p-n junction in dark. The particular solution
for the concentration profile of electrons in the quasineutral region of the p-type semiconductor and holes in
the quasi-neutral region of the n-type semiconductor isdescribed by

i
h
 eV
− x
n p ( x ) = n p0 + G τn + n p0 e kT − 1 − G τn e L N ,

pn x


0

(B.16a)

i
 eV
0
−x
= pn0 + G τp + pn0 e kT − 1 − G τp e LP .
h

(B.16b)By substituting these concentration profiles of minority
carriers into Eqs. (B.2), we obtain for the current densit-

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395

B.2. Derivation of the J-V characteristic under illumination

pletion region,iese D N n p0  qV
− x
− x
e kT − 1 e L N − e G L N e L N ,LN(B.17a)


0
0
x
qV

e DP pn0−−xx0 =e kT − 1 e LP − e G L P e LP .
LP
(B.17b)JN ( x ) =JP

In case of an ideal p-n junction the effect of recombination in the depletion region was neglected. However,the contribution of photo-generated charge carriers to
the current in the depletion region has to be taken into
account. The contribution of optical generation from thedepletion region to the current density is given by

J N | x =0 = e

Z0

−W
Z0

J P | x 0 =0 = e− G dx = −e G W,

(B.18a)− G dx 0 = −e G W.

(B.18b)−W

The total current density flowing through the p-n junction in the steady state is constant across the junction.
Under the superposition approximation we thus can determine the total current density as the sum of the electron and hole current densities at the edges of the de-

Copyright Delft University of Technology, 2014

J (Va ) = JN | x=0 + JP | x=0e D N n p0
e DP pn0  eVa=
e kT − 1+
LN
LP(B.19)− e G ( L N + LP + W )Eq. (B.19) can be rewritten as




qVa
J (Va ) = J0 exp− 1 − Jph ,kT(B.20)

where Jph is the photo-current given by
Jph = e G ( L N + L P + W ) .

(B.21)

A number of approximations have been made in order to derive the analytical expressions for the currentvoltage characteristics of an ideal p-n junction in dark
and under illumination. These approximations are summarised below:• The depletion-region approximation• The Boltzmann approximation• Low-injection conditions• The superposition principle• Infinite thickness of doped regions• Uniform generation rate

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396

B. Derivation of homojunction J-V curves

The derived expressions describe the behaviour of an
ideal p-n junction and help us understand the basic
processes behind the operation of the p-n junction.
However, they do not fully and correctly describe real
p-n junctions. For example, the thickness of a real p-njunction is limited, which means that the recombination at the surface of the doped regions has to be taken
into account. The thinner the p-n junction, the more important surface recombination becomes. The surface
recombination modifies the value of the saturationcurrent density. Further it was assumed that there are
no recombination-generation processes in the depletion
region. However, in real p-n junctions, the recombination in the depletion region represents a substantial loss
mechanism. These and other losses in a solar cell are
discussed in Chapter 10.

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C

Atomic force microscopy and statistical parameters
C.1

Atomic force microscopy (AFM)

called tapping mode in which the probe is oscillating at
its resonance frequency. From the obtained amplitude
and phase pictures the height profile of the investigatedsample can be obtained.

AFM1 is a powerful technique with which the morphology of nano-textured surfaces can be determined[128]. During an AFM measurement a tiny probe witha tip radius of several nanometers is brought so close
to the surface that atomic forces between the surface
and the probe become important. Due to these forces
the surface morphology can be determined. Under amplitude modulation the amplitude and the phase of the tip
will change during the scan depending on the Van der
Waals forces, i.e. the distance between the tip and thesample. Several modes can be used. We used the so

The height profile is not the real height profile of thesample but a convolution of the height profile and theshape of the tip. A blunt tip therefore will lead to a
blurry picture. If the tip is broken and therefore consists of two or more peaks, the features on the scan will
be doubled. Further, scanning the surface too fast willlead to contact loss between the tip and the sample at
steep features. Therefore the AFM measurements haveto be performed very carefully.

1 This appendix is taken from the PhD thesis of Klaus Jäger [53].

Figure C.1 shows a picture of a typical tip of AFM
397

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398

C. Atomic force microscopy and statistical parameters

The rms roughness in principle is the standard deviation of the height profile. It is defined asvuu
σr = tFigure C.1: The NSG 10 gold-coated single-crystal silicon tip

used throughout this project [129].

N1(z − z̄)2 ,∑N − 1 i =1 i

(C.1)

where N is the number of data points, zi is the heightof the ith datapoint and z̄ is the average height. It becomes clear from the definition that σr is insensitive
to the lateral feature sizes. Samples with very smallor large lateral features both can have the same rms
roughness.

probes. This so-called NSG 10 tip is made from antimony doped single-crystal silicon and coated with gold.
Its tip radius is typically 6 nm, with a guaranteed radius of maximal 10 nm [129].

The correlation length gives an indication of the lateralfeature sizes. Its derivation is less straightforward thenthat of σr : it has to be extracted from autocorrelation
function (ACF) and/or the height-height correlation
function (HHCF) [130]. For a discrete set of data, thetwo-dimensional ACF is given by

C.2

ACF(τx , τy ) =Statistical parameters

The AFM scan reveals three dimensional data of thenano-textured surface via the height function z( x, y),which can be used to extract statistical properties ofthe samples. To compare the different samples we use
two different parameters, the root-mean-squared (rms)roughness σr and the correlation length `c .

Copyright Delft University of Technology, 2014

N −n M−m1· ∑ ∑ z(kδ + τx , lδ + τy ) z(kδ, lδ),
( N − n)( M − m) l =1 k=1(C.2)

where δ is the distance between two data points,
m = τx /δ and n = τy /δ. For AFM scans usually the
one-dimensional ACF along the fast scanning axis ( x ) isThis copy is provided for free, for personal use only.

399

C.2. Statistical parameters

used:(a) 1500

ACFx (τx ) = ACF(τx , 0) =N M−m1
∑ z(kδ + τx , lδ) z(kδ, lδ).
N ( M − m) l∑=1 k =1

ACF (nm2 )

1200
900

ACF
fit

600

The one-dimensional HHCF is given by
HHCFx (τx ) =

300
0
0.1

1

10

τx (µm)HHCF (nm2 )(b) 3000
2400

HHCF
fit

1800
1200
600
0

(C.3)

0.1

1

10

τx (µm)Figure C.2: The ACF, HHCF and fitted Gaussian functions for

SnO2: F.

Copyright Delft University of Technology, 2014N M−m
1[z(kδ + τx , lδ) − z(kδ, lδ)]2 .
∑N ( M − m) l =1 k∑
=1

(C.4)

To determine `c , Gaussian functions can be fitted to the
ACF and the HHCF. They are given by


τx2
2
−
exp
ACFfit
(
τ
)
=
σ
,
(C.5)
x
r
x
`2c



τx2
2
1
−
exp
−
HHCFfit
(
τ
)
=
2σ
,
(C.6)
x
r
x
`2c
respectively. The correlation length then is the length atwhich the (fitted) ACF has decayed to 1/e of its highest
value. Instead of applying Eq. (C.1) directly, the rms
roughness also can be obtained by fitting Gaussianfunctions to the ACF and/or the HHCF.
Figure C.2 shows the ACF and the HHCF and fitted
Gaussian functions for the SnO2 :F sample depicted inFig. 13.3 (a). Instead of fitting to Gaussian functions,some authors also use exponential functions for fitting.
For this sample, Gaussians lead to better fits.

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400

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C. Atomic force microscopy and statistical parameters

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