Chalcogenide Photovoltaics: Physics, Technologies, and Thin Film Devices

Scheer · Schock T his ﬁrst comprehensive description of the most important material properties and device aspects clos...

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Scheer · Schock

T

his ﬁrst comprehensive description of the most important material properties and device aspects closes the gap between general books on solar cells and journal articles on chalcogenide-based photovoltaics. Written by two very renowned authors with years of practical experience in the ﬁeld, the book covers II-VI and I-III-VI2 materials as well as energy conversion at heterojunctions. It also discusses the latest semiconductor heterojunction models and presents modern analysis concepts. Thin ﬁlm technology is explained with an emphasis on current and future techniques for mass production, and the book closes with a compendium of failure analysis in photovoltaic thin ﬁlm modules. With its overview of the semiconductor physics and technology needed, this practical book is ideal for students, researchers, and manufacturers, as well as for the growing number of engineers and researchers working in companies and institutes on chalcogenide photovoltaics.

Roland Scheer, Hans-Werner Schock

Chalcogenide Photovoltaics Physics, Technologies, and Thin Film Devices

Hans-Werner Schock became head of the Institute of Technology at the Helmholtz Zentrum Berlin für Materialien und Energie (former Hahn-Meitner Institute) in the division “Solar Energy Research” in late 2004. He received his diploma in electrical engineering in 1974 and obtained his PhD in electrical engineering from Stuttgart University, Germany, in 1986. Starting in the early 70s, he has taken the development of chalcogenide solar cells from basic investigations to the transfer to a pilot fabrication plant. From 1986 to 2003 he coordinated the research on chalcopyrite based solar cells in the framework of the European photovoltaic program. From 1982 to 2004 he was head of the compound semiconductor thin ﬁlm group of the Institute of Physical Electronics at the University of Stuttgart. He is author or co-author of more than 300 contributions in books, scientiﬁc journals and published conference proceedings. For his achievements in the development of chalcopyrite based solar cells he received the prestigious “Becquerel Prize” in 2010.

www.wiley-vch.de

Chalcogenide Photovoltaics

Roland Scheer received his diploma degree in electronic engineering from the University of Applied Sciences Berlin, Germany. He joined AEG Konstanz, Germany, in 1983 as electronic engineer. In parallel, he studied physics at the University of Konstanz and at the Technical University of Berlin where he received his physics diploma in 1990. In 1994, he ﬁnished his doctoral thesis in physics and joined the Helmholtz-Centre (former Hahn-Meitner Institute) in Berlin. His R&D activities are focused on thin ﬁlm solar cells where he invented a new solar cell based on CuInS2 . In 2002, Roland Scheer was visiting scientist at the Advanced Institute for Science and Technology (AIST) in Tsukuba, Japan. He was lecturer at the University of Potsdam until in 2010 he became full professor holding the endowed chair for photovoltaics at the Martin-Luther-University in Halle-Wittenberg.

Roland Scheer and Hans-Werner Schock Chalcogenide Photovoltaics

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Roland Scheer and Hans-Werner Schock

Chalcogenide Photovoltaics Physics, Technologies, and Thin Film Devices

Prof. Dr. Roland Scheer Martin-Luther Universit¨at Halle-Wittenberg Naturwissenschaftliche Fakult¨at II Institut f¨ur Physik/Fachgruppe Photovoltaik Von Dankelmann-Platz 3 06120 Halle (Saale) Germany Prof. Dr. Hans-Werner Schock Helmholtz-Zentrum Berlin Glienicker Straße 100 14109 Berlin Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at .  2011 WILEY-VCH Verlag & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microﬁlm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Typesetting Laserwords Private Ltd., Chennai, India Printing and Binding Fabulous Printers Pte Ltd Singapore Cover Design Graﬁk-Design Schulz, Fußg¨onheim Printed in Singapore Printed on acid-free paper ISBN: 978-3-527-31459-1

V

Contents Preface XI Symbols and Acronyms XIII 1 1.1 1.1.1 1.2 1.2.1 1.3

Introduction 1 History of Cu(In,Ga)(S,Se)2 Solar Cells 1 Milestones of Cu(In,Ga)(S,Se)2 Development 3 History of CdTe Solar Cells 5 Milestones of CdTe Development 6 Prospects of Chalcogenide Photovoltaics 7

2 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.5.1 2.4.5.2 2.4.5.3 2.4.5.4 2.4.6 2.4.6.1

Thin Film Heterostructures 9 Energies and Potentials 9 Charge Densities and Fluxes 11 Energy Band Diagrams 13 Rules and Conventions 13 Absorber/Window 17 Absorber/Buffer/Window 20 Interface States 24 Interface Dipoles 29 Deep Bulk States 29 Bandgap Gradients 32 Diode Currents 36 Superposition Principle and Shifting Approximation 36 Regions of Recombination 37 Radiative Recombination 40 Auger Recombination 43 Defect Related Recombination 44 SCR Recombination 50 QNR Recombination 60 Back Surface Recombination 62 Interface Recombination 63 Parallel Processes 73 SCR and QNR Recombination 73

Chalcogenide Photovoltaics: Physics, Technologies, and Thin Film Devices. Roland Scheer and Hans-Werner Schock Copyright  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31459-1

VI

Contents

2.4.6.2 2.4.7 2.4.8 2.4.9 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.4.1 2.5.4.2 2.5.4.3 2.5.4.4 2.5.4.5 2.5.4.6 2.5.5 2.5.6 2.5.7 2.5.7.1 2.5.7.2 2.5.7.3 2.6 2.6.1 2.6.1.1 2.6.1.2 2.6.1.3 2.6.2 2.6.2.1 2.6.2.2 2.6.2.3 2.6.2.4 2.6.3 2.6.4

SCR and IF Recombination 75 Barriers for Diode Current 76 Bias Dependence 78 Non-Homogeneities 79 Light Generated Currents 80 Generation Currents 81 Generation Function 84 Photo Current 86 Collection Function 87 Absorber Quasi Neutral Region 87 QNR with Graded Bandgap 90 QNR with Back Surface Field 91 Absorber Space Charge Region 92 Buffer Layer 94 Simulating the Collection Function 95 Quantum Efﬁciency and Charge Collection Efﬁciency 96 Barriers for Photo Current 97 Voltage Dependence of Photo Current 99 Width of SCR 99 Interface Recombination 100 Photo Current Barriers 100 Device Analysis and Parameters 101 Equivalent Circuits 101 DC Equivalent Circuit 101 AC Equivalent Circuit 103 Module Equivalent Circuit 105 Current–Voltage Analysis 107 External Collection Efﬁciency 107 Diode Parameters 108 Open Circuit Voltage 112 Fill Factor 116 Capacitance–Voltage Analysis 119 Admittance Spectroscopy 122

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Design Rules for Heterostructure Solar Cells and Modules 129 Absorber Bandgap 129 Band Alignment 131 Emitter Doping and Doping Ratio 137 Fermi Level Pinning 140 Absorber Doping 142 Absorber Thickness 147 Grain Boundaries 150 Back Contact Barrier 156 Buffer Thickness 159 Front Surface Gradient 162

Contents

3.11 3.12

Back Surface Gradients 165 Monolithic Series Interconnection 171

4 4.1 4.1.1 4.1.2 4.1.3 4.1.3.1 4.1.3.2 4.1.3.3 4.1.4 4.1.4.1 4.1.4.2 4.1.5 4.1.6 4.2 4.2.1 4.2.1.1 4.2.1.2 4.2.1.3 4.2.2 4.2.3 4.2.3.1 4.2.3.2 4.2.3.3 4.2.3.4 4.2.3.5 4.2.3.6 4.2.4 4.2.4.1 4.2.4.2 4.2.5 4.2.5.1 4.2.5.2 4.2.6 4.3 4.4 4.4.1 4.4.2 4.5

Thin Film Material Properties 175 AII -BVI Absorbers 175 Physico-Chemical Properties 175 Lattice Dynamics 179 Electronic Properties 180 Practical Doping Limits 180 Defect Spectroscopy 182 Minority Carrier Lifetime 182 Optical Properties 185 CdTe 185 Multinary Phases 187 Surface Properties 188 Properties of Grain Boundaries 189 AI -BIII -CVI 2 Absorbers 192 Physico-Chemical Properties 193 Ternary Phase Diagrams 193 Multinary Phases 198 Diffusion Coefﬁcients 200 Lattice Dynamics 201 Electronic Properties 203 Single Point Defects 204 Defect Complexes 209 Defect Spectroscopy 210 Practical Doping Limits 214 Carrier Mobility 215 Minority Carrier Lifetime 216 Optical Properties 216 Ternary Semiconductors 217 Multinary Semiconductors 218 Surface Properties 220 Surface Composition 220 Surface Electronics 222 Properties of Grain Boundaries 224 Buffer Layers 226 Window Layers 226 Low Resistance Windows 226 High Resistance Windows 230 Interfaces 230

5 5.1 5.1.1

Thin Film Technology 235 CdTe Cells and Modules 235 Substrates 235

VII

VIII

Contents

5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.1.7 5.1.8 5.1.8.1 5.1.8.2 5.1.9 5.2 5.2.1 5.2.2 5.2.3 5.2.3.1 5.2.3.2 5.2.3.3 5.2.3.4 5.2.4 5.2.5 5.2.6 5.2.7 5.2.8 5.2.8.1 5.2.8.2 5.2.9 5.2.10

Window Layers for CdTe Cells 236 Buffer Layers for CdTe Cells 237 CdTe Absorber Layer 240 Activation by Chlorine Treatment 243 Inﬂuence of Oxygen 245 Inﬂuence of Copper 246 Back Contact 249 Surface Modiﬁcation 249 Primary and Secondary Contacts 250 Module Fabrication and Life Cycle Analysis 251 Cu(In,Ga)(S,Se)2 Cells and Modules 252 Substrates 253 Back Contacts 253 Cu(In,Ga)(S,Se)2 Absorber Layers 255 Co-evaporation 257 Deposition Reaction 260 Sputtering 262 Epitaxy, Chemical Vapor Deposition, and Vapor Transport Processes 262 Inﬂuence of Sodium 262 Inﬂuence of Gallium 265 Inﬂuence of Sulfur 265 Inﬂuence of Oxygen 266 Buffer Layers of CIGS 268 Chemical Bath Deposited CdS 268 Alternative Buffer Layers 270 Window Layers of CIGS 272 Module Fabrication 273

6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.1.5 6.1.6 6.1.7 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.4.1 6.2.4.2

Photovoltaic Properties of Standard Devices CdTe Device Properties 277 Solar Cell Parameters 277 Diode Currents 277 Collection Functions 279 Device Anomalies 280 Transient Effects and Metastability 281 Device Model 282 Stability 285 AI -BIII -CVI 2 Device Properties 286 Solar Cell Parameters 286 Diode Currents 288 Collection Function 290 Transient Effects and Metastability 292 Relaxed State 292 Models for Relaxed State 293

277

Contents

6.2.4.3 6.2.4.4 6.2.4.5 6.2.4.6 6.2.4.7 6.2.4.8 6.2.4.9 6.2.5 6.2.6

Red Light Effect 294 Forward Bias Effect 295 Blue Light Effect 295 White Light Effect 296 Reverse Bias Effect 296 Models for Metastability 297 Implications for Module Testing Device Model 299 Stability 302

7 7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.2 7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.4.6 7.5 7.5.1 7.5.2

Appendix A: Frequently Observed Anomalies 305 JV Curves 305 Roll Over Effect 305 Cross Over 306 Kink in Light JV Curve 306 Violation of Shifting Approximation 307 Solar Cell Parameters 308 Reduced Jsc but High Voc 308 Reduced Voc but High Jsc 308 High Jsc but Low FF 309 Diode Parameters 309 Diode Parameter A > 2 309 Activation Energy Ea < Eg,a 309 Diode Quality Factor Illumination Dependent 310 Diode Quality Factor Temperature-Dependent 310 Quantum Efﬁciency 311 High Jsc but Low EQE 311 Low Jsc but High EQE 311 Low Blue Response in IQE 311 Low Red Response in IQE 312 Quantum Efﬁciency Low at All Wavelengths 312 Apparent Quantum Efﬁciency 313 Transient Effects 313 Voc Time-Dependent with dVoc /dt > 0 313 Voc Time-Dependent with dVoc /dt < 0 314

8

Appendix B: Tables References 321 Index 361

315

298

IX

XI

Preface The subjects of this book are thin ﬁlm solar cells based on sulﬁde, selenide, or telluride semiconductors. We call them the chalcogenides. The most prominent members – at least today – are CdTe and CuInSe2 . These solar cells have been in production since the beginning of the twenty-ﬁrst century, and production volume is growing rapidly. More companies are entering the ﬁeld, each company expecting efﬁciency and cost advantages from their speciﬁc cell design or production process. This commercialization period, however, had a long forerunner. Chalcogenide photovoltaics has been the subject of research and development for four decades. The hope was that the use of polycrystalline ﬁlms – instead of single crystal materials – would allow the fabrication of low-cost solar cells. Counting from 1980, there have been over 80 international scientiﬁc events dealing with this material group and its application in solar cells. The authors do remember enthusiastic periods when exploitation appeared just ahead and then again pragmatic periods when more research was shown to be needed. However, constantly important ﬁndings have been made – several of them producing unexpected results. The method was empirical science where in the best case theoretical understanding was the second step. The ﬁndings brought out several so-called magic steps which now are executed in production. Today it is fair to say that the achievements have been remarkable: chalcogenide-based photovoltaic devices achieve the highest energy conversion efﬁciencies among the thin ﬁlm technologies – more than 20%. This value we still consider as a miracle, given that so many basic material properties are not well understood. The topics in this book are the physics and technology of chalcogenide-based photovoltaics. Considerable space is given to heterostructures (stacks of different semiconductor materials) which build the working core of the solar cell. CdTe- and CuInSe2 -based solar cells are heterostructures but the concepts are valid for any type of heterostructure. Important topics are band diagrams, diode currents, and photo currents in heterostructures where we discuss state of the art models. We provide only a limited section on device analysis which treats with speciﬁc aspects of non-ideal solar cells. Non-ideal solar cells are also the subject of Chapter 7, Appendix A, entitled Frequently observed anomalies, which gives a compendium of phenomena and the models which help to explain these anomalies. Chalcogenide Photovoltaics: Physics, Technologies, and Thin Film Devices. Roland Scheer and Hans-Werner Schock Copyright  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31459-1

XII

Preface

Material aspects of the chalcogenides for photovoltaics are treated with focus on II–VI and I–III–VI2 materials. The reader will ﬁnd tables and ﬁgures on cell-relevant material properties related to solar cells which normally are found in a handbook. Within Chapter 5, entitled Thin ﬁlm technology, the clear focus is on CdTe and CuIn1–x Gax Se1–y Sy , the chalcogenides currently most exploited. We discuss every technological step and every relevant chemical impurity. We hope that the reader will ﬁnd enough information to make his own highly efﬁcient chalcogenide solar cell. In Chapter 6, we then describe the photovoltaic properties of CdTe and CuIn1–x Gax Se1–y Sy standard devices also regarding the question of metastability and stability. We develop electronic models and band diagrams for the cells – admittedly based partly on secured knowledge and partly on the authors’ opinion. This book is the result of ﬁve years’ work. During this period, the authors sought discussions with several scientists all over the world. The feedback from those colleagues was very valuable. We want to thank Dr Abou-Ras, Prof. Burgelman, Prof. Cahen, Dr Eisenbarth, Prof. Ferekides, Dr Fr¨anzel, Dr Igalson, Prof. J¨agermann, Prof. Klein, Prof. M¨onch, Prof. Romeo, Prof. Rau, Prof. Siebentritt, B.A. Schubert, Dr Visoly-Fischer, H. Wilhelm, and Prof. W¨urfel. We further thank our families for their encouragement and their forbearance. Roland Scheer Hans-Werner Schock

XIII

Symbols and Acronyms

Symbol

Unit

A AM CBM Cc Dn Dp Dd dtot , dδ

cm –2 eV−1 cm2 s –1 cm2 s –1 cm –3 eV –1 cm

Ec,δ

eV

Ea EF EFn EFp Eg,δ

eV eV eV eV eV

Ei

eV

En,δ

eV

Ep,δ

eV

Ed

eV

Ev,δ

eV

En

eV

Ep

eV

EQE En , Ep

eV

Diode quality factor Air mass Conduction band minimum Band gap gradient defect factor Electron diffusion coefﬁcient Hole diffusion coefﬁcient Defect distribution Total thickness of heterostructure and thicknesses of absorber, buffer, window layers (δ = a, b, w) Conduction band edge of absorber, buffer, window layers (δ = a, b, w) Activation energy of Jo Equilibrium Fermi energy Quasi Fermi energy of electrons Quasi Fermi energy of holes Energy bandgap of absorber, buffer, window, or interface (δ = a, b, w, IF) Intrinsic position of the Fermi energy with respect to a reference level Absolute value of ζn in absorber, buffer, window (δ = a, b, w) Absolute value of ζp in absorber, buffer, window (δ = a, b, w) Defect level referenced to the valence band maximum Valence band edge of absorber, buffer, window layers (δ = a, b, w) Fermi energy referenced to the conduction band or quasi-Fermi energy of electrons Fermi energy referenced to the valence band or quasi-Fermi energy of holes External quantum efﬁciency Demarcation levels

Chalcogenide Photovoltaics: Physics, Technologies, and Thin Film Devices. Roland Scheer and Hans-Werner Schock Copyright  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31459-1

XIV

Symbols and Acronyms Symbol

Unit

EFn EFp Ec,back , Ev,back

eV eV eV

Ec,front , Ev,front

eV

Ec b,a , Ev b,a

eV

Ec w,b , Ev w,b

eV

F FF FLP G GB HRW J J0 J00 JCBM

V/cm

Jsc Jζ

A cm –2 A cm –2

LO LRW n

cm –3

Nv Nc Nd NIF NP q QIF QNR Rbc SCR SRH Sn0 , Sp0 TO U UHV V VBM

cm –3 s –1

A cm –2 A cm –2 A cm –2 A cm –2

cm –3 cm –3 cm –3 cm –2 As As cm –2

cm s –1 cm –3 s –1 V

Discontinuity of the electron quasi Fermi energy Discontinuity of the hole quasi Fermi energy Bandgap variation at the back surface of a layer realized by a variation of the conduction band edge or valence band edge, respectively Bandgap variation at the front surface of a layer realized by a variation of the conduction band edge or valence band edge, respectively Conduction band and valence band offset, respectively, at the buffer/absorber interface (see sign convention in Section 2.3.1) Conduction band and valence band offset, respectively, at the buffer/window interface (see sign convention in Section 2.3.1) Electric ﬁeld Fill factor Fermi level pinning Generation rate Grain boundaries High resitance window Current density Saturation current density Reference current density Current density due to a gradient in the conduction band minimum Short circuit current Current density due to a gradient in the reduced chemical potential Longitudinal optic Low resistance window Electron density Refractive index Effective density of states in valence band Effective density of states in conduction band Bulk defect density Interface defect density Nitric–phosphoric acid Elementary charge Interface charge density Quasi neutral region Back contact reﬂectivity Space charge region Shockley, Read, Hall (recombination) Surface or interface recombination velocity Transversal optic Recombination rate Ultrahigh vacuum Voltage Valence band minimum

Symbols and Acronyms

Symbol

Unit

vn vp Voc wtot , wδ

cm s –1 cm s –1 V cm

x y α χ

K –1 eV

φb p φb n

As V –1 cm –1 V V

η η(V)

%

ηc (z,V)

µn µp τn0 τp0 ζn , ζp

cm2 V –1 s –1 cm2 V –1 s –1 s s eV

ζn , ζp

eV

ζn0 , ζp0

eV

Electron thermal velocity Hole thermal velocity Open circuit voltage Total width of SCR and SCR of absorber, buffer, window layers (δ = a, b, w) Stoichiometry deviation [Eq. (4.5)] Molecularity deviation [Eq. (4.5)] Thermal expansion coefﬁcient Electron afﬁnity; corresponds to the absolute value of ζn0 Dielectric constant Hole barrier Electron barrier Charge collection efﬁciency in an electron beam induced current experiment (EBIC) Solar cell efﬁciency External collection efﬁciency; describes the dependence of the photo current on the voltage Collection function; describes the collection probability of charge carriers generated in the solar cell Electron mobility Hole mobility Minimum lifetime of the electron Minimum lifetime of the hole Reduced chemical potential of electrons and holes; the part of the chemical potential which reﬂects the number density of species Chemical potential of electrons and holes, respectively Concentration independent parts of the chemical potential of electrons and holes; quantities which reﬂect the chemical environment of the species

XV

1

1 Introduction The subject of this book is sketched in Figure 1.1. It is a heterostructure solar cell based on polycrystalline thin ﬁlms comprising a metallic back contact, a semiconducting absorber layer, and some kind of emitter which is transparent for most of the solar spectrum. Typical ﬁlm thicknesses are around 1–3 µm – enough to gather most of the photons incident from the sun. The ﬁlm thickness of the absorber layer is in the size range of the polycrystalline grain and the absorption length of sunlight. All ﬁlms are deposited by astonishingly simple methods, given that chalcogenide based cells today have the highest efﬁciency among the thin ﬁlm solar cells. In the following, we give a historical digest on solar cell development. We try to label some milestones of progression and understanding. Certainly, the prospect of these cells is more difﬁcult. Hence, we only outline some facts and useful estimations. Figure 1.1 Sketch of a polycrystalline thin ﬁlm cell with window layer and absorber layer. The boundaries of the large grains of the absorber layer are outlined. The thickness of the window and absorber layers are dw and da . wa is the width of the space charge layer within the absorber layer. The letters (a) and (b) indicate perpendicular (a) and parallel (b) grain boundaries with respect to the space charge layer.

a Emitter

Back contact

Absorber

b

y -da

-wa

x z

1.1 History of Cu(In,Ga)(S,Se)2 Solar Cells

The history of Cu(In,Ga)Se2 (CIGSe)-based solar cells began in 1975 when Bell laboratory scientists achieved 12% solar energy conversion efﬁciency with a cell where a layer of CdS was evaporated onto a CuInSe2 single crystal [1]. Stimulated by this Chalcogenide Photovoltaics: Physics, Technologies, and Thin Film Devices. Roland Scheer and Hans-Werner Schock Copyright  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31459-1

1 Introduction

result, the development of CuInSe2 thin ﬁlms started. At the University of Maine (see Figure 1.2), thin ﬁlms were deposited by dual source evaporation of CuInSe2 and Se [2]. Small area devices in the substrate conﬁguration were built with an efﬁciency in the range of 5%. In the early 1980s, companies started developing CIGSe solar cells (see Figure 1.2). At this stage, two different preparation concepts for CIGSe were pursued. Boeing used a co-evaporation process where Cu, In, and Se were deposited from separate evaporation sources. The Boeing group was the ﬁrst to demonstrate a more than 10% efﬁcient thin ﬁlm device [3]. Arco Solar, in contrast, developed a deposition–reaction process using Cu and In metallic precursors, with H2 Se as the reacting chalcogen source. Surprisingly, for a period of time the deposition–reaction technology gave better device efﬁciencies than the co-evaporation approach (see Figure 1.2). The reason was that, during the reaction step, the absorber layer was doped with Na – a fact that was overlooked by the rest of the research community. In 1995, Na-containing glasses became standard and both technologies – co-evaporation and deposition–reaction – gave almost similar efﬁciencies of around 14% on the cell level. The deposition–reaction technology, however, was much easier to scale up: sputtering of metal ﬁlms was well established in the glass industry. The limitation was rather in the width of available quartz tubes for the selenization reactor. Accordingly, the ﬁrst modules developed by Arco Solar were only 1 ft (305 mm) wide. Parallel selenization of up to 20 substrates was a batch process with around 6 h process time.

NREL

ZSW

20 Efficiency [%]

2

EuroCIS SSI Arco (3)

15

(4) Uni Johbg Showa

(2) SSG

10 (1)

5

Boeing

Wuerth

SSI

Univ. of Maine

1980

1990 Year

Figure 1.2 Solar energy conversion efﬁciency of thin ﬁlm cells and modules based on Cu(In,Ga)(S,Se)2 (CIGSSe). Full circles – cells by co-evaporation, open circles – full modules by co-evaporation, full squares – cells from deposition–reaction technology, open squares – full modules from deposition–reaction. Abbreviations: Arco – Arco Solar, SSI – Siemens Solar Industries (USA), EuroCIS – University

2000

2010

Stuttgart and University Uppsala, SSG – Siemens Solar AG (Germany), Wuerth – Wuerth Solar (Germany), Showa – Showa Shell (Japan), Uni Johbg – University of Johannisburg (South Africa), NREL – National Renewable Energy Laboratories (USA), ZSW (Germany). Labels (1) to (4) refer to milestones of (CIGSSe) development which are discussed in the text.

1.1 History of Cu(In,Ga)(S,Se)2 Solar Cells

The headstart achieved by co-evaporated solar cells (not modules) began with employing more complicated evaporation processes. The so-called three-stage process invented in 1994 by the National Renewable Energy Laboratories (NREL) group [4] provided the basis for an efﬁciency of 20%. Around 2000, an effort was made at Wuerth Solar to bring the co-evaporation technology into production. Wuerth Solar designed line-evaporation sources that were able to deposit on substrates 2 ft wide. As depicted in Figure 1.2, co-evaporated module efﬁciency could rapidly compete with modules from the deposition–reaction technology. In 2008, several laboratories around the world had the experience to prepare a CIGSe solar cell with an efﬁciency above 19%. Concerning large area modules (>0.4 m2 ), both co-evaporation and deposition–reaction technologies reached above 12% efﬁciency. While writing these lines, submodules of Cu(In,Ga)(S,Se)2 formed by the deposition–reaction process have been reported with module efﬁciencies approaching 16% [5]. Both process schemes are still in competition and may further be pursued in parallel. Today, a large number of subvariants of these process schemes are being investigated by research laboratories and companies. They all have the medium-term goal of a thin ﬁlm module with more than 15% efﬁciency produced with costs below 0.5 ¤/Wp . 1.1.1 Milestones of Cu(In,Ga)(S,Se)2 Development

As the ﬁrst evaporated CuInSe2 ﬁlms (from CuInSe2 and Se evaporation sources) suffered from poor control of the ratio of copper to group III element, Cu/In (in early days the group III element was only indium), the ﬁrst milestone, labeled (1) in Figure 1.2, was to apply a triple-source evaporation geometry in which the copper and indium rates during deposition and thus the Cu/In ratio in the ﬁlm could deliberately be adjusted. Soon it was recognized that the best cell efﬁciencies were achieved by a copper-poor composition, that is, Cu/In < 1. Around 1988, two innovations, summarized under milestone (2), led to devices with ∼12% efﬁciency. The ﬁrst innovation was the use of a thin CdS layer prepared in a chemical bath. This thin buffer layer (see Figure 1.3) rendered a reduction of the optical absorption and a better coverage of the polycrystalline CuInSe2 surface. Due to the complete coverage of the absorber by a CdS layer ∼50 nm thick, shunting i-ZnO/n-ZnO 0.25 – 1 µm Buffer

50 nm

Cu(In,Ga)(S,Se)2 1 – 3 µm Mo 2 µm

0.5 – 1 µm

Glass, metal, polymer

Figure 1.3 Schematic outline of a substrate-type solar cell based on Cu(In,Ga)(S,Se)2 (CIGSSe).

3

1 Introduction

paths could be eliminated. Chemical bath deposition of the buffer at that time became standard, and today turnkey equipment for this process step is available. The second innovation was the employment of a Cu-rich growth period during CuInSe2 ﬁlm formation. Although CuInSe2 ﬁlms require a Cu-poor composition in the end, their morphology and electronic properties are superior if they are partly grown under an excess of Cu. CIGSe ﬁlms grown by the deposition–reaction process pass the Cu-rich growth regime automatically. In the beginning of the 1990s, it became clear that CuInSe2 solar cells perform better if the cells are prepared on sodium (Na)-containing glass and if part of the indium is replaced by gallium (Ga). (As a speciality of the deposition–reaction process, also a part of the selenium was replaced by sulfur.) Thus, the absorber became a multinary compound semiconductor. This established milestone (3) which allowed demonstrating efﬁciencies of around 14%. Today, Na and Ga doping is an issue for all types of CIGSe preparation. With the introduction of Ga (and S), the CuInSe2 cell became a CIGSSe cell with a material stack as shown in Figure 1.3 – a molybdenum back contact, a Cu(In,Ga)(S,Se)2 absorber layer, a CdS buffer layer, and ZnO formed by two separate layers. We note, however, that the principle outline of the cell (Mo/absorber/CdS/ZnO) was invented in 1987 by the ARCO Company [6]. Although in the mid-1990s there was still room for improvement for all types of CIGSe preparation techniques, it was a very particular innovation which yielded around 18% devices and brought up milestone (4) at the end of the 1990s. The use of a complicated three-stage co-evaporation process which until then had only been realized in the laboratory. The principle of this process is explained in Section 5.2.3. Subtle parameter variations and the optimization of the contact scheme for single 0.5 cm2 cells brought up a cell of 20% air mass (AM) 1.5 efﬁciency [7]. Figure 1.4 shows the current–voltage curve of such a highly efﬁcient CIGSe solar cell (redrawn from [8]). This laboratory cell shows the enormous potential of CIGSe

Current density [mA cm−2]

4

0 −10 −20

(2) 20

−30 −40 0.0

%

(1) 0.2

0.4 0.6 Voltage [V]

0.8

Figure 1.4 Current density versus voltage curve of (1) a high efﬁciency Mo/Cu(In,Ga)Se2 /CdS/ZnO solar cell and (2) a high efﬁciency BC/CdTe/CdS/Zn2 SnO4 /Cd2 SnO4 solar cell (redrawn after Ref. [8]).

1.2 History of CdTe Solar Cells

solar cells. Also, cells made by the deposition reaction process have been improved in the past decade due to parameter optimizations (Figure 1.2). 1.2 History of CdTe Solar Cells

The development of CdTe solar cells was initially motivated by their potential space application in communication satellites. In 1963, the ﬁrst thin ﬁlm solar cell of the type n-CdTe/p-Cu2 Te was demonstrated by a General Electric Research Laboratory [9]. The cell had an efﬁciency of 6%, but device instability (caused by Cu diffusion) led to the quest for an alternative heteropartner for CdTe. In 1972, Bonnet and Rabenhorst presented a p-CdTe/n-CdS heterojunction solar cell which also had 6% efﬁciency [10]. The CdTe layer was deposited by vapor transport deposition (cf. Section 5.1.4), a method that today is used in mass production. These cells were of the substrate type, having a Mo back contact and a CdTe thickness of more than 10 µm [10]. Ten years later, Tyan et al. [11] presented a superstrate thin ﬁlm cell with more than 10% efﬁciency (Figure 1.5). This cell was grown by closed-space sublimation – the leading method of later champion devices. Encouraged by this achievement, several companies started inhouse developments. The goal was to ﬁnd the best low-cost deposition method that allowed high throughput and that could deliver more than 10% efﬁcient modules. Surprisingly, this goal was reached using very diverse deposition methods such as vapor transport deposition, electrodeposition, spray pyrolysis, and screen printing (cf. Table 5.2). CdTe was 18 (4)

USF

16

NREL

Efficiency [%]

NREL (3)

14

BP Solar

Photon Energy

12

BP Solar

Kodak 10 8 6

(2) (1)

AMETEK

BP Solar

Matsushita Batteries

FS

BP Solar Matsushita

Batelle Inst. 1980

1990 Year

2000

Figure 1.5 Solar energy conversion efﬁciencies of thin ﬁlm cells (full marker) and modules (open markers) based on CdTe. Abbreviations: USF – University of South Florida, NREL – National Renewable Energy Laboratories (USA), FS – First Solar.

2010

5

6

1 Introduction

considered as an extraordinary forgiving material. Nevertheless, it was not earlier than 2002 that CdTe modules became a mass product on basis of vapor transport deposition while the so-called low-cost methods such as spray pyrolysis and screen printing have been ruled out – at least for the time being. Reasons for the stepback of early industrial players were probably rather in strategic decisions than in technological obstacles. On the cell level, the NREL (USA) in 2001 achieved the long-standing record of 16.5% efﬁciency [8]. They used close-space sublimation for the absorber preparation. Nowadays, CdTe modules are produced on the GWp /year level and currently are the cost leader in the photovoltaic industry. In a sense, the CdTe technology exempliﬁes the cost advantage of thin ﬁlm photovoltaics over wafer-based technologies – an advantage that has been claimed for quite some time. 1.2.1 Milestones of CdTe Development

Soon after realization of the ﬁrst CdS/CdTe device it was recognized that this type of cell is more efﬁcient if formed in the superstrate conﬁguration. Use of this conﬁguration (Figure 1.6) was the ﬁrst milestone, as shown in Figure 1.5. Interdiffusion between CdS and CdTe which can easily happen in the process sequence was believed to be a reason for the superior performance of the superstrate conﬁguration. The second milestone in 1982 was the incorporation of oxygen during ﬁlm deposition. Several effects of oxygen on ﬁlm properties are discussed in Section 5.1.6. Use of chlorine may have been implicit for several preparation techniques such as screen printing and electrodeposition. But it was the third milestone when researchers after 1990 became aware that the activation of the CdTe by chlorine treatment substantially increases the device efﬁciency. Since that time, chlorine activation using speciﬁc and critical process parameters has always been part of the process. The result of such parameter optimization were laboratory-scale solar cells with efﬁciencies exceeding 15% – ﬁrst demonstrated by Ferekides et al. [12]. It took almost 10 years before the fourth milestone led to a 16.5% device. We identify this Glass

1 µm

TCO

0.5 – 1 µm

CdS

100 nm

CdTe

2 – 8 µm

Back contact

Figure 1.6 Schematic outline of a superstrate-type solar cell based on CdTe with a ZnTe:Cu/Ti back contact.

1.3 Prospects of Chalcogenide Photovoltaics

milestone with the insertion of a high mobility window layer of Cd-stannate. The current–voltage (JV) curve of such a highly efﬁcient device is shown in Figure 1.4. Highest module efﬁciencies are around 11% since the champion cell technology has not yet been completely transferred to production. 1.3 Prospects of Chalcogenide Photovoltaics

If chalcogenide photovoltaics are to substantially contribute to future energy supply, we speak of some terawatts of installed capacity. Currently, chalcogenide module production is in the gigawatts/year range, mostly driven by one CdTe module producer (First Solar). We want to address the question about possible hurdles for future expansion. Zweibel assessed future cost reduction potentials of thin ﬁlm photovoltaics [13]. Table 1.1 condenses this work and reveals that both technologies [CdTe and Cu(In,Ga)(S,Se)2 ] have about the same cost perspective for large scale production of around 0.5 $/Wp . Already today (2009) CdTe module production is reported with costs of 0.85 $/Wp at an efﬁciency of 11%. Thus, it appears realistic that future cost reduction and efﬁciency increase will lead to the ﬁgures in Table 1.1. With an anticipated module lifespan of 30 years and the listed costs for ground-mounted and rooftop installations, these numbers translate into about 0.06–0.09 $/kWh AC electricity costs in sunny climates with around 1800 kWh m−2 year−1 [13]. Even lower-cost ﬁgures are derived in a study by Keshner and Arya (Hewlett Packard) [14]. In the climate referenced above, the energy payback time of today’s CdTe modules is around 1.1 years [15]. Thus, we see no hurdles for CdTe and Cu(In,Ga)(S,Se)2 technology expansions in terms of costs and energy payback time. A chalcogenide thin ﬁlm module contains, depending on the chalcogenide ﬁlm thickness, some grams per square meter of metals and chalcogens (see Table 1.2). Among the materials listed in Table 1.2, indium and tellurium are considered most critical in terms of availability. In order to estimate the maximum installation possible based on each material, two different approaches are pursued: (i) use of published material reserves and (ii) extrapolation of current annual material Cost projections of CdTe and Cu(In,Ga)(S,Se)2 module production according to Ref. [13] assuming gigawatts/year production level. Table 1.1

CdTe/glass CIGSSe/glass

Projected efﬁciency (%)

Module costs ($/Wp )

Systems costs for large ground mounted systems ($/Wp )

Systems costs for large commercial roof top systems ($/Wp )

14 15.5

0.47 0.51

1.12 1.18

1.61 1.67

7

8

1 Introduction Table 1.2 Material consumption for chalcogenide modules with absorber thickness 1 µm and material resources after United States Geological Survey data for 2008. Maximum possible installations of CdTe and CuIn0.7 Ga0.3 (S,Se)2 single junction solar cells assuming efﬁciencies of 14 and 15.5%, respectively, a module lifespan of 30 years, and complete use of the resource.

Cd Te In Ga Se

g/m2

Reserves (106 kg)

Reserve base (106 kg)

Maximum installation (TWp ) according to Anderson [16]

Maximum installation (TWp ) according to Schubert et al. [17]

Maximum installation (TWp ) according to Zweibel [13]

3.2 3.3 1.5 0.27 2.4

490 22 11 No data 86

1200 48 16 >1000 172

12.5 1.51 3.8 No data 12.5

– – 1.29 – –

– 15 8.5 – 150

extraction. Both approaches are reﬂected in Table 1.2. Anderson [16] uses reported reserves for indium and tellurium and derives 1.51 and 3.8 TWp maximum installation for CdTe and Cu(In,Ga)(S,Se)2 , respectively. About half this number for Cu(In,Ga)(S,Se)2 is obtained by Schubert et al. [17] but there only about 50% use of indium for PV is assumed. The original ﬁgures of Anderson need correction because indium reserve estimations increased by a factor of ∼2.5 (!) between 2007 and 2008. Interestingly, for 2009 no data are given by the U.S. Geological Survey, reﬂecting the fact that the reserves data for indium contain large uncertainties. This is why Zweibel [13] uses a different approach. Today, indium is exploited from Zn ore (although also contained in copper, lead, tin, and tungsten ores). Extrapolating the Zinc mining capacity with 1% annual growth gives an alternative estimation for indium reserves. This approach (and the corresponding one for tellurium) is used by Zweibel in Ref. [13] and in the respective column of Table 1.2. After correction to 1 µm absorber thickness (for the sake of comparability) we arrive at around 10 TWp for each technology. Thinning the chalcogenide absorber to 0.5 µm simply doubles these values. (Today, the world average power demand is around 10 TW.) We may conclude that – based on the current estimates – chalcogenide thin ﬁlm solar cells based on CdTe and Cu(In,Ga)(S,Se)2 can substantially add to the world power supply. However, it is questionable whether the reserves are sufﬁcient to make them the only photovoltaic technologies of the future.

9

2 Thin Film Heterostructures Today, thin ﬁlm chalcogenide photovoltaic devices are fabricated in the form of heterostructures. An example is p-type CdTe in contact with n-type CdS and n-type SnO2 . CdTe is the absorber, CdS the buffer, and SnO2 the window. We call this device an absorber/buffer/window heterostructure although there are even more components to mention, namely the metal contacts and interfaces. Slightly simpler is an absorber/window heterojunction. In this chapter we present the physical concepts to describe the band diagram and the electrical transport phenomena in heterostructure solar cells. In the last section, we discuss particular analysis aspects. We use device simulation extensively – since many physical relationships can be explained more easily by graphs than by formulas. For the simulation which is accomplished mostly in one dimension, we designed a generic absorber/buffer/window device. Table 8.1 gives the default values of the physical quantities of this device. Deviations from the default values are highlighted in the text.

2.1 Energies and Potentials

First, we brieﬂy introduce deﬁnitions for energies and potentials in as far as they are used throughout this book. For a more detailed description, the reader is referred to other works [18, 19]. Very generally, Figure 2.1 depicts the valence band edge Ev and conduction band edge Ec of a non-homogeneous semiconductor under non-equilibrium condition. The non-equilibrium condition may be induced by illumination, carrier injection or both. The reference energy is the potential energy E = 0. (In the heterostructure’s energy band diagram, we assign the zero of energy to the lowest lying valence band edge of all semiconductors of the heterostructure.) The absolute values of the band edges are not known experimentally (see Ref. [20] for a detailed discussion). Note, that we do not deﬁne a vacuum level inside the semiconductor. If band offsets at a heterostructure interface are required, we can use calculated or measured values of Ec and Ev . Chalcogenide Photovoltaics: Physics, Technologies, and Thin Film Devices. Roland Scheer and Hans-Werner Schock Copyright  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31459-1

10

2 Thin Film Heterostructures E=0 Ec

-qϕ -qϕ

En

EFn Ec

Ev

ζn0

Ei EFp

ζ′n

ζ′p

ζp0 χ

EFn

ζn Eg ζp

EFp Ep

Ed Ev

E z

Figure 2.1 Schematic band diagram of a semiconductor. In order to demonstrate a very general case, we allow the semiconductor to have a varying electron afﬁnity and varying energy gap. Furthermore, the semiconductor is in non-equilibrium as revealed by the energy separation of the electron and hole quasi Fermi levels.

As for the reference level, the absolute value of the electrostatic potential ϕ is also not known. In band diagrams we plot it on a separate energy scale. Analytically, we only make use of variations of the potential energy −q dϕ/dz. These variations in −qϕ can be due to space charge within the semiconductor or due to an external electric ﬁeld. The semiconductor in Figure 2.1 exhibits a bandgap energy Eg and an electron afﬁnity χ, both varying as a function of position. The physical reason may be that the semiconductor exhibits varying material properties such as element composition or strain. Gradients of Eg and χ are independent of the gradient in the electrostatic potential. Since the semiconductor is depicted under non-equilibrium condition, each carrier has its own quasi Fermi level, EFn and EFp . In the case of thermodynamic equilibrium, both quasi Fermi levels join to the equilibrium Fermi energy EF . The quasi Fermi levels can be identiﬁed with the electrochemical potentials of electrons and holes. This is reﬂected in the relations EFn = −qϕ + ζ n

(2.1)

EFp = −qϕ − ζ p

(2.2)

as can be inferred from Figure 2.1. The quantities ζ n and ζ p are the chemical potentials of electrons and holes, respectively. If the chemical potentials are related to the band edges, we write the nonprimed variables ζn and ζp which we denote as reduced chemical potentials. In the case that −ζn and −ζp are larger than 3kT, that is, if the semiconductor is nondegenerate, we can write ζn = kTln{n/Nc } and ζp = kTln{p/Nv }. From these expressions we immediately see that the reduced

2.2 Charge Densities and Fluxes

chemical potentials ζn and ζp depend on the particle number densities n, p, and the density of states in the conduction and valence band, Nc and Nv . Thus, the reduced chemical potentials reﬂect those parts of the chemical potential which are concentration dependent. However, there are those parts ζn0 and ζp0 which depend on the chemistry of the material and not on the concentrations. That is ζn = ζ n − ζn0

ζp = ζ p − ζp0

(2.3) (2.4)

The absolute value of ζn0 is denoted as electron afﬁnity. The reduced chemical potentials ζn and ζp both have negative values as reﬂected in the orientation of the arrows in Figure 2.1. Thus, it is convenient to introduce the absolute quantities En = −ζn and Ep = −ζp . Their deﬁnition is sketched on the right hand side of Figure 2.1 with arrows going up (for positive energies). In non-equilibrium, En and Ep are the quasi Fermi energies referenced to the bound edges. In equilibrium, En and Ep denote the same Fermi energy but referenced to the conduction band and valence band, respectively. The semiconductor may contain deep or shallow impurity levels of density Nd . Their electronic transition energies are quoted as Ed in general and ED and EA in the case of donors and acceptors. Note, that we use the term electronic transition energy instead of energy level in agreement with the deﬁnition given in Ref. [21]. The transition energy of a single donor state, for instance, describes the position of the Fermi energy at which the donor changes from D0 to D+ . Transition energies of all energies are given with respect to the valence band maximum. With the deﬁnitions above, the quasi Fermi levels become EFn (z) = −qϕ(z) − χ(z) − En (z)

(2.5)

EFp (z) = −qϕ(z) − χ(z) − Eg (z) + Ep (z)

(2.6)

z is the space coordinate as depicted in Figure 2.1.

2.2 Charge Densities and Fluxes

Three sets of equations are sufﬁcient to describe the transport of carriers within the semiconductor or within heterojunctions: the transport equations, the continuity equations, and the Poisson equation. The transport equations for electrons and holes are given by dEFn (z) (2.7) dz dEFp (z) Jp (z) = µp p(z) (2.8) dz where the electric current densities Jn (z) and Jp (z) are expressed in units of A cm−2 . n, p are the carrier densities of electrons and holes and µn and µp are their mobilities. Jn (z) = µn n(z)

11

12

2 Thin Film Heterostructures

With the aid of Figure 2.1, it may be instructive to verify the sign convention in this book. Due to the orientation of the coordinate z in Figure 2.1, dEFn /dz and dEFp /dz are positive and thus the current densities Jn and Jp are positive. Electrons are moving toward smaller values of EFn (to the left) and holes are moving toward larger values of EFp (to the right). The total electric current density J(z) = Jp (z) + Jn (z) is positive if ﬂowing from left to right. Using Eqs. (2.5) and (2.6), the current densities can be expressed as dϕ(z) dχ(z) dEn (z) − − (2.9) Jn (z) = µn n(z) −q dz dz dz dϕ(z) dχ(z) dEg (z) dEp (z) Jp (z) = µp p(z) −q − − + (2.10) dz dz dz dz In Eqs. (2.9) and (2.10) we assume that the mobilities of electrons and holes, µn and µp , are constant and do not change with position. Furthermore, we do not allow a variation of temperature within the semiconductor. (For the discussion of thermoelectric effects see Ref. [18]). Equations (2.9) and (2.10) state that the electric current in a semiconductor device such as a solar cell is driven by gradients of the electrostatic potential, gradients of the electron afﬁnity and bandgap, and gradients of En and Ep (the absolute of the reduced chemical potentials). Equations (2.9) and (2.10) are also valid for degenerate semiconductors. In the case of nondegenerate semiconductors (En,p > 3kT) we can write dEn kT dn(z) kT dNc (z) =− + dz n(z) dz Nc (z) dz

(2.11)

dEp kT dp(z) kT dNv (z) =− + dz p(z) dz Nv (z) dz

(2.12)

From Eqs. (2.11) and (2.12) we recognize that electric current can be driven by gradients of the carrier density and the density of states. In a homojunction solar cell, the current is only driven by gradients of En and Ep (in the quasi neutral region, QNR) and a gradient of −qϕ (in the ﬁeld zone). However, in the heterojunction solar cells, gradients of the state density and the band edges give rise to additional currents. These gradients are sometimes denoted as effective force ﬁelds [22]. Later we have to quantify the different sources of current ﬂow as included in Eqs. (2.9−2.12). The continuity equations for electrons and holes are given by ∂n(z) 1 dJn (z) = Gn (z) − Un (z) + ∂t q dz

(2.13)

1 dJp (z) ∂p(z) = Gp (z) − Up (z) − ∂t q dz

(2.14)

Here we have introduced the particle ﬂux densities of electrons and holes, Jn /q and Jp /q. The continuity equations (Eqs. (2.13) and (2.14)) describe particle number densities and not charge densities. Gn (z), Gp (z), Un (z), and Up (z) are the generation and recombination rates of electrons and holes. In the case of recombination

2.3 Energy Band Diagrams

and generation via trap states, the generation rates as well as the recombination rates can be different for electrons and holes. In the absence of traps, however, generation of an electron into the conduction band necessarily leaves a hole in the valence band. In the latter case, generation of electrons and holes take place simultaneously and we can write Gn = Gp as well as Un = Up . Note that we differentiate the external rate of generation by photons and the internal, thermal generation rate. The continuity equations (Eqs. (2.13) and (2.14)) describe dynamic processes. Throughout this book, however, we mostly deal with the stationary state where we can put ∂n(z)/∂t = ∂p(z)/∂t = 0. The Poisson equation relates charge densities and the electrostatic potential. The most general form is derived from the Maxwell equation d/dz · D(z) = ρ(z) with D(z) = −ε(z)d/dzϕ(z). Then the Poisson equation reads −ρ(z) dln{ε(z)} dϕ(z) d2 ϕ(z) = + dz dz ε(z) dz2

(2.15)

Here, we allow the dielectric constant to be dependent on position. In the case of ε = const, Eq. (2.15) simpliﬁes to d2 ϕ(z)/dz2 = −ρ(z)/ε. The space charge ρ(z) is given by ρ(z) = q/ε(z) · (p(z) − n(z) + ND + (z) − NA − (z))

(2.16)

where p(z), n(z) represent the mobile charges and ND + (z), NA − (z) the ﬁxed charges in the form of charged donors and acceptors. Within the scope of this book, we are interested in the short circuit current density, in the open circuit voltage, and in the ﬁll factor of the chalcogenide solar cell. Only implicitly, these quantities are included in Eqs. (2.9), (2.10), (2.13), (2.14), and (2.16).

2.3 Energy Band Diagrams

The energy band diagram of an energy converting heterostructure is very important as it deﬁnes the carrier transport, carrier recombination, and Fermi level splitting. In this section, we need to introduce the deﬁnitions of band offsets, discuss the distribution of the electrostatic potential and consider the effects of interface charge, interface dipoles, and deep bulk states. Furthermore, bandgap gradients shall be taken into account. 2.3.1 Rules and Conventions

In general, we consider solar cells consisting of the back contact (bc), the absorber layer (a), the buffer layer (b), the window layer (w), and the front contact (fc). Between each component there will be an interface (IF). But for the beginning, let

13

14

2 Thin Film Heterostructures

(1) Ec EF Ev

∆Ec

(2) Ec EF

∆Ev Ev

(3)

(4)

Figure 2.2 Schematic steps of a four-step recipe on how to construct a heterojunction band diagram (see text).

us discuss a simple heterojunction between the two semiconductors absorber (a) and window (w). In order to construct the band diagram, we need to know the band offsets Ev and Ec as well as the equilibrium Fermi level positions Ep,a and En,w where the subscript characters a and w refer to absorber and window, respectively. The heterojunction in equilibrium is deﬁned by two conditions: (i) the band offsets remain preserved on a relative scale and (ii) the equilibrium Fermi level is ﬂat. With aid of an example, we show the four steps of a simple recipe how to construct the band diagram in Figure 2.2: • The basis is a picture of Ec , Ev , and EF for both semiconductors before contact, together with their band offsets as given by vertical bars. • In the next step a ﬂat Fermi energy is drawn. Far from the junction, Ev and Ec have their positions (with respect to the Fermi energy) as it was before contact formation. • The band offsets are placed depending on the doping ratio of the two semiconductors. If the n-type semiconductor is much higher doped (as assumed in the example), the offset Ec remains close to the Fermi energy. Vice versa, if the p-type semiconductor is much higher doped, the offset Ev remains close to EF . If the doping levels are about identical, the offsets are placed between the two extreme cases, that is, both offsets have about the same distance to EF . • The band edges are connected with convex shape for the p-type semiconductor and concave shape for the n-type. Thereby, it is recommended to draw the majority carrier related band edge ﬁrst, that is, the valence band of the p-type and conduction band of the n-type semiconductor. The procedure may be extended to a heterostructure with three semiconductors. Figure 2.3a shows the band edges of three semiconductors plus two metals and their Fermi energies before contact. Here, it is assumed that each semiconductor

2.3 Energy Band Diagrams

-qϕ -qϕ(-wa) = qVbi -qϕ(0) -qϕ = (0)

E -qϕ ∆ECa,b

∆ECb,w

EC

E

EC

Ep,a

(fc)

z=0

(bc)

EV absorber buffer window z

(a)

Figure 2.3 Schematic of the heterostructure formation process between a window layer, w, a buffer layer, b, and an absorber layer, a. The heterostructure is contacted by metallic front, fc, and back, bc, contacts. Before contact formation (a), the band edges are aligned according to experimental or theoretical band offsets. Fermi energies differ while the electric potential is constant. After

E=0 (b)

EV -da

-wa 0 db db+dw

z

contact (b), the valence band and conduction band offsets are preserved and appear as band discontinuities. Charge transfer between different layers has led to the equilibration of the Fermi energy. Note that the electric potential does not exhibit any discontinuities since the heterostructure’s interfaces are free of electric dipoles.

is homogeneous with respect to effective densities of states, doping densities, static dielectric constant, bandgap, and electron afﬁnity. If it comes to the band offsets, we use the following sign convention: the bandgap difference Eg is a positive quantity if we write at each interface Eg = Eg l − Eg s where the superscripts denote large (l) and small (s) bandgaps. If we deﬁne for the band offsets Ev = Ev s − Ev l and Ec = Ec l − Ec s , it is at each semiconductor/semiconductor interface Eg = Ec + Ev

(2.17)

It is common practice to speak of positive and negative band offsets. With the above deﬁnitions, a band offset is positive if a charge carrier has to spent kinetic energy in order to overcome an energy barrier from the small gap to the large gap semiconductor. Thus, the example in Figure 2.2 includes positive offsets for both the conduction band and the valence band. Vice versa, a carrier gains kinetic energy at a negative offset. Band offsets are also classiﬁed by the terms spike (for the positive barrier) and cliff (for the negative barrier). In Figure 2.3, electrons coming from the absorber must pass a conduction band spike while entering the buffer layer and must pass a conduction band cliff while entering the window layer. And ﬁnally, there is a convention to denote different types of heterojunctions. If both band extrema (valence band maximum and conduction band minimum) are located in one semiconductor, this interface is referred to as a type I interface. In Figure 2.2 this is the case and in Figure 2.3 this is the case for the buffer/absorber interface. A type II interface is found for the buffer/window interface of Figure 2.3.

15

16

2 Thin Film Heterostructures

Here the valence band maximum is in one semiconductor and the conduction band minimum is in the other. The band line up is a property of the two semiconductors forming an interface. The valence band offset, Ev , for a particular interface is obtained from experiment or is calculated from theory. Linear models have been proposed which assume that the band offsets are suspect to a transitivity rule: Knowledge of Ev for the heterojunctions AB as well as BC (A, B, C are different semiconductors) allows to calculate Ev for the AC heterojunction. These linear models, however, are still under debate (see Ref. [20]). In the case that chemical intermixing occurs at an interface, the band offset of the heterojunction of interest may differ from the one found in a particular experiment or from the theoretical band offset. In Table 4.15 a compilation of experimental valence band offsets for chalcogenide materials is given. Knowing the valence band offset Ev , Eq. (2.17) may be applied to calculate the conduction band offset Ec , provided the bandgaps of the semiconductors at the interface are known. In Figure 2.3a we have drawn the band edges according to the band offsets at each particular interface. This is also the case for the semiconductor/metal interfaces where in order to describe the band alignment, we need to know the difference between the semiconductor valence band edge and the metal Fermi energy from experiment or theory. In the example of Figure 2.3a, all Fermi energies before equilibration are on different energy levels. The establishment of equilibrium requires mobile carriers to be transferred across each newly formed interface. In the example in Figure 2.3: 1) Electrons from the window to the front contact form a narrow depletion region of tunneling length. 2) Electrons from buffer to window and from buffer to absorber lead to complete depletion of the buffer layer. The buffer layer acts as a dielectric in the speciﬁc example of Figure 2.3. 3) Holes from absorber to buffer form the absorber depletion region. 4) Holes from absorber to back contact form a back surface band bending which, if too large, acts as a barrier for the diode current. Ionized impurities as well as free carriers form the space charge which enters Eq. (2.16). For the densities of mobile carriers we have to write Ep (z) Ev (z) − EF = Nv (z)F1/2 − p(z) = Nv (z)F1/2 kT kT EF − EC (z) En (z) = Nc (z)F1/2 − (2.18) and n(z) = Nc (z)F1/2 kT kT where F1/2 {η} is the Fermi–Dirac integral of order 1/2 √ ∞ βdβ 2 F1/2 {η} = √ π 0 1 + exp{β − η}

(2.19)

Even for heterostructures of nondegenerate semiconductors, the band offsets can bring the Fermi level very close to the band edges. Therefore, the Boltzmann

2.3 Energy Band Diagrams

approximation is not necessarily applicable and a correct description of the position dependent carrier concentrations is only possible through Eq. (2.18).1) According to Poisson’s equation (Eq. (2.15)), the space charge is connected with the electrostatic potential ϕ(z) and the dielectric function ε(z) which may change at the semiconductor interfaces. Inserting Eq. (2.16) into Eq. (2.15) gives dln{ε(z)} dϕ(z) d2 ϕ(z) + dz dz dz2 Ep En − − Nc (z)F1/2 − (2.20) = −q N+ D (z) − NA (z) + NV (z)F1/2 − kT kT Equation (2.20) is the screening equation for heterostructures with varying band edges Ec,v (z) and electric potential ϕ(z). It is a nonlinear differential equation for ϕ(z). As the complete heterostructure of volume V will be charge neutral, the boundary condition for Eq. (2.20) is

db +dw

−da

ρ(z)dz = 0

(2.21)

Equation (2.20) together with the boundary condition Eq. (2.21) is best solved numerically by a device simulation program. However, simple cases can be solved analytically as will be shown in the next section. In Figure 2.3b we see the electric potential ϕ(z) as a function of position. It is a smooth function which does not include discontinuities due to band offsets. In contrast, the functions Ec,v (z) are subject both to band offsets and to the varying potential. Finally, we want to comment on the orientation of our band diagrams. In contrast to other publications, here the light enters the cell from the right hand side. Thus, the window layer in a band diagram is on the right hand side and the absorber on the left hand side. The reason is that, with this convention, we are able to write the diode equation in such a way that the electric diode current is positive under forward bias and the photo current is negative; at the same time we can respect Eqs. (2.7) and (2.8). Furthermore, by regarding this convention we can speak of a positive electric current if ﬂowing from left to right in an energy band diagram. 2.3.2 Absorber/Window

In the following, we consider a device where the emitter only contains the window layer. We are interested in the potential distribution within such a device. For a heterojunction between two semiconductors without interface charge, an analytical 1) We note, however, that at present time

the Fermi–Dirac integral may not be implemented in a simulation software. Then the integral in Eq. (2.18) is nevertheless

approximated by the exponential of the Boltzmann statistic. This can be a source of (small) deviations in device simulation.

17

18

2 Thin Film Heterostructures

-qϕ -qϕ(-wa)= qVbi -qϕ(0) -qϕ= 0

-qϕ -qϕ(-wa) =qVbi -qϕ =0 Ec

E

Ec

E

(bc) EV Ep,a

(fc)

EV

(bc)

z= 0

(fc) Ep,a

z=0

E= 0 (a)

absorber -da

E =0

window

-wa0

dw

z

(b)

absorber window -da

-wa0

dw

z

Figure 2.4 Schematic of absorber/window heterojunction band diagrams with (a) symmetric doping, that is, εa NA,a = εw ND,w and (b) asymmetric doping, that is, εw ND,w εa NA,a .

solution of Eq. (2.20) can be derived [23]. The following approximations are employed: 1) Charge contributions due to mobile carriers are omitted. 2) Impurity atoms are completely ionized in the space charge regions (SCRs) (depletion approximation). 3) The dielectric functions are constant within each layer, that is, dεm /dz = 0 with m = a, b, w. We consider the heterojunction between the n-type window and the p-type absorber of Figure 2.4. The built-in voltage, Vbi , of these structures is derived from the balance qVbi = Eg,a − Ep,a − En,w + Ec

(2.22)

where Eg,a is the bandgap of the absorber. Note that Vbi shall be a positive quantity. The moduli of the reduced chemical potentials of holes in the absorber (Ep,a = EFp − Ev,a ) and electrons in the window (En,w = Ec,w − EFn ) are deﬁned in Figure 2.1. The conduction band offset Ec may be positive (type I) or negative (type II) or zero as in the present example. We put the interface at the position z = 0. The potential drop in the window layer is given by −ϕ(0) = Vbi εa NA,a /

(2.23)

where we set = εa NA,a + εw ND,w . As becomes apparent in Section 2.6.3, a critical quantity for the performance of a heterostructure solar cell is the type inversion of the absorber at the absorber/emitter interface. Type inversion in a p-type absorber means that, in the cell’s equilibrium, the quantity Ep,a is small in the bulk and is large at the interface. In the best case this would mean that Ep,az=0 approaches Eg,a . In general, we want to use the quantity Ep,az=0 for the degree of absorber

2.3 Energy Band Diagrams

inversion. An equivalent notation used in other works is the hole barrier φp . For a heterojunction of an n-type window and p-type absorber it is Ep,az=0 (V) = Ep,a + q(Vbi − V)εw ND,w /

(2.24)

We see from Eq. (2.24) that Ep,az=0 becomes large if εw ND,w is much larger than εa NA,a . Through Vbi in Eq. (2.22), Ep,az=0 increases with increasing Ec . A large value of Ep,az=0 is realized in Figure 2.4b. This is the nonsymmetric n+ p-heterojunction. The depletion region mostly extends into the absorber. The interface between window and absorber may contain interface charge. For the beginning, we assume that the interface charge density is independent of the Fermi energy at the interface. Thus, the interface charge density QIF = qNIF is ﬁxed by the assumption of NIF charged interface states. In reality, this is not the case since the Fermi energy dictates the occupation of the states NIF . Nevertheless, for the time being it is instructive to discuss ﬁxed charge densities. Moreover, the absolute interface charge density may be the measured quantity of a capacitance experiment [24]. The neutrality condition of an absorber/window heterostructure with interface charge is −qNA,a wa + zqNIF + qND,w ww = 0

(2.25)

where wa and ww are the depletion widths of absorber and window. It is z = +1 for a positive interface charge and z = −1 for a negative one. Equation (2.25) is valid as long as free carriers can be neglected. With the neutrality condition in Eq. (2.25), an analytical solution of Eq. (2.20) can be obtained and we can calculate the Fermi level position at the absorber surface

qε ND,w qN2IF Ep,az=0 (V) = Ep,a + W (Vbi − V) − εw ND,w − εa NA,a 2 2 2 zqNIF 2 (V ε N ε N − V) − N + w D,w a A,a bi IF 2 q

(2.26)

The ﬁrst two terms of Eq. (2.26) are identical to Eq. (2.24) while the last two terms describe the inﬂuence of a positive or negative charge at the interface. In Figure 2.5 we investigate an n+ /p heterostructure where the negative interface charge has been varied. If not otherwise stated the device parameters are as given in Table 8.1. Due to the higher doping of the window, the Fermi level of the absorber at the interface with low interface charge is close to the conduction band, that is, Ep,az=0 > Eg,a /2. A density of negatively charged states of NIF = 1012 cm−2 can shift Ep,az=0 from close to the conduction band up to close to the valence band. Negative interface charge removes the inversion. The inﬂuence of the interface states, however, depends on the window doping: the higher the window doping, the less sensitive is the heterostructure to interface states. The width of the absorber’s depletion region is given by

2 2

zεa NIF εa εw ND,w q (Vbi − V) − NIF + (2.27) wa (V) = NA,a 2

19

2 Thin Film Heterostructures -Qi [C cm−2] 10−7

+Qi [C cm−2]

10−8

8 6

1.2

4

2

10−8

8 6

4

2

2

4

6 8

10−7 2

4

6 8

1019 1018

z =0

[eV]

1.0

Ep,a

20

0.8

NA,a =1016 cm−3

ND,w = 1017 cm−3

0.6 1017

0.4 0.2 0.0

1018 8 6

1012 (a)

4

2

8 6

1011

4

2

2

1010 −2

-zNi [cm ]

Figure 2.5 Calculated Fermi level position at the absorber surface Ep,az=0 as a function of the interface charge density of an absorber/window heterostructure. In (a) it is NA,a = 1016 cm−3 and ND,w is the varying

4

6 8

1010 (b)

2

1011

4

6 8

1012 −2

+zNi [cm ]

parameter. In (b) it is ND,w = 1016 cm−3 and NA,a is the varying parameter. Conduction band offset Ec = 0. Other parameters as in default Table 8.1.

where again it is = εa NA,a + εw ND,w and z gives the charge state of the interface state density NIF . For NIF = 0 and ND,W NA,a , we have 2(Vbi − V)εa wa (V) = (2.28) qNA,a For completeness we give the window’s depletion layer width

2 2

zεw NIF εa εw NA,a q (Vbi − V) − NIF + ww (V) = − ND,w 2

(2.29)

2.3.3 Absorber/Buffer/Window

Next we turn to a heterostructure of three semiconductors of the kind n-type window/n-type buffer/p-type absorber which was introduced in Figure 2.3a,b. We again put the focus on the Fermi level position at the absorber surface and will vary the doping densities and interface charge. This holds Ep,az=0 = Eg,a + Ec w,b + Ec b,a − En,w + qϕ(0)

(2.30)

where Ec w,b and Ec b,a are the conduction band offsets at the buffer/window and buffer/absorber interface. In the example of Figure 2.3 it is Ec w,b < 0 and Ec b,a > 0. ϕ(0) is the potential drop in the window and buffer layer (qϕ(0) < 0 in

2.3 Energy Band Diagrams

Figure 2.3). If we put χ = Ec w,b + Ec b,a , Eq. (2.30) indicates that χ must be large in order to obtain a large Ep,az=0 . The quantity ϕ(0) is best found by numerical solution of Eq. (2.20). For the example in Figure 2.3, however, ϕ(0) may be approximated by an analytical solution. Approximations (1–3) are as given in Section 2.3.2, but in addition the window doping (4) is much higher than the buffer and absorber doping. Therefore, the potential drop in the window layer is negligible. Also, the buffer layer (5) is completely depleted of mobile charge carriers; and the Fermi energy (6) is ﬂat within all layers. Thus, ϕ(0) can be calculated by integration of the charge density per area in the window qND,w ww and in the buffer qND,b wb ϕ(0) = −

q q ND,b w2b − ND,w wb ww 2εb εb

(2.31)

where wb , ww are the widths of the depletion regions in buffer and window and ND,b , ND,w are the donor densities of the respective layers. If we add up the energies from the left side of the buffer/absorber interface, we obtain a second equation for Ep,az=0 Ep.az=0 = Ep,a +

q2 NA,a w2a 2εa

(2.32)

where wa is the width of the absorber depletion region and NA,a the acceptor density in the absorber. Together with the charge neutrality condition, we have three equations with the three unknowns: Ep,az=0 , wa , and ww . According to approximation (5), wb is given by the total width of the buffer layer, that is, wb = db . The band diagram in Figure 2.6a depicts a heterostructure without interface charge in equilibrium. We have selected the special case of ND,b = NA,a = 1016 cm−3 . Also given in Figure 2.6a are carrier densities, electric ﬁeld, and volume charge. Due to the high n-type doping of the window layer of ND,w = 1018 cm−3 the band bending in the window indeed is negligible. If we inspect the density of free carriers in the buffer layer in Figure 2.6a, we ﬁnd that electron and hole concentrations are below 1015 cm−3 everywhere in the buffer layer. It are the band and Eb,a offsets, Ew,b c c , which predeﬁne the conduction band of the buffer layer and support the buffer depletion. Therefore, even for ND,b = NA,a the buffer is fully depleted and all the approximations listed above can be applied. The built-in voltage of the cell in Figure 2.6a is given by qVbi = Eg,a − Ep,a − En,w + χ

(2.33)

The charge neutrality condition of the cell in Figure 2.6a without interface charge may be expressed as qND,w ww + qND,b wb − qNA,a wa = 0

(2.34)

Equation (2.34) is only valid as long as wa does not extend up to the SCR of the back contact. With the approximations listed above we can solve Eqs. (2.30–2.32)

21

2 Thin Film Heterostructures

22

EF

Ec

Ec

Ev

EA,d EF Ev

p

1015 1010 105

p n

n 2

+ 1017

+

14

10 10

(a)

−

(b)

−

(c)

−

+ (d)

−

1012 + 1010 1012

−

17

+

+

−0.6 −0.4−0.2 0.0 0.2

−0.6−0.4−0.2 0.0 0.2

−0.6 −0.4 −0.2 0.0 0.2

−0.6−0.4 −0.2 0.0 0.2

Area Charge [cm−2]

0 −2× 105

−

Volume Charge [cm−3] F [V cm−1] n,p [cm−3]

E [eV]

4 3 2 1 0

Position z [µm]

Figure 2.6 Simulated solution curves of an absorber/buffer/window heterostructure: (a) without interface charge, (b) with area charge of +8 × 1011 q cm−2 at the buffer/absorber interface, (c) with area charge of 2 × 1012 q cm−2 at the buffer/window interface, and (d) with acceptor states of density 5 × 1017 cm−3 at EA,d = 0.6 eV (dashed-dotted line) in the absorber layer. Other parameters as in default Table 8.1. Note that (a) shows the default device from Table 8.1 in equilibrium. From bottom to top: total electric volume charge

S(z) according to Eq. (2.16) on a logarithmic scale (left axis) and electric area charge Qss (right axis). Area charge is given as a full line. The zero line was set to a concentration of 1014 cm−3 for the volume charge and 1010 cm−2 for the area charge. Electric ﬁeld F(z). Small jumps in the electric ﬁeld in (a) indicate differences in the dielectric constant. Densities of mobile electrons and holes. In the absorber, the densities exhibit a crossing point of n = p which locates the electronic junction of the heterostructure.

and (2.34) for Ep,az=0 thus

Ep.az=0 (V) = Eg,a 1 − κ−1 IF +

κ−1 IF

q(Vbi − V) + Ep.a−νIF z1

q2 ND,b d2b q2 NA,a db + − wa (V) 2εb εb

(2.35)

where κIF = 1 and νIF = 0 in the case of absent interface charge. The width of the absorber SCR, wa (V), is given by

2 2N

εa d b 2 d q εa db 2ε D,b a b wa (V) = − q(Vbi − V) + + + 2 + νIF z2 εb κIF εb κIF q NA,a κIF 2εb (2.36)

2.3 Energy Band Diagrams 1.2

Ep,a z = 0 [eV]

1.0 0.8

1015 cm−3

1015 1016

1015

0.6 0.4

1016 10

NA,a = 10

17

(a)

ND,b = 1017 cm−3

1016 ND,b

cm

−3

cm

0.2 0.0 1015

16

−3

[cm−3]

1017

1015 (b)

1016 NA,a

Figure 2.7 Fermi energy referenced to valence band maximum, Ep,az=0 , at the buffer/absorber interface of an absorber/buffer/window heterostructure without interface charge (NIF = 0). The data shown are: simulated values (symbols) and calculated values after Eq. (2.35) (lines). (a) Data

[cm−3]

1017

NA,a = 1017 cm−3

10−7 (c)

10−6

10−5

10−4

db [cm]

shown as a function of buffer donor density with parameter NA,a . (b) Data shown as a function of the absorber acceptor density with parameter ND,b . (c) Data shown as a function of the buffer thickness with ND,b = 1015 cm−3 and with parameter NA,a . Other parameters as in default Table 8.1.

and the electric ﬁeld at the buffer/absorber interface is given by

qNA,a db 2 2NA,a qNA,a db q2 ND,b d2b Fz=0 (V) = − + + + νIF z2 q(Vbi − V) + εb κIF εb κIF εa κIF 2εb (2.37) Figure 2.7a gives values of Ep,az=0 calculated after Eq. (2.30) as a function of the buffer donor concentration for different absorber acceptor concentrations (full line). The calculated values of Ep,az=0 are compared to values for Ep,az=0 obtained by device simulation (markers). In Figure 2.7b, Ep,az=0 is plotted as a function of the absorber acceptor concentration for different buffer donor concentrations. The following trends for achieving a large value of Ep,az=0 may be observed in Figure 2.7a,b: At low buffer doping of ND,b = 1015 cm−3 and ﬁxed thickness of the buffer layer, Ep,az=0 only depends on the absorber doping. The buffer acts as a dielectric and the potential drop in the buffer is determined by the charge in the window and absorber. This charge increases with increasing absorber doping and as a result Ep,az=0 decreases. However, the potential drop in the (lowly doped) buffer layer also depends on the buffer thickness. The formula of the parallel plate capacitor, V = qdb /εb ε0 , shows that the potential drop linearly increases for a larger buffer thickness. For a larger buffer thickness, the buffer starts to counterbalance the complete absorber charge and the value of Ep,az=0 can be approximated according to the simple absorber/emitter heterojunction model of Eq. (2.24). If the buffer doping approaches the absorber doping, Ep,az=0 starts to rise in Figure 2.7a. Now the charge in the buffer layer becomes sufﬁcient to partly counterbalance the charge in the absorber. Thus, if we aim for a low value of Ep,az=0 due to device reasons, a higher value of NA,a is tolerable for high ND,b . Certainly, Ep,az=0 becomes larger if the absorber acceptor concentration is

23

2 Thin Film Heterostructures 120 × 10−6 100

ND,b = 1017 cm−3 1016 cm−3

80 Wa [cm]

24

1015 cm−3

60 40 20 0 1015

1016

1017

NA,a [cm−3]

Figure 2.8 Width of absorber space charge region calculated after Eq. (2.36) (lines) and simulated (symbols) as a function of absorber acceptor density for ﬁxed buffer donor densities of ND,b = 1015 , 1016 , and 1017 cm−3 .

decreased. The conclusion of Figure 2.7a,b can be expressed in the rule that ND,b has to be larger than NA,a for a large Ep,az=0 . Following this rule transforms the absorber/buffer/window heterostructure in Figure 2.6 into a simple heterojunction between an n-type emitter and a p-type absorber. Based on these considerations, we understand that in Figure 2.6a the inversion is not complete since the condition ND,b > NA,a is not fulﬁlled. Although in equilibrium electrons are still majority carriers at the absorber surface, this will change for small electric forward bias. The volume charge within the window layer is high but restricted to a small region. As a result, the potential drop in the window layer can be neglected [which was conditional for Eq. (2.31)]. The small offsets of the electric ﬁeld at the buffer/window and buffer/absorber interfaces are due to differences in the dielectric constant. Figure 2.8 depicts the width of the SCR calculated after Eq. (2.36) as well as simulated data for wa . Calculated and simulated values are in good agreement. 2.3.4 Interface States

Interface states may add additional charge to the charge neutrality equation and thus may change the potential distribution. If interface states are charged or not, depends on the position of the Fermi energy at the respective interface. Before we take this into account we want to simplify matters and to consider the effect of interface charge QIF only. We do this by assuming that a given density of interface states NIF is fully charged – either negative or positive. For interface charge at the buffer/window or the buffer/absorber interface the charge neutrality condition reads qND,w ww + qND,b wb + zqNIF − qNA,a wa = 0

(2.38)

2.3 Energy Band Diagrams

−QIF [C cm−2] 10−7 1.2

4

2

10−8

8 6

4

2

2

4

6 8

10−7 2

4

6 8

1015

1.0

Ep,a z= 0 [eV]

+QIF [C cm−2]

10−8

8 6

25

0.8 1016 NA,a = 1017 cm−3

0.6 0.4 0.2 0.0

8 6

1012 (a)

4

2

8 6

4

2

1011

2

1010 −2

−zNIF [cm ]

Figure 2.9 Device as described in caption of Figure 2.7 but with (a) negative charge density −qNIF and (b) positive charge density +qNIF at the buffer/absorber interface. Values of Ep,az=0 obtained by device simulation (symbols) are given as a function of

1010 (b)

4

6 8

2

4

1011

6 8

1012 −2

+zNIF [cm ]

the charged interface state density for absorber acceptor densities of 1016 , 5 × 1016 , and 1017 cm−3 . Other parameters as in default Table 8.1. Solid lines represent Ep,az=0 calculated according to Eq. (2.35).

In the case of charge at the buffer/absorber interface, Eq. (2.35) is the solution for Ep,az=0 , if we insert as parameters κIF = 1, νIF = q2 NIF db /εb (where NIF is given in units of cm−2 ). We further put z1 = z2 = +1 in the case of charged interface donors and z1 = z2 = −1 in the case of charged interface acceptors. Figure 2.9 shows simulated and calculated values of Ep,az=0 as a function of negative (a) and positive (b) charge densities at the buffer/absorber interface. Note that NIF on the abscissa of Figure 2.9 is the density of ﬁxed charge divided by the element charge. Thence, NIF is the density of charged interface states. In Figure 2.9, we ﬁnd that Ep,az=0 can strongly be inﬂuenced by a charged state density exceeding ±1011 cm−2 . In absolute terms, Ep,az=0 changes if the interface charge density becomes comparable to the buffer and absorber charge per area, the latter given by the product of doping density and depletion width. Negative charge at the buffer/absorber interface has the same effect as a higher absorber acceptor concentration: It decreases Ep,az=0 and reduces the inversion. Vice versa, a positive charge resembles a higher buffer donor concentration. It increases Ep,az=0 and reinforces the inversion. Ep,az=0 values calculated after Eq. (2.35) follow the simulated data points as long as the condition ND,b ≤ NA,a is fulﬁlled. Also the charge at the buffer/window interface can inﬂuence the band diagram. In Figure 2.10, we plot simulated values of Ep,az=0 as a function of negatively and positively charged interface state densities. A negative charge at the buffer/window

2 Thin Film Heterostructures −QIF [C cm−2] 10 1.2

6 5 4 3

z =0

[eV]

1.0

Ep,a

26

2

10−7 76 5 4 3

2

2

3

4 5 67

2

3 4 5 6

1015

0.8 1016

0.6 0.4

NA,a = 1017 cm−3

0.2 0.0

4 3

2

76 5 4 3

1012 (a)

+QIF [C cm−2]

−7

2

2

1011

−zNIF [cm−2]

Figure 2.10 Device as described in the caption of Figure 2.7 but with (a) negative charge density −qNIF at the buffer/window interface and (b) positive charge density +qNIF . Values of Ep,az=0 as obtained by

1011 (b)

3

4 5 67

2

3 4

1012 +zNIF [cm−2]

device simulation are given as a function of the charged interface state density with NAa as parameter. Other device parameters as in default Table 8.1.

interface has the same effect as negative charge at the buffer/absorber interface. It decreases Ep,az=0 and reduces the inversion. Compared to the buffer/absorber interface, however, a higher charge density is required at the buffer/window interface. This is due to the fact, that the positive charge density in the window layer is higher than in the buffer layer. No inﬂuence on the absorber inversion has a positive charge at the buffer/window interface (in contrast to positive charge at the buffer/absorber interface). Such a positive charge is screened by electrons in an accumulation region of the window layer. In the case of buffer/window interface charge, Ep,az=0 cannot be calculated by use of Eq. (2.35) since the potential drop in the window layer can no longer be omitted. In Figure 2.6b, the band diagram, carrier densities, and so on, in the case of interface charge at the absorber/buffer interface are depicted. While without interface charge (Figure 2.6a) the absorber inversion was not complete, it becomes complete in the case of 1 × 1012 cm−2 positive charge density (see right axis) at the buffer/absorber interface. Now the electron concentration at the absorber surface exceeds the hole concentration by more than 15 orders of magnitude. Electrons remain majority carriers at the absorber surface even for not too large a forward bias. The electric ﬁeld shows a jump at the absorber/buffer interface due to the ﬁxed interface charge. The electric ﬁeld region mostly is located within the absorber. An opposite effect has a negative charge either at the absorber/buffer or at the buffer/window interface: It decreases the inversion of the absorber surface. In Figure 2.6c, we give the example for negative charge at the buffer/window interface. Looking at the volume charge, we see that a positive volume charge in window and buffer screens the negative area charge. As a result of charge balance, less space charge is allocated in the absorber and the potential drop in the absorber is lower. The absorber is not in inversion and holes are majority carriers at the absorber

2.3 Energy Band Diagrams

surface. Thus, in an absorber/buffer/window heterostructure, the buffer/window interface can be extremely important. Now, we consider the more realistic case that the Fermi energy at the interface determines the charge in a given interface state density. The interface charge is given by EA,IF − EF QIF = −qNA,IF F1/2 KT ED,IF − EF and QIF = +qND,IF [1 − F1/2 (2.39) KT in the case of acceptor and donor states of density NA,IF and ND,IF . EA,IF and ED,IF are the energy levels of the interface acceptor and donor states and F1/2 is the Fermi–Dirac function. Thus, the charge neutrality condition for discrete interface states is qND,w ww + qND,b wb + qIF − qNA,a wa = 0

(2.40)

Due to the exponential term in the Fermi–Dirac function (and also in the Boltzmann approximation), Eqs. (2.30–2.32) and (2.40) cannot be solved analytically. In the following, we continue to focus on the absorber surface and its degree of inversion. As an example, we introduce discrete interface states of acceptor and donor type and of varying density at the buffer/absorber interface of a heterostructure as depicted in Figure 2.6. Two critical cases are investigated: First, acceptor states at different energies for a heterostructure with ND,b > NA,a . As we learn from Figure 2.7, without interface states this heterostructure has an inverted absorber. And second, donor states at different energies for a heterostructure with noncomplete inversion due to ND,b < NA,a . Figure 2.11 gives Ep,az=0 for (a) acceptor and (b) donor states of different energy levels as a function of the interface defect density, respectively. The defect states are located at 0.9, 0.7, and 0.5 eV above the valence band of the absorber. We select different ratios of absorber/buffer doping in order to illustrate the effects of acceptor and donor interface states. In Figure 2.11a Ep,az=0 is large or the inversion is strong with 1010 cm−2 acceptor states present at the interface. Acceptor states of density > 1011 cm−2 reduce the inversion of the absorber and Ep,az=0 decreases. A different starting point is chosen in Figure 2.11b where, due to the absorber/buffer doping ratio, the inversion is weak and Ep,az=0 is ∼0.6 eV. Now donor states located close to the conduction band enhance the inversion. The closer their energy level to the conduction band, the stronger is the inversion. As an example, for a state density of 1013 cm−2 the Fermi energy is more than 1 eV above the valence band at the interface while the donor defect is 0.9 eV. Thus, even though the donor states are only partially ionized (empty) the positive interface charge is sufﬁcient to substantially add to the positive charge of buffer and window. As a result, the absorber SCR is enlarged and the potential drop in the absorber becomes higher. Another variation explains the phrase ‘Fermi level pinning’ for a heterostructure with interface states. For a ﬁxed density of discrete interface states energetically

27

28

2 Thin Film Heterostructures

Ep,a z = 0 [eV]

−QIF [C cm−2]

(a)

10−5 10−6 10−7 10−8 1.2 1.0 EA,IF = 0.9 eV 0.8 0.7 eV 0.6 0.5 eV 0.4 0.2 0.0 1014 1013 1012 1011 1010 NA,IF

[cm−2]

QIF [C cm−2] 10−8 10−7 10−6 10−5 0.9 eV 0.7 eV

No IF states

EA,IF = 0.9 eV 0.7 eV 0.5 eV

ED,IF = 0.5 eV

ND,b =5× 1016 cm−3

1010 1011 1012 1013 1014 (b)

ND,IF

Figure 2.11 Device as described in the caption of Figure 2.7 but with varied buffer and absorber doping. (a) ND,b = 5 × 1016 cm−3 and NA,a = 1 × 1016 cm−3 and acceptor states NA,IF at the buffer/absorber interface. The acceptor states are at different discrete energy levels Ec − EA,IF in the bandgap of the absorber. (b) Buffer and absorber

[cm−2]

1015 (c)

1016 NA,a

1017

[cm−3]

doping of ND,b = 5 × 1016 cm−3 and NA,a = 1 × 1017 cm−3 and interface donor states ND,IF at different energy levels. (c) Buffer doping of ND,b = 5 × 1016 cm−3 and varying absorber acceptor doping with discrete acceptor states of density NA,IF = 1013 cm−3 at different energy levels in the absorber bandgap.

located at a certain energy, a variation of the absorber acceptor concentration has no inﬂuence on the Fermi level position at the interface or, in other words, no inﬂuence on Ep,az=0 . This is visible in Figure 2.11c, were we plot Ep,az=0 for discrete acceptor levels of density 1013 cm−2 located at EA,IF = 0.5, 0.7, and 0.9 eV. We see that with decreasing absorber acceptor concentration, the inversion cannot further be increased. Instead, the Fermi level is ‘pinned’ at the acceptor interface state energy. Like in the bulk of a semiconductor, an interface may exhibit different states of both donor and acceptor type. Unlike in the bulk, their occupation is not easy to predict and depends on the doping of the adjacent layer. Consider, for instance, the case of ND,b > NA,a and acceptor states at EA,IF = 0.5 eV in Figure 2.11a. Due to an acceptor state density of NA,IF = 1013 cm−2 the Fermi energy lies at Ep,az=0 = 0.4 eV. Higher lying donor states of density comparable with the acceptor will improve the situation and will pin the Fermi energy at a larger value of Ep,az=0 due to their positive charge. In contrast, for the other case ND,b < NA,a (Figure 2.11b) the same combination of acceptor and donor states has no inﬂuence on the Fermi level position. In that case, the Fermi energy stays at a large distance from the conduction band since the charge in the donor states is not sufﬁcient. The case of more than one donor (acceptor) level at different energies at the interface is more simple. Usually, it is the donor (acceptor) energetically located nearer to (farer from) the conduction band edge which determines the band bending and thus the value of Ep,az=0 . Lower (higher) lying donors (acceptors) may not be charged and, therefore, may not contribute to the overall charge balance. The extreme variant of multiple interface states of different ionization character (acceptor and donor) is a continuous interface defect density (per area and energy). If this state density derives from the complex band structure [25], it exhibits

2.3 Energy Band Diagrams

acceptor character close to the conduction band and donor character close to the valence band. Depending on the position of the Fermi level, the interface either exhibits a net positive charge if the Fermi level lies below a certain charge neutrality level or exhibits a net negative charge if the Fermi level lies above a charge neutrality level. If the Fermi level coincides with the charge neutrality level, the net charge in the interface states is zero. The shape of the interface state density as a function of energy in the gap may be ∪-type with increasing density toward the band edges. In the vicinity of the charge neutrality level it may be approximated by a constant density. A constant interface state density with charge neutrality level has been assumed in Ref. [26]. With the parameters νIF = q2 NIF wb /εb , κIF = 1 + νIF , z1 = φn − Eg,a , and z2 = φn − Ep,a Eq. (2.35) can be used to calculate the quantity Ep,az=0 for charge neutrality levels φn located at different energy positions in the bandgap. φn is measured from the valence band edge. The results are similar as for discrete interface states: at a sufﬁcient large density of interface states, the Fermi level is pinned at the charge neutrality level, regardless of the doping ratio between buffer and absorber which dictates the position of the Fermi level at the interface without interface states. In conclusion, acceptor states at the buffer/absorber (or buffer/window) interface push the Fermi energy downwards and diminish the inversion. In contrast, donor states energetically located close to the conduction band can enhance the inversion. Certainly, interface states not only modify the band bending but also can contribute to recombination. For instance, there could be a trade off between beneﬁcial donor states in terms of inversion and detrimental donor states in terms of recombination. 2.3.5 Interface Dipoles

Interface dipoles can change the band offset at an interface. They can be introduced deliberately by use of ‘band offset engineering’ [20] or they may be the result of differences in the fabrication of an interface. To date chalcogenide heterostructures exhibit isovalent or pseudo-isovalent interfaces. As pointed out by Franciosi and van de Walle [20], isovalent interfaces are difﬁcult to engineer and only small deviations from the canonical band line up are to be expected. The implication of a possible interface dipole may easily be conceived based on the value of Ec at the buffer/absorber interface in Figure 2.3. A dipole with negative charge at the buffer side and positive charge at the absorber side would increase the value of Ec . Thereby, a type II interface could be inverted to a type I interface. 2.3.6 Deep Bulk States

The semiconductors of our heterostructure may contain defect states within the bandgap. If the defect states become charged, they add to the space charge of the heterostructure, similar as interface states. Thus, they inﬂuence the potential

29

30

2 Thin Film Heterostructures

drop in the layers. Certainly, defect states in the bandgap also give rise to carrier recombination. In equilibrium, the charge of the defect states is governed by the Fermi level. Donor-like (acceptor-like) defect states sufﬁciently above (below) the Fermi level contain positive (negative) charge. Under non-equilibrium, that is, voltage or light bias, the occupation of defects states is governed by the kinetics of electron and hole capture. In the following, we investigate the inﬂuence that deep states in the p-type absorber layer have on the band diagram. As an example, we select the standard heterostructure of window, buffer, and absorber, as depicted in equilibrium in Figure 2.6a. Without deep states, the absorber is in inversion. Now we introduce deep acceptor states of density NA,d = 5 × 1017 cm−3 (σp mp ∗ /σn mn = 103 ) in the absorber layer (Figure 2.6d). Due to the position of the Fermi energy, the deep acceptor states are not charged in the QNR. Thus, the hole concentration about equals the doping concentration of NA,a = 1016 cm−3 at z = da . At the buffer/absorber junction, however, the deep acceptors become charged due to absorber band bending. Consequently, they add to the space charge of the absorber. Band bending in the absorber is enforced close to the junction but in total the inversion is reduced. A higher interface charge would be required in order to preserve inversion of the absorber surface. In total, the deep acceptor states have modiﬁed the band diagram at the buffer/absorber junction. The SCR of the absorber in Figure 2.6d is divided into two subregions of volume charge density: ρ/q = −5 × 1017 cm−3 and −1 × 1016 cm−3 . In order to solve the screening equation (Eq. (2.20)), the absorber would require to be calculated as two different layers. Thus, Eqs. (2.35) and (2.36) can no longer be applied. Meyer et al. developed [27] a differential model to calculate small variations of a band diagram caused by a differential supplement charge in the absorber and emitter layer. Using this model, voltage and light induced effects on the diode current and on the space charge capacitance can be described. Subregions of different volume charge can also be the effect of compensation doping. Consider the case with NA,a = 5 × 1017 cm−3 and ND,d = 4.9 × 1017 cm−3 . In the vicinity of the buffer/absorber junction, the deep donors are not charged and do not compensate the high p-type doping of NA,a = 5 × 1017 cm−3 . The band diagram would be similar to Figure 2.6d. How could the cases of charged deep traps experimentally be diagnosed? The absorber width of the SCR with deep states is only slightly reduced to about 0.2 µm compared to the case without deep states (see Figure 2.6a). Thus, the zero voltage capacitance of all devices would be rather similar. Capacitance–voltage measurements, however, would reveal a change in slope of a C−2 (V) plot indicating two regimes of different doping concentrations [28]. A possible compensation doping can be determined from the temperature dependence of the dark carrier concentration [29]. Furthermore, deep trap levels can be analyzed by deep-level transient spectroscopy [30]. Compared to the case without deep states, the electric ﬁeld of the device in Figure 2.6d is much enlarged at the buffer/absorber interface. This can have a beneﬁcial effect on the Voc since the effective recombination zone in the SCR

2.3 Energy Band Diagrams

is reduced (see Section 2.4.5.1). However, a ﬁeld strength of 2 × 105 V cm−1 can cause tunneling enhancement of recombination (see Section 2.4.5.1). The tunneling would increase the diode current under solar cell operation and thereby reduce Voc . Certainly, deep states may also act as efﬁcient recombination centers (RCs) (depending on the capture cross sections) and thereby reduce Voc . Cases where the occupation of deep states is illumination and bias dependent is treated in Section 2.4.8. In the dark, the occupation of deep states in the bandgap of a semiconductor is governed by the Fermi–Dirac function. Under non-equilibrium conditions such as voltage and light bias, however, the occupation of deep defects depends on transition rates between the defects and the conduction and valence bands. The transition rates are proportional to the capture cross sections. If the capture cross sections for electrons and holes are largely different (which is normally the case) a steady state occupancy with one charge carrier type may occur. The trapped photo generated charge carriers then may alter the band diagram with respect to the case without illumination. An example is given in Figure 2.12 where the absorber of 1016 cm−3 p-type doping contains 5 × 1017 cm−3 deep acceptors energetically located at midgap energy. If the deep acceptors exhibit equal capture cross sections (σn = σp = 10−15 cm−2 ), the band diagram under illumination and in short circuit condition resembles the case in the dark (see Figure 2.6d). If it is σp σn , however, holes photo generated within the absorber layer will become trapped in the deep acceptor states thus reducing the net negative charge close to the buffer/absorber junction. As a result, the charge density in the absorber close to the buffer/absorber junction is reduced and the band diagram under illumination approaches the case without deep states (see Figure 2.6a). As a second example of the deep states’ inﬂuence on the band diagram we introduce deep acceptors in the buffer layer. We assume that the buffer layer is compensated. The net n-type doping of 1015 cm−3 is realized by a background doping with ND,b = 5 × 1018 cm−3 donors and a density of NA,d = 4.999 × 1018 cm−3 compensating deep acceptors. The band diagram in the dark is depicted in Figure 2.13a. The net space charge of ρ/q = +1015 cm−3 in the buffer is not sufﬁcient to balance the negative absorber space charge. Thus, the buffer acts as a dielectric and, due to the missing interface charge, the absorber is only in slight inversion. The majority of deep acceptor states in the buffer is occupied or, in other words, is negatively charged in the dark. This may change, however, under illumination where photo generated holes may discharge the deep acceptor states. If the capture cross section of holes largely exceeds the capture cross section of electrons, under steady state conditions the deep acceptors become empty. The compensation is reduced and the positively charged donors add to the space charge. This case is reﬂected in Figure 2.13b. Under illumination, the buffer space charge can partly balance the absorber space charge. As a result, the band bending in the buffer layer is reduced. If we consider the electron barrier φb n from the window to the absorber in Figure 2.13, we see that this barrier is reduced for the case σp σn . Such a barrier reduction can be the origin of a cross over between JV curves measured in the dark and under illumination (see Section 7.1.2).

31

2 Thin Film Heterostructures

4 E [eV]

3

Ec EFp

EA,d EFn

EFn

EFp

2 1 Ev

0

F [V cm−1]

n,p [cm−3]

p

Volume Charge [cm−3]

32

1015

p

1010

n

n

105

2 0 −2 × 105

1017 10

+

+

14

−

−

1017 −0.6

−0.4

(a)

−0.2

0.0

0.2

Position z [µm]

Figure 2.12 Solution curves to show the inﬂuence of capture cross sections of deep acceptor states in the absorber. Device under AM1.5G illumination in short circuit. (a) Equal capture cross sections of electrons and holes of σp = σn = 10−15 cm−2 . (b) Hole capture cross section σp = 10−12 cm−2

−0.6 (b)

−0.4

−0.2

0.0

0.2

Position z [µm]

and electron capture cross section σn = 10−15 cm−2 . Deep acceptor state density NA,d = 5 × 1017 cm−3 . Other parameters as in default Table 8.1. For comparison, Figure 2.39a gives the default device (without deed acceptor states) under short circuit.

2.3.7 Bandgap Gradients

Bandgap gradients may occur in any semiconductor constituting the chalcogenide solar cell. They may occur as gradients in the conduction band, in the valence band or in both bands. In this section, we present some possible bandgap gradients in the absorber layer. By aid of simulated examples we want to exemplify Eqs. (2.9) and (2.10) which state that bandgap gradients can add to the current density of the device. Figure 2.14 sketches absorbers with normal and double bandgap gradients. Only linear gradients occurring within a limited depth of the absorber are considered.

2.3 Energy Band Diagrams

Ec

E [eV]

4

Ec

φbn

φbn

3 2

EA,d

EA,d

Ev

Ev

1

Volume Charge [cm−3]

F [V cm−1]

n,p [cm−3]

0

(a)

1015

p

1010 105

n

2 0 −2 × 105 +

1017 1014

+

−

−

1017 −0.4

−0.2

0.0

0.2

Position z [µm]

Figure 2.13 Solution curves to show the inﬂuence of capture cross sections of deep donors in buffer. Default heterostructure but with high degree of compensation in the buffer layer. Background doping in the buffer is ND,b = 5 × 1018 cm−3 .

0.4

−0.4

−0.2

(b)

Position z [µm]

0.0

0.2

0.4

Compensating acceptors in the buffer of density 4.999 × 1018 cm−3 at midgap with capture cross section for electrons and holes of σp = 10−12 cm−2 and σn = 10−18 cm−2 , respectively, (a) in the dark and (b) under AM1.5G illumination in short circuit.

Normal bandgap gradients widen the energy gap toward the back surface (a,b) or toward the front surface (c,d) of the semiconductor. A V-shaped bandgap is the origin of a double bandgap gradient (e,f). If we allow only one change in slope of the bandgap located at dmin , the number of possible combinations of front and back surface gradients in the conduction and valence bands amounts to 15. Not all combinations, however, are practical and useful. The bandgap gradients in Figure 2.14 can be described by the parameters Ec , Ev , and dmin . The effect of a back surface gradient is to decrease minority carrier recombination at the back contact. This leads to an increased photo current and decreased diode current. The impact of a back surface gradient becomes larger for a smaller distance between back contact and collecting junction. The relevant quantity for the discussion of bandgap gradients is the current density of the minority carriers.

33

34

2 Thin Film Heterostructures

∆Ec,back ∆Ev,back

Ev (a)

(b)

-dmin

∆Ec,front (c)

(d)

(e)

(f)

E

∆Ev,front

-dmin

-dmin,f z

(g)

-dmin,b

(h)

Figure 2.14 Eight examples (a–h) of bandgap gradients in absorber layers with light entering from the right side. dmin is referenced to z = 0 at the right hand side of the absorber and is a positive quantity.

In the p-type absorber this is the electron current density Jn . According to Eq. (2.9) (for a one-dimensional problem), Jn is composed of the ﬁeld term Jn,E (z) = −µn (z)n(z) dqϕ(z)/dz, the effective force ﬁeld term Jn,χ (z) = −µn (z)n(z)dχ(z)/dz and a term which depends on the chemical potential of the electrons Jn,ζ (z) = −µn (z)n(z) dEn (z)/dz. For a nondegenerate semiconductor, dEn (z)/dz is given by Eq. (2.11). If the effective state density, Nc , in the conduction band is constant, the term of the reduced chemical potential becomes a pure diffusion term of −kTµn (z) dn(z)/dz. Equation (2.9) then reads Jn = Jn,E (z) + Jn,CBM (z) + Jn,ζ (z). In Figure 2.15 we show examples of cells (a) without and (b) with a back surface gradient under short circuit condition. The different current contributions according to Eq. (2.9) have been plotted separately. At the back surface at x = 1.5 µm the default device (Table 8.1) exhibits a high surface recombination velocity of Sn0 = Sp0 = 107 cm s−1 . Without back surface gradient, the cell collects electrons from about −0.6 µm. The starting point of the collection region is indicated by a sign reversal of the current density Jn . For z < −0.6 µm, photo generated minority electrons ﬂow to the back contact and recombine. The photo current is composed out of a ﬁeld component, Jn,E , and a diffusion component, Jn,ζ . With aid of a back

2.3 Energy Band Diagrams

E [eV]

4

35

EFn

3

EFp

J [A cm-2]

n [cm-3]

2 1 1013

109 + 10−1

10−1

10−1

n

n

Jn

Jn

10−5

+ 103 J [A cm-2]

n

1011

Jn Jp

Jp

Jp Jn,z

Jn,z

Jn,z

10−5 10−1 103

−1.5 −1.0 −0.5 (a)

Jc

Jn,E

Jn,E

0.0

0.5

−1.5 −1.0 −0.5 (b)

Figure 2.15 Solution curves for devices (a) without bandgap gradient, (b) with back surface gradient (Ec,back = 0.2 eV, dmin = 0.3 µm), and (c) with front surface gradient (Ev,front = 0.2 eV, dmin = 0.3 µm). AM1.5

z [µm]

Jn,E

0.0

0.5

−1.5 −1.0 −0.5 (c)

illumination under short circuit condition. Semiconductor parameters as listed in default Table 8.1. Different current contributions are calculated according to the terms in Eqs. (2.9) and (2.10).

surface gradient (Figure 2.15b) the position of sign reversal of Jn shifts almost up to the back contact. Thus more minority carriers are collected at the junction. The driving force for the extended current collection is not an electric ﬁeld as the gradient of the electric potential is zero outside of the SCR. Instead, it is the gradient of the chemical potential and, to be more speciﬁc, of the concentration independent part ζn,0 = −χ. This leads to the current contribution Jn,χ . Note also that the diffusion component is changed. Diffusion toward the back surface sets in at larger z. Thus, the diffusion component partly diminishes the positive effect of the bandgap gradient. This is the reason why the additional ﬁeld of a back surface gradient is not simply given by Ec /q(wa − dmin ) but must be calculated from the gradient of EFn . The increase in bandgap toward the back surface of the absorber can also be realized by a slope in the valence band edge (Ev,back < 0) as sketched in Figure 2.14b. If the p-type doping concentration of the absorber is constant in depth, the gain in solar cell parameters is identical to the case of Ec,front > 0. In that case the graded region of the absorber contains a constant electric ﬁeld.

0.0

0.5

36

2 Thin Film Heterostructures

The beneﬁt of a bandgap gradient at the back surface of the absorber in terms of improved cell performance depends on Eg,back and dmin . Rules for the design of a back surface gradient are presented in Section 3.11. Bandgap widening at the front surface aims at reducing the diode current in the graded region of the absorber while maintaining the photo current in the nongraded region. Figure 2.14c sketches bandgap widening at the front surface by an increased conduction band edge and (Figure 2.14d) by a decreased valence band edge. A numerical example for Ev,front = −0.2 eV(dmin = 0.3 µm) is given in Figure 2.15c. For devices with a dominant interface recombination or interface-near recombination, the introduction of a front surface gradient can shift the recombination from positions close to the interface to positions within the bulk. We will see in Section 3.10 that in this case the front surface gradient can improve the cell. For a cell without interface recombination, the beneﬁt of a front surface gradient will be small. The example in Figure 2.15c shows that the bandgap gradient leads to a positive gradient of EFn below z = −0.77 µm (the sign reversal of Jn is at slightly smaller z than in Figure 2.15a). This positive slope of dEFn /dz > 0 reduces the photo current and ﬁll factor of the device, in particular for small lifetimes. It also occurs for a surface bandgap gradient realized by a surface gradient of the conduction band. Finally, Figure 2.14e,f sketches the cases of double bandgap gradients. In certain cases, the beneﬁt for the cell performance may be larger than with front or back surface gradients alone.

2.4 Diode Currents

In this section, we discuss different recombination processes which may dominate the diode current of chalcogenide solar cells. We want to answer the questions: 1) What are the parameters J00 , Ea , and A for each recombination processes. 2) How does the cell behave if parallel processes are active. 3) What is the inﬂuence of barriers and non-homogeneities. Although recombination processes determine the diode current of the cell, we speak of diode current and not of recombination current. The reason is that recombination also reduces the photo current. We discriminate the diode current Jdiode from the photo current Jph which is treated in Section 2.4.9. The motivation behind this discrimination is to apply the superposition principle wherever possible. Other expressions for the diode current are bucking current or reverse current. 2.4.1 Superposition Principle and Shifting Approximation

If the solar cell is voltage biased in the dark, a pure diode current is ﬂowing. This diode current can easily be determined but it may be different from the

2.4 Diode Currents

diode current if the cell is under illumination. Reasons for this difference, among others, may be an illumination dependent carrier lifetime, a large series resistance changing the boundary condition for QNR recombination [31], a change from low to high injection condition under illumination. In order to understand the limitations of the solar cell we are mainly interested in the diode current under illumination. Knowing this current allows calculating the Voc of the device. In practice, one often analyzes both cases, with and without illumination. The superposition principle states that the total current can be written as a sum of diode and photo currents. Also the photo current may be voltage dependent. Jlight (V) = Jdiode (V) + Jph (V)

(2.41)

It is justiﬁed to apply the superposition principle as long as we make sure that the diode current is the one ﬂowing under illumination. Experimentally, the diode current under illumination shall be obtained by subtracting the (voltage dependent) photo current Jph (V) from the light current, Jlight (V). The problem of evaluation of the light JV curve is discussed in Section 2.6.2. The superposition principle shall be discriminated from the shifting approximation [31] which reads Jlight (V) = Jdark (V) + Jsc

(2.42)

This shifting approximation is rarely valid in chalcogenide solar cells. This is unfortunate as the validity of the shifting approximation would allow to measure two simple items to describe the complete device: the dark curve Jdark (V) and the short circuit current Jsc . 2.4.2 Regions of Recombination

The diode current of a heterostructure can be based on different recombination paths which are depicted in Figure 2.16. Recombination may take place in the SCR (path 4 in Figure 2.16), in the QNR (path 2) and at the back contact (path 3). Speciﬁc for heterostructure solar cells is the recombination path (1) at the absorber surface or, in other words, at the interface to the emitter. At this interface, recombination active defects will be present due to lattice mismatch or segregation of impurities. The possibility of recombination path (1) clearly is a disadvantage of a heterostructure solar cell against a homojunction. But there is also an advantage: Due its large bandgap recombination in the window and in the buffer layer (path 5) can be neglected for the calculation of the diode current. This is why in the following we deal with recombination in the absorber or at its surfaces. The diode current density comprises the hole current and electron current according to Jdiode = Jn (z) + Jp (z)

(2.43)

where Jn and Jp are given by Eqs. (2.9) and (2.10). Jn and Jp are dependent on z but Jdiode is independent of z. Since the recombination rate is highest in the absorber

37

38

2 Thin Film Heterostructures

Ec

(a) 2

4

3

φbn

EF Ev

Back contact

1

φbp

(b) 5 4* 1*

Ep,a

z=0

z=0 Figure 2.16 (a) Schematic presentation of the most critical recombination paths in a heterostructure solar cell. Recombination (1) at the absorber surface, (2) in the absorber bulk, (3) at the absorber back contact, and (4) in the absorber space charge region. Also, a recombination path in the buffer

layer is indicated (5). (b) Sketch of tunneling enhanced recombination (1∗ ) for interface and (4∗ ) SCR recombination. By tunneling, the density of holes which contribute to a particular recombination path may be increased.

layer and since the absorber layer is a p-type semiconductor, we can identify Jdiode with the electron (minority carrier) current density at the absorber/emitter interface, Jnz=0 . As an example, we refer to Figure 2.46a where Jdiode , Jn (z), and Jp (z) are given for the default device under forward bias. Thus, the following discussion concentrates on the electron current density. The starting point for the calculation of Jn (z) is the continuity equation, Eq. (2.13). If we integrate over the SCR of the absorber which extends from −wa to 0, we obtain 0 0 dJn (z) = q (Un (z) − G0 (z))dz + Jdiode,lF (2.44) −wa

−wa

Here, the thermal generation rate Gn = G0 has been introduced. The integral, although including the position z = 0, does not account for the interface absorber/emitter. Interface recombination is treated separately and accounted for by Jdiode,IF . The current density related to recombination in the QNR is given by Jnz=−wa . Here, in the ﬁeld free region, we can make use of Eqs. (2.9) and (2.11) in order to write the well known expression for the diffusion current dn(z) (2.45) Jnz=−wa = qDn dz z=−wa As recombination in the QNR and at the back contact requires transport of carriers to the recombination sites by diffusion, one may also speak of diffusion limited current. However, diffusion is governed by the carrier density gradient which in turn depends on the recombination rates in the ﬁeld free regions.

2.4 Diode Currents

Finally, the total diode current is given by

Jdiode = Jnz=0 = qDn

dn(z) +q dz z=−wa

0 −wa

(Un (z) − G0 (z))dz + Jdiode,lF (2.46)

The right hand side of Eq. (2.46) shows terms due to recombination in the QNR, in the SCR, and at the interface which we denote as Jdiode,QNR, Jdiode,SCR , and Jdiode,IF , respectively. It turns out that the general mathematical form of any contribution to the diode current reads qV Jdiode = J0 exp −1 with AkT −Ea J0 = J00 exp AkT

(2.47) (2.48)

J0 is the saturation current density which is activated by the activation energy Ea . Ea describes the temperature dependence of the saturation current density. J00 may be called reference current density which is only weakly temperature dependent. The diode quality factor A moderates the voltage dependence of the current density. Several recombination mechanisms may be operative in parallel, each carrying a certain share of the current. The magnitude or, in other words, the contribution of a particular process depends on its parameters J0 and A. If, for instance, a recombination process has a low J0 and a low A, it may not carry much current at low voltage. At higher voltage, due to its strong voltage dependence (small A), this process may dominate the diode. Parallel recombination processes are discussed in Section 2.4.6. Experimentally, the diode quality factor can be determined from the voltage dependence of the diode current. However, the exponential character of the JV relationship in Eq. (2.47) originates from the recombination rate and its voltage dependence. For all recombination processes considered here, the exponential voltage dependence ﬁnally results from, or can be expressed by, the voltage dependence of the carrier densities. We make use of this observation in order to derive the diode quality factors for different diode currents from the voltage dependence of the carrier densities. Recombination can be classiﬁed with respect to the three particles which can take on the recombination energy. These particles are photons (radiative recombination), electrons (Auger recombination), and phonons (phonon recombination). Another classiﬁcation scheme discriminates between band to band and defect related recombination. Which of these classiﬁcations is useful for recombination in chalcogenide devices? Recombination by the emission of phonons requires defects in the bandgap to be present. It turns out that current chalcogenide materials contain enough defects to dominate the diode current. This is why we treat defect related recombination separately from radiative and Auger recombination.

39

40

2 Thin Film Heterostructures

2.4.3 Radiative Recombination

The inverse process of optical absorption is radiative recombination. Here, we mean direct recombination of electrons and holes. It is this process which cannot be avoided and which deﬁnes the Shockley–Queisser limit of an ideal pn junction [32]. The net recombination rate under non-equilibrium conditions is given by R = Un − G0 = B(np − n2i )

(2.49)

where B is the radiative recombination constant. In equilibrium, according to the principle of detailed balance, the equilibrium generation rate G0 equals the equilibrium recombination rate U0 . Consider radiative recombination in the SCR of an asymmetric absorber/window heterostructure with ND,w NA,a . We want to use Eq. (2.46) in order to derive the JV characteristic. First, we assume that each quasi Fermi level is ﬂat from the injecting contact through the majority carrier regions and throughout the SCR.2) Only in the minority carrier regions, the quasi Fermi levels shall be allowed to vary. (As an example, see the simulated quasi Fermi levels in Figure 2.46a.) In the complete SCR the splitting of the quasi Fermi levels equals the applied bias voltage and it holds

np =

n2i

EFn − EFp exp kT

=

n2i

qV exp kT

(2.50)

With Eqs. (2.49) and (2.50), the recombination rate in the SCR, RSCR , is known and we can apply Eq. (2.44) (without the term Jdiode,IF ) to calculate the diode current density qV R Bn2i dz = q Jdiode,SCR = q − 1 dz exp 2 2 kT wa nr wa nr,a −Eg,a B qV = qwa (V) 2 Nc,a Nv,a exp exp −1 (2.51) nr,a kT kT The relative refraction index nr,a accounts for the fact that, due to total reﬂection, not all photons leave the cell [33]. A comparison of Eq. (2.51) with Eq. (2.47) shows that the diode quality factor for radiative recombination in the SCR is A = 1. Furthermore, it is Ea = Eg and J0 = qwa (V)Bn−2 r,a Nc,a Nv,a exp{−Eg,a /kT}. The width of the absorber SCR is given in Eq. (2.28). The weak voltage dependence of wa can safely be neglected against the exponential voltage dependence. From Eq. (2.51) we can learn that the recombination rate is constant in the SCR. Accordingly, the diode current increases with increasing wa . 2) The assumption of ﬂat majority carrier

quasi Fermi levels up to the point of maximum recombination at ﬁrst appears to be in contradiction to Eqs. (2.7) and (2.8). Without a gradient of the quasi Fermi levels

there would be no current ﬂow. However, since we speak of majority carriers, which normally exhibit high densities, very small gradients of EFn and EFp can drive high current densities.

2.4 Diode Currents

Since, however, wa normally is small compared to the diffusion length in the limit of radiative recombination, Jdiode,SCR is very small and does not limit the diode current of the solar cell. Concerning our aim to derive a general routine for the determination of the diode quality factors for different recombination mechanisms, we conclude that the integration step in Eq. (2.51) does not add a voltage dependence stronger than the exponential one which originates from the voltage dependence of the carrier densities. This forms the motivation to generalize Eq. (2.50) as Eg qV qV 2/A 1/A 1/A = (Nc Nv ) exp − exp (np) = ni exp AkT AkT AkT (2.52) where we use the diode quality factor in order to describe a possible exponent of the product of the carrier densities. If the recombination rate is proportional to the product np, it holds A = 1 as in Eq. (2.51). If for other recombination processes the recombination rate depends on (np)κ , we can directly read out the diode quality factor from κ = 1/A. In order to read out the voltage characteristic of the recombination rate, it is often convenient to consider the case of qV kT. In that case the product np is much larger than n2i , which arises from the generation rate, the term n2i in Eq. (2.49) can be neglected. For radiative recombination in the QNR, we have to calculate the ﬁrst term on the right hand side of Eq. (2.46). Thus, we have to ﬁnd the carrier density and its derivative as a function of voltage. In order to do so, we want to anticipate Section 2.5.4 where we introduce the collection function ηc . Between the collection function and the carrier density there is a reciprocity relationship expressed in Eq. (2.148). Using this relationship, we can write the carrier density at the position z (−da < z < −wa ) as n(z) − n0 = ηc (z)(n(−wa ) − n0 )

(2.53)

n0 is the dark electron concentration which is the minority carrier concentration. n0 shall not be position dependent. The carrier density under applied bias at the edge of the SCR of a p-type semiconductor can be obtained from Eq. (2.50) which is divided by p. This gives n2 qV qV = n0 exp (2.54) n(−wa ) = i exp NA,a kT kT with the equality p = NA,a and n0 = n2i /NA,a . Equation (2.54) again assumes that the quasi Fermi levels are ﬂat throughout the SCR. This is denoted as the ‘quasi equilibrium assumption.’ The assumption is valid for not too large recombination in the SCR and at the interface. One can argue that if IF or SCR recombination is so strong that the quasi Fermi levels are not ﬂat, then recombination in the QNR will not be the dominant recombination path. Only when the splitting of the quasi Fermi levels EFn − EFp at the position −wa equals the applied voltage, recombination in the QNR can be a relevant path and contribute to the diode

41

42

2 Thin Film Heterostructures

current. Assuming the validity of Eqs (2.53) and (2.54) then reads qV − 1 ηc (z) + n0 n(z) = n0 exp kT with the derivative dη(z) qV dn(z) = n − 1 exp 0 dz z=−wa kT dz z=−wa

(2.55)

(2.56)

This expression introduced in Eq. (2.45) gives the diode current for radiative recombination in the QNR as well as for any other type of recombination in the QNR n2i qV dη(z) Jdiode,QNR = qdn (2.57) exp −1 NA,a kT dz z=−wa In order to specify for the radiative recombination limit, we use the expression for the electron diffusion length L2n = Dn τrad and replace Dn in Eq. (2.57). Next, let us assume that the absorber thickness is much larger than the electron diffusion length. Then, the quasi Fermi levels fall off linearly in the QNR and the collection function is given by z + wa ηc (z) = exp (2.58) Ln An example for a collection function in the QNR can be seen in Figure 2.36. The a is the solution of a linear differential equation for η such as Eq. term exp z+w c Ln dη(z) z+wa 3) = 0. exp Ln indicates that the electron (2.150) with the condition dz z=−da quasi Fermi level and thus the splitting of the quasi Fermi levels decays linearly from the point of carrier injection which is the end of the SCR, −wa . With the dη(z) = L−1 derivation n , the diode current for radiative recombination in the dz z=−wa QNR becomes qLn n2i qV −1 (2.59) exp Jdiode,QNR = τrad NA,a kT We are interested in the diode quality factor and see by comparison with Eq. (2.47) that for radiative recombination in the QNR also it is A = 1. Obviously, the variation of the quasi Fermi level splitting and thus the collection function does not add a further voltage dependence. This also holds for more realistic conditions, for instance, if the diffusion length approaches the ﬁlm thickness. Then the derivative of the collection function is given by Eq. (2.153). Does radiative recombination limit the performance of chalcogenide solar cells? In order to answer this question, we need to know the radiative lifetime τrad . For a 3) The fact that we can solve a linear

differential equation for minority carriers injected into the QNR is an outcome of linearity of the recombination rate. This is not trivial and not valid for all

conditions of non-equilibrium. In the case of low injection it holds p ≈ NA,a , and the recombination rate is given by R = BNA,a (n − n0 ). Thus, the rate is linear in n.

2.4 Diode Currents

p-type absorber, the constant B is connected with the lifetime by 1/τrad = BNA,a . It further holds [29] U0 =

2πn2r c2 h3

∞ Eg

α(E)E2 dE E − 1) (exp kT

(2.60)

where nr is the refractive index. With B = U0 /n2i the radiative lifetime τrad = n2i /U0 NA can be calculated if the absorption coefﬁcient is known as a function of photon energy. Using the absorption coefﬁcients as plotted in Figures 4.6 and 4.20, the carrier lifetimes in the limit of radiative recombination were calculated after Eq. (2.60).4) The values of τrad are given in Tables 4.1 and 4.8 (see also Ref. [33]). It can be seen that the listed chalcogenide thin ﬁlms exhibit radiative lifetimes in the range of 10−6 s. As the measured lifetimes of current chalcogenide devices at room temperature (RT) are <10−6 s [35, 36], radiative recombination may not be the limiting process for the diode current of present chalcogenide solar cells. However, improved material quality may bring future cells to the limit of radiative recombination. As shown in Table 4.4, carrier lifetimes exceeding 100 ns have been measured for chalcopyrite ﬁlms. 2.4.4 Auger Recombination

Similar to radiative recombination, Auger recombination also is an intrinsic recombination process which cannot be avoided and which is active in chalcogenide devices. The net recombination rate for Auger recombination is given by R = U − G0 = Cp (p2 n − p0 2 n0 ) + Cn (n2 p − n0 2 p0 )

(2.61)

where Cp and Cn are the Auger coefﬁcients for the two possible particle excitations: Cp for the excitation of a hole and Cn for the excitation of an electron (the excitation energy originates from the recombination of the electron hole pair). For a p-type absorber with p ≈ p0 ≈ NA which is exposed to small carrier injection or medium excitation, the Auger excitation probability of an electron is very small due to the small number of electrons being minority carriers. Then, the Auger-limited lifetime is determined by hole excitation and it holds

R = U − G0 = Cp N2A n 4) For the calculation of R0 , the sub bandgap

absorption due to band tails for the chal copyrite absorbers in Figure 4.20 has not been evaluated. Instead, the absorption proﬁles have been ﬁtted by a semi-

(2.62) empirical formula as given in Ref. [34]. These corrected proﬁles for the CuIn1−x Gax Se2 were evaluated by use of Eq. (2.60).

43

44

2 Thin Film Heterostructures

with n = n − n0 . The lifetime of minority carriers in the Auger limit then is given by τAug = (Cp N2A )−1

(2.63)

From Eq. (2.63) we see that Auger recombination is the limiting process in the case of high doping concentrations. This can be the case for high efﬁciency Si solar cells [37]. Band to band Auger recombination cannot be avoided but can be reduced if the doping density is kept below a limiting value. This requires a compromise since on the other hand, high doping densities are desired as they increase the open circuit voltage (see the numerical example in Section 3.5). Experimental data for the Auger coefﬁcients of chalcogenide materials do not exist to the best of our knowledge. In order to estimate the relative importance of Auger recombination for chalcogenide absorbers in thin ﬁlm solar cells, we follow the argument of Bube who gives the lower limit for the Auger coefﬁcient of C = 10−28 cm−6 s−1 [38]. Using Eq. (2.63) we can calculate the Auger limited lifetimes for the doping densities of NA = 1015 , 1016 , 1017 , 1018 cm−3 . The result is τAug = 10−2 , 10−4 , 10−6 , 10−8 s, respectively. We see that Auger limited recombination would become critical for doping densities in the range of 1018 cm−3 . As already stated, the situation may be different if the Auger coefﬁcient would come out larger. For Si, the band to band Auger recombination coefﬁcients are below 10−30 cm−6 s−1 [39]. For GaAs, the Auger recombination coefﬁcients are also below 10−30 cm−6 s−1 [40]. Auger recombination may be enhanced by Coulomb attraction [41]. 2.4.5 Defect Related Recombination

Recombination via defect states is the complement to band to band recombination. It is an avoidable process since defect centers can, in principle, be removed from the semiconductor lattice. In practice this may be a difﬁcult task. A priori, the energy released in defect related recombination can be transferred to photons, electrons, and phonons. Since the energy released by a band to defect transition is smaller than in a band to band transition, phonon processes dominate the energy transfer [42]. First, we are interested in the lifetime of a minority carrier in the case of defect related recombination. The minimum lifetime of an electron being the minority carrier in a p-type absorber with a single defect level of density Nd may be described by τn0 = (σn vn Nd )−1

(2.64)

where vn is the thermal velocity of the electron, σn is the cross section of the defect recombination. It is the minimum lifetime because all trap centers Nd in the bandgap are assumed to be non-occupied by electrons. If the defect has a capture cross section of σn = 10−12 cm−2 , a thermal velocity of the electron of vn = 107 cm/s, and a defect density of Nd = 1013 cm−3 , a minimum lifetime

2.4 Diode Currents

of 10−8 s is calculated. Comparing this number with radiative recombination (τrad ≈ 10−6 s) and with Auger recombination (τAug > 10−6 s for NA < 1017 cm−3 ) it becomes evident that, for state of the art chalcogenide thin ﬁlm absorbers, defect recombination is the limiting process. Only to a minor extend radiative recombination may contribute to the total recombination rate. Auger recombination only dominates at high doping levels (see Section 3.5). Defect related recombination can take place at different sites within the heterostructure (see Figure 2.16): In the absorber bulk (2), in the absorber SCR (4), and at the surfaces of the absorber (1) and (3). Defect related recombination also takes place in the buffer (5) and window layer, but we show below that there it adds little to the diode current. Processes (1), (3), and (4), which include transport of carriers over energy barriers, can be enhanced by tunneling. We may further discriminate between single defect states with well deﬁned energy levels in the bandgap, multiple defects, and defect distributions with a quasi continuous state density in the bandgap. Defect related recombination via a single defect state of density Nd has been analyzed by Shockley and Read [43] and Hall [44]. We refer to this type of recombination as Shockley, Read, Hall (SRH) recombination. Figure 2.17 shows the principal transitions involved in SRH recombination: the capture of an electron (I) and hole (II) and the re-emission of an electron (III) and hole (IV). If we write the density of occupied defect states as nd , the capture and emission rates can be expressed as (I)

n/τc,n = n(Nd −nd )σn vn ,

(II)

p/τc,p = pnd σp vp ,

(III)

nd /τe,n = nd σn vn NC exp{−(Eg −Ed )/kT},

(IV)

(Nd −nd )/τe,p = (Nd −nd )σp vp NV exp{(−Ed )/kT}

p-type Ec

n-type

Ec EFn

En

En

I

Demarkation levels

EFn Ed EFp Ec

(2.65)

Ep

(a) Figure 2.17 Semiconductors (a) p-type and (b) n-type band edges in non-equilibrium with a defect level Ed . The demarkation energies Ep and En are indicated by arrows. Defects within the shaded energy regions are efﬁcient recombination centers. Defects

III EFp Ed

Ep

II

IV Ec

(b) above are electron traps and below are hole traps. Process I is the capture of an electron by the defect level Ed and III the re-emission of an electron. Process II is the capture of a hole and IV the re-emission of the hole.

45

46

2 Thin Film Heterostructures

where σn,p are the capture cross sections and τc , τe are the capture and emission time constants, respectively. We discriminate recombination states and trap states as two possible characters of the defect states. The character can change with the bias condition. Criteria for the discrimination between recombination and trap states can be formulated by aid of so-called demarcation levels En and Ep . These demarcation levels are derived from the following consideration: they give the energy at which a defect must be energetically situated in order to provide the equal probability for a carrier to be re-emitted or to be annihilated by the opposite carrier type. In other words, for a defect at the demarcation level, process III in Figure 2.17b has the same rate as process II (p-type semiconductor) and process IV has the same rate as process I (n-type semiconductor). This brings up the equations for En and Ep [38] σp m∗p Ep = En + kT ln and σn m∗n σn m∗n En = Ep + kT ln (2.66) σp m∗p The character of a defect state Nd at energy Ed in the bandgap can be derived from these conditions: • Ed < Ep : hole trap (HT) which communicates with the valence band. • Ed > Eg − En : electron trap (ET) which communicates with the conduction band. We want to visualize these conditions by aid of Figure 2.17. Consider a p-type semiconductor where holes are majority carriers even under non-equilibrium conditions. What is the energetic range where we ﬁnd efﬁcient RCs? According to Figure 2.17a, efﬁcient RCs lie between Ep and Eg − En . Centers which lie above Eg − En act as ETs, which means that it is more likely that an electron is re-emitted to the conduction band than that a hole completes the recombination process and annihilates the electron. If Ep is smaller, then even more holes exist to complete the recombination process and accordingly En is smaller. This makes it plausible that for the calculation of the electron demarcation level, we employ the hole quasi Fermi level and vice versa – a fact which, at ﬁrst glance, may be surprising. Defects which lie below Ep act as HTs. Thus in a p-type semiconductor, those defects are efﬁcient RCs which lie between Ep and Eg − En . The analog considerations hold for an n-type semiconductor in Figure 2.17b. In Figure 2.17 we have shaded the region of efﬁcient recombination. Due to the correction terms in Eq. (2.66), the demarcation levels are not simply mirrored at the intrinsic Fermi level position or, in other words, En does not equal Ep and Ep does not equal En . From the rates in Eq. (2.65), the fundamental expression for the net recombination rate R can be derived [45]: R = U − G0 =

np − ni 2

γp (n + n∗ ) + γn p + p∗

(2.67)

2.4 Diode Currents

Equation (2.67) is valid for electrons and holes. It describes the complete recombination process. The quantities n∗ = NC exp{−(Eg − Ed )/kT} and p∗ = NV exp{(−Ed )/kT} are auxiliary densities of carriers which would be realized if the Fermi energy is located at the defect level Ed . These auxiliary densities stand for carrier emission from the trap states. The quantities γp and γn in Eq. (2.67) at the moment are placeholders. In the case of bulk recombination without tunneling, γp and γn equal τn0 and τp0 , the minimum lifetimes of electrons and holes. γn,p have other meanings in the case of tunneling enhanced recombination and interface recombination. The minimum lifetime τn0 (τp0 ) can be measured if electrons (holes) are injected into a highly p-type (n-type) semiconductor with the same defect density as the semiconductor under consideration. The concentrations n and p in Eq. (2.67) represent the non-equilibrium carrier concentrations which may be realized by voltage bias or light bias. In equilibrium, it is np = ni 2 . Then, the thermal generation rate equals the recombination rate and, accordingly, the net recombination rate becomes zero. The net recombination rate sometimes is simply referred to as ‘recombination rate’ which can lead to misunderstandings. The energetic position of a defect center within the bandgap of the semiconductor can be from close to the valence band up to close to the conduction band. Above, we explain that a RC is efﬁcient if it communicates with both bands, in other words, if Ed lies between the majority carrier quasi Fermi level and the minority carrier demarcation level. Let us investigate if this condition for efﬁcient recombination is also reﬂected in Eq. (2.67). The equation indeed predicts that the net recombination rate becomes small when n∗ or p∗ are large, that is, if the defect level is close to one of the band edges. In contrast, the recombination rate can become large if n∗ and p∗ are small, that is, if the recombination level is close to midgap. In Figure 2.17, we have shaded the regions of efﬁcient recombination. In the next step, it may be asked at what position within the device the recombination rate has its maximum. We use the relationship between n and p under voltage bias, np = ni 2 exp{qV/kT}. Tunneling may be excluded, thus it is γn,p = τn0,p0 . Insertion into Eq. (2.67) yields − 1 n2i exp qV kT R = U − G0 = (2.68) n2i ∗ τp0 n + τp0 n∗ + τn0 n exp qV p + τ n0 kT The ﬁrst notion is that the net recombination rate depends on the intrinsic carrier concentration n2i = Nv Nc exp{−Eg /kT}. Thus, in a heterostructure solar cell the layer with the smallest bandgap will generate the main part of the diode current. Normally, this is the absorber layer. Only to a minor part, the diode current results from recombination in the window or buffer layer. We can turn this argument around in order to point out that regions in the solar cell with a bandgap smaller than the absorber layer carry the highest recombination trafﬁc. If, for instance, the absorber layer in the solar cell is highly defective, bandgap ﬂuctuations may reduce the effective bandgap of the absorber. This increases the diode current (see Section 2.4.9).

47

2 Thin Film Heterostructures

Ed1-Ed2

−1

−1

τn 0 n = τp−01 p

τn 0 n = τp−01 p EFn Ec

Energy

48

Ev (a)

Position z

(b)

EFp Position z

Figure 2.18 Energy region of efﬁcient recombination centers within the bandgap of a heterostructure at two different values (a, b) of forward voltage bias.

In the bulk of the absorber, maximum recombination can occur either in the SCR or in the QNR. In order to determine the maximum of recombination we form the derivation of R with respect to n qV 2 −n exp − 1 i kT n2i qV dR = τ − τ exp 2 p n 2 dn n kT n2i ∗ τp0,a n + τp0,a n∗ + τn0,a n exp qV kT + τn0,a p (2.69) dR/dn vanishes if the bracket on the right hand side of Eq. (2.69) becomes zero, that is, for τn0,a −1 n = τp0,a −1 p. (We see this if we replace n2i exp{qV/kT} by the product np.) Thus, where the condition τn0,a −1 n = τp0,a −1 p

(2.70)

is fulﬁlled, the maximum of recombination takes place. Certainly, Eq. (2.70) only in that case marks the position of maximum recombination if τn0,a and τp0,a are approximately constant within the absorber layer.5) If we assume equal lifetimes for electrons and holes, τn0,a ≈ τp0,a , the location of maximum recombination is at the position of n = p. For a reasonably doped absorber, the position of τn0,a −1 n = τp0,a −1 p is within the SCR. Thus, the position of maximum recombination normally is within the SCR. Below we show, however, that the position of maximum recombination does not necessarily dominate the diode current of the device. Although having a small maximum in the SCR, substantial recombination can also take place in the QNR. With the knowledge that the maximum recombination takes place at τp0,a n = τn0,a p, we may once more consider the region of effective RCs within the bandgap. Figure 2.18 schematically exempliﬁes this region for two different voltage biases together with the position of τn0,a −1 n = τp0 −1 p. (Note that the position of τn0 −1 n = τp0 −1 p changes with voltage bias.) The upper limit of the region of effective RCs 5) Particularly for polycrystalline thin ﬁlm

absorbers, the lifetime may be depth dependent. Since the grain size increases

with ﬁlm thickness, regions, which have been deposited in a later state of the growth process, may exhibit lower defect densities.

2.4 Diode Currents

is the demarcation level of electrons where it is p > n and the electron quasi Fermi level where it is n > p. The lower limit is the hole quasi Fermi level (p > n) and the demarcation level of holes (n > p). For small forward bias, the region of efﬁcient RCs at the position of τn0 −1 n = τp0 −1 p is very narrow. A defect level Ed1 generates only a small diode current. At higher forward bias, however, the region of efﬁcient recombination at n = p is much wider. Both defects Ed1 and Ed2 bring approximately the same current. If there would be a quasi continuous defect density in the bandgap, the density of states which contribute to recombination change with voltage bias. This inﬂuences the shape of the JV curve in the dark and under illumination. We can summarize that if the RC is not within the region of efﬁcient recombination at the position τn0 −1 n = τp0 −1 p, the recombination can still have a maximum at τn0 −1 n = τp0 −1 p, but this maximum is small and the total diode current is small. Finally, we come back to the question if recombination at τn0,a −1 n = τp0,a −1 p necessarily dominates the diode current of the device. A simple simulation of an absorber/window heterostructure can exemplify this question. In Figure 2.19, 4

Ec EFn

3 E [eV]

Ec

EFn

EFp

Ev

EFp

2 1 Ev

0

n

p

n

1 × 10−2

100

0.5

10−3

0

10−6

30 20

Jtot

Jtot

Jp

10 0 −1.5

(a)

p

1016 1014 1012 1010

JN −1.0

−0.5

JN −1.0

0.0

Position z [µm]

(b)

−0.5

Jp 0.0

Position z [µm]

Figure 2.19 Simulation of an absorber/window device with carrier lifetimes of (a) 10−8 s and (b) 10−11 s for electrons and holes in the absorber and window. The devices are under electric forward bias where the bias voltage corresponds to the Voc under illumination. Simulation was performed using SCAPS [925]. Other device parameters as in Table 8.1.

0.5

Eff. Rec. [cm−3 s−1]

J [mA cm−2]

Cum. Rec. [cm−2 s−1]

n,p [cm−3]

1018

49

50

2 Thin Film Heterostructures

recombination rates in the dark for devices with τn0,p0 = 10−8 s (a) and τn0,p0 = 10−11 s (b) are given as a function of position. The devices are under forward bias where the bias voltage is selected to match the Voc of the device under illumination. For both carrier lifetimes, recombination in the absorber is by far dominating over recombination in the window layer. For τn0,p0 = 10−11 s (Figure 2.19b), the recombination rate has a distinct maximum in the SCR of the absorber. However, it is very small in the QNR. The cumulative recombination rate conﬁrms that, for such a short lifetime, the SCR is the main recombination zone. In there, the location of n ≈ p and its surroundings carry the main recombination trafﬁc. The recombination rate drops off exponentially for n > p and p > n. In the case of τn0,p0 = 10−8 s (Figure 2.19a), the recombination rate is approximately constant in the QNR. The recombination maximum in the SCR is small. The cumulative recombination rate shows that more than half of the recombination takes place in the QNR. However, both recombination types are active in parallel. Thus, the predominance of QNR recombination and SCR recombination changes with carrier lifetime. The recombination regimes and their dominance can be differentiated by the voltage and temperature dependencies of the diode current. It is the diode quality factor A which expresses this voltage dependence. In the following, we deal separately with defect recombination in the SCR, in the QNR, and at the IF. 2.4.5.1 SCR Recombination As for all defect related recombination, the SRH formula Eq. (2.67) governs the recombination rate in the SCR. In the preceding section, we show that recombination in the SCR has a distinct maximum at the position τp0 n = τn0 p. We can make use of this equality in order to rewrite Eq. (2.52) as

n2/A = (τn0 Nv Nc /τp0 )1/A exp{−Eg /AkT}exp{qV/AkT} or p2/A = (τp0 Nv Nc /τn0 )1/A exp{−Eg /AkT}exp{qV/AkT}

(2.71) κ

κ

If now the recombination rate can be written as a function of n or p , the diode quality factor can be directly deduced from the exponent of the carrier concentrations, that is, from κ = 2/A. For instance, let the recombination rate depend on p or n where the exponent is κ = 1, then the diode quality factor is 2. We start our consideration of SCR recombination with the (historical) case where the absorber layer has a defect level approximately at midgap, Ed2 ≈ Ei , as depicted in Figure 2.20a. Such a defect level is within the energetic region of efﬁcient recombination for all voltage biases. Close to the point of maximum recombination it holds n, p n∗ , p∗ . Thus re-emission of trapped carriers can be neglected and Eq. (2.67) can be simpliﬁed to R = np − n2i (τp0,a n + τn0,a p)−1

(2.72)

At sufﬁcient forward bias of qV kT the term arising from equilibrium generation can be neglected. Directly at the point of maximum recombination τp0,a n = τn0,a p the recombination rate is n2i

Rm = p/2τp0,a

(2.73)

2.4 Diode Currents

Ed1 Ed3 Ed2

Ed1 Ed3 Ed2

EFn EFp

En

Ec

Nd0 Ed1

EFn

Ed3 Ed2

Ev

EFp

Nd2 Nd1

0

z=0

Ev

z=0

(a)

(b)

(c)

(d)

Figure 2.20 Defect related recombination via defect states Ed1 and Ed2 (a) in the SCR, (b) in the QNR, and (c) at the IF of an absorber/window heterojunction. (d) Quasi continuous defect density with maximum at Ec .

Comparing the exponential factor of the hole density with the one in Eq. (2.71) we see that A = 2. If SCR recombination is by far the dominant process, the diode quality factor for the particular device is 2. Next, we will show once more that calculating the diode current does not add a voltage dependence stronger than the exponential one of Eq. (2.73). Equation (2.72) gives the maximum of the recombination rate at the position −zm in the SCR. In the vicinity of −zm , the electron and hole concentrations vary exponentially with distance to −zm , that is n(z ) = nm exp{−qFm z /kT} and p(z ) = pm exp{qFm z /kT}

(2.74)

where z = −(z + zm ). The exponential variation of n and p can be seen in Figure 2.19. Fm is the electric ﬁeld at the position of maximum recombination. Thus, the recombination rate may be written as R(z ) =

np qF z qFm z τp0,a nm exp − m + τn0,a pm exp kT kT

(2.75)

Here, we assume that the product np is independent of z since, as stated in Section 2.4.5, the product is approximately constant up to (and also sufﬁciently beyond) the point of maximum recombination: This is equivalent to the assumption that the quasi Fermi levels are ﬂat within the SCR. The current density is Jdiode,SCR = q

wa −zm

R(z )dz

(2.76)

−zm

With the substitution u =

τ

n0,a pm τp0,a nm

1/2

−1/2 R(u) = np τn0,a pm τp0,a nm

qFm z exp kT

1 u−1 + u

51

it results

(2.77)

Ec

52

2 Thin Film Heterostructures

Integration of R(u) from u(−zm ) to u(wa − zm ) or, which is a good approximation, from 0 to ∞ yields the diode current ∞ ∞ kT R(u) np 1 kT Jdiode,SCR = du = −q du (2.78) 2 Fm u Fm τn0,a pm τp0,a nm 0 u + 1 0 The integral is listed as π/2 and with nm = n, pm = p, and τp0,a n = τn0,a p we obtain

Jdiode,SCR =

π/ kT Nc,a Nv,a 1/2 Eg qV 2 exp exp − Fm τn0,a τp0,a 2kT 2kT

(2.79)

We notice in Eq. (2.79) that indeed the diode quality factor is A = 2. Thus, the main voltage dependence of Jdiode,SCR arises from the recombination rate Rm in Eq. (2.73). The term πkT/qFm can be interpreted as an effective width of the recombination zone. Often, Fm is approximated by the maximum ﬁeld which, in the case of an asymmetric heterojunction, reads Fm ≈ Fz=0 = 2(Vbi −V)/wa

(2.80)

With increasing forward bias, Fm decreases and, accordingly, 1/Fm increases. This additional voltage dependence is the reason why for experimental solar cells which are limited by SCR recombination, the diode quality factor is slightly smaller than 2. A compilation of diode quality factors for different recombination processes is given in Table 2.1. In the following, we derive most of them in a similar fashion as above. A diode quality factor of A ≈ 2 is diagnosed when the current density in a semi-log plot varies with qV/2kT. The origin is that the recombination rate varies with exp{qV/2kT}. But the physical reason for A = 2 may be rationalized as follows: the quasi Fermi levels at the position τp0,a n = τn0,a p vary symmetrically with respect to the defect level. Since the splitting of the quasi Fermi levels is given by qV, each Fermi level varies as qV/2. Whenever a diode quality factor A = 2 (or close to 2) Table 2.1 List of diode quality factors 1/A for different recombination processes in the space charge region (SCR), the quasi neutral region (QNR), and at the interface (IF) of an absorber/window heterojunction. For an absorber/buffer/window heterostructure see Eq. (2.113).

Recombination region

Defect

Without tunneling

With tunneling

SCR

Ec,a − Ed ≈ Eg,a /2 Nd (E) Ec,a − Ed ≈ Eg,a /2 Nd (E) Ec,a − Ed ≈ Eg,i /2 Nd (E)

1/2 1/2 1 + TT∗ 1 1 1−θ 1 − θ + θ TT∗

1/2 [1 − ] 1/2 1 + TT∗ − – – (1 − θ) (1 − ) (1 − θ) (1 − ) + θ TT∗

QNR IF

2.4 Diode Currents

is diagnosed for a solar cell device, we may consider the quasi Fermi levels of electrons and holes moving symmetrically with respect to the dominating defect level under applied voltage bias. In the case that there are more than one deep defect levels in the bandgap, the recombination rates are added up as well as their contributions to the current densities. Since all current components due to deep defect levels exhibit the same voltage dependence, the A factor is still approximately 2. Thus, the density of defect states located within the zone of efﬁcient recombination can be added up in order i to write one rate equation for the recombination. It is τn0,p0,a −1 = νn,p i σn,p Nid for i defect levels. In the SCR, a defect level may also be located close to one of the band edges, as depicted by the defect Ed1 in Figure 2.20a. Then at the position τp0,a n = τn0,a p it holds n∗ n, p, p∗ and Eq. (2.67) becomes R = np/τp0,a n∗

(2.81)

By application of Eq. (2.52) it becomes apparent that A = 1. Since the product np is constant in the SCR, the recombination rate is also constant. Integration of Eq. (2.81) over the SCR is equivalent to multiplication with wa . Figure 2.21 shows the simulated JV curve for a ﬂat defect level Ed1 . In agreement with Eq. (2.81), the diode quality factor for the ﬂat defect Ed1 is close to unity. A diode with a ﬂat defect (A = 1) exhibits a stronger voltage dependence of the diode current as compared to a diode with a deep defect (A = 2). As pointed out in Section 2.4.5, however, the level Ed1 is not within the region of efﬁcient recombination. Thus, the total recombination rate of a ﬂat defect is small. Accordingly, recombination in an absorber having a combination of ﬂat and deep defect levels is dominated by the deep defect. As long as the forward bias is not too large, the diode quality factor will be A = 2.

Current density [A cm−2]

103 100 10−3 10−6

A = 1.78 Ed2 Ed1 Ed3

A = 1.1

10−9 10−12 0.0

0.2

0.4 0.6 Voltage [V]

0.8

1.0

Figure 2.21 Simulated JV curves of an absorber/window heterojunction assuming recombination via single defect levels Ed1 , Ed2 , and Ed3 as shown in Figure 2.22a. The diode quality factors where obtained by curve ﬁtting. After Ref. [46].

53

54

2 Thin Film Heterostructures

If the defect level is at an intermediate energy depth as depicted by Ed3 in Figure 2.20a, the diode factor will not lie between 1 and 2 as one could initially assume. Figure 2.21 shows that the JV curve for a diode with defect Ed3 continuously changes slope from 1.1 up to 1.78. The reason is that with increasing voltage bias, the region of efﬁcient recombination widens and ﬁnally envelopes the intermediate defect Ed3 . Thus, a change in slope of a log J(V) curve can indicate a defect at intermediate depth in the energy gap. After discussing single defects and defect combinations, we may consider the case where there is a continuous defect density in the bandgap. This opens the possibility for numerous variations. The defect distribution may be (among other possibilities) constant within the bandgap, linearly decreasing from the band edges, or exponentially decreasing. We are particularly interested in such a defect distribution that can produce a voltage independent diode quality factor. Since the carrier concentrations at the point of maximum recombination vary exponentially with voltage bias, it is an exponential defect distribution which has a strong impact on the diode quality factor [46]. A defect density which is constant or a defect distribution which is linearly decreasing from the band edges will lead to a voltage dependence of the diode factor. Examples of exponential defect distributions are shown in Figure 2.20d. Without loss of generality, the defect distribution may have its maximum Nd0 at the conduction band edge and decay into the bandgap with the characteristic energy kT∗ . Such an exponential defect distribution can be written as dNd (E) = Nd0

exp

−(Ec −E) kT∗ ∗

kT

dE (2.82)

where Nd (E) is in units of cm−3 . Then, by neglecting the term γn p∗ = τn0,a p∗ in Eq. (2.67) the recombination rate becomes c −E N (Ec ) d exp − EkT np Ec R(E) ∗ dNd (E) = R= dE kT∗ Ev τp0,a n + τn0,a p + τp0,a Nc exp − EckT−E Nd (Ev ) Nd0 (2.83) Note that the absolute number of RCs is factored in the values of τn0,a and Ec τp0,a . It holds τ−1 n0,p0,a = vn,p Ev σn,p (E)Nd (E)dE. We make use of the substitution ν = γp Nc (γp n + γn p)−1 exp{−(Ec − E)/kT} and obtain [47] ∗−1

T/T∗ −1

T/T∗ ν(Ec ) νT/T T R = np ∗ τp0,a n + τn0,a p τp0 Nc dν (2.84) T 1+ν ν(Ev ) It can be shown that also for an exponential defect distribution, the maximum recombination is located at τp0,a n = τn0,a p. The parameter 1/T∗ denotes the damping of the exponential defect distribution. A defect distribution with T∗ > T is the interesting one since it noticeable extends into the region of efﬁcient recombination. An example is given as dNd2 /dE in Figure 2.20d. To the contrary, a defect distribution dNd1 /dE with T∗ < T may hardly exceed the demarcation levels. Such a fast decaying distribution does not

2.4 Diode Currents

induce efﬁcient recombination – similar to a discrete level close to the band edges. We consider the case T∗ > T and ﬁnd by application of Eq. (2.70) R=

T 1 T∗ 2τp0,a

T/T∗

2 Nv,a

pT/T

∗+1

∗

πT/T

sin πT/T∗

(2.85)

By use of Eq. (2.71) the diode quality factor can be derived from the exponent of p which is κ = 1 + T/T∗ . Thus, 1/A = 1/2(1 + T/T∗ ). This is an important result as it delivers diode factors between 1 and 2. It is A = 2 for T∗ T and A → 1 for T∗ → T. A separate evaluation of Eq. (2.84) shows that A ≈ 1 for T∗ ≤ T. The calculation of the diode current density is similar as for a discrete level where again we make use of an exponential decay of the carrier densities around the point of maximum recombination. The result is [47]

Jdiode,SCR

T/2T∗ 2 τp0,a T Nc,a Nv,a 1/A , 1 τn0,a τp0,a Nc,a Nv,a T∗ −Eg,a qV × exp exp AkT AkT

kT = Fm

(2.86)

∗

with 1/A = 1/2(1 + T/T ). The integral (ϑ, ξ) is deﬁned by

(ϑ,ξ) =

πϑ sin(πϑ)

∞

0

(ϑ−1) u−1 + u du uξ

(2.87)

Formally, Eq. (2.86) can be written as Eq. (2.47) which allows to identify J00 =

kT Fm

Nc,a Nv,a τn0,a τp0,a

1/2

τp0,a Nc,a τn0,a Nv,a

T/2T∗

T/T∗ , 1

(2.88)

and J0 = J00 exp{−Eg /AkT}. The function (ϑ, ξ) must be evaluated numerically but has been plotted as a function of T/T∗ in Ref. [47] (curve E00 /kT∗ = 0). Equation (2.86) reveals that beside the exponential voltage dependence there is again only the linear voltage dependence of Fm which causes slightly reduced A factors. The activation energy of J0 is Eg . J00 exhibits only a weak temperature dependence. Figure 2.22 shows calculated values of J0 as a function of NA,a for an asymmetric absorber/window heterostructure (ND,w NA,a ). The defect distribution decays with T∗ /T = 3 at RT. With increasing NA,a , J0 becomes smaller. The mathematical reason is the increased electric ﬁeld Fm , but the physical reason is the steeper drop of n and p close to the point of maximum recombination. This reduces the zone of highest recombination within the SCR upon an increase of NA,a . As a comparison, Figure 2.22 gives J0 for a single defect level at Ed = Ei . The defect volume density is identical to the defect distribution. A higher saturation current is an outcome of the fact that the deep defects all are within the region of efﬁcient recombination. This is not the case for the defect distribution with a relatively low value of T∗ . However, the principal drop of J0 with increasing

55

2 Thin Film Heterostructures

Current Density J0 [mA cm−2]

56

10−2

τn0,p0 = 10−8 s with Tunneling

10−4 10−6 10−8 10−10 1013

1014

1015

1016

1017

1018

NA,a [cm−3] Figure 2.22 Calculated saturation current densities for an asymmetric heterostructure of the absorber/window type. Device parameters as in default Table 8.1. Full lines describe J0 as a function of NA,a for recombination via a deep defect without [Eq. (2.79)] and with tunneling enhancement [Eq. (2.47)] with 1/A from Table 2.1 and

J00 from Table 2.2). Dashed lines are for an exponential defect distribution without [Eq. (2.86)] and with tunneling enhancement [Eq. (2.47)] with 1/A from Table 2.1 and J00 from Table 2.2). The tunneling enhancement factor is F = 3.6 × 105 V cm−1 . The decay parameter of the exponential defect distribution is T∗ = 900 K.

NA,a is identical for both defect scenarios. The inﬂuence of NA,a on the solar cell parameters is discussed in Section 3.5. In the SCR, recombination may be enhanced by tunneling. The principal idea is depicted as path (4∗ ) of Figure 2.16: if a strong electric ﬁeld is present, the density of carriers available for recombination at a certain location within the SCR is increased due to the ﬁnite probability of carriers for tunneling into the bandgap. At location z, the tunneling enhanced carrier densities are pt (z) = p(z)(1 + T(E)exp{E/kT}) and nt (z) = n(z)(1 + T(E)exp{E/kT}). T(E) is the tunneling probability and E is the energy of the barrier. Hurkx and coworkers [48] introduced the tunneling enhancement factors n,p in order to write nt = n(1 + n ) and pt = p(1 + p ). Since re-emission from the traps is also enhanced by tunneling, the auxiliary functions in Eq. (2.67) become n∗t (z) = n∗ (1 + n ) and p∗t (z) = p∗ (1 + p ). Now, the recombination rate at a certain location is governed by the tunneling enhanced densities nt , n∗t , pt , and p∗t . Thus, we need to write the SRH equation (Eq. (2.67)) in the enhanced densities. Since, however, the carrier transport equations [Eqs. (2.9) and (2.10)] use the local densities n, n∗ , p, p∗ it is necessary to rewrite R(z) in the local densities. It is simple mathematics to show that this introduces the expressions γn = τn0,a (1 + n )−1 and γp = τp0,a (1 + p )−1 in the SRH Eq. (2.67). Although recombination enhancement by tunneling is an outcome of an increased density of carriers contributing to the recombination trafﬁc at a certain location, it can formally be viewed as due to locally reduced carrier lifetimes. For a deep defect level it is n = p = [48]. For not a too large electric ﬁeld (Fm < 9 × 105 V cm−1 ), Hurkx et al. gave an analytical expression for [48]. Here, however, we use the form developed by Rau

2.4 Diode Currents

[49] for parabolic band bending √ |F| qVb (z) E200 (z) = 2 3π exp F kT 3(kT)2

(2.89)

1

with F = (24 m∗ (kT)3 ) 2 /q¯h as the critical electric ﬁeld and |F| as the electric ﬁeld at the position of maximum recombination. As often, |F| is approximated by Fm = (2qNA,a Vb (z)/εa )1/2 .Vb (z) is the local band bending and E00 = (q¯h/2)(NA,a /m∗ εa )1/2 is the characteristic tunneling energy. As an example, can reduce the carrier lifetimes of electrons and holes by a factor of >10 if the electric ﬁeld exceeds 2 × 105 V cm−1 (see Figure 2.23). For an asymmetric heterojunction the establishment of such a high electric ﬁeld requires that the doping density in the absorber exceeds a value of about NA = 2 × 1016 cm−3 . The use of Eq. (2.89) is signiﬁcant since it can be implemented in device simulation (for instance, see program PC1D [50]). An important input parameter for tunneling enhanced recombination is the tunneling effective mass. Hurkx gave m∗t = 0.25 × mE . A lower limit may be m∗t = 0.1 × mE which has been used in Figure 2.23. The simulated inﬂuence of tunneling on the open circuit voltage of an absorber/window type solar cell is discussed in Section 3.5. The local band bending Vb (z) = Vbi (z) − V(z) can be expressed by the relation of carrier densities exp{−qVb (z)/kT} = p(z)/p0 where p0 is the hole concentration in the absorber bulk. Thus, by adapting the abbreviation = E00 2 /3(kT)2 ﬁrst introduced by Rau et al. in Ref. [47] it is convenient to write √ |F| (zm ) = 2 3π F

p p0

105

(2.90)

1019

104

Γ

1017

102 101

1016 100 10−1

1015 2

4

6

8 ×105

Electric Field F [V cm−1]

NA,a [cm−3]

1018 103

Figure 2.23 Tunneling enhancement factor and absorber doping NA,a for an asymmetric heterojunction (ND,w NA,a ). The enhancement factors are according to Eq. (7) in Ref. [48] (dashed line) or according to Eq. (2.89) (dotted line). The effective tunneling mass was ∗ assumed to be mt = 0.1 me . Other device parameters are εa = 10 and Eg = 1.2 eV.

57

58

2 Thin Film Heterostructures

where zm is the position of maximum recombination. For a discrete trap at Ed = Ei we can directly make use of Eq. (2.73) and for qV kT it is Rm = p/2γp = p(1+)/2τp0,a If tunneling dominates, that is, 1, we may write √ 3π |F| p0 p Rm = τp0,a F p

(2.91)

(2.92)

The maximum recombination rate for tunneling enhancement is located at (1 + )τn0,a p = (1 − )τp0,a n. Thus, we can again use Eq. (2.71) in order to read out the diode quality factor from the exponent of the carrier density. The exponent of p is 2/A = 1 − and accordingly 1/A = (1 − )/2. Tunneling enhanced recombination obviously can explain diode quality factors larger than 2. In the case of vanishing tunneling energy, that is, → 0, the diode factor becomes A = 2 as for SCR recombination by a deep defect level without tunneling enhancement. The current density in the case of tunneling enhanced recombination via a deep defect level can be written in the standard form Eq. (2.47) with 1/A = (1 − )/2 and J00 as listed in Table 2.2. The function (T/T∗ , 1 + ) is given in Eq. (2.87) and has been plotted in Ref. [47] for different values of . Theoretically, there exists the limit where it is ≈ 1. There, the current density due to tunneling enhanced recombination has the same magnitude as the current density due to thermally activated recombination described by Eq. (2.78). If we denote the former as Jdiode and the latter as Jdiode , the current density in a semi-log plot has the slope dln(Jdiode + Jdiode )/dV = q/AkT with 1/A = (1 + /2)/2. The A factor is smaller than 2. As can be seen in Figure 2.23, it is ≈ 1 for NA,a ≈ 1016 cm−3 . Tunneling enhanced recombination can also occur via an exponential defect distribution. The determination of the diode quality factor follows the same route as for a discrete level. In fact, Eq. (2.85) is already the appropriate expression for the maximum recombination rate. In the limit 1 we can insert /τp0,a for 1/γp and ∗+1− with Eq. (2.90) we obtain the dependency R ∼ PT/T . Thus by comparison with Eq. (2.71), the reverse diode quality factor for tunneling enhanced recombination via an exponential defect distribution reads T 1 E200 1 + ∗ (2.93) = 1− A 2 T 3(kT)2 The compilation of the inverse diode quality factors in Table 2.1 reveals the additive character of the different components to 1/A. Note again, however, that Eq. (2.93) is only valid for > 1. In the limit E00 → 0 (i.E., < 1) and with T∗ > T, Eq. (2.93) describes the inverse A factor for SCR recombination via an exponential defect distribution without tunneling enhancement. In the limit T∗ → ∞, Eq. (2.93) stands for tunneling enhanced recombination via midgap states. Finally, in the limit T∗ → ∞ and E00 → 0 Eq. (2.93) reduces to the A factor of SRH recombination via a deep trap.

Table 2.2 Collection of reference current densities J00 for defect related recombination in different regions of an n-type window/p-type absorber heterojunction with εw ND,w > εa NA,a and Eg,w Eg,a . According to the text, the abbreviation = E00 2 /3(kT)2 has been used. For the function (ϑ, ξ) see Eq. (2.87).

Recombination region

Defect

Without tunneling

SCR

Ec,a − Ed ≈ Eg,a /2

kT Fm

Nc,a Nv,a τn0 τp0

Nc,a Nv,a τn0 τp0

With tunneling

1/A 1/A

π 2

τ2 p0 Nc,a Nv,a

Nd (E)

kT Fm

QNR

Ec,a − Ed ≈ Eg,a /2

D N N q n Nc,a v,a dη(z) dz z=−w A,a a

IF

Ec,a − Ed ≈ Eg,i /2

qSp0 NA,a

Nd IF(E)

kT F

T/2T∗

T ,1 T∗

√ 2 3π

Nc,w Nv,a 1/A ND,w NA,a 1/A N N qSp0 NA,a N c,w Nv,a D,w A,a T T*

c,min Nc,w

Nc,a Nv,a τn0 τp0

1/A

1/A √ p Nc,a Nv,a 2 3π τ 0 τ τ p0 n0 p0 T × *,1+

(0, 1 + ) T/2T* τ2 p0 Nc,a Nv,a

×

–

ND,w Nc,min

× T/T*

∗ ν(Ec ) νT/T −1 ν(Ev ) 1+ν dν

N √ Nc,w Nv,a 1/A qSp0 NA,a 2 3π FFm Nc,min ND,w NA,a c,w 1/A √ N N × qSp0 NA,a 2 3π FFm N c,w Nv,a D,w A,a N N T/T* ∗ ν(Ec ) νT/T −1 D,w × T* Nc,min ν(Ev ) N 1+ν dν c,w T

c,min

2.4 Diode Currents

×

T

p0 τp0

kT F

N

59

60

2 Thin Film Heterostructures

The reference current density in the case of tunneling enhanced recombination via an exponential defect distribution has been derived by Rau [49]. It is listed in Table 2.2. In Figure 2.22, the saturation current densities for tunneling enhanced recombination via a deep trap (full lines) and via an exponential defect distribution (dashed lines) is plotted as a function of NA,a . Note that for all curves in Figure 2.22 the volume density of defect centers is identical. For the deep level and defect distribution, J0 for tunneling enhanced recombination increases for NA,a > 1017 cm−3 . (Below NA,a ≈ 1017 cm−3 , in the case of tunneling J0 is constant due to the missing factor Fm .) At the position of NA,a ≈ 1016 cm−3 tunneling and thermally activated recombination induce about the same current density. Approximately at this level of absorber doping, the tunneling enhancement factor reaches = 1. Here, the total saturation current density formally has to be written from the sum of conventional and tunneling enhanced recombination. Figure 2.22 conﬁrms that tunneling enhanced recombination may well limit the performance of certain chalcogenide solar cells. 2.4.5.2 QNR Recombination For a long carrier lifetime, the main part of the injected carriers pass the SCR without recombination and enter the QNR (see Figure 2.19a). There they are minority carriers and are subject to diffusion and recombination via defect states (process 2 in Figure 2.16). Again, we can discriminate between recombination via deep and ﬂat defects as well as via an exponential defect distribution. The principal band diagram in the QNR of a p-type absorber under forward bias is shown in Figure 2.20b. The QNR extends up to the back contact. In principle, the diode current due to QNR recombination can be calculated from Eq. (2.57). But, in order to discriminate QNR recombination from back surface recombination we consider the special case of a p-type absorber where it holds da Ln . Then, the minority carriers do not reach the back surface of the absorber (the back contact) and back surface recombination can be neglected. Instead of solving Eq. (2.57), we can make use of the temperature dependence of carrier densities according to the formalism in Eq. (2.52). Then, we can read out 1/A immediately. The diode current due to recombination in the QNR can – similar as in the SCR – be calculated from the continuity equation for electrons Eq. (2.13) in the integral form

Jdiode,QNR (−wa ) = q

−wa

−da

(Rn − G0 )dz + J( − da )

(2.94)

In general, application of Eq. (2.94) is inconvenient since we have to calculate Jdiode,QNR (−da ) separately. However, in the special case of da Ln it holds Jdiode,QNR (−da ) ≈ 0. In addition, if we consider the case of large forward bias (qV kT) only, we can neglect the generation rate G0 in Eq. (2.94) and let U = Rn − G0 ≈ Rn .

2.4 Diode Currents

For the recombination rate we employ the SRH equation [Eq. (2.67); without generation term] U=

τp0,a (n +

n∗ )

np

+ τn0,a p + p∗

(2.95)

where τn0,a and τp0,a are the bulk lifetimes. A deep defect Ed2 or a ﬂat defect Ed1 may be present in the absorber. The ﬂat defect Ed1 shall be above the electron demarcation level at Eg − En . Thus, Ed1 is located beyond the region of efﬁcient recombination. For the deep defect Ed2 it holds p p∗ , n, n∗ , and p ≈ NA,a . For the ﬂat defect Ed1 it is n∗ p, p∗ , n. For the deep defect the recombination rate can be approximated by U=

np τn0,a NA,a

(2.96)

In the case of a ﬂat defect, the denominator has to be replaced by τp0,a n∗ . The voltage dependence for a ﬂat and deep defect is identical. A diode quality factor of 1 is obtained in both cases as the exponent of the product np equals 1. Since n∗ may be larger than NA,a (the defect is beyond the demarcation levels), the recombination rate for a ﬂat defect will be smaller than for a deep defect. We continue to discuss the deep defect recombination. The product np is not constant in the QNR but depends on the coordinate z. Since diffusion and recombination governs the decay of the minority carrier density, we can again make use of the collection function ηc (z) which will be discussed in depth in Section 2.5.4. It holds np = n2i ηc (z)exp{qV/kT}

(2.97)

simply because p is constant in the p-type QNR and n obeys the reciprocity relation Eq. (2.53). Then, Eq. (2.96) becomes U(z) =

n2i ηc (z) qV exp τn0,a NA,a kT

(2.98)

In case of a ﬂat defect, we only have to replace the denominator by τp0,a n∗ . The saturation current density is calculated after J0 = q

n2i τn0,a NA,a

−wa −da

ηc (z)dz

in the case of a deep defect and J0 = q τ

(2.99) n2i p0,a n

∗

−wa −da

ηc (z)dz in case of a ﬂat defect.

Here, we remind that these solutions are only valid in the limit da Ln , that is, without back surface recombination. In view of the identical diode quality factors for deep and ﬂat defects in the QNR, it is not surprising that also for a defect distribution it results that A = 1. The physical reason for this diode factor is that for small and medium voltage bias only the quasi Fermi level of the minority carriers (EFn in the case of a p-type absorber) varies with voltage. With respect to any defect

61

62

2 Thin Film Heterostructures

level this variation is proportional to exp{qV/kT}. The recombination rate for an exponential defect distribution is given by Ec −E Ec exp − ∗ np kT dE U= kT∗ Ev τ p + τ N exp − Ec −E n0,a p0,a c kT

(2.100)

Again the case of T∗ > T is the interesting one. With the substitution ν = τp0,a Nc exp − EckT−E the integration over the bandgap can be executed. The τn0,a NA,a activation energy of J0 is Eg . J00 is listed in Table 2.2. These solutions of Eq. (2.98) are not valid for large applied voltage bias. Consider the case of a deep defect with Ed = Ei where the applied voltage is such that n = p. This means that p does no longer equal NA,a but also depends on the voltage. Now, the quasi Fermi levels move symmetrical with respect to the defect level and Eq. np p = . Comparison with Eq. (2.73) (2.95) becomes U ≈ Rn = τn0,a +τp0,a n

τn0,a +τp0,a

suggests that for such a large voltage bias, the diode quality factor reaches the value of 2. In practice this regime may be shielded by series resistances which may limit the current density at higher voltage. So far, we have analyzed the special case of QNR recombination in the limit da Ln where the current density at the back contact approaches zero. For many thin ﬁlm solar cells, it will be da ≈ Ln and we cannot neglect J(−da ) in Eq. (2.94). Therefore, it is necessary to apply the general Eq. (2.57) where the derivative of ηc (z) at the position z = −wa is used. If Ln is known and, in addition, the back surface recombination velocity of electrons, Sn0,bc , is known, we can use Eq. (2.152) dηc (z) . The saturation current density for defect related in order to calculate dz z=−wa recombination in the QNR reads Dn n2i dηc (z) (2.101) J0 = q NA,a dz z=−wa Any details about the type of defect (ﬂat, deep, continuous) in the QNR are included in the function ηc (z). In the case of a continuous defect distribution, the collection function will become temperature dependent. 2.4.5.3 Back Surface Recombination As a limiting scenario of the QNR recombination we may ask for the diode current in the case where back contact recombination is the dominant process, that is, if the bulk lifetime is very large and the interface recombination velocity is very small. Then the minority carrier electrons diffuse to the back contact and recombine with the abundant holes. We use Eq. (2.101) together with Eq. (2.153). In the limiting case that Ln,a da − wa we can approximate sinh([da − wa )/Ln,a ] ≈ (da − wa )/Ln,a and cosh[(da − wa )/Ln,a ] ≈ 1 and arrive at

Dn,a Dn,a (da − wa ) + Sno,bc L2n,a dηc (z)

≈ 2 (2.102) dz z=−wa Ln,a Sno,bc (da − wa ) + Dn,a

2.4 Diode Currents

which for Sn0,bc (da − wa ) Dn,a and Sn0,bc L2n,a Dn,a (da − wa ) reduces to dηc (z) n.a . Thus, J in the case of dominant back surface recombina≈ dD−w 0 ( a a) dz z=−wa tion scales with the inverse thickness of the absorber. An example for the ηc (z) curve may be found in Figure 2.33b (Ln = 10 µm). 2.4.5.4 Interface Recombination The absorber/emitter interface may exhibit a large number of interface states. Under circumstances these interface states form the main recombination channel of the device. This is particularly the case if the interface bandgap is smaller than the absorber bandgap. What we mean by interface bandgap is exempliﬁed in Figure 2.24. Note that generally we focus on simple absorber/window heterostructures. We refer to absorber/buffer/window type structures where appropriate. In Figure 2.24a, a type I interface is depicted where Ec is positive (type I). Here, the interface bandgap equals the absorber bandgap. Recombination at the interface takes place between electrons from the absorber conduction band and holes from the absorber valence band (we do not depict the interface states). This process is indicated by the arrow in Figure 2.24a. In contrast Figure 2.24b shows a type II interface where Ec is negative. Here, we can deﬁne the interface bandgap as Eg,IF = Ec,w − Ev,a = Eg,a + Ec . There are two possible recombination paths for this type of interface. The ‘straight’ recombination with electrons and holes from the absorber. But in addition, cross-recombination where electrons from the emitter conduction band recombine with holes from the absorber valence band. Cross-recombination is depicted by the arrow in Figure 2.24b. Following the SRH recombination model, a type II interface with its reduced (interface) bandgap may well be a high recombination zone. The general form of the SRH equation in Eq. (2.67) is unchanged for the case of interface recombination. However, the net recombination rate R in Eq. (2.67) is given in units of cm−2 s−1 . The diode current density due to interface Ec Ev

(a)

Ep,a z= 0

0

z

Ep,a z =0

absorber (b)

Figure 2.24 Energy band diagrams at the interface between a p-type absorber and an n-type window (without buffer) for (a) Ec > 0 and (b) Ec < 0. The interface bandgap is given by the minimum of the conduction band and the maximum of the

EF

emitter

valence band. In (a) it is Eg,IF = Ec,a − Ev,a and in (b) it is Eg,IF = Ec,w − Ev,a . The arrow in (a) denotes the ‘straight’ recombination path and in (b) the cross-recombination channel. Note that the quantity Ep,az=0 is smaller in (b).

63

64

2 Thin Film Heterostructures

recombination is given by Jdiode,IF = qR

(2.103)

For interface recombination via defect states without tunneling, we identify γn = Sn0 −1 and γp = Sp0 −1 in Eq. (2.67). Sn0 and Sp0 denote the nominal interface recombination velocities.6) In the case of a discrete interface state of (area) density Nd,IF with deﬁned capture cross sections σn,p , the recombination velocities become Sn,p = Nd,IF σn,p vn,p where vn,p are the thermal velocities of electrons and holes. Similar as for SCR recombination, interface recombination may be enhanced by tunneling. In that case, it is γn = [(1 + )Sn0 ]−1 and γp = [(1 + )Sp0 ]−1 . The interface recombination velocity grows directly with the interface defect density. However, it has an upper limit deﬁned by the thermal velocity. There is no lower limit of Sn0,p0 . But one may ask for an equivalence value of Sn0,p0 , assuming an interface defect density equivalent to the three-dimensional bulk defect density Nd . Let bulk and surface RCs have the same capture cross sections. Then, the equivalent surface recombination velocities may be given by Sn0,p0 =τn0,p0 −1 Nd −1/3

(2.104)

For Nd = 1013 cm−3 and τn0,p0 = 10−8 s this results in Sn0,p0 ≈ 5 × 103 cm s−1 . If one assumes that the area defect density at the interface is not lower than the (area) density in the bulk, one may consider this value as the lower limit for Sn0,p0 . The same holds for the grain boundary recombination velocities Sn0,gb , Sp0,gb and the back contact recombination velocities Sn0,bc , Sp0,bc . In order to calculate the diode current for interface recombination, a number of cases have to be discriminated. Figure 2.25 shows ﬁve levels of parameters which can be combined. The ﬁrst level is the conduction band offset which can be negative (including zero) and positive. The second level distinguishes if both quasi Fermi levels at the interface, En,az=0 and Ep,az=0 , can change with electric bias (left) or if one of the quasi Fermi levels is ﬁxed (right case). In other words, if Fermi level pinning is absent or present (see Section 2.3.4). These cases depend on the interface charge density. If the Fermi energies are not pinned, we further have to discriminate three cases: nonsymmetric heterojunctions with εw ND,w > εa NA,a or εw ND,w < εa NA,a and a symmetric heterojunction with εw ND,w = εa NA,a . In level 4, we discriminate the type of RCs at the interface: These may be discrete or energetically distributed. Finally, recombination at the interface may take place with negligible inﬂuence of tunneling ( 1) or by tunneling enhancement ( 1). This manifold of cases 6) Note that the actual interface recombina-

tion velocity depends on the occupation of the interface states Nd,IF . This occupancy in turn depends on the densities of electrons and holes. The nominal interface recombination velocity Sn0 only gives an upper limit of recombination of electrons

assuming that all interface states NIF with capture cross section σn are empty of electrons. Accordingly, Sp0 gives an upper limit of recombination velocity of holes assuming that all interface states with capture cross section σp are empty of holes.

2.4 Diode Currents

Level 1

Level 2

Level 3

Level 4

∆Ec ≤0

I

∆Ec >0

NA,I ≈ND,w /ww ND,I ≈ NA,a/wa

NA,a / wa

εwND,W >εaNA,a

III

εaNA,a = εwND,w

Ed

εaNA,a > εwND,w

Dd(E) II

Level 5

Γ 1

Figure 2.25 Case discrimination for interface recombination at a heterojunction between a n-type window and a p-type absorber. The ﬁrst level discriminates the conduction band offsets. The second level distinguishes if the charge in interface states is sufﬁcient to pin one of the quasi Fermi energies or if this is not the case. Level 3

Γ 1

refers to different doping ratios with the cases (1–3) denoted in the text. Different defect distributions are differentiated in level 3. Finally, the recombination may ( 1) or may not ( 1) be enhanced by tunneling. Three groups of cases are discussed within this book.

we want to treat in groups I–III as indicated in Figure 2.25. Groups I and III are for the different band line ups but only treat the discrete interface state without tunneling. Group II extends the case of a negative Ec for an interface defect distribution and tunneling enhancement but will only consider an asymmetric heterojuction without Fermi level pinning. Group I: ∆Ec ≤ 0, Interface State at Ed Let us begin with the case Ec ≤ 0 (Figure 2.24b) and the other parameters of group I in Figure 2.25. We assume that deep defects at the interface of an absorber/window heterojunction are present. In that case the auxiliary functions n∗ z=0 and p∗ z=0 can be neglected and Eq. (2.67) becomes for qV 3kT nwz=0 paz=0 R≈U= (2.105) Sp nwz=0 +Sn paz=0

Cross recombination is already taken into account by use of nwz=0 . Only the case of Ec = 0 is ambiguous since then the max(nwz=0 , naz=0 ) has to be used. In the

65

66

2 Thin Film Heterostructures

case of a ﬂat electron quasi Fermi energy across the interface, that is, EFn = 0 (see Figure 2.29), this maximum depends on the highest density of states, that is, max(Nc,w , Nc,a ). We ﬁrst assume that the density of charged interface states is small, that is, no Fermi level pinning occurs in the meaning of level 2 of Figure 2.25. The voltage dependence of the interface electron and hole concentrations can be written as q (Vbi − V) nwz=0 = ND,w exp − θ kT q (Vbi − V) (2.106) paz=0 = NA,a exp − (1 − θ) kT Vbi is the built-in voltage which is given by qVbi = Eg,a − En,w − Ep,a + Ec . In Eq. (2.106), we introduced the variable θ = εa NA,a /(εa NA,a +εw ND,w )

(2.107)

which can be rewritten as 1 − θ = εw ND,w /(εa NA,a + εw ND,w ). Equation (2.106) assumes that the quasi Fermi levels are ﬂat from the injecting contacts up the point of recombination, which is the interface. In other words, EFn and EFp shall be close to zero. This assumption is not trivial. As pointed out by Pauwels and Vanhoutte [51] for a high interface recombination velocity of S > 105 cm s−1 and a mobility of <100 cm2 (Vs)−1 there is a small drop of <100 meV of the quasi Fermi levels at the interface. Such a low mobility is typical for polycrystalline thin ﬁlms. The variation EFn schematically is depicted in Figure 2.29. Nevertheless, in Eq. (2.106), we neglected this drop of EFn and EFp by letting the carrier concentration depend exponentially on qV. In the case εw ND,w > εa NA,a , paz=0 at the interface is smaller than nwz=0 which allows to approximate Eq. (2.67) by R ≈ U = paz=0 Sp0

(2.108)

By use of Eq. (2.106) and with Vbi as given above, we obtain for the current density Eg,IF En,w + Ep,a Jdiode,lF = qNA,a Sp0 exp (1 − θ) exp − (1 − θ) kT kT qV × exp (1 − θ) kT (2.109) 1−θ E +E N N The term exp n,wkT p,a (1 − θ) can be rewritten as N c,w Nv,a in a temperD,w A,a

ature range where complete ionization of the dopants can be assumed. Then, J0 becomes Eg,IF Nc,w Nv,a 1−θ (2.110) exp − J0 = qNA,a Sp0 (1 − θ) ND,w NA,a kT The activation energy of the saturation current density is Eg,IF = Eg,a + Ec [52]. Thus, for Ec < 0, the activation energy is smaller than the absorber bandgap

2.4 Diode Currents

energy. From Eq. (2.109) we see that the inverse diode quality factor for interface recombination via a single defect in the middle of the interface bandgap is 1/A = 1−θ

(2.111)

The case εw ND,w < εa NA,a leads to the equation R ≈ U = γ−1 nz=0 and a diode quality factor of 1/A = θ. Thus, we can describe the limiting cases where the heterojunction is highly nonsymmetric, that is, εw ND,w εa NA,a or εw ND,w εa NA,a . Then, it generally holds A = 1. An absorber/buffer/window heterostructure can be treated in the special case that the buffer layer is fully depleted and all other conditions introduced in Section 2.3.3 are valid. For the case of naz=0 paz=0 we can use Eq. (2.108) and the expression paz=0 = Nv,a exp{Ep,az=0 (V)/kT} to describe the recombination rate as a function of voltage. Developing Ep,az=0 from Eq. (2.35) in a Taylor series [53] around Voc gives Ep,az=0 (V) = Ep,az=0 (Voc ) 



  εa db −  1 − ε b  εa

εb

1 2 db

+

qεa ND,b d2 b ε2 NA,a b

+

2εa q(Vbi q2 NA,a

− Voc )

   q(V − Voc )   (2.112)

and leads to the expressions for diode quality factor    A= 1 −  1+ and activation energy

−1

1 εa ND,b ε2 NA,a b

+

2ε2 b q(V bi 2 q εa NA,a d2 b

− Voc )

Ea (V) = qVoc + A Eg,a − qVoc + q2 NA,a db −A· εb

εa db εb

Ea,b c

2 +

+

Eb,w c

εa ND,b d2b ε2b NA,a

    

(2.113)

q2 q2 εa + ND,b d2b + 2 NA,a d2b 2εb εb

+

2εa q(Vbi − Voc ) q2 NA,a

(2.114)

Both A and E depend on the buffer layer thickness [53]. Moreover, Ea is reduced for negative band offsets both at the absorber/buffer and buffer/window interface. Now, we return to an absorber/window heterostructure and consider the limiting case of a symmetrical junction, that is, εw ND,w = εa NA,a and θ = 1 − θ = 1/2. For this case the recombination rate in Eq. (2.105) is calculated using both terms of Eq. (2.106) and the current density becomes ' q Nc,w Nv,a Sn0 Sp0 Eg,IF qV exp (2.115) exp − Jdiode,lF = Sp0 + Sn0 2kT 2kT

67

2 Thin Film Heterostructures

(a)

(b)

10−4 10−6 10−8 10−10 1.0

2.0

0.9

1.8

0.8

1.6

0.7

1.4

0.6

1.2

0.5

Diode quality factor A

J0 [mA cm−2]

10−2

Ep,a z= 0 /Eg,a

68

1.0 8

0.01

2

4

6 8

2

0.1

4

6 8

1

εaNA,a / εwND,w Figure 2.26 (a) Saturation current density for a deep defect at the interface of an absorber/window heterojunction with default parameters from Table 8.1. Further parameters are Ec = 0 eV, Sn0 = Sp0 = 107 cm s−1 and ND,w = 1018 cm−3 . Thus, in this example is Eg,IF = Eg,a . The full line represents calculated data according to Eq.

(2.109), the marked value was calculated for NA,a = ND,w after Eq. (2.115). The broken line gives J0 data extracted from ﬁtting of simulated JV curves. (b) Values of Ep,az=0 /Eg,a calculated after Eq. (2.24) and diode quality factors calculated using Eqs. (2.107) and (2.111).

The activation energy of J0 is unchanged but the diode factor becomes A = 2. This diode factor can be understood in view of the variation of both quasi Fermi levels at the interface upon applied electric bias. In the case of a symmetrical heterojunction, both quasi Fermi levels move symmetrically with respect to the deep interface defect level. In summary, if both Fermi levels at the interface are not pinned, J0 has an activation energy of Eg,IF . The diode quality factor is between 1 and 2 depending on the doping ratio. J0 strongly varies with the doping ratio. This can be seen in Figure 2.26 where simulated and calculated values of J0 are plotted for the case Ec = 0. For εw ND,w = εa NA,a , J0 reaches a value of 10−1 mA cm−2 while for εw ND,w = 100 × εa NA,a J0 approaches 10−10 mA cm−2 . The underlying reason is the availability of charge carriers. If Ep,az=0 /Eg,a (see Figure 2.16) approaches 1, the number of holes at the interface is very low. Now, the hole concentration limits the recombination rate and the interface defects are ﬁlled with electrons. If Ep,az=0 /Eg,a approaches 1/2, the densities of electrons and holes at the interface become equal

2.4 Diode Currents

and the interface states become a very efﬁcient recombination channel. That Ep,az=0 (or Ep,az=0 /Eg,a ) is the decisive quantity for the saturation current density becomes apparent by rewriting J0 from Eq. (2.109) as Ep,az=0 J0 = qNv,a Sp0 exp − kT

(2.116)

where we made use of Eqs. (2.22) and (2.24). J0 exponentially depends on Ep,az=0 as is already suggested by Figure 2.26. For an absorber/buffer/window heterojunction we ﬁnd the same principal dependence of J0 on Ep,az=0 : J0 is small if Ep,az=0 is large. We emphasize that Eq. (2.116) shall not be misinterpreted. In the case without Fermi level pinning, Ep,az=0 is not the activation energy of J0 since the quantity Ep,az=0 is temperature dependent. Instead the activation energy is the interface bandgap [52]. Now consider the case that the charge in the interface states is sufﬁcient to induce Fermi level pinning. It becomes apparent that the activation energy of J0 can then be smaller than the interface bandgap. Let us assume that EFn is pinned above the middle of the interface bandgap, that is, En,az=0 < Eg,IF /2. This pinning position allows the electron concentration nwz=0 to largely exceed the hole concentration paz=0 . However, recombination active interface defects may be deep enough as to allow n∗ , p∗ nwz=0 .7) Thus, recombination only depends on the interface hole concentration and Eq. (2.108) is valid. The carrier concentrations at the interface are En,wz=0 and nwz=0 = Nc,w exp − kT p Ep,az=0 φ qV paz=0 (V) = Nv,a exp − = Nv,a exp − b exp (2.117) kT kT kT We see in Eq. (2.117) that naz=0 is independent of electric bias since EFn is pinned at En,wz=0 . Moreover, if the interface charge density is high enough, En,wz=0 is also independent on temperature. If we set φb p = Eg,IF − En,wz=0 the saturation current density can be written as φb p J0 = qNv,a Sp0 exp − kT

(2.118)

as pointed out by Rau et al. [47]. Now, φb p is the activation energy of J0 since φb p is not dependent on temperature. Again, this is only the case for pinning of the 7) Note that the recombination active inter-

face defects are not necessarily identical with the interface states responsible for the Fermi level pinning. If the pinning

states exhibit a small capture cross section and they are close to one of the interface band edges, recombination may take place via deeper defects.

69

70

2 Thin Film Heterostructures

Fermi energy by a sufﬁcient number of interface states. The diode ideality factor comes out as A = 1 as only the hole quasi Fermi level moves under applied bias. The Fermi energy may also be pinned at an energy close to the valence band of the absorber by interface acceptor states. This case is symmetrical to the former one and one obtains En,wz=0 as the activation energy which also is smaller than the interface bandgap. Group II: ∆Ec ≤ 0 and εw ND,w > εa NA,a In order to discuss the inﬂuence of a defect distribution and the effect of tunneling enhancement, we exemplary consider the case Ec ≤ 0 and εw ND,w > εa NA,a as presented by group II in Figure 2.25. Similar as in the absorber bulk, interface defects may occur as a continuous defect distribution, Nd,IF (E). As in the case for bulk recombination, an exponential distribution is the interesting one because it inﬂuences the diode quality factor.8) For the case εw ND,w > εa NA,a a defect distribution with a maximum at Ec similar as in Eq. (2.82) is the most effective one. We can write

R(V) ≈ U =

=

Nd,IF (Ec ) Nd,IF (Ev )

npSp0 kT∗

U(E,V) dNd,IF (E) Nd0,IF

Ec Ev

With the substitution ν =

c −E exp − EkT ∗ dE (2.119) q(Vbi −V) ND,w exp − kT θ + Nc,w exp − EckT−E exp − Ec −∗E kT q(Vbi −V) θ , ND,w exp − kT

Eq. (2.119) can be solved simi-

lar as Eq. (2.83) for SCR recombination. The result is an activation energy of Eg,IF , a reference current density as given in Table 2.2 and an inverse diode quality factor of 1/A = 1 − θ(1 − T/T∗ ). Since recombination via a defect distribution is only effective for T∗ > T (see Section 2.4.5.1) and since we assumed εw ND,w > εa NA,a , the diode quality factor is between 1 and 2. The main temperature dependence of the saturation current density arises from the exponential term exp{−Eg,IF /AkT}. The residual temperature dependence of J00 (see Table 2.2) from ∗ the term T/T∗ (ND,w /Nc,w )T/T and from the integral is much weaker. As for SCR recombination, the interface recombination rate via a defect distribution may be enhanced by tunneling. This requires to replace γn−1 = Sn0 (1 + n ) and γp−1 = Sp0 (1 + p ) in Eq. (2.67). The tunneling enhancement factor in Eq. (2.90) √ q(Vbi −V) is modiﬁed to = 2 3π F|F| exp (1 − θ) where = E00 2 /(3kT)2 [47]. kT √ The reference current density in Table 2.2 besides the factor 2 3π F|F| is form

8) We remember that a constant or linear

defect distribution is too weak to compete with the exponential dependence of the carrier densities at the point of maximum

recombination (the interface) and thus too weak to noticeably change the shape of the JV curve.

2.4 Diode Currents

invariant to the case without tunneling. The diode quality factor comes out as 1/A = 1 − θ(1 − T/T∗ ) − (1 − θ). The activation energy of the saturation current density furthermore is Eg,IF . Thus, tunneling enhancement of the recombination at the interface does not alter the activation energy of the saturation current density. Rather, the activation energy is determined by the interface bandgap for all parameter combinations of group II in Figure 2.25. For an absorber/buffer/window heterostructure the activation energy also depends on the buffer layer thickness and the buffer/window band line up (see Eq. 2.114). Group III: ∆Ec > 0, Interface State at Ed The last group of interfaces are highlighted as group III in Figure 2.25. The decisive parameter of this group is Ec > 0. An example of an absorber/window heterojunction exhibiting a conduction band spike is sketched in Figure 2.24a. In order to solve Eq. (2.67) (with n∗ z=0 nz=0 and p∗z=0 pz=0 ) we have to decide on the relation between εa NA,a and εw ND,w according to the second level of parameters in Figure 2.25. Let ﬁrst be εw ND,w εa NA,a (very high nonsymmetry). Then, the electron concentration governing the interface recombination rate can be approximated by nz=0 = ND,w . Thus, we assume that the band bending in the window corresponds to Ec /q. The hole concentration is given by −Eg,a + En,w + qV paz=0 = Nv,a exp (2.120) kT

, the recombination rate Since at the interface it is for all voltages nz=0 pz=0 Ep,a −1 becomes R = γp pz=0 . Regarding Nv,a = NA,a exp kT , the current density for a single defect at the interface midgap energy then reads Jdiode,lF = qSp0 NA,a

Nc,w Nv,a ND,w NA,a

1/ A

Eg,a exp − kT

qV exp kT

(2.121)

The activation energy for Ec > 0 in the case of θ ≈ 0 is Eg,a and the diode quality factor is A = 1. We ﬁnd that Eq. (2.121) is identical as for a heterojunction with Ec = 0 and θ ≈ 0 being described by Eq. (2.109). Thus, in case of a highly nonsymmetric heterojunction with εw ND,w εa NA,a and arbitrary Ec , the activation energy of J0 can be written as Ea = Min(Eg,a , Eg,a + Ec ). In other words, the activation energy corresponds to the interface bandgap. The diode quality factor is A = 1. In case that εw ND,w = εa NA,a , the carrier concentrations at the interface have to be modiﬁed to En,az=0 qV nz=0 = Nc,a exp − exp kT 2kT Ec q (Vbi − V) ND,w Nc,a exp exp − = Nc,w kT 2kT q (Vbi − V) pz=0 = NA,a exp − 2kT

(2.122)

71

72

2 Thin Film Heterostructures

where we again made use of Eqs. (2.22) and (2.24). Insertion into Eq. (2.67) with nz=0 ∗ nz=0 and pz=0 ∗ pz=0 and multiplying with −q yields the current density 1/2 N N qNA,a N c,w Nv,a Eg,a − Ec qV D,w A,a Jdiode,IF = exp − exp (2.123) Nc,w Ec 2kT 2kT S−1 + S−1 p0 exp n0 Nc,a kT which in the case of exp{Ec /kT} Nc,w /Nc,a and with Sp0 −1 ≈ Sn0 −1 becomes ' Eg,a + Ec qV Jdiode,IF = qSp0 Nc,w Nv,a exp − exp (2.124) 2kT 2kT Thus, the activation energy for the case Ec > 0 and εw ND,w = εa NA,a of an absorber/window heterojunction exceeds Eg,a and is given by Eg,a + Ec . Cases between εw ND,w = εa NA,a and εw ND,w εa NA,a are difﬁcult to treat analytically and are best analyzed by device simulation. The last case of group III in Figure 2.25 is εa NA,a εw ND,w . This is rather a pathological case where interface recombination is so large that the condition EFn = 0 at the interface (for deﬁnition of EFn see Figure 2.29) is not fulﬁlled. Tables 2.1 and 2.2 list the diode quality factors and reference current densities for the most relevant recombination paths and most relevant parameters of absorber/window heterojunction solar cells. Together with the activation energies in Table 2.3, these values deﬁne the saturation current density which is a measurable quantity. In Section 2.5.7, we show how to extract the diode parameters A and J0 from the experiment and how to interpret these in view of the recombination path. In Section 3.4 we show the inﬂuence of interface recombination on the Voc of a model device. The interface bandgap may be larger than the absorber bulk bandgap due to bandgap gradients or an additional thin layer between emitter and absorber. In that case the activation energy for the saturation current would correspond to the interface bandgap (case without Fermi level pinning) or to the pinning position with respect to the actual band edge at the interface (case with Fermi level pinning). A widening of the absorber bandgap at the interface can thus strongly reduce the interface recombination (see design rules in Section 3.10) Table 2.3 Collection of activation energies for defect related recombination in different regions of an absorber/window heterojunction.

Recombination region

Condition

Activation energy (Ea )

SCR QNR IF

– – No FLP FLP

Eg,a Eg,a Eg,i φb p

2.4 Diode Currents

Grain boundaries in a polycrystalline absorber ﬁlm can as well be considered as interfaces. The SRH recombination rate can be calculated after Eq. (2.105) where Sn0,p0,gb is the grain boundary recombination velocity Sn0,p0 = Ngb σn,p vn,p . In Section 3.7, the effect of horizontal and vertical grain boundaries on the device performance is discussed. 2.4.6 Parallel Processes

In practical heterostructure solar cells, all recombination paths formerly discussed are active in parallel. Normally, one recombination mechanism will be dominating. However, the dominating mechanism may be changing under changing electrical bias. Consider N recombination mechanisms. The total diode current density is N ( qV (2.125) Jdiode = J0i exp Ai kT i=1 If all processes exhibit the same diode quality factor, the recombination path with the highest J0 dominates the diode current at any bias. From Table 2.1 we know, however, that the A factors can vary between 1 and 2 and beyond this range. In view of the exponential dependence in Eq. (2.125), this is a large variation. 2.4.6.1 SCR and QNR Recombination For typical heterojunction solar cells which are not dominated by interface recombination, SCR recombination, and QNR recombination take place in parallel. We ﬁrst ask which recombination mechanism is the dominant one as a function of the carrier lifetime. We simply consider absorber bulk recombination via a single deep defect. The reference current densities J00,QNR and J00,SCR , given in Table 2.2, both depend on the carrier lifetime. If we assume equal density of states and equal lifetimes for electrons and holes, J00,SCR has a 1/τn0,p0 dependence. J00,QNR , in addition is inﬂuenced by ηc (z) which also includes the carrier lifetime. Consider the limiting case that the absorber thickness is much larger than the diffusion length. Then Eq. (2.109) can be used for J00,QNR and the integral in this equation be0 comes −da ηc (z)dz = Ln,p . (For recombination at vertical grain boundaries, Ln,p,eff ' has to be used.) With Ln,p = Dn,p τn0,p0 the reference current becomes J00,QNR ∼ √ 1/ τn0,p0 . Thus, the slope of J00,SCR is larger than that of J00,QNR . This is apparent in Figure 2.27b. In Figure 2.27a, Eg,a /q − Voc is calculated in the limit of SCR and QNR recombination. We ﬁnd a certain lifetime which marks the transition from SCR limited to QNR limited recombination. In the example of Figure 2.27 this border lifetime is at τn0,p0 ≈ 10−9 s. Note that, close to this border lifetime, recombination in the SCR and QNR is about equally efﬁcient. The border lifetime is largely independent of the absorber bandgap, the effective density of states and the absorber doping. Figure 2.19 shows the accumulated recombination rate for the devices with τn0,p0 = 10−8 and τn0,p0 = 10−11 s. For absorber layers with da < Ln,a , the correct expression for J00,QNR has to be used. Then, the integral in Eq. (2.152) strongly depends on the back surface

73

2 Thin Film Heterostructures

L [µm] 0.1 0.2 0.5 1.0 2.0

Eg,a /q - Voc [V]

0.2

SCR

0.3 0.4 0.5

QNR

0.6

J00 [mA cm−2]

0.7 (a) 0.8

109

QNR

107 105

SCR

3 (b) 10

(dVoc /dT) T=RT [mV/K]

74

2.4 2.2 2.0 1.8 1.6 1.4 10−11

(c)

10−10

10−9

10−8

10−7

τn,p [S]

Figure 2.27 (a) Calculated values of Voc for recombination in the QNR (dashed line) and in the SCR (solid line) as a function of the carrier lifetime. Device parameters as in Table 8.1. No buffer layer and no interface recombination have been included. The line of extended dashes has been calculated by a two-diode model. Input parameter for all

calculated Voc are simulated values of Jsc . Data points () give simulated Voc values for comparison. (b) Lifetime dependence of J00 for SCR ad QNR recombination. (c) (dVoc /dT)T=RT calculated at room temperature for SCR (solid line) and QNR (dashed line) recombination.

recombination velocities Sn0,bc and Sp0,bc . Imagine that the back surface would be perfectly passivated, that is, Sn0,bc = Sp0,bc = 0. Than ηc (z) would be constant in the 0 QNR and −da ηc (z)dz = da . The reference current density J00,QNR would depend on da and the remaining dependence on carrier lifetime is 1/τn0,p0 . This would shift the border lifetime to slightly higher values and would increase the Voc in the regime of QNR recombination. QNR recombination is advantageous over SCR recombination in view of its temperature dependence. This becomes apparent in Figure 2.27c. The reason is the smaller A factor of QNR recombination.

Jrec [mA cm−2]

2.4 Diode Currents

105 100 SCR QNR IF

10−5

0.0

0.2

0.4

0.6

0.8

1.0

Voltage [V] Figure 2.28 Diode current densities for different recombination paths of an absorber/window solar cell. Device parameters as in default Table 8.1. Solid line: SCR recombination calculated after Eq. (2.79). Dotted line: QNR recombination calculated after Eq. (2.99) with ηc (z) = 1. Dashed line: IF recombination calculated after Eq. (2.109).

The dominance of SCR or QNR recombination also depends on the voltage bias. We selected a device with τn0,p0 in the absorber and plotted the different diode currents as a function of electrical bias (Figure 2.28). Due to the diode quality factor A = 2, the Jdiode,SCR has the smallest voltage dependence. The voltage dependence of Jdiode,QNR is larger since QNR recombination type exhibits A = 1. The latter is also true for IF recombination (for θ → 0 – see Eq. (2.107)) which is given in addition in Figure 2.28. The example in Figure 2.28 shows (the well known result) that SCR recombination dominates at small bias and QNR recombination limits the total current for large forward bias. The total current density will be the maximum curve in Figure 2.28. In the example, the curve will change slope at 0.6–0.7 V electric bias. As the device parameters underlying Figure 2.28 are those of the default device, we note that the Voc of this device is limited by QNR recombination. 2.4.6.2 SCR and IF Recombination For the parameters selected in Figure 2.28, IF recombination does not dominate at any bias. However, we know from Eq. (2.116) that IF recombination critically depends on Ep,az=0 . In fact, J0 for IF recombination increases by 1 order of magnitude for a decrease of Ep,az=0 by 60 meV. Ep,az=0 decreases either by a conduction band cliff, by a too small doping ratio or by Fermi level pinning (FLP). With a conduction band cliff of Ec = −0.4 eV, IF recombination would be dominating in the complete voltage interval in Figure 2.28. For Ec = −0.25 eV, the total diode current after Eq. (2.125) is dominated by IF recombination for V > 0.5 V. As an effect, the open circuit voltage is limited by IF recombination. This aspect is discussed in Section 3.2. The limitation of the Voc from the different recombination mechanisms is discussed in Section 2.6.2.3.

75

76

2 Thin Film Heterostructures

2.4.7 Barriers for Diode Current

So far, we have assumed that the SCR is the only barrier for current transport in the device. There may be, however, other barriers which limit the diode current. Energy barriers can form at the interfaces of a device or, to a limited extent, within a layer exhibiting bandgap gradients. Figure 2.29 depicts two relevant examples for interface barriers. Figure 2.29b shows an electron barrier φb n which is due to a negative buffer/window conduction band discontinuity and band bending within the buffer layer. It holds φb n = En,az=0 + Ec b,a − EFn where En,az=0 is given by Eq. (2.35) and EFn is the drop of the electron Fermi level at the interface. According to Figure 2.6c, also negative charge at the buffer/window interface can lead to an electron barrier φb n . To some extent the electron barrier can be voltage dependent. In the example of Figure 2.29b we have the voltage dependent part En,az=0 − EFn and the voltage independent part +Ec b,a . Due to the voltage dependent part, the electron barrier can be considered as a nonlinear electronic element which is in series with the main diode. In Figure 2.29a, we depict an example for a barrier at the absorber back contact. The back contact represents a Schottky diode of opposite polarity as the main diode (heterojunction). It is a hole barrier φb p which can impede the injection of holes into the p-type absorber. Thus, it can block the diode current. In the following, we will assume that the main diode and the back contact diode are independent and the depletion regions do not overlap. Otherwise one speaks of a reach-through diode [54, 55]. Charge transport over both barriers in Figure 2.29 may be described by thermionic emission (TE) theory. Quantum mechanical effects at the energy barrier such as reﬂection or tunneling of carriers, are not considered. In the TE theory, the saturation current density is given by Jb0 = A∗ T2 exp{−φb /kT} ≈ 1/4qvn,p Nc,v exp{−φb /kT}

(2.126)

where A∗ is the effective Richardson constant and Nc,v is the density of states in the energy band of the majority carrier [56].

Φbn = En,b z=0

En,a z=0 ∆EFN

Φbp E (a) z =-da

(b)

z

z=0

Figure 2.29 Barriers for the diode current. (a) Barrier for hole injection φb p . (b) Barrier for electrons φb n depicted at small forward bias.

2.4 Diode Currents

For the electron barrier in Figure 2.29b, the total current across the barrier is given by the current density of electrons from the window to the absorber minus the current density from the absorber to the buffer [57]. −En,bz=0 Eb,a 1 c + En,az=0 JTE = qvn,p Nc,w exp − Nc,a exp − (2.127) 4 kT kT (The index TE refers to thermionic emission.) With En,bz=0 = En,az=0 + Eb,a c + EFn 9) we arrive at −En,az=0 EFn Nc,w exp + −1 (2.128) JTE = J0,TE exp Nc,a kT kT with J0,TE

1 Ec b,a = qvn,p Nc,a exp − 4 kT

(2.129)

EFn increases with electric forward bias. In contrast, En,az=0 decreases with bias provided that there is no FLP. The size of EFn depends on the value of Ec b,a . In our default device (Ec b,a = 0)EFn is below kT. This renders the bracket in Eq. (2.128) very small. However, J0,TE is large as long as Ec b,a + En,az=0 is not too large. Therefore, it is well possible to pass a diode current even in the presence of a limited conduction band barrier. If Ec b,a , however, is large, the total diode current becomes limited by the electron barrier. In that case, we may observe a very high Voc and a very low Jsc (see example in Figure 3.5). The barrier φb n in Figure 2.29b may be illumination-dependent as shown in the example of Figure 2.13. An illumination dependent barrier height φb n can lead to cross over of the dark and light JV curves. An electron barrier which is not (or hardly) voltage dependent will lead to a complete blocking of the forward current (leakage effects not considered). In that case, we will ﬁnd a roll over of the illuminated JV curve. In the case of the blocking absorber back contact in Figure 2.29a, the total current density depends on the voltage drop at the back contact Vb by Jb = −Jb0 (exp{−qVb /kT} − 1). (The minus signs in front of Jb and Vb reﬂect the opposite polarity of the back contact junction.) The externally applied voltage, V, distributes over the main and back surface junction. It is V = Vm + Vb , where we use the subscript m for the main junction. The current density through the main junction and the back contact barrier must be equal, that is J = J0 (exp{qVm /AkT} − 1) = Jb0 (1 − exp{−qVb /kT})

(2.130)

At the moment, we neglect the effects of series and shunt resistances. As long as it is J0 (exp{qVm /AkT} − 1) Jb0 , the main voltage drop is at the main junction. Then, the ﬂow of the diode current is not impeded by the back surface barrier. If the diode current, in contrast, approaches Jb0 , the applied voltage starts to drop at the 9) In Figure 2.29, EFn is positive since

dEFn /dz > 0.

77

78

2 Thin Film Heterostructures

back contact barrier. This leads to current saturation and a JV curve with a roll over anomaly (see Section 7.1.1). Note, that if the device acts as a photovoltaic generator the back contact barrier is forward biased. Thus a moderate back contact barrier does not impede the ﬂow of photo current at short circuit [58]. This is exemplarily quantiﬁed in Section 3.8.

2.4.8 Bias Dependence

In Section 2.3.6 we discussed how deep states can modify the equilibrium band diagram due to their supplementary charge. In Section 2.4.5, we saw how deep states can become hole or electron traps dependent on their capture cross sections and on the position of the quasi Fermi levels. The decisive quantities are the demarcation levels for deep states. Thus, here it remains to mention some possible effects of a band diagram changed by supplementary charge due to voltage or light bias. We may place positive and negative charge in the absorber layer, at the buffer/absorber interface or in the buffer layer. Due to the normally much higher doping, we leave out the window layer. If under voltage or light bias negative charge is added to the SCR of a p-type absorber this can lead to:

1) An increased electric ﬁeld, Fm , at the position of τn0,a −1 n = τp0,a −1 p. As a result the zone of effective recombination πkT/qFm in the absorber SCR is reduced. This leads to a lower diode current (see also Table 2.2). 2) Tunneling enhancement of SCR recombination if Fm becomes so large that it approaches F (see Section 2.4.5.1). 3) A change of the effective doping ratio between buffer and absorber (or emitter and absorber). As a result En,az=0 may be increased with the effect of an increased interface recombination.

Certainly, all effects can occur simultaneously. The net effect on the solar cell device will be observable as a gain or loss in the Voc . An increased negative charge at the absorber/emitter interface can lead to the opposite effects as described above. Similar can be the effects of negative charge in the buffer layer or at the buffer/window interface. If the diode saturation current is voltage dependent, we observe a strong violation of the superposition principle even if we correct the voltage dependence of the photo current. An example is the cross over of light and dark JV curves [59]. If charging and discharging of the deep states exhibits very large time constants, the recombination processes can become dependent on the experimental history in terms of minutes or hours [27, 60].

2.4 Diode Currents

2.4.9 Non-Homogeneities

So far, we have calculated diode currents in heterostructures formed out of homogeneous and isotropic layers. However, within the three-dimensional absorber there may be distinct zones of particular high defect densities: grain boundaries and regions of high strain, accumulated impurities, or stoichiometry deviations. These regions can induce a very high recombination rate provided they are located close to the point of τn0,a −1 n = τp0,a −1 p. As a result a large diode current ﬂows independently of the defect density in the absorber host. A grain boundary is a good example which can stand also for other high recombination zones. Grain boundary recombination is discussed in Section 3.7. Here, we only note that limited regions of high defect density can strongly reduce the Voc while the Jsc can be less affected. In the same sense, the absorber/emitter interface may be considered as a high recombination zone. Besides the possibility of distinct regions of high defect densities, the defect density may (systematically) ﬂuctuate on a lateral scale. Equation (2.48) shows that the saturation current density depends on J00 . According to Table 2.2 the reference current J00 for all considered recombination processes is proportional to τn0,p0 −1 or sn0,p0 , thus is proportional to the bulk or interface defect ' density. Furthermore, J00 directly or indirectly (for SCR recombination Fm ∼ NA,a ) depends on the doping density in the absorber. Both the defect density and the doping level can ﬂuctuate on a lateral scale. Let us assume that, as a result, the reference current density varies by orders of magnitude and obeys a log normal distribution PLn according to dPLn (J00 ) =

2 − lnJ00 − lnJ00 dJ00 exp √ 2σn J00 σn 2π

(2.131)

where σn is the dimensionless standard deviation [61]. Then, J0 will obey the same log normal distribution and will be inﬂuenced by regions of high defect density. In order to understand the impact of a ﬂuctuating J00 let us investigate the J00 −related term which limits the Voc of a device. This is −AkT/qln(J00 /Jsc ). It is valid for all recombination mechanisms discussed in this book (see Section 2.6.2.3). If we describe J00 by the log normal distribution, the term is modiﬁed to −AkT/qln (J00 /Jsc ). For a log normal distribution the median J00 is larger than the position of the distribution’s function maximum. Thus, small areas with large J00 limit the Voc much stronger than large areas with small J00 . Intuitively we do expect such a behavior. In the other case that J00 is only Gaussian distributed, the impact on Voc will be much smaller. Besides high defect densities, also a reduced activation energy for the recombination process can be the origin of a high recombination zone. Considering Eq. (2.48), it is clear that a variation of Ea has a super linear inﬂuence on the diode current. For bulk recombination in the absorber, the activation energy equals Eg . If Eg is locally reduced, a locally higher saturation current may be the result. An example for distinct regions of reduced bandgap are inclusions of a secondary

79

80

2 Thin Film Heterostructures

phase. Other reasons for bandgap ﬂuctuations are variations in stoichiometry, alloy composition and strain. In the case of interface recombination, the activation energy is the interface bandgap (no Fermi level pinning) or the hole barrier (Fermi level pinning). Both quantities can ﬂuctuate on a micro- and mesoscopic scale. Rau and Werner [33, 62] have proposed a general approach to describe the inﬂuence of a ﬂuctuating activation energy. They assume Ea to obey a Gauss-normal distribution of

2 − E a − Ea dEa (2.132) exp dPG (Ea ) = √ 2σEa σE 2π a

Here, Ea is the mean of the distribution and σEa is the standard deviation (with the dimension of an energy). Insertion in Eq. (2.48) gives the saturation current density for a distributed Ea : ∞ Ea dPG (Ea ) J0 = J00 exp − (2.133) AkT −∞ 10)

Performing the integration with aid of tabulated integrals we ﬁnd σ2 Ea Ea − J0 = J00 exp − AkT 2(AkT)2

(2.134)

Thus, the activation energy for the diode current decreases with increasing σEa . The cell recombination process may be described by an effective activation energy Ea,eff = Ea −

σ2 Ea 2AkT

(2.135)

Obviously, Ea,eff is smaller than Ea . The impact of Ea,eff on the Voc can be understood by considering the expression Voc = Ea,eff /q − AkT/q ln(J00 /Jsc ) which is rationalized in Section 2.6.2.3. We see that a reduced Ea,eff directly decreases Voc . Equation (2.134) is valid for any type of recombination which is thermally activated. Rau et al. showed that the effect of ﬂuctuating J0 on the cell performance can be limited by a small series resistance [61]. We note in passing that if a cell with ﬂuctuating activation energy is analyzed according to Eq. (2.48) instead of according to Eq. (2.134), a temperature dependent diode quality factor may be diagnosed (see Section 7.3.4).

2.5 Light Generated Currents

In this section, we discuss generation and recombination of charge carriers by photon absorption. The section’s logic follows from the principle of superposition

dPG (Ea ) = 1, integration is performed between −∞ and ∞;

10) In order to fulﬁll

although negative bandgaps have no physical meaning.

2.5 Light Generated Currents Jgenmax

Joptloss

Jgen

Jelloss(V)

Figure 2.30 Schematic presentation of photo currents in a solar cell. The generation current Jgen max is the ideal current density obtained in the absence of recombination and optical losses. The part Jgen can be collected in the solar cell structure while internal recombination amounts to Jel . The external current is Jph (V).

Jph(V)

of diode and photo currents. A failure of the superposition principle, often observed in chalcogenide solar cells, will be alleviated by allowing the photo current to be voltage dependent. Figure 2.30 gives an overview of the currents treated in this section. The generation current Jgen is lower than the maximum generation current Jgen max due to an optical loss of Jopt loss . The photo current is lower than the generation current due to an electronic loss of Jel loss . In this book we calculate Jph (V) as a negative quantity. One can easily verify by aid of Figure 2.15 that the ﬂux of photogenerated electrons is from left to right and thus – in this case – represents a negative Jsc . 2.5.1 Generation Currents

The maximum generation current is obtained if we integrate over the solar photon spectrum and neglect all loss mechanisms. Indeed, it is instructive to look at the maximum possible generation current, Jgen max , as a function of the bandgap of the absorbing layers. The solar cell may consist of an emitter and absorber of bandgaps Eg,e and Eg,a , respectively. It may be homogeneous in the direction x, y thus deﬁning a one-dimensional problem. A photon ﬂux density djγ (E)/dE per unit area and unit energy [63] shall be impinging on the solar cell. The energy of the photons is given by ¯hω. Figure 2.31a shows the AM1.5G and AM0 equivalent photon ﬂux densities as a function of photon energy. Alternatively, we can present the spectra in values of the energy ﬂux density, djE , where it holds that djγ (E) = djE (E)/E

(2.136)

We write the energy ﬂux density of the AM1.5G spectrum as jE 100 indicating the energy ﬂux of 100 mW cm−2 . In the ideal case, only the absorber bandgap limits the absorbed ﬂux density. Thus, we construct an ideal absorbance Aid (E) which is 0 for E < Eg,a and 1 for E ≥ Eg,a . This implies that the absorber is sufﬁciently thick to exclude optical transmittance, that is, T(E) = 0 for E ≥ Eg,a . Further it assumes that the reﬂectance R(E) is zero for Eg,a < E < ∞. The integral over the spectral photon ﬂux density for the absorber/emitter combination reads

81

dJγ /dhν [s−1 m−2 eV−1]

2 Thin Film Heterostructures

4×1021 3 2 1 0 0

0

−2

−20

(b)

−40 −60

Jsc [mA cm−2]

(a)

Jsc [mA cm−2]

82

−4

Jsc,a

Eg,e

−6

Eg,a

Jsc,e

−80

−8 −10

−100 1.0

2.0 E [eV]

3.0

Figure 2.31 (a) Spectral photon ﬂux density as a function of energy for AM1.5G (solid line) and AM0 (dashed line). The spectra have been recalculated after tabulated spectra of dje (λ)/dλ and are subject to a small

Jmax gen Eg,a = −q

or Jmax gen λg,a = −q

∞

Aid (E)

∞

Aid (λ) 0

numerical error. (b) Integrated photon ﬂux densities for AM1.5G (solid line) and AM0 (dashed line) representing the maximum short circuit current density achievable for an absorber with bandgap Eg = E.

djγ (E) dE

0

4.0

djγ (λ) dλ

dE = −q

∞

djγ (E)

Eg,a

dE

λg,a

djγ (λ)

dλ = −q 0

dλ

dE dλ

(2.137)

In the ﬁrst and second part of Eq. (2.137) we expressed Jgen max by aid of energy and wavelength dependent functions, respectively.11) Numerical integration of djγ (E)/dE gives the maximum generation current density as a function of the absorber bandgap (Figure 2.31b). It is clear that Jgen max (Eg,a ) increases with decreasing Eg,a . Jgen max (Eg,a ) is larger for the AM0 spectrum than for the AM1.5G spectrum due to the higher photon ﬂux density. 11) It shall be remembered that djγ (λ) dλ

djγ (E) dE =

− hc2 E

where the minus sign indicates the different slopes of the spectra.

2.5 Light Generated Currents

Figure 2.31b gives an upper bound for an ideal solar cell. The ﬁgure can also be used to read out the share of maximum generation current achievable from the absorber and emitter separately. Optical losses reduce Jgen max by the quantity Jopt loss . Optical loss may have several origins. We start with an ambiguous case which, in principle, could also be considered as an electronic loss. The window layers of a current chalcogenide solar cell exhibit low electronic quality and do not add to the photo current. Thus, all electron–hole pairs generated in the window layer are lost.12) We can write the photo current loss due to absorption in the window layer as

Jloss,w Eg,w = −q opt

∞

djγ (E)

Eg,w

dE

dE

(2.138)

We can directly use Figure 2.31b in order to read out the optical loss due to window absorption. For a window bandgap of about Eg,w = 3.5 eV and an assumed AM1.5G spectrum, Jopt loss,w amounts to 0.5 mA cm−2 . For the AM0 spectrum this loss is 1.4 mA cm−2 . With the assumption of a complete loss in the energy range of the window absorption, we concentrate on the remaining energy interval Eg,a < E < Eg,w . In this interval, photonic loss is caused by reﬂectance R(E) = 0, transmittance T(E) = 0, and by parasitic absorption in the window layer Apar (E) = 0. These losses can be quantiﬁed as = −q Jloss,O opt

Eg,w Eg,a

O(E)

djγ (E) dE

dE

(2.139)

where O(E) stands for R(E), T(E), or Apar (E). Jopt loss,T is due to photons in the interval Eg,a < E < Eg,w which pass the absorber and reach the back contact where they are not reﬂected but absorbed. This loss can be minimized by a sufﬁcient absorber thickness or by a back contact reﬂector. Jopt loss,R is due to the total optical reﬂectance of the heterostructure. It can be minimized by matching of the indices of refraction of the different layers or by adding interference layers. Jopt loss,A is due to absorption of the window layer in the interval Eg,a < E < Eg,w . A typical example of Jopt loss,A is free carrier absorption in the highly doped window layer. The functions R(E), T(E), and Apar (E) can either be measured for the completed cell or can be generated from the optical constants of the constituting layers and by application of an appropriate matrix formalism [64]. The generation current which ﬁnally arises from the generation of electron–hole pairs in the photoactive layers amounts to Jgen = Jgen max − Jopt loss,w − Jopt loss,T − Jopt loss,R − Jopt loss,A . If we deﬁne the absorbance of the photoactive layers of the cell as A(E) = 1 − T(E) − R(E) − 12) Looking from the absorber side, Jopt loss,w in

fact is an optical loss. Certainly, the loss in the window layer could also be expressed by a zero collection function. Later, for

illustration purposes we will indeed set the collection function to zero in the range of the window layer.

83

84

2 Thin Film Heterostructures

Apar (E), we can also write the generation current as

Jgen = −q

Eg,w Eg,a

A(E)

djγ (E) dE

dE = −q

λg,a

λg,w

A(λ)

djγ (λ) dλ

dλ

(2.140)

with A(λ) = 1 − T(λ) − R(λ) − Apar (λ). We note that in a heterostructure consisting of i photoactive layers the total absorbance is A(λ) = i Ai (λ). The absorbance in the mth layer depends on the optical properties of all other layers. 2.5.2 Generation Function

While the absorbance only allows to calculate the density of photons per unit of time absorbed in the complete heterostructure or within the complete thickness of a single layer of a heterostructure, we are interested in a function which describes the position dependent generation rate. This function is the generation function G(z). It results from the absorption of all wavelengths at a given position z. In Table 2.4, we see that the generation current can be given as an integral over the position dependent generation function for an absorber/emitter heterostructure. The integration limits obey the following arguments: If the emitter consists of window and buffer, integration is performed over the buffer and absorber thickness.13) If the emitter only consists of a window layer, the higher integration limit is zero, thus corresponding to the surface of the absorber. As explained in 2.5.1, we leave out the window layer since the loss in this layer was already quoted as an optical loss. The relation between the absorbance in a particular layer and the generation function is mediated by the spectral generation rate dG(λ, z) (see auxiliary functions in Table 2.4). Note that this rate is not normalized but depends on the incoming photon ﬂux. One can introduce the normalized generation rate Gn (λ, z) = dG(λ, z)/djγ (λ)

(2.141)

For an absorber/window heterojunction, the spectral generation rate in the wavelength range λg,w < λ < λg,a can be approximated by dG(λ, z) = djγ (λ)T(λ)(1 − R(λ))αa (λ)exp{αa (λ)z}

(2.142)

where R(λ) should be obtained from optical measurement, αa (λ) is the absorption coefﬁcient of the absorber and T(λ) is the transmittance of the window layer. Equation (2.142) expresses the well known fact that light which arrives at the absorber generates charge carriers with an α(λ)exp{α(λ)z} depth dependence.14) For 13) As usual in this book, the zero position

shall be at the absorber/emitter interface. This is why we integrate from −da to db . 14) Note that here the propagation direction of photons is anti-parallel to the z-axis.

Therefore, the generation rate decreases toward negative z. This is why the reader may miss the minus sign in the exponential function in Eq. (2.142).

2.5 Light Generated Currents Table 2.4 Wavelength and position dependent formulations of the current densities introduced in this section. The integral boundaries are de , db , and −da for the thicknesses of window, buffer, and absorber layer, as well as λg,a and λg,w for the band

Wavelength dependent Jmax gen

−q

λg,a

djγ (λ) dλ

0

dλ

edge equivalent wavelengths of absorber and window. The auxiliary functions allow the transition between wavelength and position dependent descriptions. The functions are explained in the text.

Position dependent

–

Auxiliary functions

– A(λ) =

Jgen

−q

λg,w

Jph (V)

−q

λg,a

A(λ)

λg,a

λg,w

A(λ)

djγ (λ) dλ

djγ (λ)

× IQE(λ, V)dλ

dλ

dλ

−q

−db

−q

da

G(z)dz

G(z) =

da −db λg,a

λg,w

Gn (λ, z) = da

−db

G(z)ηc (z,V)dz

dG(λ,z) dλ dz dλ djγ (λ) dG(λ,z) dλ dλ

dG(λ,z) djγ (λ)

EQE(λ,V) = IQE(λ,V)A(λ) EQE(λ,V) = da dG(λ,z) ηc (z,V)dz −db djγ (λ)

an absorber/buffer/window heterostructure the approximation in the wavelength range λg,b < λ < λg,a reads dG(λ, z) = djγ (λ)T(λ)(1 − R(λ))exp{−αb (λ)db }αa (λ)exp{αa (λ)z}

(2.143)

where αb (λ) and db are the absorption coefﬁcient and thickness of the buffer layer. These formulas neglect the effect of multiple reﬂection and interference. More precisely, dG(λ, z) is calculated by dividing each layer of the heterostructure in a large number of sublayers. Then the absorbance in each sublayer is calculated. Integration of a sublayer absorbance multiplied by the spectral photon ﬂux density gives the spectral generation rate. This calculation is best performed by aid of an optical matrix calculation program. Knowing dG(λ, z), we can calculate the local generation rate of charge carriers excited by a spectral photon ﬂux density djγ (λ) in the device from λg,a dG(λ, z) G(z) = dλ (2.144) dλ λg,w Figure 2.32 gives an optical generation function which is exemplary for a solar cell based on a strongly absorbing chalcogenide layer. The majority of charge carriers are generated within the ﬁrst 500 nm of the absorber. The decline of G(z) is not purely exponential due to the integration over various dG(λ, z) functions with different α(λ). Knowing the generation function is essential for the simulation of

85

2 Thin Film Heterostructures

16× 1018 G(z) [cm−3 s−1]

86

-da

0 db

Figure 2.32 Optical generation function of the default absorber/buffer/window solar cell. Due to the smallest bandgap of the absorber with thickness da , most of the electron–hole pairs are generated therein. Note that the decline of the generation rate in the absorber is supra-exponential.

dw

12 8 4 0

−1500

−1000

−500 z [nm]

0

500

quantum efﬁciency (QE) measurements and for seeking the collection function (see below). By use of the functions – generation function, spectral generation function or normalized spectral generation function – the generation current can be written as Jgen (V) = −q = −q

db −da

G(z)dz = −q

λg,a

λg,w

djγ (λ) dλ

db

db λg,a

G(λ,z)dλdz

−da λg,w

Gn (λ, z)dzdλ

(2.145)

−da

An overview of the functions applied gives Table 2.4 2.5.3 Photo Current

Not all photons absorbed in the photoactive layers contribute to the photo current. Charge carriers which undergo recombination are responsible for the loss term Jel loss . This loss current can be voltage dependent. Table 2.4 shows that for the calculation of the photo current density from position dependent functions we introduced the collection function ηc (z, V). Before we proceed to discuss this collection function, let us brieﬂy describe the connection between the collection function and the recombination rate. The photo current density in an illuminated heterojunction can be calculated by integration of the continuity equations (Eqs. (2.13) and (2.14)) for electrons and holes. We consider a thin ﬁlm solar cell as depicted in Figure 2.2 where the junction normal is in the z direction and which is of the absorber/emitter type. Photons impinge at the front surface of the cell at z = dE . Integration of the steady state continuity equation yields de Jph (V) = −q (G(z) − U(z, V))dz (2.146) −da

2.5 Light Generated Currents

In order to allow a voltage dependent photo current, the recombination rate is written as U(z, V) while the generation rate G(z) is assumed to be voltage independent. For zero applied voltage, Jph (0) equals the short circuit current, Jsc . The local recombination rate U(z, V) depends on the carrier density and therefore on the generation rate. Therefore, it is not a unique function of the device but changes with changing excitation. However, since there is the charge collecting junction in the solar cell we are able to measure a collection current. The collection current which is induced by any excitation source is given by the product of the generation function and the collection probability. It is de G(z)ηc (z, V)dz (2.147) Jph (V) = −q −da

The position dependent collection function ηc (z, V) is a unique function of the device. It is independent of the generation rate (nonlinear effects excluded) and valid for the particular device geometry. The collection function of the absorber only depends on the position of the collecting junction. There is no simple connection between U(z,V) and ηc (z, V). When the collection function is known, the complementary function (1 − ηc (z, V)) can be used to calculate the electronic loss term Jel loss (V) by a similar equation like Eq. (2.147). Certainly, Jel loss (V) can also be obtained from Jgen − Jph (V). 2.5.4 Collection Function

The internal collection function is the probability for a generated charge carrier to be collected at the contacts [65] and thereby to contribute to Jph (V). ηc (z, V) is valid for photonic excitation as well as for impact excitation by electrons or other particles. Moreover, ηc (z, V) is valid for carrier injection and limits the diode current in the diffusion limit (see Section 2.4). In a device as shown in Figure 2.4 the sought collection function describes recombination in the emitter as well as absorber including their interfaces and depletion regions. Note, however, that the photo current loss due to absorption in the window layer was already being accounted for as optical loss. In the following, we discuss the collection function in the QNR with or without grading regions, in the SCR and in the buffer layer. 2.5.4.1 Absorber Quasi Neutral Region Charge carriers generated within the QNR of the absorber (wa < −z < da ) either diffuse to the SCR at position −wa (see Figure 2.33) or recombine. Their collection function is position dependent and can be calculated using the reciprocity theorem for charge collection [66].15) The reciprocity theorem states that the local collection 15) The reciprocity theorem can also be derived

from the principle of detailed balance [65]. It is independent of the carrier statistics (Boltzmann or Fermi–Dirac) and also valid for nonlinear recombination.

87

88

2 Thin Film Heterostructures

function in the QNR is identical to the normalized excess minority carrier concentration of a voltage-biased device in the dark [65]. As an equation, this identity reads ηc (z) =n(z)/n(−wa )

(2.148)

−wA is the location of the carrier injecting SCR edge being identical to the charge-collecting junction. The advantage of using the reciprocity theorem is that this function is obtained from an homogeneous differential equation [67]. If ηc also depends on x and y in a three-dimensional device, the quantity ηc (r) equals n(r)/n(r ). For a one-dimensional device an analytical expression for ηc (z) in the QNR of the absorber can be derived. For a p-type absorber in the limit n p (low injection or medium excitation) as well as p∗ p (trap level at mid-gap), the net recombination rate in Eq. (2.67) is given by U − G0 = (n(z)p0 − n0 p0 )/τn0,a p0 = n(z)/τn0,a . Thus, the net recombination rate is linear in n(z).16) The differential equation for n(z) reads Dn

d2 n(z) n(z) − = −G(z) dz2 τn0,a

(2.149)

where Dn is the diffusion constant for electrons and τn0,a is the lifetime of the electron in the absorber. The reciprocity theorem relies on the linearity and symmetry of the differential operator deﬁning this continuity equation. According to the reciprocity theorem, the differential equation for ηc (z) is homogeneous and reads Dn

d2 ηc (z,V) ηc (z,V) − =0 dz2 τn0,a

(2.150)

The boundary conditions are ηc (−wa ) = 1,

dηc (z) Sn0,bc = ηc (z) dz z=−da Dn

(2.151)

where da is the absorber thickness and −wa is the position of the SCR edge. With the ansatz ηc (z) = C1 exp{−z/Ln,a } + C2 exp{z/Ln,a }, Eq. (2.150) can be solved leading to S a sinh −z−d − Dn0,bc Ln,a n,a ηc (z,V) = S n0,bc 1 a (V) a (V) + Ln,a sinh da −w cosh da −w Dn,a Ln,a Ln,a 1 cosh Ln,a

−z−da Ln,a

16) This is very important since it allows to

solve the continuity equation for n in the QNR of an absorber. Furthermore, it allows to deﬁne a diffusion length as the

(2.152)

logarithmic decrement of the collection probability.

2.5 Light Generated Currents

1.2 1.0 ηc

0.8

Ln,a = 10, 1, 0.5, 0.1 µm

Sp0,bc = 102 cm s−1

0.6 0.4

107 cm s−1

0.2 -da -(wa+Ln,a) (a)

-wa0 z [µm]

Figure 2.33 Collection function in the absorber as a function of the position z. Diffusion constant of electrons Dn,a = 1 cm2 s−1 . The charge collecting junction is located at z = −wa . In (a), the back surface recombination velocity is varied at

de

-wa0

-da (b)

de

z [µm]

ﬁxed diffusion length of Ln,a = 1 µm. Note that the curves of Sn0,bc = 106 cm s−1 and Sn0,bc = 107 cm s−1 are almost identical. In (b), the diffusion length of electrons is varied at ﬁxed Sn0,bc = 106 cm s−1 .

Here, Ln is the bulk diffusion length for electrons which is calculated from L2n,a = Dn,a τn0,a . Figure 2.33a shows an example of ηc (z) for different values of the back surface recombination velocity, Sn0,bc . The parameters da , Dn,a , and Ln,a were selected in order to mimic a realistic absorber layer (see Figure 2.33). For the cases Sn0,bc = 106 and 107 cm s−1 , ηc is very small near the back contact. For Sn0,bc = Dn,a /Ln,a (Sn0,bc = 104 cm s−1 in our example), ηc exponentially declines and the collection function is similar as in a semi-inﬁnite absorber. If Sn0,bc is smaller than Dn,a /Ln,a , the back surface is passivated. This can be due to a minority carrier mirror by some kind of back surface gradient (see Sections 3.8 and 3.10). As explained in Section 2.4, the diode current in the limit of diffusion is calculated by use of the derivative of ηc (z) at the position z = −wa . From Eq. (2.152) we ﬁnd dηc (z) = dz z=−wa

Sn0,bc 1 sinh(y) + Dn,a cosh(y) Ln,a Ln,a 2 ;y Sn0,bc 1 sinh(y) + Ln,a cosh(y) Dn,a

=

da − wa Ln,a

(2.153)

The collection function ηc (z) can be experimentally determined from a QE measurement or an electron beam induced current (EBIC) experiment. Knowing ηc (z), one can extract the parameters Ln,a and Sn0,bc by curve ﬁtting [68]. The example in Figure 2.33 shows that both Ln,a and Sn0,bc inﬂuence the shape of the collection function. The inﬂuence of Sn0,bc becomes larger for larger Ln,a as is shown in Figure 2.33b. Only for Ln,a being much smaller than the absorber thickness, the effect of Sn0,bc may be neglected. In a thin ﬁlm polycrystalline cell where z is the direction of the heterojunction, the collection function will vary with x and y due the effect of grain boundaries. We can use the reciprocity theorem in Eq. (2.148) to obtain plots of ηc (z) either from three-dimensional device simulators or from analytical approximates. Dugas solved the equation for the excess minority carrier density for a columnar grain

89

2 Thin Film Heterostructures

1.0 0.8 Ln,eff [µm]

90

0.6 0.4 0.2 102

g = 0.3, 1, 3 µm 103

104

105

106

107

−1

Sgb [cm s ] Figure 2.34 Effective diffusion length under the inﬂuence of grain boundary recombination calculated after Eq. (2.154) for Ln,a = 1 µm and Dn,a = 1 cm2 s−1 . Approximation of square shaped grains without back surface recombination. With increasing grain size g, the effect of Sgb on Ln,eff becomes reduced.

on a square shaped base by a Fourier-like decomposition of the excess minority carrier density [69, 70]. Examples for the collection function in the presence of grain boundary recombination can be found in Ref. [65]. Brendel and Rau derived an approximate expression for an effective diffusion length for the case that the absorber is sufﬁciently thick to neglect back surface recombination Ln,eff = (Ln,a −2 +2Sgb /Dn,a g)−1/2

(2.154)

Here, g denotes the grain size. Plots of Ln,eff versus Sgb in Figure 2.34 show that the effective diffusion length is sensitive to grain boundary recombination if the grain size is in the range of the bulk diffusion length Ln,a . 2.5.4.2 QNR with Graded Bandgap The collection of electrons being minority carriers in the QNR of a p-type absorber may be supported by a gradient in the electron afﬁnity χ, that is, −dχ/dz > 0. In this case, the expression for the current density in Eq. (2.9) includes the second and third term and the one-dimensional diffusion equation (Eq. (2.149)) has to be modiﬁed to

Dn,a

d2 χ n(z) d2 n(z) µn,a dχ dn(z) µn,a − n(z) 2 − − = −G(z) (2.155) 2 dz q dz dz q dz τn,a

If we only allow a constant gradient of χ, the third term in Eq. (2.155) can be neglected. We can apply the generalized reciprocity theorem [71] which allows solving the homogeneous equation for ηc instead of solving Eq. (2.155): d2 ηc (z,V) ηc (z,V) 1 dχ dηc (z,V) − − =0 dz2 kT dz dz Dn,a τn,a

(2.156)

2.5 Light Generated Currents

1

8 6

ηc

4 2

-(Ln,a / kT)× (dχ / dz) = 0,1,2,5,10

0.1

8 6

-da

-wa0

de

z [µm] Figure 2.35 Collection function in the absorber with Ln,a = 1 µm and Sn0,bc = 107 cm s−1 . The back surface gradient of the electron afﬁnity (given in units of 1/Ln,a ) is varied.

The boundary conditions are identical with Eq. (2.151), we ﬁnd the solution ηc = S

n0,bc Dn,a

S −k2 exp{k1 (wa + z)}− Dn0,bc − k1 exp{k2 (wa +z)}exp{(k1 − k2 )(wa − da )} n,a S S n0,bc n0,bc − k − k − exp{(k1 − k2 )(wa − da )} 2 1 Dn,a Dn,a (2.157) )

with k1,2 =

1 dχ 2kT dz

±

dχ

2

+L12 dz n,a The effect of a back surface gradient is to support carrier collection from the bulk of the absorber. This is particularly effectful for devices with small Ln,a (see Section 3.10). Figure 2.35 shows ηc (z) calculated after Eq. (2.157) for different effective force ﬁelds induced by a gradient in χ. The value of −(Ln,a /kT)(dχ/dz) = 10 corresponds to Ec = 0.4 eV for an absorber with thickness da = 1.5 µm. In Figure 2.35 we see the positive effect on the collection function and we may expect an increased Jsc . However, due to the bandgap gradient the generation function has to be modiﬁed since the bandgap gradient induces an effective bandgap enhancement. This is a negative effect on Jsc . The trade off between both effects is discussed in Section 3.10. 1 (2kT)2

2.5.4.3 QNR with Back Surface Field If the absorber bulk includes a back surface gradient in the form of, for example, a back surface ﬁeld (gradient of the electric potential), the collection function in the remaining QNR and in the gradient region will have to be modiﬁed. Equation (2.152) points out the necessary changes in the QNR: in the collection function, we have to replace the absorber thickness da by an effective thickness da − wbsg where wbsg is the width of the back surface gradient region (BSGR). Further, we set Sbsg to a small value, provided the back surface gradient is large enough to establish a sufﬁcient minority carrier mirror. This deﬁnes the collection function

91

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2 Thin Film Heterostructures

at the position da − wbsg , that is, at the beginning of the BSGR

Sbsg da − wbsg − wa da − wbsg − wa −1 ηc da − wbsg = Lp,a sinh + cosh Dp,a Lp,a Lp,a (2.158) If the back surface gradient is nearly constant in the BSGR, the collection function will take on a constant value as given by Eq. (2.158). Close to the back contact, ηc (z) will drop again due to back surface recombination of electrons. 2.5.4.4 Absorber Space Charge Region Due to the small density of recombination partners, recombination of photo generated carriers in the SCR can often be neglected. In this case the collection function in the SCR is close to unity. For a high defect density and thus a small carrier lifetime, this approximation is not valid and recombination in the SCR cannot be neglected. The exact differential equations for electrons and holes in the SCR are derived from Eqs. (2.9−2.14) and read

d2 n(z) dn dF + qµn,a n = R(z) − G(z) + qµn,a F 2 dz dz dz dp dF d2 p(z) − qµp,a p = R(z) − G(z) Dp,a − qµp,a F 2 dz dz dz

Dn,a

(2.159)

These equations require numerical solutions as they are nonlinear in the case of SRH recombination. For photonic excitation Reichman calculated the wavelength dependent external collection efﬁciency [72]. If the drift term and the gradient of J can be neglected, one can use the Hecht equation [73]. However, we will use a different approach and will make use of Eq. (2.147). Using device simulation and a -function like generation proﬁle, we can derive η(z) in the SCR for a generation density similar as under AM1.5 condition (see details below). We note that the validity of Eq. (2.147) does not imply that the reciprocity theorem is valid in the SCR. We further note that due to the nonlinearity of Eq. (2.159), a simulated η(z) is only valid for the selected generation density (AM1.5 equivalent). Figure 2.46a shows simulated functions of η(z) (AM1.5) for different absorber lifetimes where the interface recombination velocity is ﬁxed to Sn0,p0 = 105 cm s−1 . The width of the SCR, that is, the position −wa has been extracted from device simulation but is in good agreement with the value obtained after Eq. (2.36). The ﬁgure shows that for a lifetime of about 10 ns, the assumption of η(z) = 1 in the SCR approximately is correct. For decreasing lifetime values, ﬁrst there is a deviation from unity at the edge of the SCR (−wa ) and ﬁnally also at the absorber/buffer interface (z = 0). The curve with τn0,p0 = 10 ps is much smaller than unity in the complete SCR. Thus, the photo current generated in the SCR, which is calculated by folding η(z) with the generation function according to Eq. (2.147), is much smaller than in the case of τn0,p0 = 10 ns. This shows up as an external quantum efﬁciency (EQE) curve which is much below 100% in the complete wavelength range (see Section 2.5.5).

2.5 Light Generated Currents

The edge of the SCR deﬁnes the boundary condition for calculation of η(z) in the QNR. We see in Figure 2.46a that for small lifetime η(−wa ) = 1 is not fulﬁlled. This must be considered in the calculation of the collection function in the QNR, for example, by using a scaling factor. A further impact on carrier collection in the SCR has the absorber/emitter interface. We consider the band diagram under short circuit as given in Figure 2.39a. In the p-type absorber layer with inverted surface, holes are minority carriers near the heterojunction. They are driven by the electric ﬁeld toward the edge of the depletion region where they become majority carriers but also by their concentration gradient toward the heterojunction where they recombine. In Figure 2.39a, we see Jp having a small negative peak directly at the absorber/buffer interface. This negative peak would increase upon decreased absorber doping and increased interface recombination velocity. Since recombination of holes at the interface is a loss mechanism, the collection function bends downwards close to the interface as is depicted in Figure 2.36b. The larger Sn0,p0 , the smaller is η(z) close to the interface at z = 0. The effect on the photo current Jph (V) is strong as the generation rate close to z = 0 is high (see Figure 2.32). 1.0 τ n0/p0,a =10 ns 0.8 ηc (z)

1 ns 0.6 0.4 0.2

τn0/p0,a =10 ns 100 ps 10 ps τn0/p0,a =100 ps

(a) 0.0 (c) 1.0

-wa(0 V)

0

z

ηc (z)

0.8 0.6

Sn0,p0 =103, 105, 107 cm s−1

0.4 0.2 0.0

(b)

0

-wa z

Figure 2.36 (a) Collection function of an absorber/buffer/window heterostructure in the SCR where the carrier lifetime is the parameter and the interface recombination velocity at the absorber/buffer interface is ﬁxed at Sn0,p0 = 105 cm s−1 . (b) Collection function in the SCR and in the buffer layer where the interface recombination velocity at the absorber/buffer interface is the parameter

and the lifetime is ﬁxed at τn0/p0,a = 10 ns. (c) Collection function at ﬁxed Sn0,p0 = 105 cm s−1 and at τn0/p0,a = 10 ns, 100 ps for zero forward bias (full line) and 0.5 V forward bias (dotted line). All other device parameters as in the default Table 8.1. Data from device simulation as described in Section 2.5.4.

93

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2 Thin Film Heterostructures

Naturally, the electric ﬁeld under forward bias also has an effect on η(z). The trivial effect is that in forward bias the SCR becomes smaller. This is seen in Figure 2.36c for the example of τn0/p0,a = 10 ns. For shorter lifetimes, however, also the maximum of η(z) becomes reduced. This shows up in the JV curve of the cell as a small ﬁll factor (FF). 2.5.4.5 Buffer Layer In a heterostructure of n-type buffer/window/p-type absorber, holes are minority carriers in the buffer layer. The collection function in the buffer layer is described by the collection probability of these holes. Principally, they can undergo bulk recombination as well as interface recombination. Since buffer layers normally are very thin and may also be depleted from majority carriers, interface recombination may well dominate over bulk recombination. Rothwarf proposed a simple formula which relates the interface recombination velocity and the drift velocity of minority carriers at the interface of a heterojunction [74]. Originally, this formula has been developed for interface recombination in a non-inverted absorber. In Figure 2.37 we depict an absorber/buffer/window heterostructure with interface states. A current density Jgen is due to minority carrier holes generated in the buffer. The fraction Jel loss of this current is lost via interface recombination while Jph can pass the interface. A collection probability ηc can be deﬁned according to

ηc (V) =

Jph (V) Jph (V) + Jloss el (V)

(2.160)

The external current density can be calculated from TE theory according to Jph (V) = qµp,b pb (V)Fz=0 (V). The loss current Jel loss is obtained from Eq. (2.105) with RIF from Eq. (2.67) and reads Jel loss = qSp0 pa (V). Thus, ηc (V) can be written as µp,b Fz=0 (V) ηc (V) = (2.161) µp,b Fz=0 (V) + Sp0 ηc (V) is a constant within the buffer layer. A model which takes into account also a position independent bulk recombination probability has been presented Ec

Ev

Jelloss

Jph

E

Jgen 0 db Position z

Figure 2.37 Schematic band diagram of an absorber/buffer/window heterostructure containing interface states under illumination in short circuit condition. Jph denotes the photo current due to holes generated in the buffer layer or interface near zone of the absorber, Jel loss describes the current loss due to interface recombination.

2.5 Light Generated Currents

in Ref. [75]. Figure 2.36b gives examples of the collection function in the buffer (0 < z < db ) for an absorber/buffer/window heterostructure. This being our default device exhibits interface recombination both at the absorber/buffer interface as well as at the buffer/window interface. Thus, Eq. (2.161) certainly is by far to simple. Nevertheless, we ﬁnd that ηc (V) in Figure 2.36b has a small plateau in the buffer and only bends down if approaching the buffer/window interface. The value of µp,b Fz=0 in this device is 1.1 × 106 cm s−1 . Accordingly we ﬁnd that, for Sn0,p0 = 107 cm s−1 , ηc (V) is very small in the buffer and, for Sn0,p0 = 105 cm s−1 , ηc (V) has moved toward larger values. If the buffer layer thickness can be varied in a wide range without changing the buffer layer properties, Eq. (2.161) can be a means to estimate the interface recombination velocity. 2.5.4.6 Simulating the Collection Function The collection function can be simulated using a device simulation program and with aid of Eq. (2.147). We apply a speciﬁc generation function of the type G(z) = G0 + (z − z ), where the delta function in practice is a narrow distribution at the point z and G0 is a constant generation rate similar as under AM1.5 illumination. Introducing the speciﬁc generation function into Eq. (2.147) yields

Jph (V) = −q

db −da

G0 ηc (z,V)dz − q

db

−da

δ(z − z )ηc (z,V)dz

(2.162)

Identifying Jph (V) with the integral over G0 ηc (z, V) Eq. (2.162) allows to calculate ηc (z , V) = Jph (V) − Jph (V). The currents Jph (V) and Jph (V) result from two separate simulations. Figure 2.38 shows the simulated collection function of the default device of absorber/buffer/window type at zero voltage bias. The function ηc (z) falls off to zero in the ﬁrst nanometers of the window layer due to small lifetime therein and the ﬁnite interface recombination velocity at the buffer/window interface.

1.0

ηc (z)

0.8 0.6 0.4 0.2 0.0

-da

-wa

0 db

db +dw

z

Figure 2.38 Simulated collection function of an absorber/buffer/window device with parameters as in default Table 8.1.

95

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2 Thin Film Heterostructures

2.5.5 Quantum Efﬁciency and Charge Collection Efﬁciency

For the case of photonic excitation, we want to derive a correlation between the collection function ηc (z, V) and EQE(λ, V). Both quantities can be voltage dependent. By use of the normalized generation function Eq. (2.147) reads λg,a djγ (λ) db n Jph (V) = −q G (λ,z)ηc (z,V)dzdλ (2.163) dλ −da λg,w In contrast, the QE is deﬁned as the change in the charge carrier ﬂux density −dJph (λ, V)/q per change of photon ﬂux density at a given wavelength. EQE(λ, V) = −dJph (λ, V)/qdjγ (λ)

(2.164)

Equation (2.164) describes the wavelength dependent response of the solar cell: how many charge carriers are driven toward the external circuit for a certain number of incident photons. The deﬁnition in Eq. (2.164) can be rewritten as λg,a djγ (λ) EQE(λ, V)dλ (2.165) Jph (V) = −q dλ λg,w The QE can be expressed as a convolution of the normalized generation rate and the collection function d b Gn (λ,z)ηc (z,V)dz (2.166) EQE(λ,V)= −da

We remind that the EQE and internal quantum efﬁciency are related via the absorbance, that is EQE(λ, V) = IQE(λ, V)A(λ)

(2.167)

If the excitation of charge carriers in a solar cell is accomplished by electron bombardment, we speak of EBICs. Excitation by an electron beam is done, for example, for testing purposes. A charge collection efﬁciency can be deﬁned as de Gn (E,z)ηc (z,V)dz (2.168) (E,V) = −da

n

where G (E, z) is the spatial dependent generation rate for a given electron beam energy (Eq. (2.141)). We see that again the property of the device to collect charge carriers is described by ηc (z, V). The QE in Eq. (2.166) and the charge collection efﬁciency in Eq. (2.168) can be considered as the Laplace transform of the collection function. If the generation functions are known and EQE(λ) or (E) have been measured at a certain bias voltage V , an inverse Laplace transformation will render the collection function η(z, V ). This has been demonstrated for optical [76] and electron beam excitation [77]. The concept of the collection function is particularly useful if both QE measurement and EBIC measurement have been performed on the same sample [78].

2.5 Light Generated Currents

2.5.6 Barriers for Photo Current

Similar as a barrier for the diode current (see Section 2.4.7), the ﬂow of photo current may be impeded by transport barriers. A back contact barrier does not block the photo current because this barrier diode is put under forward bias by the main diode. However, at the front contact there may be a barrier for electron transport imposed by a band discontinuity, a bandgap gradient or a speciﬁc doping proﬁle. In the case of a conduction band discontinuity, the band diagram may be similar as in Figure 2.29b, however, the drop in EFn will be negative, that is, EFn < 0. The photo current generated in the absorber layer has to surmount the energy barrier Ec b,a . Replacing φb n in Eq. (2.126) by En,az=0 + Ec b,a and using naz=0 = Nc,a exp{−En,az=0 /kT}, it follows J0,TE = 1/4qνn Nc,a exp{−(Ec b,a + En,az=0 )/kT} = 1/4qνn naz=0 exp{−Ec b,a /kT} (2.169) as the limiting current density [57]. Basically, exp{−Ec b,a /kT} is a Boltzmann term for the TE over the conduction band spike. As long as the generation current17) is much smaller than the limiting current given by Eq. (2.169), the barrier will have only negligible inﬂuence on the photo current. For a given energy barrier, the free parameter of Eq. (2.169) is the carrier density naz=0 . The photo current can pass the absorber/buffer interface exhibiting a certain energy barrier Ec b,a in case that naz=0 is sufﬁciently high. This means for a p-type absorber that the electrons must be majority carriers at the interface. (For design rules, see Section 3.3.) If the generation current Jgen reaches the limiting current J0,TE , carrier transport is impeded and minority carriers accumulate at the absorber surface. This can be seen in Figure 2.39b. Due to the accumulation of electrons at the interface, the concentration gradient in the absorber is negative at all positions in the absorber bulk. This changes the collection function in the absorber bulk and leads to higher recombination in the bulk. We see that the electron related current density in Figure 2.39b is negative in the QNR of the absorber. A barrier for the photo current may also be established by a gradient of the bandgap at the front of an absorber. If the bandgap gradient is realized by lifting the conduction band edge close to the interface as in Figure 2.14c, the case is very similar as for a conduction band spike: only electrons with sufﬁcient kinetic energy can pass this barrier. If the bandgap gradient realized by lowering the valence band edge of the absorber toward the interface (cf. Figure 2.14c), a region of dEFn /dz < 0 will form as described in Section 2.3.7. This will also reduce the photo current although we may not relate it to a photo current barrier. 17) For this argument we take the generation

current as a place holder for a ‘‘possible photo current.’’ What we mean is the photo

current which could ﬂow without a photo current barrier.

97

98

2 Thin Film Heterostructures

-qϕ

1

-qϕ

Energy E [eV]

0 −3

-qϕ

-qϕ0

EFn

Ec

−4 −5

Ev EFp

−6

J [mA cm−2]

n,p [cm−3]

−7 1018 1016 1014 1012 1010 20 10 0 −10 −20 −30

p n p n

Jn Jn Jtot Jn

−1.5 −1.0 −0.5 (a)

p n

Jtot

Jp Jtot 0.0

Jp

Jp 0.5

−1.5 −1.0 −0.5 (b)

0.0

Position z [µm]

Figure 2.39 Solution curves for absorber/buffer/window heterostructures (default device from Table 8.1) under AM1.5 illumination. (a) Without photo current barrier and (b) with a photo current barrier of 0.5 eV due a conduction band offset at the

0.5

−1.5 −1.0 −0.5

0.0

0.5

(c)

buffer/absorber interface; and (c) with a p+ layer in the absorber front. Note the different scales of the current density. The potential energy without illumination, −qϕ0 , is given for comparison.

Finally, we treat the case that the absorber exhibits a p+ region next to the heterointerface [79]. In Figure 2.39c the large negative charge in the p+ layer leads to a large potential drop in the buffer and a small potential maximum in the absorber. If we introduce additional donor states at the buffer/p+ absorber interface, we can avoid the potential drop in the buffer – the principal barrier effect, however, would be similar. The p+ layer can be viewed as a front surface barrier for minority carriers. Such a barrier impedes the ﬂow of electrons to the collecting interface. Electron current is only positive in the very interface near region but becomes negative for larger z (see Figure 2.39c). A solar cell with such kind of barrier has a high Voc – due to the large barrier for electron injection – but a small Jsc due to high recombination. The JV curve under illumination exhibits a kink which marks the transition from full photo current under reverse or moderate forward bias to a barrier reduced photo current for more positive bias. (In the example in Figure 2.39c, the kink is found at reverse bias.) If the kink is found under forward bias it leads to a low ﬁll factor of the cell.

2.5 Light Generated Currents

2.5.7 Voltage Dependence of Photo Current

In the preceding sections, we have allowed the collection function and the QE to be voltage dependent. A voltage dependent collection function will cause the photo current to be voltage dependent. This is particular of importance in the fourth quadrant of the JV curve, that is, under forward bias. A reduction of the photo current in the forward direction decreases the FF. Up to now, we encountered several mechanisms as possible origins for a voltage dependent photo current. These are variations with voltage of the SCR width, interface recombination, and photo current barrier. 2.5.7.1 Width of SCR A typical collection function of the absorber is close to unity in the SCR and declines in the QNR (see Figure 2.36). The collection width of photo induced carriers, Lcoll , may be approximated by wa + Ln,a , where wa is the SCR width and Ln,a the electron diffusion length. Due to an applied forward voltage, wa decreases [see Eq. (2.36)]. Generally, the photo current increases with reverse bias and decreases with forward bias. If the diffusion length is very small, then Lcoll ≈ wa and the inﬂuence of the applied voltage is large. Vice versa, if Ln,a wa , the changing SCR width has almost no inﬂuence on Lcoll and the voltage dependence of Jph is small. Thus, knowing the SCR width at zero bias and knowing the collection length, one can estimate the contribution of the SCR width on the photo current. A voltage dependent width of the SCR is diagnosed by voltage dependent QE measurements. For the absorber/buffer/window heterostructure Eq. (2.166) can be written as d b EQE(λ,V) = ηc (z)αb (λ)exp[ − αb (λ)z]dz T(λ)[1 − R(λ)] 0 0 −wa + αa (λ)exp[ − αa (λ)z]dz + ηc (z)αa (λ)exp[ − αa (λ)z]dz (2.170) −wa

−da

where the integrals give the contributions of the buffer layer, the absorber SCR (assuming ηc = 1) and the absorber QNR, respectively. We select a wavelength for which absorption in the buffer is about zero and assume Ln,a < da . In that case, the collection function after Eq. (2.152) can be approximated by ηc (z) = exp{(z + wa )/Ln,a } and Eq. (2.170) reduces to exp{ − αa (λ) wa (V)} EQE(λ,V) = T(λ)(1 − R(λ)) 1 − (2.171) 1 + αa (λ)Ln,a This is the well known G¨artner formula [80]. We consider the wavelength dependence of the EQE as a result of voltage dependent SCR variation. For short wavelengths with αa (λ)wa (V) 1, the EQE after Eq. (2.171) approaches unity and thus does not depend on the voltage. For longer wavelength, the variation of EQE with voltage increases. Thus, measuring the EQE at different voltages and

99

2 Thin Film Heterostructures

EQE/EQE (0V)

100

1.2

−0.5 V

1.0

+0.5 V

0.8 400

600

800

1000

Wavelength [nm] Figure 2.40 Simulated ratios of the external quantum efﬁciency (EQE) as a function of wavelength of an absorber/window heterostructure with Ln,a = 10−10 s in the absorber. Other parameters as in the default Table 8.1. Solid line: ratio at reverse bias, that is, EQE(−0.5 V)/EQE(0 V). Dashed line: ratio at forward bias, that is,

EQE(+0.5 V)/EQE(0 V). Charge carriers generated by deeply penetrating photons (long wavelength) exhibit a relatively higher collection probability at reverse bias due to the extended SCR width. Vice versa, at positive bias charge carriers generated by red photons suffer relatively more from the reduced SCR width.

normalizing to the EQE at zero voltage gives wavelength dependent functions as demonstrated in Figure 2.40. 2.5.7.2 Interface Recombination We consider an absorber/buffer/window heterostructure. The example collection function in Figure 2.36 is inﬂuenced by interface recombination in two ways. Holes photogenerated in the inversion region of the absorber and holes photogenerated in the buffer layer will partially undergo interface recombination. With applied forward bias, the electric ﬁeld at the junction decreases and interface recombination increases. Thus, the photo current decreases with forward bias and increases with reverse bias. A good example of a voltage dependent collection function due to interface recombination is a p+ /n absorber/window heterostructure as the Cux S/CdS solar cell. Here, we ﬁnd a large En,az=0 and high interface recombination velocity Sn0,p0 but it is Ec a,w = 0. In this cell there is competition between interface recombination and current sweep through the junction region [74]. If at higher forward bias, the electric ﬁeld is reduced, interface recombination increases. 2.5.7.3 Photo Current Barriers A photo current barrier can lead to a voltage dependent photo current. We consider a conduction band barrier such as the one depicted in Figure 2.39b. In this example, Ec b,a is so large (0.5 eV) that the generation current exceeds J0,TE in Eq. (2.169) already at zero voltage. J0,TE was limited by a too small electron concentration naz=0 for the large value of Ec b,a . If the band diagram is such that Ec b,a is slightly smaller, the concentration naz=0 at the interface may be sufﬁcient to allow photo current transport at zero bias. But the electron concentration at the interface decreases with forward bias. Consequently, the photo current may become voltage dependent.

2.6 Device Analysis and Parameters

All mechanisms described above have in common that the photo current increases with reverse bias and no voltage for which the photo current saturates can be given. Thus, we cannot deﬁne a unique reverse bias voltage where, for all types of cells, the photo current show a maximum. In order to deﬁne a voltage dependent external collection efﬁciency η(V) it is suggested to set η(0) = 1. Thus, we have Jph (V) = Jsc η(V)

(2.172)

Comparison with Eq. (2.147) reveals that the relation between the external collection efﬁciency η(V) and the collection function ηc (z, V) is given by d b 1 ηc (z,V)G(z)dz (2.173) η(V) = −Jsc −da

2.6 Device Analysis and Parameters

In this section, we discuss some aspects of device analysis which are most relevant for heterojunction solar cells. The goal is to determine the basic device parameters from a limited number of electric measurements. Therefore, it is instructive to consider the equivalent electric circuits of heterostructure solar cells at the beginning of this section. 2.6.1 Equivalent Circuits

The thin ﬁlm solar cell and module can be described by equivalent circuits of electric components. A circuit model shall reﬂect the response of a solar cell upon voltage and light stimulation. It is useful to discriminate between DC and AC stimulation. While DC stimulation includes the normal solar cell operation, AC stimulation is merely for the purpose of device analysis. First, we concentrate on single heterojunction solar cells. Module aspects are discussed in the last part of this section. 2.6.1.1 DC Equivalent Circuit An ideal heterojunction solar cell which obeys the principle of superposition is described by only two electric components: An ideal current source which delivers the photo current and a diode. The current source is ideal in the sense that the current delivered does not depend on voltage. The real solar cell deviates from the ideal cell in several respects. First, the photo current may depend on voltage (see Section 2.5.7). In order to preserve the superposition principle, we allow the current delivered by the current source to depend on voltage in a nonlinear (but known) fashion. In Figure 2.41a this is expressed as Jph (V). Also the diode may be not ideal – in the sense that parallel processes are carrying the recombination current. In Figure 2.41a this would

101

102

2 Thin Film Heterostructures Vm Jph(V)

I

III

Vb

GJ

DBC D

GBC

Rs J

Gsh

SCLC

Gsh

Rs

xcell SCLC

V

IV

x

(a)

y

(c)

I RBC

RBC

GPI

Cell n

Interconnect

(b)

II

Rw,BC

RTCO

(d)

CBC

CJ

II Cell n +1

II

Figure 2.41 (a) Equivalent circuit for DC bias conditions comprising a network of a voltage dependent current source, diode, shunt resistance, series resistance, and space charge current limiter. (b) Equivalent circuit of a heterostructure solar cell under AC operation where the main diode and a possible diode at the back contact are simply replaced by conductances and capacitances. Note that the main diode may need to be described by a much more complicated equivalent circuit as depicted in

(e)

RBC

Figure 2.44. (c) Schematic plan view on part of two stripes of a solar module showing the three scribe lines [interconnect scheme (a) in Figure 2.43]. Note that the image is not drawn to scale. The dashed rectangle shows one module element. (d) Module element with distributed series resistances of window and back contact, Rw and RBC . Series integration introduces the interconnect elements Rw,Bc and GP1 . (e) Parallel circuit of module elements in direction y contacted with the series resistor elements RBC .

mean that we have to replace the diode by a parallel combination of two or more diodes. The DC equivalent circuit of a non-ideal solar cell in Figure 2.41a is completed by a shunt conductance Gsh and one or more series elements. The shunt conductance is assumed to be ohmic. Possible series elements are depicted in the box in Figure 2.41a. An ohmic series resistance is a lumped quantity including ohmic losses in the QNR or at the contacts. Often also the sheet conductances of window and back contact are included in the series resistance. Here, we take a different point of view in that we consider distributed series effects due to sheet conductance as part of the module circuit. Therefore, Figure 2.41a only represents a small solar cell being the building block of a large network (the module). We call this building block, labeled as I in Figure 2.41, an elementary diode. Another series element may be the effect of current limitation by space charge limited current (SCLC). The SCLC can be expressed as JSCLC = kVB m where k and m are constants which can depend on the trap density in the SCR [81]. Thus,

2.6 Device Analysis and Parameters

the SCLC is nonlinear in the voltage. The SCLC element can dominate the JV curve of a device at high forward bias [82]. Finally, a series element can be a back contact diode which has the opposite polarity as the main diodes. Certainly, two or more series elements may occur in parallel and/or in series and may dominate at different voltage ranges. If a voltage V is applied to the circuit in Figure 2.41a, the part Vb drops at the element in series. It holds V = Vb + Vm where Vm is the voltage at the main junction. The total current is ( qVm N J= − 1 + Vm Gsh + Jph (Vm ) J0i exp (2.174) Ai kT i=1 where N is the number of parallel recombination paths (see Section 2.4.6). For practical purposes, often only one diode is considered. In case of an ohmic series resistance, the voltage drop at the series element is simply Vb = Rs J. As a result of the series element, Eq. (2.174) becomes implicit. Extraction of the diode parameters from experimental JV curves is discussed in Section 2.6.2. The case of a back contact diode has been discussed in Section 2.4.7. 2.6.1.2 AC Equivalent Circuit The AC equivalent circuit in Figure 2.41b is derived from the DC model. It does not include the current source (because of its inﬁnite internal resistance) and describes the shunting diode as a combination of one capacitance and one conductance. In the case of a simple pn junction, the conductance stands for the dominant recombination path of the diode current. It can be expressed by the saturation current density as qJ0 qV dJ = exp (2.175) GJ (V) = dV AkT AkT

A similar expression as Eq. (2.175) can be formulated for the conductance of the back contact diode. The capacitance stands for charging and discharging of shallow states at the edges of the SCR. For a heterostructure solar cell, the AC equivalent circuit may be more complicated than the one shown in Figure 2.41b. We give examples of absorber/window and absorber/buffer/window heterostructures in Figure 2.42. A simple absorber/window heterojunction without interface states (Figure 2.42a) may be described by two conductances and capacitors each in parallel. Since recombination is limited to the absorber side, the conductance Gw expresses the conductance of the window depletion layer [83] while Ga is given by Eq. (2.175). In the case of an n+ p heterojunction, Gw and Cw can be neglected. Also, the absorber/buffer/window heterostructure (Figure 2.42e) can be described by a parallel circuit. Thus, measuring the admittance Y(ω) of the cases a,e allows to identify Re(Y) with G and Im(Y) with ωC, where G and C are the total conductance and capacitance, respectively. Charging and discharging of an inversion layer (Figure 2.42b) will add a further capacitance to the equivalent circuit. Its value depends on the reduced chemical

103

104

2 Thin Film Heterostructures

Gw

Ga

Ww G−1 w =

(a)

Ca

Cw

Ga

Gw

0

Ga =

CINV (b)

Cw

Ca

CINV =

Ca

Gd

(d)

q kT

Ca =

Ga (c)

Cd =

Cd

Ga

Gw GIF

Ca

CIF

σn,w (z)−1 dz

qJ0

exp

AkT

qVa AkT

2NCkT εIF exp

− En,a

z=0

2kT

ε0εa

Wa

q2 Ndf(Ed)(1 − f(Ed)), ω kT

ω0

CIF = qNIF , ω ω0

Cw db

Ga

Gb

G−1 b =

Ca

Cb

Cb =

σn,b (z)−1 dz 0

(e)

Ga

Gb

(f) Ca

Wb

Gw

GIFa,b

GIFb,w

CIFa,b

CIFb,w

Cb

ε0εb

Cw Gb

Ga

GIFa,b CIFa,b

Ca (g) Gd

Cd

Cb

Figure 2.42 Band diagrams and corresponding equivalent circuits for heterostructures exhibiting (a) voltage drop in window and absorber, (b) an inversion layer in the absorber, (c) a singular deep trap level in the absorber, (d) interface states,

(e) a buffer layer, (f) interface states at the buffer/window and buffer/absorber interfaces, and (g) a combination of bulk and interface defects. The right column lists some mathematical descriptions for conductances and capacitors.

2.6 Device Analysis and Parameters

potential of the electrons at the interface, En,az=0 (see Figure 2.42) [83]. An inversion layer in the absorber can be the outcome of a type I band alignment as depicted in Figures 2.42b. Even for high window doping, large band bending on the window side will impede charge transport to the interface. Thus, the conductance Gw is needed in the equivalent circuit. A discrete trap level at given energy Ed in the SCR of the absorber can be modeled by introducing the series connection of a capacitance Cd and conductance Gd . In the example of Figure 2.42c, we assume an n+ p heterojunction. The elements Cd and Gd are connected in parallel to the absorber conductance. For a discrete trap level, charging and discharging takes place at the location of intersecting trap level and hole quasi Fermi level. The capacitance Cd will depend on frequency and temperature. An equivalent circuit for a trap density energetically distributed in the absorber bandgap can be found in Ref. [84]. Interface states (d,f) can add a further capacitance to the equivalent circuit of a heterojunction. The kinetic limitation of their charging and discharging is accounted for by the conductance GIF . The capacitance simply depends on the energetic density of interface states if the interface state density varies slowly with energy [85]. Inversion layer, trap level, interface states, as well as a back contact diode will render an Im(Y) which is frequency dependent. Thus the equivalent circuit describing the response of the solar cell to an AC voltage stimulation will be frequency dependent. This is the basis of admittance spectroscopy. 2.6.1.3 Module Equivalent Circuit In a thin ﬁlm module the elementary diodes of Figure 2.41a are interconnected. Thus, the module is characterized by a further DC equivalent circuit model. This model includes distributed series resistances and the integrated series connection. An AC equivalent module may be constructed in a similar fashion as the DC module by interconnecting the building blocks III in Figure 2.41b. Possible schemes for the integrated series connection are depicted in Figure 2.43. All schemes introduce a series resistance Rw,Bc . In addition, schemes Figure 2.43a,b introduce the shunt conductance GP1 into the module equivalent circuit. Due to its most widely use, we concentrate on scheme (a). Figure 2.41c depicts a schematic close-up view of a module where the current is ﬂowing in the x direction. The three lines denote the scribe lines P1–P3 (from left to right). Each module cell has a width, xcell , and a length, ycell . A dashed rectangle in Figure 2.41c marks a single element of a module cell. We call this element (which is labeled as II in Figure 2.41) a cell element. The equivalent circuit of the cell element II is depicted in Figure 2.41d. We see that besides the elementary diodes I, the cell element shows distributed series resistances of the window and back contact. The inﬂuence of Rw is larger than that of RBC due to the normally lower conductivity of the window layer. Rw is one of the parameters deﬁning the optimum length of the cell. Design rules for the cell width are presented in Section 3.12. The cell element also includes the contact resistance and shunt conductance of the interconnection scheme.

105

106

2 Thin Film Heterostructures

Window

P1

P2

P3 Absorber Backcontact

GP1

Rw/BC

GP1

(a)

Rw/BC

(b) RTCO

Rw/BC Filler (c)

(d) Contact strip

Substrate

Grid

(e) Figure 2.43 Possible schemes for the series connection of thin ﬁlm solar modules. The monolithical interconnects (a–c) are realized by different scribe and ﬁlling lines. (a) Separated scribes P1–P3, (b) overlapping P1 and P2, (c) P1 scribe with isolation ﬁlling. All interconnect schemes will exhibit contact

resistance Rw,bc . In addition, a shunt conductance GP1 can result within the P1 scribe line of schemes (a) and (b). The shingle type interconnection scheme in (d) is suitable for conductive substrates. In (e) the classical wafer interconnect has been realized using contact strips.

In the y direction, we can also think of a rectangle deﬁning a different cell element (rectangle IV in Figure 2.41c). This cell element IV includes elementary diodes connected by distributed series resistance similar as in Figure 2.41d but without interconnection. Thus in a module, each elementary diode is connected to other diodes in two dimensions by virtue of series resistances in the window and back contact. The diodes form a diode array which may exhibit a ﬂuctuation of electronic properties in two dimensions [86] (the inﬂuence of weak diodes on the module performance is discussed in Section 2.4.9). If there is no series interconnection, the array is cylinder symmetric. A weak elementary diode would be decoupled from other diodes in all directions by the relatively low conductance of the window layer. Due to the series interconnection, however, the module is not cylinder symmetric. Figure 2.41e gives an equivalent circuit of a series of cell elements II which are connected in the direction y by distributed series resistances of the back contact. Due to the normally higher conductance of the back contact, the inﬂuence of a locally weak diode on the diode array is larger than if there is no series interconnection.

2.6 Device Analysis and Parameters

2.6.2 Current–Voltage Analysis

Including series resistances and shunt conductances, the total current as a function of voltage under illumination is given by Jlight (V) = J0 (exp{q(V-Jlight Rs )/AkT}-1) + (V-Jlight Rs )Gsh + Jsc η(V) (2.176) Here, J0 is the saturation current density for the recombination mechanism under illumination. η(V) is the external collection efﬁciency. 2.6.2.1 External Collection Efﬁciency The external collection efﬁciency is an important function. Without knowing this function, it is not possible to apply the superposition principle which is so important for device analysis. The external collection efﬁciency is deﬁned in Eq. (2.172). Initially we allow the external collection efﬁciency to be both voltage and illumination dependent. In the end, we hope that the external collection efﬁciency is only dependent on voltage at least on the relevant illumination level (100 mW cm−2 ). The photo current Jph (V, jE ) as a function of voltage and photon ﬂux is a construct which renders the superposition principle applicable. A necessary condition for the following procedure is that the short circuit current density is linear in jE . Fortunately, this is fulﬁlled for most chalcogenide thin ﬁlm solar cells in the range jE 10 − jE 100 (remember that jE x denotes an energy ﬂux of density x mW cm−2 ). Thus, Jph (V, jE ) = Jsc η(V, jE ) where Jsc is given by Eq. (2.165). η(V, jE ) is the external collection efﬁciency. Possible reasons for the external collection efﬁciency being voltage dependent are discussed in Section 2.5.7. Jsc is easily measured and it remains the problem of how to determine η(V, jE ). We show how to solve this problem under the prerequisite that η(V) does not depend on jE . (If there is a barrier for the photo current this assumption may not be valid; see Section 2.5.6.) Consider two JV curves, Jlight (V, jE n ) and Jlight (V, jE m ), measured at two different energy ﬂuxes, jE n and jE m [87]. By use of Eq. (2.179) the difference current reads

Jlight (V, jE n ) − Jlight (V, jE m ) = [Jdiode (V, jE n ) − Jdiode (V, jE m )] + [Jsc n η(V) − Jsc m η(V)]

(2.177)

and a tentative external collection efﬁciency may be calculated from ηnm (V) =

Jlight (V, jnE )−Jlight (V, jm E) Jnsc −Jm sc

(2.178)

The mathematical form of Eq. (2.178) provides that η(V = 0) = 1.18) Certainly, ηnm (V) only is the ‘true’ external collection efﬁciency if Jdiode (V, jE n ) = Jdiode (V, jE m ). 18) In [88], η = 1 has been set at the maxi-

mum of Jlight (V, jnE ) − Jlight (V, jm E ). For some chalcogenide cells which show a reverse

bias shunt under illumination, the deﬁnition η(0) = 1 is more useful.

107

108

2 Thin Film Heterostructures

Otherwise, ηnm (V) reﬂects both the variation of the voltage dependence of the collection efﬁciency and the voltage dependence of the difference Jdiode (V, jE n ) − Jdiode (V, jE m ). By measuring at energy ﬂuxes, jE m,n,p,q , one may ﬁnd an illumination range where plots of ηnm (V) and ηqp (V) are congruent. In this illumination range it can be assumed that ηnm (V) = ηqp (V) = η(V) and therefore Jdiode (V, jE n ) = Jdiode (V, jE m ). Possibly, this is at a high illumination level (close to AM1.5G) where the recombination mechanism has switched from dark to illuminated. It may be worthwhile to determine the collection function with jE 100−x , j100 , and jE 100+x where E x is small. The determination of η(V) is unaffected by a shunt conductance as long as the shunt is illumination independent. If η(V) is not independent on jE , that is, η = η(V, jE ), we will not likely ﬁnd a combination of energy ﬂuxes jE m,n,q,p where ηmn (V) and ηpq (V) are congruent. 2.6.2.2 Diode Parameters Current versus voltage measurements at different temperatures are the primary means to elucidate recombination processes and the energy band diagram of a solar cell. In Section 2.4 it was shown that all diode currents discussed in this book can be described by the three diode parameters: A, Ea , and J00 :

• The size of the diode quality factor, A (see Table 2.1), gives information about the location of dominant recombination. • The activation energy, Ea , should approximately equal the minimum bandgap in the absorber including its interfaces unless the Fermi energy is pinned at the absorber/emitter interface (Table 2.3). • The reference current density, J00 , is related to the carrier lifetimes (Table 2.2). J00 and Ea are combined in the saturation current density J0 . Further device parameters which are critical for the device performance are series resistance and shunt conductance. In this section, we show how to extract the diode parameters from experiment and how to avoid possible artifacts and pitfalls. In principle, one is interested in the cell parameters which are derived from an illuminated JV curve. This is because only parameters governing the device under AM1.5G illumination are relevant for the solar cell operation. The total current as a function of voltage measured under AM1.5G illumination and at a ﬁxed temperature can be written as the sum Jlight (V, jE 100 ) = Jdiode (V, jE 100 ) + Jph (V, jE 100 ) + (V − Jlight Rs )Gsh

(2.179)

−2 where j100 E is the solar (or solar simulated) energy ﬂux density of 100 mW cm 100 100 with an AM1.5G spectral characteristic [89]. Jdiode (V, jE ) and Jph (V, jE ) are the diode and photo currents as a function of voltage. Jdiode (V, j100 ) contains the wanted E information about the recombination mechanism. For the sake of generality, we allow Jdiode (V) = Jdiode (V, j100 ). If Jdiode is not illumination dependent, that is, the E recombination mechanisms in the dark and under illumination are identical, evaluation of the dark current Jdark = Jdiode (V) + (V − Jdark Rs )Gsh is preferable.

abs(J)

100

(a)

1.0 h(V)

10−1

h(V) 0.8

10−2

Jdiode, Jlight - Jsch(V)

Jdiode, Jlight - Jsch(V)

0.6 10−3 0.4

10−4

0.2

10−5 10−6

109

−0.5

0.0

−0.5

0.5

V [V]

Figure 2.44 Simulated JV curves for absorber/window heterostructures with carrier lifetimes (a) τn0,p0,a = 10−7 s and (b) τn0,p0,a = 10−10 s. Dark and AM1.5G illuminated JV curves (––) as well as a Jsc (Voc ) plot (open circles, left axis). The data point

(b)

0.0

0.5

0.0

V [V]

for jE 100 is given as a full circle. Calculated external collection efﬁciency η(V) (right axis). A series resistance of Rs = 1 cm2 and a shunt conductance of Gsh = 1 mS cm−2 have been included. Other device parameters as in the default Table 8.1.

With the knowledge of η(V), we can replace Jph (V, j100 ) in Eq. (2.179) by Jsc η(V), E which leads to Jdiode (V, jE 100 ) = Jlight (V, jE 100 ) − Jsc η(V) − (V − Jlight Rs )Gsh

(2.180)

Now, we can extract the recombination parameters by curve ﬁtting. That this is not an easy task is illustrated in Figure 2.44. Here, we plotted simulated abs(J) versus V curves under illumination and in the dark for two devices with different absorber carrier lifetimes. The simulated Jlight have been corrected by −Jsc η(V). Both devices are simulated with a 1 mS cm−2 shunt conductance and 1 cm2 series resistance. For the smaller lifetime of τn0,a = 100 ps, the drop of η(V) is more pronounced. Obviously, for both devices the curves (Jlight + Jsc η(V)) and Jdark are subject to the series resistance Rs . There is almost no linear part of the ln[Jlight + Jsc η(V)] and lnJdark curves which can be used to extract A and J0 . From Figure 2.44 it is apparent that a correction of the (Jlight + Jsc η(V)) or Jdark data is necessary. So far, Rs and Gsh are unknown in Eq. (2.176). The ﬁrst step is to determine the shunt conductance in the dark and under illumination from dη(V) dJ dJ Gsh,D = dark and Gsh,L = L − Jsc (2.181) dV dV dV V=0

V=0

V=0

Note that dη(V)/dV is negative. Also Jsc is a negative quantity. If the term dη(V) is omitted, the shunt conductance under illumination comes out Jsc dV V=0 larger than in the dark. This is an artifact. That Gsh is independent of illumination can be veriﬁed by comparing the two values of Gsh after Eq. (2.181). The next step is to determine Rs . Several techniques to determine the series resistance have been developed [90, 91]. Exemplarily, we follow the procedure

η(V)

abs(Jscη) [A cm−2]

2.6 Device Analysis and Parameters

110

2 Thin Film Heterostructures

proposed by Werner [92]. A corrected current density is calculated according to Jlight corr = Jlight (1 + Rs Gsh ) − Gsh V − Jsc η(V). In ﬁrst order, we assume Rs = 0. A small signal conductance G = dJlight corr /dV is calculated from the experimental data. After some algebra, we obtain Rs G q 1 − G (2.182) = Jcorr AkT 1 + Rs Gsh L Thus, a plot of G/Jlight corr versus G gives a straight line with abscissa intercept (1 + Rs Gsh )/Rs and ordinate intercept q/AkT. Then, a second order approximation calculates Jlight corr including Rs . This procedure also applies to the dark JV curve. The impact of series resistances can completely be avoided by measuring pairs of Jsc and Voc at different jE . For V = Voc , Eq. (2.176) can be rewritten as −Jsc η(Voc ) = J0 exp{qVoc /AkT} + Voc Gsh

(2.183)

The simulated and corrected data [−Jsc η(Voc )] in Figure 2.44 closely follow the (Jlight − Jsc η(V)) curve except for the series resistance effect. The function ln(−Jsc η(V)) versus Voc can be evaluated by curve ﬁtting where J0 , A, and Gsh are ﬁt parameters. Graphically, the saturation current density is obtained from the extrapolation of the ln[−Jsc η(Voc )] curve at * Voc toward Voc = 0, that is, according to d ln[ − Jsc (Voc )η(Voc )] lnJ0 = ln[ − Jsc (Vˆ oc )η(Vˆ oc )] − Vˆ oc dVoc ˆ oc V = ln[ − Jsc (Vˆ oc )η(Vˆ oc )] −

qVˆ oc AkT

(2.184)

Vˆ oc lies on the linear part of the ln(−Jsc η(V)) curve. The other possibility is to calculate a corrected current density according to −Jsc corr = −Jsc η(V) − Voc Gsh and prepare a plot ln(−Jsc corr ) versus Voc . This plot yields a straight line with slope corr dln −Jsc q = AkT and with ordinate intercept lnJ0 .19) Figure 2.45a shows A factors dVoc determined in the dark and from Jsc (Voc ) plots for the two devices in Figure 2.44. Device 1 with τn0,p0 = 100 ps is limited by SCR recombination and accordingly it has A ≈ 1.9. We remind readers that, for SCR recombination, the deviation of the A factor from the value of 2 (derived in Section 2.4.5.1) is due to the electric ﬁeld varying with applied voltage. Device 2 with τn0,p0,a = 100 ns shows a noticeable deviation between the diode quality factors from the dark JV curve and the Jsc (Voc ) plot. Due to the changing slope of the lnJdiode (V) curve, this curve is not easy to model (although it has been simulated and was not subject of experimental errors). Before we proceed to evaluate J00 and Ea , we investigate the error in the determination of the diode quality factor if the voltage dependence of the external collection 19) If the recombination mechanism changes

at high voltage bias (see, e.g., Figure 2.29), this change may not be reﬂected in the due to the limited range Jsc η(Voc ) curve

of Vˆ oc jE100 . It is recommended to use

solar simulated energy ﬂuxes larger than j100 E in order to determine the slope at Vˆ oc jE100 . This slope yields the diode parameters which are decisive for the solar cell operation.

2.0 1.8

100 ps

1.6 1.4 1.2

100 ns

−20

100 ps

−30 100 ns

−40 −50 −60

1.0 200

(a)

In(J0 /A cm−2)

Diode quality factor A

2.6 Device Analysis and Parameters

240

280

320

T [K]

20 (b)

30 40 50 (AkT)−1 [eV]−1

60

Figure 2.45 (a) Simulated diode quality factors as a function of temperature for dark JV curves Jdiode (V) (––) and −Jsc (Voc ) curves (❡) for devices as in Figure 2.48. (b) Calculated values of ln(J0 ) versus (AkT−1 obtained by ﬁtting simulated Jdiode (V) (––) and -Jsc (Voc ) (❡) curves.

efﬁciency is not taken into account. Let A and A be the diode quality factors obtained from the slopes of ln(Jlight + Jsc )(V) and ln[−Jsc (Voc )] curves determined at the positions Vˆ and Vˆ oc , respectively. In other words, A and A are from illuminated JV and −Jsc (Voc ) curves not corrected by η(V). It is not difﬁcult to show [93] that these tentative factors A and A need to be corrected. It is AkT d ˆ ln η (V) and A = A η V 1− q dV ˆ V AkT d ln η (V) A = A 1 − (2.185) q dV Vˆ oc Typically, it holds that A > A > A ; and A deviates more strongly from A than A . This is the reason why evaluation of noncorrected Jlight + Jsc gives misleading results but a −Jsc (Voc ) evaluation without η(V) correction sometimes is sufﬁcient. An exception is a cell with a back contact barrier where A becomes much smaller than A (see Section 6.1.6). With the knowledge of J0 and A determined at different temperatures it is possible to determine J00 and Ea by aid of Eq. (2.47). For the sake of generality, we assume that the diode quality factor could be temperature dependent. Table 2.1 reveals that this can be the case for recombination via a defect distribution or by tunneling enhanced recombination. Rewriting Eq. (2.47) yields ln(J0 /J00 ) = Ea /(A(T)kT)

(2.186)

All reference currents in Table 2.2 are only weakly temperature dependent. That is, none of the expressions for J00 does exhibit an exponential temperature dependence. Thus, a plot of ln J0 versus (A(T)kT)−1 20) approximately has the slope of the activation 20) Note that the correction of the kT scale by

A(T) appears necessary. A plot of AlnJ0 versus (kT)−1 is not appropriate for the derivation of Ea in case that A is temperature d dependent. Then, it is −1 A lnJ00 = 0. d (kT)

111

112

2 Thin Film Heterostructures

energy Ea and the ordinate intercept ln (J00 ). In Figure 2.45b we see that regardless of the carrier lifetime, all curves of ln (J0 ) exhibit the same slope corresponding to the bandgap of the absorber. This is expected since Ec is selected to zero and Fermi level pinning is excluded. Finally, we may summarize what we consider to be an appropriate procedure how to determine the device parameters of chalcogenide solar cells particularly in the case where η(V) shows a noticeable voltage dependence: 1) Determine the external collection efﬁciency η(V) from appropriate pairs of Jlight n (V), Jlight m (V). 2) Measure −Jsc (Voc ) at different jE and calculate −Jsc (Voc )η(Voc ) with η(Voc ) from step (1). 3) Determine Gsh from −Jsc (Voc )η(Voc ) or from Jdiode (V). 4) Obtain the light values of J0 and A from a ﬁt of ln[−Jsc η(Voc )]. 5) Determine Rs from Jlight (V)η(V) or from Jdiode (V). 6) Obtain the dark values of J0 and A from a ﬁt of Jdiode (V) 7) Determine A(T), J0 (T), Voc (T) from steps 1–6 performed at different temperatures. 8) Obtain Ea and J00 from a ﬁt of lnJ0 versus [A(T)kT]−1 . It is instructive to ﬁnally compare the dark and illuminated values of Ea , A, and J00 . If these values greatly differ, the recombination mechanism may be illumination dependent. 2.6.2.3 Open Circuit Voltage By deﬁnition, the open circuit voltage Voc is the voltage the cell develops at zero load. Figure 2.46b,c shows different heterostructures under open circuit condition. Although, the partial current densities of electrons and holes can be large, the total current density Jtot (z) = Jn (z) + Jp (z)