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) is zero everywhere in the device. The value of Voc is given by the difference between the quasi Fermi energy of holes at the absorber contact and the quasi Fermi energy of electrons at the emitter contact. Thus, it is

Voc = 1/q[EFn (de ) − EFp (−da )]

(2.187)

Within the absorber, the splitting of the quasi Fermi energies may be even larger than at the contacts. This can be recognized for the device in Figure 2.46c which exhibits the highest splitting of the quasi Fermi energies within the absorber. This maximum splitting of EFn and EFp cannot be translated into a measurable Voc . The electron quasi Fermi energy drops off toward the absorber/buffer interface due to high interface recombination. The splitting of the quasi Fermi energies can be sensed by evaluation of the spectral luminescence using Planck’s generalized law [94]. For not too high illumination (excess majority carrier concentrations being small compared to the dark concentrations), Voc can alternatively be expressed as L (dϕ/dz − dϕ0 /dz)dz (2.188) Voc = 0

2.6 Device Analysis and Parameters

Ev

1

(a)

1015 1010

Ev

1 0

p

n

Jtot

0.4

Jp

0.2

Jn

0.0 −1.5 −1.0 −0.5

J [mA cm−2] n,p [cm−3]

J [mA cm−2] n,p [cm−3]

0

2

EFp

EFn

0.0

Position z [µm]

1015 n 1010 Jn 20 10 Jtot 0 Jp −10 −20 −1.5 −1.0 −0.5 0.0

(b)

Figure 2.46 (a) Default device at +0.6 V forward bias. (b) Default device under AM1.5G illumination at Voc . Device parameters as in Table 8.1. (c) Device with Ea,b c = −0.3 eV under AM1.5G illumination

3 2

EFp Ev

1

Position z [µm]

(c)

p 1015 n 1010 20 J Jtot p 10 0 −10 Jn −20 −1.5 −1.0 −0.5 0.0 Position z [µm]

at Voc . Other device parameters of (c) as in default Table 8.1. The steps in Jn,p at the interface are due to the dominating interface recombination limiting the Voc of the device in (c).

where ϕ and ϕ0 are the electric potentials under illumination and in the dark. Fonash [22] showed that, by use of Eqs. (2.5−2.12) and with the assumption of position independent mobilities, Eq. (2.188) can be rewritten as L dEg 1 L σn dEc σ 1 L σp dEc F0 dz − dz − − dz Voc = − σ q 0 σ dz q 0 σ dz dz 0 L L σp dlnNv σn dlnNc kT dσn 1 dσp kT − dz + − dz − q 0 σ dz σ dz q 0 σ dz dz (2.189) where F0 is the electric ﬁeld under equilibrium, σn = qµn n and σp = qµp p are the excess conductivities at Voc . Equation (2.189) tells that there are different contributions to the open circuit voltage of a device. The ﬁrst term on the right hand side of Eq. (2.189) is the contribution of the electrostatic potential. The second to fourth terms are the so-called effective force terms. These are contributions to the open circuit voltage due to variations of conduction band energy, energy gap, and density of states within the device. The ﬁfth term is a diffusion term and is denoted as the Dember voltage [22].21) 21) The term ‘‘Dember voltage’’ may be mis-

leading since here it does not require

Ec

0 p

J [mA cm−2] n,p [cm−3]

2

3

4

Ec

EFn Energy E [eV]

Energy E [eV]

EFp

3

4

Ec

EFn

Energy E [eV]

4

113

differences in the electron and hole mobilities.

2 Thin Film Heterostructures

-qϕ

1 Energy E [eV]

0

EFn

-qϕ0

3.0 EFp

2.0 1.0

F, F0 [V cm−1]

0.0 0 −2

F

F0

−4 −6 × 104

∆σn /σ

1.0 ∆σn,p / σ

∆σp /σ 0.5 0.0 105

d∆σp /dz

+ 100 ↑ 100 ↓

−

± d∆σn,p /dz [S/cm]

114

105 −0.8

d∆σn/dz −0.4 0.0 Position z [µm]

Figure 2.47 Default heterostructure of the absorber/buffer/window type under open circuit condition. Plots of potential energy, band edges, electric ﬁeld, excess conductivities, and carrier gradients have been

0.4

obtained from device simulation. Device parameters as in the default Table 8.1. Note that positive and negative values of dσp /dz and dσn /dz are printed upwards and downwards on a log scale.

In order to evaluate the integrals in Eq. (2.189), it is necessary to know the electric ﬁeld in the dark and the excess conductivities. However, σn and σp depend on the voltage which has been developed due to illumination, that is, σn (Voc ) and σp (Voc ). Thus Eq. (2.189) is an implicit function and its use is limited for contemplation rather than for prediction of the open circuit voltage. In Figure 2.47, relevant quantities for Eq. (2.189) are plotted for a typical heterostructure solar cell. Since this heterostructure exhibits step-like variations of Eg , one can initially

2.6 Device Analysis and Parameters

assume that the Voc has a considerable contribution of the third term in Eq. (2.189). However, this is not the case. Due to the overlap of σn,p and F0 , the ﬁrst term in Eq. (2.189) is the dominant one. Also the Dember term has a noticeable size due to the large values of dσn and dσp . The effective force ﬁeld (the third term) amounts to only some millielectronvolts, since dEg /dz (which is in the range of 108 eV cm−1 ) is limited to the heterointerfaces. Extended bandgap gradients which may be implemented in the absorber to promote carrier collection, also contribute very little to the open circuit voltage. The reason is that either these gradients of the band edges are small compared to the gradient of the electric potential (for instance in the case of a front surface gradient) or they do not coincide with high values of σn,p (for instance in the case of a back surface gradient). A second (and more useful) approach to describe the open circuit voltage developed by a solar cell under illumination employs the principle of superposition (see comment in Section 2.4) which is formulated in Eq. (2.176). Under open circuit condition, where J is zero, it is Voc =

Ea AkT −Jsc η(Voc ) Voc Gsh − + ln q q J00 J00

(2.190)

A shunt conductance of 10−3 S cm−2 is 102 times smaller than the quotient Jsc /Voc of a typical cell. Thus, under one-sun illumination, the second term in the bracket can safely be neglected, giving Ea AkT −Jsc η(Voc ) Voc = + ln q q J00

(2.191)

with the derivative Ak J00 dVoc = − ln dT q −Jsc η(Voc )

(2.192)

Here, it is assumed that J00 and −Jsc η(V) are not temperature dependent in the temperature range of interest. Furthermore, the derivative dEg /dT is neglected. As discussed in Section 2.4, the activation energy, Ea , of the diode current depends on the dominant recombination process. Equation (2.191) shows that Ea can be obtained from a plot of Voc versus T. The intercept with the ordinate (T = 0 K) of a tangent line to the Voc versus T curve is interpreted as Ea /q according to Ea /q = Voc

T=0

= VocT=T0 −T0

dVoc = T0 dT T

(2.193)

where T0 can be around RT. Application of Eq. (2.193) is straightforward as long as the diode quality factor and the reference current density are temperature

115

116

2 Thin Film Heterostructures

independent.22) In case of a temperature dependent diode quality factor, Eq. (2.193) reads

VocT=0 =

Ea dA kT0 2 -Jsc η(V) − ln q q J00 dT T=T0

(2.194)

Even if A = A(T) can be determined from analysis of the JV curve, evaluation of Eq. (2.194) is difﬁcult since it requires knowledge of J00 . Thus, in the case of a temperature dependent A factor, evaluation of J0 (T) is recommended (see Section 2.6.1). The temperature dependence of the open circuit voltage as described by Eq. (2.192) is very important for the actual module efﬁciency under operating conditions. (dVoc /dT)T=RT (where RT = room temperature) has been calculated and plotted in Figure 2.27c. The full line describes the regime of SCR recombination while the dashed line of QNR recombination. (dVoc /dT) decreases with increasing carrier lifetime. Also of importance under operating conditions is the dependence of Voc on the illumination intensity. Again, this can be described by Eq. (2.191). Figure 2.48 shows examples of calculated and simulated values of Voc for QNR and SCR recombination. It is the A factor which causes a faster decline of Voc for SCR recombination. Note that no change in the solar spectrum due to lower sun inclination has been taken into account. A shunt conductance smaller than 10−4 S cm−2 (Rp > 10 kcm2 ) has no noticeable inﬂuence on the Voc (PI ) curves in the range from 100 to 1mWcm−2 . Also, a shunt conductance of 10−3 S cm−2 appears to be tolerable as long as the solar device is designed for energy harvest in an outdoor application. For indoor use, however, the data in Figure 2.48 need a shunt conductance smaller than 10−4 S cm−2 . It is noted that the illumination dependence of Voc and Jsc can be used to determine the saturation current density and diode quality factor of a device (see Section 2.6.1). 2.6.2.4 Fill Factor The cell delivers the maximum output power at a voltage Vmpp and a current density Jmpp (MPP =maximum power point). The ﬁll factor describes the relationship between Voc , Jsc , and the power at the MPP

FF = Vmpp Jmpp /Voc Jsc 22) Device simulation programs may use tem-

perature dependent material parameters such as τn,0 (T), τp0,a (T), Nv,a (T), or Nc,a (T). According to semiconductor theory, Nv,a and Nc,a vary with temperature as Nv,a = N0v,a (T/T0 )3 and Nc,a = N0c,a (T/T0 )3 . In a

(2.195) device simulation program this power law may be referenced to T0 = 300 K. It increases the slope of the Voc (T) curve by about 3kT0 in the case of SCR and QNR recombination (Burgelman, M., personal communication).

2.6 Device Analysis and Parameters

QNR

0.7

Voc [V]

0.6

SCR

0.5 0.4 Rp

0.3 2

0.1

4

6

2

4

6

1

2

4

6

10

100

−2

PI [mW cm ] parameter used is the parallel resistance of Rp (k cm2 ) = 1, 10, and [∞]. Calculation for a AM1.5G spectrum of variable integral intensity. Experimentally this corresponds to the use of gray ﬁlters.

Figure 2.48 Calculated values of Voc for QNR (τn0,p0 = 10−8 s) and SCR recombination (τn0,p0 = 10−10 s) limited devices at 300 K as a function of illumination intensity according to Eq. (2.190). The

For any current–voltage relationship, the following equation holds dJ/dVmpp = −Jmpp /Vmpp

(2.196)

Equation (2.196) describes a speciﬁc property of the MPP. This property can be used to calculate Vmpp for a cell which obeys the superposition principle J = J0 (exp{qV/AkT} − 1) + Jph (V)

(2.197)

Here A is the diode quality factor, J0 is the saturation current density, and Jph (V) is the voltage dependent photo current density. With Eq. (2.196) we obtain qVmpp dJph (V) Jmpp q = J0 (2.198) − exp + Vmpp AkT AkT dV V=Vmpp

with Jmpp = J0 (exp{qVmpp /AkT}) + Jph (Vmpp ) Equation (2.198) can be rewritten as qV mpp −Jph (Vmpp ) − J0 exp AkT Vmpp = qV dJ qJ0 ph mpp exp + dV AkT AkT

(2.199)

(2.200)

V=Vmpp

Due to the implicit character of Eq. (2.200), it must be solved numerically in order to determine Vmpp . A requisite for the calculation is the knowledge of Jph (V) which ﬁnally is related, according to Eq. (2.165), to a voltage dependent QE(λ, V). (Jph is a negative quantity.) For the ideal case that dJph /dV = 0 and Jph = Jsc , Eq. (2.200) reduces to q(Voc − Vmpp ) AkT Vmpp = exp −1 (2.201) q AkT

117

2 Thin Film Heterostructures

1.0 0.9

Eg,a =1.8 eV Eg,a =0.8 eV

0.8 FF

118

A=1.0 A=1.1 A=2.0

0.7 0.6 0.5 0.4 0.0

0.5

1.0

1.5

2.0

Voc [V] Figure 2.49 Calculated ﬁll factor values for A = 1.0, 1.1, and 2.0 according to Eq. (2.202). The markers give ﬁll factors as obtained by device simulation for a the absorber/window default device but with varying absorber bandgap energy (Eg,a = 0.8–1.8 eV).

Equation (2.200) can be approximated by Vmpp = Voc − AkT/q ln(qVoc /AkT + 1) [19, 95]. Accordingly, Jmpp can be calculated after Eq. (2.199). The result for the ﬁll factor in this ideal case reads qVoc qVoc AkT − ln AkT + 1 (2.202) FF = qVoc AkT + 1 In Figure 2.49, the ﬁll factor as a function of Voc is plotted as obtained from Eq. (2.202) for diode factors of A = 1.0, 1.1, 2.0. The ﬁll factor increases with increasing Voc . Highest ﬁll factors are obtained for A = 1. The accuracy of the approximation in Eq. (2.202) is tested by comparison to simulated data of a absorber/window heterojunction (simulation parameters are given in the ﬁgure caption). In the simulation, the bandgap, Eg,a , of the absorber is varied from 0.8 to 1.8 eV. In this bandgap range, Voc linearly increases with Eg,a . All simulated JV curves exhibit a diode factor A of 1.1. It is apparent that the simulated data indeed resemble the calculated curve for a diode factor A = 1.1. The FF(Voc ) dependence as depicted in Figure 2.49 is taken into account if the solar energy conversion efﬁciency is calculated as a function of the optimum bandgap. Apparently, high bandgap materials are favored from the ﬁll factor point of view. However, this effect cannot overcompensate the decline of the photo current density at higher bandgaps. Looking at Eq. (2.200), we may ask for the inﬂuence of a voltage dependent QE on the ﬁll factor. There is, however, no simple relation between dJph (V)/dV and FF since Jph is nonlinearly dependent on the voltage, that is, dJph (V)/dV = f (V). An instructive example is the inﬂuence of a voltage dependent SCR. In the SCR the collection function for charge carriers ηc is unity, that is, all generated charge carriers are collected (consider the graphs in Figure 2.33 as representative for the nonbiased case). If the applied voltage bias changes, the width of the SCR changes.

2.6 Device Analysis and Parameters

In the ideal case, the width of the SCR is proportional to (V)1/2 . Since the collection function in the SCR is unity but declines beyond SCR, a variation of the SCR changes the collection function in a nonsimple fashion. As a result, the derivative dJph (V)/dV depends on the voltage. 2.6.3 Capacitance–Voltage Analysis

Measurement of capacitance as a function of voltage is a simple and widely used technique [96]. The capacitance is derived from the imaginary part of the measured admittance Y, that is, C = Im(Y)/ω. Identifying this capacitance with the SCR capacitance directly assumes that the correct equivalent circuit is a capacitance and a conductance in parallel. However, as revealed by Figure 2.42, the equivalent circuit can be more complicated due to capacitance contributions from deep defects, interface defects and additional resistance elements. We have to select conditions for the C–V measurement which allow to simplify the equivalent circuit and to facilitate the interpretation of the capacitance. The geometrical capacitance of the depletion region of an absorber/buffer/ window heterostructure is given by 1 C= w a/ + wb/ + ww/ εa εb εw

(2.203)

In the speciﬁc case of a highly doped window layer (ww ≈ 0), a fully depleted buffer layer (wb = db ), and without deep bulk states in the absorber, the quantity wa is given by Eq. (2.36). We want to make use of this Eq. (2.36) in order to show how to obtain the built-in potential Vbi and the doping density NA from a C–V measurement. For the speciﬁc case, Eq. (2.203) becomes 1 db κIF −1 db 2 2 q2 ND,b db 2 = + + 2 [q(Vbi − V)] + + vIF z2 C εb κIF εb κIF q NA,a εa κIF 2εb (2.204) In the Mott–Schottky approach 1/C2 is plotted versus V. Thus, we have to consider the expression 2(Vbi − V) db 2 1 εb ND,b κIF − 1 2 2νIF z2 1 = + 2 + + + 2 C2 qNA,a εa κIF εb κIF κIF 2 εa NA,a q NA,a εa κIF q2 ND,b d2b db κIF − 1 2 db 2 +2 + 2 q (Vbi − V) + + νIF z2 εb κIF εb κIF q NA,a εa κIF 2εb (2.205) The ﬁrst term on the right hand side of Eq. (2.205) resembles the C–V dependence of a simple n+ p heterojunction where the SCR only extends into the p-type absorber. The second term accounts for the geometrical capacitance contribution of the buffer layer. The third term represents the inﬂuence of interface charge. In the case of a ﬁxed charge density at the buffer/absorber interface, it is κIF = 1

119

120

2 Thin Film Heterostructures

1/C2

NA,a

db, NIF+

Vbi

V

Figure 2.50 Schematic 1/C2 versus V plot illustrating the effect of increasing absorber doping concentration, NA,a , and a potential drop at the buffer layer or positive interface charge, N+ IF . The case of negative interface charge is not depicted.

(see Section 2.3.2), and the fourth term on the right hand side of Eq. (2.205) vanishes. Then, 1/C2 becomes linear in V which allows to extract NA,a from the slope of the 1/C2 versus V curve according to + −1  1 2 d C 2   NA,a = − (2.206) qεa dV Obviously, a large slope of 1/C2 indicates a small value of NA,a and vice versa (see example in Figure 2.50). If we allow the (shallow) doping density NA,a to vary in the SCR, Eq. (2.206) can be used to determine the doping proﬁle NA,a (z) by evaluating the slope of 1/C2 at different voltages. This gives NA,a (V), which is translated to NA,a [wa (V)] by aid of Eq. (2.203) and the measured relation C(V). Equation (2.205) is also valid for a continuous charge density at the buffer/absorber interface. Such a density dNIF /dE makes the coefﬁcient κIF = 1 + νIF and leads to a nonlinear and curved 1/C2 versus V plot. The deviation from linearity is understandable since upon applied voltage bias, the interface charge density changes, leading to a varying potential distribution in the different layers. We return to the case of κIF = 1 (ﬁxed interface charge) where the built-in potential Vbi can be inferred from db 2 d 1+ νIF z2 εb ND,b + 1+ Vbi + 2 − C2 εb dV εa NA,a q + 1+ 2 C+ + V = V 1 C2 = 0 = (2.207) d 1 2 − dV C From the extrapolation of 1/C2 to 1/C2 = 0, Vbi can be obtained provided that the parameters of the buffer layer and the interface charge are known. A buffer layer as well as positive interface charge leads to an extrapolated voltage value which is larger than the built-in potential (see Figure 2.50). Accordingly, a negative interface charge leads to a smaller voltage value. For our default device with a 50 nm buffer

2.6 Device Analysis and Parameters

Ec Ed,A Ev -wa

-zd

0

Position z Figure 2.51 Schematic illustration of the effect of a deep acceptor trap in the bandgap of an absorber with energy Ed . The states of density Nd,A add to the negative space charge in the range zd < −z < 0. This space charge is not considered in Eq. (2.36) and thus distorts the measured NA,a . As a

second inﬂuence, charging and discharging of the deep states at position −zd leads to an additional admittance contribution and thus to the measured capacitance. The latter effect may be circumvented by freezing out the response of deep states.

layer, the second term in Eq. (2.207) amounts to below 5% of Vbi and therefore is in the range of the accuracy limit of the C–V method. Note, however, that due to the parabolic dependence on db the second term rises fast with increasing db . If Vbi has been obtained from a C–V measurement, the band offsets can be derived provided that the dopant concentrations in the different layers are known. The basis for this procedure are the Eq. (2.22) for the absorber/window and Eq. (2.33) for the absorber/buffer/window heterostructures. In the last step, we have to account for the inﬂuence of deep states on the C–V measurement. Consider an absorber with deep acceptor states as shown in Figure 2.51, which is part of a pin+ structure (absorber/buffer/window). As described in Section 2.3.6, the charge in the deep traps will inﬂuence the potential distribution in the device and, in particular, the SCR width of the absorber. This is the static part of the inﬂuence of deep traps. In addition, charging and discharging of the traps with the frequency of the measurement, ωAC , can add to the measured capacitance. This is the dynamic part of the inﬂuence of deep traps. As explained in the next section, the defect response is maximal at the position where it is Ed = Ep,a (z). This is at the location −z = zd . Three quantities are important to discriminate different measurement regimes: (i) the emission rate of the deep traps ω0 = 2/τe where τe is the emission time constant, (ii) the frequency of the AC voltage ωAC , and (iii) the sweep rate of the DC bias voltage ωDC . In a C–V measurement, we normally have ωAC > ωDC . In addition, we must always guarantee ωAC < τ−1 d = σa /εa where τd is the dielectric relaxation time and σa is the conductivity. Otherwise, the measured capacitance corresponds to the sample geometry. The different regimes for a C–V measurement are: 1) ωAC > ωDC > ω0 . The deep trap can follow neither the AC frequency of the capacitance measurement nor the loading/disloading of the DC bias change. In that case, the slope of the 1/C2 curve gives the shallow absorber doping density NA,a [97].

121

122

2 Thin Film Heterostructures

2) ωAC > ω0 > ωDC . Due to the high AC frequency, the dynamic response of the deep state does not add to the capacitance and can be neglected. From the slope of the 1/C2 curve we obtain a larger doping concentration with value NA,a + Nd,A zd /wa provided that both state densities are homogeneous ) in the Ed −Ep,a zd SCR. This is the static inﬂuence of deep states. If we replace wa = 1 − q(V −V) bi

[98] and measure the 1/C2 proﬁle at different temperatures, NA,a and Nd,A can be obtained by curve ﬁtting [99]. 3) ωT > ωAC > ωDC . If the AC frequency is low enough to charge/discharge the deep states at the position −zd , evaluation of the 1/C2 curve after Eq. (2.206) gives the sum of shallow doping concentration and deep state density, that is, NA,a + Nd,A . The task now is to ﬁnd the appropriate measurement conditions for one of the cases described above. This can be accomplished by aid of admittance spectroscopy. 2.6.4 Admittance Spectroscopy

Spectroscopy of the device admittance is a technique for the measurement of doping density, deep defect levels, and other electronic properties. By measuring the small-signal AC admittance of the junction under different conditions, for example, with frequency and temperature as parameters, defect states in the SCR [100] and at the interface [85] can be analyzed. First, we consider a single defect state in the SCR of the absorber. We refer to the Gaussian acceptor defect in the absorber of the default device (Table 8.1) with parameters Nd = 1014 cm−3 , Ed = 0.4 eV, and σd = 0.05 eV. The defect can communicate with the valence band by the transactions illustrated in Figure 2.17. As the state is in the lower part of the band gap, communication with the conduction band can be neglected. Charging and discharging of the state will give a certain admittance contribution. As for the C–V measurement, we interpret the imaginary part of the admittance in terms of a capacitance. In the capacitance are lumped the contributions of the SCR and of the defect. Figure 2.42c reveals that the capacitance Cd of a defect state in the absorber SCR is in parallel with the SCR capacitance, Ca . We note, however, that if there is a large series resistance, the equivalent circuit has to be extended [101] and the capacitance has to be extracted by mathematical manipulations. Charging and discharging of the defect state is governed by the rate expressions in Eq. (2.65). At the position where the Fermi level crosses the defect state, rates II and IV of Eq. (2.65) will be equal and defect charging and discharging will be most efﬁcient. In Figure 2.51, we have marked the position where it holds τc,p = τe,p = τp0 by −zd . For small enough frequency, the defect occupation is governed by the corresponding quasi Fermi level and can be charged and discharged with frequency ω. However, for too large a frequency charging and discharging of the defect can no longer follow the AC frequency. The border between the low frequency and high frequency regime is denoted as ω0 .

28.8 CSCR ω0

28.7 103

104

105

Frequency ω

(a)

Nd [cm−3 eV−1]

CSCR + Cd

-dC/dInw [a.u.]

Capacitance C [nF cm−2]

2.6 Device Analysis and Parameters

3 × 1014

Figure 2.52 (a) Simulated capacity of the default absorber/buffer/window device from Table 8.1 as a function of frequency and the capacity derivative. For frequency ω < ω0 , charging and discharging of the defect state

1 0 0.0

106 [s−1]

2

(b)

0.5

1.0

E [eV]

represents a capacity contribution, Cd , which adds to the capacity of the SCR, CSCR . (b) Evaluation of the defect density Nd according to Eq. (2.214).

As shown in Figure 2.52a, C(ω) represents a step-like function. The characteristic frequency ω0 can be determined by a maximum in the plot −dC(ω)/dlnω versus ω. For Nd NA,a it is [102] Cd (ω) ∝

ω20

ω20 + ω2

(2.208)

Furthermore, the derivative dC/dlnω is a measure of the defect density. ω0 is related to the emission time constant τp0 according to [102] ω0 = 2/τp0 = 2ν0 exp{−Ed /kT}

(2.209)

where ν0 is the attempt to escape frequency, ν0 = Nv,a vp σp . σp is the (hole) capture cross section of the defect. With the temperature dependences of vp ∝ T1/2 and Nv ∝ T3/2 we can write ν0 = σp vp Nv,a = ξ0 T2

(2.210)

which yields ω0 = 2ξ0 T2 exp{−Ed /kT}

(2.211)

With the deﬁnition of the effective density of states and vp = (3kT/mp )3/2 , the emission factor ξ0 for emission in the valence band comes out as ξ0 =

√ √ A∗ m p k2 6π4π 3 σp = 6π σp h q

(2.212)

with A∗ , the effective Richardson constant. Thus, from an Arrhenius plot of ω0 /T2 versus T−1 , the defect energy Ed can be determined. This is shown in Figure 2.53d for the bulk defect (). We remind that in this work the defect energy Ed is related to the valence band. Thus, in the case of a defect in the lower part of the band gap, the activation energy of the ω0 /T2 versus T−1 plot directly gives Ed . We note, however, that in principle an admittance experiment gives the same response for a

123

124

3.5 3.0

En,az=0

EF Ev

2.5 2.0 (a)

200 K

28

27

0.0

Position z [µm]

0.2

0

1 2 × 1010 −2

(b) NIF [cm

−1

eV ]

Ea=Ed 10−1

T 100 K

103 (c)

100

10−2 26

−0.4 −0.2

101

29 ω0 /T2 [s−1 K−2]

Capacitance density [nF cm−2]

Energy E [eV]

4.0 Ed

Ea=En,a z=0

300 K

Capacitance density [nF cm−2]

32

Ec

104

105

106

Frequency ω [Hz]

(d)

T

−1

−1

[K ]

+V 28 −V 26

24 103

4 6 × 10−3

2

200 K

30

(e)

105

Frequency ω [Hz]

Figure 2.53 Figure 2.53(a) Band diagram of the absorber/buffer/window default solar cell but with a constant density of donor states at the absorber/buffer interface. Ed marks the Gaussian-broadened acceptor-type defect state in the absorber band gap. (b) Interface state density NIF . (c) Simulated capacitance density spectra for different temperatures. The capacitance spectra exhibit a frequency dispersion due to the capacitance contribution of the interface states () and of the bulk state (◦). Markers give the characteristic frequencies. (d) Arrhenius plot of ω0 . The slope Ea corresponds to the intersection of EF with the interface state density, that is, to En,az=0 , and to the defect energy Ed . (e) Capacitance spectra simulated at 200 K, but with different bias voltage. The shift of the inﬂection point ω0 upon eletric bias reﬂects the variation of the electron quasi Fermi level at the interface.

2 Thin Film Heterostructures

4.5

2.6 Device Analysis and Parameters

HT and an ET. The activation energy of ω0 is simply the energetic distance between the band edge of the carrier involved and the location of the defect level in the band gap. Discrimination between ETs and HTs is possible by deep-level transient spectroscopy. The characteristic frequency, ω0 , deﬁnes the border between a low frequency capacity CLF and a high frequency capacity CHF . In Figure 2.52a it is CLF = Cd + CSCR and CHF = CSCR . The difference is the capacity of the defect response. Besides from a deep defect state, a capacitance step can also arise from an energy barrier at a contact [103], from the dielectric relaxation in a bulk layer, and from interface states (see below). Fermi level pinning resulting from high concentrations of deep-level defects can distort the measured activation energy and the apparent capture cross section [104]. Walter et al. [102] generalized the admittance theory of Losee [100] to a continuous defect densities. Now, the Fermi level intersects the defect density in an extended region which depends on the type of band bending. For a p-i-n diode where the built-in voltage drops linearly in the i layer of thickness w, the defect density can be calculated after [102] Nd (Ed ) =

dC ω Vbi 2

wa qVbi − Ep,a − Ed dω kT

(2.213)

In contrast, one ﬁnds for a pn+ diode with parabolic band bending in the p layer 3/2

Nd (Ed ) = −

2Vbi dC ω ,

dω kT wa q qVbi − Ep,a − Ed

(2.214)

Thus, evaluating the −dC/dlnω curve in Figure 2.52a yields the defect density as a function of energy. In Figure 2.52b, the ordinate results from Eq. (2.214) [102] while the abscissa is obtained from E = ln(ω/2ν0 ). Here, ν0 includes the unknown quantity ξ0 which, however, can be ﬁtted by bringing Nd (E) curves of different temperatures into congruency. This is shown in Figure 2.52b where a plot of Nd as a function of E gives a defect band of width 2σ = 0.1 eV and with a maximum at 0.4 eV – in accordance with the Gaussian defect in the default device (Table 8.1). Also interface states can be charged and discharged by an AC stimulation and thence can contribute to the cell’s capacitance. A prerequisite is that the Fermi energy intersects the interface state density. The equivalent circuit of a device with interface states is depicted in Figure 2.42d. The equivalent circuit of a device with both deep bulk defect and interface defect states is a combination of Figure 2.42c and d. In the fortunate case where the bulk defects are deeper than the interface defects, the equivalent circuit is again reduced to Figure 2.42d. In the example of Figure 2.53, we add a continuity of donor type interface states to our default absorber/buffer/window heterostructure. At sufﬁciently low frequency, electrons are captured and emitted by interface states located at the Fermi energy and thus contribute to the cell’s capacitance. If the AC frequency exceeds the emission rate, τ−1 e,n of the interface states at En,az=0 , the interface states can no longer follow.

125

126

2 Thin Film Heterostructures

The limiting frequency again is interpreted using Eq. (2.211) where Ed equals the quantity En,az=0 . From an Arrhenius plot we obtain Ea = En,az=0 , as shown in Figure 2.42e. In case that the Fermi level is not pinned, the capacitance step CLF − CHF is related to the interface state density – in the simplest case it is CLF − CHF = CIF = qNIF . However, in the case where the Fermi level is pinned the capacitance step has a geometrical meaning, as explained by Niemegeers et al. [79] and cited in the following. In the low frequency range, electrons are supplied from the window through the buffer to the buffer/absorber interface. The AC potential builds up between the buffer/absorber interface and the edge of the absorber SCR. The low frequency capacitance density is given by CLF =

εa wa

(2.215)

If the interface states can no longer follow the applied AC signal (ω > ω0 ), the AC potential builds up between the edge of the emitter SCR (for high window doping this is close to the buffer/window interface) and the edge of the absorber SCR. That is 1 (2.216) CHF = w a/ + we/ εe εa where the index e refers to the emitter (window + buffer, or only window). The step in the capacitance frequency curves thus is related to the space charge width we 1 1 we ≈ εe − (2.217) CHF CLF In case of a highly doped window, a depleted buffer, and large NIF the quantity we about equals the buffer thickness. Measurement of En,az=0 by admittance spectroscopy can be obscured by the dielectric relaxation effect in the least conductive layer of the heterostructure. In the example in Figure 2.53 this can be the buffer layer since it may be depleted from carriers. In this case, ω0 is determined by the inverse of the dielectric relaxation time τd of the electrons in the buffer layer [79] ω0 = ωd = 1/τd = qµn n/εb . With n = Nc,b exp{−(En,b )} it is obvious that ω0 is still temperature activated and we may detect frequency dispersion of the capacitance. An Arrhenius plot of ω0 yields the quantity En,b for the buffer layer. In the case where the band offset Ec a,b is known, we can calculate the difference between conduction band and Fermi energy in the absorber from En,az=0 = En,b − Ec a,b . The parameters in Figure 2.53 are chosen, however, to make sure that the buffer is sufﬁciently conductive and ωd is very large. Bulk defects are continuous in space but may show a narrow energy distribution. In contrast, interface state densities are discrete in space and (often) continuous in energy. How can we discriminate between an homogeneous bulk defect and a conﬁned interface defect? If the bias voltage on the sample is changed, the

2.6 Device Analysis and Parameters

bulk defect state crosses the Fermi energy only at a different position within the SCR – but the limiting frequency remains the same. However, due to applied bias, the interface state density is intersected by the quasi Fermi energy at a different energy level – provided that the Fermi level is not pinned. Thus, the limiting frequency of a device with a nonpinned Fermi level changes. This is exempliﬁed in Figure 2.53f. In conclusion, if the limiting frequency ω0 is bias independent, the frequency dispersion of the capacitance may be due to a bulk defect or due to interface states plus Fermi level pinning.

127

129

3 Design Rules for Heterostructure Solar Cells and Modules The solar conversion efﬁciency of a heterostructure cell is determined by the material parameters of the single layers as well as by device parameters. By device parameters we mean band alignment, doping ratios, ﬁlm thicknesses, and other parameters. These device parameters have in common that they determine the layout of the band diagram. In this chapter, we conduct design studies based on calculated or simulated solar cell parameters. We consider a related sequence of examples which then give rules for the heterostructure design. It is not so much the absolute number of a solar cell parameter which is important, but how the cell performance depends on this parameter in general. The speciﬁc band diagram of a heterostructure inﬂuences all four recombination paths of Figure 2.46. The Fermi level position at the interface affects interface recombination. The size of the electric ﬁeld inﬂuences space charge region (SCR) recombination. Bandgap gradients play a role in quasi neutral region (QNR) recombination. Electric ﬁeld and ﬁlm thickness determine the impact of back surface recombination. An extreme example is recombination path (1) which, on the one hand, can strongly deteriorate the cell performance but, on the other hand, can largely be deactivated by a proper design of the band diagram – even in the presence of interface states. In order to minimize recombination and maximize performance, speciﬁc rules for the design of the band diagram have to be regarded. Here, we discuss these rules based on the different degrees of freedom for the design of a heterostructure band diagram. Table 3.1 depicts the study logic which (with exceptions) is followed in the next sections. Starting with a simple absorber/window heterostructure, the design study step by step adds further device parameters. From the choice of a buffer layer thickness onward, the device is of the absorber/buffer/window type. Note that the ﬁnal outline of the cell is not identical to the default device in all design aspects.

3.1 Absorber Bandgap

The absorber bandgap determines the maximum short circuit current density of the heterostructure (see Figure 2.31b). Decreasing the bandgap increases the 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

130

3 Design Rules for Heterostructure Solar Cells and Modules Table 3.1 List of design studies and derived design rules. Settings denotes device parameters which have been set in accordance with the output of a design study.

Design parameter

Main impact

Rule

Setting

Absorber bandgap

Generation current, maximum splitting of EFn − EFp Interface recombination

1 eV < Eg,a < 1.6 eV

1.2 eV

Ec ≥ 0

Ec = 0 eV Sn0,p0,IF = 105 cm s−1 ND,w = 1018 cm−3 No FLP

Band alignment Emitter doping Fermi level pinning (FLP) Absorber doping

Absorber thickness Grain boundaries (GB) Back contact barrier Buffer thickness

Interface recombination, optical loss Interface recombination

ND,w NA,a

Collection length, recombination region, interface recombination Recombination rate, collection length Effective diffusion length, saturation current

1015 < NA,a < 1017 cm−3

Impedance, recombination rate Photo current, optical reﬂection Interface recombination

Front surface gradient Back surface gradient Volume recombination, back surface recombination

NA,a = 1016 cm−3

da = 1.5 µm Sn0,p0,bc = 107 cm s−1 Sn0,p0,gb ≤ 103 cm No GBs s−1 no horizontal GBs p p ϕb < 0.3 eV ϕb = Ep,az=0 db = minimal Eg, IF > Eg, a , dmin minimal dmin = f (da )

db = 50 nm ND,b = 1015 cm−3 dmin,front = 10 nm, Ev,front = 0.2eV dmin,back = 0.5 µm, Ec,back = 0.3eV

photo current; but, Voc decreases, see Eq. (2.191). It is clear that a plot of solar cell efﬁciency versus absorber bandgap exhibits a maximum. But, is this maximum dependent on the electronic parameters of the absorber or is it a unique function? As examples, we selected two limiting scenarios (Figure 3.1). The ﬁrst scenario only includes radiative recombination (Shockley–Queisser limit) and the second uses recombination in the SCR. Both scenarios have in common that optical absorption in the emitter is neglected. The Voc of the devices have been calculated by use of Eq. (2.191) with J00 for radiative recombination only (see Eq. 7 in Ref. [33]) and for SCR recombination (Table 2.3). The ﬁll factor (FF) has been calculated after Eq. (2.202). Jsc has been obtained from Eq. (2.137) for both scenarios. Thus, even for SCR recombination the Jsc is assumed to be ideal. Such a cell requires excellent light trapping.1) 1) However, we found that if we calculate

Jsc (Eg ) by considering the effect of a limited collection probability in the absorber, the

shape of the Jsc (Eg ) curve in Figure 2.7b is largely unaffected.

3.2 Band Alignment 35 SQL

η [%]

30 25

SCR

20 15 10 0.8

1.0

1.2

1.4

1.6

1.8

2.0

Eg [eV]

Figure 3.1 Calculated maximum solar cell efﬁciencies for AM1.5G (solid lines) and AM0 (dashed lines) in the limit of radiative recombination only (Shockley–Queisser limit, SQL) and for recombination in the SCR with

absorber lifetime τn0,p0 = 10−10 s. For both cases, the full Jsc (Eg ) according to Figure 2.31b was assumed. Calculation without reﬂection loss and without absorption in the emitter layers.

In the Shockley–Queisser limit, maxima of η(Eg ) at 1.15 and 1.35 eV can be discerned. Thus, optimum efﬁciency can be expected for an absorber bandgap of 1.4 eV. (The ripples in the η(Eg ) curve stem from the absorption bands in the AM1.5G spectrum.) The curve for the AM0 spectrum has a single maximum at 1.3 eV. The picture, however, changes if the absorber is in the limit of SCR recombination. Now, the maximum of η(Eg ) is shifted to larger values of Eg . The reason is the larger loss in Voc due to the term −q−1 AkTln(J00 /Jsc ) in Eq. (2.191). with decreasing electric ﬁeld Fm . As Fm in According to Table 2.3, J00 increases √ the absorber is proportional to Vbi [combine Eqs. (2.36) and (2.80)], J00 for SCR recombination is larger for lower bandgap absorbers. Thus for absorbers with poor electronic quality, the optimum bandgap is higher than for high quality material. In particular, for an absorber bandgap below 1 eV, it is very important to have high quality material. This picture does not change if we in addition let the Jsc be reduced in the SCR scenario. Since the optimum bandgap of an absorber depends on the electronic quality, we may only state the well known rule: Rule 1: The absorber of a single junction heterostructure should have a bandgap of 1 eV < Eg,a < 1.6 eV. We note again that this rule does not take into account optical and electronic losses in the emitter. If the actual wavelength range which contributes to Jgen becomes smaller, the efﬁciency maximum is shifted to smaller band gap values. 3.2 Band Alignment

The band alignment of a heterostructure is determined by the offsets of valence and conduction band at the interface. These offsets are approximately ﬁxed for a

131

3 Design Rules for Heterostructure Solar Cells and Modules

Ec Energy E [eV]

132

(a)

EF

Ev

(b)

(c)

(d)

Position z Figure 3.2 Schematic examples of band alignments which are critical for the function of a heterostructure solar cell. In (a), the band alignment in an absorber/window may be varied. In the absorber/buffer/window heterostructure, the band offset at the

absorber/buffer interface (b), at both interfaces buffer/window and buffer/absorber (c), and at the buffer/window interface (d) may be varied. Example values for the Voc are given in Figure 3.5.

given combination of heteropartners. Only dipole layers, in principle, may change the band offsets. The dipole concept of interface engineering [20], however, has not yet been realized in chalcogenide solar cells. Thus, the established degree of freedom for interface engineering is the choice of the heteropartners. In order to emulate this choice in the following section, we consider devices with layers of ﬁxed bandgap but different band offsets; and we ask: (i) which solar cell parameter is mostly affected by the band alignment, and (ii) is there a rule for the design of the band alignment? A survey of the band alignments to be discussed is given in Figure 3.2. First, we discuss the band alignment at the interface of an absorber/window heterojunction. It is assumed that the window layer is highly n-type doped with ND,w = 1018 cm−3 . The absorber is moderately doped p-type with NA,a = 1016 cm−3 . The interface between window and absorber contains interface states. Without restriction in generality, we assume these states to be located in the middle of the interface bandgap, that is, at (Ec,min + Ev,a )/2 where Ec,min is the minimum of window and absorber conduction band edge. As explained in Section 2.4.5.4, we further assume that cross recombination at the interface can take place. Electrons from the lowest lying conduction band edge can recombine with holes from the highest lying valence band edge. We vary the interface quality by varying the nominal interface recombination velocity S = Sn0,p0 = NIF σn,p vn,p . The interface states are of neutral character and do not inﬂuence the band diagram.2) 2) Defect states are called ‘‘neutral’’ if their

state density is small and they do not noticeable inﬂuence the band diagram. Their charge can be neglected if compared to the charge density in the SCR. They may

nevertheless induce carrier recombination if their capture cross sections are large. Here, we use ‘‘neutral’’ defect states in order to decouple interface recombination and Fermi level pinning.

3.2 Band Alignment

In Figure 3.3 it can be seen that for a nominal interface recombination velocity of Sn0,p0 = 103 cm s−1 , efﬁciencies of above 20% are obtained for conduction band offsets in the range −0.2 eV < Ec w,a < 0.4 eV. Thereby, all solar cell parameters are largely independent of Ec . Only for Ec w,a > 0.4 eV, Jsc and FF strongly decline. (We come back to this point below.) The tolerance of the solar cell parameters against variation of Ec gradually vanishes with increasing interface recombination. However, even for very high nominal interface recombination velocity of S = 107 cm s−1 , there is a range of conduction band offsets (0.1 eV < 0.8

Voc [eV ]

0.7 0.6 0.5

Sn0,p0

0.4 0.3

Jsc [mA cm−2]

36 35 34 33 84

FF [%]

80 76 72 24

η [%]

20 16 12 8

−0.4

−0.2

0.0

0.2

0.4

∆Ecw,a [eV ] Figure 3.3 Simulated solar cell parameters as a function of Ec w,a for an absorber/window heterostructure. Other device parameters according to default Table 8.1. Nominal interface recombination velocities

are Sn0,p0 = 103 , 104 , . . . , 107 cm s−1 (direction of arrows) realized by interface defects energetically located in the middle of the interface bandgap.

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3 Design Rules for Heterostructure Solar Cells and Modules

Ec w,a < 0.4 eV) where Voc is independent of Sn0,p0 and only the values of Jsc are slightly reduced. In this range of Ec w,a , the diode current obviously is small and photo generated electrons can pass the highly defective absorber/window interface without signiﬁcant recombination. This surprising result can be explained by regarding the carrier concentrations, as depicted in Figure 3.4b and by the aid of Eq. (2.116). Interface states communicate with the valence and conduction bands of both the window and the absorber layer. Thus, the kinetic of occupation of the interface states is related to the concentration of carriers in four bands. Figure 3.4b shows that the electron concentration at the surface for all values of Ec is much larger than the hole concentration in the absorber. (We omit the hole concentration at the window surface, since this concentration is very small.) For positive Ec , it is the electron concentration at the absorber surface which dominates the occupation of interface defects. Thus, the interface defects are largely occupied by electrons being the majority carriers at the interface. Interface recombination is governed by the concentration of free holes at the absorber surface. In the case of Ec w,a > 0, the hole concentration is small both for zero bias (Figure 3.4b, dashed line) and for forward bias (full line). This is the reason why the diode current is small (high Voc ) and the photo current is high (high Jsc ). The saturation current density J0 for interface recombination is expressed by Eq. (2.116). According to this equation, J0 becomes small for a large value of Ep,az=0 (or a small value of En,az=0 ). This is also shown in Figure 2.26. Ep,az=0 corresponds to p the hole barrier φb . A requisite for Ep,az=0 being large is that Ec is not negative, that is, Ec ≥ 0. Then, the saturation current, J0 , due to interface recombination can be negligible even for a large value of Sp0 . If the J0 for interface recombination is small, the device will be limited by bulk recombination. Obviously, interface recombination can be switched off by a proper band alignment provided that the doping ratio ND,w /NA,a is high. This allows: Rule 2: The band alignment in a heterojunction solar cell should be such as to allow Ep,az=0 ≈ Eg,a . Rule 2 basically demands that the absorber is in inversion. Inversion means that the minority carriers in the bulk of the absorber are majority carriers at the site of possible recombination – here at the interface.3) Rule 2 is valid for an emitter consisting of a single window layer as well as for an emitter consisting of a buffer/window combination. For a negative Ec , Voc declines proportional to −Ec . The onset of this decline depends on Sn0,p0 . The onset marks the point where interface recombination starts to dominate over bulk recombination. The decline can be explained by the decreased value of Ep,az=0 (Figure 3.4a). Physically, it is due to the increased 3) Another

formulation is that the pn-junction (the location of n = p) should be located within the absorber layer. It can be considered as a buried junction. This term however is difﬁcult to quantify. Even for slightly higher emitter doping,

the absorber would host the pn-junction at zero bias. For a certain forward bias, however, the location of n = p would move to the interface and the cell would become interface limited.

3.2 Band Alignment

Ep,az = 0 [eV]

1.2 1.0 0.8 0.6 (a) 1020

n,p [cm−3 ]

1018 nw

1016

z= 0

14

naz = 0

10

1012

paz = 0

10

10

108

−0.4

−0.2

0.2

0.0

0.4

∆Ecw,a [eV ]

(b)

Ep,az = 0 [eV ] 1.20

1.10

1.00

0.90

0.80

0.70

0.4

0.5

0.8

Voc [ V ]

0.7 0.6 0.5 0.4 Sn0,p0 0.3 0.0 (c)

0.1

0.2

0.3

En,a z= 0 [eV ]

Figure 3.4 (a) Simulated values of Ep,az=0 as a function of the conduction band offset Ec w,a . Absorber/window type heterojunction solar cell as in Figure 3.2a. (b) Carrier densities at the two sides of the absorber/window interface under AM1.5G illumination at zero bias (dashed lines) and at 0.7 V forward bias (solid lines). Hole concentration

at the absorber side, pa , electron concentration at the absorber side, na , and at the window side, nw , with interface recombination velocity Sn0,p0 = 10 cm s−1 . (c) Voc values from Figure 3.3 plotted as a function of En,az=0 and Ep,az=0 for different values of the nominal interface recombination velocity 103 = Sn0,p0 , . . . , 107 cm s−1 .

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3 Design Rules for Heterostructure Solar Cells and Modules

hole concentration at the absorber surface at forward bias (Figure 3.4b). For highest interface recombination velocity, the decline of Voc already sets in for Eg,a − Ep,az=0 > 0.05 eV. In Section 3.10 we show that a cell with negative conduction band offset can be ‘cured’ by the introduction of a bandgap gradient at the front of the absorber. Thereby, the hole barrier b p or Ep,az=0 is increased. Next, we want to address the question why, for all Ec w,a < 0.4 eV, the inﬂuence of interface recombination on the Jsc is small. This can be understood by inspecting the hole concentration at zero bias under AM1.5G illumination. This concentration is much smaller than the electron concentration. Thus, photo generated electrons representing the photo current in the cell can pass the interface without signiﬁcant recombination – even for Sn0,p0 = 107 cm s−1 . Note that Jsc becomes strongly affected by interface recombination if the doping ratio ND,w /NA,a is smaller. This point is addressed in Section 3.4. For Ec w,a > 0.4 eV, Jsc and FF markedly decline, deﬁning a cell efﬁciency below 10%. The reason is the strong spike in the conduction band which forms a barrier for the photo current (see Section 2.5.6). This current barrier is intuitively understood and has often been invoked as an argument against any kind of spike in the conduction band. We see, however, that current ﬂow is not impeded for Ec w,a < 0.4 eV where the full Jsc can pass the interface. The decisive quantities, as has been derived in Section 2.5.6, are the Boltzmann term for the thermionic emission over the conduction band spike, exp{−Ec w,a /kT}, and the absorber electron concentration at the surface, naz=0 . Below Ec w,a = 0.4 eV the electron concentration is sufﬁcient and the spike in the conduction band does not impede ﬂow of photo generated electrons. For a too large value of Ec w,a , the electron concentration (naz=0 ) is not sufﬁcient to overcompensate the small Boltzmann term. In the example of Figure 3.4, this is the case for Ec w,a > 0.4 eV. For the absorber/window heterostructure we can summarize: (i) a positive conduction band offset is required for large Sn0,p0 and (ii) the short circuit current is large as long as the absorber is in inversion and the conduction band spike is not too large. In principle, both mechanisms are also valid for an absorber/buffer/window heterojunction with varying alignment at the buffer/absorber interface according to Figure 3.2b. Nevertheless curve b in Figure 3.5 reveals that for the absorber/buffer/window heterostructure, the achieved open circuit voltage is smaller than for an absorber/window heterojunction (curve a). The reason is that here we assumed a low buffer doping. Therefore, the buffer acts like a dielectric and induces a further potential drop as is sketched in Figure 3.2b. Only for very large Ec b,a , Voc approaches the value of the absorber/window heterojunction. Case c in Figures 3.2 and 3.5 varies the conduction band offset buffer/window as well as buffer/absorber by letting the sum equal zero, that is, χ = Ec w,b + Ec b,a = 0. The drop of Voc for Ec b,a is similar as for case b but there is a strong increase of Voc for Ec b,a > 0.3 eV. This is due to a barrier for the diode current. We regret that such cells exhibit a low Jsc due to the concomitant large barrier for the photo current. Finally we consider the band alignment at the buffer/window interface while keeping Ec b,a = 0 eV. The dependence of Voc for case d in Figure 3.5 is similar as

3.3 Emitter Doping and Doping Ratio

1.4 1.2 c Voc [V ]

1.0 0.8

b

a

0.6 0.4

d

0.2 0.0

−0.4

−0.2

0.0

0.2

0.4

∆Ec [eV] Figure 3.5 Simulated values for Voc as a function of the conduction band offset for the variations as sketched in Figure 3.2 Other device parameters as in default Table 8.1.

for case b. The strong increase of Voc for Ec w,b < −0.3 eV is due to the barrier for the diode current. It comes alongside with a kink in the JV curve and low values of Jsc and FF. For the following sections, we choose Ec = 0 at the absorber surface.

3.3 Emitter Doping and Doping Ratio

The emitter doping or, to be more precise, the absorber/emitter doping ratio also has a strong impact on interface recombination. It turns out that a high doping ratio helps to reduce interface recombination. We again consider the junction between an n-type window and a p-type absorber as depicted in Figure 3.2a. The conduction band offset is zero, that is, Ec w,a = 0. There is no interface charge and the absorber doping is NA,a = 1016 cm−3 . By varying the window doping ND,w between 1015 and 1018 cm−3 , we can change Ep,az=0 according to Eq. (2.24).4) We assume energetically discrete interface states located in the middle of the interface bandgap. As in Section 3.1, we vary the inﬂuence of the interface by varying the nominal interface recombination velocity Sn0,p0 = NIF σn,p vn,p . Simulated values for Ep,az=0 as a function of the doping ratio ND,w /NA,a are given in Figure 3.6b. For ND,w /NA,a = 0.1 we ﬁnd the Fermi level at the interface close to the valence band edge of the absorber, that is, Ep,az=0 < Eg,a /2. Most of the built-in potential drops in the window layer. In contrast, for ND,w /NA,a = 100 the 4) Note that according to Eq. (2.22), a change

in the window doping concentration with a ﬁxed absorber concentration, slightly

changes the built-in potential. For the presented numerical example, however, this will remain a minor effect.

137

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3 Design Rules for Heterostructure Solar Cells and Modules

Ep,az =0 [eV]

Ep,az =0, Ep,a z =0 [eV]

0.4 0.6 0.8 1.0 1.1

Sn0,p0

30 20

0 80

0.8 En,az=0

Ep,a

z=0

0.4 0.0 0.01

0.1

1 ND,w / NA,a

10

100

1016

70 60 50

1012 108 104 100

20

η [%]

1.2

(b)

10

na,pa [cm−3]

FF [%]

Jsc [mA cm −2]

Voc [ V ]

0.35 0.8 0.7 0.6 0.5 0.4 0.3

paz= 0

naz= 0 na0z= 0 0.4

0.8

1.0

Ep,az=0 [eV]

(c)

15

0.6

0

pa z= 0

10 5 0 0.01

(a)

0.1

1 ND,w / NA,a

10

100

Figure 3.6 (a) Simulated solar cell parameters as a function of the doping ratio ND,w /NA,a of window and absorber layer. Window/absorber heterostructure as depicted in Figure 3.2a with conduction band offset Ec w,a = 0 eV. Other device parameters as in default Table 8.1. Nominal interface recombination velocities Sn0,p0 = 103 , 104 , . . . , 107 cm s−1 (direction of arrows) realized by a single interface defect

energetically located in the middle of the absorber bandgap at the interface. (b) Electron and hole reduced chemical potentials En,az=0 , Ep,az=0 as a function of the doping ratio ND,w /NA,a simulated at equilibrium. (c) Electron and hole concentrations at the absorber side of the heterojunction under AM1.5G illumination at zero electric bias (na o z=0 , pa 0 z=0 ) and at forward bias of 0.7 V (naz=0 , pa ). z=0

potential drop is mostly in the absorber layer since Ep,az=0 > Eg,a /2. The absorber is in inversion and the hole barrier is large. With an interface recombination velocity of 103 cm s−1 , the cell efﬁciencies are well above 18%, irrespective of the value of Ep,az=0 (see Figure 3.6a). The devices are mainly limited by bulk recombination. Whether the larger portion of the band bending is found in the window layer (Ep,az=0 < 0.6 eV) or in the absorber layer (Ep,az=0 > 0.6 eV), approximately has no inﬂuence on Voc , Jsc , and

3.3 Emitter Doping and Doping Ratio

FF.5) With increasing interface recombination velocity the device performance starts to become sensitive to the distribution of the band bending and thus to Ep,az=0 . The Voc ﬁrst exhibits a minimum at about Ep,az=0 = 0.6 eV. With increasing Sn0,p0 , this minimum becomes deeper and directly translates into a minimum of the solar cell efﬁciency. From Figure 3.6c the reason for the minimum in Voc becomes apparent: for Ep,az=0 = Eg /2, the carrier densities directly at the interface become approximately equal at forward bias. According to Eq. (2.67), this is the condition for a maximal recombination rate provided that Sn0 ≈ Sp0 .6) In the case of Ep,az=0 = Eg,a /2, one of the carriers becomes a minority carrier at the interface, recombination decreases and a higher Voc can be obtained. For not too large Sn0,p0 , the Voc is symmetric in Ep,az=0 . A different behavior is observed for Jsc . In the case of band bending exclusively in the absorber, that is, Ep,az=0 > Eg,a /2, Jsc is largely independent of Sn0,p0 . If, on the one hand, the band bending is mostly in the emitter, that is, Ep,az=0 < Eg,a /2, Jsc becomes very sensitive to Sn0,p0 . Thus, Jsc is non-symmetric with respect to Ep,az=0 = Eg,a /2 and, as a consequence, the efﬁciency η is also non-symmetric. The reason can be found in the interface recombination of electrons which are photo generated in the absorber layer. In the extreme case of Ep,az=0 < 0.4 eV, photo generated electrons arrive at the interface as minority carriers (naz=0 and paz=0 in Figure 3.6c). Since holes are abundant at zero bias, recombination can become efﬁcient at high Sn0,p0 . At forward bias, on the other hand, the difference between naz=0 and paz=0 at Ep,az=0 = 0.4 eV again is very large. Now, minority carrier (electron) injection from the emitter into the base is difﬁcult due to the large band bending in the emitter. This is why Voc starts to recover for Ep,az=0 < 0.4 eV. The small asymmetry of Voc with respect to Ep,az=0 = Eg,a /2 originates from the asymmetric Jsc .7) Figure 3.6 reveals that the concept of a solar cell where the electric ﬁeld mostly extends in the absorber is the preferred one. Even for high nominal interface recombination velocity of Sn0,p0 = 107 cm s−1 , the cell efﬁciency can be large provided that Ep,az=0 ≈ Eg,a , that is, if the absorber is fully inverted. Then electrons arrive at the interface as majority carriers both as the diode current at forward bias and as the photo current at zero bias. In a fully inverted absorber, recombination 5) Certainly, this conclusion is only valid if the

7) A solar cell where the electric ﬁeld mostly

carrier lifetime is high and the collection region is not limited to the SCR. Otherwise Jsc and FF would decline for higher Ep,az=0 and thus for a smaller SCR being the main collection region. 6) Not for any type of heterojunction the minimum in Voc is found at Ep,az=0 = Eg /2. The exact position of the minimum of Voc depends on the density of states. Further on, it depends on the ratio of the nominal interface recombination velocity. For Sn0 < Sp0 (Sn0 > Sp0 ), the position where Voc becomes minimal shifts to Ep,az=0 > Eg /2 (Ep,az=0 < Eg /2).

extends within the emitter is referred to as injection solar cell. Photo generation takes place in one semiconductor and carrier separation by the electric ﬁeld takes place in the other semiconductor. The principle of an injection solar cell is realised in the Cu2 S/CdS heterojunction [105]. Requisites for high performance are the carrier collection by diffusion in the absorber layer (since there is no SCR) as well as an interface with very low interface recombination velocity.

139

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3 Design Rules for Heterostructure Solar Cells and Modules

of the electrons is limited by the availability of holes. Without Fermi level pinning, inversion of the absorber surface is accomplished by a high absorber/emitter doping ratio. This allows: Rule 3: In order to induce strong inversion of the absorber surface, the doping ratio between emitter and absorber should be high. Rule 3 is also valid for an absorber/buffer/window heterostructure. It also applies to heterostructures with Ec = 0. In fact, Rule 3 has already been regarded in the discussion of Section 3.1. If there is a barrier for the photo current, that is, Ec w,a or Ec b,a > 0, the requirement of Ep,az=0 ≈ Eg,a is even more important (see Section 2.5.6) since it also provides a high carrier density for thermionic emission. In case that there is no Fermi level pinning and Ec w,a = 0 (absorber/window type), the activation energy of J0 would correspond to the absorber bandgap. Only the diode quality factor would depend on the doping ratio. It would change from A ≈ 2 for Ep,az=0 ≈ Eg,a to A ≈ 1 for Ep,az=0 = Eg,a /2 according to Eq. (2.111). Principally, the same holds for the absorber/buffer/window heterostructure where A is given by Eq. (2.113). Complying with Section 3.12, the emitter should also be highly doped because of its function as the front electrode of the solar cell or module. With a required sheet conductance of 5 /sqr. and typical mobilities in the range of 40 cm2 V−1 s−1 , carrier concentrations exceeding 1020 cm−3 are necessary. In the following, we assume that the window layer doping is ND,w = 1020 cm−3 .

3.4 Fermi Level Pinning

So far, we have assumed that the position of the Fermi level at the interface is completely determined by the doping ratio of absorber and emitter. In Section 2.3.4, however, it is shown that the Fermi level position at the interface can likewise be determined by interface charge. If the interface charge is high enough, the doping ratio between window and absorber has no inﬂuence on Ep,az=0 . This situation is denoted as Fermi level pinning. Here, for clarity, we assume two types of interface states at the absorber/window interface. Neutral states which cause recombination of carriers and charged states which induce Fermi level pinning. It may be argued that Fermi level pinning is not a solar cell design instrument since the interface density is not in the hand of the experimentalist. However, Fermi level pinning has large ramiﬁcations and may become controllable in the future. The principle dependence of the solar cell parameters on Ep,az=0 or En,az=0 in the case of Fermi level pinning is the same as depicted in Figure 3.6. The smaller Ep,az=0 , the smaller is Voc in case of high Sn0,p0 . A numerical example is given in Figure 3.7 where we calculated Voc after Eq. (2.191). We see that if the Fermi level position at the absorber surface is ﬁxed by acceptor-like interface states, the Voc can become lower than if it is ﬁxed by the absorber/window doping ratio. This is so, although the hole barrier is identical.

3.4 Fermi Level Pinning

Ep,az=0 [eV] 1.20 0.7

1.10

1.00

1.90

1.80

0.2 0.3 En,az=0 [eV]

0.4

Voc [V ]

0.6 0.5 0.4

∆Ec=0, NIF =0

0.3

∆Ec=0, NIF−>0 ∆Ec<0, NIF =0

0.2 0.0

0.1

Figure 3.7 Calculated values for Voc as a function of the Fermi level position, En,az=0 , at the absorber surface of an absorber/window heterostructure. Nominal interface recombination velocity is Sn0,p0 = 107 cm s−1 . The upper scale gives the size of the hole barrier φb p = Ep,az=0 . The graph compares three types of cells where a certain value of En,az=0 has been realized using

different conﬁgurations: (solid line) by the doping ratio without pinning states, (dotted line) by Fermi level pinning with aid of acceptor like states (ND,w /NA,a = 100), and (dashed line) by a conduction band offset with ﬁxed ND,w /NA,a = 100. Voc calculation using Eq. (2.191) with material parameters as found in the default Table 8.1.

The physical reason is the different voltage drop upon forward bias. In the pinned case, forward bias is identical with the reduction of the hole barrier. In the non-pinned case, forward bias results in a smaller reduction of the hole barrier because the electron barrier reduces also. Thus, the hole concentration at the interface – the rate limiting quantity for interface recombination – rises faster with forward bias in the pinned case. The mathematical reason is the different activation energy Ea which enters the expression for Voc [Eq. (2.191)]. For the case of Fermi level pinning we have Ea = φb p . In the absence of Fermi level pinning we have Ea = Eg,a for Ec w,a = 0 and Ea = Eg,IF for Ec w,a < 0. Thus, Fermi level pinning in combination with interface recombination can be diagnosed by an activation energy of J0 which is smaller than Eg,a ; much like the case of Ec < 0. Fermi level pinning can compensate for an unfavorable doping ratio. This is so if the charge of donor-like interface states adds to the space charge of the window layer. The Fermi level must be pinned at high Ep,az=0 . This, however, is only possible if a density of donor-like states close to the conduction band is present. Positively charged interface states, thus, can compensate for the unavoidable potential drop in a lowly doped buffer layer. In contrast, low lying acceptor states can even increase the potential drop in the buffer layer. This leads to a decrease in Ep,az=0 . (For the relation between interface state energy and doping ratios we refer to Figure 2.11.) Also negatively charged acceptors at the buffer/window interface can decrease Ep,az=0 (see Figure 2.10). Figure 2.5 shows that a simple absorber/window heterostructure can be made less

141

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3 Design Rules for Heterostructure Solar Cells and Modules

susceptible to negative interface charge by an increased window doping. The higher window doping simply prevents a potential drop in the window. Also the inﬂuence of negative charge at the buffer/window interface of an absorber/buffer/window heterostructure can be suppressed by a high window doping. We note, however, that this is not possible for negative charge at the buffer/absorber interface. For the following design studies, we assume (except for indicated cases) that the defect density at the surface can lead to surface recombination but not to Fermi level pinning.

3.5 Absorber Doping

In the former section it is shown that type inversion of the absorber surface is a condition for suppressed interface recombination. This type inversion requires the absorber doping being much smaller than the emitter doping. However, there are further implications of the absorber doping on the device performance. It determines: the (i) SRH recombination rate in the SCR and in the QNR of the absorber, (ii) the probability of tunneling enhanced recombination in the SCR, (iii) the collection probability of photo generated carriers, and (iv) the probability of Auger recombination. High doping, in addition, is known to induce bandgap narrowing and bandgap tails. It can also reduce the mobility and lifetime. Let us consider an absorber/window type heterojunction where we vary the absorber doping in the range of 1014 cm−3 < NA,a < 1018 cm−3 . The window doping is ND,w = 1020 cm−3 , thus much higher than any value used for NA,a . This fact together with a zero conduction band offset ensures that interface recombination is switched off, provided that there is no Fermi level pinning. We consider absorber layers with different lifetimes of τn0,p0 = 0.1, 10, 1000 ns. Bandgap narrowing at high doping density is neglected. Also, the carrier lifetimes are independent of the doping concentration. The doping efﬁciency is 100%, that is, the hole concentration is p = NA,a . First, we discuss the case without tunneling and without Auger recombination (dashed lines in Figure 3.8). The main observation is that at least in a certain interval all Voc curves rise by 60 mV per decade with increasing doping density NA,a . This slope of Voc is indicated as the dotted line in Figure 3.8a. In the case of the smallest lifetime of τn0,p0 = 0.1 ns, the diode current is mainly controlled by recombination in the SCR of the absorber. For an absorber/window heterojunction, there is a simple relationship between absorber doping and Voc . This can be seen by aid of Eq. (2.37) where we put db = 0 and νIF = 0. Then Fz=0 is proportional √ to NA,a . According to Table 2.3 (SCR recombination via Ed without tunneling), J00 is inversely proportional to Fm , the electric ﬁeld at the position of maximum recombination. Fm can be approximated by Fz=0 , the ﬁeld at the interface. Thus,

3.5 Absorber Doping

Voc [V ]

1.0 τ n0,p0 = 1 µs 0.8

10 ns

0.6

100 ps

Jsc [mA cm−2]

0.4 40 30 20

FF

0.8 0.6 0.4

η [%]

30 20 10 1014

1015

1016

1017

−3

NA,a [cm ] Figure 3.8 Simulated solar cell parameters as a function of the absorber doping NA,a . Absorber/window heterojunction with conduction band offset of Ec w,a = 0 eV. Other device parameters according to default Table 8.1. The carrier lifetimes in the absorber are

τn0,p0 = 0.1, 10, and 1000 ns in direction of arrow. Cases without (dashed line) and with (solid line) tunneling enhanced recombination plus Auger recombination. Simulation by PC1D [50].

we expect the relationship , + qVoc ≈ AkT ln NA,a N0A,a

(3.1)

where NA,a 0 is a reference doping concentration. Equation (3.1) tells that the Voc of an absorber/window heterojunction should increase by about 60 mV per decade of absorber doping concentration. This is indeed observed in Figure 3.8a.

143

144

3 Design Rules for Heterostructure Solar Cells and Modules

Besides this mathematical explanation, the dependence of Voc on the absorber doping for SCR recombination may also be rationalized by physical arguments. In Figure 2.46, we ﬁnd that next to the position of highest recombination the difference between n and p rises quickly. Only at positions of n ≈ p, recombination occurs at a high rate. In the case of NA,a = 1017 cm−3 , the difference between n and p rises much faster than in the case of NA,a = 1015 cm−3 . Thus, the region of high recombination is narrower for higher doping. According to Shockley [45], the recombination rate falls off exponentially with a characteristic length of kT/qFm . Thus, the width of the effective recombination zone is inversely proportional to the ' electric ﬁeld in the SCR, Fm . With Fm being proportional to NA,a , this brings us back to the formal explanation given above for the case of SCR recombination. For a lifetime τn0,p0 = 1000 ns, recombination takes place mainly in the QNR of the absorber. For not too low NA,a then Voc increases as qVoc = kTln(NA,a /NA,a 0 )

(3.2)

Thus, for recombination in the QNR there is an increase of Voc identical as for SRH recombination in the SCR. Descriptively, Voc increases with NA,a since the recombination in the QNR region decreases with the dark minority carrier concentration, which is given by ni 2 /NA,a . For τn0,p0 = 1000 ns and NA,a < 1016 cm−3 , Voc becomes independent of the doping concentration. This is an effect of the limited absorber thickness and the anticipated passivation of the back surface of the absorber. The diode current becomes independent of the doping concentration NA,a but is only dependent on lifetime, diffusion constant, and absorber thickness da . The reference current ' density for this case reads J00 = (τn0 + τp0 )−1 da q Nc,a Nv,a . This is similar to the case of very high forward bias as discussed at the end of Section 2.4.5.2. Deviations from the simple 60 mV per decade behavior is also found for the simulated solar cells with small lifetime of τn0,p0 = 0.1 ns. We ﬁnd that the dashed line is below the dotted line for very low absorber doping. The reason is the low collection current as expressed in the Jsc curve in Figure 3.8b. The Jsc for τn0,p0 = 0.1 ns in general is well below the currents of high lifetime cells. In addition, it is further reduced for small NA,a . Inversion of the interface combined with low doping and low carrier lifetime leads to recombination of carriers which are generated in the front region of the absorber. In a quantum efﬁciency measurement, front region recombination is manifested as a low blue response (see Section 7.4.2). Since the region of high recombination at the front further extends with increasing forward bias, the FF of the device with τn0,p0 = 0.1 ns and NA,a = 1014 cm−3 becomes very poor. For high doping the dashed Jsc curve with τn0,p0 = 0.1 ns is also strongly reduced. This is due to the small diffusion length in the QNR region of the absorber. Since the SCR narrows with increasing NA,a , the collection length becomes small. This is perceivable as a low red response in a quantum efﬁciency measurement (see Section 7.4.4).

3.5 Absorber Doping

Now, we include the effect of tunneling enhanced recombination and Auger recombination (full lines). Auger coefﬁcients of Cp = 10−28 cm6 s−1 8) and a tunneling mass of 0.1 × me 9) are assumed. Thus, the critical electric ﬁeld for tunneling is F = 2.4 × 105 V cm−1 (see Section 2.4.5.1). The Voc curves display a maximum at about NA,a = 1017 cm−3 . For τn0,p0 = 0.1 and 10 ns, the reduction of Voc at higher NA,a is due to tunneling enhanced recombination and the inﬂuence of Auger recombination is negligible. For τn0,p0 = 1 µs, the reduction of Voc is due to Auger recombination and the inﬂuence of tunneling enhanced recombination is negligible. Certainly, there is an intermediate carrier lifetime where both effects (Auger recombination, tunneling recombination) are of the same magnitude. Although the example in Figure 3.8a is for a simple absorber/window heterojunction, the principle relationship between Voc and (NA,a /NA,a 0 ) also holds for an absorber/buffer/window type of heterostructure. Then Fm grows as predicted by Eq. (2.37). The combination of all effects listed at the top of this section tailor the curve of the efﬁciency as a function of NA,a in Figure 3.8. Summing up this curve, we have: Rule 4: The absorber doping in a heterojunction solar cell shall not exceed a value of NA,a = 1017 cm−3 . A doping concentration below NA,a = 1015 cm−3 shall only be selected for carrier lifetimes exceeding 10 ns. Since the parameters selected for Auger recombination and tunneling enhanced recombination are rather pessimistic, the NA,a = 1017 cm−3 limit in Rule 4 is conservative. However, as stated above, we have neglected the effect of bandgap narrowing. Bandgap narrowing at high doping concentrations would decrease Voc and increase Jsc . The effect has been studied for III–V [108] and element semiconductors [109]. In p-GaAs, for instance, bandgap narrowing has been found to reduce the optical bandgap by 2.4 × 10−8 p1/3 . This empirical expression gives a bandgap reduction of 23 meV for NA,a = 1018 cm−3 . Thus, the effect of bandgap narrowing is small against the effects of tunneling enhanced recombination and Auger recombination for the doping concentrations considered for a heterojunction absorber layer. Rule 4 is roughly valid for absorber bandgaps in the range of 1 eV < Eg < 1.7 eV. This can be seen in Figure 3.9 where devices of different absorber bandgap but with identical lifetimes of τn0,p0 = 10 ns have been simulated. For all bandgaps we ﬁnd strongly decreasing solar cell parameters for NA,a > 1017 cm−3 due to tunneling enhanced recombination and Auger recombination. In addition, there is a general trend of decreasing η with increasing NA,a for large bandgap absorbers. This is due to the relative importance of effects (1) and (3) in the effect listing at the top of this paragraph. The 60 mV per decade gain in Voc due to increasing NA,a is relatively less 8) In accordance with Section 2.4.4 we se-

lected maximum values for the Auger coefﬁcients. 9) The tunneling mass is still a quantity of debate. Hurkx gives a value of 0.1 × me for silicon [48]. Since the tunneling electrons

and holes have energies much lower than the band edge energies, the tunneling mass is smaller than the band edge mass [106]. Values as low as 0.01 × me for silicon can be found in the literature [107].

145

3 Design Rules for Heterostructure Solar Cells and Modules

1.4

Voc [ V ]

1.2

Eg =1.72 eV 1.51 eV

1.0 1.32 eV 0.8 0.6

1.15 eV 1.02 eV

Jsc [mA cm−2]

40

30

20

0.9

FF

0.8 0.7 0.6 0.5 22 20 η [%]

146

18 16 14 1014

1015 1016 NA,a [cm−3]

1017

Figure 3.9 Simulated values of Voc , Jsc , FF, and η for an absorber/window heterojunction with ND,w = 5 × 1019 cm−3 without interface recombination as a function of the absorber bandgap Eg . Case with tunneling enhanced recombination plus Auger recombination.

The conduction band offset is Ec w,a = 0 eV. Other device parameters according to default Table 8.1. The absorber bandgaps are Eg = 1.02, 1.15, 1.32, 1.51, 1.72 eV in direction of arrow.

3.6 Absorber Thickness

important for larger bandgaps. Yet, the Jsc loss due to decreasing SCR is relatively more important for larger bandgaps. As was discussed in Section 2.3.6, deep defects can become charged within the depletion region and thus can alter the band diagram. Consider, for instance, acceptor like deep states in the absorber. Since they become ionized in the depletion region of the p-type absorber, they add to the space charge and increase electric ﬁeld and band bending. Thus deep acceptor states must be counted in addition while applying Rule 4. If these deep defect induce a strong lattice relaxation, they may be persistently charged by electrons and thus induce metastable effects (see Section 6.2.4). For the subsequent design studies (see Table 3.1), we set the absorber doping to NA,a = 1016 cm−3 .

3.6 Absorber Thickness

The absorber thickness can inﬂuence the cell parameters in various ways. Upon reduced thickness: 1) 2) 3) 4)

The collection length can reduce thus deteriorating Jsc . The impact of back contact (BC) recombination can increase. The width of the recombination zone can be reduced, thus increasing Voc . In the case of a highly reﬂecting BC the local generation rate may be increased, thus increasing Voc .

Fabricating a chalcogenide solar cell with a minimum thickness of the absorber layer has two potential advantages: it can reduce production costs (lower material consumption, higher throughput) and it can decrease recombination losses. In a thoughts experiment, we start with a thick absorber layer which develops the full solar performance deﬁned by its bulk and interface properties. By gradually reducing the absorber thickness, we may ﬁnd the minimum thickness where ﬁrst, the photo current generation is still high, and second, the minority carrier recombination at the BC is still negligible. An overview of optical and electronic processes inﬂuencing the photo current generation in a chalcogenide solar cell is given in Figure 3.10. What is the optical inﬂuence of the BC? Photons can either be absorbed within the BC layer or they can be reﬂected. The reﬂected photons again are either absorbed during the second light pass through the absorber layer or escape the device by transmission through the emitter (multiple reﬂectance is another possibility). Thus, reﬂection and absorption describe the optical inﬂuence of the BC. Naturally, they depend on the reﬂectance and absorbance of the BC layer. Reﬂection and absorption can as a ﬁrst approach be considered as independent from reﬂection and absorption in the emitter. Below, we differentiate 0 and 100% reﬂection (transmission through the BC is neglected). The electronic inﬂuence of the BC can be expressed by the BC recombination velocity. This recombination velocity inﬂuences both the diode current and the

147

Window

Buffer

Absorber

3 Design Rules for Heterostructure Solar Cells and Modules

Back contact

148

Back contact reflection Emitter reflection

AM 1.5G

Parasitic absorption Recombination

Photo current

Figure 3.10 Loss mechanisms reducing the photo current in an absorber/buffer/window solar cell (after [34]).

photo current. In case that there is no band bending at the BC we have to deal with the recombination velocity of the minority carriers, here electrons. We differentiate low and high back surface recombination velocities, that is, Sn0,bc = 103 cm s−1 and Sn0,bc = 107 cm s−1 . In order to demonstrate the inﬂuence and interrelation of optical and electronic BC properties, we show simulated solar cell parameters as a function of the absorber thickness.10) For Sn0,bc = 103 cm s−1 and RBC = 100% (Figure 3.11a), Voc indeed increases with decreasing da except for the case τn0,p0 = 1 µs. In the case of τn0,p0 = 100 ps, the increase in Voc is found at very low da where the absorber is completely depleted and band bending is deﬁned by the inverse ﬁlm thickness. As explained in Section 3.5, SCR recombination decreases and thus Jdiode,SCR decreases with steeper bands and thus with reduced ﬁlm thickness. In the case of τn0,p0 = 10 ns and Sn0,bc = 103 cm s−1 , recombination in the QNR is dominant and becomes reduced with reduced absorber thickness. This is so in despite of the reducing photo current. Finally, in the case of τn0,p0 = 1 µs, the diode current is limited by BC recombination for all thicknesses [see Eq. (2.152)]. Here, Voc decreases with decreasing da due to the reduction in Jph . The short circuit current density, is found to decrease for da < 1 µm. This can be rationalized by the average absorption length of band edge photons of 1/α. Thus, for the absorption coefﬁcient α0 = 104 cm−1 typical for several chalcogenide 10) Note that the simulated cells in Figure 3.11

exhibit a highly inverted absorber due to the high doping ratio. The nominal recombination velocity at the absorber/window

interface is Sn0,p0 = 105 cm s−1 . With Ec = 0 eV, interface recombination is not the dominant process.

3.6 Absorber Thickness

149

Voc [V]

0.8 0.7 Sn0,bc =103 cm/s

Sn0,bc =103 cm/s

RBC =100%

RBC =0%

Sn0,bc = 107 cm/s

0.6

Sn0,bc = 107 cm/s

Jsc [mA cm−2]

40 35 30 25 20

RBC = 0%

RBC = 100%

FF [%]

15 85 80 75 70 65

η [%]

25 20 15 10 0.1 (a)

2

4 6

1

da [µm]

2

0.1 (b)

2

4 6

1

da [µm]

Figure 3.11 Solar cell parameters for an absorber/window heterojunction. For the material parameters see default Table 8.1. The carrier lifetime as the family parameter is τn0,p0 = 100 ps (solid line), 10 ns

2

0.1 (c)

2

4 6

1 da [µm]

2

0.1 (d)

2

4 6

1

da [µm]

(dashed line), and 1000 ns (dotted line). (a) Sn0,bc = 103 cm s−1 , RBC = 100%, (b) Sn0,bc = 103 cm s−1 , RBC = 0%, (c) Sn0,bc = 107 cm s−1 , RBC = 100%, (d) Sn0,bc = 107 cm s−1 , RBC = 0%.

absorbers (see Table 4.5), a limiting absorber thickness of 1 µm may be identiﬁed. Such a value has been encountered in several publications [110–112]. Nevertheless, we refrain from formulating a design rule for the limiting absorber thickness since, according to Figure 3.11, it strongly depends on Sn0,bc and RBC . A drastic example is a cell with Sn0,bc = 107 cm s−1 and τn0,p0 = 10 ns in Figure 3.11c,d where the efﬁciency already declines for da < 3 µm, irrespective of the BC reﬂectance. For the design discussion in the following sections, we continue to anticipate an absorber thickness of 1.5 µm. The example in Figure 3.11 reveals that a reduced absorber thickness must be compensated by an improved BC reﬂectance and a reduced BC recombination. At very low absorber thickness, it is advised to apply light trapping structures at the BC (for an overview see Ref. [65]). Recombination may be reduced by a band edge gradient (see Section 3.10) or a back surface ﬁeld (see Section 3.10).

2

150

3 Design Rules for Heterostructure Solar Cells and Modules

3.7 Grain Boundaries

In Section 2.4.5.4 we treat the absorber surface, that is, the absorber/emitter interface as a special zone where the (volume averaged) recombination rate can be higher than in the absorber bulk. It is similar with the absorber BC. In a polycrystalline absorber also grain boundaries (GBs) can exhibit a recombination rate higher than in the bulk. In addition, these GBs may be charged and induce band bending. GBs are 2-dim recombination zones which can be oriented more parallel or more perpendicular to the main junction [see markers (a) and (b) in Figure 1.1]. They are characterized by a nominal GB recombination velocity, Vn,p = Ngb σn,p Sn0,p0,gb , much like the absorber/emitter interface. For given values of Sn0,gb and Sp0,gb the recombination rate can be calculated after Eq. (2.67), provided that the carrier concentrations at the GB are known. Green developed upper bounds for GB effects assuming the carrier densities n,p in the presence of a GB to be identical to the case without GB [113]. This overestimates GB effects since GB recombination reduces the carrier concentrations with respect to the GB-free case. Using 3-dim device simulation, Gloeckler et al. was able to quantify the impact of vertical and horizontal GBs on solar cell performance [114, 115]. Much is learned from an horizontal 2-dim GB which is located at a distance zgb from the main junction, that is, parallel to the junction between emitter and absorber. Initially, the GB may not be charged (neutral GB) and it may exhibit Sn0,gb = Sp0,gb = Sgb due to equal capture cross sections for electrons and holes. Note that we consider the recombination states, which deﬁne Sgb , to be independent from charged states which induce band bending. We are interested in the position zgb of maximal impact on the device performance. Figure 3.12a shows contour plots of normalized solar cell parameters for a neutral GB. For a given Sgb , the parameters Voc , FF, and η show a distinct minimum if the GB is located at zgb ≈ 90 nm. Figure 3.13b reveals that this position coincides with the location of n ∼ p at forward bias (V = 0.65 V). Precisely spoken, a horizontal GB has the strongest impact if it is located at the position Sn0,gb na = Sp0,gb pa . Up to Sgb = 105 cm s−1 , a neutral GB at the edge of the SCR reduces the efﬁciency by maximal 5%. Located deep in the QNR, the inﬂuence of a horizontal GB is even lower. Note that beyond the position of the horizontal GB – deeper in the absorber – the splitting of the quasi Fermi levels is largely unaffected. It may be surprising that Jsc is only marginally affected by a horizontal GB located in the SCR. The reason is that at zero bias, the carrier densities at the location of Sn0,gb na = Sp0,gb pa are small. This can be seen in Figure 3.13a. Only for very high Sgb , Jsc tends to be reduced. Then, the minimum of Jsc also is for a GB at the location of Sn0,gb na = Sp0,gb pa . If the GB is located in the QNR, the loss of Jsc can be modeled by an effective lifetime obtained from 1/τeff ,n = 1/τn0 + Sn0 /da (and accordingly for 1/τeff ,p ).

3.7 Grain Boundaries

107 0.75

0.9

0 .6 5

0.85

106

5

105

0.95

104

Voc

103 107

85

0.85

0.65

0.

85

0.95

0.

106

0.95 105 Jsc

103 107

0.8

5

0.55

106

0.95

Sgb [cm/s]

104

0.95

105

0.85

0.95

104

FF

103

103 (a)

η 10−6

10−5 zgb [cm]

0.65

0.9

0.95

0.95

5

0.75

0.85

0.8

5

104

5 0.6

105

3 0.

5

7 0.

106

0.3

5

107

5

10−4

Figure 3.12 Normalized solar cell parameters Voc , Jsc , FF, and η (from top to bottom) from device simulation for (a) neutral and (b) charged grain boundary located at zgb in the absorber layer. The positive charge density is 2 × 1011 q cm−2 . Material parameters

10−6 (b)

10−5 zgb [cm]

10−4

as in default Table 8.1. The grain boundary was realized by an equivalent layer of thickness dgb = 2 nm with Nd = NIF /dgb recombination centers. Capture cross sections of 10−12 to 10−8 cm2 mimic grain boundary recombination velocities of 103 –107 cm s−1 .

151

3 Design Rules for Heterostructure Solar Cells and Modules

Energy E [eV]

4

3 EFn

EFp

2

n,p [cm−3]

1015

p n

1010 GB rec. 1022

Rn [cm−3 s−1]

152

1020 1018 1016 −0.4

(a)

0.0 −0.2 z [µm]

Figure 3.13 Band edge energies and quasi Fermi energies, carrier densities, as well as electron recombination rate, R, as a function of depth in an absorber/window heterostructure. Material parameters as in default Table 8.1 with Sgb = 105 cm s−1 . The neutral grain

−0.4

0.2 (b)

0.0 −0.2 z [µm]

0.2

boundary at dgb = 90 nm was realized, as described in the caption of Figure 3.12. (a) V = 0 V and (b) V = 0.65 V roughly corresponding to the simulated Voc of this device. The position of dgb is characterized by a peak in R.

Thus, the effective lifetime model mimics the GB states evenly distributed within the absorber layer. The model is a good approximation for Sn0,p0 up to 105 cm s−1 [114]. Similar as for interfaces, the GB states may be charged. Negative charge can be provided by acceptor states close to Ev . They would induce upward band bending and an electron-repulsive barrier (Figure 3.14a). In a p-type absorber, GB acceptor states of very high density would be needed since, due to a small Ep,az=zgb , their occupation would be low. Thus, a negatively charged GB is unlikely in p-type absorbers. A positively charged GB (more likely in a p-type absorber) leads to an hole-repulsive and electron-attractive barrier (Figure 3.14b). It would be due to high lying charged donor states. Since the positively charged GB attracts electrons, we expect it to enhance GB recombination [113]. In Figure 3.12b we ﬁx the area charge density of the GB to 2 × 1011 q cm−2 and vary the location of the GB. In the QNR this area charge induces a band bending

3.7 Grain Boundaries

Ec

Ep,a

z=zgb

Ep,a

EF

−qVbigb

Ev qVbi

(a)

(b)

gb

(c)

qVbigb +∆Ev (d)

Figure 3.14 Sketch of the band diagram perpendicular to a grain boundary showing (a) upward band bending due to negative GB charge, (b) downward band bending due to positive charge, (c) a layer with increased bandgap realized by a valence band offset, and (d) combined accumulation and depletion zones [116]. gb

of qVbi = 0.11 eV. We see that the principle dependence of the cell parameters on zgb is not altered: maximum impact is on Voc and FF. The minimum in η is shifted to slightly larger zgb . This is due to the extended SCR in the presence of positive charge. Figure 3.12 reveals that the impact of a charged GB for a given Sgb is larger than for a neutral GB. Note, however, that the GB band bending induced by the ﬁxed charge is much smaller in the SCR than in the QNR due to lower carrier concentration. Calculating the solar cell parameters in Figure 3.12 assumed that the horizontal GB is exactly parallel to the main junction. Certainly, this is an artiﬁcial assumption since in reality ‘horizontal’ GBs may show some inclination with respect to the junction plane. Nevertheless, we may draw the conclusion: Rule 5: Horizontal GBs in the SCR should be avoided. As seen above, a GB located outside of the SCR has a minor impact on the device performance as long as Sgb is not too large. Rule 5 may favor substrate-type solar cells over superstrate-type solar cells: In substrate-type solar cells, absorber material from the initial growth phase exhibiting a high GB density is at the BC. Thus, substrate-type solar cells may exhibit less GBs in the SCR. We note that if low Voc and FF but high Jsc are diagnosed for a particular thin ﬁlm solar cell, a high density of horizontal GBs could be the reason. This is similar to interface recombination. Recombination at horizontal GBs in the SCR will lead to a diode quality factor close to 2. The saturation current is activated by the GB bandgap, normally, Eg,a . Thus, device analysis does not give a hint for GB recombination, besides the discrepancy of high Jsc and low Voc (see Section 7.2.2). The impact of vertical GBs can now be understood based on the knowledge about horizontal GBs. If the vertical GB runs through the complete absorber, recombination is highest in the range of Sn0,gb na = Sp0,gb pa , thus where the vertical GB passes the SCR. Furthermore, Voc and FF is more affected than Jsc . This is

153

3 Design Rules for Heterostructure Solar Cells and Modules

Voc , Jsc , FF, η (norm.)

1.0

A

154

Jsc 0.9

FF

0.8

Voc

0.7 0.6

η

0.5 2.0

A

1.5 1.0

102

103

104

105

106

107

Sgb [cm s−1] Figure 3.15 Neutral grain boundary recombination on columnar grain boundaries. Data from 2-dim device simulation. The grain size (distance between two grain boundaries) is 1 µm. All parameters are normalized to a cell without grain boundaries with absorber electron lifetime of 2 ns. After Gloeckler et al. [115].

visualized in Figure 3.15, obtained by 2-dim device simulation with 1 µm grain size [115]. The efﬁciency loss at high Sgb is mainly due to Voc and FF. Certainly, for a given Sgb the reduction in η for a vertical GB in Figure 3.15 is less than for a horizontal GB located in the SCR (Figure 3.12) simply due to geometrical reasons. The example of Figure 3.15 reveals that, for 1 µm grain size, high solar cell performance can only be achieved if the recombination velocity of a vertical GB does not exceed Sgb = 103 cm s−1 with τn0,p0 = 0.2 ns. This may be formulated as: Rule 6: The effective GB recombination velocity for 1 µm absorber grain size shall not exceed the bulk equivalent velocity [see Eq. (2.104)]. We use the term effective Sgb since band bending and band offsets at the GB can modify the nominal GB velocity. We admit, however, that Sgb eff is not well deﬁned. For comparison, some GBs in multicrystalline Si have recombination velocities Sn0,p0 above 106 cm s−1 [117]. If the vertical GB in a p-type absorber is positively charged (hole-repulsive band bending), the GB attracts electrons and recombination increases – exactly like in a horizontal GB. At large band bending, the GB inverts and the pn junction extends from the absorber/emitter interface along the grain surfaces deep into the absorber layer. This can cause shunting of the cell by an n-type channel (see Figure 3.16) [115]. Strong inversion, in contrast, can reduce GB recombination at zero bias due to hole depletion. For an inverted GB, the recombination rate at zero bias reads Rgb = pa Sp0,gb . Since pa is low, a large hole-repulsive GB exhibits a large Jsc . Due

3.7 Grain Boundaries

0.0 (a) 0.5

p~n

n >> p

1.0

z [µm]

1.5 p >> n

V=0 0.0 (b) 0.5

p~n 4 1.0

0

n>p

−4 1.5

p >> n V = 0.5 V 0.2

0.4

0.6

0.8

x [µm] Figure 3.16 Log(n/p) along a vertical GB by JE100 with spectrum equivalent AM1.5. from 2-dim device simulation: (a) at V = 0 V Sgb = 105 cm s−1 . Absorber doping NA.a = and (b) V = 0.5 V forward bias. GB lo2 × 1016 cm−3 , Eg,a = 1.15 eV. Gray scale cated at x = 0.5 µm and absorber/emitter represents log(pa /na ). From Ref. [115] with junction located at z = 0 µm. Illumination permission.

to the collecting properties of the n-type channel, Jsc can even exceed the GB free case. However, at forward bias the inversion is reduced due to partial occupation of the (high lying) donor states. The hole concentration increases up to the condition pa ∼ na . Thus, a strong hole-repulsive GB alone is not appropriate to deactivate GB recombination. A means to stabilize a hole-repulsive barrier at the GB is to reduce the valence band edge at the GB. As a consequence the majority carrier concentration is reduced by the factor exp{Ev /kT}. This has been studied by Gloeckler et al. [115] and Taretto et al. [118] using 2-dim device simulation. A lowered valence band edge may be combined with a positive band bending. A sketch of a GB cross section with band bending and valence band offset is shown in Figure 3.14c. Provided that the sum of Ev + qVbi gb is large enough, recombination at a vertical GB can be largely suppressed. The result is a solar cell which is hardly affected by vertical

155

156

3 Design Rules for Heterostructure Solar Cells and Modules

GBs. Figure 3.14d gives the GB band diagram which combines an electron barrier with hole depletion [116].

3.8 Back Contact Barrier

At the back surface of an absorber, a contact barrier may form due to Fermi level pinning or due to the particular band line-up with the BC band structure (see Figure 2.26b). Thus in principle, we are concerned with a Schottky barrier of barrier height φb p where the index p denotes a barrier for holes. In the equivalent circuit, a BC barrier can be described by a BC diode (see Section 2.6.1) as long as the BC barrier does not interfere with the main diode. That is, as long as the depletion regions of the two diodes do not overlap. The BC diode is of opposite polarity than the main diode. The photo current ﬂows through the back diode as a forward majority carrier current. In order to admit this current ﬂow, the BC diode needs to be forward biased. This bias voltage is supplied by the main diode which thereby itself becomes forward biased. The bias voltage is Vm = −Vb where m and b denote the main and back diode, respectively. Figure 3.17a shows a heterojunction with BC diode at short circuit. The diode equation for the BC diode yields −Vb = kT/q ln(1 + Jph (Vb )/Jb0 )

(3.3)

where Jb0 is the saturation current of the back diode. Thus, −Vb increases with decreasing Jb0 . If we assume thermionic emission limited transport over the BC barrier, we can use Eq. (2.126) to see that −Vb increases with increasing φb p : the larger the barrier height, the larger is the voltage drop on main and back diode under short circuit. In Figure 3.17a, Vb equals the difference in the electric potentials ϕ − ϕ0 taken in the center of the absorber where ϕ0 stands for the non-illuminated case. Using device simulation, we investigate the principle inﬂuences of a BC barrier on cell parameters. In Figure 3.18, we differentiate between different absorber thicknesses. First, we see an increase of Jsc with increasing da . This is an independent effect of the absorber thickness which is only due to an increased charge collection. However we ﬁnd that for our default device, the effect of φb p on Jsc is negligible: Although the main cell is forward biased under short circuit (see Figure 3.17a), the effect of voltage dependent current collection is too small to depress Jsc . This is different for the other solar cell parameters. The FF starts to degrade for φb p > 0.3, independent from the absorber thickness. Now, this is indeed the effect of voltage dependent charge collection. And as the forward bias of the main junction is similar for all thicknesses in Figure 3.18, the FF loss is independent of da . Considering the inﬂuence of φb p on Voc , we have to discriminate between different absorber thicknesses. For thick absorbers, Voc is hardly inﬂuenced even for φb p = 0.4 eV. The reason is that the main and BC diode can be considered as two independent devices: At Voc , no current is passing through the combination of two anti-parallel diodes. No current can be blocked by the BC

3.8 Back Contact Barrier

−qϕ0 1

1

− qj

Ec

−4 −5

Ev EFp

−7 1018

1016 n

1012 1010 Jn

−20

Jp

Jtot

−30 −1.5

1014 1012

20 J [mA cm−2 ]

J [mA cm−2 ]

−10

1016

1010

0

(a)

−5

−7

1014

−qϕ0

−4

−6

p

−qϕ

−3

−6

1018 n,p [cm−3]

Energy E [eV]

−3

0 EFn

n,p [cm−3]

Energy E [eV]

0

−1.0

−0.5

0.0

Position z [µm]

0

Jtot

−10 −20 −1.5

0.5

Jn

10

−1.0

(b)

JP −0.5

0.0

0.5

Position z [µm]

Figure 3.17 Solution curves for an absorber/window heterojunction with back contact barrier (a) under short circuit condition and (b) at open circuit (AM1.5). Device parameters as in default Table 8.1 but with −qφb = 0.5 eV as back contact barrier height. qϕ0 gives the potential energy for the non-illuminated case.

diode and the recombination of light generated carriers is only governed by the main diode. For our default device (see Table 8.1) we have Ln,a = 1.13 µm and wa = 0.33 µm. Thus the criterion is approximately da /(Ln,a + wa ) ≥ 3. For thinner cells, φb p = 0.3 eV already leads to some loss in Voc . A larger BC barrier height further decreases the cell performance. Comparing Figure 3.17b with Figure 2.19b (no BC barrier), the slope of the electron quasi Fermi level toward the BC is much stronger. The BC barrier leads to enhanced minority carrier transport to the BC and thus to higher recombination both in the bulk and at the BC [54, 55]. In other words, the main diode and the BC diode are not independent. A BC barrier together with minority carrier recombination at the BC can lead to a cross over between dark and light JV curves already for moderate barrier heights

157

3 Design Rules for Heterostructure Solar Cells and Modules

0.4

4

0.73

4

0.7

0.73

8

0. 73 2

0.3 0.2

36.6

0.0 0.3

78.8

78.4

78.2

78

77.6

77.8 79

36.8

Jsc [mA cm−2]

35 .8

0.2 0.1

Voc [ V ]

36.4

36

35.4

0.

36.2

0.0 0.3

6

73

0.1

φbp [eV]

78.6

79. 2

0.2 0.1

21 .6

21.2

21.4

20

0.0 0.3

FF [%]

0.1 0.0

.8

0.2 η [%]

20

158

2

4

6 da [µm]

8

Figure 3.18 Simulated solar cell parameters of an absorber/window heterojunction with back contact barrier of barrier height φb p . Contour plot showing the inﬂuence φb p and the absorber thickness da . Other material parameters as in Table 8.1.

[119]. Large BC barriers may be diagnosed by a roll over in the JV curve or, which is equivalent, by a saturation of the current density under forward bias. If the forward current approaches the reverse saturation current of the back diode, Jb0 , any further increase in V drops at the back diode, that is, V = Vb . Thus Vm remains constant and the current saturates. If the barrier height φb p is small, the rollover effect is only observed at low temperature. Can we formulate a general rule for the maximum tolerable barrier height at the BC? Figure 3.18 suggests that a barrier height of <0.3 eV may be tolerated without strong efﬁciency loss regardless of the absorber doping concentration. Now, we have to ask for the inﬂuence of the absorber bandgap. In principle, we expect that an efﬁciency decrease upon an increasing barrier height is smaller for a large bandgap absorber. This is because the voltage effect will be smaller in relative terms. The simulation results in Figure 3.19 indeed reveals this trend: the decline of η with increasing φb p is smaller for larger Eg,a . Nevertheless, the threshold value for the decline in η is rather similar for all bandgaps. Thus a general rule may be formulated which states: Rule 7: The barrier height at the BC of the absorber should not exceed 0.3 eV. This rule has been derived for an absorber valence band effective density of states of Nv,a = 2 × 1018 cm−3 . According to Eq. (2.126), the limiting value in Rule 7 may

3.9 Buffer Thickness

0.9

1.6

1.4

1.2 1.1 0.0

0.2

0.4 0.6 φbp [eV ]

0. 3

0.7

1.3

0. 5

0.98

Eg,a [eV]

1.5

0.8

1.0

Figure 3.19 Simulated normalized solar cell efﬁciencies as function of absorber bandgap Eg,a and back contact barrier height φb p . Simulation of an absorber/window heterojunction. Other material parameters as in Table 8.1. Dashed line gives the barrier height corresponding to zero band bending.

be slightly larger for a higher effective density of states in the valence band of the absorber.

3.9 Buffer Thickness

The emitter of a chalcogenide solar cell may be composed of different layers. Common is the buffer/window concept where the window layer itself may consist of two or more layers (typically, a doped and a non-doped transparent oxide). In this section, we want to emphasize the optical and electronic implications of the buffer layer on the device performance. The buffer layer can act as a matching layer for a difference in the refractive index of window, nw , and absorber, na . If it holds that nw < nb < na in the wavelength range between λEg,w and λEg,a , then introducing a buffer layer causes an optical gain. This gain is due to a decreased emitter reﬂection (denoted as window reﬂection in Figure 3.10). In contrast, the optical bandgap of the buffer layer, Eg,b , may be smaller than that of the window layer, Eg,w , such that Eg,w > Eg,b > Eg,a . Thus, absorption of photons with wavelengths between λEg,w and λEg,a in the buffer layer may take place. Depending of the collection function of the buffer layer this may induce a collection loss. The collection loss can be calculated as a current loss by use of Eq. (2.147) if we replace ηc (z) by [1 − ηc (z)]. Let us assume that the collection function in the n-type buffer layer is small, that is, photo generated holes recombine in the buffer volume or at the buffer/absorber interface. Then, the collection (recombination) loss scales with the buffer layer thickness. There may be a buffer thickness up to which the optical gain compensates the collection loss. This thickness will depend on the speciﬁc optical parameters of window, buffer, and absorbers. Thus, a general rule cannot state a particular buffer layer thickness.

159

3 Design Rules for Heterostructure Solar Cells and Modules

1.0

∆Jsc [mA cm−2]

160

opt

∆Jsc

0.0 ∆Jsc −1.0 opt

∆Jsc −2.0 0

20

40

60

80

100

CdS Thickness [nm] Figure 3.20 Simulated values of the optical gain Jsc opt , the collection loss Jsc col and the total variance Jsc = Jsc opt + Jsc col as a function of the thickness of a buffer layer in a CuIn0.7 Ga0.3 Se2 /CdS/ZnO heterostructure (after [34]). Refractive indices and other device parameters are similar as in Table 8.1.

It may be instructive to consider the example in Figure 3.20 as developed by Orgassa [34]. For the material parameters chosen, a buffer layer of up to 40 nm hardly induces a total optical loss. About this buffer layer thickness is indeed realized in typical chalcogenide solar cells. Certainly, it is a requisite that the thickness is appropriate to serve other functions of the buffer layer, namely as a surface passivation and a diffusion barrier. In Section 3.3, we develop Rule 3, which states that the emitter should be much more highly doped than the absorber. What is the meaning of this rule if the emitter consists of a combination of layers? In principle, Rule 3 applies to the layer in direct contact to the absorber layer, which is the buffer layer. Thus, the n-type buffer layer should be doped higher than the p-type absorber layer. However, this may not be feasible due to experimental limitations. In addition, there is evidence that a thin resistive buffer layer improves the performance of cells with lateral ﬂuctuations of the diode current [61]. We come back to this point. For the rest of this section let us assume that doping in the buffer layer is low and that the buffer rather acts as a dielectric layer. Then, as explained in Section 2.3.3, there is a potential drop within the buffer layer, much as sketched in Figure 3.2c. This potential drop leads to a decrease in Ep,az=0 (see Section 2.3.3). For the case of NA,a = 1016 cm−3 we can read out a decrease of about 0.1 eV in Ep,az=0 from Figure 2.7c for db = 40 nm. Figure 3.6a indicates that such a decrease in Ep,az=0 combined with a high nominal interface recombination velocity can induce a noticeable decrease of Voc , even for a high absorber/window doping ratio. This consideration underlines that the thickness of a buffer layer with low doping is critical for the device performance as long as there is no Fermi level pinning. In the case of Fermi level pinning close

3.9 Buffer Thickness

to En,az=0 = 0 by donor-like states, the potential drop in the buffer will have no inﬂuence. However, even without Fermi level pinning there exists the possibility to compensate the buffer layer potential drop by a positive sum of the conduction band offsets, that is, χ = Ec w,b + Ec b,a > 0. All of this is expressed by Eq. (2.35). A further possibility to compensate for the potential drop in the buffer layer is to widen the front bandgap of the absorber. This is the subject of Section 3.10. As mentioned above, a lowly doped buffer layer can cure the impact of mesoscopic ﬂuctuations of the diode current. These ﬂuctuations may well be present at least in industrially fabricated cells [86] (see also Section 2.4.9). Rau et al. [61] showed that adding a local series resistance to the cell limits the inﬂuence of spatially non-homogeneous diode currents. Series resistances up to 1 cm2 are appropriate depending on the level of non-homogeneity. The buffer layer can act as a local series resistance of the cell. Consider, for instance, a layer of 40 nm thickness, carrier mobility of 1 cm2 V−1 s−1 and ND,b = 1015 cm−3 . With a calculated series resistance of 50 m cm2 , this layer is appropriate to partly decouple regions of different diode current. Within our study logic (see Table 3.1) we now continue to investigate heterostructures of the absorber/buffer/window type. The buffer layer in the simulated or calculated examples shall have a thickness of 50 nm and a doping concentration of ND,b = 1015 cm−3 . Thus according to Figure 2.7, the buffer acts like a dielectric and there is a potential drop on the buffer. The energy band diagram of this cell is depicted in Figure 3.21a. Ep,az=0 of this cell does not approach Eg,a and accordingly the cell’s Voc is not optimal. A solution for this problem is outlined in Section 3.10.

Ec

Ec

Ev

Ev

−wa (a)

−wa

0db Position z

(b)

Figure 3.21 (a) Energy band diagram of an absorber/buffer/window cell with potential drop in the buffer layer and without front surface gradient. This cell is the starting point for the following design studies

0db Position z and will be further modiﬁed. (b) Similar cell as in (a), but with Ep,az=0 ≈ Eg,a due to Fermi level pinning. With a bulk lifetime of τn0 = 10 ns, the efﬁciency of cell (b) will be limited by bulk recombination.

161

3 Design Rules for Heterostructure Solar Cells and Modules

3.10 Front Surface Gradient

A front surface bandgap gradient aims at increasing the open circuit voltage of a solar cell while retaining the short circuit current. If the bandgap at the location of highest recombination is increased, the diode current will decrease and the open circuit voltage will rise. First, we discuss the case of optimizing a thin ﬁlm cell the Voc of which is limited by IF recombination. According to Figure 3.4, IF recombination increases for Ep,az=0 Eg,a . The absence of absorber inversion with Ep,az=0 Eg,a can be a consequence of the potential drop across the lowly doped buffer layer [see Eq. (2.35)] or of an unfavorable FLP position. In the example of Figure 3.22a, the cell has En,az=0 = 0.33 eV or Ep,az=0 = 0.87 eV due to the potential drop in the buffer layer. The band diagram of this cell is illustrated in Figure 3.21a. The cell efﬁciency is limited to 17.6% by interface recombination (Voc = 0.66 V, Jsc = 35.2 mA cm−2 , FF = 75.9%) as here we put

(a) 0.9 0.85

Voc [ V ]

0.2

0.8 0.75

0.1 0.0 0.3

0.7

(b)

0.2

26

28

−2

Jsc [mA cm ]

30

0.1

32

35

72

68 70

0.1

FF [%]

(d)

0.2

η [%]

0.1

20

0.0 5

16 17

19

6 7 89

18

0.3

64

78

0.2 0.0

66

0.3

76

(c)

34

79

0.0

74

0.3

−∆Ev,front [eV ]

162

18

2

3

4 5 6 7 89

0.1

1 dmin [µm]

Figure 3.22 Simulated solar cell parameters Voc , Jsc , FF, and η (from a–d) for a linear front bandgap gradient in the absorber as a function of Ev,front and dmin (for deﬁnition of dmin see Figure 2.14). Device parameters according to default Table 8.1 but with Sn0,p0 = 107 cm s−1 . The bottom line

cell has a potential drop in the buffer layer and therefore exhibits En,az=0 = 0.33 eV. This cell is mainly limited by interface recombination with the following parameters Voc = 0.657 V, Jsc = 35.2 mA cm−2 , FF = 75.9%, η = 17.6%.

3.10 Front Surface Gradient

again Sn0,p0 = 107 cm s−1 and thus limit the cell by interface recombination. Now, we want to introduce a front surface gradient (FSG) of the type d in Figure 2.14 and to derive a rule for dmin and Ev . Solar cells of the absorber/buffer/window type with different combinations of Ec and dmin are simulated and the JV curves analyzed. Figure 3.22 shows contour plots of simulated values of Voc , Jsc , FF, and η as a function of dmin and Ev . We ﬁnd that the efﬁciency of a cell which is mainly limited by interface recombination can be largely improved by a FSG. With −Ev,front = 0.2 eV and dmin = 50 nm, the bottom line cell can be improved from 17.6 to 20.4% efﬁciency. The improvement is mostly due to an enhanced Voc . For larger dmin , however, the cell efﬁciency even declines below the bottom line value. Obviously this gives: Rule 8: In order to fully or partially deactivate interface recombination, the bandgap of the p-type absorber should be widened by lowering the valence band edge (Ev,front < 0). The region of Ev,front < 0 should be narrow (some tunneling lengths). The FSG increases Ep,az=0 or, which is the same, the hole barrier φb p at the buffer/absorber interface. We note that front gradients of the conduction band are not helpful in order to deactivate interface recombination (unless they completely change the band alignment). The reason is that a conduction band gradient does not alter the interface bandgap. The hole barrier will remain unchanged. For a p-type absorber, design concepts (e) and (g) of Figure 2.14 are not valuable. A cell which is limited by bulk recombination hardly beneﬁts from a front surface bandgap gradient. To the contrary, it can suffer from loss in Jsc due to a positive slope of the quasi Fermi level dEFn /dz > 0. In the logic of this chapter, we realize the bulk limited cell by inducing strong Fermi level pinning at the buffer/absorber interface at a position of En,az=0 ≈ 0. In addition, we reduce the interface recombination velocity to Sn0,p0 = 103 cm s−1 . Thus, the cell with a bulk absorber lifetime of 10 ns has a Voc which is limited by bulk recombination and has solar cell parameters of Voc = 0.74 V, Jsc = 35.0 mA cm−2 , FF = 79.3%, and η = 20.6%. The energy band diagram of this cell is sketched in Figure 3.21b. Before we look at the impact of a FSG, we have to consider the impact of the effective bandgap. A bandgap gradient, modiﬁes the effective bandgap of the absorber layer. According to Figure 3.1 a change of the bandgap is connected with a change of the maximum efﬁciency. If, for instance, the reference cell has a bandgap below the point of maximum efﬁciency in Figure 3.1, a bandgap widening will theoretically and (in the best case) experimentally increase the efﬁciency not only as a result of the bandgap gradient but also as a result of the η(Eg ) dependence. Thus, it is recommended to quantify the beneﬁt of a graded absorber by comparing to a cell which has the same open circuit voltage but no bandgap gradient. Following Gloeckler [114], the normalization can be accomplished by X = X(E, dmin ) − X(const Eg , same Voc )

(3.4)

163

3 Design Rules for Heterostructure Solar Cells and Modules −2

∆Jsc

−1

0.2 0.1

∆FF

−4

−6

0

2 0

0.2

−2 ∆η

0.4

−3

.5 −2

0.3

−1

−1

0.0

−0.5

0.1

1

0.2

−8

0

0.0 0.3

−1.5

−0.5

−2

0.3

−∆Ev,front [eV ]

164

−1.5

0.1 0.0

0

5 6 7 89

2

3

4 5 6 7 89

0.1

1 dmin,front [µm]

Figure 3.23 Simulated normalized solar cell parameters Jsc (mA cm−2 ), FF (%), and η(%) for a linear front bandgap gradient in the absorber as a function of Ev,front and dmin (for deﬁnition of dmin see Figure 2.14).

Device parameters according to default Table 8.1. Strong inversion at the buffer/absorber interface realized by FLP at En,az=0 = 0 eV. Reduced interface recombination velocity of Sn0,p0 = 103 cm s−1 .

where X stands for a solar cell parameter Jsc , FF, and η. dX is the gain of a parameter relative to a cell with identical Voc . This normalization is necessary for cells limited by bulk recombination. Figure 3.23 gives normalized values of Jsc , FF, and η. For the bulk limited cell there is a small efﬁciency gain of 0.4% with the parameter combination EV,front = 0.15 eV and dmin ≈ 0.2 µm. This efﬁciency gain results from a combination of a loss in Jsc and enhancements of FF. The loss in Jsc has been explained in Section 2.3.7. It is due to the positive slope of the electron quasi Fermi level dEFn /dz > 0 in part of the absorber. The small gain in FF is due to a reduction of SCR recombination due to the larger bandgap in the SCR. For dmin > 0.4 µm, all FSGs result in an efﬁciency deterioration mostly due to a reduction in FF. As that part of the absorber which has dEFn /dz > 0 increases with voltage, current collection becomes voltage dependent – leading to a reduced FF. In summary, a FSG is hardly useful to optimize a cell which is limited by bulk recombination. This is even more true if we consider that a bandgap gradient can, under circumstances, induce electronic defects (see next section) – a fact that we neglected hitherto. Finally, we come back to our study logic. The small Ep,az=0 caused by the buffer layer introduced in Section 3.9 can be ‘cured’ by a FSG of −Ev,front = 0.2 eV and dmin,front = 10 nm. This reduces interface recombination and improves the solar cell parameters to Voc = 0.75 V, Jsc = 34.8 mA cm−2 , FF = 77.9%, and η = 20.36%. The energy band diagram of this cell is depicted in Figure 3.24.

3.11 Back Surface Gradients Figure 3.24 Sketch of an absorber/buffer/ window cell with potential drop in the buffer layer and with front surface gradient of dmin = 50 nm and −Ev,front = 0.2 eV.

Ec

Ev

−wa

dmin

0db

Position z

3.11 Back Surface Gradients

The motivation to implement a back surface gradient (BSG) is to reduce the photo current loss and to decrease the diode current. Back surface bandgap gradients have been studied, for example, in Refs. [120–123]. A BSG can optimize the cell in two respects: (i) It can minimize bulk recombination, and (ii) minimize back surface recombination. Whether bulk recombination or back surface recombination is the limiting process depends on the reduced absorber thickness da = (da − wa )/Ln,a . A sensitivity analysis shows that the limiting process is mainly bulk recombination for da > 1 and mainly back surface recombination for da < 1. In Section 2.3.7, we show that a BSG of the conduction band adds a current contribution, JCBM , and thereby increases the photo current. An alternative explanation is that the BSG increases the collection function (see Section 2.5.4.2). A BSG in the absorber can be realized by a chemical gradient (change in chemical composition) in a mixed crystal system. An example is the isopleth system CuInSe2 –CuGaSe2 where the bandgap widens with increasing [Ga]. If this chemical gradient leads to a gradient in the lattice constant, defects may be formed. The total defect density Nd may then depend on the gradient Ec,back /(da − dmin,back ). As a ﬁrst approach we may assume a linear dependence, that is, Nd = Nd0 + Cc Ec,back /(da − dmin,back ) where Cc is a material parameter. Thus, the lifetime may become τ(dmin,back , Ec,back ) =

τ0 Nd0 Ec,back Nd0 + Cc da −dmin,back

(3.5)

In a lattice matched mixed crystal system, Cc is negligible. Thus for the beginning, we assume that the region of the absorber exhibiting a BSG has the same carrier lifetime as the ungraded region.

165

166

3 Design Rules for Heterostructure Solar Cells and Modules

75

0.4

0.7

0.3

0.77

0.3

35.6

0.1

∆Ec,back [eV]

77

77.6

0.2 0.1

77.4

0.0

20.8

77.2 1 21.

0.4 2 0.3 1.2 0.2

−0

0.3

0

0.2

0.4

0.1 0.0 0.3 0

21.2

.6

0.6

0.2

21.1

0.1 20.5 5 6 78 9

2

3

4

5 6 7 89

0.1 (a)

.2

0.4

0.4

0.1 0.0

0

0.0 0.4

.4

77

0.3

0.5

0.1 35.6

0.0 0.4

1

0.2

0

0.2

0.4

1

.4

35

35.2

0.3

0.755 1

0.0 0.4

0

0.1

0.2

0.765 0.76

35.8

0.2

∆Ec,back [eV]

0.75

In the following, we discuss the impact of a BSG for two absorbers with thicknesses of da = 3.0 and 0.5 µm. Thus, we deviate from our default absorber thickness of 1.5 µm for a moment but come back to this thickness at the end of the section. The BSG is realized by a gradient of the conduction band in the range Ec = 0–0.5 eV. With the default electron lifetime of 10−8 s, the absorber has a diffusion length of 1.1 µm. Both cells are not limited by interface recombination as we introduced the FSG with dmin,front = 10 nm and −Ev,front = 0.2 eV, as shown in Figure 3.24. First, we consider the cell with the thick absorber (da = 3.0 µm). The SCR width wa equals 0.36 µm. With the parameters given above, this translates into a reduced absorber thickness of da = 2.4. The reference cell with a homogeneous bandgap of 1.2 eV, that is, without BSG, exhibits an efﬁciency of 20.5%. Thus, the reference cell is already highly efﬁcient. The dominating limitation of the cell is carrier recombination in the absorber bulk. Therefore, in the ﬁrst place we may improve the cell by reducing the bulk recombination. Figure 3.25 gives the results of simulated solar cell parameters as contour plots on an absolute scale (a) and after normalization (b) to a cell with identical Voc .

0.0

2

1 dmin,back [µm]

Figure 3.25 Simulated solar cell parameters Voc (V), Jsc (mA cm−2 ), FF (%), and η (%) (from top to bottom) for an absorber/buffer/window heterostructure with linear back surface bandgap gradient of varying Ec,back and dmin,back (for deﬁnition see Figure 2.14). Absorber thickness 3 µm.

0.2 5 6 7 89

2

0.4 3

4 5 6 7 89

0.1 (b)

2

1 dmin,back [µm]

Device parameters according to default Table 8.1 but with front surface gradient of dmin,front = 10 nm, Ev,front = 0.2 eV (see band diagram in Figure 3.24). (a) Absolute values. (b) Values normalized to a cell without bandgap gradient but with the identical Voc [see Eq. (3.4)].

3.11 Back Surface Gradients

With a BSG, the efﬁciency increases from 20.5 to 21.1%. This absolute efﬁciency increase is also found for the normalized parameters. It results from of a gain in all cell parameters. From the contour plot of the efﬁciency we see that optimally the bandgap gradient extends across the QNR of the absorber and induces a bandgap widening of 0.2 ± 0.1 eV. Thus, the absorber has a dmin,back which corresponds to wa . Why are BSGs with a larger dmin,back not as efﬁcient? The answer is that the limiting recombination takes place in the bulk of the absorber. The larger dmin,back , the less photo generated carriers can beneﬁt from the additional force provided by the bandgap gradient. Using slightly different cell parameters, Gloeckler derived the same rule for a BSG in a thick absorber [114]: Rule 9: For a cell limited by bulk recombination, a back surface bandgap gradient should extend through the QNR of the absorber. Certainly, this rule is based on the assumption that the gradient does not induce further defects and thereby reduce the carrier lifetime [124]. If the material parameter Cc is not negligible, the beneﬁt of a back surface bandgap gradient becomes smaller and will eventually disappear. In Figure 3.26, we simulate this case using Cc = 1010 cm−2 eV−1 . We see that in principle Rule 9 remains valid. The beneﬁt of the BSG, however, is reduced by the reduced carrier lifetime in the grading region. Next, we ask for a bandgap gradient in a thin absorber layer but therefore return to a perfectly lattice matched system (Cc = 0). An example has been calculated in Figure 3.27. The absorber thickness was reduced to 0.5 µm while all electronic parameters are identical as for the device calculated in Figure 3.25. Thus, the reduced thickness of the absorber is da = 0.12. With a back surface recombination velocity of 107 cm s−1 , this cell is limited by back surface recombination. The efﬁciency of the bottom line cell with a homogeneous bandgap of 1.2 eV is only 17.2%, that is,

4e - 09

6e - 09

0.33

0

0

8e-09

.2

−0

0.3 0.2

0

(b)

0.4 0.3 0.2 0.1 0.0 0.4 0.3 0.2 0.1 0.0 5

09 2e-

∆Ec,back [eV]

(a)

6 7 89

2

3

4 5 6 7 89

0.1

2

1 dmin,back [µm]

Figure 3.26 Device as in Figure 3.25 but with gradient dependent carrier lifetime. (a) Carrier lifetime in the grading region for a gradient with Ec,back and dmin,back . Without bandgap gradient, the absorber lifetime is 10−8 s (default value). With increasing

Ec,back and dmin,back the gradient becomes larger and the carrier lifetime is reduced according to Eq. (3.5). (b) Efﬁciency normalized to a cell without bandgap gradient but with the identical Voc (see Section 3.10).

167

0.7 9

0.4

0.78

3 Design Rules for Heterostructure Solar Cells and Modules

168

Voc [V]

0.3 0.2

0.77 0.76 0.74

0.1

0.75

0.0

Jsc [mA cm−2] 32.5

0.1

33

FF [%]

0.3 0.2 0.1

77.5 76.5

0.0 0.3

0.3 0.2 0.1

∆FF [%] 1

1.5

1

0.0

η [%]

0.15

0.2

19.5 18.5

19

0.20

0.25

0.30

dmin,back [µm]

∆η [%]

1.5

0.3 20

0.1

(a)

0.4

0.4 19

0.2 0.0 0.10

∆Jsc 2.5 [mA cm−2]

1

0.5

0.0 ∆Ec,back [eV]

0.4

2

0.2

0.0 77

∆Ec,back [eV]

0.1

1.5

0.3

32

0.2

0.4

1

0.4

0.5

0.3

5 31.

33.5

0.4

2

0.1 0.35

0.0 0.10

0.40

(b)

Figure 3.27 Similar device as in Figure 3.25 but with absorber thickness of 0.5 µm. Linear back surface bandgap gradient in the absorber with dmin,back = 0–0.5 eV and Ec,back = 0.1–1.4 µm. Back surface recombination velocity Sn0,bc = 107 cm s−1 .

1

0.5

0.15

0.20

0.25

0.30

0.35

0.40

dmin,back [µm]

Other device parameters according to default Table 8.1. (a) Graphs showing Voc (V), Jsc (mA cm−2 ), FF (%), and η(%) (top to bottom). (b) Graphs showing Jsc , FF, and η normalized to a cell without bandgap gradient but with the identical Voc (see text).

3.2% lower than the thick bottom line cell in Figure 3.25. In Figure 3.27, the BSG was varied in the ranges Ec,back = 0–0.5 eV and dmin,back = 0.1–0.4 µm. It can be seen that with a bandgap widening of 0.25 ± 0.1 eV and dmin,back = 0.4 eV, the absolute cell efﬁciency rises to simulated 20.2%. The efﬁciency gain is caused by a gain in Jsc and Voc . Thus, the optimum gradient, as in the case of the thick absorber, uses a bandgap widening of about 0.3 eV. But for the low thickness absorber which is limited by back surface recombination, the gradient does not extend through the absorber but is restricted to a range near the BC. The same conclusion can be drawn from plots of the relative cell parameters on the right hand side of Figure 3.27. In order to show the systematic for the optimum location of dmin,back in absorbers of different thicknesses, the value of dmin,back has been varied for different absorber thicknesses but constant Ec,back = 0.3 eV. Figure 3.28 gives the efﬁciency for absorbers with thicknesses between 0.9 and 2.0 µm (reduced thicknesses of 0.5, 0.8, 1.0, 1.2, 1.5 µm). The plot of the cell efﬁciencies points out that the best value of dmin,back (giving the highest efﬁciency) moves from the back side of the absorber to the front side of the absorber when the absorber thickness is increased.

3.11 Back Surface Gradients

Efficiency η [%]

wa 21.2 20.8 20.4 20.0 0.0

d'a = 0.5, 0.8, 1.0, 1.2, 1.5 0.5

1.0 dmin,back [µm]

1.5

Figure 3.28 Simulated solar cell efﬁciency for a linear back surface bandgap gradient of Ec,back = 0.3 eV in the absorber as a function of dmin,back (for deﬁnition of dmin,back , see Figure 2.14). Different reduced absorber

2.0

thicknesses. Device parameters according to default Table 8.1 with front surface gradient (dmin,front = 10 nm, Ev,front = 0.2 eV). For a typical band diagram see Figure 3.29.

Ec

EF Ev

−da

−dmin,back

0

Position z Figure 3.29 Energy band diagram of an absorber/buffer/window cell optimized with a front surface gradient to reduce interface recombination and with a back surface gradient to reduce volume and back contact

recombination. dmin,back = 0.5 µm was selected according to Figure 3.28 where it is da = 1.17. This cell represents the optimized device of this study. Other parameters as in Table 8.1.

Thus for cells being limited mainly by back surface recombination (da < 1) we can formulate: Rule 10: For a cell limited by back surface recombination, the back surface bandgap gradient should be restricted to the back surface near region of the absorber. Rule 10 remains valid if the gradient leads to a reduction in carrier lifetime in the grading region. The phenomenological reason for the inferior function of a BSG extending through the complete thin absorber with da < 1 is the lower short circuit current due to the higher bandgap in average. This higher bandgap is not

169

3 Design Rules for Heterostructure Solar Cells and Modules

fully compensated by an increased Voc . If the BSG is restricted to the back surface near region, the effective ﬁeld is very high near the BC, which acts as an electron mirror. The default absorber thickness of this study is da = 1.5 µm, which gives a reduced thickness of 1.17 µm (Ln,a = 1.1 µm). According to Figure 3.28, the optimum dmin,back is about 0.5 µm. Thus, the optimized cell has a double grading structure (as illustrated in Figure 3.29) which gives an efﬁciency of 20.4% with parameters Voc = 0.787 V, Jsc = 33.76 mA cm−2 , and FF = 76.8%. It may be instructive to look at the different contributions for this efﬁciency in more detail. In Figure 3.30 we see that without a bandgap gradient, the device performance is limited to about 17%. (For the band diagram of this cell, see Figure 3.21a.) Introduction of

Voc [ V ]

0.80 0.75 0.70

FF [%]

Jsc [mA cm−2]

0.65 34.8 34.4 34.0 33.6 77 76

21 η [%]

20 19 18

Figure 3.30 Simulated solar cell parameters for an absorber/buffer/window heterostructure with a thin absorber of da = 0.5 µm. Material parameters as in default Table 8.1. Case discrimination without bandgap gradi-

FSG / BSG

FSG

BSG

17 No Gradient

170

ents, with back surface bandgap gradient of dmin,back = 0.1 µm, Ec,back = 0.2 eV (BSG), with a front surface bandgap gradient of dmin,front = 0.01 µm, Ev,front = 0.2 eV (FSG) and with a combination of BSG and FSG.

3.12 Monolithic Series Interconnection

a back surface bandgap gradient (BSG) alone decreases the current but increases the FF. Introduction of a FSG alone improves the open circuit voltage due to the suppression of interface recombination. We also see a slight increase in FF. With the FSG, the cell is ‘cured’ from interface recombination. Adding to this cell the BSG leads to the highest efﬁciency and to the band diagram as in Figure 3.29. With the given absorber carrier lifetime this cell can only be improved by appropriate light trapping structures [65]. If indeed the absorber carrier lifetime could be increased we could reduce the bulk recombination rate. It is interesting that thereby, we could bring the cell back to the situation where its open circuit voltage is limited by interface recombination. As stated in Section 2.4.6, it is the combination of J00 , A, and Ea values which determines the dominant recombination process limiting the open circuit voltage. Similar to a back surface bandgap gradient is the function of a back surface ﬁeld [125]. It can be realized by an absorber layer of increased doping concentration near the BC. However, it can also be realized by an accumulation layer at the absorber/BC interface. The result is an electric ﬁeld which reduces the photo current loss and decreases the diode current. Both a p+ layer and an accumulation layer lead to an additional potential drop at forward bias. In order to calculate the beneﬁt of a back surface ﬁeld, we refer to Ref. [126]. We are interested in the relative beneﬁt of a back surface ﬁeld for different absorber thicknesses. The electric ﬁeld is realized by an accumulation layer at the BC which is due to contact formation and is characterized by the barrier height φb − φb 0 where φb 0 is the barrier height under ﬂatband condition. For φb − φb 0 < 0, an electron barrier has been formed at the BC. At the same time we can inspect the opposite case: a depletion layer with electron barrier φb − φb 0 > 0 (see Section 2.4.7). In Figure 3.31 we see that the beneﬁt of a back surface ﬁeld (φb − φb 0 < 0) is modest for reduced absorber thicknesses of da > 1. This complies with the ﬁnding of Figure 3.11 which shows that recombination at the BC hardly reduces the cell parameters for τn0,p0 = 10 ns and large absorber thickness. For da < 1, the positive effect of a back surface ﬁeld becomes obvious. However, a hole barrier reduces the efﬁciency to 14.8%. If such a hole barrier cannot be avoided due to the band alignment at the absorber/BC interface, a back surface bandgap gradient is required in order to partly compensate for the efﬁciency loss.

3.12 Monolithic Series Interconnection

Monolithic series interconnection is one of the cost relevant advantages of thin ﬁlm technologies over wafer-based technologies. Frequently used interconnection schemes are shown in Figure 2.43. For all interconnection schemes, the cell length xcell , as deﬁned in Figure 2.41c, has to be optimized. This optimization has a number of variables as given in Table 3.2. We see that there is always a trade-off between FF losses and Jsc losses.

171

3 Design Rules for Heterostructure Solar Cells and Modules

φb –φb0 [eV ]

172

0.2 0.1 0.0 −0.1 −0.2 0.2 0.1 0.0 −0.1 −0.2 0.2 0.1 0.0 −0.1 −0.2 0.2 0.1 0.0 −0.1 −0.2

2 0.7 0.73

Voc [V] 0.74

0.745

0.75

0.76

32

Jsc [mA cm−2]

33 34

35

35.5

74

75

FF [%] 76 77

17

η [%]

18 19

2

20 20.8

3

20.5

4 5 6 7 89

2

1 da′ [µm] Figure 3.31 Contour plot of the solar cell parameters (from top to bottom) Voc , Jsc , FF, and η as a function of the absorber thickness and the back contact barrier φb . Device parameters as in default Table 8.1. Front surface gradient as discussed in

Section 3.10 and as shown in Figure 3.24 has served to deactivate interface recombination. A hole (electron) barrier is given for φb > 0 (φb < 0). Further parameters: RBC = 0 and Sn0,p0,bc = 107 cm s−1 .

Table 3.2 Parameters for the optimization of the monolithic series interconnection given as interrelated couples which deﬁne a trade-off between FF and Jsc losses. The interrelated parameters are in italics.

Couple number

FF loss

Jsc loss

1

Window layer sheet resistance, R , w Shunt conductance across P1, GP1 Contact resistance between window and back contact layer, Rw,BC Cell width, xcell

Window layer absorbance, 1 − Rw − Tw Width xP1 of the interconnection region Width xP2 of the interconnection region

2

3

Number of cells per module, Ncell = xmod /xcell

η [%]

0.96

14

(b) 2 10 Rw [Ω/sqr.]

4

6 8 100

(η-ηMod) /η [%]

6 8

16

24 8 22

7

20

6

18

5 20

(c) Figure 3.32 (a) Optical transmission of ZnO ﬁlms (after [128]) (markers) together with a ﬁt to the data according to the algebraic function T = T0 − (Rw,0 /Rw )m where T0 = 1, R0 = 0.28, and m = 0.753 has been used. (b) Cell efﬁciency η as a function of short circuit current density of simulated absorber/window heterostructures (simulation parameters see Table 8.1). Module efﬁciency

Xcell [mm]

0.88 4

(a)

20 18

0.92

ηMod,

Transmission, Tw

3.12 Monolithic Series Interconnection

25 30 Jsc [mA cm−2]

35

ηMod simulated for an optimized cell width xcell using the program Design of M. Burgelman and A. Niemegeers [129]. Optimization parameters are Rw = 8 /sqr., Tw = 0.92, Rw,BC = 10−3 cm2 , and interconnect loss 400 µm. (c) Relative efﬁciency loss due to series interconnection and optimum cell width xcell .

The ﬁrst trade-off concerns the window layer, which is characterized by an optical transparency Tw (λ) at a given sheet resistance, R,w : due to free carrier absorption and due to high impurity concentrations, the transparency reduces with lower sheet resistance. This effect is wavelength-dependent. Thus, in order to derive an empirical function Tw (R,w ), the optical transmission should be weighted by the normalized internal quantum efﬁciency of a cell with particular absorber bandgap [127]. Figure 3.32a gives an example of the transparency of experimental ZnO thin ﬁlms for an absorber bandgap of 1.15 eV as a function of R,w . Increasing the window layer ﬁlm thickness reduces R,w and thus reduces the loss in FF, but also increases 1 − Tw and thus the loss in Jsc . Trade-off 1 in Table 3.1 is interrelated with trade-off 3 since the cell length xcell plays a role. We note in passing that increasing the window layer thickness also increases the production costs. For the second trade-off, optimization of xcell depends on the contact resistance of window and BC Rw,BC as well as the shunt conductance GP1 (see Figure 2.41d). GP1 depends on the sheet conductance of the absorber and the width xP1 . Rw,BC depends on the speciﬁc contact resistance and the width xP2 . Thus, the free parameters are xP1 and xP2 . GP1 and Rw,BC decrease with increasing width of P1 and P2. The price to be paid is a higher area loss and therefore a lower Jsc . The third trade-off for the cell width concerns the relative area loss due to the interconnection and the distributed series resistance: if the cell length is too small,

173

174

3 Design Rules for Heterostructure Solar Cells and Modules

the area loss is too large thus diminishing Jsc . If the cell length is too large, the distributed series resistance diminish the FF. An optimum of xcell occurs which depends on the Jsc of the elementary diode and thus on the absorber bandgap. If material cost aspects are taken into account, a cell width somewhat smaller than the optimum width may be selected. In order to compare different absorber bandgaps, we assume that sheet conductance and optical transparency of the window layer are ﬁxed. This is a simpliﬁcation that slightly discriminates large bandgap absorbers since they hardly suffer from free carrier absorption in the window layer. Nevertheless, some valid conclusions can be drawn from the simulation. We use the software package Design of M. Burgelman and A. Niemegeers [129] in order to obtain plots of module efﬁciency ηmod and xcell versus the absorber bandgap, Eg,a . Figure 3.32b reveals that the module efﬁciency is around 20% below the cell efﬁciency. This is due to area loss, resistance loss, and transmission loss. The efﬁciency loss due to monolithic series interconnection is a structural problem. Besides optimizing the parameters in Table 3.1, it is proposed to add a contact metal grid on top of the window layer of each interconnected cell [130]. This allows a reduction in the window thickness, thereby reducing optical losses at constant resistance losses. A metal grid, however, requires an additional (and not simple) process step. Figure 3.32c reveals that the optimum cell length increases with decreasing Jsc and thus with increasing absorber bandgap. Also, the relative loss in module efﬁciency is smaller for lower Jsc . In reality, this effect is even larger since transparency losses due to free carrier absorption are overestimated for the larger absorber band gap. Due to the large sum of input parameters for the monolithic interconnection, it is not possible to give a design rule. A rough guide may be the cell width as a function of bandgap, which has been plotted in Figure 3.32c. Table 3.2 already suggested that the efﬁciency loss by monolithic series interconnection is due to reductions in FF and Jsc . Indeed, the Voc,mod of the module is simply given by Ncell × Voc,cell . Depending on the application, a particular Voc,mod may be required for a given module size. This would introduce a constraint on Ncell and would require to re-assess window layer thickness, interconnection width and cell length.

175

4 Thin Film Material Properties Within this chapter we give an account of absorber, buffer, and window properties in as far as they are relevant for the function of chalcogenide cells. Since several properties of the respected materials are still not completely known, we emphasize that this account is a snapshot of the current understanding rather than a completed data base. For preparation aspects we refer to Chapter 5. Nevertheless, here we ﬁrst want to rationalize why several chalcogenide ﬁlms can be obtained through relatively easy ﬁlm synthesis methods – a fact which makes chalcogenide ﬁlms suitable for thin ﬁlm photovoltaics and which is one of the reasons for their high solar cell efﬁciencies.

4.1 AII –BVI Absorbers

Compound semiconductors of the type AII –BVI form a large group of materials with a variety of bandgaps (Figure 4.1). Within this group, only CdTe has a bandgap appropriate for a single junction solar cell (1.0 eV < Eg < 1.5 eV). This is why the following section focuses on CdTe material properties and only touches other binary phases and solid solutions as far as they are relevant for CdTe-based solar cells. 4.1.1 Physico-Chemical Properties

CdTe crystallizes in the zincblende (ZB) structure with space group F43m (Figure 4.2a). It is the only compound in the Cd1−x Tex phase diagram (Figure 4.3a). If grown on hexagonal substrates, thin layers of CdTe may also be stabilized in the wurtzite structure [131]. CdTe exhibits a high formation enthalpy (heat of fusion) of about −100 kJ mol−1 (see Table 4.1), which makes it thermodynamically very stable. The CdTe melting temperature of 1365 K is much higher than that of Cd (594 K) and Te (723 K). CdTe sublimates and evaporates congruently via the equilibrium reaction CdTe ⇔ Cd + 1/2Te2 . The high thermodynamic stability together with its congruent evaporation allows CdTe to be formed by various preparation methods. 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

4 Thin Film Material Properties 2.8 CdS Band gap [eV]

176

ZnTe

2.4

2.0 CdSe

CdTe

1.6

1.2 5.6

5.8

6.0

6.2

6.4

6.6

Lattice parameter [Å]

Figure 4.1 Bandgap vs lattice constant of selected II–VI semiconductors of hexagonal (hexagons) and cubic (squares) crystal structure.

AIBIIICVI2

AIIBVI

(a)

AII BVI

(b)

AI

BIIIAI + 2VAI

AI2BIICIVDVI4

(c)

AI

(d)

AI

BIII

BII

BIII

CVI

CIV

VAI

DVI

BIIIAI CVI

Figure 4.2 Crystal structures of (a) zincblende (ZB), (b) chalcopyrite (CP), (c) kesterite, (d) chalcopyrite structure with defect complex of BIII AI –2VAI [134].

McCandless has discussed the thermochemical aspects of CdTe processing [132, 133]. Table 4.1 gives the chemical diffusion coefﬁcients at 500 ◦ C for some impurity elements which are relevant for the solar cell process. These values are measured on monocrystalline CdTe where the effect of grain boundary diffusion is negligible. Among the impurities listed, Cu and Na have the highest reported diffusivity

4.1 AII –BVI Absorbers

Liquidus Solidus

1600 1365 K

1200 z

1000 800

z+w

w

778 K

600

(a)

1.6 1.5

1.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x x (b) (c)

Figure 4.3 Atmospheric pressure pseudobinary phase diagrams of (a) Cd1−x Tex (after Jianrong et al. [135]), (b) CdTe1−x Cl2x (after Tai et al. [136]), and (c) CdTe1−x Sx (after Ohata et al. [137]) (z – zinc blende, w – wurtzite). The calculated energy bandgap of CdTe1−x Sx is given as a dashed line on the right axis [138].

outperforming the diffusion of Cl and In. The Cu diffusivity appears to be further promoted by grain boundaries as the diffusivity is higher in thin ﬁlms than in crystals [157]. S has the lowest diffusion constant. It was found, however, that under a CdCl2 : O2 : Ar vapor pressure, the diffusion coefﬁcient of S is increased by 3 orders of magnitude [158]. The large chemical self-diffusion coefﬁcient DCdTe guarantees the growth of homogeneous CdTe ﬁlms. The pseudobinary phase diagram of CdTe1−x Sx (Figure 4.3c) is so important as chemical intermixing at the device CdTe/CdS interface has been observed. The stable crystallographic structure of Te-rich CdTe1−x Sx is the ZB structure and that of S-rich CdTe1−x Sx is the wurtzite structure. The phase diagram in Figure 4.3c shows a large miscibility gap between CdTe and CdS. This miscibility gap becomes smaller at high temperatures. At 673 K it ranges from x = 0.06 to 0.97 [138]. However in a metastable state, thin solid solution ﬁlms with all values of x in the ZB structure can be formed [159]. Within each structure type, the lattice parameter of CdTe1−x Sx follows Vegard’s rule [138]. The pseudobinary phase diagram of CdTe1−x Sex (not shown) has been studied experimentally in the temperature range 1070–1570 K [160]. It exhibits a ZB phase ﬁeld on the CdTe side and a wurtzite phase ﬁeld of the CdSe side in accordance with the crystal structures of the pure binaries. At 1070 K, the border between ZB and wurtzite is at around x = 0.5. Between the two phase ﬁelds there is a narrow (3%) two-phase region at equilibrium. Other publications report on a more extended two-phase region where the minimum bandgap occurs for the ZB structure [161]. Theoretical calculations point out a positive mixing enthalpy at T = 0 [162] and a miscibility gap below 450 K which widens toward lower temperatures [163].

Eg [eV]

T [K]

1400

177

178

4 Thin Film Material Properties Table 4.1

CdTe physico-chemical and opto-electronic properties.

Property

Symbol

Value or range

References

Melting point Heat of fusion (300 K) Entropy (300 K) Space group Lattice constant Thermal expansion coefﬁcient (300 K) Density Optical band gap (300 K) Temperature dependence of Eg,a Electron effective mass Hole effective mass

Tm H0f S0 – a α ρ Eg dEg /dT mn ∗ /m0 mlp ∗ /m0 mhp ∗ /m0 Nc Nv ni µn µp ε/ε0 ε∞ ε0 n Lα0 U0 τrad D at 773 K (cm2 s−1 )

1365 K −100 kJ mol−1 95(1) J K−1 mol−1 F43m 6.48 A˚

[139] [140] [140] – [141] [142] [141] [143–145] [145] [146] [147] – Calculated Calculated Calculated [147] [147] [147] [147] [147, 148] Calculated Eq. (2.60) Calculated [149] [150] [151] [152] [153] [154] [155] [156]

Effective density of states Intrinsic carrier concentration (300 K) Electron mobility Hole mobility Static dielectric constant Optical dielectric constant Refractive index (600 nm) Absorption length Radiative recombination rate Radiative lifetime Bulk diffusion coefﬁcients

5.3 × 10−6 K−1 5.86 g cm−3 1.49(1) eV −4 × 10−4 eV/K 0.098 0.145 0.82 8 × 1017 cm−3 2 × 1019 cm−3 1.1 × 106 cm−3 1050 cm2 V−1 s−1 104 cm2 V−1 s−1 10 7.1–7.3 3.1 1.2 × 10−4 cm 28 cm−3 s−1 1 × 10−6 s CdTe: 3 × 10−7 Zn: 8 × 10−12 Cu: 5 × 10−9 S: 4 × 10−15 Cl: 8 × 10−11 In: 2 × 10−11 Na: ∼10−9 Au: 6 × 10−12

Alloying of CdTe and ZnTe can lead to an absorber layer with larger band gap which may be used in a top cell of a tandem device. CdTe and ZnTe both crystallize in the ZB structure. Thus, no separated phase ﬁelds are expected but it remains the question of a miscibility gap. Such a gap has been pointed out by Haluoi et al. [164] to extend up to around 570 K. Random ordered thin ﬁlms, in contrast, have been prepared in the complete solid solution system [148, 165, 166]. The minimum temperature of miscibility thus may depend on the preparation method. Due to the important role of the chlorine treatment for CdTe solar cell, we are interested in the CdTe1−x Cl2x phase system. A typical chlorine treatment temperature is 670 K. The equilibrium phase diagram after Ref. [136] is shown in

4.1 AII –BVI Absorbers

Figure 4.3b. A eutectic point at x = 0.74 is depicted with a melting temperature of 778 K. In the presence of oxygen, the melting temperature may be even lower [167]. Presently, it is not clear if the chlorine treatment leads to a liquid phase recrystallization of the CdTe ﬁlm at 670 K. 4.1.2 Lattice Dynamics

Raman spectroscopy is a powerful tool for material characterization. The vibrational spectrum of ZB crystals with point group T2d consists of two three-fold degenerate zone-center vibrational modes of symmetry F2 [168]. Of these, three are acoustic modes. The optic mode is split into transversal optic (TO) and longitudinal optic (LO) contributions. Thus for randomly oriented ﬁlms, the Raman spectrum of CdTe exhibits two lines at around 143 cm−1 (TO) and 167 cm−1 (LO). In Raman backscattering geometry, the dipole selection rules are as follows: from 100 LO scattering is allowed but TO is forbidden, from 110 LO is forbidden but TO is allowed, and from 111 both LO and TO are allowed. Nevertheless, the forbidden phonon LO mode from 110 can be observed for resonant Raman scattering [169]. The phonon dispersion curve for CdTe has been calculated, for example, in Refs. [170, 171]. The LO mode exhibits of slope with dE/dk < 0 around the point. Hence, we expect that in the case of high defect density and a resulting phonon conﬁnement, the A1 mode becomes asymmetric toward smaller wavenumbers – in accordance with the experiment [172]. An indication for crystal quality of CdTe material is the number of higher-order LO lines which can well be on the order of 5 [166]. A collection of Raman lines from secondary phases is given in Table 4.2. For single crystals of CdTe and its alloys, Raman spectroscopy often reveals Te precipitates by the occurrence of the Te A1 mode at around 122 cm−1 [169, 173]. The solid solution systems Cd1−x Znx Te [166, 174, 175], Cd1−x Mnx Te [174], CdTe1−x Sex [176], and CdTe1−x Sx [177] all exhibit a two-mode behavior. Table 4.2 Vibrational modes of CdTe and main vibrational modes of related compounds detected by Raman scattering.

Phase/references

Vibrational modes (cm−1 )

CdTe Te [173, 178, 179] CdTeO3 [180] TeO2 [181] CdO [182]

143 (TO), 167 (LO) 92, 127, 143 650a, 760a 148, 393, 648 270, 400, 480, 960

a

Broad band.

179

180

4 Thin Film Material Properties

4.1.3 Electronic Properties

The electronic band structure of ZB CdTe has been calculated by Chelikowsky and Cohen [183]. The valence band maximum at the point of CdTe is four-fold degenerate and splits into two bands at k = 0. The heavy hole band with an effective mass m∗ hp and the light hole band with an effective mass m∗ lp . At the point the two-fold degenerate spin-orbit split-off band is 0.96 eV below the valence band maximum. Thus, only the heavy and light hole bands need to be considered here. Table 4.1 gives the effective masses of electrons and holes for CdTe. In the limit of parabolic bands we make use of 2πm∗n,p kT (4.1) Nc,v = 2 h2 in order to calculate the effective density of states of conduction and valence band (Table 4.1). With these values, we calculated the intrinsic carrier concentration. For crystalline CdTe an electron mobility of 1050 cm2 V−1 s−1 and a hole mobility of 104 cm2 V−1 s−1 have been reported [147]. For polycrystalline thin ﬁlms of device quality, hole mobilities of ∼5 for Cu-doped ﬁlms and ∼50 cm2 V−1 s−1 for undoped ﬁlms have been measured. These values are lower than the peak values listed in Ref. [147] – an observation that has been ascribed to carrier scattering at charged grain boundaries [184]. Thus, using 40 cm2 V−1 s−1 for the hole mobility and 320 cm2 V−1 s−1 for the electron mobility in device simulation appears reasonable [185]. In our general simulations of a chalcogenide device in Chapter 1 we will use even more conservative values (see default Table 8.1 ). 4.1.3.1 Practical Doping Limits CdTe is recognized as an amphoteric semiconductor which can be doped n-type and p-type. Electron concentrations and hole concentrations of up to 1018 cm−3 have been achieved by arsenic doping [186] and chlorine doping [187], respectively. In polycrystalline CdTe absorber layers of up to date solar cells, however, hole concentrations are only in the range of 1015 cm−3 [188]. Figure 4.4 gives calculated defect transition energies in the bandgap of CdTe [189]. The error of calculated transition energies is about 0.1 eV. We differentiate between intrinsic defects and impurity levels. Intrinsic defects in the form of interstitials, vacancies, and antisites can act as donors or acceptors provided that their transition energies are shallow. Figure 4.4 reveals that most of the intrinsic defects have large transition energies and therefore may lie between the demarcation levels for recombination (Section 2.3.6). The single ionized Cd vacancy is the shallowest acceptor energetically situated at Ev + 0.13 eV. We conclude that based on the calculated energies in Figure 4.4 intrinsic p-type doping requires a Cd-poor preparation. The hole concentration will be limited by the large defect transition energy of VCd (0/−) . In principle, impurities in CdTe can provide more shallow defect levels. The character of the substitutional impurities in Figure 4.4 is in accord with this rule of

4.1 AII –BVI Absorbers

Ec

1.5

CdTe

Cui

AlCd lnCd

Nai

(2+ /0)

(+/0)

GaCd TeCd

ClTe

(2+ /0)

TeCd

(2+ /+)

VTe

(2+ /0)

Tei

(0 /2−)

VCd VCd

(−/2−) VCd + Cui

1.0 Energy [eV ]

(+ /0)

Tei

(+/0)

0.5

0.0 Figure 4.4

Ev

(0/−)

VCd + ClTe

CuCd AuCd AgCd NaCd

(0/−)

Sb Te As Te P Te N Te

(0/−)

Theoretical values of electronic defect levels in CdTe. After Ref. [189].

thumb. Substitutional impurities with an electron excess are donors and with an electron deﬁcit are acceptors. Shallow acceptors are NaCd as well as NTe and PTe . (+/0) Since, however, the defect transition energy of Nai is very similar to the one (0/−) of NaCd , acceptor-type NaCd defects may be compensated by donor-type Nai defects [189]. Chlorine treatment (Section 5.1.5) and back contact formation (Section 5.1.8) are important process steps for high efﬁciency devices and are known to change the hole concentration in the absorber. They can lead to a Te and the in-diffusion of O, Cl, and Cu atoms. Cu and Cl can form defect complexes with VCd provided that the material is Te-rich. Indeed, the VCd –ClTe defect complex has a relatively small activation energy (see Figure 4.4). Cu atoms can behave as amphoteric impurities, that is, Cu is a donor in the interstitial site (Cui ) or an acceptor when substituting Cd (CuCd ). The defect complex VCd –Cui is an acceptor, the complex CuCd –Cui mentioned in Ref. [190] would be neutral and could account for the low doping efﬁciency of Cu atoms. Nevertheless, Cu is believed to be a p-type dopant in CdTe [184, 191, 192]. While oxygen is not expected to form a shallow dopant, there are several reports on the doping effect of O2 used as ambient during CdTe deposition [11, 193]. The hole concentration with oxygen doping can reach 1015 cm−3 while without oxygen it is 1 order of magnitude lower. Since the oxygen concentration in the ﬁlms is 1019 –1020 cm−3 [194] we may not speak of a typical doping effect. The role of

181

182

4 Thin Film Material Properties

oxygen in CdTe is discussed in Section 5.1.6. It was reported that the p-type doping also increases with substrate temperature by creation of native defects [195]. 4.1.3.2 Defect Spectroscopy Defect levels in CdTe solar cells have been investigated by use of admittance spectroscopy (AS) and deep-level transient spectroscopy (DLTS). We concentrate on these methods as they are appropriate to determine absolute defect densities. Table 4.3 gives selected defect energies found in device quality CdTe absorbers in the solar cell. Note that more defects have been found in samples with non-optimum CdCl2 treatment [196], without nitric–phosphoric acid (NP) etching step [197], or in cells with a Cu treated back contact [198]. For cells which may exhibit a large back contact barrier (e.g., not etched cells), impedance data should be interpreted based on an equivalent circuit which includes the back contact depletion layer (see Ref. [199] and Section 2.6.4). A further inﬂuence factor on the measured defect density is the chemical environment of the CdCl2 treatment. In Ref. [200], different defects have been discriminated for cells being CdCl2 treated in air or in vacuum. The ﬁrst observation is that their number is large although Table 4.3 presents only a selection of defects. Only one tentative assignment to a microscopic defect has been given. The second observation is that the defect densities appear to be rather small. Values exceeding 1014 cm−3 have been reported for cells which had no CdCl2 treatment [196]. Otherwise, the defect densities are in the range of 1013 cm−3 . Using AS, no defect density was found in Ref. [196] for a cell with good CdCl2 treatment. In order to estimate the impact of the found defects on the device performance, the capture cross sections for electrons and holes must be known. So far, however, these have not been measured with ample precision; instead the capture cross sections are rather ﬁtting parameters. Burgelman et al. conclude that defect densities by DLTS are generally measured too small and introduced deep acceptors of 1015 cm−3 in order to simulate the measured JV curves [200]. A clear relation between measured defect densities and device performance has yet not been found. Thus, in order to simulate a state of the art CdTe solar cells one should rather use measured carrier lifetimes as input parameters for the simulation. 4.1.3.3 Minority Carrier Lifetime The lifetime of minority carriers in the bulk of the absorber can determine the open circuit voltage of the cell (as long as the device is not limited by interface recombination). In Section 2.4.6 it is shown that a lifetime of τ ≈ 10−9 s roughly marks the border between recombination in the SCR and recombination in the QNR as the limiting processes for the diode current. Therefore, knowledge about the lifetime is very important for a comprehensive understanding of a device. Minority carrier lifetimes have been investigated using photoluminescence (PL) decay measurements or photoconductivity transients. Specimens were either bare chalcogenide absorbers or ﬁnished solar cells. Often, the decay curve does not

Defect energies measured using admittance spectroscopy (AS) and deep-level transient spectroscopy (DLTS) in device quality CdTe solar cells. Hole or electron trap assignment as derived from DLTS measurement.

Table 4.3

AS

–

–

–

0.51 [197]

0.43 [197]

–

DLTS

0.76 [201]

0.74 [200]

0.72 [200]

0.5 [200]

0.44 [200]

0.42 [200]

Trap Treatment

h CdCl2 in O2

h CdCl2 in vacuum [200]

h CdCl2 in air [200]

h CdCl2 in vacuum [200]

e CdCl2 in air [200]

e CdCl2 in air [200]

Assignment

–

Density (cm−3 )

–

–

4 × 1012 [200]

–

5 × 1013 [200]

–

3 × 1011 [200]

–

6 × 1012 [200]

–

4 × 1012 [200]

– 0.27–0.35 [201, 202] e,h HgTe : Cu [202]

–

7 × 1012 [201]

0.18 [197] 0.185 [200] h CdCl2 in air [200] no NP etch [197] –

2 × 1012 [200]

0.13 [196, 198] 0.13 [200]

0.11 [200] h CdCl2 in air [200]

–

3 × 1012 [200] 4.1 AII –BVI Absorbers

h CdCl2 in air [200] Poor CdCl2 [196] V− Cd [198] + V− Cd − ClTe [198] 1 × 1011 [200] 2 × 1013 [196]

–

183

184

4 Thin Film Material Properties

follow a simple mono-exponential time dependence [35, 36, 203, 204]. Instead, it needs to be described by at least two exponentials where the ﬁrst exponential can ﬁt the initial decay and the second the late decay. Under the best conditions, the PL decay would reﬂect bulk recombination and grain boundary recombination. Both quantities are aggregated in an effective lifetime, τeff , which limits the device performance. Unfortunately, other effects can reduce the measured lifetime such as surface recombination and charge carrier separation at internal electric ﬁelds. If surface recombination is large on the bare chalcogenide absorber but is much smaller in the solar cell, the lifetime of a bare absorber would be measured erroneously small. The same is the case for charge carrier separation in a depletion layer. Charge separation quenches the radiative recombination and the measured lifetime comes out too small [205]. Therefore, an assignment of the PL decay to the bulk carrier lifetime is not straightforward. Empirically, however, it was found that lifetime values calculated from the PL decay time constants are correlated with the device Voc both for chalcopyrite (CP) [206] as well as for CdTe based devices [36]. This may justify to label measured decay times as the limiting bulk minority carrier lifetimes. Table 4.4 gives measured PL decay times for CdTe absorbers. The initial PL decay of CdTe samples or solar cells was described by a mono-exponential decay function [36]. The derived lifetime values are at best 1 ns. They show a dependence on CdCl2 treatment, sulfur diffusion [207], oxygen concentration [36], and back contact deposition [208]. According to Figure 2.27, we expect up to date CdTe solar cells to be limited by SCR recombination.

Table 4.4 Photoluminescence decay times for various chalcogenide absorbers and solar cells. All decay times have been obtained from ﬁts to mono- or bi-exponential functions. For the excitation conditions, see references.

Absorber Cu(In1−x Gax ) Se2 , x = 0.3 − 0.4 Cu(In1−x Gax ) Se2 , x = 0.3 − 0.4 CuInSe2 Cu(In1−x Gax ) Se2 , x = 0.25 − 0.35 Cu(In1−x Gax ) Se2 , x = 0.25 − 0.35 CdTe CdTe

Emitter – CdS/ZnO – – – SnO2 CdS/SnO2

Remark

τ1 (ns)

τ2 (ns)

Reference

Three-stage process Three-stage process Sequential process Three-stage process Three-stage process Non-activated Activated

0.7–3

5–15

[209]

2–10

10–50

[209]

Not evaluated 7–55

[206]

0.2–1

[35]

Not evaluated

250

–

[210]

<0.1 ns 0.4–0.8

– –

[207] [36]

4.1 AII –BVI Absorbers

4.1.4 Optical Properties 4.1.4.1 CdTe In the ZB structure, the fundamental transition 15 → (see Figure 4.5) marks the minimum optical bandgap. For CdTe thin ﬁlms, the exact value of Eg at room temperature is still a matter of debate. A value of 1.49(1) eV has been reported in recent publications [143, 145, 159]. Earlier references give values down to 1.45 eV [22, 211]. The bandgap variation with temperature is −4 × 10−4 eV K−1 (see Table 4.1). The tabulated absorption coefﬁcient of bulk CdTe as a function of energy can be found in Ref. [213]. Some authors report on an excitonic peak in the absorption curve of CdTe [213, 214]. Figure 4.6 shows: (a) optical constants n, k plotted from data of Paulson et al. [148], and (b) the calculated absorption coefﬁcient. The optical constants have been obtained by spectral ellipsometry on ﬁlms prepared by CdTe evaporation at 325 ◦ C, thus from polycrystalline ﬁlms of nearly device quality. The absorption data in Figure 4.6 largely correspond to other absorption curves reported in the literature (see comparison in Ref. [188]). A ﬁt of α(hν) according to the semi-empirical expression ) hν − Eg (hν − E1 ) + α1 exp (4.2) α = α0 kT B1

gives α0 = 8 × 103 cm−1 with Eg = 1.49 eV. The ﬁrst term in Eq. (4.2) is the fundamental equation for a direct bandgap semiconductor [42]. From α0 , an CIS-like Γ6

Zinc blende Γ1

CGS-like

Γ1

Γ6

Γ1

Γ6

Γ7

Γ4 Γ5

Γ7

Γ4 Γ7

Γ15

Γ6 Γ5 Γ7

∆so

∆cf >0

∆cf <0

Figure 4.5 Schematic derivation of energy bandgap states of CuInSe2 (left, CIS) and CuGaSe2 (right, CGS) from a hypothetical zincblende structure without spin-orbit splitting. Solid (dashed) arrows represent transitions allowed by symmetry in E⊥c (EIIc) polarization. After Ref. [212].

∆so

185

4 Thin Film Material Properties λ [nm]

λ [nm] 300

3.8 3.6

2.0

106

1.5

105

1.0

3.2 3.0

k

3.4

α [cm−1]

1200830 600500 400

n

186

1200 830

600 500

400

104

0.5

103

0.0

102

2.8 1

2

3

4

hν [eV]

(a)

1.0

1.5

(b)

2.0

2.5

3.0

3.5

hν [eV]

Figure 4.6 (a) Optical constants n, k for a CdTe thin ﬁlm as obtained by evaporation. (b) Absorption coefﬁcient calculated after α = 4πk/λ (markers). Simulated absorption curve (line) after Eq. (4.2) using the semi-empirical parameters of Table 4.5. Data reprinted from Ref. [148].

Table 4.5

Parameters for the ﬁt of absorption coefﬁcients to the semi-empirical Eq. (4.2).

Material

α0 (104 cm−1 ) Eg (eV) α1 (103 cm−1 ) B1 (eV) E1 (eV) EU (meV) Reference

CdTe CuInSe2 CuIn1−x Gax Se2 (x = 0.23) CuIn1−x Gax Se2 (x = 0.51) CuIn1−x Gax Se2 (x = 0.78) CuGaSe2 CuInS2

0.8 0.98 0.86

1.49 1.025 1.148

5 6.01 6.42

0.377 0.363 0.404

1.67 1.574 1.545

– 24 67

This book [34] [34]

0.8

1.32

5.94

0.410

1.575

83

[34]

0.79

1.525

4.98

0.396

1.645

79

[34]

0.84 1.04

1.738 1.49

3.59 6.5

0.357 0.25

1.765 1.53

61 30

[34] This book

estimate of the absorption length Lα0 = 1/α0 can be obtained which for CdTe amounts to Lα0 = 1.2 × 10−4 cm. The second term in Eq. (4.2) accounts for interband transitions off the Brillouin zone center and for higher interband transitions at in an empirical fashion. Fit parameters for CdTe according to Eq. (4.2) are given in Table 4.5. In order to estimate the radiative lifetime limit of CdTe we use Eq. (2.60) and assume a doping concentration of NA = 1016 cm−3 . We obtain a value of τrad = 1 × 10−6 s which is similar to the lifetime of the Cu(In, Ga)(S, Se)2 CPs (see Table 4.8). The similar radiative lifetime reﬂects the similar absorption coefﬁcient of Cu(In, Ga)(S, Se)2 close to the bandgap.

4.1 AII –BVI Absorbers

4.1.4.2 Multinary Phases CdTe can form solid solutions with other AII BVI semiconductors. Solid solutions between AII BVI and AI BIII CVI semiconductors are discussed in Section 4.4. The most important solid solution for the solar cell is CdTe1−x Sx . Elemental diffusion at the interface of the CdS/CdTe heterojunction can lead to CdTe1−x Sx interlayers. On the CdS side of the junction an interdiffused layer with large x and on the CdTe side with small x. In order to understand the device impact of interdiffusion, we have to consider the optical properties of the solid solution CdTe1−x Sx . The compositional dependence of the bandgap can be described by

Eg = Eg A x + Eg B (1 − x) − bx(1 − x)

(4.3)

which is a general formula for random solid solutions. In the case of CdTe1−x Sx the bowing coefﬁcient is b ∼1.8 eV [159, 214]. The bandgap as a function of x is shown in Figure 4.3c (right axis). The Eg (x) curve is valid for metastable as well as equilibrated ﬁlms. It has a minimum of Eg = 1.39 eV at around x = 0.3. Theoretical calculations indicated that the bowing coefﬁcient is not constant but may be a function of x [162]. The bandgap minimum at CdTe0.7 S0.3 is a consequence of the large bowing parameter: if the derivation dEg /dx|x=0 = (Eg A − Eg B ) − b becomes negative, a solid solution ﬁlm will exhibit a smaller bandgap than the pure binary phases. Due to interdiffusion between CdS and CdTe, both heteropartners exhibit bandgaps at the interface being smaller than in the bulk. This effect becomes noticeable in a quantum efﬁciency (QE) curve: (i) The QE onset expected at the bandgap of CdTe (∼830 nm) is shifted to longer wavelengths by a few tens of nanometers. (ii) The QE drop expected at the bandgap of CdS (∼500 nm) is also shifted to longer wavelengths. The in-diffusion of CdS into CdTe does only change the bandgap but has no principle impact on the steepness of the absorption coefﬁcient [214]. Thus, also CdTe(S) forms an absorber with very high absorption above the bandgap. With a bandgap of 1.7 eV, CdSe is a potential candidate for the absorber of the top cell in a tandem device [215]. Upon mixing with CdTe, its bandgap can be reduced. The bandgap as function of mixing parameter in the random solid solution of CdTe1−x Sex exhibits a theoretical bowing parameter of 0.75 eV [162]. Experimental values are from 0.4 eV [216] up to 0.7 eV, with some clustering around 0.65 eV. Thus, the bowing parameter is larger than the bandgap difference and accordingly, there is a bandgap reduction for small x. The bandgap minimum of 1.39 eV is at x = 0.4 [217]. Upon mixing with ZnTe (Eg = 2.29 eV) the bandgap of CdTe can be widened. Both endpoint semiconductors have a direct bandgap and so has the solid solution. Paulson et al. determined a bowing coefﬁcient, b ∼ 0.2 eV [148]. (In Ref. [148], the CdTe bandgap of 1.52 eV is slightly larger than in Table 4.1.) Thus, the bowing coefﬁcient is smaller than the bandgap difference of 2.29 − 1.52 = 0.77 eV. This means that at low x the bandgap increases with x in contrast to the case of CdTe1−y Sy and CdTe1−y Sey . In order to increase the bandgap of CdTe, mixing with ZnTe appears preferable since the mixing parameter can be kept small for

187

188

4 Thin Film Material Properties

a noticeable bandgap widening. A smaller mixing parameter means less lattice distortion and less defects. 4.1.5 Surface Properties

In many respects the surface properties of AII –BIII absorber materials including CdTe are well known. A wealth of information is available for the (110) cleavage surface of CdTe [218]. The surface of cleaved CdTe crystals are assumed to be stoichiometric [218]. However, it is possible to stabilize Te-rich surfaces of CdTe by acidic etching [219] and by Cd desorption at a temperature above 200 ◦ C [220]. By etching in alkaline solution, a Cd-rich surface can be achieved. Oxidation of the CdTe surface leads to the formation of CdO and TeO2 or a ternary compound such as Cdx Tey Ox [221]. These oxides can be removed in ultrahigh vacuum by mild Ar-sputtering [222]. The surface Fermi level position Epz=0 as a function of the surface composition or as a function of the absorber doping has not systematically been studied. P-type thin ﬁlms which underwent CdCl2 activation followed by sputter cleaning reveal a Fermi level position close to the valence band, that is, no distinct band bending at the bare surface [222]. Other reports on CdCl2 -activated surfaces give values of Epz=0 = 0.5–0.85 eV, depending on the activation time and temperature [223]. Schulmeyer et al. point out that the surface Fermi level position of CdTe surfaces rather reﬂects the bulk doping than the interface band bending. It may be concluded that the surface state density in the bandgap of CdTe is not very high [224, 225] but certainly this may depend on the type of surface. Currently, we do not know which lattice planes preferentially form the interface to the buffer contact in a solar cell or if there is a preferred surface at all. The Fermi level at CdTe interfaces with CdS, In2 O3 , ZnO, SnO2 as well as several metals has been investigated in detail [226]. A common pinning position of Epz=0 ∼ 1 eV was found. A similar result was found for the system CdTe/In [227]. The Fermi level pinning (FLP) for the CdTe/CdS system can be annihilated by CdCl2 activation [226]. For contact formation, the zero charge transfer barrier height (charge neutrality level or branch point energy) is important. The value for CdTe is between 0.85 and 1.12 eV with respect to the valence band maximum [228] which is in the upper half of the bandgap (Table 4.6). With the zero charge transfer barrier p height of bp = 0.85 eV, a slope parameter SX = 0.37 eV per Pauling unit and the interpolated Pauling electronegativity Xs = (XCd XTe )1/2 = 1.89 we calculate the Schottky barrier height according to [228]. p

Bp = bp − Sx (Xm − Xs )

(4.4)

We obtain for the p-CdTe/Au contact (XAu = 2.54 Pauling units) a barrier height of Bp = 0.6 eV which roughly agrees with published data [119]. There are reports on a lower barrier height in Schottky contacts for the Te-rich surface [219]. Upon

4.1 AII –BVI Absorbers Table 4.6

Branch-point energies for chalcogenide semiconductors.

Compound

Eb (eV)

Compound

Eb (eV)

ZnO ZnS ZnSe ZnTe CdS CdSe CdTe MgS MgTe

3.04 2.05 1.48 1 1.93 1.53 0.85–1.12 1.97 1.19

CulnS2 CulnSe2 CulnTe2 CuGaS2 CuGaSe2 CuGaTe2 CuAlSe2 AglnSe2 AgGaSe2

1.47 0.75 0.55 1.43 0.93 0.61 1.25 1.11 1.09

a After

Ref. [228].

aging, the barrier height for the p-CdTe/Au contact approaches the zero charge transfer barrier height [229]. 4.1.6 Properties of Grain Boundaries

A polycrystalline CdTe thin ﬁlm typically has a grain size of around 1 µm. This is in the range of the important quantities absorption length, diffusion length, and ﬁlm thickness. Thus, grain boundaries (GB) in CdTe could, in principle, severely inﬂuence the solar cell performance. Yet, it was found that efﬁcient devices (η ∼ 12%) can be made regardless of whether the grain size is 2.0 or 0.2 µm [132]. Hence, the GB recombination velocity must be well below 106 cm s−1 (see Figure 3.15). Further, champion CdTe solar cells still suffer from non-optimum values of Voc and FF – a fact, which could well be assigned to the inﬂuence of grain boundaries. Let us start with a general remark on grain boundaries. A useful classiﬁcation scheme for grain boundaries is the coincidence site lattice notation [230]. The coincidence site lattice is constructed as follows (see the 2-dim example in Figure 4.7): at the grain boundary, where two lattices of different orientations join, we think the second lattice virtually extended into the volume space of the ﬁrst lattice. Then, there will be lattice sites which coincide. These coincident sites will again form a periodic structure, however, with larger elementary cell than of the crystal lattice. The volume ratio between the coincidence site lattice and the point lattice of the crystal gives the value of the grain boundary. Figure 4.7 sketches a 3 grain boundary in CdTe. The larger the value, the larger in general is the number of broken and distorted bonds. Polycrystalline CdTe cells with an efﬁciency of 16% and epitaxial GaAs cells with an efﬁciency of 25% both have about the same bandgap energy. Sites et al. assign two-thirds of this 9% efﬁciency difference to the polycrystallinity of CdTe

189

190

4 Thin Film Material Properties

60° (111) [111] [110]

(110)

[111] [110] (a)

AII BIV

(b)

Figure 4.7 Formation of a coincidence site lattice from two rotated sphalerite lattices in (a) [111] projection and (b) [110] projection [231]. The coincidence site lattice is marked by circles. The volume ratio of the

coincidence unit cell and the crystal unit cell gives the number . In the present example this volume ratio is 3 characterizing a 3 grain boundary.

[232]. They infer that polycrystallinity mainly affects the diode current and thus Voc and FF. Losses in Jsc are considered to be small [232]. Indeed, the latter ﬁnding matches with an important observation which is the lateral homogeneity of current collection: using electron beam-induced current (EBIC) measurements in front-wall geometry, Durose and coworkers showed that under low injection conditions grain boundaries in CdTe solar cells just behave as the grain interior [233]. They are nearly not visible in an EBIC image. It appears as if grain boundaries in CdTe ﬁlms are not particularly recombination active under short circuit. This is in contrast to CdTe crystals where grain boundary recombination is visible in the EBIC experiment [234, 235]. However, these crystals where not intentionally contaminated with Cl, S, O, or Cu. A dark grain boundary contrast on thin ﬁlms has only been found in EBIC images recorded in back-wall geometry. This was interpreted as due to a higher doping of the grain boundaries – thus to a smaller depletion and collection region of the main junction [233]. What type of grain boundaries have been identiﬁed in CdTe thin ﬁlms? Very common in the CdTe lattice are 3 grain boundaries [236]. Although having the same value, 3 grain boundaries can differ in the number density of defects. Yan has shown that the lamellar twin boundary does not contain broken bonds, while the double-positioning twin boundary does contain broken bonds [237]. Both are 3 boundaries. At the location of the lamellar twin boundary, which is very abundant in CdTe, the formation energy of Te anti-site donors may be increased [238]. This may lead to a compensation of the p-type conductivity. A very high density of lamellar twins can induce a local wurtzite ordering with the consequence of internal band offsets [238]. Also double-positioning twin boundaries have been found experimentally. It has been argued that the double-positioning twin boundary can be taken as

4.1 AII –BVI Absorbers

a prototype for grain boundaries with broken bonds, thus for grain boundaries with higher . According to theoretical calculations, the defects can partly be passivated by Cl, S, and O atoms where the most perfect passivation is achieved by Cl addition [239]. Also, Cl and Cu co-passivation has been suggested [240]. Virtually all the candidate elements for passivation appear to segregate at CdTe grain boundaries. For device quality CdTe thin ﬁlms, the introduction of chlorine is indispensable (see Section 5.1.5). Secondary ion mass spectrometry (SIMS) [241] and energy dispersive X-ray spectroscopy (EDS) [242] maps show that a high concentration of chlorine is accommodated at the grain boundaries. The chlorine concentration is highest near the back contact [132]. Close to the CdS, the CdTe grain boundaries are sulfur-rich [132]. Sulfur atoms, coming from the CdS heterocontact, diffuse via grain boundaries [158, 243]. If the CdTe ﬁlm on top of CdS has been grown with oxygen, the sulfur diffusion appears to be reduced [244], which indicates that oxygen is also present at grain boundaries. Finally, grain boundaries provide pathways for Cu diffusion [157]. Thus, those chemical elements which are potentially suited for grain boundary passivation are indeed abundant at grain boundaries. The energy band diagram across a grain boundary still is a matter of debate [188]. According to the ﬁndings on bare surfaces discussed in Section 4.1.5, we may state that there is no distinct pinning position of the Fermi level. However, the various contaminations may induce a certain type of band bending at the grain boundaries. The main question is whether grain boundaries induce accumulation or depletion (inversion) zones. Metzger and Gloeckler showed that in the case of an inverted grain boundary, the grain boundary core can attract charge carriers and thus can promote carrier collection [245] (see also Section 3.7). Existence of an accumulation zone was concluded from EBIC experiments in front-wall geometry: if the beam current is high, a bright EBIC contrast is found at grain boundaries. Or, vice versa, a dark EBIC contrast is found in the grain interior [246]. As the dark contrast could arise from high injection, it was argued that the hole concentration in the grain interior is lower than at the grain boundary [247]. This favors a band diagram as in Figure 3.14a, that is, an accumulation zone at the grain boundary with a minority carrier barrier. The higher doping of the grain boundary region would also explain the dark grain boundary contrast in the back-wall experiment. Higher doping means shorter depletion width and shorter collection width at the grain boundary. However, there are also experimental results which point toward a depletion or inversion zone. Electric transport measurements show an activated transport of majority carriers or, in other words, a barrier for holes [116, 248–250]. The band diagram would be as in Figure 3.14b. Grain boundary charge collection has been deduced from scanning probe experiments in back-wall geometry [251, 252]. (Unlike in the EBIC back-wall experiments, the samples had not been polished but are macroscopically rough.) Grain boundary charge collection could be the outcome of an inversion zone. In addition, the unusual large diffusion length of 4 µm measured in CdCl2 -treated cells [246] could be explained by the inversion

191

4 Thin Film Material Properties

zone. Woods tried to solve the contradiction between an accumulation and inversion zone by proposing a mixed band diagram with an inversion core and a high doping zone, as depicted in Figure 3.14d. This band diagram could satisfy both the bright EBIC contrast and the transport experiments. It would lead to a large barrier for holes which could partly be traversed by tunneling [253]. Device simulations by Metzger and Gloeckler [245] showed that, for a strong grain boundary inversion, the short circuit current is high due to grain boundary charge collection but the Voc is reduced. This matches the general observation for polycrystalline CdTe solar cells. There are no indications for an increased bandgap at the grain boundaries as in Figure 3.14c. 4.2 AI –BIII –CVI 2 Absorbers

In this section, we again focus on those AI –BIII –CVI 2 semiconductors with C=S1 Se1 Te which are suitable as absorber layers in solar cells. The bandgap range of absorbers for a single junction solar cell (1.0–1.5 eV) includes the ternary phases CuInSe2 , CuInS2 , CuInTe2 , CuGaTe2 , and AgInSe2 . This is shown in Figure 4.8 where a selection of CP phases is represented in a Eg vs lattice parameter plot. Furthermore, this bandgap range includes a variety of CP solid solutions. All Heterocontact Lattice Parameter [Å] ln2S3 ZnS 3.0

AgGaS2

ZnSe

CdS

ZnTe

AglnS2

AgGaSe2

CuAlSe2

CuGaS2

2.5

Eg [eV]

192

2.0 CuGaSe2 1.5

AgGaTe2 CulnS2

AglnSe2 AglnTe2

1.0

CuInTe2

CulnSe2 CuGaTe2 5.2

5.4

5.6

5.8

6.0

6.2

6.4

Lattice Parameter (2a + c)/4 [Å]

Figure 4.8 Bandgap versus lattice constant of selected I−III−VI2 semiconductors. Dashed lines reveal experimentally conﬁrmed miscibility within the Cu(Al, Ga, In)(S, Se, Te)2 system. Dotted lines show miscibilities

between Cu(Al, Ga, In)(S, Se, Te)2 and Ag(Al, Ga, In)(S, Se, Te)2 systems. A miscibility gap is depicted by a gap in the lines as, for example, for the solid solution between CuGaSe2 and AgGaSe2 .

4.2 AI –BIII –C2VI Absorbers

these phases crystallize in the CP structure. We note in passing that Cu-BIII -O2 in contrast have a layered delafossite structure with the space group R3m. 4.2.1 Physico-Chemical Properties

Locally, the crystal structures of AI BIII CVI 2 contain tetrahedra of the type AI n BIII 4−n (n = 1 − 4) which are formed around a common C atom. Wei et al. have shown that energy minimization can be realized if n = 2. In that case the octet rule is fulﬁlled [254]. The CP structure with point group D12 2D has a unit cell which is twice as large as that of the ZB structure (Figure 4.2). It contains eight atoms instead of four. Due to the chemical difference between the two cations, the symmetry is reduced and two new parameters are introduced for structure description: the tetragonal distortion parameter η = c/2a and the anion displacement parameter u = 1/4 + (RAC 2 − RBC 2 ) a−2 η. For the ZB structure it holds η = 1 and u = 1/4. Table 4.7 gives the anion displacement parameters for several CP phases. 4.2.1.1 Ternary Phase Diagrams Figure 4.9 shows a generic phase triangle between the three elements of a ternary compound. In addition to the three element phases, there are binary and ternary phases. The phase triangle can be sectioned by composition-temperature planes. These so called isopleths are deﬁned by ﬁxed composition ratios. Examples are isopleths with I 2 CVI 2 CVI A − 1 = 0, z = − 1 = 0 x = III − 1 = 0, y = I B A + 3 BIII AI + BIII (4.5)

The isopleth may contain lines of two-phase equilibria (tie lines) and lines of invariant three-phase equilibria (critical tie lines). An isopleth is denoted as pseudobinary if all critical tie lines lie within the plane and all tie lines of adjacent two-phase equilibria lie parallel to the plane [260]. G¨odecke revealed that in the system Cu–In–Se, the isopleth with y = 0 (full line in Figure 4.9) is a pseudobinary isopleth. This isopleth connects the phases Cu2 Se and In2 Se3 . It is often assumed that all ternary Cu- and Ag-chalcopyrites have a pseudobinary isopleth between AI 2 CVI and BIII 2 CVI 3 . What is experimentally veriﬁed is that many Cu- and Ag-based phase triangles exhibit tie lines along A2 CVI and B2 CVI 3 at room temperature [261]. This is why we draw a full line between AI 2 CVI and BIII 2 CVI 3 in Figure 4.9. Formally, the chemical composition of all phases on the isopleth with y = 0 can be written as AI 1−x BIII 1+x/3 CVI 2 . Since the phase system of Cu–In–Se has been investigated in most detail, we exemplarily depict in Figure 4.10 the isopleths In2 Se3 –Cu2 Se (Figure 4.10a) and InSe–CuSe (Figure 4.10b). A thorough description of the Cu–In–Se diagram has been given by G¨odecke et al. [260, 262, 263]. A crystallographic compendium of the binary and ternary phases in the Cu–In–Se phase system can be found in Ref. [264].

193

194

Structure data and relative unit cell volumes of selected Cu- and Ag-chalcogenides and related compounds.

Compound

Structure

˚ a (A)

˚ c (A)

Anion sublattice

u (anion)

Unit cells

Volume (A˚ 3 )

Volume ratio to CuBIII CVI 2

CuGaS2 Cu2−x S(x = 0.2) CuS CuInS2 Cu2−x S(x = 0.2) CuIn5 S8 β-In2 S3 NaInS2 CuAlSe2 β-Cu2−x Se(x = 0.2) CuSe CuGaSe2 β-Cu2−x Se(x = 0.2) CuGa3 Se5 CuGa5 Se8 α-Ga2 Se3

Tetragonal Cubic Hexagonal Tetragonal Cubic Cubic Cubic Trigonal Tetragonal Cubic Hexagonal Tetragonal Cubic Tetragonal Tetragonal Monoclinic

0.025 – – –0.036 – – – – 0.01 – – 0 – – – –

1 2 – 1 2

– 1 2 – 1 2 1 1 1

299.4 347.5 – 339.1 347.5 305 312.3 – 344.0 380.2 – 347.4 380.2 330.4 326.4 491.12

1 1.16 – 1 1.02 0.90 0.92 – 1 1.10 – 1 1.09 0.95 0.94 –

Tetragonal Cubic Tetragonal Hexagonal Hexagonal Trigonal

10.46 – 16.33 11.13 – – – 6.98 10.98 [256] – 17.25 11.022 – 10.95 [257] 10.91 [257] c = 6.66, β = 108.12◦ 11.6188 [258] – 11.5359 [258] 32.78 [259] 19.38 20.89

fcc fcc – fcc fcc fcc fcc – fcc fcc – fcc fcc fcc fcc fcc-like

CuInSe2 β-Cu2−x Se(x = 0.2) CuIn3 Se5 CuIn5 Se8 γ-In2 Se3 NaInSe2

5.35 5.58 3.79 5.52 5.58 10.69 [255] 10.77 3.80 5.60 5.75 3.95 5.614 5.75 5.49 5.47 a = 6.66, b = 11.65 5.7815 5.75 5.7581 4.04 7.12 3.972

fcc fcc fcc – – –

–0.026 – – – – –

1 2 1 1 – –

388.36 380.2 382.48 – – –

1 0.98 0.98 – – –

1 4 1 4

4 Thin Film Material Properties

Table 4.7

4.2 AI –BIII –C2VI Absorbers

CVI

∆y > 0 ∆y < 0 ∆z > 0

AICVI

AIBIIICVI2

III

B

VI

2C 3

BIIICVI

∆z < 0 AI2CVI

AI

∆x > 0

∆x < 0

Figure 4.9 Generic phase triangle of group I (AI ), group III (BIII ) and group VI (CVI ) atoms. Tie lines connect phases which – at a given temperature – are in equilibrium. For all Cu-chalcopyrites and Ag-chalcopyrites, the ternary phase AI BIII CVI 2 exists along the line

BIII

AI 2 CVI –BIII 2 CVI 3 . More ternary phases may be present. As an example, we depict an additional ternary phase between AI BIII CVI 2 and BIII 2 CVI 3 . In addition, more tie lines may be present in the speciﬁc phase triangle.

The CuInSe2 CP phase in Figure 4.10 is denoted as α. According to Table 4.8, this phase has a formation enthalpy at 0 K of −49.8 kJ mol−1 (−2.03 eV) which is in good agreement with calculated values (−2 eV [134] and −1.8 eV [265]). Other ternary phases along the In2 Se3 –Cu2 Se isopleth are CuIn3 Se5 (β), CuIn5 Se8 (χ), and δ. The β phase is a defect CP, built by ordered arrays of defect complexes 2VCu –InCu on the cation sublattice [134]. This phase can be derived from the CP CuInSe2 by having one 2VCu –InCu defect complex in every 20-atom Cu5 In5 Se10 unit cell [134]. Often, the β phase is referred to as ordered defect compound (ODC). A structure model of the ODC phase is still a matter of debate [258, 266–270]. Both, the α and the β structures exhibit an fcc Se sublattice. The existence range of the β-phase extends from a Cu content of 16.0 to 10.5 at%. The δ phase is the high temperature modiﬁcation of both CuInSe2 and CuIn3 Se5 . It is of sphalerite type, that is, it does not show cation ordering. The third ternary phase along the isopleth of y = 0 is CuIn5 Se8 . Its stable structure is hexagonal while the tetragonal form is metastable [271]. For 7.5 at% < Cu < 10.5 at% CuIn5 Se8 and CuIn3 Se5 coexist in a two-phase region. As a metastable phase, CuInSe2 and other CPs can also occur in the CuAu ordering. The CuAu ordering P42m is most closely related to the chalcopyrite ordering as it also exhibits AI n BIII 4−n tetrahedra with n = 2. The formation energies of CP and CuAu differ only by 2 meV/atom. Other structures exhibiting n = 1 and 3 tetrahedra, CuPt ordering R3m or Y2-like ordering, are thermodynamically less stable. Early reports on their existence in CuInSe2 thin ﬁlms [282] have not yet been conﬁrmed. It shall be emphasized that all polytypes are only different with respect to

195

4 Thin Film Material Properties

196

L

1000

1000 900

+

600 β

Temperature [°C]

Temperature [°C]

α α

δ

+ γ

α+ Cu2Se(HT)

400 β

300

α+ β

10

700

L +δ +α

600 500

15

20

25

at. % Cu

30

L4 +α InSe +α

300 200

35

L3 +α

+ Cu2Se(HT)

α+ Cu2Se(RT)

L3 +δ

L +δ

400

200

(a)

δ

800

700

100

L3

β+δ

800

500

L

δ

900

α 0

10

(b)

20

α +γ-CuSe

30

40

50

at. % Cu

Figure 4.10 Pseudobinary equilibrium phase diagrams of (a) In2 Se3 –Cu2 Se and (b) InSe-CuSe. After Refs. [260, 262, 263].

their cation sublattice but have an identical fcc chalcogen sublattice. The existence of the CuAu ordering in CuInSe2 and CuInS2 can be inferred from diffraction measurements [283] and from Raman spectroscopy [284]. The metastable phases of CuInSe2 are not included in the phase diagram of Figure 4.10. The end members of the phase cut in Figure 4.10b are In2 Se3 and Cu2 Se. In2 Se3 exists in four modiﬁcations depending on composition and temperature. The βand the γ-phase can co-exist between 470 and 1080 K. β-In2 Se3 is a rhombohedral layered structure with slight selenium deﬁciency. The γ-In2 Se3 phase is of the defect wurtzite-type [285]. In order to satisfy the octet rule for sp3 hybridization, one-third of cation sites in γ-In2 Se3 are vacant. Both structures, β- and γ-In2 Se3 , do not exhibit an fcc anion sublattice. Cu2 Se exists as a low temperature α-phase (stable below 396 K) and as a high temperature β-phase (stable up to 1403 K). The β-phase has the anion fcc sublattice and thus is closely related to the CP structure. It can be formed by introducing anti-sites and interstitials of the type CuIn and Cui into the cation sublattice of CuInSe2 . In equilibrium, the CuInSe2 α-phase has a small but measurable existence range. Along the In2 Se3 –Cu2 Se pseudobinary cut (Figure 4.10b), CuInSe2 is found as

4.2 AI –BIII –C2VI Absorbers

the single phase with a Cu content from 24.0 to 24.5 at% (0.94 < [Cu]/[In] < 0.97. Samples with a Cu content of 25 at% (i.e., with the nominal composition of CuInSe2 ) are not single phase but include Cu2 Se as a secondary phase. Samples with a Cu content of <24 at% include the δ-phase (CuIn3 Se5 ). That there is an extended existence range of the α-phase has been explained by the unusual stability of the 2VCu -InCu defect complex. By the formation of this defect complex a Cu-deﬁcit in the structure can be accommodated. Figure 4.2c shows the defect complex in the CuInSe2 unit cell as derived from energy minimization [134]. The presence of 0.1 at% of Na or the replacement of In by Ga by some at% widens the existence range of the α-phase toward In- and Ga-rich compositions [286]. The effect of adding Ga has been explained by the reduced ordering energy of the 2VCu -GaCu defects [287]. The existence range of the α-phase further widens at higher temperature. At 800 K, a ﬁlm with 22 at% Cu ([Cu]/[In] = 0.82) lies within the single-phase region of the α-phase. Upon cooling, the ﬁlm moves into the two-phase α + β region. As the composition of typical Cu(In, Ga)Se2 thin ﬁlms is [Cu]/[In] + [Ga] ∼ 0.9 and the typical growth temperature is around 800 K, segregation of the β-phase upon cooling will be expected. The segregation may take place at the surface, as shown in Section 4.2.5. It was found that under non-equilibrium conditions, the existence range of the α-phase is even larger [260]. This means, a ﬁlm with [Cu]/[In] + [Ga] ∼ 0.9 which has been quenched to room temperature may still be of single-phase CuInSe2 . In other words, the segregation of the β-phase strongly depends on the sample preparation. Since the β-phase has a larger electronic bandgap than the α-phase, this may inﬂuence the device performance. Next we consider the isopleth with z = 0 in Figure 4.10b. At room temperature, the α-phase is in equilibrium with InSe and CuSe. Along the isopleth with z = 0, the existence range of the α-CuInSe2 phase extends toward Cu < 25 at% and not toward Cu > 25 at%. At 649 K there is a phase transition from CuSe → Cu2 Se + Seliq . The stabilization of the Se liquid (Sliq ) on the surface of a growing ﬁlm will depend on the Se vapor pressure. Above 795 K, the α-phase is in equilibrium with a Cu-Se liquid denoted as L3 in Figure 4.10a. Thus in case of high selenium pressure, the vapor growth of Cu-rich CuInSe2 may take place in the presence of a Cu-Se liquid as proposed by Klenk et al. [288]. Such vapor–liquid–solid (VLS) growth mechanism is known to reduce defect densities due to high mobility of atoms in the liquid. It should be noted, however, that ion mobility under Cu-rich conditions can also be realized along the y = 0 isopleth: the high temperature phase β-Cu2 Se is an ambipolar ion conductor. The diffusion of In3+ in β-Cu2 Se at 650 ◦ C was estimated as 4 × 106 cm2 s−1 which is a value typically found for diffusion in liquids [289]. Thus, even the presence of the Cu2 Se secondary phase during CuInSe2 growth may lead to a VLS type of growth process and thus to a low defect density. The phase diagram of Cu-In-Se shows some peculiarities which are also found for other Cu- and Ag-chalcopyrites: 1) In the Cu-In-Se system, the homogeneity range of the AI BIII CVI 2 phase extends toward x < 0 (x deﬁned in Eq. (4.5)). This is also the case for CuInS2

197

198

4 Thin Film Material Properties

[290], CuGaSe2 [291], CuInTe2 [261], CuGaTe2 [292], and AgInTe2 [261]. The homogeneity range of CuGaSe2 is smaller than that of CuInSe2 [287]. The homogeneity range of CuAlSe2 may be on the Cu-rich side of the y = 0 isopleth (i.e., x > 0) [256]. 2) In the Cu–In–Se system, a ternary phase of the chemical formula AI BIII 3 CVI 5 is stable at room temperature. This phase is a result of ordering of defect complexes of the type 2VA –BA [134]. The AI BIII 3 CVI 5 phase is also known for the Cu–Ga–Se system. One earlier paper by Ganbarov et al. reports on CuIn3 S5 and CuIn3 Te5 [293]. 3) A ternary phase of the chemical formula AI BIII 5 CVI 8 exists in many systems such as Cu–Ga–Se, Cu–Ga–S, Cu–In–S, Cu–Ga–Te, Cu–In–Te, Ag–Ga–Te, Ag–In–Se (see Table 4.7). Its structure is either cubic, tetragonal, or hexagonal at room temperature. Since binary or ternary phases can be concomitant during or after the growth of CPs, crystallographic relations between the CP and its secondary phases are important. Table 4.7 lists some crystallographic data of relevant phases. It also gives relative volumes of some binary and ternary phases. Growth of Cu(In, Ga)CVI 2 under Cu-rich conditions (or as for CuInSe2 even under stoichiometric conditions) will inevitably lead to the formation of a Cu-CVI secondary phase which, at growth temperature, is Cu2−x CVI . The couple Cu(In, Ga)CVI 2 and Cu2−x CVI can show an epitactic or topotactic relationship [294]. Table 4.7 reveals that, for the system Cu2−x Se/CuInSe2 with x = 0.2, the volume ratio at room temperature is close to unity. Thus, the volume mismatch is small and we may expect a small strain within the layers. The volume ratio systematically increases in the order In → Ga → Al. It becomes even larger for x → 0 and at higher temperature due to the thermal expansion coefﬁcient of Cu2 Se (2.5 × 10−5 K−1 ) [295] being larger than that of CuBIII Se2 . Growth of Cu(In, Ga)CVI 2 under Cu-poor conditions may lead to the formation of a secondary Cu1−x (In, Ga)1+x/3 CVI 2 phase. In Table 4.7 we see that also the CuIn3 Se5 /CuInSe2 volume ratio is more close to unity that the CuGa3 Se5 /CuGaSe2 volume ratio. With respect to the volume mismatch to its secondary phases, α–CuInSe2 is in a privileged state compared to other CuBIII Se2 -chalcopyrites. 4.2.1.2 Multinary Phases AI BIII CVI 2 semiconductors of the system (Cu, Ag)(Al, Ga, In)(S, Se, Te)2 exhibit several isopleths of complete miscibility. For solar cells, the miscibility of different ternaries is important because it allows tailoring a desired electronic bandgap in the multinary system. Figure 4.8 depicts a selection of ternary phases in the Eg versus (2a + c)/4 parameter space. Those ternary isopleths for which complete miscibility has been shown experimentally are depicted as dashed lines in Figure 4.8. It includes the isopleths between most of the Cu-chalcopyrites. However, there appears to be no complete miscibility between some Cu-chalcopyrites and Ag-chalcopyrites [296] (marked as dotted lines in Figure 4.8). The reason for the miscibility gap may be the large difference in the ion radii between Ag and Cu.

4.2 AI –BIII –C2VI Absorbers

Of technological importance is the ternary phase NaInSe2 . No miscibility between CuInSe2 and NaInSe2 has been reported. The reasons are that NaInSe2 exhibits a layered structure in its stable state (space group R3m) and that the mixing enthalpy for the reaction xNaInSe2 + (1 − x) CuInSe2 ⇔ Nax Cu1−x InSe2 is positive [297]. Vegard’s law states that the lattice parameter of a solid solution depends linearly on the concentration of the substituting elements [298]. This law is approximately valid in the systems Cu(In1−x Gax )Se2 [299] and CuInSe2−2× S2x [300]. Deviations from Vegard’s law are found, for example, for Cu1−x Agx InSe2 and other Ag-containing systems. The approximate validity of Vegard’s law can be used to tailor pentanary CP compounds without tetragonal distortion [301] and therefore without crystal ﬁeld splitting. Tetragonal AI BIII CVI 2 compounds can also form solid solutions with binary II VI D E compounds. (Here DII and EVI denote the group II and group VI elements of the binary.) One source of interest in these solid solutions is a possible partial replacement of In (and Cu) by a less expensive element. Another source of interest in AI BIII CVI 2 –DII EVI systems is the possible intermixing taking place at the interface between the ternary and the binary. The system CuInS2 + 2CdSe ⇔ CuInSe2 + 2CdS has been investigated by Olekseyuk et al. [302, 303]. Figure 4.11 shows the isothermal section at 620 K. There exists a solid solution range between CuInSe2 and the two binaries in the CP structure. The δ phase is of the sphalerite type – the high temperature modiﬁcation of CuInSe2 . (Note that here we use phase symbols different from those in the work of Parasyuk et al. [302].) Thus, by the addition of DII EVI to AI BIII CVI 2 , the sphalerite crystal structure can be stabilized down to room temperature. Finally, at large amounts of DII EVI the hexagonal η-phase is formed. Large two-phase ﬁelds are found in between α, δ, and η. If the lattice constants of each structure are transformed into an effective cubic lattice constant, then a linear variation according to Vegard’s law is observed [304]. For VI VI all CuInCVI = S, Se, Te, the ZB structure is found for 2 –ZnC 2 systems with C large x and the CP structure for small x [305]. There is a miscibility gap (two-phase ﬁeld) for intermediate x. Since all ZnCVI 2 crystallize in the cubic structure, no solid solution η phase is formed. CulnS2

2CdS

α+δ

η

δ+η α

CulnSe2

δ

2CdSe

Figure 4.11 Isothermal section of the reciprocal system CuInSe2 + 2CdS ⇔ CuInS2 + CdSe at 620 K. After Ref. [302]. Marked are the phase ﬁelds of α : chalcopyrite, δ : sphalerite, and η : wurtzite as well as the two-phase ﬁelds.

199

200

4 Thin Film Material Properties

4.2.1.3 Diffusion Coefﬁcients The knowledge about the chemical diffusion coefﬁcients in monocrystalline CPs is far less comprehensive than for CdTe. A few more data are available for polycrystalline ﬁlms. Since thin ﬁlm data depend on grain boundary density and thus are speciﬁc for each investigated ﬁlm, these data may be used with caution. Cu is assumed to be the most mobile species in Cu-chalcopyrites. Comparison between the self-diffusion constant of Se and Cu in CuInSe2 crystals (Table 4.8) supports this assumption. Cu ions were found to migrate under an electric ﬁeld in the form of Cu+ [276]. As VCu are acceptors in CuInSe2 (see Section 4.2.3.1) and Cui are donors, migration of Cu can locally change the doping type. Electromigration of ions has been applied for the formation of pn junctions in bulk CuInSe2 [276] and thin ﬁlms [306, 307]. The diffusion constant of indium is only known for CuInS2