Nano- and Micromaterials

advances in materials research 9 advances in materials research Series Editor-in-Chief: Y. Kawazoe Series Editors: M...

Author: Kaoru Ohno | Kaoru Ohno | Masatoshi Tanaka | Jun Takeda | Yoshiyuki Kawazoe

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advances in materials research

9

advances in materials research Series Editor-in-Chief: Y. Kawazoe Series Editors: M. Hasegawa

A. Inoue

N. Kobayashi

T. Sakurai

L. Wille

The series Advances in Materials Research reports in a systematic and comprehensive way on the latest progress in basic materials sciences. It contains both theoretically and experimentally oriented texts written by leading experts in the f ield. Advances in Materials Research is a continuation of the series Research Institute of Tohoku University (RITU). 1

Mesoscopic Dynamics of Fracture Computational Materials Design Editors: H. Kitagawa, T. Aihara, Jr., and Y. Kawazoe

2

Advances in Scanning Probe Microscopy Editors: T. Sakurai and Y. Watanabe

3

Amorphous and Nanocrystalline Materials Preparation, Properties, and Applications Editors: A. Inoue and K. Hashimoto

4

Materials Science in Static High Magnetic Fields Editors: K. Watanabe and M. Motokawa

5

Structure and Properties of Aperiodic Materials Editors: Y. Kawazoe and Y. Waseda

6

Fiber Crystal Growth from the Melt Editors: T. Fukuda, P. Rudolph, and S. Uda

7

Advanced Materials Characterization for Corrosion Products Formed on the Steel Surface Editors: Y. Waseda and S. Suzuki

8

Shaped Crystals Growth by Micro-Pulling-Down Technique Editors: T. Fukuda and V.I. Chani

9

Nano- and Micromaterials Editors: K. Ohno, M. Tanaka, J. Takeda, and Y. Kawazoe

Kaoru Ohno Masatoshi Tanaka Jun Takeda Yoshiyuki Kawazoe (Eds.)

Nano- and Micromaterials With 204 Figures

123

Professor Dr. Kaoru Ohno Professor Dr. Masatoshi Tanaka Professor Jun Takeda Yokohama National University, Graduate School of Engineering, Department of Physics Tokiwadai, Hodogaya, Yokohama 240-8501, Japan E-Mail: [email protected], [email protected], [email protected]

Professor Dr. Yoshiyuki Kawazoe Tohoku University, Institute of Materials Research Katahira, Sendai 980-8577, Japan E-Mail: [email protected]

Series Editor-in-Chief:

Professor Yoshiyuki Kawazoe Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

Series Editors: Professor Masayuki Hasegawa Professor Akihisa Inoue Professor Norio Kobayashi Professor Toshio Sakurai Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

Professor Luc Wille Department of Physics, Florida Atlantic University 777 Glades Road, Boca Raton, FL 33431, USA

ISSN 1435-1889 ISBN 978-3-540-74556 Springer Berlin Heidelberg New York Library of Congress Control Number: 2007938884

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springer.com © Springer Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data prepared by SPi using a Springer LATEX macro package Cover: eStudio Calmar Steinen Printed on acid-free paper

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Preface

In nanotechnology to date, much emphasis is placed on the creation of the nanostructures by means of micro- and atomic manipulations. This research ﬁeld has been highly respected and promoted by the society, polytics, and economics. Rapid progress in this ﬁeld has been greatly stimulated by more fundamental study on nano- and micromaterials. In this respect, the scientists and engineers in diﬀerent ﬁelds of physics, chemistry, materials science, and information technology including experimentalists, theorists, and also researchers doing computer simulations have collaborated to form a new interdisciplinary ﬁeld. This book covers the recent advances in this growing research ﬁeld, in particular, those developed mainly in the interdisciplinary research project named “Materials science for nano- and microscale control: Creation of new structures and functions,” which was formed in 2004 in the Graduate School of Engineering of Yokohama National University in collaboration with the Institute for Materials Research, Tohoku University and other universities. The topics described in this book are as follows. In computational materials design, ﬁrst-principles calculations and simulations can give reliable guidelines for structural and functional controls of nanomaterials. In this respect, the development of new computational methods, in particular, for the excited states of materials, is highly desirable to investigate atomic and electronic dynamics on the nano- and microscales. The state-of-the-art GW and T -matrix calculations, transport calculations, and lattice dynamics calculations will be explained in detail in this book. From experimental point of view, in particular from the viewpoint of structural controls, the use of the self-organization of surface or local nanostructures controlled by light or heat is described in detail, in which a variety of useful structures appear through grain boundary motions on submicron scales. Such novel nanointegration technologies are particularly useful to create quantum dots or quantum well devices, and such applications are also described in detail. Also a variety of interesting optically controlled chemical or catalytic reactions, and phase transitions are described in detail for

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particular interesting systems. Moreover, the functionalities of quantum dots, the creation of micro-/nanomachines using microstereolithography, and the development of new techniques of laser spectroscopy to observe dynamical processes related to optic functionalities are described in detail. We hope that this book would be beneﬁt to not only the scientists or engineers in this ﬁeld but also the researchers in other ﬁelds to see what is going on in the researches of nano- and micromaterials. Finally, we would like to thank C.E. Ascheron and his coworkers at Springer-Verlag in Heidelberg for their continuous help in completing this book. Yokohama, Sendai January 2008

Kaoru Ohno Masatoshi Tanaka Jun Takeda Yoshiyuki Kawazoe

Contents

1 General Introduction K. Ohno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Nanometer-Scale Structure Formation on Solid Surfaces M. Tanaka, K. Shudo, and S. Ohno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Atomic Layer Etching Processes on Silicon Surfaces . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Real-Time Optical Measurements . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Adsorption of Halogen Atoms: Sticking Coeﬃcient and Potential Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Site-Selective Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Desorption of Silicon Halides and Restoration of the DAS Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Nanoscale Fabrication Processes of Silicon Surfaces with Halogens . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Scanning Tunneling Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Thermal Desorption Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Cluster Alignment by Passive Fabrication . . . . . . . . . . . . . . . . 2.3.5 Active Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Self-Organized Nanopattern Formation on Copper Surfaces . . . . . . . 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Novel Phenomena on Cu(001)–c(2×2)N . . . . . . . . . . . . . . . . . . 2.4.4 Nanopattern Formation at Vicinal Surfaces . . . . . . . . . . . . . . 2.4.5 Strain-Dependent Nucleation of Metal Islands . . . . . . . . . . . . 2.4.6 Strain-Dependent Dissociation of Oxygen Molecules . . . . . . . 2.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 21 24 26 34 39 48 50 50 53 56 62 68 76 77 77 78 79 79 82 85 88 89

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3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials J. Takeda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2 Femtosecond Optical Kerr Gate Luminescence Spectroscopy . . . . . . 97 3.2.1 Time-Resolved Luminescence Spectroscopy: Up-Conversion Technique vs. Opical Kerr Gate Method . . . 97 3.2.2 Femtosecond OKG Method: Experimental Setup and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Femtosecond Transient Grating Spectroscopy Combined with a Phase Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.3.1 Principle of Transient Grating Spectroscopy . . . . . . . . . . . . . . 105 3.3.2 Transient Grating Spectroscopy Combined with a Phase Mask: Experimental Setup and Results . . . . . . . . . . . . . . . . . . 107 3.4 Femtosecond Real-Time Pump-Probe Imaging Spectroscopy . . . . . . 109 3.4.1 Principle of Real-Time Pump-Probe Imaging Spectroscopy . 109 3.4.2 Experimental Demonstrations of Real-Time Pump-Probe Imaging Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4 Defects in Anatase Titanium Dioxide T. Sekiya and S. Kurita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2 Growth of Anatase Single Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3 Control of Defect States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3.1 Heat Treatment Under Oxygen Pressure . . . . . . . . . . . . . . . . . 123 4.3.2 Heat Treatment Under Hydrogen Atmosphere . . . . . . . . . . . . 124 4.4 Properties of Anatase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.4.1 Absorption Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.4.2 Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.4.3 EPR Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.4.4 Electric Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.5 Carrier Control by Photoirradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.5.1 Photoconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.5.2 EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5 Organic Radical 1,3,5-Trithia-2,4,6-Triazapentalenyl (TTTA) as Strongly Correlated Electronic Systems: Experiment and Theory J. Takeda, Y. Noguchi, S. Ishii, and K. Ohno . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Crystalline Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

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Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3.1 Paramagnetic Susceptibility and Electron Spin Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.3.2 Reﬂectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.3.3 Photoinduced Magnetic Phase Transition . . . . . . . . . . . . . . . . 151 5.4 Electronic Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.4.1 Results Within the LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.4.2 Breakdown of the LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.4.3 T -Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.4.4 Results in the T -Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . 164 5.4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6 Ab Initio GW Calculations Using an All-Electron Approach S. Ishii, K. Ohno, and Y. Kawazoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 Many-Body Perturbation Theory and GW Approximation . . . . . . . . 172 6.3 Choice of Basis-Set Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.4 Application to Clusters and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.4.1 Alkali-Metal Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.4.2 Semiconductor Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.4.3 Gallium Arsenide Clusters and Crystal . . . . . . . . . . . . . . . . . . 180 6.4.4 Benzene Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.4.5 Why Are LDA Eigenvalues of HOMO Level Shallower Than Experiments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.5 Self-Consistent GW vs. First Iterative GW (G0 W0 ) . . . . . . . . . . . . . . 184 6.6 Appendix: Proof of WT Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7 First-Principles Calculations Involving Two-Particle Excited States of Atoms and Molecules Using T -Matrix Theory Y. Noguchi, S. Ishii, and K. Ohno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.2 Methodology: T -Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.3 Double Electron Aﬃnity of Alkali-Metal Clusters . . . . . . . . . . . . . . . . 193 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.3.2 Eﬀect of the Coulomb Interaction in the DEA Spectra . . . . . 193 7.3.3 Short-Range Repulsive Coulomb Interaction Within the T -Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.4 Double Ionization Energy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.4.2 Two-Valence-Electron Systems . . . . . . . . . . . . . . . . . . . . . . . . . 198

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7.4.3 Inert Gas Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.4.4 CO and C2 H2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5 Two-Electron Distribution Functions and Short-Range Electron Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.5.3 Ar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.5.4 CO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.5.5 CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.5.6 C2 H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.7.1 Fourier Transformation of Green’s Function . . . . . . . . . . . . . . 213 7.7.2 Fourier Transformation of K-Matrix . . . . . . . . . . . . . . . . . . . . 214 7.7.3 Fourier Transformation of T -Matrix . . . . . . . . . . . . . . . . . . . . . 215 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8 Green’s Function Formulation of Electronic Transport at Nanoscale A.A. Farajian, O.V. Pupysheva, B.I. Yakobson, and Y. Kawazoe . . . . . . . 219 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.2 Landauer’s Transport Formalism: The Green’s Function Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.2.1 Multichannel Landauer’s Formula . . . . . . . . . . . . . . . . . . . . . . . 220 8.2.2 Surface Green’s Function Matching Method . . . . . . . . . . . . . . 221 8.2.3 Scattering Matrix and Transport Properties . . . . . . . . . . . . . . 223 8.2.4 Alternative Formulation of the Total Conductance . . . . . . . . 226 8.3 Carbon Nanotube Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.3.1 Conductance of Nanotubes with Vacancy or Pentagon–Heptagon Defects . . . . . . . . . . . . . . . . . . . . . . . . . 227 8.3.2 Doped Nanotube Junctions: Rectiﬁcation and Novel Mechanism for Negative Diﬀerential Resistance . . . . . . . . . . . 230 8.3.3 Eﬀects of Random Disorder on Transport of Nanotubes . . . . 234 8.4 Functional Molecule Between Two Metallic Contacts . . . . . . . . . . . . 235 8.4.1 Transport Through Xylyl-Dithiol Molecule Attached to Two Gold Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.4.2 Transport Through Benzene-Dithiol Molecule Attached to Two Gold Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

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9 Self-Assembled Quantum Dot Structure Composed of III–V Compound Semiconductors K. Mukai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.2 Control of QD Structure by Growth Condition . . . . . . . . . . . . . . . . . . 244 9.2.1 Control of Growth Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.2.2 Closely Stacked QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 9.2.3 QD Buried in Strained Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 9.3 Growth Process of QD Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 9.4 Analysis of QD Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.4.1 Grazing Incidence X-Ray Scattering . . . . . . . . . . . . . . . . . . . . . 256 9.4.2 Scanning Tunneling Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.5 Summary and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 10 Potential-Tailored Quantum Wells for High- Performance Optical Modulators/Switches T. Arakawa and K. Tada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10.2 Parabolic Potential Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 10.3 Graded-Gap Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.4 Asymmetric Coupled Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 10.5 Intermixing Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 11 Thermodynamic Properties of Materials Using Lattice-Gas Models with Renormalized Potentials R. Sahara, H. Mizuseki, K. Ohno, and Y. Kawazoe . . . . . . . . . . . . . . . . . . . 275 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 11.2 Scheme of the Potential Renormalization . . . . . . . . . . . . . . . . . . . . . . . 276 11.3 Application of the Potential Renormalization . . . . . . . . . . . . . . . . . . . 278 11.3.1 Application to Melting Behavior of Si . . . . . . . . . . . . . . . . . . . 278 11.3.2 Application to Cu–Au Phase Diagram . . . . . . . . . . . . . . . . . . . 282 11.3.3 Application to Transition and Noble Metals . . . . . . . . . . . . . . 286 11.3.4 Order–Disorder Phase Transition of L10 FePt Alloy Using the Renormalized Potential Combined with First-Principles Calculations . . . . . . . . . . . . . . . . . . . . . . . 287 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 12 Optically Driven Micromachines for Biochip Application S. Maruo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 12.1.1 Two-Photon Microstereolithography for Production of 3D Micromachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

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Contents

12.1.2 Assembly-Free, Single-Step Fabrication Process of Movable Microparts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 12.2 Optically Driven Micromachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 12.2.1 Optical Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 12.2.2 Optical Driving Method of Multiple Micromachines . . . . . . . 298 12.2.3 Optimization of Time-Divided Laser Scanning . . . . . . . . . . . . 300 12.2.4 Cooperative Control of Micromanipulators . . . . . . . . . . . . . . . 302 12.2.5 Optically Driven Micropump . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 12.2.6 Concept of All-Optically Controlled Biochip . . . . . . . . . . . . . . 307 12.3 Conclusion and Future Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 13 Study of Complex Plasmas M. Shindo and O. Ishihara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 13.1 Overview of Complex Plasma Research . . . . . . . . . . . . . . . . . . . . . . . . 311 13.2 Charging of a Dust Particle in a Plasma . . . . . . . . . . . . . . . . . . . . . . . 312 13.3 Measurements of the Charge of Dust Particles Levitating in Electron Beam Plasma [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 13.4 Various Approaches to Plasma-Aided Design of Microparticles System in Ion Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 315 13.4.1 Analysis of Ion Trajectories Around a Dust Particle in Ion Flow [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 13.4.2 Wake Potential Formation to Bind Dust Particles Aligned Along Ion Flow . . . . . . . . . . . 318 13.4.3 Attractive Force Between Dust Particles Aligned Perpendicular to Ion Flow [30] . . . . . . . . . . . . . . . . . . . . . . . . . . 320 13.5 Simulation Study of Cluster Design of Charged Dust Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 13.6 Complex Plasma Experiment in Cryogenic Environment [38] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

List of Contributors

Taro Arakawa Department of Electrical and Computer Engineering Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected]

Yoshiyuki Kawazoe Institute for Materials Research Tohoku University 2-1-1 Katahira, Aoba-ku Sendai 980-8577, Japan [email protected]

Amir A. Farajian Department of Mechanical Engineering and Materials Science Rice University Houston, TX 77005, USA [email protected]

Susumu Kurita Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokoham 240-8501, Japan

Osamu Ishihara Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected] Soh Ishii Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected]

Shoji Maruo Department of Mechanical Engineering Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan PRESTO Japan Science and Technology Agency 5 Sanbancho, Chiyoda-ku Tokyo 102-0075, Japan [email protected]

XIV

List of Contributors

Hiroshi Mizuseki Institute for Materials Research Tohoku University Sendai 980-8577, Japan [email protected]

Kohki Mukai Department of Solid State Materials and Engineering Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected]

Yoshifumi Noguchi Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan Research Fellow (DC2) of Japan Society for the Promotion of Science Yokohama National University Yokohama 240-8501, Japan Computational Materials Science Center (CMSC) National Institute for Materials Science (NIMS) 1-2-1 Sengen, Tsukuba Ibaraki 305-0047, Japan [email protected]

Kaoru Ohno Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected]

Shin-ya Ohno Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokoham 240-8501, Japan [email protected] Olga V. Pupysheva Department of Mechanical Engineering and Materials Science Rice University Houston, TX 77005, USA [email protected] Ryoji Sahara Institute for Materials Research Tohoku University Sendai 980-8577, Japan [email protected] Takao Sekiya Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokoham 240-8501, Japan [email protected] Masako Shindo Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected] Ken-ichi Shudo Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokoham 240-8501, Japan [email protected]

List of Contributors

Kunio Tada Graduate School of Engineering Kanazawa Institute of Technology 1-3-4 Atago, Minato-ku Tokyo 105-0002, Japan [email protected] Jun Takeda Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokohama 240-8501, Japan [email protected]

XV

Masatoshi Tanaka Department of Physics Graduate School of Engineering Yokohama National University 79-5 Tokiwadai, Hodogaya-ku Yokoham 240-8501, Japan [email protected] Boris I. Yakobson Department of Mechanical Engineering and Materials Science Rice University Houston, TX 77005, USA [email protected]

1 General Introduction K. Ohno

In a fundamental part of the ﬁeld of nano- and microscale science, revolutional progress has been made since last two decades, in a way highly respected by the society, politics, and economics. In this stream, scientists and engineers from diﬀerent ﬁelds of physics, chemistry, materials science, and information technology, including experimentalists, theorists, and researchers doing computer simulations, have collaborated to form a new interdisciplinary ﬁeld called nanotechnology. In the ﬁeld of electronics, for example, since the invent of the transistor by Shockley, Brattain, and Bardeen in 1940s, downsizing of the electronic devices has been continued. According to the so-called Moore’s law, the density or the number of transistors per unit area on an integrated circuit is doubled every 2 years; in other words, the size of transistors decreases by a factor of 1/8 every decade starting from 1 cm in 1950, and it is certainly ∼ 50 nm in 2007 as shown in Fig. 1.1. Figure 1.2 shows the atomic structure of the interface between Si and SiO2 [1, 2]. For example, a titanium deposition on top of silicon surfaces (Fig. 1.3) [3] is considered as a way to increase the mobility of the electronic devices. A lot of experimental and theoretical eﬀorts have been devoted to these and many related but diﬀerent systems. However, it is anticipated that the fabrication of electronic devices based on the present-day semiconductor technology will soon face the technical limit, and the use of nanolithography or self-organization controlled by light or heat (see Chap. 2), or the use of new idea such as quantum dots or molecular devices is highly expected. As a related topic, microstereolithography (Chap. 12) will be useful to manipulate micromachines. When the size or the dimension of materials decreases, a variety of new phenomena which have never been expected in bulk materials will appear. It would be a tremendous idea to use them as the future devices. First of all, when the size decreases, the quantum eﬀect becomes, in general, dominant as pointed out by Kubo in 1962 [4], and this is often called as the Kubo eﬀect. Consider for example metals. Near the Fermi level, metals have continuum spectra and the splitting between adjacent quantum levels is quite small and

2

K. Ohno

Fig. 1.1. Moore’s law of the minimum size of transistor used in the integrated circuit

Fig. 1.2. Si/SiO2 interface

(a)

(b)

Fig. 1.3. Ti on Si (001) surface. (a) Pedestal site and (b) dimer vacancy site [3]

1 General Introduction

3

Fig. 1.4. Ionization potential (IP) and electron aﬃnity (EA) and their relation to the energy levels

negligible. However, in clusters made of small number of atoms, the splitting between adjacent quantum levels is ﬁnite, and in general this splitting increases when the number of atoms in the cluster decreases or equivalently when the size of the cluster decreases. This is true not only for semiconductor clusters but also metal clusters. For neutral clusters and molecules, the electron aﬃnity (EA) is deﬁned as the maximum energy gain to attach an electron from inﬁnitely apart to the lowest unoccupied molecular orbital (LUMO), and the ionization potential (IP) is deﬁned as the minimum energy required to detach an electron from the highest occupied molecular orbital (HOMO) to inﬁnitely apart. The absolute values of the LUMO and HOMO energies correspond EA and IP, respectively, and the IP minus EA gives the energy gap; see Fig. 1.4. There is a general tendency that the energy gap increases when the size of the cluster decreases although there are exceptions due to the irregular geometries of the bond between atoms. For the GW approximation, see Chap. 6. Experimentally, quantum levels of bulk samples are measured by the photoemission or inverse photoemission experiment. The photoemission determines the quantum levels of the occupied states from the absorbed photon energy minus the emitted excited electron, while the inverse photoemission determines those of the empty states from the absorbed electron energy minus the emitted photon energy. For clusters, the mass of charged clusters is separated by the time-of-ﬂight (TOF) method, which uses the acceleration proportional to e/m under an applied electric ﬁeld. Simultaneously, by photoirradiation, the electron aﬃnity (EA) is measured as the threshold value of a photon energy with which the negatively charged clusters is photodetached and neutralized. Irrespective to bulk or cluster, optical absorption spectra is diﬀerent from the photoemission and inverse photoemission spectra. This is because, in the optical absorption process, the excited electron does not go away from the

4

K. Ohno

cluster but still trapped inside the cluster, forming an electron–hole pair called exciton. Due to the binding energy of the Coulomb attraction between the electron and hole, the threshold energy of the optical absorption is generally smaller than the energy gap. For semiconductor clusters, the phenomenon that the photoluminescence energy is smaller than the optical absorption energy, i.e., the photoluminescence has longer wavelength than the optical absorption, is called the Stokes shift. This phenomenon occurs because the relaxation of atomic geometry takes place in each process. Then because the wavelength of the photoluminescence is diﬀerent from the incident light, it can be detected distinctly from everywhere the cluster exists. Moreover, the color of the photoluminescence depends on the cluster size. The clusters showing strong luminescence are therefore useful to mark particular biomolecule, for example, since the luminescence with diﬀerent wavelengths is controlled by the cluster size. In this respect, CdSe clusters are often used in biomedical experiments. Since the zero-dimensional system inside which charged carriers and excitations are conﬁned is called the quantum dot, these clusters are often called quantum dots. (More commonly the term “quantum dot” is used in electron transport problems explained later.) Stable structures and optical absorption spectra of small CdSe clusters (Fig. 1.5) have been calculated from ﬁrst principles [5–7]. For passivated nonstoichiometric CdSe clusters, the result of the state-of-the-art ﬁrst-principles calculation solving the Bethe–Salpeter equation for the twoparticle Green’s function is compared with the result of the time-dependent density functional theory in [8]. The wavelength of the absorption peaks is strongly size dependent and monotonically increases as the size of the cluster decreases. The majority of the clusters have a series of dark transitions before the ﬁrst bright transition. This may explain the long radiative life times observed experimentally. For an example of metal clusters, FePt clusters have attracted considerable interest because it can be used for magnetic thin ﬁlms with high coercivity. Figure 1.6 shows the structure of FePt cluster with a diameter of 17 nm at 3,000 K determined by a fcc-lattice Monte Carlo simulation using the total

Fig. 1.5. Most stable structure of (CdSe)13 and (CdSe)34 . After Noguchi et al. [5] and Kasuya et al. [6]

1 General Introduction

5

Fig. 1.6. Structure of FePt cluster with a diameter of 17 nm at 3,000 K

energies determined by a ﬁrst-principles calculation (see Chap. 11) [9, 10]. Au clusters are also of much current interest because it was found to exhibit catalytic behavior [11, 12]. To control the energy gap in p–n junction has been crucially important in semiconductor technology. This idea may be directly used to create the high performance solar battery. The tuning of the optical absorption spectra to the spectra of sun light is basically possible by combining diﬀerent sized clusters. Another example is the photosynthesis in chlorophyll or light-harvesting property in dendrimers (see Chap. 3). Figure 1.7 is a π-conjugated dendrimer, star-shaped stilbenoid phthalocyanine (SSS1Pc) with oligo (p-phenylenevinylene) peripheries, which shows a light-harvesting property [13–16]. The calculated wavefunctions [16] are shown in Fig. 1.8, in which the levels are clearly separated to those belonging to the peripheries (P) and those belonging to the core (C). When an electron is selectively excited in the periphery (P), electron and hole transfer from the periphery to the core through π-conjugated network as shown in Fig. 1.9. From dynamics simulation [16], it has been found that the one-way electron and hole transfer occurs more easily in dendrimers with planar structure than in those with steric hindrance because π-conjugation is well maintained in the planar structure. This results well explain the experiments by Akai et al. [13,14] and Takeda et al. [15]. Another example of the gap control is a photocatalysis. In this respect, metal oxide such as anatase TiO2 (see Fig. 1.10) has been widely investigated (see Chap. 4). In particular, the doping of transition metal impurity is quite important in controlling the energy gap suitably.

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K. Ohno

Fig. 1.7. Structure of (a) SSS1Pc-1 and (b) SSS1Pc-2. In both (a) and (b), upper ﬁgures show the front view and lower ﬁgures show the side view. The structure of SSS1Pc-2 (b) is three-dimensional due to steric hindrance between the peripheries

Fig. 1.8. (Color online) Amplitude of the wave function at the ground state of SSS1Pc-2. The cubes are the unit cells. For each level, the points at the center and the upper right side show the core and the periphery, respectively. Gray and black areas denote the positive and negative values of the wave function, respectively

For example, the dissociation of H2 O by solar energy would be one of the wonderful applications in the photocatalytic reaction. Figure 1.11a shows the absorption of H2 O on the surface of anatase crystal. Figure 1.11b, c shows the geometry and wavefunction of the most stable, adsorbed ground state [17].

1 General Introduction

7

Energy eigenvalue (eV)

0 −1

Periphery (P)

Core (C)

P-LUMO

−2

C-LUMO

−3

hv C-HOMO

−4

P-HOMO

−5

Fig. 1.9. The energy eigenvalues of SSS1Pc-2. Black and white circles denote electrons and holes, respectively. First, an electron is excited from the (almost doubly degenerate) P-HOMO levels to the (doubly degenerate) P-LUMO levels on the periphery side (solid line with an arrow ). Then, the electron is transferred from the P-LUMO levels to the (doubly degenerate) C-LUMO levels (dashed line with an arrow ), and the hole is transferred from the P-HOMO levels to the C-HOMO level (dotted line with an arrow )

(a)

(b)

Fig. 1.10. (a) The unit cell of anatase (TiO2 ) crystal. Large and small circles correspond, respectively, to oxygen and titanium atoms. (b) The supercell for treating the surface of anatase (TiO2 ) crystal

Figure 1.12 shows a schematic diagram of the dissociation of H2 O molecule by photocatalyst. Figure 1.12a is the simplest scheme, in which four holes created by light absorption induce a reaction 2H2 O → 4H+ + O2 + 4e− and produce an oxygen molecule, while two electrons induce a reaction 2H+ + 2e− → H2 and produce a hydrogen molecule. Figure 1.12b is a combination of two independent reactions using diﬀerent catalysts can induce oxygen and hydrogen molecules separately, called the Z scheme [18].

8

K. Ohno

(a)

(b)

(c)

Fig. 1.11. Reaction between H2 O molecule and the surface of anatase crystal

Transition metal oxides are well-known strongly correlated systems, whose electronic structure is hardly treated by the standard band structure calculation. Similar and related topic is a synthetic metals and organic Mott insulators. For example, the high-temperature phase of an organic radical 1,3,5-trithia-2,4,6-triazapentalenyl (TTTA) crystal exhibits a Mott insulator phase [19] (see Chap. 5). The state-of-the-art T -matrix calculation solving the Bethe–Salpeter equation can handle the multiple scattering and short-range correlations between electrons, and enables us to evaluate the on-site Coulomb energy U of this material (see Chaps. 5 and 7). By this method, it is demonstrated that the so-called “Coulomb hole” plays a very important role in the problem of short-range electron correlations. Quantum dots are more commonly considered in the electron transport problem in conﬁned area (see Chaps. 8 and 9). The electron transport through a quite small structure such as quantum dots is governed by the quantum eﬀects. For example consider a spherical particle with a diameter of d embedded in the medium of dielectric constant ε. Then the capacitance of this particle is given by C = 2πεd,

(1.1)

and the energy required to charge up this particle with one excess electron (or hole) is given by E=

e2 e2 = . 2C 4πεd

(1.2)

(If we consider a Cooper pair in a superconductor, e2 should be replaced by 4e2 .) When the size d (and therefore the capacitance C) of this cluster becomes

1 General Introduction

O

O

H+

H

H

O

H+

H

H

H

H

Pt

O H

hν

H

H+

Fe e+

3+

H

hν

e+

2+

A

H O

O

hν

e−

e+ e−

H

H

H H+

9

Fe

e−

B

3+

Fe

Fe

H

H

H H+ O

H+

H

Pt

hν H+

H

H

O

H H+

H

H

e+ e−

O

H

A

H

Fe3+

H

H

O

H

(a)

A H H

H hν 2+

e− Fe

H2 H

3+

O2

O O O O

O H

H

Fe

B

H Pt H

O O

O

e+

Fe2+

e+

O2

O

hν

e–

H O

2+

B

H2

(b)

Fig. 1.12. (a) The simplest scheme of the dissociation of H2 O by photo-catalyst. Four holes created by light absorption induce a reaction 2H2 O→ 4H+ +O2 +4e− and produce an oxygen molecule, while two electrons induce a reaction 2H+ + 2e− →H2 and produce a hydrogen molecule. (b) Two independent reactions using diﬀerent catalysts can induce oxygen and hydrogen molecules separately

extremely small, this energy E becomes large and exceed the thermal energy kB T . In this case, the electron transfer (i.e., the conductance) is blocked. This phenomenon is called “Coulomb blockade.” Then, according to the bias voltage, the electric current jumps up stepwise. This anomalous conducting behavior can be observed in nanometer-sized metal clusters embedded in the oxide tunnel junction sandwiched by metal conductors at quite low temperature. The origin of E in (1.2) is the electron–electron repulsive interaction inside the quantum dot, and this problem is related to the problem of strongly correlated electrons. Such an problem can be treated by the T -matrix theory (see Chaps. 5 and 7). When the conﬁned area is two-dimensional, the structure is called “quantum well” (see Chap. 10). The density of states in two-dimensional materials is much sharper than in three-dimensional materials, and therefore quantum

10

K. Ohno

Fig. 1.13. TTTA crystal in which the electronic charge distribution shaded by blue clouds is restricted inside each molecule in the HT phase or between the dimerized molecules in the LT phase

wells are widely used as diode lasers. They are used also for the heterostructure ﬁeld eﬀect transistor (HFET) which is called also as the high electron mobility transistor (HEMT). Related but completely new idea in the physics of nanotechnology is based on the wavefunction control instead of the energy gap control. One example is the quantum computing, which uses, for example, the quantum spin states | ↑ and | ↓ called “qubits.” Although there are still many problems to be solved, quantum dot may be used as a qubit in the future. Qubits may be also realized by constructing three Josephson junctions in a superconducting circuit (see Fig. 1.14) [20, 22]. In the classical von Neumann-type computer, this information is used just as 0 or 1. In contrast, in the quantum computer, the mixture of the two quantum states is also used, and certain problems such as integer factorization is expected to be solved exponentially faster than the classical computer. Another example is the use of the Aharonov–Bohm (AB) eﬀect. A well-known example of the AB eﬀect is as follows: The wavefunction of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic ﬁeld though the magnetic ﬁeld is zero in the region through which the particle passes. As is seen in this example, the electron wavefunction may become physical quantity and may be used to develop completely new electric devices in the future.

1 General Introduction

11

(b)

(a)

Fig. 1.14. (a) Josephson junction of 800 nm wide and (b) superconducting loop including three Josephson junctions working as a qubit. Both of them are made of aluminum. Courtesy of Shimazu [20]

(a)

(b)

(c)

Fig. 1.15. Structure created in dust plasma. (a) is the side view, while (b) and (c) are the cross section view. After Ishihara [21]

A completely diﬀerent but very interesting topic is a complex plasma known also as a dust plasma. It includes ﬁne particles of size ranging from nanometers to micrometers in size. What is interesting is the creation of hollow structure of dust particles despite the Coulomb repulsive interaction between dust particles. Figure 1.15 is a computer simulation image of the structure. See Chap. 13 for more details. As a new kind of low-dimensional materials, fullerenes and nanotubes made of only carbon atoms have attracted considerable interests since the discovery of C60 by Kroto et al. in 1985 [23]. By laser ablation or arc discharge experiments using a graphite rod, carbon chain molecules are aggregated in a plasma state. Fullerenes are created when the plasma is cooled down in a helium gas atmosphere. Fullerenes a hollow, closed cage structure made of

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K. Ohno

(a)

(b)

(c)

Fig. 1.16. Na insertion into carbon nanotube

spherical network of six- and ﬁve-membered rings. It is well-known that, due to mathematical Euler theorem, the number of ﬁve-membered rings is always 12. The most abundant fullerene is C60 , which has a soccer ball shape. The next abundant fullerene is C70 , which has a rugby (foot) ball shape, and there are many higher fullerenes such as C74 , C76 , C78 , C82 , C84 , C90 , C94 , . . .. On the other hand, carbon nanotubes have a hollow cylindrical tube structure formed from a rolled graphite sheet and therefore made of six-membered rings only [24]. Carbon nanotubes have very high tensile strength and elastic moduli due to the covalent sp2 bonds between adjacent carbon atoms. The encapsulation of foreign atoms or molecules inside fullerenes and carbon nanotubes has been also investigated. Figure 1.16 represents the snapshots of the ﬁrst-principles molecular dynamics simulation of inserting a sodium atom with 70 eV kinetic energy into a single-walled carbon nanotube [25], although no such experiment has been performed yet. If these materials could be created experimentally, they would be applied to a molecule-based diode or conductor as well as the gold nanowires [26]. For the calculation of transport properties of these materials, see Chap. 8. It has also been revealed that a polyyne molecule (C10 H2 ) can be put inside an open-ended single-walled carbon nanotube [27]. There is an energy gain of about 1.7 eV when C10 H2 . The bonding between C10 H2 and SWNT is due to the large area of weak overlap of the wave functions in the intermolecular region inside the SWNT [28]; see Fig. 1.17. A recent related experiment showing the molecular motion inside SWNT has been reported in [29]. The so-called endohedral fullerene, which has at least one foreign atom inside the cage of the fullerene, have attracted interested. Experimentally, it has been conﬁrmed that at least one lanthanum, yttrium, or scandium atom can be encapsulated inside C82 or C84 using arc-discharge vaporization of composite rods made of graphite and the metal oxide [30]. The creation of endohedral C60 is possible, though the creation rate is very low, by using a nuclear recoil of isotope nuclear reaction [31]. Figure 1.18 represents a snapshots of a ﬁrst-principles molecular dynamics simulation of the insertion of Po atom with 40 eV kinetic energy into C60 . It is quite amazing that such a heavy element as Po can be successfully encapsulated inside C60 with such low energy. Experimentally, the existence of Po@C60 in the solvent was certainly

1 General Introduction

13

polyyne molecule

Fig. 1.17. Wave function of the HOMO level of C10 H2 @SWNT

Fig. 1.18. Snapshots of a ﬁrst-principles molecular dynamics simulation of a Po atom insertion into C60 with 40 eV kinetic energy

conﬁrmed in the synchronized measurements using high-performance liquid chromatography and UV detector [31]. As a related topic, the electron capture (EC) decay rate of 7 Be encapsulated in C60 was measured using a reference method comparing with the rate in Be metal crystal, and it was found that the half-life of 7 Be endohedral C60 (7 Be@C60 ) decreases about 0.83% than inside Be metal crystal [32, 33].

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Fig. 1.19. Structure of the 3D polymers crosslinked by [2 + 2] cycloadditional four-membered rings

(a)

(b)

Fig. 1.20. Optimized 3D structure of peanut-shaped polymers crosslinked by eightmembered rings in a monoclinic unit cell. (a) is a side view and (b) is a view of the cross section of this structure

The decay rate is further accelerated when the 7 Be@C60 sample is cooled down at liquid helium temperature (its half-life is 1.5% shorter than Be metal) [34]. This phenomenon can be explained theoretically by the calculation of the electron density at the 7 Be nucleus position inside the C60 cage and in the Be metal crystal. The theoretical estimates are in fair agreement with the experimental observations [34, 35].

1 General Introduction

15

Fig. 1.21. Structure of polymer

Fullerene polymers made of C60 are also interesting [36]. Figure 1.19 represents C60 polymer networks crosslinked by [2+2] cycloadditional fourmembered rings, and Fig. 1.20 represents a peanut-shaped fused C60 polymer chains crosslinked by eight-membered rings, which are considered as a model for the electron beam irradiated C60 samples [37]. Owing to the overlap of wave functions as well as the hybrid networks of sp2 -like (threefold coordinated) and sp3 -like (fourfold coordinated) carbon atoms, the electronic structure is considerably diﬀerent from each other. The resulting electronic structure is either semiconductor or semimetal depending on the spatial dimensionality of materials [36]. Another interesting topic in nano- and micromaterials is soft materials like ﬂexible polymers, although they are not described in detail in this book. For example, micelle formation of AB block-copolymers can be used as a drag delivery system (DDS) in biomedical applications. Figure 1.21 represents an example of the mixture of water, oil, and amphiphilic polymers [38].

References 1. A. Pasquarello, M.S. Hybertsen, R. Car, Appl. Surf. Sci. 104/105, 317 (1996) 2. T. Morisato, K. Ohno, Y. Kawazoe, unpublished 3. B.D. Yu, Y. Miyamoto, O. Sugino, T. Sasaki, T. Ohno, Phys. Rev. B 58, 3549 (1998) 4. R. Kubo, J. Phys. Soc. Jpn. 17, 975 (1962)

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5. Y. Noguchi, K. Ohno, V. Kumar, Y. Kawazoe, Y. Barnakov, A. Kasuya, Trans. Mater. Res. Soc. Jpn. 29, 3723 (2004) 6. A. Kasuya, R. Sivamohan, Y.A. Barnakov, I.M. Dmitruk, T. Nirasawa, V.R. Romanyuk, V. Kumar, S.V. Mamyukin, K. Tohji, B. Jeyadevan, K. Shinoda, T. Kubo, O. Terasaki, Z. Liu, R.V. Belosludov, V. Sundararajan, Y. Kawazoe, Nat. Mater. 3, 99 (2004) 7. S. Botti, M.A.L. Marques, Phys. Rev. B 75, 035311 (2007) 8. M.L. del Puerto, M.L. Tiago, J.R. Chelikowsky, Phys. Rev. Lett. 97, 096401 (2006) 9. S. Masatsuji, Master Thesis, Yokohama National University, 2006 10. Y. Misumi, S. Masatsuji, S. Ishii, K. Ohno, MRS fall meeting, 2007 11. M. Haruta, N. Yamada, T. Kobayashi, S. Iijima, J. Catl. 115, 301 (1989) 12. M. Valden, Z. Lai, D.W. Goodman, Science 281, 1647 (1998) 13. I. Akai, H. Nakao, K. Kanemoto, T. Karasawa, H. Hashimoto, M. Kimura, J. Lumin, 112, 449 (2005) 14. I. Akai, A. Okada, K. Kanemoto, T. Karasawa, H. Hashimoto, M. Kimura, J. Lumin, 119–120, 283 (2006) 15. A. Ishida, Y. Makishima, A. Okada, I. Akai, K. Kanemoto, T. Karawasa, M. Kimura, J. Takeda, preprint (DPC07) 16. Y. Kodama, S. Ishii, K. Ohno, J. Phys. Condens. Matter. 19, 365242 (2007) 17. J. Shiga, S. Ishii, K. Ohno, unpublished 18. H. Kato, M. Hori, H. Sugihara, K. Domen, Chem. Lett. 33, 1348 (2004) 19. K. Ohno, Y. Noguchi, T. Yokoi, S. Ishii, J. Takeda, M. Furuya, Chemphyschem 7, 1820 (2006) 20. Y. Shimazu, J.E. Mooij, in Towards the Controllable Quantum States, ed. by H. Takayanagi, J. Nitta (World Scientiﬁc, Singapore, 2003) pp. 353–358 21. O. Ishihara, J. Phy. D: Appl. Phys. 40, R121 (2007) 22. J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, S. Lloyd, Science 285, 1036 (1999) 23. H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl, R.E. Smaley, Nature 318, 162 (1985) 24. S. Iijima, I. Ichihashi, Y. Ando, Nature 356, 776 (1992) 25. A.A. Farajian, K. Ohno, K. Esfarjani, Y. Maruyama, Y. Kawazoe, J. Chem. Phys. 111, 2164 (1999) 26. Y. Kondo, K. Takayanagi, Science 289, 606 (2000) 27. D. Nishide, H. Dohi, T. Wakabayashi, E. Nishibori, S. Aoyagi, M. Ishida, S. Kikuchi, R. Kitaura, T. Sugai, M. Sakata, H. Shinohara, Chem. Phys. Lett. 386, 279 (2004) 28. R. Kuwahara, Y. Kudo, T. Morisato, S. Ishii, K. Ohno, unpublished 29. M. Koshino, T. Tanaka, N. Solin, K. Suenaga, H. Isobe, E. Nakamura, Science 316, 853 (2007) 30. H. Shinohara, H. Sato, M. Ohkohchi, Y. Ando, T. Kodama, T. Shida, T. Kato, Y. Saito, Nature 357, 52 (1992) 31. T. Ohtsuki, K. Ohno, Phys. Rev. B 72, 153411-1 (2005) 32. T. Ohtsuki, H. Yuki, M. Muto, J. Kasagi, K. Ohno, 33. H. Pllcher, Nature 431, 412 (2004) 34. T. Ohtsuki, K. Ohno, T. Morisato, T. Mitsugashira, K. Hirose, H. Yuki, J. Kasagi, Phys. Rev. Lett. 98, 252501 (2007), to be published online on 20 June 2007

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35. T. Morisato, K. Ohno, T. Ohtsuki, K. Hirose, Y. Kawazoe, Phys. Rev. B, in submission 36. S. Ueda, K. Ohno, Y. Noguchi, S. Ishii, J. Onoe, J. Phys. Chem. B 110, 22347 (2006) 37. J. Onoe, T. Ito, S.-I. Kimura, K. Ohno, Y. Noguchi, S. Ueda, Phys. Rev. B 75, 233410 (2007) 38. N. Nakagawa and K. Ohno, in AIP Proceedings on the 5th International Workshop on Complex Systems (IWCS2007), in print

2 Nanometer-Scale Structure Formation on Solid Surfaces M. Tanaka, K. Shudo, and S. Ohno

2.1 Introduction Nanostructured materials have been extensively studied for more than 10 years because of tremendous potential to application in a variety of technology, such as electronics, materials science, and biotechnology. Although a large part of these studies concerns nanostructures in three dimensions, this section focuses on nanostructures in rather lower dimensions, on solid surfaces. Moreover, “nano-” usually means the range of a few to hundreds of nm; however, we concentrate on the structures one order smaller than usual, in other words, the structures in “atomic scale” rather than “nanometer scale,” especially those formed on well-characterized surfaces under ultrahigh vacuum (UHV) conditions. Even in this scale, nanostructures can be formed both by selforganization and by ultraﬁne machining. Before we present our latest studies, some categories of this kind of nanostructures are introduced in this section. We do not attempt to present a detailed review with reference to huge number of articles, but give a few examples of each category with emphasis on the initial works or fundamental studies. Self-organization process can potentially produce uniform nanostructures in wide area. It is more attractive in fabricating nanodevices if the controllability is achieved. Self-organized nanostructures are classiﬁed into some categories, for instance, those on metal surfaces are diﬀerent from those on semiconductor surfaces [1]. Surface reconstruction on a (110) surface of face-centered-cubic metals (Ni, Cu, Pd, Ag, Ie, Pt, Au) is probably oldest known self-organized nanostructures on surfaces [2]. Added row and missing row reconstructions are found on diatomic gas molecule-adsorbed surfaces and on alkali-metal adsorbed surfaces as well as on clean surfaces [3]. The driving force of this kind of reconstruction is simply thought to be surface energy; however, the reconstruction is as a result of subtle energy balance between electronic energy and surface energy. Very low coverage of gas molecule or alkali-metal induces the reconstruction, which means that local coordination number and nonlocal

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charge distribution of transferred charge play important roles of the reconstruction. Prototype of nanopatterned metal surfaces is observed on vicinal Au(111) and nitrogen-covered Cu(001), and the pattern formation is explained by the elastic continuum model [4]. Long-range elasticity is dominant to form not only these prototypes but also all kinds of adlayers, and can be a tool for self-organization [5]. These structures are used as templates for growing one-dimensional (1D) and two-dimensional (2D) structures. Metal epitaxy on metal surfaces, such as Cu/Ru(0001) and Au/Ni(111), exhibits also nanopatterns [6]. Self-organized nanostructure formation on semiconductor surfaces, especially IV group semiconductors [7] and III–V group semiconductors [8], has been more extensively studied than that on metal surfaces because of potential applications in industries. As for IV group semiconductors, elongated Ag islands with aspect ratios greater than 50:1 were formed on Si(001) and the formation of this 1D structure was a result of elastic relaxation of the strained layer [9]. The 1D structures are found also in other systems. Self-assembled Ge nanowires were grown on Si(113) by molecular beam epitaxy [10]. Bi line structures were formed on Si(001) in the vicinity of its desorption temperature [11]. The 2D structures, such as the growth of Ge layers on Si surface, have been a subject of greater interest than these 1D structures. Ge nanoislands were formed by taking advantage of the Stranski–Krastanow (SK) growth mode [12,13]. Coherent SK growth was explained in terms of elastic deformation around the islands. The island size and spacing grow progressively more uniform, when Si layers and Si0.25 Ge0.75 layers are formed alternately on these nanoislands [14]. An approach toward nanointegration through the control of self-organization processes of surface structures – such as surface reconstruction, atomic steps, and the phase boundaries of reconstructed domains – was proposed [15]. On the other hand, as for III–V group semiconductors, the ﬂat (211), (311), and (111) GaAs surfaces break up into regular facets and make superlattices with lateral corrugation of the interfaces during multilayer molecular beam epitaxy [16]. Quantum dots have been developed since dislocation-free strained In0.5 Ga0.5 As islands were found during the growth on a GaAs(001) substrate [17]. More practical methods to obtain highly uniform dot size and density were proposed [18, 19]. The prototype of the ultraﬁne machining is the method using Ga+ focused ion beam (FIB) with a diameter of only 100 nm [20]. Signiﬁcant progress has been made in FIB technology and it is now a powerful tool in lithography, etching, deposition, doping, and even 3D nanostructures [21]. However, the dimension of nanostructures produced with this method is still beyond the scale focused in this section. Best spatial resolution in ultraﬁne machining is achieved with a scanning tunneling microscopy (STM) which can manipulate atoms one by one. Atom manipulation was ﬁrst demonstrated by sliding Xe atoms on Ni(110) at 4 K to form an “IBM” logo where each letter was written by a collection of atoms [22]. Anther example is excision of S atoms from MoS2 surface by ﬁeld

2 Nanometer-Scale Structure Formation on Solid Surfaces

21

evaporation to form characters at room temperature (RT) [23]. Nanometerscale modiﬁcation of H-passivated Si(111) surface in air was also reported [24]. Under UHV condition, nanoscale patterning of H-passivated Si(100) surface was achieved by local desorption of hydrogen due to tunneling current, and only the patterned area was subsequently oxidized [25]. Tunneling electrons not only manipulate an atom, but also form an eﬀective excitation source for inducing chemical reaction. The concept of bond-selective chemistry using this mechanism was proposed with the examples of single O2 molecule dissociation on Pt(111) and displacement of Si adatoms on Si(111) [26]. Fe(CO) molecules were formed starting from Fe atoms and CO molecules adsorbed on a Ag(110) surface [27]. The feasibility of inducing all the steps of a surface chemical reaction by using the STM tips was shown by the synthesis of biphenyl molecules starting from iodobenzene adsorbed on Cu(111) [28, 29]. Although spatial resolution of surface modiﬁcations using STM is perfect, it cannot be applied directly to the production of devices. As a more practical way, possibility of nanostructuring the surface by inelastic processes, induced by electrons or photons, has been widely discussed. Modiﬁcation of materials by electronic excitation is becoming attractive due to recent advances in laser and synchrotron radiation [30]. In this section, we present our latest studies on nanometer-scale structure formation on solid surfaces since 2002 in the following sections. In Sect. 2.2, the fundamental processes of layer-by-layer etching of a Si(111) surface are described. Nanostructures in the lateral direction are also found: Halogen atoms are adsorbed at selective sites, and clusters are formed during the desorption process. In Sect. 2.3, how to control silicon surface at the atomic scale is described with regard to the dynamic processes, for example, passive fabrication due to thermal process to align nanoclusters and active fabrication via nonequilibrium reaction pathways due to electronic excitation. In Sect. 2.4, an example of self-organization on a metal surface is introduced: Nitrogen adsorption on Cu(001) surface induced strain and forms patch patterns which are used as a template for nanoscale arrangements.

2.2 Atomic Layer Etching Processes on Silicon Surfaces 2.2.1 Introduction Etching with halogen gases is the widely used process to fabricate semiconductor surfaces, and the usual industrial process is thermally induced etching in ambient chlorine gas. To understand the process, adsorption and desorption of chlorine at silicon surfaces have been studied [31, 32]; but because the current technology requires ultrahigh precision [33], the importance of study at the atomic scale is growing. Halogen etching is also a promising candidate method for atomic layer etching, which is one of the basic techniques

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to fabricate nanometer-scale structures [34]. Reaction between halogens and semiconductor surfaces has therefore attracted much attention in the recent years. Halogen etching consists of several stages; adsorption of halogen atoms on the surface, desorption of silicon halides, and reconstruction of the clean surface. Understanding of the atomic-scale mechanisms of these fundamental processes will be useful to optimize etching conditions and necessary for future development of atomic-scale etching. Although atomic-scale etching itself is a kind of ultraﬁne machining, our studies on the fundamental processes of the etching have revealed that they involve self-organizing processes, for instance, halogen atoms can be adsorbed at selective sites to form adsorbate patterns and nanoclusters can be formed by the thermal treatment of halogen-covered surface. In this section, the atomic-scale mechanisms of these fundamental processes are elucidated mainly by means of real-time optical measurements. Etching of the Si(001) surface is preferentially studied in connection with industrial applications, but etching of the Si(111) surface is also of interest, because the dimer-adatom-stacking fault (DAS) structure [35] has a variety of sites with diﬀerent chemical reactivity [36]. The DAS model is illustrated in the left half of Fig. 2.1. STM has greatly improved our understanding of chemical processes at the atomic scale, and most studies have focused on the electronic states or the morphology mainly of the Si adatoms. However, another type of dangling bond on the rest-atoms which are not accessed by STM must have some role in surface reactions. The static properties at each stage in the fundamental processes of halogen etching have been revealed by a variety of methods. It is known that chlorine atoms ﬁrst react with adatom sites to form monochlorides and remove dangling bond states near EF at low coverage [37, 38], while further exposure produces SiCl2 and SiCl3 species [39, 40]. These polychlorides tend to be formed on the center adatom sites [41]. On the other hand, there is little direct evidence for the presence of polybromide species, although their presence is generally accepted. In the right half of Fig. 2.1, the chloride species are schematically shown. When a dichloride is formed, the back-bond of the adatom is broken and a new dangling bond appears on the rest-atom, as illustrated in Fig. 2.1. With further chlorine, the second back-bond is broken to form a trichloride. Annealing at about 700 K for a Cl-saturated surface and at 500–650 K for a Br-saturated surface removes Si adatoms, and the rest-surface (the surface consisting of rest-atoms; see Figs. 2.37 and 2.41) covered with halogen atoms takes a 1×1 structure as in Fig. 2.27 [37–39, 42, 43]. Ultraviolet laser irradiation also produces this kind of rest-surface [41, 44]. X-ray photoelectron spectroscopy [39, 40] and surface-enhanced X-ray absorption ﬁne structure [45] studies conﬁrmed that only monochloride species remain after annealing above 673 K. In accordance with the above observations, a thermal desorption spectroscopy (TDS) study revealed that polychlorides species are desorbed as a peak at 690 K, and a laser-induced thermal desorption (LITD) study showed that the SiCl3 species almost disappears above 630 K [46]. On

2 Nanometer-Scale Structure Formation on Solid Surfaces Clean

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Adsorption Emerging dangling bond Relaxing

Si restatoms (with native d.b.)

Si restatoms (bonded to the adatom originally)

Fig. 2.1. (a) Structural model of clean (left half ) and Cl-adsorbed Si(111)–7×7 DAS surfaces (right half ). a and r indicate the dangling bonds on Si adatoms and Si rest-atoms, respectively. Mono-, di-, and trichlorides are marked as M , D, and T , respectively. Hatched Cl atoms terminate the adatom dangling bonds (N ) or the newly emerging dangling bonds (E) at the rest-atoms. (b) Dichloride formation from a Si monochloride. The back-bond of the adatom is broken and a new dangling emerges at the rest-atom

the 1×1 Br-terminated rest-surface, many bilayer islands and clusters are found [43, 47]. When the halogen-covered rest-surface is heated above 900 K, halogen atoms are desorbed mainly as SiCl2 [40,46] and SiBr2 [48], as shown by a TDS study. As for the desorption mechanism, an STM study indicated that spontaneous Br etching of Si(111) at 700–900 K results in step retreat [43]. In this way, one Si layer is taken oﬀ, and a clean 7×7 DAS structure is subsequently restored. In spite of these studies, the dynamic processes of the desorption of silicon chlorides and the reconstruction to form the 7×7 DAS structure are still poorly understood, and need to be established before the relative reactivity of halogens on the Si(111) surface can be discussed.

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2.2.2 Real-Time Optical Measurements Real-time in situ observation is essential to investigate the kinetics. Optical methods are superior to others, because they are noninvasive, nondestructive, and capable of very rapid response. The optical responses of the surfaces are related to the surface electronic states [49–53]. Studies on halogen-etching processes on Si(111) introduced in this section were investigated by means of two optical methods: surface diﬀerential reﬂectivity (SDR) spectroscopy and second harmonic generation (SHG). These optical methods were combined with TDS which gives the total halogen coverage. These experimental techniques are not so popular compared with standard techniques for surface analysis such as electron spectroscopy. Accordingly, principles of these techniques are brieﬂy introduced and their experimental procedures are described. Surface Diﬀerential Reﬂectivity Spectroscopy SDR spectroscopy was proved to be a powerful tool for the real-time study of hydrogen adsorption on Si(111) [49]. Diﬀerential reﬂectivity is deﬁned as ∆R/R ≡ (Ra − Rc )/Rc , where Ra and Rc are the reﬂectivities of the Hcovered and clean surfaces, respectively. Spectral features of adsorption on adatom dangling bonds and breaking of adatom back-bonds were identiﬁed from the calculation of the ∆R/R spectrum for the hydrogenated 7×7 surface [54, 55]. These spectral features arise from the surface states of the clean surface, so that the SDR spectrum is considered not to depend on the adsorbate. These features develop with time during adsorption processes, whereas in the desorption processes, they decay with time as the clean surface structure is restored. Magnitudes of the SDR spectral features is interpreted to be proportional to the densities of saturated dangling bonds and broken bond breakage. The schematic diagram of the experimental setup is shown in Fig. 2.2 [56]. Measurements reported in this section were performed in an UHV chamber at a base pressure of 2 × 10−8 Pa. The 7×7 structure of the clean surface Powermeter

Q - Switched N d : YAG Laser λ/2

Imaging Assembly Monochro mator PDA

F1 LS1

Quartz Plate

SHG F3

AgCl Electrochemical Cell

Monochromator

F4

P1 LS2

P2 F2

Controller

PM

PM

Preamplifier

SDR

Si(111) Digital Oscilloscope

PC QMS

PC

Fig. 2.2. Experimental setup for SDR and SHG. See [56] for detail

2 Nanometer-Scale Structure Formation on Solid Surfaces

25

was conﬁrmed by low-energy electron diﬀraction. Halogen gas was generated in the vacuum with a AgX (X = Cl, Br) electrochemical cell doped with CdX2 (5% wt) [57]. The electrochemical cell produces more atoms than molecules [58]. The setup for the SDR measurement is as follows. Light from a halogen tungsten lamp (LS1) or a deuterium lamp (LS2) was polarized horizontally with a Glan-Taylor prism (P1), and separated into a probe beam (90%) and a reference beam (10%). The p-polarized probe beam was introduced into the vacuum chamber and incident on the surface at an angle of 70◦ from the surface normal. The specularly reﬂected probe beam and the reference beam were introduced via optical ﬁbers to a grating spectrograph with an imaging assembly correcting astigmatism. The spectra of both beams were detected by a dual photodiode array, and the intensity of the reﬂected spectrum was normalized with respect to the reference spectrum. Photoinduced electrons in the diode array were accumulated at each pixel for 10 s to improve the signal-to-noise ratio. Second Harmonic Generation SHG has been employed more extensively than SDR to observe the kinetics of adsorption [51, 59] and desorption [60, 61] on Si(111). When the fundamental wave of a Nd:YAG laser is used as a pump laser, the two-photon energy of the fundamental wave is resonant to the S3 –U1 transition [52], where S3 and U1 states are attributed to the adatom back-bond and the adatom dangling bond, respectively. The nonlinear susceptibility χ(2) then decreases linearly with the coverage at low coverage. The adsorption process on adatom dangling bonds at low coverage is therefore detected more sensitively by SHG, and vice versa at high coverage. However, SDR is superior rather than complementary to SHG as a tool of real-time measurement, because SDR reveals the adsorption process in the full exposure range and provides information about not only the adsorption on adatom dangling bonds, but also the breaking of adatom back-bonds. In the SHG measurement, the fundamental wave (1,064 nm, 8 ns) of a Qswitched Nd:YAG laser was used as a pumping laser. The duration of the light pulse of the fundamental wave was 8 ns. The laser radiation polarized along the [211] direction with a half-wave plate was incident on the surface of the specimen with an incident angle of about 20◦ . The reﬂected second harmonic (SH) signal was passed through another polarizer, puriﬁed with a bandpass ﬁlter (F3) and a monochromator, and detected by a photomultiplier and gate integrated with a digitizing oscilloscope. Part of the incident light was directed to a quartz plate which produced strong SH signal used as a reference signal. The SH intensity was numerically obtained from the reﬂected signal divided by the reference signal.

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Coverage Evaluation by Thermal Desorption Spectroscopy The total halogen coverage was determined by means of TDS. TDS spectra were measured with a quadrupole mass spectrometer located in front of the specimen. The target mass was 63 and 107 for SiCl+ and SiBr+ ions, respectively. These species are considered to be cracked from SiCl2 and SiBr2 by the electron ionizer in the mass spectrometer. It was found that about 10% of the X atoms are desorbed around 600–700 K and about 90% are desorbed around 1,000 K when the coverage is above 60% of the saturation [46,48]. The former component arose from polyhalides, the amount of which could not be evaluated quantitatively. Accordingly, the specimen was annealed at 743 K for 2 min (Cl) or at 673 K for 3 min (Br) before the TDS measurement to eliminate polyhalides. After the annealing, all polyhalides including Si adatoms are desorbed [48] and the adatom layer disappears [43, 47]. Instead, the Xterminated rest-surface appears together with clusters including halogenated adatoms [43, 47], and this surface partially reconstructs a X-terminated 1×1 bulk-like surface [38]. In the measurements of isothermal desorption (SDR, SHG), desorption from this “1×1” rest-surface was observed. TDS was measured also from this surface with typical heating rate of 10 K s−1 . The total coverage was calculated from the area of the SiX+ . TDS spectrum between 850 and 1,200 K (Cl) or 770 and 1,170 K (Br), as the additional 10%, is taken into account when the coverage is above 60% of the saturation. The saturation of TDS is normalized to 1.35 ML (= 66/49), where 36 halogen atoms are on the adatoms and 30 halogen atoms are on the rest-atoms. 2.2.3 Adsorption of Halogen Atoms: Sticking Coeﬃcient and Potential Barrier The adsorption processes of chlorine and bromine on Si(111)–7×7 were investigated by temperature dependence of adsorption on adatom dangling bonds and breaking of adatom back-bonds [56, 62]. The process was observed by means of real-time SDR spectroscopy and SHG measurements. The interpretation of SDR spectra is based on the results of the calculation of optical responses [63]. The kinetics of the adsorption process is discussed from the point of view of the direct adsorption of atoms. The analysis yields the sticking probability on an adatom dangling bond and the breaking probability of an adatom back-bond. Temperature dependence of these probabilities reveals the adsorption process and the breaking process with regard to potential energy. The diﬀerence in the reactivity of chlorine and bromine on Si(111) is discussed in terms of the interaction between adsorbates. Surface Diﬀerential Reﬂectivity Spectroscopy Figure 2.3 shows the relative variation of p-polarized SDR spectra for several Cl exposures at 300 K. The vertical axis corresponds to ∆R/R ≡ (Ra − Rc )/Rc ,

2 Nanometer-Scale Structure Formation on Solid Surfaces

27

0.01 1L

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Fig. 2.3. Variation of p-polarized reﬂectance spectra for several Cl exposures at room temperature. Data are taken from [56] 0.02 0.01

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6

Fig. 2.4. Decomposition of the ∆R/R spectrum at 60 L in Fig. 2.3 (gray line) into two component spectra SA (solid line) and SB (dashed line). See [56] for detail

where Ra and Rc are the reﬂectances from the Cl-covered and clean surfaces, respectively. The ∆R/R spectrum at the exposure of 1 Langmuir (L) (1 L = 1.33 × 10−4 Pa s) has only a negative peak A located at 1.8 eV. At 8 L, a shoulder B at 2.4 eV becomes apparent as well as small structures above 3.3 eV. The peak A is almost saturated at 8 L, whereas B is saturated above 60 L. Each spectrum in Fig. 2.3 is reproduced by a linear combination (aA SA + bA SB ) of two component spectra SA and SB representing the structures A and B, respectively. For example, as shown in Fig. 2.4, the experimental plot (gray line) at 60 L is well reproduced by the sum of SA (solid line) and SB (dashed line). The magnitudes of SA and SB are determined so as to reproduce the spectra at 60 L with aA = 1 and aB = 1.

28

M. Tanaka et al. adatom

(a) Clean(Si21H27)

(b) Cl - adsorbed (Si21H27Cl)

H

Si

Cl

Fig. 2.5. Geometric structure of the Si(111) clusters: (a) clean surface cluster (Si21 H27 ), (b) Cl-adsorbed surface cluster (Si21 H27 Cl), and (c) dichloride surface cluster (Si21 H27 Cl2 ) (d) Calc. (H) 1.4eV

2.1eV

2.8eV

∆R/R

(c) Exp. (H)

2.8eV

(b) Calc. (Cl) 2.2eV 1.55eV

(a) Exp. SA (Cl)

1

2 3 4 Photon Energy (eV)

Fig. 2.6. SA component of the SDR spectrum for the monochloride Si(111) (a) and the spectrum obtained by calculation (b). The experimental (c) and calculated (d) spectra of monohydride Si(111) [54] are plotted for comparison

The optical responses of clean and Cl-adsorbed Si(111) surfaces have been calculated using time-dependent density functional theory (TD-DFT). The electronic states of three model clusters were calculated after optimizing the geometry. The geometric structures of the clusters are shown in Fig. 2.5. They represent the clean surface, and the surfaces with monochloride and dichloride. The SDR spectrum was obtained from the reﬂectance derived from the transition probability of each model with the aid of an extrapolation to the photon-energy range dominated by the bulk transition. Figure 2.6 compares SA component of Fig. 2.4 in (a) with that of the calculated spectrum in (b). The spectrum in (b) is obtained by subtracting the reﬂectivity for Fig. 2.5a

2 Nanometer-Scale Structure Formation on Solid Surfaces

29

(d) Calc. (H)

∆R/R

(c) Exp. (H)

(b) Calc. (Cl)

(a) Exp. SB (Cl) 1

2 3 Phoeon Energy (eV)

4

Fig. 2.7. SB component of the SDR spectrum for the dichloride Si(111) (a) and the spectrum obtained by calculation (b). Experimental (c) and calculated (d) spectra for the dihydride [55] are also shown for comparison 1.2

aA , bA

1.0 aA

0.8 0.6

Br Cl

bA

0.4 0.2 0.0

0

1

2

3

4

5

6

7

8

9

10

11

Ni

Fig. 2.8. Development of the coeﬃcients aA and bA vs. Ni at room temperature. Solid symbols are for Br adsorption, whereas open symbols are for Cl adsorption. Solid and dashed lines are the best-ﬁt curves for direct adsorption of atoms

from that for Fig. 2.5b. The results for hydrogen adsorption are also plotted in (c) and (d) [54]. The calculated feature at 1.55 eV is assigned to the transition from adatom back-bond states to antibonding adatom dangling bond states. The same comparison for the SB component is shown in Fig. 2.7. The spectrum in (b) is obtained by subtracting the reﬂectivity for Fig. 2.5b from that for Fig. 2.5c. All the spectra in Fig. 2.7 contain a negative peak at around 2.6 eV or about 3.0 eV. This peak can be assigned to the loss of the transition at adatoms owing to the missing of adatom back-bonds. Above assignments allow us to commonly interpret that SA and SB represent the adsorption on the adatom dangling bonds and the breaking of adatom back-bonds, respectively. The coeﬃcient aA and bA are plotted against the accumulated number of atoms impinging on the area of the 1×1 unit, Ni , in Fig. 2.8. Open and closed symbols represent the coeﬃcients for Cl adsorption and Br adsorption,

30

M. Tanaka et al.

respectively. Figure 2.8 reveals the development of adsorption on the adatom dangling bonds and the development of breaking of the adatom back-bonds. It is known that the halogen gas produced by the electrochemical cell involves more atoms than molecules [58]. In the case of direct adsorption of halogen atoms without migration, the rate equation is as follows: dnA = αni (1 − nA ). dt

(2.1)

Here, nA is the normalized density of adatoms bonded to at least one halogen atom, α is the sticking probability of halogen atoms on the adatom dangling bond, and ni is the number of halogen atoms impinging on the area of the 1×1 unit cell per second. Solutions of (2.1) ﬁt to aA are shown by dashed and solid lines in Fig. 2.8 for Cl and Br, respectively. On the other hand, the adatom back-bond is assumed to be broken only when the adatom dangling bond is terminated by at least one halogen atom. In the case of direct adsorption of atoms without migration, the rate equation is as follows dnB = βni (NB nA − nB ), dt

(2.2)

where β is the breaking probability of adatom back-bond by halogen atoms, nB is the normalized density of broken adatom back-bond, and NB is the saturated number of nB . NB should be smaller than 2 because adatoms are not removed by the halogen exposure. Solutions of (2.2) ﬁt to bA are shown in Fig. 2.8. Thus obtained sticking probability on the adatom is αSDR = 0.7 and 1.20 ± 0.2 for Cl and Br adsorption, respectively, and the breaking probability is βSDR = 0.2 and 0.49 ± 0.04 for Cl and Br adsorption, respectively. These results at room temperature are summarized in Table 2.1. The sticking probability αSDR for Br adsorption is 1.7 times as large as αSDR for Cl adsorption. Larger αSDR for Br adsorption can be at least partially explained by the larger atomic radius of the Br atom (0.115 nm for Br and 0.100 nm for Cl) [64], because the cross section of the collision to the adatom dangling bond is larger for larger impinging atoms. However, the ratio of the atomic radius, Table 2.1. Summary of SDR results for the adsorption process of halogens on Si(111)–7×7

Sticking probability on the adatom dangling bond (feature A) Breaking probability of the adatom back-bond (feature B) Activation energies Adsorption on the adatom dangling bond Breaking of the adatom back-bond

Cl

Br

0.7

1.2 ± 0.2

0.2

0.49 ± 0.04

−8 ± 14 meV −38 ± 7 meV

−2 ± 16 meV −39 ± 19 meV

2 Nanometer-Scale Structure Formation on Solid Surfaces

31

1 SDR

SHG

SDR

0.1 2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

1000/T (1/K)

Fig. 2.9. Logarithmic plots of the sticking probability αSDR (solid square), αSHG (open square) and the breaking probability βSDR (solid circle) for Cl adsorption. Dashed line, dash-dotted line, and solid line are the best-ﬁt straight lines for αSDR , αSHG , and βSDR , respectively

1.15, is smaller than the ratio of αSDR , 1.7, so that this explanation seems insuﬃcient. The breaking probability βSDR for Br adsorption is 2.5 times as large as βSDR for Cl adsorption. This means that the back-bond is broken more easily by bromine than by chlorine. In other words, polybromide species are produced more easily than polychloride species. More detailed information will be obtained in the following section by means of extended SDR studies combined with TDS. The ∆R/R spectra for Cl adsorption were measured between 310 and 423 K, and αSDR and βSDR are determined at each temperature. Figure 2.9 shows the logarithmic plots of and against 1/T . Nominal activation energies determined from the slopes of the straight lines are −8 ± 14 meV for the adsorption on adatom dangling bond and −38 ± 7 meV for the breaking adatom back-bond as shown in Table 2.1. SDR results for Br adsorption are quite similar to those for Cl adsorption. Activation energies for the adsorption on adatom dangling bond and for the breaking adatom back-bond are determined as −2 ± 16 and −39 ± 19 meV, respectively [65]. These activation energies agree well with those for Cl adsorption, so that the process of Br adsorption is essentially the same as that of Cl adsorption. However, this is not true as shown in the following section. Second Harmonic Generation The nonlinear susceptibility χ(2) decreases linearly with the coverage at low coverage when the fundamental wave of a Nd:YAG laser is used as a pump laser. In the case of H adsorption on Si(111), the linearity holds up to 0.15 ML (1 monolayer (ML) is deﬁned as 49 atoms per unit cell, namely 1 ML = 7.81 atoms cm−2 ), and χ(2) becomes nonlinear above 0.15 ML even if adatom back-bonds are not broken [61]. The relationship between the SH

32

M. Tanaka et al.

IS H G (arb.unit)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Coverage (ML)

Fig. 2.10. Dependence of SH intensity on Cl coverage determined from TDS. Solid line is determined so as to ﬁt a linear combination of two exponential functions to the experimental plot

Coverage (ML)

0.2

0.1

0.0

0

1

2

3

Exposure (L)

Fig. 2.11. Development of the Cl coverage determined from SH intensity at room temperature. Solid line is the best-ﬁt curve for direct adsorption of atoms

intensity and the coverage can therefore be determined with the aid of TDS before the results of the SHG measurement are analyzed. Figure 2.10 shows the coverage dependence of the SH intensity. There are at least two coverage regimes: linear and nonlinear. A linear combination of two exponential functions is therefore used for approximation. A solid line shown in Fig. 2.10 is determined to ﬁt the function to the plot of χ(2) data vs. coverage. The SH intensity is then transformed to the coverage by using this curve. The development of coverage against chlorine exposure at 300 K is shown in Fig. 2.11. The solution of (2.1) is ﬁt to the plot in Fig. 2.11, which yields the sticking probability αSHG = 0.51. The SH intensities were measured between 310 and 423 K. From a logarithmic plot of αSHG against 1/T , nominal activation energy was determined as −9 ± 8 meV. This is nearly the same as −8 ± 14 meV obtained from SDR. Adsorption on Adatom Dangling Bonds and Breaking of Adatom Back-Bonds The activation energy for the halogen adsorption on the adatom dangling bond is found to be almost zero. This indicates that the adsorption is not thermally activated, and there is no potential barrier in the adsorption. This is reasonable because adsorption of atoms on dangling bonds does not require the breaking of any chemical bond. The above experiments present direct evidence

2 Nanometer-Scale Structure Formation on Solid Surfaces

33

V(z)

~44meV

z E*

Ea

ka

k*

kd

Fig. 2.12. Potential energy for the breaking of adatom back-bonds

for this expectation. On the other hand, the activation energy for the breaking of adatom back-bonds is determined as about −40 meV from the SDR. Negative activation energy indicates that the breaking of adatom back-bonds is not thermally activated, and that there is no potential barrier for the bondbreaking. In the analysis related to (2.2), the adatom back-bond has been assumed to be broken only when the adatom dangling bond is terminated by at least one halogen atom. This assumption means that the potential barrier for the breaking of adatom back-bonds is high compared with thermal energy when the adatom dangling bond is present, and that the barrier becomes much lower when the adatom dangling bond is removed due to the halogen adsorption. The experimental result indicating no potential barrier is compatible with this assumption. Moreover, negative activation energy reveals that the potential curve has a small hump, and that a metastable state exists at the position where the bond length is slightly larger than that of the chemisorption state. The potential energy for the atom incident on the adatom terminated by a halogen atom is schematically shown in Fig. 2.12 [56]. The horizontal axis corresponds to the distance from the surface. The breaking probability of back-bonds via the metastable state is determined from three rate constants: the adsorption rate from the metastable state to the chemisorption state (ka ), the restoring rate from the chemisorption state to the metastable state (kd ), and the desorption rate from the metastable state (k ∗ ) [66]. At the initial stage of the breaking, the restoration from the chemisorption state can be ignored. The initial breaking probability is therefore evaluated without kd , namely β = ka /(k ∗ + ka ). Since ka and k ∗ are expressed as ka0 exp(−Ea /kB T ) and k0∗ exp(−E ∗ /kB T ), respectively, using the height of the hump Ea and the depth of the metastable state E ∗ , Ea − E ∗ is obtained from the logarithmic plot of βSDR /(1 − βSDR ). The nominal activation energy for Cl adsorption, Ea − E ∗ , was determined to be −4 ± 48 meV, and this corresponds to the energy at the top of the hump.

34

M. Tanaka et al.

A theoretical study predicted the adsorption energy for Cl on Si(111), deﬁned as the diﬀerence between the total energy at the adsorption site and the total energy at a distance far from the surface [67]. A ﬁrst-principle calculation for the adsorption process and the breaking of the back-bond process indicates that both processes of adsorption on the dangling bond and breaking of the back-bond are barrierless [68]. The calculation also shows the possibility of a metastable state when a Cl atom is incident on the surface along the surface normal. It is reasonable to conclude that a small hump originating from the energy consumed to break back-bonds can appear in the potential curve for the Si–X bond breaking process. 2.2.4 Site-Selective Adsorption The adsorption process, especially the adsorption site preference, is focused in this section, which may be available for atomically controlled surface modiﬁcation, site-selective etching, and so on. On Si(001), it has been already reported that halogen atoms have site preference in the adsorption process. For example, a patterning of larger halogen (Br or I) adsorbates was found in the form of stable c(4×2) structure at 0.5 ML [69, 70]. This phase involves adsorption on nonneighboring dimers under certain conditions at elevated temperature. At high coverage in Br adsorption, a (3×2) structure in which Si dimer rows alternate with atom vacancy lines is favored as a result of desorption of volatile SiBr2 [71, 72]. The roughening – under which dimer vacancies, dimer vacancy lines, pits, and Si regrowth are observed – occurs at temperatures below the threshold for SiX2 (X = Cl, Br) desorption [73–75]. Si epitaxial growth on Br–Si(001) produces ordered Si overlayer chain [76]. The results of these STM studies were interpreted in terms of repulsive interaction both experimentally and theoretically [77, 78]. However, this simple picture is not enough because the inﬂuence of adsorption on the properties of the underlying substrates should be taken into account. Patch formation on Cl–Si(001) was then explained by an attractive interaction between anticorrelated bare dimers on Si(001) [79]. However, interaction between adsorbates has not been well studied on Si(111). The STM study on halogen molecule adsorption at room temperature [80] showed that a Cl2 molecule with 0.05 eV translational energy tends to be adsorbed on center adatoms of the DAS structure to form a single chloride or a pair of chlorides. The neighboring pair of adsorbates seemingly suggests an attractive interaction between adsorbates. On the other hand, signiﬁcant I–I interaction was seen at high coverage as the binding energy decreases in X-ray photoemission spectra [81]. Furthermore, in “Adsorption on Adatom Dangling Bonds and Breaking of Adatom Back-Bonds” section, we compared adsorption processes of Br atoms with that of Cl on Si(111), and found that the Br process yields a higher sticking probability on adatom dangling bonds and a higher breaking probability of adatom back-bonds. These results suggest

2 Nanometer-Scale Structure Formation on Solid Surfaces

35

a repulsive interaction between Br adsorbates on Si(111). Thus, interactions of opposite directions were reported so far. The underlying interaction in the adsorption on Si(111) may be diﬀerent from that on Si(001) because of diﬀerent surface structure. The distance between center adatoms on Si(111) is 0.69 nm and much longer than 0.38 nm of the distance between dimers on Si(001). The latter is rather close to the distance between the adatom and the bare rest-atom on Si(111). The rest-atoms having dangling bonds are also reactive and the reaction of 6 rest-atoms in the unit cell cannot be negligible compared with that of 12 adatoms. However, Jensen et al. [80] assumed that the adatoms are the exclusive adsorption sites, and they proposed dissociative adsorption on the adatom–rest-atom pair contrarily in their previous paper [82]. The adsorption on the rest-atoms is therefore crucial to discuss interaction between adsorbates on Si(111). Restatom dangling bonds on Si(111) can hardly be accessed by an STM. The reaction of the rest-atoms is often neglected in studies on surface reactions, and the reactivity of rest-atoms has not yet been clariﬁed. The question to be addressed is whether or not there is site preference as regards adsorption on the adatom and the rest-atom of Si(111) and, if there is a preference, whether or not the site preference in Br adsorption is diﬀerent from that in Cl adsorption. Chemical trend of the site preference is expected because diﬀerent halogens will react with a semiconductor surface in diﬀerent ways because of diﬀerent ionic radii and diﬀerent electron aﬃnities; for example, the sticking probabilities, the desorption rates, and their temperature dependences will be diﬀerent. Understanding the chemical trend of halogen reactivity is crucial to optimize etching conditions. In this section, studies on the adsorption site preference on Si(111) [83, 84] by means of SDR spectroscopy and TDS [56] are introduced. Densities of Saturated Dangling Bonds and Broken Back-Bonds The coeﬃcients aA and bA in Fig. 2.8 are proportional to the densities of saturated dangling bonds and broken back-bonds, respectively. There are 12 adatom dangling bonds in the 7×7 unit cell, so that aA is normalized to 0.24 ML (= 12/49), whereas bA is normalized to 0.49 ML (= 24/49) because two of three adatom back-bonds for each Si adatom are breakable. At each exposure, the SDR spectrum was ﬁrst measured, and coeﬃcients aA and bA were determined. The TDS spectrum was then measured so as to determine the total coverage θ. The densities of saturated dangling bonds and broken back-bonds for Br and Cl adsorption are plotted against the total coverage in Fig. 2.13a, b, respectively. Open squares represent the density of saturated dangling bonds, whereas solid squares represent the density of broken back-bonds. Apparently, Br and Cl adsorption follows diﬀerent lines, which reveal a chemical trend in the adsorption processes. It has already been found that both the sticking probability on adatom dangling bonds and the breaking probability

36

M. Tanaka et al. 0.7 0.6

(a) Br

Density (ML)

0.5 0.4 0.3 0.2 0.1 saturated dangling bond broken back bond

0.0 0.7 0.6

(b) Cl

Density (ML)

0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Total Coverage (ML)

Fig. 2.13. Density of saturated dangling bonds aA (open squares) and that of broken back-bonds bA (solid squares) determined from SDR vs. total coverage determined from TDS. Part (a) is for Br and (b) is for Cl. Gray lines are shown merely as visual guides

of adatom back-bonds for Br adsorption are higher than those for Cl adsorption [62]. With the aid of TDS, the development of the adsorption process can be seen more quantitatively in relation to the total coverage in Fig. 2.13. This ﬁgure shows how many Br atoms are adsorbed on adatom dangling bonds and how many Br atoms break adatom back-bonds, when a speciﬁed number of Br atoms is adsorbed. Partial Coverages on the Adatoms and the Rest-Atoms A new dangling bond appears at the rest-atom upon breaking of the adatom back-bond as shown in Fig. 2.1b. Hereafter, we call it “emerging dangling bonds at the rest-atoms” which should be distinguished from “native dangling bonds at the rest-atoms” on the clean surface. A new dangling bond also appears at the adatom, however, it adsorbs the halogen atom immediately, because there is no evidence for asymmetric polyhalide in the STM observations [38]. Consequently, the partial coverage on the adatoms is evaluated as aA + bA . The partial coverage on the rest-atoms is then calculated as the diﬀerence between the total coverage and the partial coverage on the adatoms, θ − (aA + bA ). If all the adatoms form trihalides at saturation, the partial coverages on the adatoms and the rest-atoms can be as large as 12×3/49 = 0.73 ML and (6+12×2)/49 = 0.6 ML, respectively. Thus obtained

2 Nanometer-Scale Structure Formation on Solid Surfaces

37

Partial Coverage (ML)

1.0 (a) Br 0.8 0.6 0.4 0.2

adatom rest atom

0.0 1.0 Partial Coverage (ML)

(b) Cl 0.8 0.6 0.4 0.2 0.0 0.0 0 .2

0.4 0 .6

0.8 1 .0

1.2

1.4

1.6

Total Coverage (ML)

Fig. 2.14. The partial coverage on the adatoms aA + bA (open squares) and the partial coverage on the rest-atoms θ − (aA + bA ) (solid squares) vs. total coverage. Part (a) is for Br and (b) is for Cl

partial coverage on the rest-atoms cannot be estimated with STM, and the present SDR–TDS method is the only available means to evaluate it. The partial coverages of Br are plotted against the total coverage in Fig. 2.14a. The result of Cl adsorption is shown in Fig. 2.14b. Open squares represent the partial coverage of the adatoms, whereas solid squares represent the partial coverage of the rest-atoms. Error bars correspond to the sum of the errors of densities of saturated dangling bonds and broken back-bonds shown in Fig. 2.13. Adsorption Site Preference The experimental results shown in Figs. 2.13 and 2.14 establish the adsorption site preference in the adsorption process. For Br adsorption, the slope of the partial coverage on the adatoms at the ﬁrst stage is almost 1.0 and that on the rest-atoms is nearly 0, which means that all adsorbed Br atoms sit on the adatoms, and none on the rest-atoms. On the other hand, the slope of the partial coverage on the adatoms for Cl adsorption is almost 2/3 and that on the rest-atoms is nearly 1/3. This means that Cl atoms are adsorbed on both the adatoms and the rest-atoms with equal probability, because there are 12 dangling bonds at the adatoms and 6 native dangling bonds at the rest-atoms in a clean 7×7 unit cell. Cl atoms impinging to the surface will be adsorbed at the site where the collision occurs, no matter whether the target is the adatom or the rest-atom. The Cl adsorption on the rest-atom is thus suggested to be also barrierless because the adsorption on the adatom dangling bond is barrierless. As for the Br adsorption on the rest-atoms, both interaction between halogen adsorbates and interaction between halogen atoms and the

38

M. Tanaka et al.

rest-atoms should be taken into account at high coverage, however, only the latter is eﬀective at low coverage. Since no Br atom sits on the rest-atoms even at very low coverage, there must be a potential barrier for Br atom to be adsorbed on the rest-atoms. In other words, there is repulsive interaction between Br atoms and the Si rest-atoms. At 0.1 ML in Br adsorption, about ﬁve adatoms per unit cell or 40% of the adatoms have adsorbed Br, while 60% of the adatoms have the dangling bonds. Nevertheless, the breaking of back-bonds begins. An electronstimulated desorption (ESD) study [85] reported that the desorption of SiBr+ 2 ions, suggesting polybromide formation, was apparent even at coverage as low as 0.1 ML for Br-covered Si(111), though no ion-containing Si was detected from Cl-covered Si(111) at such low coverage. This result agrees well with ours. There are two possibilities for the breaking of back-bonds at such an early stage. In case I, the SiBr species at the adatoms hinders other SiBr species at the adatoms, and one SiBr2 species is formed with a barrier lower than that to form an adjacent pair of SiBr species. A stronger repulsive interaction between Br adsorbates plays an essential role. In this case, a patterning in which Br atoms are adsorbed on every other adatom is expected. In case II, adsorption to the center adatoms is diﬀerent from that to the corner adatoms as suggested for the adatom with low electron density to be favored [80]. Since the interaction with the adatom of low electron density is eﬀectively attractive, the barrier for the process is expected to be low. If one SiBr2 species at the center adatoms is energetically preferred to the conﬁguration with one SiBr species on the center adatom and the other SiBr species on the corner adatom, the breaking of back-bonds begins after six center adatoms in the 7×7 unit cell (0.12 ML) are adsorbed. Interaction between Br adsorbate and the Si adatom plays an essential role. In this case, a patterning decorated with adsorbates on the center adatoms is expected. In both cases, underlying interactions suggest patternings of adsorbates on the Si(111) surface. On the other hand, the onset of back-bond breaking in Cl adsorption is at 0.3 ML, i.e., about 15 atoms per unit cell. Back-bond breaking begins only after about 80% of dangling bonds at the adatoms and the rest-atoms have adsorbed Cl. We can see little trace of interaction between Cl adsorbates or interaction between Cl adsorbate and the Si adatom. The adsorption behavior above the onset of back-bond breaking is quite diﬀerent from that below the onset. In the range of 0.1 < θ < 0.3 ML in Br adsorption, the rest-atoms remain intact (Fig. 2.14a) and the slopes of the densities of saturated dangling bonds and broken back-bonds are nearly equal (Fig. 2.13a). This means that about a half of impinging Br atoms are adsorbed on the dangling bonds of the adatoms and the other half breaks the adatom back-bonds. In other words, 50% form monobromide and 50% form dibromide at the adatoms, but none is on the rest-atoms. In the range of 0.3 < θ < 0.6 ML in Br adsorption, about 40% of newly adsorbed Br atoms are on the adatoms and about 60% are on the native or emerging dangling bonds at the rest-atoms (Fig. 2.14a). On the other hand,

2 Nanometer-Scale Structure Formation on Solid Surfaces

39

in the range of 0.3 < θ < 0.6 ML in Cl adsorption, almost all the impinging Cl atoms break the adatom back-bonds and adsorbed on the adatoms (Fig. 2.13b). In other words, newly adsorbed Cl atoms preferentially form dichlorides at the expense of the Cl atoms on the rest-atoms, and the emerging dangling bonds on the rest-atoms remain intact. Terminating the emerging dangling bond seems to be more diﬃcult than breaking the other back-bond. The breaking the back-bond is known to be barrierless, whereas di- or trichloride at the adatom may hinder the intrusion of the next chlorine. The intrusion requires the distortion of the adatom chlorides and probably involves a potential barrier. This kind of repulsive interaction between Cl adsorbates was seemingly strong in this range. Therefore, the repulsive interaction is not simply determined by the geometric size of the atom (0.115 nm for Br and 0.100 nm for Cl) [64], but it is determined by the total energy. The stronger repulsive interaction is a result of increase of the total energy due to distortion energy. This study provides the ﬁrst direct evidence for adsorption site preference and suggests a pattern formation on a Si(111) surface. As mentioned in Sect. 2.3.2, STM study showed that, at coverage less than 0.03 ML, adsorbed bromine atoms were rarely isolated, while chlorine atoms showed a greater tendency to be adsorbed separately. Center adatoms had higher reactivity than the corner adatoms. As for the underlying interaction, interaction between Br adsorbate and the Si adatom with low electron density [80] (case II) seems plausible at the coverage range. In the case II, the rest-atoms have higher electron density, and hardly adsorb halogen atoms. Back-bond breaking by Br atoms occurs at lower coverage because back-bonds of Br-adsorbed Si adatom are more weakened than that of Cl-adsorbed Si adatom and SiBr2 can be formed more easily than SiCl2 , as shown in Si(001) [77]. 2.2.5 Desorption of Silicon Halides and Restoration of the DAS Structure The desorption process of silicon chloride and bromide from Cl- and Brterminated Si(111) rest-surfaces, respectively, was investigated in real time by means of SDR spectroscopy for the ﬁrst time and SHG [62, 86, 87]. One should note that the structure of the rest-surface on which the desorption takes place is diﬀerent from the 7×7 DAS structure on which the adsorption takes place. The chlorine and bromide desorption and subsequent restoration of the DAS structure were examined between 873 and 923 K. The time courses of the recoveries of the dangling bonds, the back-bonds, and SH intensity yield the rate constants, the order of reaction, and the activation energies in the desorption process. The kinetics of desorption and restoration processes are evaluated here from the bond density data. The cluster formation on the terrace is suggested from the consideration of the order of reaction. The difference in the reactivity of Cl and Br on Si(111) is discussed in terms of the interaction between adsorbates.

40

M. Tanaka et al.

Surface Diﬀerential Reﬂectivity Spectroscopy The thick lines in Fig. 2.15 show the relative variation of p-polarized reﬂectance spectra during the isothermal desorption at 903 K from the Cl-covered Si(111) surface. The vertical axis is ∆R/R = (Ra − Rc )/Rc , where Ra and Rc are the reﬂectances from the Cl-covered surface and the clean surface at 903 K, respectively. The spectrum at 20 s has a negative peak B at around 2.4 eV, and a negative shoulder A at around 1.7 eV. There are positive peaks at around 3.4 and 4.3 eV. The feature A is depressed faster than B , and almost disappears at 400 s, whereas B remains even at 800 s. These spectral features are quite similar to those observed for Cl adsorption at room temperature, except for the relative magnitude of the feature A [56, 88]. The transitions relevant to the features A and B are related to missing surface states of the clean 7×7 DAS structure. The decay of these features therefore corresponds to the restoration of the structure of the clean surface. Each ∆R/R spectrum in Fig. 2.15 is reproduced by a linear combination (aD SA + bD SB : shown by thick lines) of two component spectra, SA and SB , representing the features A and B , respectively. Thin lines represent the decomposition of each spectrum into two component spectra: aD SA and bD SB . Thus obtained aD and bD are plotted in Fig. 2.16a, b, respectively. They are normalized so that the values ﬁtted by (2.5) described below are 1 at 0 s. The decay of aD represents the recovery of the adatom dangling bond, whereas the decay of bD represents the recovery of the adatom back-bond.

903 K

0.01

20 s

0.00

SA

SB

∆R / R

A

B

100 s

0.00

200 s

0.00

400 s

0.00

800 s

0.00 -0.01 1

2

3

4

5

6

Photon Energy (eV)

Fig. 2.15. Variation of p-polarized reﬂectance spectra during the isothermal chloride desorption at 903 K (thickest lines). Thick lines are the best-ﬁt curves of aD SA + bD SB for the experimental plots, whereas thin lines represent component spectra aD SA and bD SB

2 Nanometer-Scale Structure Formation on Solid Surfaces

41

1.0

aD

0.8 0.6 0.4 873 K 903 K

0.2 933 K 0.0

0

200

400

600

Time (s)

1.0

bD

0.8

873 K

0.6 903 K 0.4 933 K 0.2 0.0

0

200

400

600

Time (s)

Fig. 2.16. Time courses of the coeﬃcients: (a) aD and (b) bD at 873 K (solid squares), 903 K (solid triangles), and 933 K (solid circles). Solid lines are best-ﬁt curves for ﬁrst-order kinetics

The decay rate Rd is expressed in terms of the rate constant κ(1) for a ﬁrst-order process as Rd = −

dx(t) = κ(1) x(t), dt

(2.3)

where x stands for aD and bD . In the case of a thermally activated process, the temperature dependence of κ(1) can be expressed in terms of an activation energy Ed Ed (1) (1) , (2.4) κ = κ0 exp − kB T where kB and T are the Boltzmann’s constant and the temperature, respectively. The solution of (2.3) is written as x(t) = x0 e−κ

(1)

t

,

(2.5)

where x0 is the initial value of x. The solid lines in Fig. 2.16 are best-ﬁt curves obtained by using (2.5). These curves reproduce well the overall features of the decay of aD and bD . The ﬁt with higher order was worse than the ﬁt with ﬁrst order, indicating that these processes are dominated by ﬁrst-order kinetics. The decay rates κa and κb were obtained from these ﬁts at each temperature.

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Fig. 2.17. Logarithmic plots of the decay rates κa (solid circles) and κb (open circles). Solid lines are best linear ﬁts to the plots Table 2.2. Summary of SDR results for the desorption process of halogens on Si(111)–7 × 7

(a)

(b)

Cl

Br

Recovery of the dangling bonds (feature A ) Order Rate constant at 903 K (s−1 ) Activation energy for the desorption (eV)

First 0.008 2.3 ± 0.3

First 0.01 1.8 ± 0.4

Recovery of the back-bonds (feature B ) Order Rate constant at 903 K (s−1 ) Activation energy for the reconstruction (eV)

First 0.004 2.8 ± 0.5

First 0.003 3.5 ± 0.6

κa corresponds to the recovery rate of the dangling bond, whereas κb corresponds to the recovery rate of the back-bond. The temperature dependences of κa and κb are shown in Fig. 2.17 with solid and open circles, respectively. The solid lines are the best linear ﬁts to the plots. The activation energies in the recovery processes of the dangling bond and the back-bond were determined from the slopes of the solid lines as 2.3 ± 0.3 and 2.8 ± 0.5 eV, respectively. General feature of the desorption process on Br-covered Si(111) surface is almost the same as that of chloride [62]. The activation energies in the recovery process of the dangling bonds and the back-bonds for the Br-saturated surface are determined as 1.8 ± 0.4 and 3.5 ± 0.6 eV, respectively. These rate constants and activation energies are summarized in Table 2.2. Second Harmonic Generation Figure 2.18 shows the time courses of the SH intensities of the Cl-terminated Si(111)–“1×1” surface in the isothermal desorption process. The horizontal axis corresponds the time (t) after the indicated temperature (T ) had

2 Nanometer-Scale Structure Formation on Solid Surfaces

43

been achieved. On the vertical axis, the intensity, I(T ), is referred to the SH intensity of the clean 7×7 DAS surface, I0 (T ) at the respective temperature. Note that the intensity I0 (T ) decreases as the sample temperature increases, because the resonant wavelength of SHG shifts owing to lattice expansion and electron–phonon interaction [89]. The initial increase of I(T )/I0 (T ) within around 80 s at 943 K or 500 s at 873 K is named “fast” recovery. The subsequent very gradual increase in intensity will be referred to as “slow” recovery hereafter. If the slow recovery is ignored, Fig. 2.18 can be taken as showing only fast recovery to a speciﬁed SH intensity, I0 (T ) = γI0 (T ), where γ = 0.82 at 943 K, for example. The Cl coverage was evaluated under the condition that SH intensity saturates at I0 (T ) when the chlorides are desorbed out. Figure 2.19 shows that a decay of the coverage is quite similar to isothermal

0.8 0.6

SHG Intensity I(T)/I0 (T)

0.4

873K

0.2 0.0 0.8 0.6 903K

0.4 0.2 0.0 0.8 0.6

943K

0.4 0.2 0.0

0

200

400

600

800

Time (s)

Fig. 2.18. Time courses of SH intensity in isothermal chloride desorption. A fast recovery in the intensity is followed by a very slow recovery. The gray curves are the best-ﬁt product of two exponential decays 0.20

Coverage (ML)

(a) 966K 0.15

(b) 913K (c) 893K

0.10

(d) 873K

0.05 (d) 0.00

(a) 0

(c)

(b) 100

200

300 400 Time (s)

500

600

Fig. 2.19. Time courses of the Cl coverage during the isothermal desorption at four temperatures. The coverages were obtained from Fig. 2.18 using the relation in Fig. 2.10. Each curve can be well ﬁt with a single exponential curve (gray curve)

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desorption monitored by SDR. We take the desorption process to be ﬁrst order, and ﬁt the time course in Fig. 2.19 with an exponential function exp(−κ1 t),

(2.6)

where κ1 is the decay rate of the Cl coverage. Form an Arrhenius plot of κ1 , the barrier height for the chloride desorption is Ed = 2.1 eV. After chlorides have been substantially desorbed, the surface is reconstructed to the 7×7 DAS structure. The slow component therefore corresponds to the surface reconstruction. In fact, with the laser wavelength we employed, SHG is capable of monitoring the reconstruction back to 7×7 [90]. Accordingly, each curve in Fig. 2.18 is ﬁtted by the product of two exponential functions (gray curves) {1 − exp(−κ1 t)}{1 − exp(−κ2 t)},

(2.7)

where κ1 and κ2 are the SH recovery rates by the desorption (fast component) and the reconstruction (slow component), respectively. It is assumed that the reconstruction produces an exponential recovery in SH intensity. The barrier height for the reconstruction step can be estimated from the Arrhenius plot for κ2 , and we estimate the value to be 2.4 eV. Recovery of the Dangling Bond The results of the SDR and SHG experiments for the chloride desorption are summarized in Table 2.3 together with the results of other isothermal desorption studies on Cl-saturated Si(111). The order of the process, the rate constant at 903 K, and the activation energy are listed. For the recovery of Table 2.3. The order of the process, the rate constant, and the activation energy for the isothermal desorption determined by several methods Method

Order

Rate constant at 903 K (s−1 )

Activation energy (eV)

(a) Recovery of the dangling bond SDR (feature A) UPS (Cl 3p) [91] AES (Cl LMM) [92] SHG (fast)

First First – First

8 × 10−3 7 × 10−3 7 × 10−3 2 × 10−2

2.3 ± 0.3 2.2 – 2.1

(b) Recovery of the back-bond SDR (feature B) SHG (slow)

First First

4 × 10−3 2 × 10−4

2.8 ± 0.5 2.4

2 Nanometer-Scale Structure Formation on Solid Surfaces

45

the dangling bond, the rate constants determined from SDR (feature A ) and SHG (fast component) are compared with those from ultraviolet photoelectron spectroscopy (UPS) [91] and Auger electron spectroscopy (AES) [92]. UPS and AES detect the density of the chlorine atom remaining on the surface, whereas SDR (feature A ) and SHG detect the density of the dangling bond. The result of SDR (feature A ) agrees well with the results of other methods. This agreement means that the same microscopic process is detected by the three diﬀerent techniques. UPS and AES directly respond to the desorption of chlorides, which produce the dangling bond on the rest-surface. If the dangling bond on the rest-surface is also reﬂected in the SDR spectrum, the feature A can decay with the same rate as those of UPS and AES before the adatom is formed. Actually, the peak energy diﬀerence between the dangling bond states of the relaxed 1×1 surface [93] and the 7×7 DAS structure [55] estimated to be less than 0.4 eV is smaller than the band width of the feature A of 1.2 eV [49]. Consequently, the appearance of the dangling bond on the rest-surface yields nearly the same decay of A as the appearance of the adatom dangling bond does. Meanwhile, the recovery rate of the dangling bond determined from SHG is two or three times larger than that from other methods. The SHG intensity is dominated by not only the dangling bond density, but also the resonance factor. The resonance width for the SHG pumped at around 1,064 nm is about 0.3 eV [52]. SHG can therefore detect the diﬀerence between the dangling bond state of the rest-surface and that of the DAS structure, or even the diﬀerence between 5 × 5 DAS and 7×7 DAS structures. The rate constant estimated from the SHG intensity is not necessarily the same as the recovery rate of the dangling bond density. TDS studies showed that the main desorption species from the Cl-saturated Si(111) surface above 873 K is SiCl2 [39]. If randomly distributed monochloride species diﬀuse freely on the surface and two of them recombine to form the volatile SiCl2 species, the process obeys second-order kinetics. Moreover, 4.23 eV is required to extract the Si atom on the rest-surface [68]. This disagrees with the experimentally obtained activation energy, 2.3 eV. Consequently, the desorption of SiCl2 from an ideally Cl-covered rest-surface is unlikely, and defects such as steps and craters should be taken into account. Desorption at the step has already been proposed in the STM study on the bromine etching of a Si(111) surface [94]. The edge of these defects has 1D nature, and a monochloride species bound on the step inevitably recombines with other monochloride species irrespective of the Cl density on the step. The desorption rate is therefore proportional to the density on the step, which means that it is a ﬁrst-order process. The corresponding activation energy represents the energy required to form a volatile SiCl2 molecule at defects such as steps, craters, and pits, because the energy barrier for the Cl diﬀusion is estimated to be low, 0.9 eV [95]. The value of 2.3 eV is close to the calculated etching energy, 2.4 eV, which is energy diﬀerence between a surface species and a single molecule in the vacuum [67].

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SDR, SHG, and UPS detect the surface density of the dangling bond and the Cl atom, and they revealed ﬁrst-order kinetics for the isothermal desorption. On the other hand, the methods that detect desorbed species, such as LITD [96], TDS [46], and monitoring the signal of the mass spectrometer in steady-state spontaneous etching [97], revealed second-order kinetics. The LITD study on isothermal desorption is considered ﬁrst. The characteristic of second-order kinetics in isothermal desorption compared with ﬁrst-order kinetics is long-lived desorption. A signiﬁcant amounts of clusters that come from adatoms are sustained on the rest-surface after annealing at about 700 K of a halogen-saturated Si(111) surface, as observed by STM [38,47]. We therefore speculate that the desorption from the SiClx cluster lasts longer than that from the step. SDR, SHG, and UPS are highly sensitive to the surface and cannot detect halogen atoms in clusters. The signal of these methods therefore disappears when Cl atoms on the terrace are almost exhausted, although the desorption from the clusters that can be detected by the LITD study continues. Consequently, LITD can see a second-order kinetics, even when SDR, SHG, and UPS see ﬁrst-order kinetics. The results of TDS and the steady-state spontaneous etching are also interpreted by the model that the desorption from the cluster with larger activation energy lasts longer than that from the defect [62]. Thus proposed desorption model is schematically illustrated in Fig. 2.20. Chemical trend of the recovery of the dangling bond is shown in Table 2.2. The activation energy for bromide desorption, 1.8 eV, is smaller than that for the chloride desorption, 2.3 eV. The local structure of the dimer at the step edge on Si(111) is similar to that of the dimer on Si(001) [43]. According to ab initio electronic structure calculation on Si(001) [77], the desorption energy for SiBr2 (2.31 eV) is smaller than that for SiCl2 (3.11 eV). The smaller desorption energy of SiBr2 was assigned mainly to the larger strain due to the repulsive interaction between monobromide species and not to a decrease in the bond charge. On the other hand, TDS studies of bromide and chloride desorption from Si(001) showed that the activation energy for SiBr2 is 1.9 eV and that for SiCl2 is 2.4 eV [98,99]. The activation energies on Si(001) are quite similar to our results on Si(111), so that the smaller activation energy for SiBr2 on

volatile SiX2

(X = Cl, Br)

crater cluster step

Fig. 2.20. Proposed desorption mechanism. There are two types of desorption: desorption from defects on the rest-surface, such as steps and craters, and desorption from clusters. See [62] for detail

2 Nanometer-Scale Structure Formation on Solid Surfaces

47

Si(111) can also be assigned to the larger strain due to the repulsive interaction between Br adsorbates. Recovery of the Adatom Back-Bond The rate constant determined from SDR (feature B ) is about 20 times larger than the recovery rate of the slow component of SHG, as shown in Table 2.3. According to the calculated reﬂectance spectra [100], the reﬂectance spectrum for the 3×3 DAS structure exhibits a peak at around 2.5 eV due to the back-bond contribution. On the other hand, the reﬂectance spectrum for the 2×2 adatom–rest-atom structure exhibits only a broad feature in that energy region. The feature B therefore does not disappear upon adatom formation, but disappears when the DAS structure is formed. Various n×n DAS structures have been found during reconstruction from the quenched 1×1 phase and the 7×7 DAS structure is formed through the size change process [101–103]. Between 5×5 and 7×7 structures, for instance, the electronic state of Si adatom dangling bonds diﬀers by 0.4 eV [104], which is lower than the resonance width in SDR (1.2 eV) and higher than that in SHG (0.3 eV). Consequently, it is reasonable to consider that the decay of B corresponds to the formation of a DAS structure, such as 5×5, 7×7, and 9×9 structures, whereas the recovery of SH intensity corresponds to the completion of the 7×7 DAS structure. It takes a certain period of time to form the exact 7×7 DAS structure, so that the recovery of SHG intensity is much slower than that of decay of the feature B . Meanwhile, the recovery of the adatom backbond followed ﬁrst-order kinetics. This result suggests that the restoration of DAS structure is an independent event that takes place randomly on the surface, so that the recovery rate of DAS structure is proportional to the area without DAS structure. Actually, random nucleation and uniform growth of domains were observed by STM on extended terraces when the DAS structure is formed during Cl desorption [105]. The activation energy for the recovery of the back-bond corresponds to the energy required for the DAS structure formation, because the energy needed for the diﬀusion of Si clusters is estimated to be lower than 1.5 eV [106, 107]. Although the DAS structure consists of several elements – such as dimers, a corner hole, and stacking faults – these elements are inherently inseparable [101]. Accordingly, the obtained activation energy should be compared with the formation energy for a faulted half-unit cell, which is reported as 2.6 eV [106]. The activation energy for the recovery of the back-bond cannot be distinguished from 2.6 eV because of the large experimental error. On the other hand, the activation energy for the slow recovery in SH intensity is a little lower than 2.6 eV. The activation energies to form various DAS structures are reported to be between 1.7 and 2.3 eV for the observed structures [102]. This seems to correspond to the activation energy for the slow recovery of SH intensity. In the temperature range of our experiments, the surface concurrently undergoes ﬂuctuation in the DAS size and stacking-fault formation [108]. Our

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estimated value, 2.4 eV, may represent the combined eﬀect of DAS size change with about 2.0 eV and stacking-fault formation with 2.6 eV. Chemical trend of the recovery of the adatom back-bond is shown in Table 2.2. The activation energy for recovery of the back-bond after Br etching was found to be 3.5 ± 0.6 eV. This value is meaningfully larger than that for the formation energy for a faulted half-unit cell, 2.6 eV. Additional mechanisms giving higher activation energy should be taken into account for the surface reconstruction after bromide desorption. Although the nature of such a mechanism cannot be determined from the SDR results, surface morphology could aﬀect the process of DAS structure formation, as pointed out in [109]. 2.2.6 Summary As the ﬁrst step of Cl and Br etching of the Si(111) surface, the adsorption process has been investigated by means of in situ real-time SDR and SHG spectroscopy. The developments of the adsorption on the dangling bond and the breaking of back-bonds yield the sticking probability and the breaking probability. From the temperature dependence of these probabilities, the activation energy for the adsorption on dangling bonds is found to be almost zero and that for the breaking of back-bonds is approximately determined as −40 meV. Cl coverage on the adatom dangling bond was also evaluated from SH intensity with the aid of TDS. The sticking probability and its temperature dependence are almost the same as those determined from SDR. The activation energies for the adatom dangling bond and the breaking of the adatom back-bond reveal that both processes are barrierless. Moreover, it also reveals that there is a metastable state in the breaking process of the adatom back-bond, and the energy of the top of the hump in the potential energy is evaluated as about −45 meV. This hump may originate from the energy consumed to break the back-bond. The mechanism of the Br adsorption process was found to be qualitatively the same as that of the Cl adsorption, but quantitatively diﬀerent. Both the sticking probability on dangling bonds and the breaking probability of back-bonds for Br adsorption are larger than those for Cl adsorption. This chemical trend is presumed to arise from larger strain due to repulsive interaction between Br adsorbates. The site preference of halogen atoms has been quantitatively studied by means of SDR and TDS. Partial coverages on the adatoms and the rest-atoms, which cannot be estimated by other techniques, even STM, reveal the adsorption site preference of bromine atoms. At the initial stage below 0.1 ML, Br atoms are adsorbed selectively on dangling bonds of the Si adatoms, but not on those at the rest-atoms, and, at the later stage, dibromide species are formed on adatoms before monobromides reach 40% of the adatoms. On the other hand, Cl atoms are adsorbed randomly on the dangling bonds at both the adatoms and the rest-atoms. This chemical trend is well interpreted in terms of repulsive interaction between halogen adsorbates or the interaction between halogen adsorbate and the Si adatom. In the range of 0.3 < θ < 0.6 ML,

2 Nanometer-Scale Structure Formation on Solid Surfaces

49

newly adsorbed Cl atoms preferentially form dichlorides at the expense of the Cl atoms on the rest-atoms, and the emerging dangling bonds on the rest-atoms remain intact. The site preference of Cl is explained by a kind of steric hindrance due to polychloride on the adatom which prevents Cl atom from intruding to the rest-atom. As the second step of Cl and Br etching of the Si(111) surface, isothermal desorption of silicon halides from a halogen-terminated Si(111) rest-surface and subsequent restoration of the DAS structure have been investigated by means of in situ real-time SDR and SHG spectroscopy. The rate constant and the order of reaction are determined from the time courses of the dangling bond recovery, the back-bond recovery, and the SH intensity recovery. The activation energies evaluated from the temperature dependence of the rate constants are compared with the results obtained by other methods. The dangling bond contribution to the SDR spectrum recovers on the rest-surface as a result of the desorption of silicon chlorides. The time course of this decay reveals ﬁrst-order kinetics, which suggests the associated desorption of SiCl2 at defects such as steps, craters, and pits. The long-lived desorption from clusters is also suggested to explain the second-order kinetics observed by TDS, etc. The activation energy is ascribed to the energy required to form a volatile SiCl2 molecule at the defects. The activation energy for bromide desorption is smaller than that for chloride desorption, which is interpreted in terms of larger strain due to repulsive interaction between Br adsorbates. On the other hand, the back-bond contribution to the SDR spectrum recovers when the DAS structure is formed. The activation energy is ascribed to the energy required to form a faulted half-unit cell. Time course of the SH intensity involves fast recovery and slow recovery. The fast recovery corresponds to the recovery of the adatom dangling bonds. The slow recovery corresponds to the completion of the 7×7 DAS structure. The activation energy for this process is suggested to represent the combined eﬀect of DAS size change and stacking-fault formation. In this section, atomic-scale mechanisms underlying layer-by-layer etching of Si(111) surface with halogen atoms and their chemical trend have been elucidated. This should be useful information to optimize etching conditions, as well as to improve out understanding of the fundamental processes in the halogen etching of Si surfaces. Our results suggest that Br etching is superior to Cl etching, because the smaller desorption energy means better controllability of the etching process, and the larger interaction between adsorbates may be used for site-selective etching. The site preference and cluster formation suggested in this section could be utilized to achieve site-selective etching and could be applied as a template to immobilize large molecules, such as biomolecules.

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2.3 Nanoscale Fabrication Processes of Silicon Surfaces with Halogens 2.3.1 Introduction In this chapter, with regard to the dynamic process of structure control of silicon surface at the atomic scale, the recent developments of studies are reviewed. The fabrication of silicon surface is of the great importance in semiconductor technology. The processes are divided into two parts: decrease and increase of the surface component. In industrial use, the surface is often etched with halogen plasma at high temperatures, and the growth on it is done by means of deposition of such evaporated gas as silane. The former process is simply recognized as the scrape of semiconductor surface. After the process, what structure will be formed? It will lead to novel technique to make microstructures on the semiconductor surfaces if the surface structure could be controlled only by etching of the wafer. Before the construction of functional devices at nanometer scale, well-deﬁned surface is required at the atomic scale. At the microscopic aspect, we can recognize the etching process of the surface as the elemental physical/chemical phenomena for fabricating the semiconductor surface. In this section, the most typical and well-deﬁned cases are addressed mainly in terms of desorption. The most primitive and fundamental process of etching of silicon is desorption from halogenated surface, because the reactive halogen gasses are most frequently used in industrial purpose to weaken the Si–Si bond near the surface when they are adsorbed on the surfaces. The fabrication with halogens is usually promoted in energetic excitations, such as plasma state of chemical compound gas. These excitations are generally of high eﬃciency, and for the homogeneity the reaction is often held at high temperature during the gas (or plasma) exposure. However, this kind of processes are too complicated to be analyzed, and be empirically optimized in real industry. The diﬀusive process on the surface may destroy the surface structure. This is why this excitation method is not good at structure control of the surfaces at the atomic scale. Thus, to study well-deﬁned process on the surface, we introduce the halogenadsorbed silicon surfaces as the simplest model systems, which are, at the same time, applicable to the industrial purpose. The physical interpretation of the surface phenomena may open the gate to atomic arrangement of the surface structure. It is not a dream to control of the nanometer-scale structure on silicon surface. The most advanced integrated circuits are made of thinner lines than 90 nm in commercial products, and they are realized by means of lithography methods with resist layers. The optical lithography with far-ultraviolet exposure is capable to construct 45 nm lines [110], which contains only countable number of silicon atoms at the order of several hundreds (see Fig. 2.21). Along this trend, the wavelength of the light is becoming shorter, while the reduction of aberration of the microlenses is of great interest in engineering ﬁeld. On

2 Nanometer-Scale Structure Formation on Solid Surfaces

51

Fig. 2.21. Isolated 30 nm lines by means of lithography with resist at height of 90 nm [110]. The light was provided by a laser-produced plasma in xenon-fueled pinched gas jet [111], and was focused by optics with numerical aperture (NA) of 0.3

Fig. 2.22. Basic idea of the lithography and self-organized processes (illustration referred to [112]). The hole is prepared in conventional lithography, and the following heating of the sample causes the step retreat. Then, viewed in the wider scale, the steps are bunched along the hole patterns

the other hand, the limit of the method in this trend will be soon attacked by diﬀraction limit of the light and diﬀusive destruction on the surface. To overcome these limitations, several methods are proposed. Among them, the combination of such conventional lithography and self-organized process is beginning to be focused in this century [112], where the diﬀusive tendency of chemical species under reaction is positively utilized to form certain structures by spontaneous motion (see Fig. 2.22). In the combinative method, the fundamental mechanisms – such as surface diﬀusion, step motion and step bunching, reconstruction, or strain near the heterointerfaces – are important. In this review, the recent results of such fundamental mechanisms are discussed. We introduce a good example to utilize the self-organization process. Figure 2.23a shows the domain boundary on Si(111) surface oriented to [112] direction, perpendicular to the steps. This can be obtained when Si is deposited to 20 nm on the surface at ∼923 K [113]. This type of surface arrangement to construct nanopattern is addressed in the following sections, where the strain associated with surface reconstruction is discussed. Using other material such as Ge, the morphology of the clean surface can be

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M. Tanaka et al.

(a)

step

step DB

(b) DB

bend

[112]

1 µm step DB

(c)

(d)

[112] 1 µm

(e)

[112]

1 µm

step

[112]

DB

2 µm

DB step

[112]

2 µm

Fig. 2.23. Regularly arranged grid pattern on Si(111) surface observed with AFM. Steps and the domain boundaries of 7×7 DAS structure (denoted as “DB”) are aligned, to show meshed nanopattern on the Si-deposited Si(111) surface (a). The solid phase epitaxy of germanium forms clusters (bright spots) after heating at 750◦ C , and they are distributed at the step edges and domain boundaries. The size of the clusters is controlled by the deposition thickness [114]. Nanopattern reveals diﬀerent morphology dependent on the thickness of the deposited Si: (b) 1 ML, (c) 1 nm, (d) 50 nm, and (e) 100 nm. The images are taken from [115]

controlled. When Ge is deposited onto the surface at the high temperature of 1,023 K, the Ge forms clusters and they are preferentially trapped at the domain boundary and the step edges. The shape changes with the thickness of Si layer deposited prior to the Ge deposition, as shown in Fig. 2.23b–e. The asymmetric diﬀusive process bents the steps and the spacing of the boundaries to minimize the energy of the phase boundaries. In the process of halogen etching of Si(111) surface, the regular alignment of Si clusters due to the strain

2 Nanometer-Scale Structure Formation on Solid Surfaces

53

by reconstruction is found, shown later. The initial scratch made by “active” fabrication, that may be electron impact or photoirradiation with laser for example, would be utilized in the fabrication process of the artiﬁcial patterns on silicon surfaces. In this section, the thermally induced desorption process from halogenated surface is introduced ﬁrst, because the processes at high temperatures are practically very important. The temperature dependence of the desorption yield classiﬁes the desorption path, and the results rise discussions of the reaction at the atomic scale. The quantitative analysis of the desorption yield will rise the discussion of energetic interpretation of the process. Then, as processes applicable to the surface fabrication, the passive process induced by heating is discussed. At high temperature, the motion of step retreat changes the morphology. In the wide terrace, the competition of the diﬀusive process and reconstruction will cause the pattern on the surface. Finally the active process induced by electronic excitation will be overviewed. 2.3.2 Scanning Tunneling Microscopy In the fabrication technology, the ﬁndings about the microscopic aspect of the physical phenomena will contribute to the development of the combinative fabrication methods. However, the inhomogeneity of the surface structure is actively introduced. The microstructure without periodicity is diﬃcult to be observed by means of diﬀraction method and discussed in reciprocal space. Fortunately, the microscopic observation at nanometer scale is now popular, and the images may give the information about the arranged structures through self-organization mechanism. Among them, STM is capable to give atomic structure at the surface of conductors or semiconductors. Nowadays its family, so-called scanning probe microscopy (SPM), is a widespread techniques to observe the surface structure, for example, in terms of local electronic state with scanning tunneling spectroscopy (STS), and insulator structure with atomic force microscopy (AFM). The surface of Si(111) has triangular periodicity. There is a Si in the topmost layer (refereed to adatoms) supported by three Si atoms in the second layer (refereed to rest-atoms) through the back-bond (see Fig. 2.24). Due to the large electron aﬃnity of halogen atoms, the charge transfer from halogen to the rest-atom weakens the back-bonds. This is the chief reason of frequent utilization of halogen (or halogen compounds) in the etching process of silicon surface. In the early stage of the development of STM, Boland et al. [42] showed that the dangling bond of Si(111) surface disappears after termination of halogen atoms in STS, because an STM images the integration of local density of states near Fermi level. Contrarily, at the sample bias of 3 V, formation of antibonding states by the overlap between the dangling bond at the adatom Si and the adsorbed Br 4p orbitals leads to the bright appearance of the bromine adsorption [38, 43]. The STM image of Cl-adsorbed surface by less than one atomic monolayer is shown in Fig. 2.25. In Fig. 2.25a, the

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M. Tanaka et al. dangling bond rest - atom Si

halogen

−δ weaken

adatom Si

+δ

+δ +δ back - bond

Fig. 2.24. The dangling bond (a) into the vacuum from adatom Si, in the topmost layer, is terminated by halogen X atom to form Si–X bonding state (b). The dangling bond state is located energetically near Fermi level, while the Si–X is far from the level

Fig. 2.25. STM topograph of Si(111)–(7×7) exposed to bromine gas. Bromine is adsorbed 0.04 ML (after Boland). The size of the area is 15×15 nm2 . The diamondlike area in white lines is a unit cell of 7×7, in which DAS structure can be recognized. The tunneling current was 0.1 nA. Each spherical dot corresponds one adatom. (a) Sample bias was +1 V. The dark dots are bromine adsorbates, while the nonreacted sites are bright. (b) Sample bias was +3 V. Bromine-adsorbed sites are brighter than the unreacted sites

55

Density of polybromine [ML]

2 Nanometer-Scale Structure Formation on Solid Surfaces

Coverage [ML]

Fig. 2.26. Correlation of the densities of chlorine adsorbed on Si(111) surface estimated in optical SDR spectroscopy (vertical axis) and STM images (horizontal axis). Note that the ﬂuctuation of STM results is larger than the averaged density obtained by means of SDR. The data are taken from [117]

bright spheres are the dangling bonds at the adatoms, while the dark spheres indicate where the dangling bonds disappeared. The change of the electronic state near the surface can be detected optically also. When chlorine atoms impact on the Si(111)–(7×7) surface (see Sect. 2.1 for detail of DAS structure of this surface), they react ﬁrst with the Si adatom dangling bond to form monochloride (SiCl) at low coverage, while higher coverage leads to the formation of dichloride (SiCl2 ) and trichloride (SiCl3 ) [83]. It is possible to distinguish the various types of chlorides from the displacements in STM images [116]. The surface density of the polybromide species estimated from the STM images has good correlation with that from the SDR spectra, as shown in Fig. 2.26. SDR is a powerful optical tool to evaluate the density of adsorbates in real time, even during high-pressure gas exposure. This method gives quantitative information about the averaged densities of various adsorbates at diﬀerent reaction site (see Sect. 2.2 for detail of SDR). Although the density of adatom dangling bonds identiﬁed as polybromide varied by about ±20% depending on the image processing, it remains to ﬂuctuate by ∼±30% among the areas that was imaged. The densities obtained with STM coincide reasonably well with the SDR results. It is important to note that the optical response shows the averaged properties over the macroscopic area of the surface. On the other hand, the ﬂuctuation of the statistical results from the STM images indicates that the distribution of the adsorbates varies greatly at the atomic scale. The correlation of the adsorption sites with the bond breaking/formation, suggested by the optical ﬁndings, has been conﬁrmed at the atomic level by means of the microscopic method. The STM is capable to ﬁnd a sole particle on the surface. Atoms, clusters, and island play essential role in the diﬀusive process on the surface. The idea of island/cluster formation [47] and reconstruction strain [119] are proposed on Si(111) surface. The stability and instability of halogenated species formed on the surface are the clue whether the desorption of halogenated silicon

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Fig. 2.27. Island of Br-adsorbed Si(111) surface after annealing at 673 K. The 1×1 structure is seen in the plateau of the island and near the island, while 7×7 remains in the far region. The image is shadowed to enhance solidity after taken from [118]

occurs or not [67]. The thermal reactions of silicon (111) and (001) surfaces are explained in detail by Weaver et al. [120] in terms of local structure of bonding conﬁguration. They pointed out that the desorption occurs chieﬂy at the step edges on the surface, while the spectacle nucleation of migrating silicon halide may change the surface morphology. At the same time, the (111) surface reconstructs between 7×7 and 1×1 structure at high temperature (see Fig. 2.27). The clusters of silicon halides are seen on 1×1 region both on the island and outside the island. The binding energy of the clusters on 1×1 is larger than 7×7 by 0.3 eV [47]. 2.3.3 Thermal Desorption Process Adsorption and desorption at semiconductor surfaces have been widely studied, often with the aid of TDS, because this method is relatively simple. The desorption rate is measured during heating the sample with the temperature raised. The features of the TDS signal allow quantitative classiﬁcation of the reaction path [121], and TDS has often been utilized in studies of catalysis on metals, as well as semiconductor processes. However, the desorption rate is generally diﬃcult to determine precisely unless it is suﬃciently high. Generally the S/N is not necessarily enhanced even if the sensitivity is increased. For example, to obtain an activation energy, the decay of desorption is important, and so a wide dynamic range is required. We present results obtained with isothermal and TDS methods utilizing a very sensitive mass spectrometer with a wide dynamic range [48]. With this apparatus, quantitative analysis of the desorption yield from an active surface enables us to evaluate the density of the surface adsorbates. Based on the relation of the density with the desorption rate, one can discuss the mechanisms of the desorption processes in terms of activation energy and reaction order. Results of TDS from Cl-saturated Si(111) surface are shown in Fig. 2.28. The detected mass is tuned to 63 amu, corresponding to SiCl ion. The signals + of 133 and 168 amu (due to SiCl+ 3 and SiCl4 , respectively) can be assigned

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Temperature (°C) 300 400 500 600 700 800 900 1000

Desorption Rate (Arb. Units)

A SiCl+ C

B

+

SiCl2

SiCl3+

SiCl4+

500 600 700 800 900 1000 1100 1200 1300 Temperature (K)

Fig. 2.28. TDS of the Cl-saturated Si(111) surface. The ions were selected by quadrupole mass spectrometer. The shape for SiCl+ is quiet the same as that of SiCl+ 2 when vertically scaled. The plot is taken from [122]

to a mixture of SiCl3 and SiCl4 . Surface species from the Si adatoms have been classiﬁed into monochloride (SiCl) or polychlorides (SiCln with n = 2 or 3) [39, 40]. Three peaks, A, B, and C, are found. These are clearly split with the highly sensitive mass spectroscopy system. The largest peak A, which appears at the highest temperature region, has been assigned to desorption of monochlorides formed on the surface, and peak C to polychlorides formed on the surface at high levels of chloride [39, 42, 46]. Although the origins of the + SiCl+ 3 and SiCl4 signals remain unclear, these two species could be related to surface defects [122]. The desorbed species for peaks A and B is mainly SiCl2 . At 900 K and higher, it has been associated with retreat of steps where SiCl2 is generated from monochloride species [120]. It is concluded [48] that peak B is associated with surface reconstruction from 7×7 into Cl-terminated “1×1,” whose structure has been observed with an STM [47, 118]. The three processes, originating each peak, involve independent mechanisms corresponding to each temperature region. Once the Cl-saturated surface has been heated to a temperature of 600–700 K, subsequent TDS measurement from room temperature gives only peaks A and B. After this thermal treatment, peak B is missing because the structure becomes Cl-terminated 1×1, and polychlorides have been desorbed out around the temperature region for peak C. Similarly, only peak A appeared from the Cl-exposed surface after the surface had been heated to 800 K. Remaining peak(s) (A and B for 600–700 K heating; only A for 800 K heating) had the same height(s) as the peak height(s)

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Desorption (Arb. Unit)

TDS of Cl-saturated Si(111)

A

B C

As Cl-exposed

After annealing to ~800 K

400

500

600

700

800

900

1000

Temperature (K)

Fig. 2.29. Comparison of the TDS results from Cl-saturated Si(111) surface (ﬁlled circles) with that of Cl-adsorbed sample after annealing at ≈800 K for a few minutes (open and gray circles). Peaks B and C disappear once after heating to the temperature in the region of peak A. The curve of peak A from the Cl-saturated 7×7 surface precisely traces that from 1×1 surface appeared after the heating. Data are taken from [123]

obtained without such thermal treatment, as shown in Fig. 2.29 [123]. During the heating of 7×7 surface, the process of peak A must be overlapped partially by that of peak B. This means that the rate of peak A is not aﬀected by the surface periodic structure (reconstruction in peak B). And the remaining polychlorides, if any, after peak C, do not change the dominant process of peak A. We will focus chieﬂy on peak A for the energetic discussion based on the time dependence of desorption rate, so-called isothermal desorption measurement. Figure 2.30a shows the results of isothermal process for peak A of SiBr+ , which is obtained after 6-min heating at ∼800 K for removing the contribution of peaks B and C. The desorption rates were reduced to almost zero in a few hundred seconds. We used the following procedure to analyze the time dependence, based on Polanyi–Wigner’s rate equation [124, 125]. The rate of desorption, Rd , from unit surface area with an adsorbate density of N is given by the following rate equation Rd = −

dN = uN m , dt

(2.8)

where m is the order of the desorption process and u is a temperaturedependent coeﬃcient. Analytic solutions of (2.8) are given by for m = 1, uN0 e−ut , Rd = (2.9) (1−m) (1/(1−m))−1 u[(m − 1)ut + N0 ] , for m > 1, where N = N0 at t = 0. Fitting of the experimental results with Rd for m = 1 and m = 2 gives the curves in Fig. 2.30a. The curve for m = 2 agrees better with the experimental plots. From the parameters for the ﬁtted curves,

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Desorption Rate, R (arb. unit)

Isothermal desorption of Cl / Si(111)

993 K

918 K (x3) 843 K (x3) 0

20

40

60

80

100

Time (s) Temperature (°C)

ln u(T) (arb. unit)

0

782

694

619

556

−2

Ed = 2.18 eV

−4 −6 −8 −10 −12 11.0

11.5

12.0

12.5

13.0

13.5

14.0

1/(kBT) (eV)−1

Fig. 2.30. Isothermal desorption for the major process (peak A in Fig. 2.28). (a) The time course of desorption rate for each temperature. Filled circles indicate the obtained data, while the gray and thin curves show the ﬁtting for the ﬁrst-order (m = 1) and the second-order (m = 2) process, respectively. (b) Arrhenius plot of prefactor coeﬃcient for the second-order process ﬁtted to (a). The data are taken from [122]

Fig. 2.30b shows the relation between log10 u and 1/kB T (so-called Arrhenius plot ) for m = 2. When u(T ) has the Boltzmann distribution u(T ) = u0 e−Ed /kB T ,

(2.10)

where Ed , kB , T , and u0 are the desorption potential barrier, the Boltzmann’s constant, the absolute temperature, and a preexponential constant, respectively, the ﬁtting gives Ed = 2.18± ∼ 0.1 eV [126]. The desorption energy of SiBr2 is obtained in the similar method on Si(111), found to be

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Ed = 2.6 eV [48]. This desorption energy for bromine, smaller than that for SiCl2 , implies that the bromine-etching speed can easily be controlled by temperature in comparison with chlorine etching. In case of bromine-adsorbed Si(111), the value of Ed is very close to a desorption energy of 2.6 eV reported from the TDS experiment of Br/Si(001) at about 950 K [127]. The agreement suggests that the back-bond of the dimer on a Si(001) structure is similar to the back-bond of dimer at the step edge on a Si(111) surface [43, 94, 120]. To discuss the activation energy for the desorption process, the shape of the curve can provide information about the nature of the surface process, on the assumption of the ﬁrst- and second-order processes. In case of the most simple spontaneous mechanism, each TDS curve was ﬁtted to the solution to the Polanyi–Wigner’s rate equation (2.8) for a ﬁrst-order process −

dN = uN, dt

(2.11)

where u is a constant at a given temperature. However, the solution for a second-order process, −

dN = uN 2 , dt

(2.12)

representing the simple associative mechanism. In many TDS experiments, socalled Readhead analysis is often adopted to obtain the experimental values from a few TDS curves, in which merely the peak temperature is considered. For example, the desorption energy for the Cl/Si(111)–1×1 surface at ∼900 K was 2.9 eV [96]. This is diﬀerent value for our result above. The discrepancy is resolved in the estimation of whole shape of TDS spectra [126]. TDS with various parameters in Fig. 2.31a shows the coverage dependence, where chlorine exposure was changed. The surface was saturated with chlorine between 0.3 and 0.6 L. During the Cl exposure, monochloride is formed on the surface ﬁrst, and polychloride later [40]. Correspondingly, peak A appears ﬁrst at lower exposure, and is followed successively by peaks B and C. It should be noted that peak A shifts to lower temperature as the coverage is increased, while peak B shifts to higher temperature. The process leading to peak A was second order (see [126] for detail). The process resulting in peak B is too complex to allow analysis from the peak shape, because it includes reconstruction [48]. Figure 2.31b shows the TDS result when the heating rate, η ≡ dT /dt, was varied. The desorption rate is deﬁned as r(t) ≡ −σ(dN (t)/dt), where N (t) is the surface density (Cl coverage) at time t and σ is the sensitivity of the detection system. It was normalized with η, and the vertical axis is R(T )/η, where r = r(t) at time t is transformed into the desorption rate R(T ) at temperature T (t) to satisfy r(t) = R(T (t)). As all the measurements were started from a Cl-saturated surface, the total coverage at the initial time, t = 0, must be the same for all heating rates 1 ∞ 1 ∞ R(T ) Nsat = − dN = dT. (2.13) r(t)dt = σ t=0 σ T =RT η

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Temperature (°C) 300 400 500 600 700 800 900 1000

Desorption Rate, R (Arb. Unit)

(a)

A

B

C

0.6 L 0.3 L 0.10 L 0.025 L 0.010 L

500 600 700 800 900 1000110012001300

Temperature (K) Temperature (°C) 300 400 500 600 700 800 900 1000

A

Desorption Rate, R / η (Arb. Unit)

(b)

Heating Rate B

0.5 K/s

1.0 K/s 1.5 K/s 2.0 K/s 5.0 K/s C

10 K/s

500 600 700 800 900 1000 1100 1200 1300

Temperature (K)

Fig. 2.31. TDS of Cl-adsorbed Si(111) surfaces. The spectra with the Cl coverage varied (a) and TDS of Cl-saturated surface with the heating rate varied (b). Peak A shifts to the lower temperature as the coverage is increased, while peak B to higher. On the other hand, both peaks shift to lower temperature as the heating rate decreased. The plot is taken from [122]

Indeed, the heights and widths of the three peaks are similar to each other at various heating rates. To analyze the shapes of the TDS curves, we deﬁne θ(T ) as the surface Cl coverage at a temperature, T , so N (t) ≡ θ(T (t)), where T = T (t). Then, from integration of the desorption rate, θ(T ) = −

∞

T

1 dθ ≈ ση

Tmax

R(T )dT .

(2.14)

T

Numerical integration allows a discussion of the energetics. As the reaction order m = 2, the results in Fig. 2.31a were converted into a plot of ln(R/θ2 ) against 1/(kB T ), as shown in Fig. 2.31a. The plot is linear even when the

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starting coverage is varied, and the slope of the line again indicates the barrier for the desorption. The value obtained was 2.17 eV. The values obtained in the analysis for whole spectral shape have good agreement to the value of the isothermal result, Ed = 2.18 eV. Note that the plots of ln(R/θ) for the ﬁrst-order model are not straight both for Fig. 2.31a, b, suggesting that m = 1 is not the case for the desorption process. 2.3.4 Cluster Alignment by Passive Fabrication At the atomic scale, adsorption and desorption processes are fundamental stages, and the structures on the surfaces made with such processes have been well examined. For example, atomic level dynamics on silicon surface under halogen etching has been deeply elucidated, where STM was chieﬂy utilized [120]. The fundamental chemical reactions at local sites are known from the surface structures in STM images, usually observed at room temperature after such process. The local chemistry aﬀects the surface morphology. However, most of the processes are done at high temperature. Although the importance of diﬀusion has been discussed on the halogen–silicon systems [95], little results are presented experimentally about the motion of surface species. On halogen-adsorbed surface, the diﬀusing species may be silicon halide, or atomic silicon detached from the surface. During the processes, in situ observation at the high temperature will reveal what happen on the surface to form nanostructures as results of surface strain and diﬀusion. Recently, nanoclusters of silicon carbide are found to be aligned as concentric circles on oxidized Si(111) surface [128]. The bond of Si–C is apt to be formed at step edges at high temperature [129], and multiple-step holes appear nucleated around the oxygen-free pits [130]. At the every stages in the whole process, surface diﬀusion and reconstruction play essential roles. We here address formation of the regularly organized structures on silicon surface using halogen desorption. In the process, the pinning of surface species and step retreat due to the desorption are important, as can be suggested the thermal desorption process described above. There is another factor, surface strain, due to reconstruction. The silicon clusters are aligned on the terrace through halogen etching. The formation mechanisms, from the STM images observed at high temperature, will serve the methodology of self-organization techniques on silicon surfaces, which are not obtained static structure observed at room temperature. In all the images in this section, the sample was started to be processed from clean Si(111)–7×7 DAS surface to which Cl gas was exposed to the saturation coverage at room temperature. The coverage was estimated to be 1.62 ML, where 1 ML has 49 atoms in a 7×7 unit cell [40]. Figure 2.32a shows STM images observed at room temperature after 873 K heating. This image is obtained after heating for 540 s. Many steps are found. Along the diagonal line from the left bottom to the right top, there are about 30 diatomic steps, and some multiple steps among them can be only recognized in the image.

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(a)

(b)

A D

E

C

B

Fig. 2.32. (a) Nanoclusters formed at apexes of the step edges. STM image of aligned clusters on Si(111) surface observed at room temperature, obtained after 873 K heating of Cl-adsorbed sample at the saturation coverage. The area is 200×200 nm2 . Tunneling current was 0.1 nA and the sample bias was +2.5 V. (b) Clusters (near “A” and “C”) pinning the step retreat (“E”) and bunched steps (dashed bands at “D” and “B”) observed at high temperature. The process ﬁnally aligns the clusters at the apex of the bunched steps as in image (a)

This is not the average morphology of the surface due to inhomogeneity, but it was a typical area where the step density was high. At the most of apexes of the step edges, clusters are found. Typical size of the clusters is 8 nm in diameter and 2–4 nm in height. The clusters apparently grow at the edges of the multiple steps whose height was 4–6 diatomic-step height (∼1.5 nm). We do not consider that the clusters simply grow at the apexes, but clusters pin the step motion [123]. With the isothermal decay recalled (see Fig. 2.30 and Sect. 2.3.3), the main process of the desorption seems almost ﬁnished at Fig. 2.32a. That is, after the desorption, the diﬀusion of the chlorides on the terrace consists chieﬂy of movement among the clusters. We consider it,

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so far, that the surface structure reconstructs back to 7×7 with the clusters deformed, while very slight desorption occurs at the cluster edges as well as the step edges. The STM images at room temperature may not provide enough information about the diﬀusive process, because the cooling process would anneal the surface structure. The surface structure in the actual desorption process at high temperature may diﬀer from the cooled one. We observed the surface at 673 K, corresponding to peak B. However, no clusters but some double steps are seen on Cl-terminated 1×1 surface in any image of the surface during the thermal treatment up to 30 min. The shape of step curves looked changed between scans, suggesting only the dissociation or association of chlorides at the step edges. Figure 2.32b was observed at ∼160 s after the temperature rising to ∼803 K, at which process of peak A begins. At this time, the desorption is the slowest as the process for peak A. Many clusters are formed at this temperature, but some should be called islands because the height is of one double step. Deep craters are formed near mark A and B in the image. Both craters draw step bunches to the left. Larger clusters marked C and D are identiﬁed at the upper sides of the bunches near the craters A and B, respectively. Hence, a model is presented here as shown in Fig. 2.33. The surface phenomena of the cluster alignment at the apexes of the step edges as follows: in consideration that associative reaction of surface chlorides was proposed in the context of the polychloride desorption ascribed to the second-order process. (1) The clusters prevented the movement of the steps which were growing or retreating. Or, (2) the migrating chlorides are preferentially collected at the terrace edges at the upper sides of the bunches, to let the clusters grow there. Bey-shaped pit, E in Fig. 2.32b for example, may thus develop between the pinned clusters around C. Desorption has been ascribed to be related to the step structure [47, 118], and the movement of the steps at high temperatures can be assigned to desorption at the edge proposed in a step-retreat model [43,120]. Our cluster alignment occurred because some interaction regulates the seeds of clusters. Consequently, randomly distributed steps have regular spacing and the clusters are seen aligned in a line. In similar alignment of bunched steps, the pinning of step motion can be done by holes made with

Fig. 2.33. Model of pinning of the step retreat to describe the high-temperature process (Fig. 2.32b) for formation of the regularly aligned clusters seen in Fig. 2.32a

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Fig. 2.34. Nanoclusters formed on a wide terrace. Only a single step is seen. STM images of aligned clusters on Si(111) surface were observed at room temperature, obtained after 873 K heating of Cl-adsorbed sample at the saturation coverage. The area is 200×200 nm2 . Tunneling current was 0.1 nA and the sample bias was +2.5 V

photolithograph as well [112]. This mechanism is one of general methods to the fabrication of the surfaces on which diﬀusive species remain or the steps are very active. In turn, we consider the wide terrace. Figure 2.34 was obtained after 873 K heating for ∼1,200 s. In the image, only one step can be seen. Some clusters were grown at the step edges, while much more were on the terrace. The clusters were wider but their height was smaller than in Fig. 2.32. One notes that the clusters are aligned in the direction indicated by the white arrow. And the spacing between them is ∼30 nm. This suggests that there are some mechanisms, other than the step motion and pinning, to determine where the clusters grow. The surface during the high-temperature treatment is presented for the sake of examination of the cluster alignment on the terrace. In Fig. 2.35a, some 7×7 DAS areas are seen, and other areas seem structureless. The areas of 7×7 were found at the upper sides of the steps, and they grew into the terrace. It takes several minutes to reconstruct most of the surface, and the size of the structureless regions converges to a temperature-dependent fraction of the whole surface. Some part of the regions had very thin width to form the domain boundary (black line) of adjacent 7×7 regions, and they can be ﬁnally called two-dimensional dislocation. The boundary had got its width thinner as the heating time is longer, and the reduction of the width was faster as the temperature was higher. On the surface after quenched, the structureless area is indeed not periodic structure except for 5×5 region near the step. Thus we conclude that the structureless area is so-called 1×1 at high temperature,

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(b)

(c)

Fig. 2.35. (a) An STM image of Cl/Si(111) at 873 K heated approximately for 2,500 s. (b) The sample cooled from 913 to 773 K. The images were successively obtained at 773 K. The area of the image was 22×33 nm2 , sample bias was about +2.8 V, and tunneling current was 0.8 nA. (c) The sample cooled from 913 to 663 K. The area of the image is 200×200 nm2 . The 7×7 phase boundary was emphasized with black lines. The boundary had tendency to be oriented as the direction of arrows, along which the clusters are nucleated

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i.e., silicon adatoms are moving on the ﬂat 1×1 surface. The “1×1” surface reconstructs into 7×7 from the upper side of the step edge. When we observed Fig. 2.35a, temporarily 5×5 or 9×9 were seen at the front of 7×7 expansion during the high-temperature observation, indicating that the reconstruction took place into 7×7 through such periodic structures. From the observation, the lifetime of the metastable structures was about a few seconds, so that even quick scan rarely gave a clear image of the metastable structure. Next, it should be noted that no cluster was in any high-temperature images like Fig. 2.35a, so far as the temperature was monotonously raised. After the very long heating, the “1×1” area converged into a size that was smaller as the temperature was the higher. The area would be shaped into triangular. However long we keep the sample at the high temperature, no cluster appeared (in several hours of heating). To search the origin of the clusters, we quenched the sample down to 773 K successively after enough long heating time at 916 K. Then the images were obtained as in Fig. 2.35b. In the images, there is a step, whose lower side was “1×1” structure. The shape of the step changes, indicating detachment/attachment of the silicon atoms diﬀusing on the surface [131]. And the domain boundary of 7×7 regions is seen to start from the concave corner of the step into the terrace. The shape of the boundary changes from the curved, then the ﬂuctuated, and ﬁnally the straight. There is a force to make the domain boundary straight along the dimer row of DAS perimeters. At the domain boundary, there are mismatch in the stacking-fault layer and broken stabilization at the corner hole symmetry [132]. Thus, the literal anisotropy contains strain, and it is larger if the boundary is bent. This larger strain is weakened through the thermal excitation during the 7×7 reconstruction. Then one ﬁnds clusters (small protrusions) along the domain boundary in Fig. 2.35b. The height was less than a single diatomic step, indicating that they were isolated silicon atoms. While the clusters also detach from and attach to the boundary, the boundary shape was changing. So far as the silicon clusters were relatively small as seen in the ﬁgure, they changed their positions. Some horizontal ﬂashes of noise are seen near the boundary only. These are interpreted as temporary stay of silicon adatoms. No ﬂash was observed on the wider terrace, indicating that the diﬀusing velocity is much higher than the scanning probe. The potential of the lateral motion may be lower near the boundary. The 2D adatom gas was overcooled, and a part of the gaseous atoms, exceeding the density of the 2D vapor, condensated near the boundary or the step. Finally the surface structure is shown in Fig. 2.35c, observed at 663 K. The sample was heated at 913 K for enough time, and annealed to the temperature (it took several seconds to cool down to 663 K). The surface on the terrace was covered with 7×7 DAS structure. To guide the phase boundaries, segmented lines are drawn. The lines connect the protrusions about 5–8 nm in the diameter. The heights of the protrusions are all ∼0.3 nm, corresponding to the single diatomic step. As time went on, the boundary was oriented to a direction. This is caused by the mechanism, shown in Fig. 2.35b, to make the domain boundary straight

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from the step edge to the terrace. The length of the segmented straight lines was 20–40 nm, which may be the tolerance to the strain at the temperature. The size of clusters tends to larger as the temperature was lower. However, it was determined not only by the temperature but also by the width of boundary and annealing rate. After the clusters had grown larger, we did not see any case that they moved to be aligned. The larger clusters, once formed, are crystalline [123] and may smoothly match with the substrate crystal. In the recent years, the limitations of the lithography and plasma process themselves have been turned out. The ﬂuctuation of the fabricated structures is of the most serious problem (suggested by, for example, the optical and microscopic discrepancies at the atomic scale, in Fig. 2.26). On the other hand, the functional region of the semiconductor devices is of the size of a few nanometer scale. The self-organization can be introduced to control the surface structure on mere silicon surfaces, a typical system of homogeneous material, that may contain the strain near the surface due to the reconstruction described above. The heterointerfaces are much more important for the functional devices constructed by oxide insulator on the semiconductor surfaces, metal junction to the semiconductor, or semiconductor–semiconductor junction such as Ge, Alx Ga1−x As, GaN, InP, and so on. The conﬂict between materials due to the mismatch of the lattice will cause the distortion of the structure. However, such strains proved to reveal the patterning near the interface (see the example patterns in Fig. 2.36), and in the industrial use the reliability is nowadays discussed in the high performance of the electronic state-strained region of the interface layer [134, 135] as it has already been at the stage of real mass production. The high-temperature observations have shown this kind of mechanism at the atomic scale. The local strain will be soon much elucidated to utilize the relaxation and patterning in thermal treatment such as etching and growth process. The patterning at the nanometer scale on metal surfaces is presented elsewhere in Sect. 2.4. 2.3.5 Active Fabrication In this section, some results are introduced for the surface-etching process induced by electronic excitation. This kind of reaction may include usually desorption of the surface species, so that the mechanism is frequently referred to “desorption induced by electronic transition (DIET).” In the case of halogenated silicon, one could obtain the surface structure that can never been obtained by thermal process. The “active” fabrication means the surface process that one stimulates the surface with some excitation and the process goes through the path nonequilibrium. This indicates that one can obtain a surface with diﬀerent structure or morphology obtained through a high-temperature process. The structure may be controlled if the excitation is caused selectively. In consideration of the self-organized patterns like in Figs. 2.32 and 2.34, the mechanisms of diﬀusion and strain (see Figs. 2.23 and 2.36, for example) will be adopted to construct to fabricate the surface process beyond

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Fig. 2.36. TEM image of patterning at heterointerface of silicon-on-insulator structure, composed of α-Si, Ge layers on insulating SiO2 . A sample (thickness Si:Ge = 17:35 nm) after annealing at 1,373 K for (a) 2 min, (b) 20 min, and (c) 60 min, and another sample (thickness Si:Ge = 10:26 nm) after annealing for (d) 2 min, (e) 20 min, and (f ) 60 min. The images are taken from [133]

the lithography. The aligned clusters are not unique method for the template of nanopatterns. We should examine the detail of dynamic process at the atomic scale to fabricate the surface actively. The artiﬁcial scratches (wellcontrolled defects) or regularly aligned local structures can be made with electronic excitation, which obeys the selection rule. The pioneering work of the dynamics of electronic excitation is done by a Japanese on 1942 before the daybreak of surface science [136]. Since that, the desorption kinetics has been understood in terms of electronic excitation [137]. The initial excitation has been chieﬂy electron impact, which possesses the tunable energy. Now photon is also available as the coherent and tunable excitation source to elucidate the surface processes. The photoexcitation has advantage in selectivity, while electron beam can be focused into smaller area easily. These sources should be chosen according to the design of the surface processes. On Si(111) surface, Cl adsorption weakens the bonding energy of the adatom to the rest-atom. This is favorable to control the structure of the atomic layers only near the surface. The energy of the polychloride species is much weaker than that of the monochloride [67]. We show STM images that

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(a)

(b)

Fig. 2.37. An STM image of Cl-adsorbed Si(111) taken after UV irradiation of photon energy at 4.7 eV. The regions in white lines indicate unit cells of 7×7 surface. (a) The Cl exposure was ≈ 1.5 L. The sample bias voltage was +2.5 V. Some adatoms are recognized, while some are collected like clusters. Rest-atoms can be seen where the adatoms are missing. (b) The Cl was adsorbed to the saturation coverage (exposure was ≈ 10 L). There are six atoms at the perimeter of the triangle of the unit cell. The stacking-faulted and -unfaulted halves have diﬀerent brightness, corresponding to the heights of the electronic state. It should be focused that this rest-surface contains many point defects. See [44] for detail

were obtained after weak irradiation (below the ablation ﬂuence) of UV light with the photon energy of 4.7 eV (Nd:YAG pulsed laser, time duration was ∼5 ns). In Fig. 2.37a, some of rest-atoms are identiﬁed from the STM image, and many adatoms are seen as bright spheres. Note that the 7×7 unit cell is recognized in the white diamond frame and the adatoms are placed at the position close to the DAS geometry. However, some adatoms are gathered into cluster. At the same time, there are some regions in which no adatoms are seen. That is, the adatoms are desorbed and rest-atoms can be seen. In the image, the “rest-atoms” indicate six rest-atoms along the perimeter of the unit cell, which is the very conﬁguration of DAS model (Fig. 2.1). This is interpreted

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that the monochloride (identiﬁed from STM images) remains on the surface, while polychlorides are desorbed by the photon [44]. The diﬀerence from the thermally induced process is the remaining stacking-faulted 7×7 structure. When chlorine is adsorbed to the saturation coverage, the image in Fig. 2.37b shows a beautiful rest-surface (only consisting of rest-atoms in the top-most layer). At the saturation, nearly all the adatoms were polychlorinated. When the chlorination is imperfect, photoirradiation cause the surface migration due to the removal of the adatom chlorides until they are nucleated into the clusters as in Fig. 2.37a. From the STM images, reader should note the diﬀerence from Fig. 2.27; in the thermal process, no defect is seen, and the stacking fault was removed into 1×1 periodicity. However, the image in Fig. 2.37b contains the DAS periodicity, and many defects are created in the rest-surface. This suggests that the pulsed photoirradiation lets DIET to occur on Cl-adsorbed Si(111) surface. It is true that the photoirradiation may raise the temperature. The increase of temperature may induce thermal desorption. The energy of photons is injected to the system through the electronic transition, and the heat increases chieﬂy the temperature of electron system, not the lattice vibronic system. When the laser pulse has the time width of ns order, the desorption occurs on Cl-adsorbed Si(111) surface with the photon energy in the range from 2.3 to 6.4 eV, but it is indetective at 1.2 eV [138, 139]. This implies that the desorption is induced by the excited electron from the photo-absorbed substrate of the silicon bulk. In a report of the desorption closely examined with the photon energy varied from 3.8 to 5.4 eV, two peaks are found at 4.28 and 5.06 eV [140]. The optical absorption of silicon crystal has its peak at 4.28 eV to support the desorption caused by the bulk excitation. When we irradiated infrared laser (1.55 eV) with fs and ps duration, the desorption from the surface was not much [109, 122], but the change of components of the surface species was found shown in Fig. 2.38. There are apparent diﬀerences between TDS of Cl-saturated Si(111) before and after irradiation (thick and dashed curves, respectively). This spectra can never be obtained after thermal pro+ cess as in Fig. 2.27. Peak A in SiCl2 (observed as SiCl+ 2 and cracked SiCl ) is reduced at the same time peak B or C remains. The increase of peak C in SiCl4 implies the creation of the defects on the surface or the irregular chloride clusters as in Fig. 2.37a. The long-time irradiation (∼ minutes) of IR pulses may cause the desorption from the sample at room temperature, but not at 150 K [122]. This means that the desorption is coupled with the thermal excitation, but the change of the surface occurs on the sample at 150 K. Before the increase of the temperature, the modiﬁcation, known form of the surface structure, was ﬁnished. In case of laser irradiation of ps pulse at 800 nm [141], the irradiation changes TDS like Fig. 2.38. Time-of-ﬂight (TOF) measurement of the desorbed species gives the information of the energy of the desorbed species. The TOF of SiCl+ diﬀers from that of SiCl+ 2 , interpreted as the desorption of SiCl from the surface. The ratio of the desorbed species was also diﬀerent from the ratio for

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C

SiCl+

B A

C

B

SiCl2+ (X 3) SiCl3+ (X 30)

SiCl4+ (X 30)

600

800 1000 1200 Temperature (K)

Fig. 2.38. TDS spectra of the Cl-saturated Si(111) surface. The change can be seen before (solid curves) and after laser irradiation (gray curves) of IR pulses (80 fs) at the ﬂuence of ≈90 nJ cm−2 for 500 s. The repetition rate was 80 MHz. Each ions are selected by quadrupole mass spectrometer. The data are taken from [122]

the cracked species in TDS. This suggests that there are at least two species on the surface to be desorbed. The desorption ratio suggests that the two species are monochloride adatom and polychloride adatom. There is discrepancy from these results for femtoseconds [141] to the case of nanoseconds [44]. The very intensive attack from the bulk within femtoseconds may break the binding of the SiCl and SiCl2 to the surface. The desorption rate was compared with the two ions, and the desorption rate R was not proportional to the laser ﬂuence I. 3.9 , respectively. As shown in Fig. 2.39, R(SiCl+ ) = αI 4.7 and R(SiCl+ 2 ) = αI Although the meaning of the power factor (p) are not still clear, the nonlinearity indicates that the process requires multiple excitation. Two possible mechanisms were suggested for the hot carrier-induced process from experimental and theoretical approaches: desorption induced by multiple electronic transition (DIMET) [142–144] and desorption due to nonadiabatic vibronic excitation [145–148] for the process with very short pulse lasers comparable to the lifetime of the excited carriers in the conduction band [149, 150]. The pulse duration remarkably aﬀected desorbed species and rate, and especially the very short pulses are possible tool for surface fabrication other than equilibrium heating. Recalling the cases of photoinduced desorption with ns pulses (the previous paragraph), we note here that the photons with energies of 5.06 and 6.4 eV give the desorption yield proportional to the incident ﬂuence [139, 140]. However, the photons at 4.28 eV give hyperlinearly increased yield [140]. This derives that there are at least two paths to relaxation of the excited electron at the surface to lead to the desorption, linear and nonlinear mechanisms. However, these have not been related to the surface species yet. Next we change the topic with the source of excitation from photon beam (laser) to electron beam (e-beam). The electron-impacted processes have been

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104

Desorption yield (counts)

103

102

p = 4.7

4

5

SiCl+

6 7 8 910

103

102

p = 3.9

101

SiCl2+

4 5 6 7 8 910 Laser fluence (mJ/cm2)

Fig. 2.39. Desorption yield of SiCl+ and SiCl+ 2 ions with various ﬂuence of ps laser (λ = 400 nm). Note that the vertical scale is logarithmic and the relation is nonlinear, suggesting multiple electronic excitation. The data points are from [141]

one of major ﬁeld of surface science, and still keep the importance especially for semiconductor technology because electron impact is an important factor in the plasma etching/deposition and lithography is often done with e-beam. This kind of process is often referred to electron-stimulated desorption. Among them, the system of silicon with halogen is of the most importance. The desorption of bromine was measured with the primary energy of e-beam varied. When the surface coverage of bromine was not high, the desorbed species was ion of atomic bromine (Br+ ). As shown in Fig. 2.40, the desorption was increased between 200 and 300 eV, corresponding to the excitation of M shell electrons in bromine. From highly bromine-covered surface, silicon bromides are desorbed as ionic state (SiBr+ x ; x = 0, 1, 2). The kinetic energy of the desorbed ion was not so high (3–4 eV) irrespective of the primary energy. The process through the core excitation is understood in terms of the Knotek– Feibelman (KF) model where there is a transition state before the desorption. The transition state in KF model is two (or more) holes in the valence band after the relaxation of the initial core hole through Auger process. However, none can discuss further on the excitation in the bulk or at the surface unless the initial and transition states were separately discriminated [151]. In turn, we consider that the case with e-beam has no high energy. STM can inject ultimately narrow e-beam into the surface at the atomic scale. The STM current, in which the electron energy is typically within ±4 eV from Fermi level, may induce the excitation required for chemical reaction including desorption. The halogen atoms adsorbed to Si surface can be removed from the

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Fig. 2.40. The desorption yield of Br+ ion from a Br-adsorbed Si(111) surface under electron irradiation at various primary energies. The arrows show the absorption edge of the indicated electronic states. The data points are taken from [85]

40 30 20 10 0 0

10

20 (a)

30

40 nm (b)

Fig. 2.41. STM images of a Cl-saturated Si(111) surface. The sample bias was +2.5 V. (a) An image of the central area of 40 × 20 nm2 with the bias at 5.0 V. (b) A magniﬁed image of the area surrounded by the rectangle in (a). The image is taken from [155]

surface [152, 153]. This indicates that the e-beam may cut the bond between the adsorbed halogen and the silicon atom in the top-most layer. This kind of desorption is reported to have threshold bias voltages at +1.5 V for bromine and ∼+5 V for chlorine. They are ascribed to the antibonding state between the halogen and the silicon. Once the antibonding electrons are injected, the halogen atoms slide away along the adiabatic potential like classic mechanics. This picture is referred to Menzel–Gomer–Readhead (MGR) model [154]. The advantage of e-beam from STM is the polarity of the bias voltage. That is, the occupied electron in the valence state is removable from the atoms near the surface. Figure 2.41a shows the rest-surface obtained with tunneling current of 0.8 nA at the bias of +2.5 V after scanning in the inside area at −5.0 V. The inside area is found to the structure is diﬀerent from

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the outside. The magniﬁed image in the inside area is shown in Fig. 2.41b, which has obviously the same structure as in Fig. 2.37. Both images reveal the feature of electronically created rest-surface. This process, namely hole injection, may be resonant to the back-bond between adatom Si and restatom Si. However, all the three back-bonds are broken. Consider that the STM current at ∼nA supplies one hole in every 0.1 ns, which is at the same order of lifetime of a surface state on the semiconductor at longest [149, 150] and it can rarely ﬁll all the three back-bonds at the same time. Then, we have another model. We propose the inelastic scattering that enhances the local vibration [156, 157]. The energy required to desorb the surface species must have more than a single vibronic quantum. The ﬁrst excited vibration has lifetime at ∼10 ps, while higher-order excited vibration may be prolonged up to ∼ns due to the unharmonicity [158]. Then the multiple vibronic excitation accounts for the hole-injected desorption (negative bias), and this is also applicable to the electron-impacted desorption (positive bias). Close examination of the atomic desorption of hydrogen adsorbed on Si(001) surface, the desorption dependence on the bias voltage and the tunnel current concluded the multiple excitation of the local vibrational model due to the hole/electron scattering [159]. The diﬀerence of the multiple vibronic excitation from the conventional thermal excitation is whether the system near the surface (especially the atomic motion) can be said equilibrium with the electronic state (chieﬂy in the bulk) or the lattice phonon (over the substrate). The energetic distribution is described in terms of temperature, and time evolution of the temperatures of each system may be good indicator of the energy transfer. On the metal surface, the transient temperatures of each nearly independent system consistently give indeed good picture of the desorption process induced by photoexcitation [160]. In this sense, this model can be adopted not only to the e-beam process, but also to the photoinduced process. The hyperlinearity in the laser-induced cases has been explained two-hole localization due to the surface plasma vibration [161], but the multiple vibronic excitation can explain the mechanism as well. DIMET model also assumes the antibonding adiabatic potential curve like simple MGR model. However, the antibonding state is not always found, and the photon-energy dependence of the etching rate at the chlorinated surface implies that the resonant excitation is not necessary. The vibration of the surface species through the electronic antibonding state can be replaced by that by the electron–local phonon scattering at the surface. Then the diﬀerence may disappear between DIMET and the multiple excitation after some modiﬁcation. The rate of inelastic scattering at the surface species depends on the energy spacing of the vibronic steps and the energetic threshold of the injected electron in the ladder-climbing process [162]. Anyway, in the change of the bonding due to active excitation near the surface, the clue is the dissipation path of the energy and the process of the electronic relaxation. In this concept, ﬁndings about the mechanism of the dynamic reaction, desorptions typically, induced by the active excitation will

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generally serve for the technology to the controlled and precise fabrication processes on the surface. 2.3.6 Summary In this section, microscopic aspects are reviewed on the surface processes on silicon surface in association of halogens. From the viewpoint of physical phenomena on semiconductor surface, etching process is the most fundamental. The quantitative analysis of desorption by means of TDS gives rise to the good pictures to interpret the surface phenomena, and the energetic discussion becomes possible with the aid of optical results to determine the surface density of the adsorbed species. This approach is widely applicable to deposition, structural change, and surface reaction of other material, that may be semiconductor or other material, onto the silicon wafers. Self-organization processes have drawn many researchers’ interest in the recent years for their possibility in fabrication of semiconductor surfaces, because the required size for electronic devices will soon be nanometer scale, smaller than the limit of optical lithography. Scanning probe technique is one of the good crafts to control a single atom on the surface [163], as well as it is very powerful tool to analyze the surface structure, electronic state, and so on. However, the scanning probe techniques are not suitable for industrial use, in which mass production and reliability are required. This situation promotes the discussion of combinations of chemical/physical techniques such as etching, deposition, and lithography [112]. In the combination, the mechanism of the strain is important to assembly the surface structure by self-organization. No matter what material homo- or heterospecies makes on them, the product structure is ruled by the mismatch at the interfaces in heteroepitaxy. In this section, we present the simplest system, the structural growth of silicon on the surface of silicon single crystal. It is pointed out here again [123,164] that in the alignment of the silicon clusters, the desorption mechanisms (step motion and reconstruction, typically) are competing, and that regular structures emerge as the result of the balance between them. Finally, processes actively induced on silicon surfaces are addressed. Electric excitation, which used to be electron impact in the early era of surface technology, can be selectively induced today using tunable light source, typically laser and synchrotron. They are often pulsed into ultrashort time duration from fs to ns, and they are utilized to understand the ultrafast dynamics [165]. The application of such light sources is introduced in other chapters. Lastly, we comment that the detailed dynamics at the atomic scale in terms of electronic excitation [166] will be soon developed to the actual fabrication at the micrometer to nanometer scale.

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2.4 Self-Organized Nanopattern Formation on Copper Surfaces 2.4.1 Introduction As the scale of integrated circuits continues to diminish, reaching the scale of a few nanometers in lateral dimension, it is of great importance to design nextgeneration nanoelectronic devices based on a bottom-up approach, namely precise control of the atomic processes on surfaces. Thus, fabrication of low-dimensional structures such as nanoclusters [167], nanowires [168], and nanopatterns [112] has been intensively studied for decades. Formation process of these nanostructures through self-organizations is of particular interest to produce uniform structures over a macroscopic range on surfaces [169]. In this section, we review a particular system as one of the examples of such studies, focusing on nitrogen-covered Cu(001) surface (Cu(001)–c(2×2)N). It has been discovered for over a decade ago that this surface produces a novel periodic nanopattern as shown in Fig. 2.42. In this image, bright lines with the average width of about 2 nm represent clean Cu area separated with a square patches of c(2×2)N structure with the size of 5×5 nm2 . Formation mechanism of nanopattern on this surface has been investigated by STM [170– 174], spot proﬁle analyzing low-energy electron diﬀraction (SPA-LEED) [175], Rutherford backscattering spectroscopy (RBS) [176], grazing incidence X-ray diﬀraction (GIXD) [177], and ﬁrst-principles calculations [178]. The elastic eﬀects on surfaces are important to understand formation mechanism of such a nanopattern. The displacement of the Cu lattice covered

Fig. 2.42. A surface with regularly arranged c(2×2)N patches (dark area) separated by 2 nm wide Cu lines on average (bright square grids). The nitrogen coverage is 0.25 ML (the full coverage corresponds to 0.50 ML), and the image size is 100 × 100 nm2

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with N was calculated based on the elastic model, and a good accordance with GIXD results was obtained [177]. The lattice distortion near the boundary of the c(2×2)N patches was visualized by STM [174]. More recently, shrinkage of lattice constant at the clean Cu area surrounded with c(2×2)N patches was experimentally conﬁrmed by angle-resolved ultraviolet photoelectron spectroscopy (ARUPS) [179]. This study showed that upward shift of the Tamm state toward the Fermi level coincides with a slight shift of the folding point at M along the Γ –M line, and is ascribed to the strain at the clean Cu area on the grid pattern. The periodic pattern on this surface provides us with a template for fabricating periodic nanostructures on a relatively wide terrace. Formation of nanodot array and stripes was reported for magnetic metals such as Fe [180–182], Co [183], and Ni [184]. Novel magnetic properties and electronic properties were investigated with surface magneto-optical Kerr eﬀect (SMOKE) [185] and magnetic linear/circular dichroism (MLD/MCD) [186]. The inhomogeneity of electronic properties on this surface governs not only the growth process but also dissociative adsorption of molecules. It turned out that inhomogeneous strain can cause signiﬁcant change in the dissociation probability and diﬀusion rate [187]. Active role of oxygen coadsorption on the surface stress distribution has been further investigated with GIXD [188]. Fundamental understanding of these phenomena will be the basis of the bottom-up approach to create controlled nanostructures at the atomic level. Here, our recent studies on strain-induced nanopattern formation, growth process, and dissociative adsorption are summarized. 2.4.2 Experiments The experiments were carried out in an UHV apparatus, equipped with STM, LEED, AES apparatus, and ion gun. The base pressure was better than 5×10−11 Torr. We used the chemically polished Cu(001) surface of a columnshaped single crystal with the diameter of 4 mm at the surface and the height of 2 mm at the side. The STM used for this study was a commercial Omicron Vakuumphysik micro-STM. All the STM images were recorded in a constantcurrent mode with a tungsten tip at RT. The Cu(001) surface was cleaned by repeated cycles of Ar+ sputtering (500–1,000 eV) and annealing at 600 K until no sign of contamination by AES was observed. A nitrogen-covered c(2×2) surface was prepared exposing the clean Cu(001) surface to nitrogen activated by an ion gun (400–500 eV), followed by annealing up to 600–690 K. Pure iron (99.998%) was deposited on the surface at RT from a resistively heated alumina crucible at a rate of 2.0 ML min−1 . The stability of evaporation rate was conﬁrmed using the quartz crystal microbalance. The pressure during the deposition was better than 3×10−10 Torr. The surface was exposed to oxygen gas at the amount less than 300 L (1 L = 1×10−6 Torr s).

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2.4.3 Novel Phenomena on Cu(001)–c(2×2)N At low coverage of N, inhomogeneous displacement of Cu atoms from the bulk position was observed by STM [174]. Such an inhomogeneous strain may play crucial roles in the nanopattern formation and kinetic processes of adsorbates. We address here three novel phenomena observed on Cu(001)–c(2×2)N: 1. Nanopattern formation at vicinal surfaces 2. Strain-dependent nucleation of metal islands 3. Strain-dependent dissociation of oxygen molecules 2.4.4 Nanopattern Formation at Vicinal Surfaces Figure 2.43 shows STM images for a Cu(001)–c(2×2)N surface vicinal to the [110] direction prepared by annealing up to 620 K. Nitrogen coverage was 0.33±0.03 ML on average. In these images, bright lines indicate a clean Cu surface and the dark areas indicate the N-adsorbed surface. The clean Cu lines are 1.1±0.2 nm wide along the 110 direction. In Fig. 2.43a, the shape and size of the N-covered surface are not homogeneous on most of the terraces. We call this structure a “labyrinth pattern.” We also note small domains with a stripe pattern in Fig. 2.43a, such as the area indicated as S. A magniﬁed image around the stripe pattern is shown in Fig. 2.43b. The width of the terrace with the stripe pattern is always over 20 nm. In Fig. 2.43c, we show the cross-sectional line of the STM image along the line A–B in Fig. 2.43b. The apparent height of the Cu lines of the labyrinth pattern is about 0.09 nm compared with the level of the N-adsorbed surface. At the step edges along the 110 direction, bright lines with the width of 0.90±0.05 nm are observed. Their apparent height is higher than the level of the N-adsorbed surface by about 0.06 nm. Thus we consider that these step edges are made of clean Cu lines. When we exclude the area S in Fig. 2.43a, the total length of the Cu lines parallel to the [110] direction (L[110] ) is longer than that parallel to the [110] direction (L[110] ) by the ratio of L[110] /L[110] ≈ 1.5. The clean Cu lines along the 110 direction become straighter and form the stripe pattern when we increase the annealing temperature as shown in Fig. 2.44. The nitrogen coverage is the same as that for the surface with the labyrinth pattern. This STM image was obtained after subsequent annealing of the surface shown in Fig. 2.43 up to 660 K. The surface is dominantly covered with the stripe patterns. On this surface we can see two domains on the same terrace. One consists of the Cu lines parallel to the [110] direction and the other consists of those parallel to the [110] direction. The area of the former domain is about three times larger than that of the latter domain. Thus, we call the former the “major stripe domain” and the latter the “minor stripe domain.” The step of the major stripe domain is straight and aligned parallel to the [110] direction while that of the minor stripe domain is rounded. The step edges for the major stripe domain consist of clean Cu lines like those in the labyrinth pattern.

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(a)

S

[110]

[110]

(b)

A

[110]

[110]

Height (nm)

B

0.2 0.1 0.0 0 A

5

10 Distance (nm)

15

20 B

(c) Fig. 2.43. (a) An STM image of the labyrinth pattern formed on the Cu(001)– c(2×2)N surface vicinal to the [110] direction. The size of the image is 100×100 nm2 . A surrounded area (S ) indicates the stripe pattern on this surface. (b) A magniﬁed image around the stripe pattern. The size of the image is 40 × 40 nm2 . (c) The cross-sectional proﬁle along the line A–B in (b)

The average width and length of the clean Cu lines and the N-adsorbed stripes, perpendicular (⊥) or parallel ( ) to the [110] direction in Fig. 2.44a, are summarized in Table 2.4. The width of the Cu lines is almost the same for both directions while that of the N-adsorbed stripes is smaller in the minor

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G G

G [110] [110]

(a)

[110] [110]

(b) Fig. 2.44. (a) An STM image of the double-domain stripe pattern on the Cu(001)– c(2×2)N surface vicinal to the [110] direction. The size of the image is 100×100 nm2 . Surrounded areas (Gs) indicate the grid pattern on this surface. (b) A magniﬁed image at the boundary between the diﬀerent stripe domains. The size of the image is 33 × 33 nm2 Table 2.4. Widths and lengths of the Cu lines and the N-adsorbed stripes

Cu lines (⊥) Cu lines () N-adsorbed stripes (⊥) N-adsorbed stripes ()

Width (nm)

Length (nm)

1.0 ± 0.1 1.1 ± 0.1 1.6 ± 0.2 2.2 ± 0.2

<40 <80 <40 <80

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stripe domain than that in the major one. The domain boundary is aligned to the 100 direction, and the Cu lines along the 110 direction are likely to be connected with each other. The magniﬁed image (Fig. 2.44b) shows the structure of the boundary for the diﬀerent domains. There are short Cu lines at the boundary which are less than 2 nm long along the 100 direction. Among the stripe domains, we found small areas consisting of square patches with N-adsorbed surfaces, indicated as Gs in Fig. 2.44a. These patches are aligned along the 100 direction, although rotated by 45◦ with respect to the Cu lines in the stripe patterns. The average width of the square patches of the N-adsorbed surface is 5 nm. The square patches are separated by narrow Cu lines with the width of 1 nm. These features are quite similar to those of the grid pattern formed on the whole surface as shown in Fig. 2.42. The square patches on the vicinal surface are formed near the boundary of the diﬀerent stripe domains or the step edges, where the N coverage is ∼0.25 ML around the square patches. Major stripe domains, minor stripe domains, and grid domains occupy 75, 20, and 5% of the surface, respectively, in Fig. 2.44a. We demonstrate here that novel self-organized 1D patterns can be formed on vicinal Cu(001)–c(2×2)N surfaces by changing the annealing condition. The step density and direction play crucial roles in the pattern formation on this surface. 2.4.5 Strain-Dependent Nucleation of Metal Islands We investigated Fe initial growth on the grid pattern in Fig. 2.42. In Fig. 2.45a– d, we show the STM images of Fe-deposited Cu(001)–c(2×2)N surfaces. The average coverages of Fe are (a) 0.003 ML, (b) 0.015 ML, (c) 0.03 ML, and (d) 0.16 ML. Small dark spots are seen on the Cu grid as indicated by the arrows in Fig. 2.45a. These spots are not observed on the Cu(001)–c(2×2)N surfaces before the Fe deposition. We consider these as Fe inclusions. Their apparent depth is 0.015–0.03 nm, and is the same as that on the clean Cu(001) surface after Fe deposition [189]. The Fe inclusions with the size over 1 nm2 preferentially form at the Cu intersections nearby the c(2×2)N patches. At the coverage of 0.015 ML, we found that small Fe islands preferentially nucleate at the same position. This correspondence of the position suggests that Fe inclusions act as nucleation center for the growth of Fe islands. These small Fe islands grow laterally to cover the Cu intersection area. In this growth process, there are several kinetic processes after migration of deposited atoms during the growth on this surface (1) incorporation of adatom in the clean Cu surface, (2) trapping of adatoms at the inclusions or joining of a few adatoms to nucleate a small islands, and (3) trapping of adatoms at the edge of the islands. Process (1) causes inclusions which act as “nucleation centers.” Process (2) produces small islands on the surface, which is mentioned as “nucleation process.” Process (3) governs the expansion of each island, namely “initial stage of the growth.”

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(a)

(b)

(c)

(d)

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Fig. 2.45. A series of STM images showing the initial growth of Fe on the Cu(001)– c(2×2)N surfaces with the square grid pattern. (a) The coverage of Fe is 0.003 ML on average. The dark depressions pointed by arrows are Fe inclusions incorporated in the Cu grid. The coverage of Fe inclusions at the Cu grid seen in this image is less than 1 × 10−3 ML. The size of the image is 50 × 50 nm2 , taken at sample bias of −2.0 V, and tunneling current of 0.40 nA. (b) The coverage of Fe is 0.015 ML on average. Small islands with the size below 4 nm2 nucleate not only at the Cu intersections, but also at the Cu lines and the c(2 × 2)N patches. The size of the image is 60×60 nm2 , taken at sample bias of 2.0 V, and tunneling current of 0.10 nA. (c) The coverage of Fe is 0.03 ML on average. Large islands with the size over 4 nm2 are seen only at the Cu intersections, while small islands with the size below 4 nm 2 are seen everywhere on the surface. The image size is 60 × 60 nm2 , taken at sample bias of 2.0 V, and tunneling current of 0.30 nA. (d) The coverage of Fe is 0.16 ML. Fe inclusions are still seen between large islands grown at the Cu intersections. The image size is 30 × 30 nm2 , taken at sample bias of −2.0 V, and tunneling current of 0.30 nA

One possible mechanism to explain preferential formation and size distribution of the Fe inclusions on the Cu grid is the strain-dependent exchange process of the adatoms on this surface. According to the studies of elastic relaxation on the Cu(001)–c(2×2)N surface [176, 177], the strain nearby the

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boundary to the c(2×2)N patches should be larger than that at the center of the Cu lines. Existence of such inhomogeneous strain near the c(2×2)N patches was also proved by high-resolution STM study [174]. Preferential sites of the Fe inclusions correspond to the place where elastic strain is large on the substrate. In general form, Tersoﬀ [190] showed that minimization of the strain energy is the dominant driving force for surface-conﬁned intermixing in many heteroepitaxial systems. This model predicts that the size of incorporated clusters (inclusions) changes by the interface energy. The large interface energy tends to form the large inclusions. This scheme can be used to explain diﬀerence in the clustering behavior for diﬀerent species at surfaces. We apply this model to our results in the extended way that the activation barrier of mobile Fe adatoms to incorporate into the substrate should be modiﬁed with deformation of substrate Cu lattice due to the elastic strain. This can explain the preferential formation of Fe inclusions at highly strained places on this surface. There appear small Fe islands even at the c(2×2)N patches and Cu lines at the coverage of 0.015 ML. Preferential growth of large islands with the size over 4 nm2 at the Cu intersections is apparent at the coverage of 0.03 L as shown in Fig. 2.45c. In this ﬁgure, more than 80% of Fe adatoms are trapped at the Cu intersections. This means that Fe adatoms are more eﬀectively trapped by growing small islands on the Cu intersections than the rest of islands at the Cu lines or at the c(2×2)N patches. In the metal-on-metal epitaxial growth, the two-step model for the adatom trapping process has been adopted to explain the shape of islands [6]. According to this model, mobile adatoms do not completely lose their kinetic energy just after the arrival to the edge of the islands, but can move around them to ﬁnd energetically favorable kink sites. On inhomogeneous substrates such as the Cu(001)–c(2×2)N surface, we should note that the diﬀerences of the diﬀusion barrier on the surface and the sticking probability to the edge of islands for mobile adatoms depend on local structural and electronic properties on the substrate. At the Cu intersection of the Cu grid on the c(2×2)N surface, both the diﬀusion barrier and the sticking probability for Fe adatoms can be diﬀerent between the center side and the edge side because of inhomogeneous surface strain. Based on the observed preferential growth of islands, the diﬀusion barrier and/or the sticking probability are the largest at the center side of the Cu intersections. In other words, inhomogeneity of both diﬀusion process and sticking process, thus, leads to the maximum growth rate at the center side of the Cu intersections and suppression of the island growth toward the Cu lines and the c(2×2)N patches. The three-step growth process at the Cu intersection is summarized in Fig. 2.46. First, (a) Fe inclusions are formed at the side of the Cu intersections nearby the c(2×2)N patches, which act as nuclei for further aggregation. Second, (b) small islands are preferentially formed over the Fe inclusions. This process is not the essential reason for the formation of 2D array of Fe islands

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(b)

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(c)

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Fig. 2.46. Fe initial growth at the Cu intersections. (a) First, Fe inclusions are formed at the Cu intersections preferentially nearby the c(2 × 2)N patches. (b) Second, small nucleations are preferentially formed on the Fe inclusions. (c) Then, islands keep monolayer height and grow laterally to cover the Cu intersections. After the Cu intersections are fully covered, the islands grow laterally onto the Cu lines and c(2 × 2)N patches simultaneously

since we observed small islands apart from the Cu intersections at low coverage. Third, (c) the islands at the Cu intersections rapidly and laterally grow toward open space at the Cu intersections and initially not toward the Cu lines and the c(2×2)N patches. At this stage, the rest of the islands on the Cu lines and the c(2×2)N patches grow very slowly. We have conﬁrmed the same three-step formation process of the island at the Cu intersections at diﬀerent deposition rate and substrate temperature [180]. 2.4.6 Strain-Dependent Dissociation of Oxygen Molecules A microscopic understanding of the strain-dependent dissociation of molecules on metal surfaces is currently of great interest [191]. To clarify the role of strain on the dissociative adsorption on Cu(001), we have observed the distribution of oxygen atoms on clean Cu(001) surface area separated by c(2×2)N patches. Figure 2.47 shows STM images of Cu(001)–c(2×2)N after exposing 20 L of oxygen at RT. A number of depressions are exclusively found at the clean Cu area in this image. They are clearly observed in the wide range of the sample bias voltage between 0.5 and 1.5 V independently of the polarity with the tunneling current of 1–2 nA. Magniﬁed STM images are shown in Fig. 2.48a–c for the areas inside the white squares (a–c) in Fig. 2.47. The apparent depth of the depressions on the Cu grid depends on the sample bias, and is between 0.03 and 0.06 in these ﬁgures. In Fig. 2.48a, mls represent the Cu lines with the minimum line width of about 0.4 nm. A schematic model of the mls [170] is given in Fig. 2.48d. Very few depressions were found on mls as in Fig. 2.47. It seems possible to specify the fourfold hollow sites on clean Cu area of the STM images as shown in Fig. 2.48. Most of the depressions on the clean Cu surface are located on either the crossing points or the centers of the grid mesh. The results suggest that

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(a)

(b)

[010] [100]

(c)

Fig. 2.47. An STM image showing the Cu(001)–c(2 × 2)N surface after oxygen exposure of 20 L at RT. The size of the image is 50 × 50 nm2 , taken at a sample bias of 1.0 V, and tunneling current of 1.0 nA. Magniﬁed images at the white squares (a–c) are given in Fig. 2.48

the depressions are located at the fourfold hollow sites and can be identiﬁed as adsorbed oxygen atoms. We counted the number of oxygen adsorbates on the clean Cu surface in the STM image shown in Fig. 2.47. There are 20 adsorbates on the Cu intersections surrounded by four mls. The total area of these intersections is 82 nm2 , and about half of them have no adsorbed oxygen. On the other hand, 166 oxygen atoms are found on the rest of Cu intersections and wide Cu lines, the total area of which was 210 nm2 . Consequently, the oxygen coverage on the Cu intersections surrounded by four mls is 0.016 ML, which is less than one-third of that on the other wide Cu surface (0.051 ML). Density of oxygen atoms on both areas does not change during the measurement for 2 h in UHV. This indicates that the oxygen adsorbates can move across neither the c(2×2)N patches nor the mls on this surface. It has been reported that the oxygen molecules are predominantly adsorbed on clean Cu(001) surface at 300 K by the direct dissociation and not by the precursor-mediated dissociation [192]. We can neglect the diﬀusion of precursor-state oxygen molecules on the surface just before the dissociation. Thus, the observed diﬀerence of the oxygen coverage between the Cu intersections surrounded by four mls and those by the wide Cu surface is ascribed to the probability of dissociative adsorption. In the molecular beam experiments [193], the estimated change of the lattice constant along the [100] direction was 0.01% at most. Nevertheless, the initial sticking probability increases with the applied tensile stress at the translational energy of incident O2 below 173.1 meV. The results indicate the decrease of the dissociation barrier with the increase of the lattice constant. On

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(a) ml ml ml ml ml [010] [100]

(d) (1x1)Cu

c(2x2)N

ml

(b) ml

N atom Cu atom

[010] [100]

(e) (c)

Fig. 2.48. Magniﬁed STM images (a–c) of the areas shown in Fig. 2.47. (a) The size of the image is 18 × 18 nm2 , taken at a sample bias of −1.0 V, and tunneling current of 2.0 nA. The narrowest Cu lines are deﬁned as ml s. (b) The image was observed at an interval of 8 min after the scanning of (a). The size of the image is 14 × 14 nm2 , taken at a sample bias of −1.0 V, and tunneling current of 2.0 nA. (c) The image was observed at an interval of 95 min after the scanning of (a). The size of the image is 23 × 23 nm2 , taken at a sample bias of −1.5 V, and tunneling current of 2.0 nA. (d) A schematic model of the Cu(001)–c(2 × 2)N surface including a ml [170]. Dark (bright) circles are Cu (N) atoms. (e) Gridlines of the c(2 × 2) Cu lattice are superimposed on the STM image shown in (b)

the Cu grid in our sample, the lattice on the Cu intersections surrounded by the four mls is compressed, and the lattice constant can be decreased at least a few percentage from wide Cu surface [174, 176]. The present results show the same trend of the dissociation barrier modiﬁed by the lattice distortion as those of the molecular beam measurements.

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Fig. 2.49. Successive STM images for two narrow areas on the surface shown in Fig. 2.47. The numbers at the lower-left corners show the times in minutes when the scanning of the surface started. The image sizes are (a) 8.0 × 10.4 nm2 and (b) 5.6 × 5.6 nm2

Changes in the dissociation barrier and the desorption energy at the transition state are estimated for the precursor-mediated dissociative adsorption of oxygen molecules on Cu(001) surface as a function of the lattice constant using density functional calculations [194]. The barrier height increases and the desorption energy decreases with the decrease of the lattice constant. This general trend is ascribed to the downward shift of d-bands and the increase in the width of the d-bands upon the lattice compression. The above experimental results on clean Cu(001) surface agree with this picture although the direct dissociative adsorption is involved. In Fig. 2.49, two series of the successive STM images are shown to demonstrate the motion of oxygen adsorbates. By comparing these two series, we can see that those on the intersections surrounded by four mls (Fig. 2.49b) move more rapidly than those on wide areas of the Cu grid (Fig. 2.49a). In the former case, the oxygen adsorbates are sometimes imaged as depressed scratches. This is attributed their motion, and prevents us to estimate the density of the adsorbed oxygen quantitatively. The result indicates that the diﬀusion barrier for the oxygen adsorbates at compressed surface is smaller than that at less-strained surface. 2.4.7 Summary Our work conﬁrms that the surface stress on a Cu(001)–(2×2)N surface plays an important role in nanopattern formation and fundamental processes such as the growth of nanoislands and dissociative adsorption. Our results on vicinal surfaces may provide a good example of the use of step alignment to modify nanopatterns. It has been demonstrated that periodic nanoisland array is formed through multistep processes during the growth, which are signiﬁcantly

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dependent on the inhomogeneous strain distribution on this surface. We also demonstrate that dissociative adsorption of oxygen can be controlled by lattice deformation. Microscopic understanding of these phenomena would be a key factor in the future nanotechnology, utilizing inhomogeneous lattice strain on a solid surface. Acknowledgments One of the authors (S.O.) would like to thank Prof. F. Komori, Dr. K. Nakatsuji, and Mr. K. Yagyuu for collaboration of the studies on Cu(001)–c(2×2)N surfaces, and also for valuable discussions with Prof. K. Tanaka, Dr. M. Yamada, and Dr. Y. Yoshimoto.

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127. M.C. Flowers, N.B.H. Jonathan, Y. Liu, A. Morris, Surf. Sci. 343, 133 (1995) 128. V. Palermo, D. Jones, Appl. Surf. Sci. 226, 191 (2004) 129. M. Scarselli, P. Castrucci, P.D. Szkutnik, A. Sgarlata, M. De Crescenzi, Surf. Sci. 559, 223 (2004) 130. H. Watanabe, K. Fujita, M. Ichikawa, Appl. Phys. Lett. 70, 1095 (1997) 131. A. Ichimiya, Y. Tanaka, K. Hayashi, Surf. Sci. 386, 182 (1997) 132. M. Fouchier, J.J. Boland, Phys. Rev. B 57, 8997 (1989) 133. N. Taoka, A. Sakai, S. Mochizuki, O. Nakatsuka, M. Ogawa, S. Zaima, Thin Solid Films 508, 147 (2006) 134. A. Sakai, S. Zaima, Oyo-buturi (Jpn. Soc. Appl. Phys.) 75, 426 (2006) (in Japanese) and references therein 135. R. Geilenkeuser, K. Wieczorek, T. Mantei, F. Graetsch, L. Herrmann, J.-O. Weidner, Mater. Sci. Eng. B 118, 50 (2005) 136. Y. Ishikawa, Rev. Phys. Chem. Jpn. 16, 83 (1942); ibid. 16, 117 (1942) (in Japanese); Proc. Imp. Acad. Tokyo 18, 246 (1942); ibid. 19, 380 (1943) (in Japanese). 137. K. Mase, A. Nambu, Shinku (J. Vac. Soc. Jpn.) 49, 610 (2006) (in Japanese), for the outline of [136] 138. T. Iimori, K. Hattori, K. Shudo, T. Iwaki, M. Ueta, F. Komori, Appl. Surf. Sci. 130–132, 90 (1998) 139. X.H. Chen, J.C. Polanyi, D. Rogers, Surf. Sci. 376, 77 (1997) 140. H. Amasuga, M. Nakamura, Y. Mera, K. Maeda, Appl. Surf. Sci. 197–198, 577 (2002) 141. T. Kirimura, K. Shudo, Y. Hayashi, Y. Tanaka, T. Ishikawa, M. Tanaka, Phys. Rev. B 73, 085309 (2006) 142. F.-J. Kao, D.G. Busch, D. Gomes da Costa, W. Ho, Phys. Rev. Lett. 70, 4098 (1993); F.M. Zimmermann, W. Ho, Surf. Sci. Rep. 22, 127 (1995) 143. A. Prybyla, T.F. Heinz, J.A. Misewich, M.M.T. Loy, J.H. Glownia, Phys. Rev. Lett. 64, 1537 (1990); J.A. Prybyla, H.W.K. Tom, G.D. Aumiller, ibid. 68, 503 (1992) 144. J.A. Misewich, T.F. Heinz, D.M. Newns, Phys. Rev. Lett. 68, 3737 (1992) 145. H. Aizawa, S. Tsuneyuki, Surf. Sci. 377, 610 (1997) 146. J.W. Gadzuk, Chem. Phys. 251, 87 (2000) 147. R.R. Cavanagh, D.S. King, J.C. Stephenson, T.F. Heinz, J. Phys. Chem. 97, 786 (1993) 148. M. Head-Gordon, J.C. Tully, Phys. Rev. B 46, 1853 (1992) 149. J.R. Goldman, J.A. Prybyla, Phys. Rev. Lett. 72, 1364 (1994); Semicond. Sci. Technol. 9, 694 (1994) 150. J. bokor, R. Storz, F.F. Freeman, P.H. Backsbaum, Phys. Rev. Lett. 57, 881 (1986); N.J. Halas, J. Bokor, Phys. Rev. Lett. 62, 1679 (1989) 151. E. Kobayashi, K. Mase, A. Nambu, J. Seo, S. Tanaka, T. Kakiuchi, K. Okudaira, S. Nagaoka, M. Tanaka, J. Phys. Condens. Matter 18, S1389 (2006) 152. K.S. Nakayama, E. Graugnard, J.H. Weaver, Phys. Rev. Lett. 89, 266106 (2002) 153. Y. Nakamura, Y. Mera, K. Maeda, Surf. Sci. 487, 127 (2001) 154. D. Menzel, R. Gomer, J. Chem. Phys. 41, 3311 (1964); P.A. Redhead, Can. J. Phys. 42, 886 (1964) 155. K. Shudo, M. Tanaka, Shinku (J. Vac. Soc. Jpn.) 49, 22 (2006) (in Japanese)

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156. T. Komeda, Prog. Surf. Sci. 78, 41 (2005); W. Ho, J. Chem. Phys. 117, 11033 (2002) 157. K. Stokbro, C. Thirstrup, M. Sakurai, U. Quaade, B.Y.-K. Hu, F.-P. Murano, F. Grey, Phys. Rev. Lett. 80, 2618 (1998) 158. K. Stokbro, Surf. Sci. 429, 327 (1999) 159. T.-C. Shen, C. Wang, G.C. Abeln, J.R. Tucker, J.W. Lyding, Ph. Avouris, R.E. Walkup, Science 268, 1590 (1995) 160. M. Bonn, S. Funk, Ch. Hess, D.N. Denzler, C. Stampﬂ, M. Scheﬄer, M. Wolf, G. Ertl, Science 285, 1042 (1999) 161. Y. Sugita, H. Horiike, J. Kanasaki, K. Tanimura, Surf. Sci. 593, 168 (2005) and references therein 162. S.G. Tikhodeev, H. Ueba, Surf. Sci. 587, 25 (2005) 163. Example as pioneer works: D. Huang, H. Uchida, M. Aono, J. Vac. Sci. Technol. B 12, 2429 (1994); H. Uchida, D. Huang, F. Grey, M. Aono, Phys. Rev. Lett. 70, 2040 (1993) 164. K. Shudo, Y. Koike, Y. Owa, M. Koma, S. Ohno, M. Tanaka, J. Phys. Condens. Matter 19, 096010 (2007) 165. M. Mauerer, I.L. Shumay, W. Berthold, U. H¨ofer, Phys. Rev. B 73, 245305 (2006); and references therein for example for the electron dynamics of silicon surface 166. Recently, the current status of laser chemistry on variety of surfaces are reviewed in special issue of Chem. Rev. 106(10), 4113 (2006) 167. G. Renaud, R. Lazzari, C. Revenant, A. Barbier, M. Noblet, O. Ulrich, F. Leroy, J. Jupille, Y. Borensztein, C.R. Henry, J.-P. Deville, F. Scheurer, J. ManeMane, O. Fruchart, Science 300 (2003) 1416. 168. J.H.G. Owen, K. Miki, D.R. Bowler, J. Mater. Sci. 41, 4568 (2006) 169. H. Brune, Surf. Sci. Rep. 31, 121 (1998) 170. F.M. Leibsle, C.F.J. Flipse, A.W. Robinson, Phys. Rev. B 47, 15865 (1993) 171. F.M. Leibsle, S.S. Dhesi, S.D. Barrett, A.W. Robinson, Surf. Sci. 317, 309 (1994) 172. S.M. Driver, D.P. Woodruﬀ, Surf. Sci. 492, 11 (2001) 173. S. Ohno, K. Yagyuu, K. Nakatsuji, F. Komori, Jpn. J. Appl. Phys. 41, L1243 (2002) 174. S. Ohno, K. Yagyuu, K. Nakatsuji, F. Komori, Surf. Sci. 547, L871 (2003) 175. M. Sotto, B. Croset, Surf. Sci. 461, 78 (2000) 176. C. Cohen, H. Ellmer, J.M. Guigner, A. L’Hoir, G. Pr´evot, D. Schmaus, M. Sotto, Surf. Sci. 490, 336 (2001) 177. B. Croset, Y. Girard, G. Pr´evot, M. Sotto, Y. Garreau, R. Pinchaux, M. Sauvage-simkin, Phys. Rev. Lett. 88, 056103 (2002) 178. Y. Yoshimoto, S. Tsuneyuki, Surf. Sci. 514, 200 (2002) 179. D. Sekiba, K. Nakatsuji, Y. Yoshimoto, F. Komori, Phys. Rev. Lett. 94, 016808 (2005) 180. S. Ohno, K. Nakatsuji, F. Komori, Jpn. J. Appl. Phys. 42, 4701 (2003) 181. S. Ohno, K. Nakatsuji, F. Komori, Surf. Sci. 523, 189 (2003) 182. S. Ohno, K. Nakatsuji, F. Komori, Surf. Sci. 493, 539 (2001) 183. K. Mukai, Y. Matsumoto, K. Tanaka, F. Komori, Surf. Sci. 450, 44 (2000) 184. Y. Matsumoto, K. Tanaka, J. Appl. Phys. 37, L154 (1998) 185. F. Komori, S. Ohno, K. Nakatsuji, J. Phys. Condens. Matter 14, 8177 (2002) 186. P. Finetti, V.R. Dhanak, C. Binns, K.W. Edmonds, S.H. Baker, S. D’Addato, J. Electron. Spectrosc. Relat. Phenom. 114–116, 251 (2001)

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3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials J. Takeda

3.1 Introduction Observation of transient behavior of excited states at diﬀerent frequencies is essential and of great importance in studying ultrafast photochemical and photophysical processes, such as photosynthesis, energy transfer, photochromic reaction, and carrier dynamics, of nano- and micromaterials. To investigate such phenomena, femtosecond laser spectroscopy has been extensively utilized for the last two decades [1]. In this review, we pay attention to three types of new femtosecond laser spectroscopy: optical Kerr gate (OKG) luminescence spectroscopy, transient grating spectroscopy combined with a phase mask, and real-time pump-probe imaging spectroscopy, which are applicable to the observation of transient behavior of excited states and the evaluation of optical functional properties of nano- and micromaterials.

3.2 Femtosecond Optical Kerr Gate Luminescence Spectroscopy 3.2.1 Time-Resolved Luminescence Spectroscopy: Up-Conversion Technique vs. Opical Kerr Gate Method When samples emit photons as luminescence (or ﬂuorescence), measurements of time-resolved luminescence are often used for characterizing the dynamics of photochemical and photophysical processes of nano- and micromaterials in picosecond and femtosecond time domains [2, 3]. In many cases, kinetic information recorded at one or a few selected emission wavelengths might be suﬃcient, but for studying complex or successive photochemical reactions and relaxation processes, complete time- and frequency-resolved spectra are required.

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Several methods are available for measuring time-resolved luminescence spectra such as time correlated single photon counting, luminescence upconversion technique, streak camera and OKG method. These methods can be divided into two categories, depending on whether single-frequency or multifrequency data are collected [4]. In the case of single-frequency detection, the luminescence up-conversion technique is most commonly utilized in the picosecond and femtosecond time regimes since Shar and coworkers developed this technique [5–7]. Response times of less than 200 fs can be easily achieved [8–13], making the luminescence up-conversion technique the best option when temporal resolution is of primary importance. Moreover, since the wavelength and direction of the sum frequency output are far away from those of the detected luminescence and gating laser pulses, the luminescence up-conversion technique is basically a background-free detection. Time evolution of luminescence at a given frequency is thus sensitively obtained. In this technique, however, the phase-matching angle of nonlinear crystal (NLC) to generate the sum frequency output has to be varied with changing detected luminescence frequency. This brings diﬃculties to correct the relative luminescence intensity for diﬀerent frequencies and to determine the time origin for the delay of the luminescence. As the results, this technique makes it diﬃcult to obtain time-resolved luminescence spectra, which could provide us important and fruitful information on reaction dynamics and relaxation processes of materials. To overcome this diﬃculty, Gustavsson et al. [14] developed a luminescence up-conversion instrument capable of automated scanning over frequencies at ﬁxed delay times. By computer controlling of the phasematching angle of the NLC, the monochromator and the optical delay stage, they showed the ability to collect time-resolved luminescence spectra with 200 fs temporal resolution. Recently, true multifrequency detection of luminescence with a ∼100 nm spectral range was also achieved using broadband up-conversion approach [15]. Although these demonstrations were noteworthy, the limited spectral range and diﬃculty of the experiments might preclude the use of these up-conversion techniques as general-use methods to collect multifrequency data. Inherently better suited to multifrequency collection of time-resolved luminescence spectra is the OKG luminescence spectroscopy [16, 17]. The OKG method oﬀers a good temporal resolution comparable to that of the luminescence up-conversion technique coupled with a beneﬁt of wide spectral coverage. Like luminescence up-conversion technique, luminescence excited by ultrafast pump laser pulses is gated by variably delayed gating pulses. The gating pulse induces a transient birefringence in a Kerr medium, which enables momentary passage of luminescence otherwise blocked by crossed polarizers. In the OKG method, liquids or solutions such as CS2 are generally used as the Kerr media. Since the time response of the optical Kerr eﬀect for liquids/solutions is governed by rotational relaxation of molecules, the typical time resolution of the OKG method is limited to 1–2 ps so far [3]. Moreover,

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the OKG method contains substantial backgrounds coming from the scattered light and luminescence itself which survive against the crossed polarizers. Hulin et al. obtained time resolution as short as 0.5 ps in visible light region by using benzene solution as the Kerr medium [18]. However, the temporal response near UV light region was broader (∼1 ps) due to the large group velocity dispersion (GVD) between the diﬀerent spectral components of probe pulses, and the contrast ratio available was small because of the component of molecular rotational relaxation with rather long recovery time in the medium. For the above reasons, it seems diﬃcult to observe the time-resolved luminescence spectra of materials in wide spectral region with a subpicosecond time resolution by the OKG method. High refractive solid glasses are known to have a high third-order nonlinear susceptibility and have an instantaneous response because of the absence of the rotational relaxation, implying that they are good candidates as the Kerr media to investigate ultrafast photochemical and photophysical processes of materials. In this point of view, a few groups including us tried to measure the time-resolved luminescence spectra of diﬀerent materials in femtosecond time domain by use of solid glasses as the Kerr media [19–22]. In Sect. 3.2.2, we present a time-resolved luminescence spectroscopy by use of the OKG method with a subpicosecond time resolution using solid glasses as the Kerr media. 3.2.2 Femtosecond OKG Method: Experimental Setup and Results Figure 3.1 illustrates a typical schematic diagram of optical setup in the OKG method. Here, second harmonic (400 nm) and fundamental laser pulses (800 nm) from a Ti:sapphire regenerative ampliﬁer laser system with a 100 fs pulse duration and a 1 kHz repetition rate were used as excitation and gating pulses, respectively. The fundamental beams from the regenerative ampliﬁer were divided into two beams at a beam splitter (BS); one is used as the excitation pulse of sample after the second harmonic generation at a NLC and the other as the gating laser pulse to rotate the polarization of luminescence at a Kerr medium. The Kerr medium was placed between crossed polarizers (P). To reduce the GVD of the optical setup, ﬁlm polarizers with an extinction ratio of 103 –104 in visible light region and oﬀ-axis paraboloidal mirrors (OPM) were used. The luminescence or scattered light from a sample was collected by the OPMs. The use of the OPMs instead of lenses eliminates an aberration among diﬀerent wavelengths of detected luminescence/scattered light at the Kerr medium. The luminescence/scattered light from the sample after passing through the ﬁrst polarizer was focused into the Kerr medium together with gating pulses, which were variably delayed by a translation stage with a minimum step of 1 µm. The spot sizes of gating pulses and luminescence/scattered light at the Kerr medium were arranged at ∼300×300 and ∼200×200 µm2 , respectively. The time-resolved luminescence then passed through the second polarizer and was focused into an entrance slit of a monochromator with a

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Fig. 3.1. Typical schematic diagram of optical setup in the OKG method

CCD detector cooled by liquid nitrogen and a photomultiplier (PM). These detectors can be easily switched on and oﬀ by moving an internal mirror of the monochromator. Time-resolved luminescence spectra were detected by the CCD detector, while time evolution of third-order correlation function of the scattered excitation pulses was detected by the PM with a lock-in detection. To obtain a higher Kerr eﬃciency and a better time resolution, we examined two types of solid glasses with a thickness of 0.25, 0.5, and 1 mm as the Kerr media; one is a quartz plate and the other is a SFL-6 (SCHOTT GLASS) plate with nonlinear refractive indexes of 1.1 × 10−13 and 9.9 × 10−13 esu [23], respectively. The quartz is transparence above 200 nm, while the SFL-6 above 400 nm. The both Kerr media are therefore suited for time-resolved luminescence measurements in visible light region. The Kerr eﬃciency and the overall time resolution of our OKG spectroscopic system were evaluated by the Kerr signal (third-order correlation function) of the scattered excitation pulses (400 nm) from a diﬀusive glass plate placed at the sample position. The Kerr eﬃciency is deﬁned by the ratio of intensities of the Kerr signal relative to the total scattered light passed through the polarizers whose polarization direction is parallel. Figure 3.2 shows the Kerr eﬃciency of the quartz and SFL-6 plates with a thickness of 0.5 mm as a function of gate intensity in a log–log plot. Solid circles and triangles show the observed Kerr eﬃciency of the quartz and SFL-6, respectively. The Kerr eﬃciency follows a xth power law of the gate intensity as shown by the dashed lines; the value of x for the quartz is 2.4 and that for

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Kerr Efficiency ( % )

100

10

1 quartz SFL-6 0.1 10

50

500

100 2

Gate Intensity ( mJ/cm ) Fig. 3.2. Log–log plot of the Kerr eﬃciency of quartz and SFL-6 with a thickness of 0.5 mm vs. gate intensity [20]

the SFL-6 is 2.0. For same gate intensity, the Kerr eﬃciency of the SFL-6 is 60–80 times larger than that of the quartz. For all the measurements, generation of a white-light continuum or presence of any higher-order nonlinear eﬀect at the Kerr medium was not observed. A strong electric ﬁeld by the gating pulses induces an optical anisotropy and consequently a transient birefringence in an isotropic Kerr medium. The refractive index n is given by n = n0 + n2 E 2 ,

(3.1)

where n0 is the linear refractive index, n2 the nonlinear refractive index, and E the applied optical electric ﬁeld. The phase shift Φ induced by the birefringence depends on the applied optical electric ﬁeld Φ=

2πl n2 E 2 , λ

(3.2)

where l is the path length traveled in the Kerr medium by the light of a wavelength λ. The observed transmitted intensity (Kerr signal) IK then depends on the induced phase shift Φ according to 2 Φ 2 Φ 2 ∼ I0 ∝ n22 IG . (3.3) IK = I0 sin 2 2 Here, IG is the gate intensity and proportional to the square of the applied electric ﬁeld. For small values of Φ the Kerr signal is proportional to the square of the phase shift Φ, that is, the square of IG and n2 [24]. For larger values

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SFL-6 0.5 mm 620 fs

0 quartz 0.5 mm 230 fs 0 -1000

-500

0

500

1000

Delay Time ( fs )

Fig. 3.3. Time response of third-order correlation function of the scattered excitation pulses (400 nm) for quartz and SFL-6 plates. Dots show the experimental data and solid curves indicate the Gaussian ﬁts. The fwhms of the correlation function for the quartz and SFL-6 are 230 and 620 fs, respectively [20]

of n2 , IG , and l, a stronger Kerr signal is obtained. As shown in Fig. 3.2, the observed Kerr eﬃciency increases almost quadratically with the gate intensity IG , as indicated by (3.3). Because the nonlinear refractive index of the SFL-6 is nine times larger than that of the quartz, the Kerr eﬃciency of the SFL-6 is about 80 times higher than that of the quartz for the same gate intensity. Figure 3.3 shows third-order correlation function of the scattered excitation pulses as a function of delay time for the two kinds of Kerr media, the quartz and SFL-6. The gate intensity was arranged to have the same Kerr eﬃciency of 5%. The third-order correlation function measured by the OKG method is deﬁned by ∞ 2 G(3) (τ ) = IP (t)IG (t + τ )dt. (3.4) −∞

Here, IG (t) and IP (t) are the intensities of the strong gating pulse and the weak probing pulse (scattered light), respectively. Assuming that the correlation function G(3) (τ ) has a Gaussian shape, the full width of half maximum (fwhm) of the correlation function is 620 fs for the SFL-6 and 230 fs for the quartz. The time response function of the quartz is much narrower than that of the SFL-6. The SFL-6 slightly absorbs the scattered excitation pulses in the wavelength region of 400 nm, implying that the GVD of the SFL-6 around 400 nm is large. The blue part of the scattered light propagates slower than the red part of the scattered light through the SFL-6, and consequently the time proﬁle of the Kerr signal might become broader.

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The time response function as well as the Kerr eﬃciency depends on the thickness of the Kerr medium. The Kerr eﬃciency increases with increasing the path length of the Kerr medium by (3.2) and (3.3). On the other hand, the time response function becomes broader with increasing the path length due to the GVD of the Kerr medium. In the case of the quartz, for instance, the Kerr eﬃciency and the time response were 10% and 350 fs (1 mm), 5% and 230 fs (0.5 mm), and 3% and 190 fs (0.25 mm) for the same gate intensity of 200 mJ cm−2 . We obtained the Kerr eﬃciency of 5–10% and the width of the time response function of ∼250 fs when we used a quartz plate with a thickness of 0.5 mm as the Kerr medium. In the luminescence up-conversion technique, the eﬃciency of the sum frequency generation and time resolution of the system are typically ∼10% and ∼200 fs, respectively, when one uses the fundamental and second harmonic pulses as the gating and the excitation pulses, respectively, from a femtosecond laser system having a pulse duration of 100 fs [8,10]. We therefore conclude that a quartz plate with a thickness of 0.5 mm is a reasonable choice as the Kerr medium, and that the obtained Kerr eﬃciency and time resolution are enough for measurements of ultrafast carrier dynamics and relaxation processes of materials. To demonstrate performance of the OKG method, we ﬁrst measured the time-resolved luminescence spectra of β-carotene, which has two energetically low-lying singlet states; one is related to the 21 A− g (S1 ) state, which is dipole forbidden from the ground state by the parity conservation, and the other is related to the 11 Bu+ (S2 ) state, which is responsible for strong absorption in visible light region [25, 26]. The optical transition to the S2 state induces an ultrafast internal conversion to the S1 state, followed by relaxation to the ground state. The luminescence quantum yield is therefore very low (10−6 –10−4 ) [27, 28] and the lifetime of the S2 state is very short (190–250 fs) [29–31]. The purchased β-carotene was dissolved in n-hexane solution with a concentration of 2.5×10−5 M. The solution was kept under N2 saturated conditions and circulated by a peristaltic pump through a ﬂow cell with a 1 mm path length during the time-resolved luminescence measurements. For the time-resolved luminescence measurements, the second harmonic (400 nm) and the fundamental (800 nm) pulses of the Ti:sapphire regenerative ampliﬁer were used as the excitation pulses of a sample and the gating pulses, respectively. The origin of time axis (t = 0) is deﬁned by the time when the intensity of the scattered excitation pulse becomes maximum. The typical accumulation time of the CCD detector to take a time-resolved luminescence spectrum was 1–2 min. The GVD of our optical setup was evaluated by using a white-light continuum generated at n-hexane/water in a ﬂow cell which was placed at the sample position instead of the sample. The chirp of the timeresolved luminescence spectrum was then corrected by a software procedure on a microcomputer. When the time-integrated luminescence spectrum was

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104

- 340 fs

160 fs

- 160 fs

360 fs

- 100 fs

660 fs

0 fs

860 fs

0

0

0

0 450

500

550

600 450

500

550

600

Wavelength ( nm ) Fig. 3.4. Time-resolved luminescence spectra of β-carotene in n-hexane solution for diﬀerent delay times at room temperature [20]

measured, two pieces of the polarizers were set to have a parallel polarization and the gating pulses were blocked. Figure 3.4 shows time-resolved luminescence spectra of β-carotene for different delay times at room temperature measured by the OKG method. The solid curves are added as visual guides to help clarify the peak position and the fwhm of the spectra. The luminescence spectrum at t = 0 has a shoulder at ∼460 nm, which deviates from the solid curve as shown by a solid arrow. This comes from the scattered light of the excitation pulse. The peak position (∼515 nm) and the fwhm of the spectra (∼80 nm) are almost independent of the delay time. This indicates that the observed time-resolved luminescence of β-carotene comes from a single origin. The absence of the dynamic Stokes shift suggests that the intramolecular relaxation takes place within the time resolution of the system. Figure 3.5 shows time evolution of the luminescence of β-carotene measured by the OKG method. Closed circles, triangles, and squares indicate the observed data points at diﬀerent detected wavelengths 480, 520, and 560 nm, respectively. The instrumental response function obtained from the time proﬁle of the third-order correlation function is also shown by a dashed line. A solid curve is the best-ﬁt obtained by a single exponential decay of

Luminescence Intensity ( a. u. )

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instr. response 480 nm 520 nm 560 nm fit ( 210 fs )

1.0

0.5

0.0 -1000

0

1000

2000

Delay Time ( fs ) Fig. 3.5. Time evolution of luminescence of β-carotene. Closed circles, triangles and squares indicate the observed data points at diﬀerent detected wavelengths 480, 520, and 560 nm, respectively. The instrumental response function is also shown by a dashed line [20]

210 ± 20 fs convoluted with the instrumental response function. The lifetime of the S2 state was estimated to be 190∼250 fs by the transient absorption measurement [29] and the luminescence up-conversion technique [30]. The observed luminescence decay time of 210 fs is in good agreement with the estimated value from the previous measurements. We clearly observed the time-resolved luminescence spectra of β-carotene in femtosecond time regime, which has a very low luminescence quantum yield and an ultrafast luminescence decay time. This strongly shows that the OKG method is a powerful spectroscopic tool to investigate ultrafast carrier dynamics and relaxation processes of materials. In fact, after the development of the femtosecond OKG method, many ultrafast transient phenomena such as ultrafast carrier dynamics in semiconductors [32–36], ultrafast relaxation processes in carbon nanotubes [37, 38] and so on [39] have been extensively studied by the femtosecond OKG method.

3.3 Femtosecond Transient Grating Spectroscopy Combined with a Phase Mask 3.3.1 Principle of Transient Grating Spectroscopy Not only lifetime but also spatial diﬀusion of excited states is very important information to elucidate ultrafast relaxation processes and dynamics of nano- and micromaterials. Transient grating spectroscopy is a powerful spectroscopic tool to simultaneously determine the lifetime and diﬀusion rate of

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Fig. 3.6. Transient grating spectroscopy with three pulses. (a) general beam conﬁguration. (b) Producing grating ﬁelds with colinear and crosslinear polarizations of pump pulses [40]

the photoexcited carriers of materials. The principle of the transient grating spectroscopy is schematically sketched in Fig. 3.6 [40]. The ﬁrst two pump pulses with wavevectors k1 and k2 come to overlap temporally and spatially on the sample and induce a nonlinear grating in the sample. The produced grating can then be detected at variable delay time τ , by diﬀraction of the probe pulse with wavevector k3 , which appears in the background-free direction ∓k1 ∓ k2 + k3 . The character of the produced grating changes by the polarization combination of the pump pulse pair. In the case of colinear polarizations, the pump

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pulses interfere and generate a spatially periodic carrier concentration grating with a fringe spacing Λ=

λ 2π = , |k2 − k1 | 2 sin(θ/2)

(3.5)

where Λ and θ are the wavelength and crossing angle of the pump beams, respectively. In crosslinear case of pump pair polarizations, the laser electric ﬁelds do not interfere, so that the intensity in the crossing volume varies smoothly as the Gaussian transverse mode of the laser. Instead of no sinusoidal intensity modulation, the polarization of the combined electric ﬁeld is spatially modulated from right-circular polarization to left-circular polarization and vice versa if both pump pulses have the same intensities. For example, in GaAs QWs, the lowest heavy-hole (hh) exciton level is degenerate fourfold whose angular momenta J are ±1 and ±2 [41]. The rightcircularly polarized pump pulse excites excitons to the state with J = +1 and the left-circularly polarized pump to J = −1. The states with J = ±2 are the optically inaccessible states, the so-called dark states. After all, the exciton spin concentration grating is produced, where the mixing ratio of up-spin and down-spin excitons changes with a spatial periodicity. The grating spacing should be identical to that in exciton concentration grating given by (3.5). In transient grating measurement, we can therefore deduce the spin relaxation rates by probing the decay of this spin grating, while can obtain the recombination lifetime of photoexcited carriers from the decay of the carrier concentration grating. 3.3.2 Transient Grating Spectroscopy Combined with a Phase Mask: Experimental Setup and Results To simplify the optical arrangement, “phase mask,” a transmission type diﬀractive optics, to generate two pump pulses with equal intensities (+1 and −1 order diﬀracted beams) from a single pump pulse is employed [40, 42]. As shown in Fig. 3.7, the phase mask is made by a quartz substrate coated by dark ﬁeld chrome with several speciﬁed grating patterns with diﬀerent groove/space width d. Single incident pump pulse on the phase mask is diﬀracted into many orders and they interfere with each other. Only ±1 order diﬀracted beams are used as the pump pair to generate the grating at the sample, while the other orders are blocked. The ±1 order diﬀracted pulses focused on the sample creates an interference pattern. The optically delayed probe pulse, which is spatially overlapped with the pump pair at the sample, reads out the decay of the produced grating as the diﬀracted signal. By introducing the phase mask, the optical setup has the following advantages (1) spatial and temporal overlapping at the sample, and the equal intensities of the two pump pulses are automatically conﬁrmed and (2) the interference pattern spacing can be easily tuned without optical realignment by changing the grating patterns of the phase mask.

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phase mask sample

+1 pump beams

diffracted probe beam

-1

to detector Λ

variably delayed probe beam

Diffraction Intensity (a. u. )

Fig. 3.7. Transient grating spectroscopy combined with a phase mask

100

1 Λ = 3.2 µm

0 0

50

100

150

10-1 10

-2

10

-3

0

Delay Time ( ps )

50

100

150

Delay Time ( ps )

Fig. 3.8. Typical diﬀracted probe signal of ZnO epilayers as a function of delay time at room temperature in liner and logarithmic scales. Dotted lines show the ﬁtting ones with a single exponential decay [43]

As a demonstration, here, we show the experimental results on ultrafast carrier dynamics of electron–hole plasma (EHP) in ZnO epitaxial thin ﬁlms under high excitation density [43]. Figure 3.8 shows typical observed diﬀracted probe signal as a function of delay time in linear (left side) and logarithmic scales (right side). The signal was obtained by using the produced grating fringe spacing Λ = 3.2 µm. After an instantaneous electronic response at t = 0, the signal decreases exponentially with time as shown by dotted lines. The lifetime τG of the produced grating is given by the following equation: 1 4π2 D 1 = + , τG τr Λ2

(3.6)

where τr is the carrier lifetime and D is the carrier diﬀusion coeﬃcient. The ﬁrst term in the right hand of the above (3.6) indicates reduction of the grating amplitude, that is reduction of the carrier concentration, due to recombination of electrons with holes in the EHP state. On the other hand, the second term causes dissipation of the produced grating by the carrier diﬀusion. Both terms

3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials

109

0.04

1/

G

( ps-1 )

0.08

r = 23.8 ps D = 10.8 cm2/s

0

0

4

8 2

4π /

2

12

16

-2

( µm )

Fig. 3.9. Decay rate of the produced grating as a function of grating fringe spacing [43]

contribute to reduction of the grating contrast, leading to the decrease of the diﬀracted probe signal. By performing several experiments of diﬀerent grating pitches, the lifetime and diﬀusion coeﬃcient of the carriers can be obtained separately. In our experiments, tuning of the grating pitch was achieved easily by using a diﬀerent pattern of the phase mask without any optical realignment. Figure 3.9 shows the decay rate 1/τG as a function of the grating pitch. Using (3.6), the lifetime and diﬀusion coeﬃcient of the carriers in the EHP state at room temperature were determined to be 23.8 ps and 10.8 cm2 s−1 , respectively, as shown by a dashed line. The estimated lifetime is in good agreement with the previous values obtained by pump-probe technique [44] and time-resolved OKG luminescence method [32, 33] in which the carrier diﬀusion was neglected. This result implies that the carrier diﬀusion does not aﬀect the carrier relaxation process in a short carrier lifetime of a few tens picosecond in ZnO thin ﬁlms.

3.4 Femtosecond Real-Time Pump-Probe Imaging Spectroscopy 3.4.1 Principle of Real-Time Pump-Probe Imaging Spectroscopy Frequency-resolved ultrafast transient signals reveal important information about photochemical and photophysical properties of nano- and micromaterials. To obtain ultrafast transient signals, femtosecond pump-probe transient absorption spectroscopy has been generally utilized so far. In this technique, a pump pulse induces transient signals, and a variably delayed probe pulse probes the transient signals. The pump-probe sequence is thus repeated many times to cover whole time and spectral regions of interest. Consequently,

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photochemical reaction dynamics and relaxation processes of some organic and most biological materials are less explored since buildup of reaction products or structural deterioration after many repetitions of intense laser pulses strongly masks to observe the real transient signals. Although it is possible to study irreversible ultrafast processes in liquid states by ﬂowing a sample fast enough that each pair of the pump and probe pulses interrogates a new portion of the sample, such an approach is not feasible for organic and biological samples, where large sample quantities are not readily available, and the samples are easily photodegraded. To overcome these limitations, single-shot femtosecond spectroscopy has been proposed [45–47]. Although the single-shot techniques can record multiple temporal data points from a single probe pulse at a given frequency, they still require many repetitions to obtain full spectral information, which is greatly important to understand ultrafast relaxation and dynamics of materials. Very recently, we have developed real-time pump-probe imaging spectroscopy, which enables us to simultaneously map temporal and spectral data points with femtosecond time resolution, and to visualize ultrafast phenomena in real time [48–51]. In this section, we survey a new ultrafast spectroscopy, femtosecond real-time pump-probe imaging spectroscopy, and show that, using this method, one can readily map time–frequency 2D transient absorption image of biochemical materials having wide temporal and spectral ranges. The typical accumulation time becomes about two or three orders of magnitude shorter than that by the conventional pump-probe technique. Finally, we will demonstrate that the real-time pump-probe imaging spectroscopy is applicable to observation of solid-state photochemical reactions. Real-time pump-probe imaging spectroscopy is essentially similar to previously reported single-shot pump-probe technique [45] but uses a white-light continuum as the probe. Figure 3.10 shows schematic experimental apparatus for real-time pump-probe imaging spectroscopy implemented on a single-shot basis. The fundamental laser pulse from a Ti:sapphire regenerative ampliﬁer system with a repetition rate of 1 kHz, a center wavelength of 800 nm and a pulse duration of 100 fs is divided into two beams; one is used as the pump pulse after second-harmonic generation in a BBO (Type I) crystal, while the other is used to generate a white-light continuum by focusing it on a CaF2 thin plate. To avoid thermal heating, the CaF2 thin plate is constantly rotated during the measurements and resultantly a stable and intense white-light continuum is obtained in visible and UV light regions [52, 53]. The white-light continuum is also divided into two beams: one is utilized for probe pulse itself, while the other is used for reference one to normalize absorbance changes for the same spectral proﬁle. The pump beam is magniﬁed to ∼1.2 cm diameter, and then the edge of the beam is clipped by passing through a mask to make a square-shaped pump beam with a size of 6.5×6.5 mm2 and a spatially homogeneous intensity. The typical pump beam proﬁle obtained is illustrated in an inset of Fig. 3.10. The collimated pump and probe beams intersect with an angle of ∼ 21◦ and are linearly focused on a sample with cylindrical lenses

3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials

probe beam (white-light)

M

wavelength 1340 pixels

reference beam

reference beam

sample

2D CCD detector

delay

BS : beam splitter P : polarizer M : mirror

Intensity

pump beam ~ 6 mm

pump beam profile

1300 pixels

~6 mm

time

θ CaF2 plate

transient signal

slit monochromator with M' P

BS

P

111

Screen Delay time

Fig. 3.10. Experimental apparatus for real-time pump-probe imaging spectroscopy implemented on a single shot basis

and the pump beam is incident normal to the sample. Since the probe beam reaches diﬀerent portions of the sample at diﬀerent times, a time delay between pump and probe beams is spatially encoded across the sample. After passing through the sample, the probe beam is recollimated and linearly focused on an entrance slit of a monochromator coupled with a 2D CCD imaging array detector (1,340×1,300 pixels) and a shutter having a minimum time response of 8 ms. To remove scattered light from the excitation and fundamental laser pulses, we place appropriate bandpass ﬁlters in front of the monochromator. Temporal information of the probe beam is analyzed along the direction parallel to the slit, whereas the spectral information is recorded along the direction normal to the slit. Consequently, real-time 2D mapping of time- and frequency-resolved absorbance changes of materials are obtained. Under our experimental conditions, the time resolution per pixel is 12.5 fs and the whole mapping area per unit frame covers wide spectral and temporal ranges of 420–650 nm and ∼6 ps, respectively. The time resolution of the imaging spectroscopy strongly depends on the sample thickness, the interbeam angle and the laser pulse duration: for our experimental conditions with 1 mm sample thickness, the time resolution is estimated to be ∼300 fs [48, 54]. The actual time resolution of our imaging method is evaluated from a third-order correlation function for the probe beam measured by the OKG method [19, 20]. The sample is replaced by a quartz plate with a 1 mm thickness at the sample position. The two polarizers (P) in Fig. 3.10 are arranged in a crossed conﬁguration and the polarization of the pump beam is set to 45◦ against that of the probe beam. The time– frequency 2D image of the probe beam measured by the OKG method is shown in Fig. 3.11a. The time evolution of the Kerr signal (third-order correlation function) reproduced from the 2D image is also depicted in Fig. 3.11b. Judging from the fwhm of the Kerr signal, the time resolution of our imaging method is less than ∼400 fs in whole spectral range from 420 to 650 nm. The 2D image

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J. Takeda a)

b)

1300 pixels

Kerr Signal (a.u.)

= 480 nm 32 pixels = ~400 fs

0

400

1340 pixels

420 nm

650 nm

600 800 Pixel (12.5 fs / pixel )

Fig. 3.11. (a) Time–frequency 2D image of the probe beam measured by the optical Kerr gate method. (b) Temporal evolution of the Kerr signal at 480 nm reproduced from the 2D image of (a) [50]

of the Kerr signal also shows the chirp characterization due to the GVD of the probe beam as shown by dotted line in Fig. 3.11a. This 2D image is stored and used to correct the GVD of actual pump-probe imaging data. The spatial intensity proﬁle of the pump beam with a nearly homogeneous intensity is also recorded by a CCD detector and used to normalize that of the transient absorption change. 3.4.2 Experimental Demonstrations of Real-Time Pump-Probe Imaging Spectroscopy As a ﬁrst demonstration using this technique, we measured ultrafast internal conversion processes of β-carotene in solution, which plays an important role in the energy transfer during photosynthesis [55]. Before analyzing the spectral information with the monochromator, to ensure that the probed passed through the sample, we inserted a mirror M in front of the monochromator and imaged the probe beam on a 1×1 cm2 screen (see Fig. 3.10). This image contains information of the transient absorption of β-carotene. The image was taken by a digital camera with an exposure time of 1 s. Figure 3.12 shows the images of the transmitted probe beam with the pump beam oﬀ (a) and the pump beam on with diﬀerent delay times, 0 (b), 1 (c), and 2 ps (d). The delay between the pump and probe beams is tuned by a mechanical translation stage. In the image, the time-encoding direction is horizontal. When the pump beam is blocked, Fig. 3.12a shows a homogeneous color (orange) that reﬂects the transmission of the probe without the transient absorption of β-carotene. When the pump beam is on, on the other hand, Fig. 3.12b–d has a dark area that moves along the time-encoding direction with the delay. This dark area corresponds to the transient absorption of β-carotene, which attenuates the transmission intensity of the probe. This result shows that our technique successfully works to visualize the ultrafast transient absorption of materials with a wide temporal range of 5–6 ps in real-time.

3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials

(a)

(b)

(c)

(d)

pump beam off

pump beam on 0 ps

1 ps

2 ps

~5 ps

113

delay time

Fig. 3.12. Images of the probe beam transmitted through β-carotene solution with the pump beam oﬀ (a) and the pump beam on with diﬀerent delay times of 0 (b), 1 (c), and 2 ps (d)

0.15 0.0 − 0.15

Delay Time (ps)

Absorbance Change

0.3 10

5

0 − 0.3

450

500 550 600 Wavelength (nm)

650

Fig. 3.13. Time–frequency 2D image of transient absorbance changes of β-carotene in n-hexane solution with an accumulation of 20 laser shots [50]

Next, we remove the mirror M from the optical path and the image is cylindrically focused onto an entrance slit of the monochromator to measure the time–frequency 2D image of the transient absorption spectra of β-carotene in solution. The solution is kept under N2 saturated conditions and circulated by a peristaltic pump through a ﬂow cell with a 1 mm path length during measurements. Magic angle (54.7◦ ) polarization between pump and probe beams is used to remove any transient due to relaxation of anisotropy. The excitation power and area of the pump pulse at the sample are 20 µJ and 6.5×0.2 mm2 , respectively, resulting a photon density of ∼3×1015 photons cm−2 . Typical accumulation time for taking an image is less than 1 s per unit frame. Figure 3.13 shows the time–frequency 2D image of the transient absorbance changes of β-carotene in n-hexane solution. The intensity of the absorbance changes is indicated as contours. The accumulation time is only 20 ms (20 laser shots) per unit frame. The 2D image is obtained by adding several frames with diﬀerent delay times to cover the entire relaxation process of β-carotene after the GVD correction. The 2D image shows the instantaneous

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b)

a)

0.3

450 nm

− 0.3

550 nm

0.3

0 0

5 10 Delay Time (ps)

Absorbance Change

Absorbance Change

0

0

− 0.2 ps

0

0.4 ps

0

2 ps

0

10 ps

0

13 ps

400 450 500 550 600 650 Wavelength (nm)

Fig. 3.14. Temporal evolution of the absorbance change at 450 and 550 nm (a), and time-resolved absorbance change spectra with diﬀerent delay times (b) reproduced by slicing oﬀ the 2D images shown in Fig. 3.13

absorption bleaching of 11 A− g state at 420–500 nm and the transient absorption from 21 A− g to higher states at 500–620 nm with rise and decay times of ∼1 and ∼10 ps, respectively [56–61]. Figure 3.14 shows the time evolution of the absorbance changes (a) and the time-resolved absorbance change spectra with diﬀerent delay times (b) reproduced by slicing oﬀ the 2D images shown in Fig. 3.13. Although the spectra obtained by the accumulation of 20 laser shots (solid lines) are slightly noisy, not only the intensity but also the spectral shape is in quite good agreement with those obtained by the conventional pump-probe spectroscopy (dotted lines). This shows that the real-time pumpprobe imaging spectroscopy works well even with a very short accumulation time, implying that this method would be a powerful spectroscopic tool applicable to photochemical reaction dynamics in biological and organic materials which easily undergo photodeterioration after laser irradiation. In the case of photochemical reaction dynamics in solid-state, since a particular region of the sample is photoirradiated many times, the buildup of undesirable reaction products that cannot be ﬂowed away and removed considerably masks real transient signals. Therefore, the conventional pump-probe technique is not feasible for observation of photochemical reaction dynamics in solid-state organic and biological materials. On the contrary, here, we show that the imaging spectroscopy enables us to measure the real transient signals even in solid-state photochemical reaction dynamics against any backgrounds due to the undesirable reaction products. As a demonstration, we measure ultrafast excited-state dynamics of β-carotene dispersed in PMMA polymer ﬁlms with a concentration of 1.0×10−3 M, a thickness of 250 µm, and a size of 1×1 cm2 . The ﬁlm sample contains only 74 µg of β-carotene, whose quantity is two or three orders of magnitude less than that used typically for the conventional pump-probe technique.

3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials

115

0.0

OD

0.6

−0.2

PMMA n-hexane

0.4

∆ OD

0.2 0.0 300 400 500 Wavelength (nm)

−0.4 460 nm −0.6 0

2000

4000

6000

8000

10000

Number of laser shots

Fig. 3.15. Photodegradation of steady-state absorption of β-carotene in PMMA ﬁlm as a function of the number of irradiated pump laser shot. Steady-state absorption spectra of β-carotene in PMMA (solid line) and n-hexane solution (dashed line) are also shown in the inset

Before doing the imaging experiments, we evaluate the photodegradation of β-carotene in PMMA ﬁlm. Figure 3.15 shows the photodegradation of the steady-state absorption of β-carotene in PMMA ﬁlm as a function of the number of irradiated pump laser shot. The steady-state absorption spectra in PMMA (solid line) and n-hexane solution (dashed line) are also shown in the inset. Except for the ∼20 nm redshift and the spectral broadening, the steady-state absorption spectrum of β-carotene in PMMA ﬁlm looks essentially the same as that in n-hexane solution. After irradiation of 2,000 pump laser shots (2 s) with a photon density of 3×1015 photons cm−2 , the absorption at 460 nm, which corresponds to the optical transition from 11 A− g to 11 Bu+ state [56–61], disappears almost completely due to the photodegradation. After irradiation of 20 pump laser shots, on the other hands, since 95% of β-carotene in PMMA ﬁlm remains without photodegradation, we may be able to observe the ultrafast transient signals of β-carotene in PMMA ﬁlm. When performing the imaging experiments, we ﬁrst measure the time–frequency 2D image of the transient absorption of β-carotene in PMMA ﬁlm with ∆t = 0 at some area (strip (a)), which contains both real transient signals and undesirable ones due to photodegradation. Next, we measure the time–frequency 2D image with ∆t = −100 ps at the diﬀerent area from the strip (a) (strip (b)), which only contains the undesirable signals due to photodegradation. Here, ∆t is the time delay between pump and probe pulses. The imaging data obtained by subtracting the data at the strip (b) from those at the strip (a) leave only the real transient signals [62]. Figure 3.16 shows time–frequency 2D image of the transient absorbance change of β-carotene in PMMA ﬁlm with an accumulation of 20 laser shots obtained by the above procedure. The absorbance change is indicated by

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Fig. 3.16. Time–frequency 2D image of transient absorbance changes of β-carotene in PMMA ﬁlm with an accumulation of only 20 laser shots

a)

0

OD

0 0 0 0

0.2

PMMA n - hexane

− 0.2 ps

b) 570 nm 550 nm

0.1

0.4 ps 2 ps

0.0

10 ps − 0.1

13 ps

450

500

550

600

Wavelength (nm)

650

− 0.2

470 nm 450 nm 0

4 8 Delay Time ( ps)

12

Fig. 3.17. Time-resolved absorbance change spectra with diﬀerent delay times (a) and temporal evolution of the absorbance change (b) of β-carotene in PMMA ﬁlm (solid lines) and in n-hexane solution (dashed lines)

contours. The 2D image clearly shows the instantaneous absorption bleaching at 430–490 nm and the transient absorption at 520–600 nm with a fast rise time (see dotted lines in Fig. 3.16). Figure 3.17a, b shows the time-resolved absorbance change spectra with diﬀerent delay times and the time evolution of the absorbance changes of β-carotene in PMMA ﬁlm (solid lines), respectively, reproduced by slicing oﬀ the 2D images shown in Fig. 3.16. As comparison, the experimental data of β-carotene in n-hexane solution measured by the imaging spectroscopy with an accumulation of 20 laser shots are also shown by dashed lines in Fig. 3.17a, b. Except for the redshift of ∼20 nm

3 Ultrafast Laser Spectroscopy Applicable to Nano- and Micromaterials

117

in the time-resolved absorption spectra, which is also seen in the steady-state absorption spectra (see the inset of Fig. 3.15), the observed transient data of β-carotene in PMMA ﬁlm seem essentially the same as those of β-carotene in solution. Because our new imaging method is implemented on a single-shot basis and does not require many repetitions of pump-probe sequence, we can successfully image the time- and frequency-resolved transient absorption of β-carotene in polymer ﬁlms with a good signal to noise ratio. As far as we know, this is the ﬁrst observation of ultrafast excited-state dynamics of carotenoids in solid-state. In summary, we demonstrate a new scheme for real-time pump-probe imaging spectroscopy applicable to observation of photochemical reaction dynamics in organic and biological nano- and micromaterials. Because the imaging spectroscopy does not require irradiation of many laser shots and resultantly only small quantities of samples are needed, we can successfully map the 2D image of the transient absorbance change of β-carotene even in polymer ﬁlms. We believe that our new imaging spectroscopy will open the door for studying solid-state photochemical reaction dynamics in organic and biological nanoand micromaterials, for which large quantities are not readily available and/or photodegradation readily takes place.

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48. N. Furukawa, C.E. Mair, V.D. Kleiman, J. Takeda, Appl. Phys. Lett. 85, 4645 (2004) 49. N. Furukawa, C.E. Mair, V.D. Kleiman, J. Takeda, Ultrafast Phenomena XIV, Springer Series in Chemical Physics, vol. 79 (Springer, Berlin Heidelberg New York, 2005), p. 133 50. Y. Makishima, N. Furukawa, A. Ishida, J. Takeda, Jpn. J. Appl. Phys. 45, 5986 (2006) 51. J. Takeda, Y. Makishima, A. Ishida, Ultrafast Phenomena XV, Springer Series in Chemical Physics, vol. 88 (Springer, Berlin Heidelberg New York, 2007), p. 534 52. A. Brodeur, S.L. Chin, Phys. Rev. Lett. 80, 4406 (1998) 53. C. Nagura, A. Suda, H. Kawano, M. Obara, K. Midorikawa, Appl. Opt. 41, 3735 (2002) 54. J.T. Fourkas, L. Dhar, K.A. Nelson, R. Trebino, J. Opt. Soc. Am. B 12, 155 (1995) 55. H.A. Frank, A.J. Young, G. Britton (eds.), The Photochemistry of Carotenoids, Advances in Photosynthesis, vol. 8 (Kluwer, Dordrecht, 2000) 56. H. Nagae, M. Kuki, J.-P. Zhang, T. Sashima, Y. Mukai, Y. Koyama, J. Phys. Chem. A 104, 4155 (2000) 57. J.-P. Zhang, L.H. Skibsted, R. Fujii, Y. Koyama, Photochem. Photobio. 73, 219 (2001) 58. M. Yoshizawa, H. Aoki, H. Hashimoto, Phys. Rev. B 63, R180301 (2001) 59. N. Furukawa, J. Takeda, Nonlinear Opt. 29, 579 (2002) 60. D. Kosumi, M. Komukai, H. Hashimoto, M. Yoshizawa, Phys. Rev. Lett. 95, 213601 (2005) 61. G. Cerullo, D. Polli, G. Lanzani, S. De Silvestri, H. Hashimoto, R.J. Cogdell, Science 298, 2395 (2002) 62. A. Ishida, Y. Makishima, J. Takeda (in preparation)

4 Defects in Anatase Titanium Dioxide T. Sekiya and S. Kurita

4.1 Introduction Transition metal oxides have been attracting fundamental and technological interests because their various properties are inﬂuenced by many factors such as the number of d-electrons on transition metal ions, crystalline structures, oxygen defects and doped impurities. Elucidation of inﬂuence on properties from the factors will lead us to discovery of novel materials. From this standpoint of view, titanium dioxide is one of the prototypes. Titanium dioxide has no d-electron by itself, so that the number of d-electron can be controllable by doping of transition metals. Three crystalline modiﬁcations of titanium dioxide, rutile, anatase, and brookite are known well, so that structural dependence on properties can be discussed. It is expected that defects and impurities are also dominant over optical and electrical responses, such as transmittance, photoluminescence and conductivity, which is the same as conventional semiconductors. Titanium dioxide, moreover, is a material which has been used for a long time in a wide range of common and high technique applications because of its moderate price, chemical stability and nontoxicity. Recent topical application is a photocatalyst [1]. Photocatalytic reaction of titanium dioxide is a redox reaction of reactants adsorbed on the surface and it involves photogeneration, migration, and trapping of charge carriers. In these processes, the photogeneration and the migration of carriers are crucial processes to govern inherent activity of the material as a photocatalyst. It can be easily imagined that the factors in nanoscale size have some inﬂuence on the behavior of the carriers and the catalytic activity. In fact, anatase has higher photocatalytic activity than rutile because of a diﬀerence in Fermi energy [2] and the charge carrier in anatase thin ﬁlm has a higher mobility than that of rutile [3]. Many approaches to raise photocatalytic activity under visible light have been done; for example transition metal doping [4, 5], nitrogen doping [6], and oxygen defect [7]. Their results indicate that optical absorption in the visible region is controllable by doping of impurities. Among three crystalline modiﬁcations of titanium dioxide, only rutile crystals have

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been obtained by crystallization of the substance from its own melt or from a solution in a melt [8]. In contrast to extensive studies for rutile, fundamental optical and electronic properties of anatase which is low temperature modiﬁcation have not been well understood. To reveal fundamental properties of anatase titanium dioxide, it is indispensable to investigate defect states in it. Recently, anatase titanium dioxide single crystals can be grown by gas phase reaction [9]. As-grown anatase crystals generally exhibit pale blue color in spite of the wide bandgap of about 3.3 eV. This suggests the presence of some defects in the as-grown crystals. On this report, it is shown that several colors in anatase can be available by defects controlled in nanoscale, and some electrical properties are controllable by photoirradiation, which implies the possibility of nanoscale doping.

4.2 Growth of Anatase Single Crystal Rutile phase is the most stable phase of titanium dioxide. On heating anatase, it transforms irreversibly into rutile. The temperatures at which this transformation takes place varies from 400 to 1,000◦C depending on concentration of impurities in the crystals or annealing atmosphere [10]. So, crystal growth of anatase must be performed at fairly low temperatures. The chemical vapor transport (CVT) method permits a change in temperature of the crystallization zone over a wide range. The single crystals produced by this method are comparatively small but are suitable for measurement of electrical and optical properties, because the grown crystal frequently has speciﬁc crystallographic faces. It was believed that anatase single crystals could be grown by chemical transport reactions with the help of additional mineralizers like aluminum, gallium, and indium oxides as stabilizers using HCl, HBr, NH4 Cl, or TeCl4 as transport agent [11–14]. Pure anatase crystal with no additives has been provided ﬁrst by Lausanne research group in 1993 [9], here the word “pure” implies “without additives.” After that, the growth technique has been improved for a simple procedure and the use of pure raw materials (4N TiO2 ) [15, 16]. TiO2 (99.99%) and NH4 Cl powders were introduced into SiO2 glass ampoule of 18 mm inner diameter and 150 mm long with amounts of about 30 and 3 mg cm−3 , respectively. TiO2 powders were calcined at 1,400◦C for 12 h to remove adsorbed water before use. After evacuation using a diﬀusion vacuum pump, the ampoule was sealed and heated in a horizontal tube furnace equipped with two independent heaters. The temperatures of the source and growing zones were kept for 2–3 weeks at 750–800 and 650–700◦C within ±1◦ C, respectively. The transport agent NH4 Cl is thermally decomposed into NH3 and HCl in the ampoule. The transport reaction is believed to be transportation of gaseous molecules along temperature gradient of the furnace according to reversible reaction: TiO2 + 4HCl ↔ TiCl4 + 2H2 O. In the low temperature region of the furnace, TiO2 crystals precipitate on the wall of

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the ampoule. The phase, rutile, or anatase depends on temperature. Grown anatase single crystals have generally transparent pale blue color, optically ﬂat surfaces and distorted octahedral shape. The phase of the product can be easily identiﬁed by Raman spectra [17]. Niobium-doped anatase single crystals can be also grown by the CVT method [18]. First, rutile solid solution was formed by heating a mixed powder of high purity NbO2 and TiO2 (5:95) in a silica glass ampoule at 1,000◦C for 12 h because of recommendation of the use of rutile phase titanium dioxide powder as raw material [9]. The solid solution powder was added into TiO2 rutile powder at a desired composition and they were sealed in a silica glass ampoule with a transport agent of NH4 Cl. A following procedure is the same as that mentioned above.

4.3 Control of Defect States The synthetic anatase single crystals grown without additives by CVT method generally exhibit pale blue color in spite of the wide bandgap of about 3.3 eV. This suggests the presence of some defects in the as-grown crystals. A heat treatment under oxygen or hydrogen atmosphere results in the change in color of anatase single crystal. The diﬀerence in color will correspond to variety in defects which are investigated by polarized optical absorption and electron paramagnetic resonance (EPR) [19]. 4.3.1 Heat Treatment Under Oxygen Pressure An as-grown anatase crystal was polished to have (010) face. The crystal placed in a platinum crucible was put in a high pressure vessel of a sintered alumina tube and heated under oxygen pressure of 1.0 MPa at the desired temperature for several hours. Figure 4.1 shows the change in polarized optical absorption spectra measured at 77 K depending on the heat treatment under oxygen atmosphere. In the spectra of E ⊥ c and E c polarization conﬁgurations of as-grown crystal, the sharp-cut absorption starts at around 3.30 and 3.33 eV, respectively, where E and c imply the electric ﬁeld of incident light and crystalline axis vectors. A broad absorption A is observed in the lower energy region than 2.5 eV in both polarizations. A weak band B is also observed at about 2.2 and 2.3 eV in the E ⊥ c and E c conﬁgurations, respectively. The blue color of the crystal is due to the existence of the absorptions A and B. The absorptions A and B decrease in intensity with increase in the annealing temperature and disappear on the annealing at 300◦ C for 2 h. New band C at 3.0 eV for E ⊥ c conﬁguration appears on the annealing at 200◦C for 2 h. A band C is also observed near the absorption edge for E c conﬁguration. They become intense with increase in the annealing temperature up to 500◦ C. The color of the crystal changes into yellow because of the existence of the bands C and C and the disappearance of

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0 30

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2.0

2.5

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Fig. 4.1. Change in polarized absorption spectra depending on temperature of oxygen annealing, where E and c are the electric ﬁeld vector of incident light and crystalline axis, respectively. (a) and (b) correspond to the transformation of as-grown (pale blue) → yellow and yellow → colorless anatase single crystal, respectively [19]

the absorptions A and B. Further annealing in oxygen atmosphere at higher temperatures results in a decrease in the intensity of the bands C and C , as shown in Fig. 4.1. After the annealing at 800◦ C for 60 h, the bands C and C disappears and the crystal becomes colorless to have sharp absorption edge and no absorption in the visible region. The correspondence in the generation and extinction processes of band C and C suggests that they have the same origin. The crystal keeps its transparency during those heat treatments. The colorless crystal is spectroscopically pure, and density of defects in the colorless crystal will be quite low compared to the other colored crystals. It was conﬁrmed that as-grown (pale blue) crystal can be changed into yellow one by heat treatment under vacuum or nitrogen atmosphere. This indicates that the change will not be due to ﬁlling of oxygen vacancies but will be due to removal of some impurities. The colorless crystal cannot be obtained from yellow crystal by annealing under nitrogen atmosphere. This clearly denotes that the presence of oxygen is indispensable to the transformation from yellow crystal to colorless one. In this process, oxygen defects are ﬁlled up. 4.3.2 Heat Treatment Under Hydrogen Atmosphere The transparent colorless anatase crystal is assumed to be stoichiometric with few oxygen defects. The colorless anatase crystal is subjected to reduction

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650°C-16h 650°C-1h 600°C-16h 600°C-1h 550°C-1h 500°C-1h colorless

Absorption Coefficient (cm−1)

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A

D

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0 E //c 30

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20

D'

10 A'

B'

0 1.5

2.0

2.5 Photon Energy (eV)

3.0

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Fig. 4.2. Change in polarized absorption spectra depending on temperature of hydrogen annealing, where E and c are the electric ﬁeld vector of incident light and crystalline axis, respectively. The change in spectra corresponds to the transformation of colorless → pale blue → dark blue anatase single crystal [19]

procedure by putting it in the hydrogen stream. The crystal was placed in a silica glass tube. The tube was inserted into an electric tubular furnace and heated at the desired temperature for several hours in hydrogen gas stream of 300 ml min−1 . Figure 4.2 indicates the change in polarized absorption of the crystal depending on the hydrogen annealing. When the colorless crystal is heated at temperatures of 500–600◦C for 1 h under hydrogen atmosphere, the color of the crystal changes into pale blue. The crystals annealed at 500– 600◦ C show the broad absorptions which are observed in the lower energy region than 2.5 eV for both polarization conﬁgurations. The absorption curves are compared to those of as-grown crystal. The further annealing at higher temperatures up to 650◦ C results in the increase in intensity of the absorption in the lower energy region. After this heat treatment, the crystal shows transparent dark blue color. In addition, a new band D is observed near the absorption edge for E ⊥ c conﬁguration. The corresponding band D appears

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dark blue 200°C-2h 300°C-2h 400°C-2h 500°C-2h 800°C-60h

E

Absorption Coefficient (cm−1)

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C 10

D

0 E //c 30

dark blue 200°C-2h 300°C-2h 400°C-2h 500°C-2h 800°C-60h

20

10

C'

D'

E'

0 1.5

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Fig. 4.3. Change in polarized absorption spectra depending on temperature of repeated oxygen annealing, where E and c are the electric ﬁeld vector of incident light and crystalline axis, respectively. The change in spectra corresponds to the transformation of dark blue → dark green → yellow → colorless anatase single crystal, respectively [19]

at 2.8 eV for E c conﬁguration, showing the dichroism. These positions of the absorption band D and D are clearly diﬀerent from the band C and C observed in the yellow crystal, so that it reﬂects a diﬀerent defect state. To conﬁrm whether the dark blue crystal could return into colorless, the dark blue crystal was subjected to the annealing under oxygen pressure. The results of the reoxidization procedure are shown in Fig. 4.3. With increase in annealing temperature up to 400◦ C, bands D and E reduce their intensities. After this heat treatment, the crystal color becomes dark green. When the dark green crystal was heated at 500◦ C for 2 h, the color suddenly changes to yellow. The polarized absorption spectra of this yellow crystal show a strong resemblance to those of the yellow one encountered in the initial oxidation procedure from as-grown to colorless crystal (Fig. 4.1). The crystal also becomes

4 Defects in Anatase Titanium Dioxide

Absorption Coefficient (cm−1)

20

127

E⊥c

10

0 20

E //c

10

0 1.5

2.0

2.5

Photon Energy (eV)

3.0

3.5

Fig. 4.4. Absorption spectra of hydrogen-implanted anatase single crystal (solid line). The broken line exhibits absorption curve of colorless crystal which was used as starting material [19]

colorless on oxygen annealing at 800◦ C for 60 h. No shift of the absorption edge indicates that the crystal keeps anatase phase during these annealing. To elucidate what happens by the hydrogen reduction procedure, ionized hydrogens were implanted into colorless anatase single crystal with acceleration voltage of 130 kV and a total dose of 2.5 × 1017 at room temperature. After the implantation, the crystal changes its color to pale blue. The optical absorption spectra of the hydrogen-implanted crystal show a close resemblance with those of pale blue crystal, as shown in Fig. 4.4. Annealing under oxygen pressure at 200◦ C for 2 h results in the discoloration. These experimental results show that the blue-colored crystals, such as as-grown crystal, the hydrogen-implanted crystal, and the pale blue crystal obtained by hydrogen annealing, have the same optical property. This suggests large possibility for the existence of hydrogen in the three types of the blue-colored crystals. The implanted hydrogen is located near the surface and is removed easily. Figure 4.5 shows a schematic representation of the change in the characteristic ﬁve colors of the crystals controlled by the heat treatments in oxygen or hydrogen atmosphere. Figure 4.6 shows polarized optical absorption spectra of Nb-doped anatase crystal. The absorption increases in lower energy region than 2.7 eV. A broad absorption A (A ) is observed in the lower energy region than 2.5 eV in both polarization conﬁguration. A weak band B (B ) is also observed at

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H + implant

colorless O2

H2 O2

yellow

pale blue (as-grown) O2

N2

H2 O2 dark green dark blue N2 Fig. 4.5. Change in characteristic color depending on the heat treatments, where white, grayish, and black arrows represent heat treatments under hydrogen, inert and oxygen atmosphere, respectively

60

E⊥ c

50 40

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A B

20 10 0 E//c 50

300K 77K 5K

40 30 20

A' B'

10 0 1.5

2.0

2.5 Photon Energy (eV)

3.0

3.5

Fig. 4.6. Polarized optical absorption spectra of Nb-doped anatase single crystal at 300, 77, and 5 K, where E and c are the electric ﬁeld vector of incident light and crystalline axis, respectively [18] (copyright Elsevier. Reproduced with permission)

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about 2.2 eV. The crystal exhibits blue color because of the existence of these absorptions. It is noteworthy that the absorption spectra of Nb-doped crystal resembles largely to those of nondoped as-grown crystal.

4.4 Properties of Anatase 4.4.1 Absorption Edge The fundamental absorption edge of the anatase phase of TiO2 starts at around 3.30 and 3.33 eV, shown in Figs. 4.1–4.3. The absorption curves from 4.2 to 300 K, show an exponential spectral dependence near the edge, which describes Urbach’s tail [16, 20]. As shown in Fig. 4.7, each of the slopes increases with decreasing temperature up to 90 K. No change in the curve is observed in absorption measurement lower than 90 K. The exponential spectral dependence suggests that anatase is a semiconductor with a direct allowed optical transition. According to Urbach’s rule, the absorption coeﬃcient, α, near the fundamental absorption edge depends on photon energy, h ¯ ω: hω)/kT ], where E0 is a constant independent from α(¯hω) = A exp[−σ(E0 − ¯ 104 3

10

Absorption Coefficient (cm−1)

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E⊥c <90K 150K 220K 293K

101 100 10−1 103

E//c

<90K 150K

102

220K 293K

101 100 10−1 3.1

3.2

3.3

3.4

Photon Energy (eV)

3.5

Fig. 4.7. Semilogarithmic plot of absorption coeﬃcients near the absorption gap measured at several temperatures. The lines are best ﬁt for linear part of the curves hω)/kT ] using the formula, α(¯ h, ω) = A exp[−σ(E0 − ¯

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temperature. The steepness parameter σ is temperature dependent according hωp ] tanh[¯ hωp /2kT ], where σ0 is a constant. to the equation σ(T ) = σ0 [2kT /¯ Two diﬀerent models were proposed for the Urbach’s rule observed in single crystals. Dow and Redﬁeld ascribe the Urbach’s tail to the ionization of excitons, charged impurities, lattice disorder, etc. [21]. While Toyozawa and coworkers ascribe the Urbach’s tail to the inﬂuence of phonon ﬁelds on the center-of-mass motion of excitons [22–24]. The latter model applies to highly ionic crystals such as alkali halide and oxides, where the exciton–phonon interaction is large. In this case, the steepness parameter σ0 is inversely proportional to exciton–phonon coupling constant. The Urbach parameters of anatase deduced from the temperature dependent absorption curves can be accounted for the theory of the momentary localization of excitons. It is concluded that the excitons in anatase are self-trapped by deformation potential ﬂuctuations due to phonon interaction [20]. The parameter E0 of anatase has been estimated to be 3.419 and 3.460 eV in polarizations E ⊥ c and E c, respectively [16]. There are many reports on theoretical calculation for band structure of anatase and parts of them are listed in [25–28]. Figure 4.8 shows the calculated band structure of anatase by ﬁrst-principle pseudopotential method based on local density approximation in density functional theory [27]. Mikami et al. showed that the calculated minimal bandgap of 2.05 eV for anatase is indirect, where the bottom of the conduction bands is at Γ and the top of the valence bands is near X in the reciprocal lattice vector unit for the primitive cell. Asahi et al. showed that the minimal direct bandgap is at Γ and the energy diﬀerence in the valence band maxima at Γ and Z is only 2 meV. This result is consistent with experimental results from

Fig. 4.8. Calculated band structure of anatase in Teter-type potential case. The energies are relative to the top of valence bands [27]

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Fig. 4.9. Molecular-orbital bonding structure for anatase TiO2 (a) atomic levels, (b) crystal-ﬁeld split levels, and (c) ﬁnal interaction states. The thin solid and dashed lines represent large and small contributions, respectively [28]

polarized reﬂection measurements in the range of 2–25 eV using a synchrotron orbital radiation [15]. A noticeable feature from those band calculation results is that the top of the valence band and bottom of the conduction band consist of oxygen 2p and titanium 3d orbitals, respectively. Figure 4.9 shows a molecular orbital bonding diagram for anatase [28]. 4.4.2 Photoluminescence In photoluminescence spectra of anatase, a large broadband is simply observed at about 2.3–2.4 eV with a band width of 0.6 eV (FWHM) [29–31]. The Stokes shift of about 1 eV which is evaluated as the diﬀerence between the emission peak and optical absorption edge suggests that the exciton–phonon interaction is large. This indicates that the excitons are self-trapped, which corresponds to the analysis of the optical absorption spectra. The photoluminescence spectra of pure and Al-doped anatase crystal at low temperature nearly coincide with each other [30]. This suggests that the relaxation of the exciton to the selftrapped state is dominant because of the large exciton–phonon interaction and the emission is due to the recombination of the self-trapped excitons.

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4.4.3 EPR Spectra The colorless anatase crystal showed no EPR signal in temperature range of 10–300 K. The absence of EPR signal implies that few defects with paramagnetic spin are present in the colorless crystal. On the other hand, a strong signal is observed in EPR spectra of Nb-doped, as-grown (pale blue), dark blue and dark green crystals which have optical absorption bands lower than 2.5 eV [18, 19] and show some electric conduction. Figure 4.10 shows temperature dependence of EPR spectra of as-grown crystal. The signals at room temperature for H ⊥ c and H c conﬁgurations are broad and asymmetric. The signals become narrow with decrease in temperature. It is worth to notice that the asymmetric property of the spectra remains till 10 K. The asymmetric feature of the signal can be related to the conduction electrons in the crystal. The Hall eﬀect measurement indicates that the as-grown crystal shows n-type conduction [32]. It is known that the free carriers in metals and semiconductors give rise to an asymmetric EPR signal due to the skin eﬀect [33, 34]. A line shape analysis of the asymmetric signal reveals that the electron spin relaxation time T1 of as-grown anatase is inversely proportional to temperature [19].

H//c

dI / dH

H⊥c

3320

3360

3400

3440

RT

RT

150 K

150 K

80 K

80 K

40 K

40 K

10 K

10 K

Magnetic Field (Gauss)

3480 3360

3400

3440

3480

3520

Magnetic Field (Gauss)

Fig. 4.10. The change in X-band EPR spectra of pale blue (as-grown) anatase single crystals depending on temperature measured for H ⊥ c and H c conﬁgurations [19]

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1.995 1.990

g-value

1.985 1.980 1.975 1.970 1.965 1.960 0

90

180

Angle, θ (degree)

270

360

Fig. 4.11. Angular dependence of the EPR signal of pale blue anatase single crystal at 10 K. The curve represents the best ﬁt of the equation of g = 2 g⊥ sin2 θ + g2 cos2 θ with parameters of g⊥ = 1.992 and g = 1.963 [19]

Figure 4.11 shows angular dependence of the EPR signal of as-grown crystal at 10 K, which is a strong resemblance to that of Nb-doped, dark blue and dark green crystals. It should be pointed out that hydrogen-implanted crystal shows the same angular dependence. The EPR signal is frozen on a rotation in ab-plane. The angular dependence of g-value on the rotation in ac-plane clearly indicates that the signal is due to S = 1/2 species in a tetragonal symmetry. The components of the g-tensor are estimated to be g⊥ = 1.992 and g = 1.963 obtained by a least square ﬁt using the following formula: 2 sin2 θ + g 2 cos2 θ. (4.1) g = g⊥ The conduction electrons will be supplied from impurities, Nb or H atoms, in the blue-colored crystals. The donor levels formed by the impurities are quite shallow because of the asymmetric feature of EPR signals and electric conduction at low temperatures. This is consistent with extraordinary large dielectric constant of titanium dioxide. On the basis of theoretical studies for anatase TiO2 , the lowest state of conduction band is composed mainly of dxy orbital of titanium and the conduction electrons is located on the dxy orbital. The analysis for the angular dependence of EPR spectra of pale blue crystal gives the component of the g-tensor, g⊥ = 1.992 and g = 1.963. The component of eﬀective g-value, g˜, of S = 1/2 can be written using g-value of free electron, ge , and spin–orbit coupling constant, ζ, as follows: ⎛ ⎞ ζϕ0 |lp |ϕn |lq |ϕ0 ⎠, g˜pq = ge ⎝δpq − (4.2) (εn − ε0 ) n=0

where p, q = x, y, z and the subscript zero implies a ground state. Then, the component of eﬀective g-value is rewritten as; g⊥ = gxx = gyy = ge (1 − ζ/∆ε)

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and g = gzz = ge (1 − 4ζ/∆ε), which mean the tetragonal symmetry. These formulas indicate a close relation with the g⊥ component in the ab-plane rotation. While the degenerated dyz and dzx orbitals make some contribution to the g component, so that the angular dependence of the g component shows two-hold symmetry. The mean energy separation between dxy and the other d-orbitals ∆ε, and the spin–orbital coupling constant of titanium ion ζ are estimated to be about ∆ε = 3.6 eV [28] and ζ = 154 cm−1 (19.1 meV), respectively. Then calculated values, g⊥ = 1.992 and g = 1.960, can be obtained. These estimated values show a good agreement with the observed ones. This estimation can be applied to a model that an electron is localized on a substitutional Ti4+ site in anatase crystal to form Ti3+ . Because of the slight energy diﬀerence between the impurity level and the bottom of the conduction bands, the wavefunction of the impurity has almost the same properties as that of the electron located on the bottom of the conduction bands. It is assumed that the electric conduction and asymmetric signal feature are decisive of the origin of the EPR signal of blue-colored anatase crystal, that is, conduction electrons. 4.4.4 Electric Conduction The optical bandgap of titanium dioxide exceeds 3 eV, so that titanium dioxide should be regarded as insulator. But some electric conduction can be observed on the colored anatase crystals and even on colorless one. This indicates that such crystals contain some defects or impurities which largely dominate the electric conduction of the crystal. So, the electric conductivity depends on sample. The amount of oxygen defects will change depending on time, even if the same sample is used for the measurement. Figure 4.12

Fig. 4.12. Temperature dependence of the resistivity of an as-grown anatase single crystal. The inset shows the variation of the resistivity of the same single crystal after successive annealing under 1 atm of oxygen for 1, 3, and 15 days [32]

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(b)

Fig. 4.13. (a) Hall coeﬃcient vs. temperature of an anatase single crystal. The carrier density calculated from the Hall coeﬃcient is plotted vs. reciprocal temperature in the inset. The carrier density is thermally activated in the whole temperature range with an activation temperature of 50 K. (b) Hall mobility vs. reciprocal temperature in the high-temperature regime (>70 K). The inset shows the Hall mobility plotted against temperature over the entire temperature range measured [32]

shows temperature dependence and time dependence of the resistivity of an as-grown anatase crystal [32]. The successive annealing of the as-grown crystal under 1 atom of oxygen for 15 days makes its resistivity be 100 times. The absolute value of electric conductivity or resistivity is a quick guide to the amount of defects and impurities. The temperature dependence of the electrical resistance of as-grown anatase shows a typical semiconductor behavior. At temperatures of 70–300 K, where the carrier concentration saturates (the exhaustion regime), the electric resistivity increases with increasing temperature. This temperature dependence is similar to that of conventional metals. At low temperatures, below 50 K, the resistivity rises with decreasing temperature. The low-temperature behavior occurs as the thermally activated promotion of electrons to conducting states. Temperature dependences of Hall coeﬃcient, calculated carrier density and Hall mobility of anatase single crystal are shown in Fig. 4.13. The anatase crystal contains n-type carriers which are produced by thermal excitation from the shallow donor states with activation energy of 4.2 meV. The carrier density is estimated to be ∼ 1018 cm3 at room temperature. The carrier mobility above 160 K falls with increasing temperature in proportional to exp(T0 /T ), where T0 is about 850K. The comparability of T0 to the optical phonon temperature suggests scattering of the conducting electrons by optical phonons. The mobility of carriers is estimated to be 20 cm2 V−1 s−1 . Nb-doped anatase crystal with Nb content of 0.08 wt% has a resistivity of the order of 10−2 to 10−3 Ωcm and carrier density of ∼ 1019 cm−3 in temperature range of 4–300 K. The mobility of carriers is estimated to be 180 cm2 V−1 s−1 . It is believed that the doped niobium replaces titanium in anatase. Niobium has one additional valence electron compared to titanium, and it will make donor level in the bandgap. The small change in resistivity at low temperatures suggests that the donor level is quite shallow. The increase

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Fig. 4.14. The electronic track measured by time-of-ﬂight technique. The additional current induced by the carrier transfer on external circuit is converted into voltage by reading resistance

in density of carriers contributing electric conduction results in decrease in energy diﬀerence between the donor levels and the bottom of the conduction bands. The carrier mobility should be elucidated for crystals containing comparatively few defect or impurities. The application of above-mentioned technique using direct current is impossible to measure electric conductivity of such crystals which is almost insulator. The mobility of carriers can be measured by a time-of-ﬂight (TOF) method for colorless anatase crystal, as shown in Fig. 4.14. In TOF method, the length of time is measured for carriers to take to travel across a ﬁxed distance between two electrodes with a given potential. A pair of blocking electrodes is separated by insulator ﬁlms from the sample crystal. Charged carriers are induced on one surface of the crystal by irradiation of pulsed-Nd:YAG laser light and the electrostatic attraction causes electrons (or halls) to move toward the positively (or negatively) charged electrode. At the same time, additional current is induced by the carrier transfer on external circuit. Drift mobility of electrons is estimated to be ∼ 1 cm2 V−1 s−1 at room temperature. These indicate that mobility of the conduction electrons become large with increase in content of impurities and defects in crystal. When many conduction electrons are freely movable in crystal, scattering potential around ionized impurity or defect is weakened due to shielding by such electrons.

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4.5 Carrier Control by Photoirradiation Recently, interesting phenomena that photogenerated carriers have an interaction with defects or impurities in crystals were reported on conductivity and EPR measurements under irradiation of UV-light. The persistent change which has a large decay constant can be induced by photoirradiation. Such persistent phenomena indicate that properties of materials are controllable by light. 4.5.1 Photoconductivity Figure 4.15 shows change in photoconductivity of anatase and rutile crystals induced by irradiation of UV-light measured at room temperature [35]. The anatase crystal grown by CVT method is colorless and transparent. The rutile crystal is commercially obtained. In the ﬁgure, the light irradiation starts at t = tON and terminated at t = tOFF , which corresponds to the time origin. The remarkable enhancement in conductance is observed in both crystals. The photoinduced change of anatase crystal shows the gradual response to the photoirradiation, while the change in rutile crystal responds promptly. Moreover, the photoinduced change of the anatase crystal persists over 3,000 s, which can be named as persistent photoconductance. The decay of persistent photoconductance can be characterized by the stretched-exponential function: σ(t) = σ(0) exp[−(t/τ )β ], where σ(t), β, τ denote the photoconductance at t, decay exponent and decay constant, respectively. The trapping and detrapping of carriers by the shallow levels occur before the recombination, and thus

Fig. 4.15. Temporal change in the conductance of anatase (upper panel ) and rutile (lower panel ) crystals induced by illuminating with white light from a Xe lamp at 300 K. tON and tOFF denote the start and the end of illumination, respectively [35]

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the recombination rate is largely reduced. Presuming the energy distribution of donor or acceptor levels formed by defects and impurities, the persistent photoconductance can be understood. A similar phenomenon is reported on GaN [36, 37]. 4.5.2 EPR Figure 4.16 shows the temperature dependence of EPR signals of anatase single crystal under the irradiation of UV-light at 3.40 eV [38]. The UV irradiation of TiO2 results in dissociation of oxygen from its surface and the formation of a perpetual defect. The weak signal marked with an asterisk appeared on UV-light irradiation. After the irradiation, it can be persistently observed even if the irradiation of UV-light is cutoﬀ, so that the weak signal originates from the perpetual defect. The sharp signal observed at temperatures below 30 K has g matrix elements of gxx = gyy = 1.9929 and gzz = 1.9643 which are comparable to bulk Ti3+ of (gxx , gyy ) = (1.990, 1.960) [39]. Two pairs of sextuplets can be observed in the range of 3,315– 3,385 G at temperatures of 110–40 K under the UV-light irradiation, as shown in Fig. 4.16. Individual signals of the sextuplet have the same intensity and H //a

RT

* *

200K 150K

*

120K

dI/dH

* *

100K

*

90K

*

80K

* *

70K

*

50K

60K 40K

*

30K 20K 10K

3300

3320

3340

3360

3380

3400

3420

3440

Magnetic Field (Gauss) Fig. 4.16. Temperature dependence of EPR spectra of colorless anatase under UV irradiation. The crystalline a axis is parallel to applied magnetic ﬁeld. The individual spectra are oﬀset along the ordinate for clarity of presentation [38] (copyright WileyVCH Verlag GmbH & Co. KGaA. Reproduced with permission)

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interval, so that each sextuplet will arise from a paramagnetic species of S = 1/2 interacting with I = 5/2. The sextuplet is impossible to be assigned to a paramagnetic spin on titanium or oxygen atoms because of no observation of EPR signal due to the other isotopes. Angle resolved EPR spectra indicate that the paramagnetic species are occupied at a lattice site in anatase. The sextuplet will be originated from impurity atom with I = 5/2, such as Al and Mn. This indicates that the carriers induced by irradiation of UV-light are trapped on the impurities. The sextuplet signals rise just after UV irradiation and saturate for less than 5 min at temperatures of 110–40 K. The signal disappears immediately on temperature rising to more than 120 K. Decay curves of the sextuplet signals at several temperatures are shown in Fig. 4.17. The intensity of the signal decreases to one-tenth of the saturated value at 100 K for 1 h. At 90 K, the signal reduces gradually and it reaches one-forth of the saturated value. At temperatures below 80 K, the intensity of the signal keeps almost constant value for 10 h. These decay curves can be evaluated by the stretched-exponential function. The stretched-exponential function is interpreted as stack alignment

ESR Signal Intensity (a.u.)

100K 90K 80K 70K 60K

0

2

4

6

8

10

Time (hour) Fig. 4.17. Decay in intensity of EPR signals induced by UV irradiation at several temperature on colorless anatase crystal

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of exponential functions with diﬀerent relaxation times. This indicates that photoinduced spins interacts with impurities through trapping and detrapping process.

References 1. A. Fujishima, K. Honda, Nature 238, 37 (1972) 2. H.P. Maruska, A.K. Ghosh, Sol. Energy 20, 493 (1978) 3. H. Tang, K. Prasad, R. Sanjin´es, P.E. Schmid, F. L´evy, J. Appl. Phys. 75, 2042 (1994) 4. A.K. Ghosh, H.P. Maruska, J. Electrochem. Soc. 124, 1516 (1977) 5. M. Anpo, Catal. Surv. Jpn. 1, 169 (1997) 6. R. Asahi, T. Morikawa, T. Ohwaki, Y. Taga, Science 293, 269 (2001) 7. R.G. Breckenridge, W.R. Hosler, Phys. Rev. 91, 793 (1953) 8. J.R. Berkes, W.B. White, R. Roy, J. Appl. Phys. 36, 3276 (1965) 9. H. Berger, H. Tang, F. L´evy, J. Cryst. Growth 130, 108 (1993) 10. R.D. Shannon, J.A. Pask, J. Am. Ceram. Soc. 48, 391 (1965) 11. I.N. Anikin, I.I. Naumova, G.V. Rumyantseva, Kristallograﬁya 10, 230 (1965) (Sov. Phys. Crystallogr. 10, 172 (1965)) 12. G.D. Davtyan, Kristallograﬁya 21, 869 (1976) (Sov. Phys. Crystallogr. 21, 499 (1976)) 13. C.H.R. Rao, A. Turner, J.M. Honig, J. Phys. Chem. Solids 11, 173 (1959) 14. F. Izumi, H. Kodama, A. Ono, J. Crystal Growth 47, 139 (1979) 15. N. Hosaka, T. Sekiya, C. Satoko, S. Kurita, J. Phys. Soc. Jpn. 66, 877 (1997) 16. T. Sekiya, M. Igarashi, K. Ichimura, S. Kurita, J. Phys. Chem. Solids 61, 1237 (2000) 17. T. Ohsaka, F. Izumi, Y. Fujiki, J. Raman Spectrosc. 7, 321 (1978) 18. D.D. Mulmi, T. Sekiya, N. Kamiya, S. Kurita, T. Kodaira, Y. Murakami, J. Phys. Chem. Solids 65, 1181 (2004) 19. T. Sekiya, T. Yagisawa, N. Kamiya, D.D. Mulmi, S. Kurita, Y. Murakami, T. Kodaira, J. Phys. Soc. Jpn. 73, 703 (2004) 20. H. Tang, F. L´evy, H. Berger, P.E. Schmid, Phys. Rev. B 52, 7771 (1995) 21. J.D. Dow, D. Redﬁeld, Phys. Rev. B 5, 594 (1972) 22. H. Sumi, Y. Toyozawa, J. Phys. Soc. Jpn. 31, 342 (1971) 23. K. Cho, Y. Toyozawa, J. Phys. Soc. Jpn. 30, 1555 (1971) 24. M. Schrieber, Y. Toyozawa, J. Phys. Soc. Jpn. 51, 1528 (1982) 25. A. Fahmi, C. Minot, B. Silvi, M. Causa, Phys. Rev. B 47, 11717 (1993) 26. S.D. Mo, W.Y. Ching, Phys. Rev. B 51, 13023 (1995) 27. M. Mikami, S. Nakamura, O. Kitao, H. Arakawa, X. Gonze, Jpn. J. Appl. Phys. 39 L847 (2000) 28. R. Asahi, Y. Taga, W. Mannstadt, A.J. Freema, Phys. Rev. B 61 7459 (2000) 29. N. Hosaka, T. Sekiya, S. Kurita, J. Luminescence 74, 874 (1997) 30. H. Tang, H. Berger, P.E. Schmid, F. Levy, G. Burri, Solid State Commum. 87, 847 (1993) 31. M. Watanabe, S. Sasaki, T. Hayashi, J. Luminescence 87–89, 1234 (2000) 32. L. Forro, O. Chauvet, D. Emin, L. Zuppiroli, H. Berger, F. L´evy, J. Appl. Phys. 75, 633 (1994) 33. F.J. Dyson, Phys. Rev. 98, 349 (1955)

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34. G. Feher, A.F. Kip, Phys. Rev. 98, 337 (1955) 35. C. Itoh, A. Wada, Phys. Stat. Sol. C 2, 629 (2005) 36. S.J. Chung, O.H. Cha, Y.S. Kim, C.H. Hong, H.J. Lee, J.O. White, E.K. Suh, J. Appl. Phys. 89, 5454 (2001) 37. M.T. Hirsch, J.A. Wolk, W. Walukiewicz, E.E. Haller, Appl. Phys. Lett. 71, 1098 (1997) 38. T. Sekiya, H. Takeda, N. Kamiya, S. Kurita, T. Kodaira, Phys. Stat. Sol. C 3, 3603 (2006) 39. P. Meriaudeau, M. Che, P.C. Gravelle, S.J. Teichner, Bull. Soc. Chim. Fr. 13 (1971)

5 Organic Radical 1,3,5-Trithia-2,4,6-Triazapentalenyl (TTTA) as Strongly Correlated Electronic Systems: Experiment and Theory J. Takeda, Y. Noguchi, S. Ishii, and K. Ohno

5.1 Introduction Recently, organic radicals in solid state have attracted much current interests with regard to photoinduced phase transitions, Peierls transitions, molecular Mott insulators, synthetic metals, and so on. They are expected to play a major role in novel materials research because of their potential in future applications to molecular magnetic and optical devices. In photoinduced phenomena, light irradiation stimulates the macroscopic phase transition between the ground state and a metastable state, and may control various properties including, e.g., magnetic and chromic properties of materials. In this review, we focus our attention to the organic crystal [1, 2] of 1,3,5trithia-2,4,6-triazapentalenyl (TTTA) recently found, which is a heterocyclic organic radical composed of eight atoms (three sulphur, three nitrogen, and two carbon atoms). Each TTTA radical molecule has one unpaired electron on the nitrogen atom in the S–N–S moiety. The singly occupied molecular orbital (SOMO) is shown in Fig. 5.1. The TTTA crystal has two phases, high-temperature (HT) and low-temperature (LT) phases, and undergoes a ﬁrst-order phase transition with a wide thermal hysteresis loop over wide temperature range from 230 to 305 K. As shown in Fig. 5.2, the HT phase has a 1D stack of evenly spaced radicals and characterized as a molecular Mott insulator, while the LT phase consists of dimerized radicals and can be regarded as a diamagnetic Peierls insulator [2–10]. Since the HT and LT phases exhibit, respectively, antiferromagnetic and diamagnetic behaviors, there is a magnetic bistability. Both the phases have an optical gap of approximately 1.5 eV [3,4]. This phase transition can be also induced by light irradiation [3, 5, 6, 10].

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Fig. 5.1. Singly occupied molecular orbital (SOMO) of a TTTA radical molecule

Fig. 5.2. TTTA crystal in which the electronic charge distribution shaded by blue clouds is restricted inside each molecule in the HT phase or between the dimerized molecules in the LT phase

5.2 Crystalline Structure The structure of the HT phase of TTTA crystal was originally reported by Wolmersh¨ auser and Johann [1]. Later, Fujita and Awaga [2] showed that the crystal undergoes a magnetic phase transition accompanied by a large structural change around room temperature. Fujita et al. [2, 3] determined precisely the crystalline structures of both the HT and LT phases by the X-ray diﬀraction experiment. More recently, assuming the crystal symmetries, lattice constants and angles of the unit cell determined by the experiment, Furuya et al. [7] carried out an ab initio geometrical optimization of atomic positions inside the unit cell by means of the ultrasoft-pseudopotential approach (using 26 Ry cutoﬀ energy for the plane waves and 36 special k points), and found good agreement with the experiment. Here we summarize these results of the structures of the HT and LT phases of the TTTA crystal. The resulting structures are shown in Figs. 5.3 and 5.4, respectively, for the HT and LT phases. Calculated atomic coordinates are also listed in Tables 5.1 and 5.2, respectively, for the HT and LT phases. Several ◦ interatomic distances in units of A are given in Tables 5.3 and 5.4. From these data, one may conﬁrm that the position data determined experimentally [3] are well reproduced by the ab initio structural optimization ◦ within the error of 0.09 A.

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Fig. 5.3. Geometrical structure of the HT phase of the TTTA crystal in which the regular packing of radical molecules appear along the stacking direction, i.e., along the b-axis. Atomic positions and selected bond lengths are listed, respectively, in Tables 5.1 and 5.3

Fig. 5.4. Geometrical structure of the LT phase of the TTTA crystal in which the Peierls distortion along the stacking direction appears. Atomic positions and selected bond lengths are listed, respectively, in Tables 5.2 and 5.4

The crystalline structures of the HT and LT phases are characterized as follows: There is a crucial diﬀerence in molecular packing in the two phases. The molecular planes are all parallel in the HT phase (Fig. 5.3), whereas the unit cell of the LT phase includes two molecular-plane orientations (Fig. 5.4). In addition, uniform 1D stacking of the radical molecules appear in the HT phase, while, in the LT phase, strong dimerization along the stacking direction appears. The existence of the volume and symmetry changes indicates that the phase transition between HT and LT phases is a ﬁrst order.

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Table 5.1. Crystallographic data and fractional coordinates for the HT phase of TTTA Crystalline system space group

Monoclinic P 21 /c

◦

9.4420a 3.7110a 15.062a

a (A) ◦ b (A) ◦ c (A) α (◦ ) β (◦ ) γ (◦ ) ◦ V (A3 ) Z Atom C1 C2 N1 N2 N3 S1 S2 S3 a

104.630a 510.6a 4 ξa

ξb

ξc

0.31224 0.17140 0.30915 0.08479 0.33152 0.14689 0.42365 0.17859

0.75325 0.89798 0.82671 0.95223 0.69811 0.96843 0.68444 0.82505

0.40480 0.36027 0.23616 0.41582 0.49419 0.24456 0.33208 0.51904

Reference [3]

5.3 Experimental 5.3.1 Paramagnetic Susceptibility and Electron Spin Resonance Fujita and Awaga [2] measured the temperature dependence of the paramagnetic susceptibility χP for a polycrystalline sample of TTTA upon cooling (open circles) and upon heating (closed circles) by using a SQUID susceptometer (Fig. 5.5). The behavior clearly indicates a ﬁrst-order phase transition with a wide thermal hysteresis: Tc↓ = 230 K and Tc↑ = 305 K. The dashed curve is the theoretical best ﬁt of the 1D antiferromagnetic chain with intracolumn interaction [11], J/kB = −320 K, and intercolumn interaction, J /kB = −60 K. Takeda et al. [4, 5] measured the temperature dependence of the electron spin resonance (ESR) intensity for polycrystalline TTTA samples. Figure 5.6 shows (a) the derivative curves of the ESR signal with diﬀerent temperatures and (b) the ESR intensity as a function of temperature upon cooling (circles) and upon heating (triangles). With decreasing temperature, the ESR intensity decreases slowly at the beginning, then decreases rapidly below 230 K, and ﬁnally becomes very weak below 180 K. When the sample is heated from a

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Table 5.2. Crystallographic data and fractional coordinates for the LT phase of TTTA Crystalline system space group ◦

7.5310a 10.0230a 7.0240a 100.598a 96.978a 77.638a 507.2a 4

a (A) ◦ b (A) ◦ c (A) α (◦ ) β (◦ ) γ (◦ ) ◦ V (A3 ) Z Atom C1 C2 C3 C4 N1 N2 N3 N4 N5 N6 S1 S2 S3 S4 S5 S6 a

Triclinic P¯ 1

ξa

ξb

ξc

0.86186 0.78264 0.80274 0.72314 0.51699 0.90145 0.04194 0.46126 0.84229 0.98147 0.70272 0.54798 0.10410 0.48984 0.64539 0.04493

0.14149 0.28220 0.20930 0.35081 0.15397 0.36535 0.11889 0.21857 0.43455 0.18686 0.03643 0.31156 0.26917 0.37933 0.10160 0.33728

0.71448 0.69369 0.19967 0.18193 0.64867 0.71811 0.75371 0.10826 0.21163 0.24407 0.68878 0.64602 0.76548 0.12193 0.15128 0.25719

Reference [3] ◦

Table 5.3. Several calculated interatomic distances in the HT phase in units of A N5–S2 N2–N5 N5 –S2 N1–S6

3.089 3.089 4.891 5.570

N2–N5 N1–N1 N5–S2 N4 –S3

3.280 3.711 5.655 5.568

low temperature of 140 K, the ESR intensity remains very weak over the range from 140 to 310 K. Then the ESR intensity shows sudden increase and retrieves to the initial value of the intensity. This behavior clearly shows a ﬁrst-order magnetic phase transition with a wide hysteresis loop. The observed hysteresis

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Table 5.4. Several calculated interatomic distances in the LT phase in units of A N1–N4 N2–S12 S1–S4 S2–S4 S2–S5 S9–S12

3.925 3.062 3.906 3.784 3.349 3.714

N1–N4 N5–S9 S1–S4 S2–S5 S3–S6

3.234 3.053 3.256 3.821 3.713

Fig. 5.5. Temperature dependence of the paramagnetic susceptibility, χP , for a polycrystalline sample of TTTA upon cooling (open circles) and upon heating (open triangles). After Fujita and Awaga [2]

suggests that the ground state of the HT phase is more stable than that of the LT phase near 300 K, while vice versa near 200 K as shown by inset pictures in Fig. 5.6b. Because the LT phase is diamagnetic due to strong dimerization along the stacking direction, the ESR intensity of the LT phase is about two orders of magnitude smaller than that of the HT phase. Assuming that the ESR signal of the LT phase comes from the remaining radical monomers after the dimerization, we could estimate that ∼0.5% of the TTTA molecules remains as monomers. Because TTTA crystal has a bistability over the wide temperature range from 230 to 310 K, we examined optical control of the magnetic phase transition between the LT and HT phases by laser light irradiation [4]. The irradiation was performed on the LT phase at 250 K to avoid heating of the sample by the laser light irradiation. A second harmonic of an Nd:YAG laser

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Fig. 5.6. (a) Derivative curves of the ESR signal with diﬀerent temperatures and (b) temperature dependence of the ESR intensity upon cooling (circles) and upon heating (triangles) for polycrystalline TTTA samples [5]

dI / dH (a. u.)

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0.5 0.0 −0.5 −1.0

before irrad. after irrad. 3200

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Magnetic Field (G) Fig. 5.7. Derivative curves of the ESR signal for the LT phase at 250 K before/after the laser light irradiation. The irradiation was carried out by a second harmonic of an Nd:YAG laser (532 nm) [4]

(532 nm) was used as a light source, whose energy is close to the charge transfer (CT) band of the LT phase (see Sect. 5.3.2). Figure 5.7 shows the derivative curves of the ESR signal for the LT phase at 250 K. The broken curve corresponds to the signal before the laser light irradiation, while the solid curve is that after the laser light irradiation of 2 min. The signal intensity after the irradiation is about three times larger than that before the irradiation, strongly indicating that part of the LT phase changes to the HT phase by the light irradiation. The results strongly suggest the possibility of optical control of magnetic molecular bistability, and show

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Fig. 5.8. ESR intensity as a function of irradiated photon density at 290 K

that the organic radical crystal is a new type of materials appropriate to study the photoinduced phase transition. As far as we know, this is the ﬁrst time observation of the photoinduced phase transition in organic radical crystals. The increase of the ESR signals after the photoirradiation is also investigated in detail at 290 K, where the ground state of the HT phase is more stable than that of the LT phase (see a broken arrow in Fig. 5.6b) [5]. Figure 5.8 shows the ESR intensity as a function of irradiated photon density. The irradiation was carried out by a single laser shot of a second harmonic of an Nd:YAG laser (532 nm). The ESR intensity after the irradiation of a single laser shot with ∼4.0 × 1021 cm−3 photon density becomes about six times larger than that before the irradiation. We found that the growth of the ESR intensity clearly exhibits existence of a threshold photon density; the ESR intensity suddenly grows up above the critical threshold photon density of ∼1.0 × 1021 cm−3 , followed by saturation, while it never increases with the intensity below the critical threshold even after irradiation of many laser shots. The spectral shape of the ESR signal did not change during the experiments, indicating that the samples do not show any damage of degradation. The observed increase of the ESR intensity strongly suggests that part of the LT phase changes to the HT phase by the pulsed-laser irradiation. Since the irradiated photons are absorbed near the sample surface due to small penetration depth of the excitation and the irradiated portion (150×150 µm2 ) is much smaller than the whole sample area, only those molecules located near the surface within the irradiated part can change to the HT phase. The ESR intensity therefore does not reach to that of the whole HT phase. 5.3.2 Reﬂectivity The polarized reﬂection spectra for the HT and LT phases of TTTA single crystal were measured at room temperature in the energy range from 1 to

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4 eV by a conventional method [3, 4]. The polarized reﬂection spectra of the HT phase were examined on the (100) surface. In the E b polarization, the reﬂectivity peaks are observed at ∼1.8 and ∼3.2 eV, which are responsible for the purple color of the HT phase. Here, E and b denote the electric ﬁeld and the crystallographic b-axis, respectively, and the b-axis is along the direction of 1D stacking of the radical molecules. In the E ⊥ b polarization, the reﬂectivity peak at ∼3.2 eV becomes enhanced, while the strong one at ∼1.8 eV disappears and a weak reﬂectivity peak remains at ∼2.1 eV. The reﬂectivity peaks at ∼3.2 eV and around ∼2 eV are assigned to intramolecular π–π transitions and CT bands, respectively. The crystalline structure of the HT phase belongs to the P 21 /c space group and has four molecules per unit cell. Thus there exist four crystalline excited states having Ag , Au , Bg , and Bu symmetries [4]. Since the unit cell has an inversion symmetry, the g → u selection rule is applicable. According to the selection rule, two transitions, Ag → Au and Ag → Bu are allowed as Davydov components. The former is polarized parallel to the b-axis and the latter in the ac-plane. According to the above consideration, the reﬂectivity peaks located at ∼1.8 and ∼2.1 eV are assigned to the b- and ac-Davydov components, respectively. To obtain the LT phase of TTTA crystal, the sample was kept at 77 K for several hours. Then the polarized reﬂection spectra of the LT phase were measured at room temperature. The strong reﬂectivity peaks are observed at ∼1.8 and ∼2.2 eV in the E b polarization, which are responsible for the green color of the LT phase. On the other hand, in the E ⊥ b polarization, the reﬂectivity peaks are observed at ∼2.0 and ∼3.4 eV, indicating that strong dichroism remains even in the LT phase. 5.3.3 Photoinduced Magnetic Phase Transition Since the TTTA crystal shows a magnetic bistability over the wide temperature range from 230 to 310 K, Takeda et al. [5] and Matsuzaki et al. [6] independently examined the photoinduced phase transition from the LT to HT phases at room temperature near the phase boundary. To conﬁrm that the photoinduced magnetic phase transition really occurs on TTTA, Takeda et al. [5] measured Raman spectra of the HT and LT phases, and that of the LT phase after a single-shot laser irradiation with diﬀerent wavelengths and photon intensities at 290 K. The Raman spectra were measured by an FT-Raman spectrometer coupled with an excitation light source of a cw-Nd:YAG laser (1.064 µm) and an optical microscope. The laser irradiation upon the LT phase of TTTA crystal to drive the photoinduced phase transition was carried out using a second harmonic (532 nm) and a third harmonic (355 nm) of an Nd:YAG laser with a pulse duration of 10 ns, whose excitation energies locate at the CT and intramolecular π–π transition bands, respectively. The polarization direction of the irradiation laser pulses was parallel to the molecular stacking axis.

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a) HT

b) LT

c) 532 nm exc.

d) 355 nm exc.

Raman Intensity (a. u.)

0

7.2x1020 cm−3

8.5x1020 cm−3

2.8x1021 cm−3

4.3x1021 cm−3

4.3x1021 cm−3

8.5x1021 cm−3

0

0

0

400

600

800

Raman Shift

400

600

800

(cm−1)

Fig. 5.9. Raman spectra of the (a) HT and (b) LT phases, and that of the LT phase after the photoexcitation of (c) the CT band and (d) the intermolecular π–π transition band at 290 K in a frequency region between 300 and 900 cm−1 . New Raman modes and the doublet structures appeared in the LT phase are denoted by open circles and arrows, respectively. Dotted lines are the simulated spectra as described in the text [5]

Figure 5.9a, b shows Raman spectra of the HT and LT phases, respectively, in a frequency region between 300 and 900 cm−1 . The Raman spectrum of the HT phase in this region consists of mainly four vibrational modes of 434.3, 501.8, 681.1, and 779.5 cm−1 , while that of the LT phase shows a lot of vibrational modes. The observed Raman frequencies of the HT and LT phases in a whole frequency region are listed in Table 5.5. To elucidate the origin of the observed vibrational modes, Takeda et al. [5] calculated the Raman and IR frequencies of a TTTA molecule by using a Gaussian 98 package program with the exchange correlation of B3-LYP and the 6-311†g basis functions. Since a TTTA molecule consists of eight atoms, 18 vibrational modes are found as shown in Table 5.6. Although the observed Raman frequencies of the HT phase are not in good agreement with the calculated ones probably

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Table 5.5. Observed Raman frequencies of the HT and LT phases in a whole frequency region between 100 and 3,500 cm−1 [5] HT phase (cm−1 )

LT phase (cm−1 )

108.4 208.7 (w)

114.2 193.3 223.0 (w) 268.5 291.6 (w) 347.6 395.8 434.3 (s) (507.6, 517.3) (s) 555.8 (w) (669.6, 679.3) (s) 752.5 (779.5, 787.3) (s) 854.7 978.1 (w) 1012.9 (w) 1103.5 (w) 1186.4 1213.4 1319.5 (w) 1342.7 (s) (1367.7, 1375.4) (w) 1431.4 (w) 1450.6 (w)

272.4 (w) 285.9 (w) 343.7 (w) 434.3 501.8 555.8 681.1

(s) (s) (w) (s)

779.5 (s) 978.1 (w) 1107.4 (w) 1182.6 (w) 1213.4 1346.5 (s) 1369.6 1431.4 (w) 1448.7 (w)

The notation s and w denote the strong and weak Raman intensities, respectively. The Raman frequencies in each parenthesis in the LT phase represent the doublet structure

due to the diﬀerent molecular structure between solid and molecular phases in TTTA, the observed strong vibrational modes around 1,350 cm−1 might be attributed to the C = N stretching and S–N stretching modes in a planar molecular structure of TTTA, respectively. Although the assignment of the Raman modes in TTTA crystal is still disputable, some striking features are found: (1) New intense Raman modes emerge at 752.5 and 854.7 cm−1 in the LT phase (open circles) (2) Some Raman modes in the HT phase become doublet structures in the LT phase (downward arrows)

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Table 5.6. Calculated vibrational frequencies, Raman and IR intensities of a TTTA radical molecule [5] Frequency (cm−1 ) 138.6 164.8 242.8 291.8 373.1 440.4 447.6 454.5 560.1 586.1 650.0 653.6 654.3 700.3 984.2 1164.0 1423.6 1515.3

Raman intensity 0.311 0.059 0.426 5.185 17.47(s) 23.27(s) 5.227 0.028 6.949 7.298 0.167 21.20(s) 0.687 14.68(s) 1.597 2.873 28.96(s) 3.441

IR intensity 5.079 0.0003 3.354 0.540 0.286 0.425 10.91(s) 8.679 0.464 0.0000 0.0003 18.98(s) 8.762 13.43 0.668 32.60(s) 15.89(s) 1.166

Since the LT phase exhibits strong dimerization along the stacking direction, the periodicity along the stacking direction in the LT phase would become almost doubled compared to that in the HT phase, leading to the doublet structure in the LT phase due to the zone folding eﬀect. The observed doublet structure optically conﬁrmed that the phase transition from the HT to LT phase in TTTA is accompanied with the strong dimerization. Figure 5.9c, d shows Raman spectra after irradiation of a single laser shot having the 532 and 355 nm excitation wavelengths at various photon densities, respectively. The photoinduced Raman spectra were taken from the laserirradiated part of the sample. The color of the laser-irradiated part on the crystal surface changes from green to purple, and shows the same polarization dependence as the HT phase. This result strongly indicates that part of the LT phase is transformed into the HT phase by the excitation of the CT and intramolecular π–π transition bands. The Raman spectra shown in Fig. 5.9c, d seem to consist of a mixture of the Raman spectra in the LT and HT phases: for instance, at the doublet of the 779.5 and 783.3 cm−1 , the intensity of the lower frequency mode is larger than that of the higher frequency mode after the irradiation, implying that part of the lower frequency mode comes from the 779.5 cm−1 mode of the HT phase. The irradiated photon energies to drive photoinduced phase transition lie at

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the CT and intramolecular π–π transition bands, while the photon energy to measure the Raman spectra (1.064 µm) locates at a transparency energy region to prevent damage/degradation of samples. Therefore, we should detect the Raman scattering from both the photoinduced HT fractions near the surface and the remaining LT fractions at the bulk. The amount of the HT fractions is evaluated as follows: First, Raman intensities of the HT and LT phases as a function of frequency ω, IHT (ω), and ILT (ω), are assumed to consist of sum of Raman intensities of Gaussian shape vibrational modes. Second, Raman spectrum of the LT phase after the laser irradiation is expressed as αIHT (ω) + βILT (ω), with the fractions of the HT phase being given by α/(α + β). The simulated Raman spectra are shown by broken lines in Fig. 5.9c, d, which reproduces well the observed Raman spectra. Figure 5.10 shows conversion yield from the LT to HT phase as a function of irradiated photon density estimated from the Raman measurements (open and solid circles). A broken line is a visual guide for eyes. In the case of the photoexcitation of the CT band, the conversion rate strongly depends on the photon density; it increases almost linearly above the threshold photon density of ∼1.0 × 1021 cm−3 , and then abruptly grows up above ∼2.0 × 1021 cm−3 , followed by saturation. Similar threshold-like behavior is also observed in the photoinduced ESR measurement (open triangles). The existence of the ﬁrst slow and subsequent sudden growth with the photon density implies two diﬀerent nonlinear processes occurring in TTTA crystal. In the case of the photoexcitation of the intramolecular π–π transition band, on the other hand, 1.0

Raman meas. (355 nm) 1 Raman meas. (532 nm) ESR meas. (532 nm)

Conversion Yield

ESR Intensity (a. u.)

0.8 0.6 0.4 0.2 0.0

0

2 4 6 8 10 Photon Density (1021/cm3)

0

Fig. 5.10. Photoinduced conversion from the LT to HT phase as a function of irradiated photon density obtained from Raman scattering measurements (circles). Net increase of the ESR intensity as a function of photon density is also shown (triangles). The ESR intensity is scaled as it is visually comparable with the conversion rate obtained from the Raman measurements [5]

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the conversion to the HT phase seems to take place without a threshold photon density or with a small threshold photon density less than ∼1020 cm−3 within the experimental accuracy. Recently, the photoinduced phase transition from the LT to HT phase was also studied by microscopic IR measurements, and the threshold photon density for the photoexcitation of the intramolecular π–π transition band is estimated to be ∼7 × 1018 cm−3 [10]. The saturation of the conversion yield occurs when the photon density becomes ∼5.0×1021 cm−3 . Since only part of TTTA molecules near the surface can transform from the LT to HT phase by the photoexcitation, the maximum conversion rate is not unity. Because the penetration depth for the excitation light lying at the intramolecular π–π transition band is 3–4 times larger than that at the CT band, the maximum conversion rate due to the excitation of the intramolecular π–π transition band is larger than that of the CT band. ◦ The unit volume of TTTA crystal in the LT phase is 507.2 A3 and involves two TTTA molecules. Assuming that all of TTTA molecules in the photoirradiated part transforms from the LT to HT phase at the observed saturation density (∼5.0 × 1021 cm−3 ), we can estimate that the conversion eﬃciency from the LT to HT phase per photon is about ∼0.4. The observed conversion from the LT to HT phase takes place due to the optical excitation but not due to heating eﬀect by following reasons: (1) In the case of the 532 nm excitation, just below the threshold photon density of 1.0 × 1021 cm−3 , the conversion does not take place even after irradiation of many laser shots with high repetition rate of 15 Hz. (2) The conversion is also observed with the same characteristics even at low temperature (250 K), which is far from the critical temperature (310 K) of the phase transition from the LT to HT phase. (3) The observed threshold photon density much depends on the excitation wavelength, suggesting that the conversion is not due to the thermal eﬀect. These experimental results show that both the photoexcitations of the CT and intramolecular π–π transition bands can induce the diamagnetic to paramagnetic phase transition but with diﬀerent characteristics: the phase transition takes place above a threshold photon density with two diﬀerent nonlinear processes through the CT band excitation, while it occurs without a clear threshold photon density through the intramolecular π–π transition band excitation. The same experimental behavior was reported on the photoinduced ionic to neutral transformation of CT complex, TTF-CA [12, 13]. In TTF-CA, a single CT exciton cannot generate a macroscopic neutral phase domain, and formation of the precursor, which is the neutral molecular clusters generated on the 1D chain, plays an important role in inducing the structural phase transition. When excitation intensity is high enough, the neutral molecular clusters can grow into 1D macroscopic neutral phase order. The same situation might exist in the photoinduced magnetic phase transition in TTTA. Photoexcitation of the CT band ﬁrst induces the HT

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phase clusters randomly distributed in the stacking column as precursors, and subsequently, if the total size of the clusters is beyond a critical value, the clusters may collapse into the HT phase domain. The macroscopic HT order is eventually formed across the stacking columns. This may lead to the observed threshold-like behavior with two diﬀerent nonlinear processes. On the other hand, the photoexcitation of the intramolecular π–π transition band is expected to generate larger size of the HT clusters than that due to the photoexcitation of the CT band because of much excess energy, leading to having a very small threshold photon density.

5.4 Electronic Structure Calculations 5.4.1 Results Within the LDA Furuya et al. [8] performed electronic structure calculations of the TTTA crystal by means of an all-electron mixed basis approach, which uses both PWs and atomic orbitals (AOs) as a basis set. Totally 208 numerical AOs and 2,135 (for the HT phase) or 2,401 (for the LT phase) PWs corresponding to a 12 Ry cutoﬀ energy are used. The number of special k points is 33. The resulting valence band structures are shown in Figs. 5.11 and 5.12, respectively, for the HT and LT phases. Fermi surface (FS) exists in the HT phase (Fig. 5.11) and nearly 0.8 eV bandgap appears in the LT phase (Fig. 5.12). In view of Figs. 5.3 and 5.4, one can recognize that the structural phase transition from the P 21 /c (in the HT phase) to the P ¯1 (in the LT phase) symmetries takes place due to the Jahn–Teller eﬀect, because the static Jahn– Teller eﬀect occurs when the highest occupied level is degenerate and occupied

Fig. 5.11. Calculated valence band structure in the HT phase. Energy zero is set at the Fermi level. The symmetry points are represented as Γ :(0,0,0), X:(1/2,0,0), M :(1/2,1/2,0), R:(1/2,1/2,1/2) in the components of the primitive reciprocal lattice vectors, b1 , b2 , b3 . The dispersion is weak between the Γ and M points along the molecular stacking direction

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Fig. 5.12. Calculated valence band structure in the LT phase. Energy zero is set at the middle of the bandgap. The symmetry points are represented as Γ :(0,0,0), X:(1/2,0,0), M :(1/2,1/2,0), R:(1/2,1/2,1/2) in the components of the primitive reciprocal lattice vectors, b1 , b2 , b3

spontaneously to gain the energy by removing the degeneracy. In the present case, a TTTA molecule is radical and its highest occupied molecular orbital (HOMO) shown in Fig. 5.1 is half ﬁlled. Therefore, FS exists in the HT phase (Fig. 5.11) within the LDA [7]. For T < Tc (Tc is the transition temperature), this lattice distortion is characterized by the dimerization of molecules and the corresponding phase transition can be regarded as a kind of Peierls transition (due to the quasi-1D response of the 3D electronic system) between the HT pase having the uniform 1D stacking and the dimerized LT phase. This explanation is the analogy of the well-known 1D organic radical conductor, TTF-TCNQ [14]. In general, the Peierls instability in 1D electron systems is induced by the electron–phonon interaction, which can develop due to the characteristic topology of the FS with a perfect nesting. A 1D metal is unstable toward static periodic lattice distortions below the Peierls transition temperature because of the nesting property of the FS. Such a distortion opens an energy gap at the new zone boundary, and thus the total energy decreases. In the LT phase, because of the dimerization of TTTA molecules, all twofold degenerated levels including core levels split into two separate closed shell diamagnetism (for an insulator) in the LT phase. The reﬂection spectra as well as the dielectric response function of the LT phase are calculated within the random phase approximation (RPA) by Furuya et al. [8]. The frequency and wave number dependent dielectric response function is given within the RPA as ε(q, ω) = 1 + 4πχ(q, ω), (5.1) iq · r 2 2 |k, ν| |k + q, λ|e χ(q, ω) = − 2 [f0 (εk+q ,λ ) − f0 (εk,ν )], Ωq ε − εk,ν − ω − iδ k λ ν k+q ,λ (5.2)

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where the summations with respect to the level indices λ and ν run over all levels (300 levels are taken into account in the calculation), k is the wave vector inside the ﬁrst Brillouin zone, and f0 (ε) denotes the Fermi–Dirac distribution function. The factor of 2 in the prefactor is due to the spin multiplicity. There are components which couple transverse and longitudinal electromagnetic disturbances. Using he commutation relation between H and r, one may rewrite in the limit q → 0 as χαβ (0, ω) 2 k, λ|∇α |k, ν · k, ν|∇β |k, λ [f0 (εk+q ,λ ) − f0 (εk,ν )] , =− 3Ω εk+q ,λ − εk,ν − ω − iδ (εk,λ − εk,ν )2 k λ ν (5.3) where the indices α and β of χαβ represent the directions x, y, z of the induced and external electric ﬁelds. The calculated dielectric response function is presented in Figs. 5.13 and 5.14 with the electric ﬁeld E of lights parallel and perpendicular to the stacking axis, respectively. The real and imaginary parts satisfy the Kramers– Kronig relation with high accuracy. There is a signiﬁcant diﬀerence between Figs. 5.13 and 5.14, which indicates the dielectric response depends strongly on the direction of the electric ﬁeld. Figure 5.15 represents the reﬂectivity spectra in the LT phase. The thick and thin curves represent the reﬂectivities for the electric ﬁeld E of lights parallel and perpendicular to the stacking axis, respectively. If we compare these results with the experimental reﬂection data (Fig. 5.15b), we ﬁnd close similarity in the overall behavior, although there is a diﬀerence in peak positions and amplitudes: The large peak at 1.3 eV in the thick curve (E ⊥ stacking direction) corresponds to the experimental peak at 2.2 eV, while the two peaks at 1.8 and 3.1 eV in the thin curve (E stacking direction) corresponds to the

Fig. 5.13. Dielectric response function for the electric ﬁeld perpendicular to the stacking direction in the LT phase. The thick and thin curves represent, respectively, the real part ε1 (ω) and the imaginary part ε2 (ω)

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Fig. 5.14. Dielectric response function for the electric ﬁeld parallel to the stacking direction in the LT phase. The thick and thin curves represent, respectively, the real part ε1 (ω) and the imaginary part ε2 (ω)

Fig. 5.15. Reﬂectivity spectra in the LT phase. The thick and thin curves represent the results for the electric ﬁeld E of the lights parallel and perpendicular to the stacking axis, respectively

experimental peaks at 1.9 and 3.4 eV. The amplitude of the calculated reﬂectivity is overestimated about two or three times larger than the experimental reﬂectivity. This discrepancy remains to be solved in the future, although a possible explanation on this problem might be as follows: In general, the LDA underestimates the bandgap in insulators typically 30–50%, and this could explain the shift of the peaks. Since the present TTTA molecular crystal is classiﬁed undoubtedly into one of the highly correlated systems such as the Mott insulators, the many-body eﬀect due to the electron–electron interaction must be inevitably taken into account. The HT phase is, however, not optically a metal in experiments. As will be shown in Sect. 5.4.2, even a calculation based on the local spin density approximation (LSDA) does not solve this discrepancy. This problem will be only solved if one goes beyond the density functional theory (DFT).

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5.4.2 Breakdown of the LDA Figures 5.11 and 5.12 represent, respectively, the energy bands of the HT phase and the LT phase of the TTTA crystal. calculated within the local density approximation (LDA) of the (spin-independent) DFT [7]. Here the energy zero is located at the Fermi level. Obviously the HT phase is a halfﬁlled metal, while the LT phase is a band insulator. Figure 5.16 represents the electronic density of states (DOS) for the HT phase calculated with the LSDA of the (spin-dependent) DFT. This result also indicates a half-ﬁlled valence band characterizing a weakly ferromagnetic metal in contradiction to the experimental characteristic of an antiferromagnetic insulator. In reality the antiferromagnetic spin order would double the crystal periodicity and induce a bandgap at the boundary of the folded Brillouin zone [15]. However, the standard band calculations failed to reproduce this true characteristic. In this sense, the HT phase is a typical molecular Mott insulator, where each unpaired electron is localized at each molecule due to the extremely strong electron correlation (see Fig. 5.2). The on-site Coulomb energy U plays a central role in strongly correlated systems [16, 17]. It has been long demonstrated that the correct treatment of the short-range electron correlation is inevitable in the evaluation of U as ﬁrst pointed out by Kanamori [18] in his pioneering work on the metallic ferromagnetism of Ni. To evaluate U of the HT phase of the TTTA crystal, we carry out a ﬁrst-principles calculation by beginning with the GW calculation [19–22] and then taking into account the multiple scatterings between electrons by collecting all ladder diagrams up to the inﬁnite order. It is well known in the ﬁeld of nuclear physics that the short distance behavior can

Fig. 5.16. The unrealistic electronic DOS of the HT phase calculated within the spin-dependent DFT. Energy zero is set at the Fermi level. Left and right sides represent the down and up spins, respectively. After Ohno et al. [9]

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be correctly described by these diagrams [23, 24]. The details of the calculation are explained in the following section. One advantage of this method is that one can determine the two-particle wave function associated with the two-particle state. Besides the following sophisticated calculation of the on-site U , we also estimate a simple and naive expectation value of the nonlocal screened Coulomb interaction W (r, r ) between electrons, Wij = ρi (r)W (r, r )ρj (r )dr dr , (5.4) for the same one-molecule system as above or a system with two molecules inside a unit cell in the same geometry as the HT phase. Here, ρi (r) = |ψi (r)|2 denotes the LDA electron density of the (half ﬁlled) topmost valence level of the ith molecule (i = 1, 2). According to our calculation based on the all-electron mixed basis approach within the LDA and RPA, the intermolecular and intramolecular Coulomb integrals are estimated to be W12 = 1.3 eV and W11 = 5.7 eV. (They become 5.8 and 7.3 eV if we use the bare Coulomb interaction.) While the intermolecular screened Coulomb energy W12 correctly represents the binding energy of the charge-transferred exciton, the intramolecular screened Coulomb energy W11 does not represent the true on-site energy, because the RPA cannot express the short-range correlation between up-spin (↑) and down-spin (↓) electrons [18]. In an ideal electron gas system, this eﬀect of the short-range correlation is well-known as the Coulomb hole where the ↑ and ↓ electrons cannot approach each other [25, 26]. This is essentially diﬀerent from the socalled exchange hole where the ↑ and ↑ (or ↓ and ↓) electrons cannot approach each other due to the Pauli principle. The resulting naive expectation value of the on-site screened Coulomb interaction W11 should be compared with the on-site U evaluated more accurately with the T -matrix calculation that includes the eﬀect of short-range correlations (see below). 5.4.3 T -Matrix Theory To deal with the short-range correlation correctly, we carry out a ﬁrstprinciples calculation that begins with the GW approximation (GWA) [19–22] for the one-particle Green’s function beyond the DFT, and goes into the T -matrix calculation for the two-particle Green’s function that accounts for multiple scatterings between electrons up to the inﬁnite order within the ladder approximation. So far, there has been no such full T -matrix calculation except for a calculation concerning only the imaginary part [27] and our previous calculation of double ionization energy spectra of atoms and small molecules [28]. For the GWA and the T -matrix theory, see Chaps. 6

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and 7, respectively. The T -matrix obeys the ﬁeld-theoretical Bethe–Salpeter equation (BSE) [29] T = W + W KT,

(5.5)

where K is the two-particle propagator composed of 2 one-particle Green’s functions as K(1, 2|1, 2 ) = iG(1 , 1)G(2 , 2), and W is the screened Coulomb interaction in the static approximation W (1, 2) = W (r1 , r2 )δ(t1 − t2 ). We determine this W within the RPA [30]. It is convenient to Fourier transform (5.5) from t to ω and sandwich K and W by four one-particle eigenstates, i.e., ψα (r 1 )ψβ (r 2 ) from the left and ψν∗ (r 1 )ψµ∗ (r 2 ) from the right. In this representation, K is diagonal and expressed as Kαβνµ = Kνµ δαν δβµ with ⎧ 1 ⎪ ⎨ − ω − Eν − Eµ − iη for occupied ν, µ, Kνµ = (5.6) ⎪ 1 ⎩+ for empty ν, µ, ω − E − E + iη ν

µ

where Eν , Eµ are the GWA quasiparticle energies, and η is a positive inﬁnitesimal number. Then we can rewrite BSE (5.5) as a matrix eigenvalue equation Hαβνµ Aνµ (Ω) = ΩAαβ (Ω) (5.7) νµ

for the two-particle states, where Aνµ (Ω) denotes a coeﬃcient for the twoparticle wave function A∗νµ (Ω)ψν (r 1 )ψµ (r 2 ) (5.8) ψΩ (r 1 , r 2 ) = νµ

and

Hαβνµ =

fνµ − ω δαν δβµ − Wαβνµ fνµ Kνµ

(5.9)

denotes a two-particle Hamiltonian matrix element in which the ω-dependence is cancelled out. In (5.9), we deﬁned − 1 for occupied ν, µ, (5.10) fνµ = + 1 for empty ν, µ. The eigenvalues Ω represent the poles of the T -matrix or equivalently the poles of the two-particle Green’s function; they represent the double ionization energy spectra [28]. The on-site Coulomb energy U can be estimated from the relation [18], U = Ωmin + Eα + Eα , where Ωmin is the lowest eigenvalue and Eα denotes the GWA quasiparticle energy of the topmost valence level. We apply this method on the basis of the all-electron mixed basis approach to a large supercell system that contains one TTTA molecule in it.

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5.4.4 Results in the T -Matrix Theory For a single TTTA molecule, we solve the BSE (5.5) for the multiple scattering between electrons, and obtain the lowest eigenvalue Ωmin . The result is found to be Ωmin = 16.8 eV (or 20.1 eV if we use the bare Coulomb interaction for W ). Then, the on-site Coulomb energy is estimated to be U = Ωmin + Eα + Eα = 16.8 − 6.95 − 6.95 = 2.9 eV,

(5.11)

where Eα denotes the GWA quasiparticle energy of the topmost valence level. The eﬀect of the short-range correlation can be seen if we compare the above result of the on-site U with the naive expectation value of the screened Coulomb interaction W11 of (5.4). From this comparison, we ﬁnd that the intramolecular on-site energy U is signiﬁcantly reduced from the naive expectation value of the intramolecular Coulomb interaction W11 ; the original intramolecular interaction of W11 = 5.7 eV (or 7.3 eV when the bare Coulomb interaction is used instead of the screened Coulomb interaction) reduces to U = 2.9 eV. This is a surprisingly huge and drastic change and demonstrates the importance of the short-range correlation between electrons. This value, U = 2.9 eV, minus the binding energy, W12 = 1.3 eV, of the charge-transferred exciton equals 1.6 eV, which satisfactorily explains the experimentally observed optical gap of approximately 1.5 eV [3]. Let us show the two-particle wave function ψΩ (r 1 , r 2 ) of the topmost valence level of a single TTTA molecule. It is composed of π orbitals only and is antisymmetric with respect to the molecular plane as well as the oneelectron wave function ψα (r1 ) at the topmost valence level. Figure 5.17a, b shows, respectively, the two-electron probability distribution |ψΩ (r 1 , r2 )|2 and the one-electron probability distribution |ψα (r 1 )|2 , on a plane parallel to and ◦ 0.2 A oﬀ from the molecular plane. The two-electron probability distribution (Fig. 5.17a) is depicted as a function of the position of the ﬁrst electron r1 provided that the second electron is ﬁxed at the in-plane position r 2 marked by a cross near the nitrogen atom in the S–N–S moiety. We ﬁnd that the two-electron distribution (Fig. 5.17a) is considerably distorted from the oneelectron distribution (Fig. 5.17b) in particular in the vicinity of the position r2 of the second electron. That is, a major population is shifted away from the position r 2 of the second electron. This electron depletion around r2 clearly manifests the Coulomb hole. The strong Coulomb interaction inhibits quantum mechanically the two electrons’ motion to approach each other. This eﬀect reduces the value of the on-site Coulomb energy U signiﬁcantly from the naive expectation value of the intramolecular screened Coulomb interaction W11 . At the same time, this strong electron–electron repulsion becomes the origin of the localization of individual electrons. Now we discuss the energetics. According to our (spin-independent) band calculation within the DFT, the total energy diﬀerence between the HT and LT phases amounts to be 0.077 eV per molecule. However, this value is extremely large compared to the experimentally obtained enthalpy change of

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(b)

Fig. 5.17. The probability distributions of (a) the two electrons (the second electron is assumed to lie at the point marked by the cross) and (b) the single electron. The ◦ contour lines are depicted on a plane parallel to and 0.2 A oﬀ from the molecular plane. The signiﬁcant change of the distribution from (b) to (a) is due to the strong repulsion from the second electron and clearly indicates the Coulomb hole. Small solid circles indicate the projected positions of one sulphur, two nitrogen, two carbon, two sulphur, and one nitrogen atom from up to down. After Ohno et al. [9]

0.024 eV per molecule [3]. This discrepancy may be attributed to the missing ground state of the HT phase. When the electronic structure is reconstructed from the DFT metal phase to a Mott insulator phase, a certain amount of energy reduction is expected. In fact, in the DFT metal phase, electrons are equally distributed and the probability of two electrons occupying a molecule is 1/4, while this probability is zero in the real Mott insulator phase. Therefore, the potential energy decreases by an amount of U/4 per molecule when changing into the Mott phase. Then, according to the exact virial theorem [31], this change is accompanied by an increase in a kinetic energy by an amount of U/8. In addition, there is also an energy loss from the band energy because each valence level will be occupied by one electron everywhere in the whole Brillouin zone (not by two electrons below the Fermi level as in the metal phase). Speciﬁcally, the energy corresponding to the average band width B = |εk − εF |∼0.31 eV is additionally required for each electron. By summing all these energy changes, we estimate the energy reduction to be ∆E = U/8 − B∼0.05 eV per molecule, which certainly explains the abovementioned discrepancy between the DFT and experimental enthalpy changes. Although this is a rough estimate, the consistency of the values would suggest the validity of the present calculation. Next, by ﬁtting the DFT dispersion curves of the valence band of the HT phase (Fig. 5.18) to the tight-binding model, we extract the transfer integrals tij as given in Table 5.7. Figure 5.18 indicates what each symbol denotes. The largest transfer integral p = 0.123 eV arises between two adjacent molecules in

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Fig. 5.18. The transfer integrals between the TTTA molecules in the HT phase. The symbol p denotes the transfer integral in the stacking direction and q denotes the intracolumn interactions. The resulting ground state spin conﬁguration is shown by up and down arrows. After Ohno et al. [9] Table 5.7. Transfer integrals obtained by ﬁtting the LDA energy band for the HT phase to the tight-binding model. After Ohno et al. [9] Symbol

eV

p q1 q2 q3 q4 q5 q6

+0.123 +0.087 +0.002 −0.018 −0.035 −0.009 −0.005

the stacking direction. Because U p, the localization of individual electrons take place in separate molecules. Thus we conclude that the HT phase is a typical Mott insulator. Using the on-site Coulomb energy U and the transfer integrals, we can construct the Hubbard Hamiltonian [17]. From this Hubbard Hamiltonian, we can extract the spin Hamiltonian [32] by using the second-order perturbation theory starting from the strong interaction limit U → ∞. The resulting spin Hamiltonian stands for 1 H = − , (5.12) Jij si · sj − 4 i=j

where si means the electron spin at site i, and the exchange interaction is given by Jij = −

2t2ij . U

(5.13)

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Since the largest |Jij | arises in the stacking direction, we conclude that the HT phase of the TTTA crystal should have the antiferromagnetic spin order in this direction. There also exist weak interactions in diﬀerent directions. The ground-state spin conﬁguration is depicted in Fig. 5.18. N´eel temperature TN and paramagnetic susceptibility χP are roughly estimated by the transfer matrix approach of the 1D antiferromagnetic chain weakly coupled by interchain interactions. The results are TN ∼310 K and χP ∼4.0 × 10−3 emu mol−1 ; these are comparable to the experimental data [3], TN ∼360 K and χP ∼4.7 × 10−3 emu mol−1 . It is also possible to determine the diamagnetic susceptibility of the LT phase from ﬁrst principles by using the second-order perturbation theory within the DFT [33]. There are two separate contributions; the core contribution χcore and the valence-empty contribution χve . We ﬁnd χcore = −2.226 × 10−5 emu mol−1 and χve = −4.551 × 10−5 emu mol−1 . Thus the LT diamagnetic susceptibility χd is estimated as χd = −6.78 × 10−5 emu mol−1 . 5.4.5 Concluding Remarks As a conclusion, to take into account the eﬀect of short-range correlation between electrons, we carried out a ﬁrst-principles calculation solving the BSE of the T -matrix theory. We found that, in the Mott insulating (HT) phase of the TTTA crystal, the Coulomb hole plays an essential role and the resulting on-site Coulomb energy U = 2.9 eV is considerably smaller than the naive expectation value of the screened Coulomb interaction W11 = 5.7 eV. We explicitly demonstrated this behavior by showing the two-particle wave function. This strong electron–electron repulsion causes the localization of individual electrons in diﬀerent molecules. The on-site Coulomb energy U for the LT phase would become further less because of the itinerancy of the electrons in the dimerised molecules. The enthalpy change across the phase transition between the HT and LT phases was estimated from ﬁrst principles and successfully compared with the experiment. We determined also the spin alignment, the N´eel temperature and the paramagnetic susceptibility of the HT phase as well as the diamagnetic susceptibility of the LT phase. The agreement with available experimental data suggests the signiﬁcance of the short-range correlation in organic Mott insulators. Acknowledgments The authors thank K. Awaga for helpful discussions. This work has been partly supported by the Grant-in-Aids for Scientiﬁc Research B (nos. 17310067 and 17310068) and Scientiﬁc Research for Priority Area (Grant no. 18036005) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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References 1. G. Wolmersh¨ auser, R. Johann, Angew. Chem. Int. Ed. Engl. 28, 920 (1989) 2. W. Fujita, K. Awaga, Science 286, 261 (1999) 3. W. Fujita, K. Awaga, H. Matsuzaki, H. Okamoto, Phys. Rev. B 65, 64434-1 (2002) 4. J. Takeda, M. Imae, S. Kurita, T. Kodaira, Phase Trans. 75, 863 (2002) [Proceedings of PIPT2001] 5. J. Takeda, M. Imae, O. Hanado, S. Kurita, M. Furuya, K. Ohno, T. Kodaira, Chem. Phys. Lett. 378, 456 (2003) 6. H. Matsuzaki, W. Fujita, K. Awaga, H. Okamoto, Phys. Rev. Lett. 91, 17403-1 (2003) 7. M. Furuya, Y. Kawazoe, K. Ohno, Sci. Tech. Adv. Mat. 5, 689 (2004) [Proceedings of NMQP’03] 8. M. Furuya, K. Ohno, T. Morisato, Y. Kawazoe, J. Takeda, Trans. Mat. Res. Soc. Jpn. 28, 911 (2003) [Proceedings of 14th MRSJ, 2002] 9. K. Ohno, Y. Noguchi, T. Yokoi, S. Ishii, J. Takeda, M. Furuya, ChemPhysChem 7, 1820 (2006) 10. S. Oguri, O. Hanado, J. Takeda, M. Furuya, K. Ohno, S. Inoue, T. Kodaira, J. Luminescence, 112, 283 (2005) 11. W.E. Estes, D.P. Gravel, W.E. Hatﬁeld, D.J. Hodgson, Inorg. Chem. 17, 1415 (1978) 12. S. Koshihara, Y. Tokura, T. Mitani, G. Saito, T. Koda, Phys. Rev. B 42, 6853 (1990) 13. T. Suzuki, T. Sakamaki, K. Tanimura, S. Koshihara, Y. Tokura, Phys. Rev. B 60, 6191 (1999) 14. J.P. Ferraris, D.O. Cowan, V. Walatha Jr., J.H. Perlstein, J. Am. Chem. Soc. 95, 948 (1973) 15. J.C. Slater, The Self-Consistent Field for Molecules and Solids, vol. 4 (McGrawHill, New York, 1974), p. 232 16. N.F. Mott, Proc. Phys. Soc. Lond. A 62, 416 (1949) 17. J. Hubbard, Proc. R. Soc. Lond. A 276, 238 (1963) 18. J. Kanamori, Prog. Theor. Phys. 30, 275 (1963) 19. M.S. Hybertsen, S.G. Louie, Phys. Rev. Lett. 55, 1418 (1985); Phys. Rev. B 34, 5390 (1986) 20. R.W. Godby, M. Schl¨ uter, L.J. Sham, Phys. Rev. Lett. 56, 2415 (1986); Phys. Rev. B 37, 10159 (1988) 21. S. Ishii, K. Ohno, Y. Kawazoe, S.G. Louie, Phys. Rev. B 63, 155104-1 (2001); ibid. 65, 245109-1 (2002) 22. S. Ishii, K. Ohno, V. Kumar, Y. Kawazoe, Phys. Rev. B 68, 195412-1 (2003) 23. K.A. Brueckner, J.L. Gammel, Phys. Rev. 109, 1023 (1958) 24. G. Baym, L.P. Kadanoﬀ, Phys. Rev. 124, 287 (1961) 25. K.S. Singwi, M.P. Tosi, R.H. Land, A. Sj¨olander, Phys. Rev. 176, 589 (1968) 26. H. Yasuhara, Physica 78, 420 (1974) 27. M. Springer, F. Aryasetiawan, K. Karlsson, Phys. Rev. Lett. 80, 2389 (1998) 28. Y. Noguchi, Y. Kudo, S. Ishii, K. Ohno, J. Chem. Phys. 123, 144112-1 (2005) 29. See for example: A. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, (McGraw-Hill, New York, 1971) 30. S.L. Adler, Phys. Rev. 126, 413 (1962)

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31. See for example: R.P. Feynman, Statistical Mechanics: A Set of Lectures, (Westview, Oxford, 1972, 1998), p. 57 32. P.W. Anderson, in Solid State Physics, vol. 14, ed by F. Seitz, D. Turnbull (Academic, New York, 1963), pp. 99–214 33. K. Ohno, F. Mauri, S.G. Louie, Phys. Rev. B 56, 1009 (1997)

6 Ab Initio GW Calculations Using an All-Electron Approach S. Ishii, K. Ohno, and Y. Kawazoe

6.1 Introduction Investigating excited states of materials are of much interest from the viewpoint of both the theoretical and experimental physics because experiments are done in the presence of some ﬁeld to excite materials and measure some quantity. However, calculating excited states of materials using ﬁrst principles is nowadays very diﬃcult. In the present chapter, we focus on ab initio calculations to investigate excited states. Especially we focus on state-ofthe-art ﬁrst-principles calculations within the GW approximation (GWA), introduced by Hedin [1] from the viewpoint of the many-body perturbation theory (MBPT) in the quantum ﬁeld theory. we discuss clusters rather than crystals because clusters have many interesting properties such as magic number, size dependence of optical gap, and so on and some reviews of GW calculations of crystals have already published. Nowadays, the standard methods for calculating electronic structures of real materials are the local density approximation (LDA) and generalized gradient approximation (GGA) within the density functional theory (DFT) that can calculate the ground-state-properties [2–4]. Indeed, the LDA and GGAs have almost succeeded in calculating ground-state-properties such as bond lengths of molecules and crystals. First Moruzzi et al. applied the LDA to metals, succeeding in reproducing the band structures qualitatively [5]. However, there still remain some issues. For example, (a) both LDA and GGAs underestimate bandgaps of semiconductors and insulators by about 30–50% because the DFT is in principle for ground-state properties. (b) van der Waals interactions are not reproduced. (c) The so-called strongly correlated systems are not described. (d) Kohn–Sham eigenvalues of the HOMO level via Koopmans theorem underestimate ionization potentials of isolated systems. Going beyond those diﬃculties, the MBPT is one of useful approaches in which the GWA is a good approximation for the evaluation of one-particle quasiparticle (QP) energies. Of course, there are other approaches such as the time-dependent DFT [6] and conﬁguration interaction method.

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Ab initio GW calculations were ﬁrst performed by Hybertsen and Louie [7] using plane wave basis set with a pseudopotential approach, succeeding in reproducing the experimental bandgaps of typical semiconductors and insulators that the LDA and generalized gradient approaches (GGAs) within the DFT underestimate by about 30–50%. (of course, the DFT is exact and the LDA is a good approximation for the description of ground state properties). In Sect. 6.2, we explain the MBPT and GWA and our technical details. In Sect.6.3, we explain our basis (all-electron mixed basis approach). In Sect. 6.4, we concentrate on the GW QP energies of small clusters and molecules rather than crystals. We just explain the result of gallium arsenide crystal as well as its clusters. In Sect. 6.4, we discuss the relation to the Ward–Takahashi identity and the invalidity of self-consistent GW calculations. Section 6.5 is devoted to summary.

6.2 Many-Body Perturbation Theory and GW Approximation The GWA is introduced from the viewpoint of MBPT by Hedin [1]. In the MBPT, the equation to solve is the Dyson equation, which is given for inhomogeneous systems by (T + Vext + VH )ψn,k (r) + dr Σ(r, r ; En,k )ψn,k (r ) = En,k ψn,k (r), (6.1) where T , Vext , VH , and Σ are the kinetic energy operator, the external potential, the Hartree potential, and the self-energy, respectively. The primary task is to evaluate the self-energy operator in (6.1). The self-energy operator is deﬁned through the equations: W (1, 2) = v(1, 2) + d(34)v(1, 3)P (3, 4)W (4, 2), (6.2) P (1, 2) = −i

d(34)G(1, 3)G(4, 1+ )Γ (3, 4; 2),

(6.3)

d(34)G(1, 3)Γ (3, 2; 4)W (4, 1+),

(6.4)

Σ(1, 2) = i

δΣ(1, 2) G(4, 6)G(7, 5)Γ (6, 7; 3). δG(4, 5) (6.5) Here we adopt the notation 1 ≡ (x, y, z, σ, t) and so on. v(1, 2), W (1, 2), P (1, 2), G(1, 2), and Γ (1, 2; 3) denote the bare Coulomb interaction, dynamically screened Coulomb interaction, polarizability function, the one-particle Green’s function, and the three-point vertex function, respectively. So far, everything is exact. Γ (1, 2; 3) = δ(1, 2)δ(1, 3) +

d(4, 5, 6, 7)

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Here, we employ an approximation called the GWA introduced by Hedin [1]. In the GWA, Γ is approximated by Γ (1, 2; 3) = δ(1, 2)δ(1, 3).

(6.6)

P (1, 2) = −iG(1, 2+ )G(2, 1).

(6.7)

Then P becomes This is equivalent to the random phase approximation (RPA). Finally, the self-energy operator is given by Σ(1, 2) = iG(1, 2)W (1+ , 2).

(6.8)

This approximation is the GWA because of the symbolic representation for the self-energy, Σ = i GW. If one employs another approximation, Σ(1, 2) = iG(1, 2)v(1, 2),

(6.9)

and if the corresponding Dyson equation is solved self-consistently, one retrieves the Hartree–Fock approximation. After the Fourier transformation with time to ω, the self-energy operator is given by i Σ(r, r ; ω) = dω G(r, r ; ω + ω )W (r, r ; ω )eiηω , (6.10) 2π which is nonlocal in space and energy dependent. Usually, we use the undressed Green’s function G0 instead of dressed Green’s function G to evaluate the self-energy and W . LDA wave functions and LDA eigenvalues are used for constructing G0 , giving G0 (r, r , ω) =

LDA∗ ψnLDA ,k (r)ψn ,k (r ) , ω − En k ± iδ

(6.11)

n ,k

where, in the denominator, + and − should be used for empty and occupied states, respectively. W in the Fourier space is given by WG,G (q, ω) = [−1 ]G,G (q, ω)v(q + G ),

(6.12)

where v(q + G) = 4π/Ω|q + G|2 is the bare Coulomb potential in the Fourier space (Ω is the volume of the unit cell) and G,G (q, ω) is the dielectric matrix deﬁned by (6.13) G,G (q, ω) = δG,G − v(q + G)PG,G (q, ω)

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with the polarizability function within the RPA, occ emp n, k|e−i(q+G)·r |k + q, n n , k + q|ei(q+G )·r |k, n PG,G (q, ω) = En,k − En ,k+q − ω + iδ n n

k

−

emp occ n

n

n, k|e−i(q+G)·r |k + q, n n , k + q|ei(q+G )·r |k, n . En,k − En ,k+q − ω − iδ

The Fourier transform of (6.12) to real space is then W (r, r ; ω) = ei(q+G)·r WG,G (q, ω)e−i(q+G )·r .

(6.14)

(6.15)

q G,G

Finally, we move on the evaluation of the self-energy. The self-energy operator can be divided into two parts (Σx and Σc ). Σx is deﬁned by i v(r − r ) eiω η G(r, r ; ω )dω . Σx (r, r ) = (6.16) 2π The integration with ω can be done analytically and the diagonal element of Σx is given by Σx,n = ψn (r)|Σx (r, r )|ψn (r ) ψ ∗ (r)ψm (r)ψ ∗ (r )ψn (r ) n m . = − dr dr |r − r | m

(6.17)

This is the so-called Fock-exchange term. On the other hand, Σc is given by i dω eiω η G(r, r ; ω + ω )[W (r, r ; ω ) − v(r − r )] (6.18) Σc (r, r ; ω) = 2π whose diagonal element is written as Σc,n = k, n|Σc (r, r ; ω)|k, n n, k|ei(q+G)·r |n , k − qn , k − q|e−i(q+G )·r |n, k = n

×i 0

∞

dω 2π

q G,G

1 ω + ω − Ek−q,n − iδk−q,n ×

+

1 ω − ω − Ek−q,n − iδk−q,n

WG,G (q, ω ) − δG,G v(q + G) .

(6.19)

After the evaluation of the self-energy, the ﬁnal GW QP energy is given by EnGWA = EnLDA + n|{Σ(EnLDA ) − µLDA xc }|n

(6.20)

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or EnGWA = EnLDA +

1 n|{Σ(EnLDA ) − µLDA xc }|n. 1 − (∂Σ(ω)/∂ω)EnLDA

(6.21)

Here µLDA is the exchange-correlation potential of the LDA and z deﬁned by xc z = (1 − (∂Σ(ω)/∂ω)EnLDA )−1

(6.22)

is called renormalization factor. Theoretically, z should be unity when calculating the QP energy because of the Ward–Takahashi identity [8, 9] (see below). In practice, the value of z is typically about 0.8–0.9 and resulting QP energy does not change signiﬁcantly within 0.2–0.3 eV in cases of isolated systems. In the case of semiconductor crystals, z makes QP energies just shift constantly. Hybertsen and Louie [7] used generalized plasmon-pole (GPP) model for the evaluation of self-energy to bypass numerical ω integration. In the present chapter we also compare GPP model and ω integration.

6.3 Choice of Basis-Set Function There are some types of GW programs, in which plane wave (PW) basis set or localized basis set is used. Every approach has a good point and at the same time has a bad point. For example, although PW approach is easy to develop a code, the calculation of core electrons is diﬃcult in practice. Although localized orbital is suitable for the description of localized states, it is impossible to describe the continuum states that is related to excited states of systems. Here, we introduce an all-electron mixed basis approach where both PWs and atomic orbitals (AOs) are used as a basis-set function. The one-electron wave function is represented by 1 (nlm) (nlm) ψk,ν (r) = Ck,ν (G) exp[i(k + G) · r] + √ Ck,ν φ (r). N j nlm t G (6.23) As an type of AOs, we can take any type of AOs, such as gaussian. In the present study, we used the numerical AOs generated by the Herman–Skillman code [10], which is rigorous within the LDA. This approach has many advantages. For example, continuum sate, which is signiﬁcant when considering excited states, is represented by PWs and core sate represented by mainly AOs. The cusp due to electrons and nucleus, is also represented because we use numerical basis set generated by Herman–Skillman code. However, coding is complicated. For example, one has to consider the many types of matrix elements of an operator V such as PW|V |PW, PW|V |AO, AO|V |AO. Moreover, in AO|V |AO for example one has to consider s|V |s, s|V |p, s|V |d, p|V |d, and so on.

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6.4 Application to Clusters and Molecules 6.4.1 Alkali-Metal Clusters In this section, we discuss QP energies of small alkali-metal (lithium, sodium, and potassium) clusters. First, the atomic conﬁgurations of alkali-metal clusters are shown in Fig. 6.1 [11, 12]. It was found that QP energies do not depend on atomic conﬁgurations especially in sodium clusters. In other words, sodium systems are typical examples of the jellium model. First, Saito et al. performed GW calculations of sodium and potassium clusters using jellium models [13,14]. However, their method was only applicable to magic number clusters because they employed spherical potentials and resulting GW QP energies of potassium clusters were not so good compared with experiments. Onida et al. performed ab initio GW calculations of Na4 [15]. We have calculated GW QP energies using the all-electron mixed basis approach [11, 12] and obtained much better results compared with [13, 14]. For sodium clusters, we show the separate contributions to GW self-energy correction in Table 6.1. For the highest occupied molecular orbital (HOMO)

Li

2.73 Å

2.94 3.16

3.19

3.33 3.17

Na

3.21

K

Fig. 6.1. Atomic conﬁgurations of alkali-metal clusters and silicon clusters

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Table 6.1. Comparison of GW quasiparticle energies of the HOMO and LUMO levels of sodium clusters with the corresponding LDA eigenvalues and experiments in eV [11] Level

EnLDA

µLDA xc,n

HOMO −3.14 −5.45 LUMO −1.89 −4.23 Na4 HOMO −2.62 −5.30 LUMO −2.10 −4.48 Na6 HOMO −2.71 −5.65 LUMO −2.19 −4.81 Na8 HOMO −2.83 −5.74 LUMO −1.58 −4.93 Note that at least the absolute potential [16] Na2

Σx,n

Σc,n

EnGWA

Expt.

−6.66 −0.73 −5.08 −4.9328 ± 0.001 −2.02 −0.66 −0.43 −0.430 ± 0.015 −5.92 −1.01 −4.25 −4.268 ± 0.054 −2.01 −1.35 −0.98 −0.91 ± 0.15 −6.24 −0.86 −4.16 −4.118 ± 0.054 −2.12 −1.53 −1.03 −6.21 −0.85 −4.15 −4.05 ± 0.054 −2.73 −1.40 −0.78 value of the HOMO level equals ionization

level, although the absolute value of the LDA eigenvalue underestimates experimental ionization potential (IP) by about 1–2 eV, the GW QP energy is in good agreement with experiment [17]. Note that the absolute value of KohnSham eigenvalue of the highest occupied level in the DFT is identical to IP [16]. Similarly, for the lowest unoccupied molecular orbital (LUMO) level, although the absolute value of the LDA eigenvalue is larger than experimental electron aﬃnity by about 1–2 eV, the GW QP energy is in very good agreement with experiment [18]. The absolute value of exchange part Σx is larger than that of correlation part Σc . This tendency was also found in other clusters. We show the results of other alkali-metal clusters (lithium and potassium clusters) in Table 6.2. The GW QP energies agree very well to experimental data [17–20], which are not reproduced by the LDA eigenvalues. Let us discuss the validity of the GPP model in [7]. We evaluated the Σc,n either by performing the numerical integration with ω or by using the GPP model. Results are shown in Table 6.3. The GPP model reproduces the results of numerical integration within about 0.15 eV, which is within the error bar of our calculation (0.2 eV). The GPP model assumes that Im −1 GG (ω) has a single peak at the frequency of ΩGG deﬁned by (30) of [7]. Although ω must be real for each component of G, G , the GPP model gives a lot of imaginary values. It naively brings us a question if the GPP model might break down. In the case of sodium clusters, we found that such cases are about 1/3 to 1/2 of the total combinations. However, in spite of this question, by simply ignoring these terms, the GPP model quite well reproduces the results of numerical integration.

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Table 6.2. GW QP energies of lithium and potassium clusters [12] compared with experiments [17–20] and LDA eigenvalues. These values are referred to [12]

Li2 Li4 Li6 Li8 K2 K4 K6 K8

Level

LDA

GWA

Expt.

HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO

−3.17 −1.73 −2.77 −2.00 −3.03 −1.81 −2.93 −1.60 −2.55 −1.64 −2.29 −1.97 −2.51 −1.81 −2.46 −1.59

−5.06 −0.47 −4.30 −0.91 −4.25 −0.94 −4.10 −0.82 −3.84 −0.72 −3.29 −0.88 −3.30 −0.81 −3.21 −0.68

−5.14 −0.437 ± 0.009 −4.31 −4.20 ± 0.05 −4.16 ± 0.05 −4.05 ± 0.05 −0.550 ± 0.010 −3.6 ± 0.1 −1.048 ± 0.025 −3.35 ± 0.03 −1.091 ± 0.020 −3.4 ± 0.01 −0.85

Table 6.3. Correlation part of the self-energy Σc of sodium clusters evaluated using numerical integration with ω or the GPP model in eV [12]

Na2 Na4 Na6 Na8

Level

Numerical integration

GPP model

HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO

−0.73 −0.66 −1.01 −1.35 −0.86 −1.53 −0.85 −1.40

−0.71 −0.66 −0.90 −1.23 −0.69 −1.49 −0.70 −1.25

6.4.2 Semiconductor Clusters In the present section, we discuss semiconductor clusters such as silicon and germanium clusters. In these covalent bond systems, a serious problem that was not important for metal clusters occurs to reproduce the experimental QP energies. In metal clusters, there is a large screening eﬀect and the structural change is not signiﬁcant when one electron is removed or attached. In contrast, in the covalent bond systems, screening among electrons is small, giving rise to a large atomic relaxation in ionization process in particular in small systems.

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The atomic conﬁgurations of silicon clusters are shown in Fig. 6.1 [21]. (For the atomic conﬁgurations of germanium clusters, see [22].) There is a considerable diﬀerence between neutral and negatively charged clusters in particular for pentamers and hexamers. In the Ge crystal, because it is well known that semi-relativistic eﬀect plays an important role, we also evaluated this eﬀect in Ge clusters. The resulting relativistic energy correction is, however, very small and found to be less than 0.1 eV for all clusters studied in [22]. This result is very diﬀerent from the Ge crystal. We show the GW QP energies calculated for the neutral geometries in Table 6.4. Similar to alkali-metal clusters, calculated IPs are in good agreement with experiments. However, EAs of Si5 , Si6 , Ge4 , and Ge5 are far from experiments by about 0.5–1.0 eV. The reason of this discrepancy is due to the following fact. The above results were calculated for neutral clusters, and Table 6.4. Comparison GW quasiparticle energies of the HOMO and LUMO levels of silicon and germanium clusters with corresponding LDA eigenvalues [21, 22] and experiments in eV [23–26] Level

LDA

GWA

Expt.

HOMO −5.56 −7.42 −7.5 LUMO −4.50 −1.92 −2.0, −1.8 Si5 HOMO −5.86 −7.57 −7.8 LUMO −3.82 −1.17 Si6 HOMO −5.59 −7.57 −7.7 LUMO −3.39 −1.25 (−) LUMO −5.12 −2.15 −2.3 Si5 (−) Si6 LUMO −4.57 −2.29 −2.2 HOMO −5.69 −7.93 −7.97 to −8.09 Ge3 LUMO −4.49 −2.11 −2.23 ± 0.01 Ge4 HOMO −5.61 −7.75 −7.87 to 7.97 LUMO −4.40 −2.07 −1.94 to 0.05 Ge5 HOMO −5.90 −7.88 −7.87 to −7.97 LUMO −3.83 −1.48 Ge6 HOMO −5.74 −7.72 −7.58 to −7.76 LUMO −3.81 −1.61 (−) LUMO −5.30 −2.34 −2.51 ± 0.05 Ge5 (−) Ge6 LUMO −4.95 −2.24 −2.06 ± 0.05 Note that at least the absolute value of the HOMO level equals ionization potential (−)0 (−)0 and Gen denote the results obtained for the [16]. The columns denoted by Sin neutral clusters having the negatively charged geometry. The corresponding GW (−)0 QP energies are corrected by the total energy diﬀerence δE = E(An ) − E(An ) (A = Si or Ge). These values can be compared with the experimentally measured (adiabatic) EAs Si4

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correspond to the vertical EAs. However, the experimental EA is determined for negatively charged clusters as the lowest photon energy required to photodetach one excess electron and neutralize the cluster. Therefore what is calculated in the theory and what is observed in the experiment are different when there is a structural diﬀerence between neutral and negatively charged clusters. In fact, the geometry change between anions and neutral clusters of pentamers and hexamers is very large as shown in Fig. 6.1 for silicon clusters. For this reason, we have performed similar GW calculations also by using the structures of anions for the pentamers and hexamers. The resulting GW (−)0 QP energies corrected by the total energy diﬀerence δE = E(An ) − E(An ) (−) (−) (A = Si or Ge) are also shown in the Sin and Gen columns in Table 6.4. These GW QP energies correspond to the experimentally observed (adiabatic) EAs, and in fact the agreement is much better than the results using the neutral geometries [21,22]. Experimentally, to measure the vertical EAs of neutral clusters is diﬃcult. To separate the mass of clusters, one has to fabricate anions, removing one electron from each cluster. The ﬁrst peak of photoelectron spectra is interpreted as the EA of neutral systems. This assumption is valid only when the structural diﬀerence between anions and neutral clusters is small such as the alkali-metal clusters. More details are given in [21, 22]. Thus we found that the behaviors of silicon and germanium clusters are similar to each other. For pentamers and hexamers, the geometry change is signiﬁcant, and the energies observed in the photoemission and inverse photoemission processes are quite diﬀerent [21,22]. In this sense, one may say that the time-reversal symmetry is broken in these processes [21]. 6.4.3 Gallium Arsenide Clusters and Crystal The situation is, however, diﬀerent in the case of gallium arsenide (GaAs) clusters [27]. For the GaAs clusters, the structural change between neutral clusters and anions is not so large and therefore there is no large diﬀerence between the adiabatic and vertical EAs. In Table 6.5, we show the GW QP energies, EnGWA , together with the LDA energy eigenvalues, EnLDA , and the experimental IPs [28] and EAs [29] with minus signs, Enexpt. . In the ﬁrst column, Gan Asn (n = 2–4) denote neutral clusters with the most stable ground-state geom(−)0 etry, while Gan Asn denote neutral clusters with the optimized geometry of anions. The former corresponds to the vertical transition, and the latter corresponds to the adiabatic transition for EA (the absolute LUMO energy is EA). In the same table, we also show the other contributions to the GW QP energies (see (6.21)), i.e., the expectation values of the LDA exchangecorrelation potential µLDA xc , the exchange part Σx , and the correlation part Σc of the self-energy. Although Σx,n almost does not depend on the cluster size for the HOMO level, it depends on the cluster size for the lowest unoccupied

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Table 6.5. The comparison (in eV) of the GW QP energies (EnGWA ) for the HOMO and LUMO levels of gallium arsenide clusters with the LDA eigenvalues (EnLDA ) and the experimental electron aﬃnities [29] with minus signs (Enexpt. )

Ga2 As2 (−)0

Ga2 As2 Ga3 As3

(−)0

Ga3 As3 Ga4 As4

(−)0

Ga4 As4

HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO HOMO LUMO

EnLDA

µLDA xc,n

Σx,n

−5.27 −3.90 −4.92 −4.39 −5.34 −3.23 −5.32 −3.46 −5.28 −3.87 −5.08 −4.26

−12.03 −11.62 −11.95 −11.76 −12.17 −10.20 −12.15 −10.49 −12.29 −11.76 −12.19 −11.74

−14.26 −7.73 −14.18 −7.90 −14.14 −6.66 −14.13 −6.93 −14.04 −8.59 −13.81 −8.59

Σc,n (EnLDA ) EnGWA −0.79 −1.67 −0.78 −1.60 −0.53 −1.87 −0.51 −1.81 −0.81 −1.65 −0.91 −1.62

−7.99 −1.90 −7.65 −2.13 −7.61 −1.71 −7.59 −1.84 −7.57 −2.50 −7.34 −2.66

Enexpt. (−7.90a ) −2.10 ± 0.10b (−7.90a ) −2.10 ± 0.10b (−7.90a ) −2.10 ± 0.10b (−7.90a ) −2.10 ± 0.10b (−7.90a ) −2.30 ± 0.10b (−7.90a ) −2.30 ± 0.10b

HOMO and LUMO correspond, respectively, to IP and EA. An available experimental ionization potential of Gan Asn is such that 6.4 eV < IP ≤ 7.9 eV [28], whose higher bound is shown inside parentheses. The ﬁnal result EnGWA is evaluated LDA through (6.21): µLDA | n, Σx,n = n | Σx | n, and Σc,n = n | Σc | n xc,n = n | µxc are the expectation values of, respectively, the LDA exchange-correlation potential, and the exchange and correlation parts of the self-energy Σ. In the ﬁrst column, (−)0 Gan Asn and Gan Asn denote neutral clusters with the geometry optimized under neutral and negatively charged conditions, respectively. The former corresponds to the vertical transition, and the latter corresponds to the adiabatic transition for EA a Reference [28] b Reference [29]

molecular orbital (LUMO) level. The absolute value of Σx,n for LUMO level (−)0 (−)0 (−)0 of Ga3 As3 is 1.0 eV, and 1.7 eV less than that of Ga2 As2 and Ga4 As4 . However, these contributions tend to be compensated by the negative contribution from µLDA xc,n , and therefore the ﬁnal results for the GWQPEs are not (−)0

with aﬀected so much. If we compare the results of Gan Asn and Gan Asn same n, each contribution (µLDA , Σ , and Σ ) to the correction to the c,n x,n xc,n LDA eigenvalues has similar values, reﬂecting the insensitiveness of each contribution to the structural diﬀerence between neutral and negatively charged clusters. Hence, the diﬀerence in the LDA eigenvalues between Gan Asn and (−)0 seems to reﬂect directly the diﬀerence in the resulting GWQPEs. Gan Asn The resulting HOMO–LUMO gap monotonically decreases with cluster size. This tendency is not seen in other semiconductor clusters such as silicon and germanium, in which the HOMO–LUMO gap does not strongly depend on the cluster size. This tendency is somewhat similar to the alkali-metal clusters.

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When we compare our results with experimental data, we ﬁrst note that the experimental IPs of Gan Asn are such that 6.4 eV < IP ≤ 7.9 eV [28]. The data are available only for Gan Asn with odd n. The resulting GWQPEs for the HOMO level of each cluster are all consistent with this experiment. Our results for the vertical (adiabatic) EAs of dimer, trimer, and tetramer estimated from the GWQPEs are 1.90 (2.13), 1.71 (1.84), and 2.50 (2.66) eV, respectively. The diﬀerence between the vertical and adiabatic EAs in Gan Asn is small and less than 0.2 eV, although that of Ge5 and Ge6 is large, about 0.8 and 0.65 eV. We have also calculated the quasiparticle (QP) energy spectra of the gallium arsenide (GaAs) crystal by means of the same method (one-shot GW in the all-electron mixed basis approach) [27]. We calculated the QP energies at Γ , X, and L points in the ﬁrst Brillouin zone. In Table 6.6, we compare the results of our calculation and previous GW calculations as well Table 6.6. Quasiparticle energies at symmetry points of GaAs crystal in units of eV GaAs

GTOa

PAWb

Γ1v Γ15v Γ1c Γ15c

−12.69 −12.64 0.00 0.00 1.32 1.26 4.60 4.19

X1v X3v X5v X1c X3c

−10.27 −10.26 −7.16 −2.71 −2.77 2.65 1.72 2.72 2.02

L2v L1v L3v L1c L3c

−11.02 −11.04 −6.91 −6.67 −1.17 −1.19 1.92 1.53 5.65 5.04

FP-PAWc

FP-LMTOd

Present

Expt.e

0.00 1.09

0.00 1.30 4.31

−12.98 0.00 1.25 4.27

−13.21 0.00 1.52 4.61

1.65 1.99

−10.75 −7.02 −2.78 2.10 2.57

−10.86 −6.81 −2.91 1.90 2.47

−11.47 −6.87 −1.14 1.75 5.48

−11.35 −6.81 −1.41 1.74 5.45

1.64

1.53

1.55

For each symmetry point, the values of GWA results and experiments (Expt.) are listed. The symbols GTO, PAW, FP-PAW, and FP-LMTO denote, respectively, the previous theoretical results using the pseudopotential gaussian-type orbital approach [30], the projector-augmented wave approach [31], the all-electron fullpotential PAW [32], and the full-potential linear muﬃn-tin-orbital approach [33]. Present denotes our all-electron mixed basis approach a Reference [30] b Reference [31] c Reference [32] d Reference [33] e References [34, 35]

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as the experimental data. For each symmetry point, the values of GW QP energies and experiments (Expt.) are listed. The symbols, GTO, PAW, FPPAW, and FP-LMTO denote respectively, the previous theoretical results using the pseudopotential gaussian-type orbital approach, the pseudopotential projector-augmented wave approach corrected by the all-electron wave functions [31], the full-potential PAW approach [32], and the full-potential linear muﬃn-tin-orbital approach [33]. All these calculations are one-shot GWA. On the whole, our all-electron calculation seems to correspond to experiments most accurately of all these calculations, which suggests the validity of the present method within the one-shot GW. In a respect that our all-electron mixed basis GW method is applicable not only clusters but also crystals in a consistent way, it is prior to the other all-electron GW methods such as the APW- or LMTO-based methods. 6.4.4 Benzene Molecule In atoms and molecules, the Hartree–Fock Approximation (HFA) via Koopmans theorem gives good QP energies of the HOMO level. In Fig. 6.2 we show our results obtained from LDA eigenvalue, GW calculation, and HFA using Gaussian 03 package [36]. Experimental IP is 9.24 eV [37]. The HFA gives fairly good result for the HOMO level. However, the LUMO energy is far from 0 eV, corresponding to the bottom of continuum state. In the present

Fig. 6.2. Energy diagrams of benzene molecule in eV: LDA, GWA, and HFA denote local density approximation, GW approximation, and Hartree–Fock approximation, respectively. The corresponding HOMO–LUMO gaps are shown in eV

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GW calculation, we take into account oﬀ-diagonal elements of Σ − µLDA that xc reduces the unoccupied levels [38]. It is found that the QP energies of the HFA is in good agreement with experimental IP. Of course, the GW QP energies also give a good result. The HOMO–LUMO gap of the LDA, GWA, and HFA is about 5.1, 8.8, and 11.1 eV, respectively. Recently, Neaton et al. [39] performed a GW calculation of benzene molecule. They obtained slightly larger HOMO–LUMO gap than our result (10.5 eV). This may be due to the neglect of the oﬀ-diagonal elements of Σ − µLDA xc . 6.4.5 Why Are LDA Eigenvalues of HOMO Level Shallower Than Experiments? The absolute value of LDA eigenvalues of the HOMO level, which is identical to the ionization potential (IP) [16], is always smaller than experiments. Let us explain this reason brieﬂy. Similar explanation is given in [4]. In the LDA, 1 exchange part of the exchange-correlation potential is proportional to ρ 3 . LDA wave functions decay exponentially in asymptotic region. For example, consider an isolated hydrogen atom. The Coulomb potential within the LDA decays exponentially, although the exact Coulomb potential must behave like −1/r in the asymptotic region. In other words, the exchange-correlation potential decays rapidly compared with reality. This means an electron of the HOMO level can be removed more easily than experiments. That is, LDA eigenvalues of the HOMO level is shallower than experiments. For example, the eigenvalues of the HOMO level obtained using the optimized eﬀective potential (OEP) method that exactly treats exchange energy gives much better IPs compared with the LDA eigenvalue. Similarly, it is more diﬃcult to attach an extra electron to neutral systems in the LDA. In spite of this, the total energy diﬀerence of E(N ± 1) − E(N ) gives good IPs and EAs. From the above discussion, E(N + 1) is larger than exact total energy of (N + 1)-electron system. Because of that, E(N ) is larger than exact one. However, the diﬀerence of these two quantities is in good agreement with experiments.

6.5 Self-Consistent GW vs. First Iterative GW (G0 W0 ) Although the Dyson equation should be solved self-consistently, it is invalid when one employs the GWA. In this section, we discuss this issue. Recently, Kotani et al. performed all-electron self-consistent GW calculations for many crystals, reproducing the experimental bandgaps [40]. They asserted that ﬁrstiterative solutions within the GWA do not give good results compared with experiments and self-consistent GW solutions are in good agreement with

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experiments. However, as is well known, ab initio GW calculations employing pseudopotential approach gives correct solutions. Which is correct? In the GWA, the vertex function is assumed to be unity, giving good QP energies at least when one solves the Dyson equation ﬁrst iteratively (G0 W0 calculation) within the GWA. To solve Dyson equation self-consistently within the GWA contradicts with the Ward–Takahashi (WT) identity [8] that guarantees local charge conservation law that any system has to satisfy. The WT identity is written as q · Γ (p, q) − q 0 Γ 0 (p, q) = G−1 (p + q) − G−1 (p).

(6.24)

Here, we assume that the system has inﬁnitesimal translational invariance for convenience. Even without this assumption, the following discussion is satisﬁed because charge conservation law 3

∂µ J µ (z) = 0

(6.25)

µ=0

is local in space. The WT identity is exact (see Sect. 6.6). The WT identity states that, if one changes the Green’s function, one also has to change vertex function at the same time to satisfy the local charge conservation law. In other words, one must change vertex function to satisfy the WT identity when performing a self-consistent calculation. This is discussed by Takada for the case of electron gas systems [41] and by Nambu for the case of ladder approximation in electron–phonon system [42]. However, Kotani et al. calculated QP energies without vertex corrections. That is, charge conservation is not satisﬁed. Therefore, although the resulting QP energies seem in good agreement with experiments, this type of calculation is physically inadequate for the systems.

6.6 Appendix: Proof of WT Identity The Ward–Takahashi (WT) identity referred to in Sect. 6.5 guarantees local charge conservative law. In this appendix, we give a proof of WT identity. There are some equivalent approaches to prove WT identities. Here, we take Engelsberg’s approach as is given in [9]. First, we deﬁne the vertex function as (µ = 0, 1, 2, 3) † Λµ (x, y, z) ≡ T {J µ (z)Ψn (x)Ψm (y)} ≡ − Γ µ (x , y , z)G(x, x )G(y , y)dx dy .

(6.26)

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The current density 4-vector J µ is deﬁned by J 0 (z) = Ψ † (z)Ψ (z), 1 † {Ψ (z)∂ k Ψ (z) − [∂ k Ψ † (z)]Ψ (z)}, 2i which satisﬁes charge conservation law: J k (z) =

3

k = 1, 2, 3,

∂µ J µ (z) = 0.

(6.27)

(6.28)

µ=0

Under the translational invariance, Fourier transforms of G and Γ are represented by d4 p G(x, x ) ≡ G(x − x ) = G(p) exp[ip(x − x )] , (6.29) (2π)4 d4 p d4 q Γµ (x , y , z) = Γµ (p, q) exp[ip(x − y ) + iq(x − z)] . (6.30) (2π)8 ∂ † † T {J µ (x)Ψn (y)Ψm (z)} = T {∂µ J µ (x)Ψn (y)Ψm (z)} ∂xµ † + δ(x0 − y 0 )T {[J 0 (x), Ψn (y)]Ψm (z)} † (z)]}, (6.31) + δ(x0 − z 0 )T {Ψn (y)[J 0 (x), Ψm

where the δ-function are from derivative with respect to time. Using the charge conservative law, the ﬁrst term of the right-hand side (RHS) in (6.31) becomes zero, then we obtain ∂ † † T {J µ (x)Ψn (y)Ψm (z)} = −qδ 4 (x − y)T {Ψn(y)Ψm (z)} ∂xµ † + qδ 4 (x − z)T {Ψn(y)Ψm (z)}, (6.32) where we used the commutation relations: [J 0 (x, t), Ψn (y, t)] = −qΨn (y, t)δ 3 (x − y), [J

0

(x, t), Ψn† (y, t)]

=

qΨn† (y, t)δ 3 (x

− y).

(6.33) (6.34)

Putting the Fourier representation of G and Γ into (6.26) the 4-divergence of the RHS of (6.26) is ∂ µ (x, y, z) = i [q · Γ (p, q) − q 0 Γ 0 (p, q)]G(p)G(p + q) ∂z µ d4 p d4 q × exp[ip(x − y) + iq(x − z)] . (6.35) (2π)8 Since the RHS and LHS of (6.26) are equal, so are their Fourier transformation: i[G(p + q) − G(p)] = i[q · Γ (p, q) − q 0 Γ 0 (p, q)]G(p)G(p + q).

(6.36)

This relation is identical to the WT identity: q · Γ (p, q) − q 0 Γ 0 (p, q) = G−1 (p + q) − G−1 (p).

(6.37)

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6.7 Summary We have reviewed the ab initio calculations for isolated systems within the GWA, which is introduced from the viewpoint of the MBPT. We used the allelectron mixed basis approach, where both atomic orbitals and plane waves are used as a basis set. For alkali-metal clusters, GW QP energies are in very good agreement with experiments. For semiconductor clusters, it is sometimes important to take into account the diﬀerence between the atomic conﬁgurations of anions and neutral clusters to reproduce experimental EAs. For benzene molecule, oﬀ diagonal elements of Σ − µLDA play an important role, making the energy xc level upper than the bottom of continuum state change dramatically. We also emphasized that the self-consistent GW calculation without vertex corrections is physically dangerous because the exact sum rule (Ward–Takahashi identity) is not satisﬁed. Acknowledgment The authors sincerely thank Prof. S.G. Louie for the study of alkali-metal clusters. One of the authors, S.I., sincerely thanks Prof. H. Yasuhara and Y. Takada for teaching and suggesting many things. The authors thank Hokkaido University information initiative center for the support of SR11000 supercomputing facilities. This work has been partially supported by the Grant-in-Aids for Scientiﬁc Research B (no. 17310067) and for Scientiﬁc Research on Priority Area (nos. 18036005 and 19019005) from the Japan Society for the Promotion of Science and from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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7 First-Principles Calculations Involving Two-Particle Excited States of Atoms and Molecules Using T -Matrix Theory Y. Noguchi, S. Ishii, and K. Ohno

7.1 Background Since the density functional theory (DFT) was established by Horhenberg and Kohn, the ab initio methods have been rapidly developed together with the improvements of the computer facilities. The local density approximation (LDA) or generalized gradient approximation (GGA) based on the DFT, in particular, have been widely applied to various real systems and have succeeded in describing the ground state properties. However, these methods based on the DFT cannot treat the excited states because of using a variational principle. An alternative approach to treat the excited states such as those associated with ﬁrst ionization potential, electron aﬃnity, and optical absorption spectra is to use the Green’s function method based on the many-body perturbation theory beyond the framework of the DFT. In recent years, the Green’s function method based on the many-body perturbation theory, which collects a speciﬁc kind of diagrams up to the inﬁnite order, has drawn an attention as a powerful ab initio methods in the treatment of excited states of real systems. For example, GW approximation (GWA) [1] in which the one-electron self-energy operator is approximated as the product of the one-particle Green’s function G and the dynamically screened Coulomb interaction W within the random phase approximation (RPA) [2] is capable to evaluate correctly the ﬁrst ionization potential (Fig. 7.1a) or the electron aﬃnity (Fig. 7.1b) [3, 4]. Another example is the calculation of the optical absorption spectra evaluating the exciton binding energy (Fig. 7.1e). In evaluation of the exciton binding energy Ee , the Bethe–Salpeter equation (BSE) for the electron–hole two-particle Green’s function, which includes the electron–hole ladder diagrams up to the inﬁnite order, is solved as an approach superior to the RPA [5–10]. However, there are many other kinds of excited states (for example, two-particle and more excited states) that remain unsolved from ﬁrst principles. Double ionization energy (DIE) and double electron aﬃnity (DEA) shown in Fig. 7.1c, d, which are deﬁned as the energy required for removing or acquired for adding two

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Fig. 7.1. Various excited states

electrons, need the evaluation of the short-range electron correlations between particles. The satellite structure (Fig. 7.1f) below Fermi level cannot be reproduced even with the GWA because of the interactions between two holes. Moreover, there is no attempt to apply the Green’s function method to the strongly correlated systems. The common feature in Fig. 7.1c–f is that the short-range electron correlation (or repulsive Coulomb interaction) plays an important role to determine these physical quantities. One of the most eﬃcient method to treat above problems is a T -matrix theory describing the multiple scattering between electrons or between holes by means of the electron–electron (or hole–hole) ladder diagrams up to the inﬁnite order. The T -matrix gives an exact behavior when the interelectron (or interhole) distance is very small. In this study, we solve the ﬁrst-principles T -matrix theory via the eigenvalue problem and apply it to the evaluation of the two-particle excited states

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(i.e., DIE or DEA spectra) of alkali-earth atoms (Be, Mg, and Ca), inert gas atoms (Ne, Ar, and Kr), alkali-metal dimers (Li2 , Na2 , and K2 ), and molecules (CO, CO2 , and C2 H2 ) For deeper understanding of the short-range electron correlations, two-particle wave functions of small systems (Ar, CO, CO2 , and C2 H2 ) are calculated from the eigenfunctions of the T -matrix. And the Coulomb hole, which appears between antiparallel spin electrons in the short-range region where interelectron distance is very small, is discussed in the comparison of the one-particle and two-particle wave functions.

7.2 Methodology: T -Matrix Theory We employ the all-electron mixed basis approach [11–13], in which both atomic orbitals (AOs) and plane waves (PWs) are used as basis set functions. All the core AOs are generated by Herman–Skillman’s atomic code [14] using the logarithmic mesh in the radial direction within the nonoverlapping atomic spheres. Valence AOs are generated in the same way but smoothly truncated at the surface of the nonoverlatting atomic sphere. The advantage of this approach is that one can express not only spatially localized states but also extended states like free electron states above the vacuum level. The following formulation is, however, very general, and not at all restricted for the all-electron mixed basis approach but applicable to any other ﬁrst-principles approaches. After the electronic states are treated within the LDA of the DFT, we carry out the calculations using the GWA (see Chap. 6 for detail discussions of the GWA) and then proceed to the T -matrix theory. Single quasiparticle energies such as the IP and EA are obtained by solving the Dyson equation G = G0 + G0 ΣG, where Σ is the electron selfenergy operator. Here, we introduce the short-hand notation 1 ≡ (r 1 , t1 ). Within the GWA, the one-particle self-energy is given by Σ GWA (1, 1 ) = iG0 (1, 1 )W 0 (1+ , 1 ), which is divided into the exchange part Σx and the rest. Usually, the dynamically screened Coulomb interaction W 0 is evaluated within the RPA. In diagrammatic sense, the RPA is equivalent to summing ring diagrams up to the inﬁnite order. In the present study, we use the generalized plasmon-pole model [3], giving reliable one-particle excitation energies [15]. To evaluate the DIE or DEA, we introduce the two-particle Green’s function via the T -matrix, which describes the multiple scattering between two particles [16] and the Coulomb hole. The T -matrix satisﬁes the following BSE [17]: T (1, 2|3, 4) = U (1, 2)δ(1 − 3)δ(2 − 4) + U (1, 2) K(1, 2|1, 2 )T (1 , 2 |3, 4)d1 d2 ,

(7.1)

where U (r1 , r 2 )δ(t1 −t2 ) denotes the bare Coulomb interaction. The schematic representation of this equation is shown in Fig. 7.2. Here, K represents the

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Fig. 7.2. BSE for T -matrix (square). The dotted line represents the bare Coulomb interaction U . The solid lines with arrows represent the one-particle Green’s function G

disconnected part of the two-particle Green’s function deﬁned by K(1, 2|1, 2 ) = iG(1 , 1)G(2 , 2). The Fourier transformation of this function is given by K(r1 r2 |r1 r 2 ; ω) = − +

occ ψν∗ (r 1 )ψν (r 1 )ψµ (r 2 )ψµ∗ (r 2 )

νµ emp νµ

ω − Eν − Eµ − iη ψν (r 1 )ψν∗ (r 1 )ψµ∗ (r 2 )ψµ (r 2 ) , ω − Eν − Eµ + iη

(7.2)

where Eν is the GWA quasiparticle energy and η is a positive inﬁnitesimal number. The relation between the T -matrix and the two-particle Green’s function (G2 ) is given by (7.3) T (1, 2|1, 2 )K(1 , 2 |3, 4)d1 d2 = U (1, 2)G2 (1, 2|3, 4). We use the LDA wave functions ψν (r) instead of the true excited wave functions throughout the calculation because they diﬀer negligibly except when Eν approaches the vacuum level [18]. The BSE (7.1) can be rewritten by introducing matrix elements sandwiched by the LDA wave functions as a matrix eigenvalue problem. In this representation, the disconnected part of the twoparticle Green’s function K becomes diagonal and expressed as Kνµ δαν δβµ . Then the BSE reads fνµ δαν δβµ − Uαβνµ fνµ fνµ Kνµ (ω)Tνµγδ (ω) = Uαβγδ , (7.4) Kνµ (ω) νµ where we deﬁne fνµ as fνµ = −δνocc δµocc + δνemp δµemp, and δνocc (δνemp ) is equal to unity if ν is occupied (empty) and zero otherwise. We solve the eigenvalue problem as follows: Hαβνµ Aνµ (Ω) = ΩAαβ (Ω). (7.5) νµ

Here, Hαβνµ are the two-particle Hamiltonian matrix elements that are independent of ω.

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Hαβνµ ≡

193

fνµ − ω δαν δβµ − Uαβνµ fνµ Kνµ (ω)

(7.6)

From the eigenvalues Ω and eigenfunctions Aνµ (Ω), one can construct the T -matrix associated with both the hole–hole and electron–electron Green’s functions. The spectra of eigenvalues Ω, which are the poles of the T -matrix, directly provide the DIE spectra. On the other hand, the eigenfunctions are the component of the two-particle wave functions ΨΩ (r, r ) as follows: A∗νµ (Ω)ψν (r)ψµ (r ). (7.7) ΨΩ (r, r ) = νµ

The present T -matrix treats accurately the short-range part of the Coulomb interaction [16, 19] that is inevitable in the determination of the DIE, DEA, or Coulomb hole. If we neglect the interaction between the two particles, the DIE or DEA spectra simply become the sum of two one-particle energies, −Eν − Eµ , obtained from the GWA.

7.3 Double Electron Aﬃnity of Alkali-Metal Clusters 7.3.1 Introduction The problem of electron correlations is particularly important for small clusters because of the conﬁned geometry. So far, the single ionization (quasiparticle) energy spectra of small alkali-metal clusters have been calculated with the state-of-the-art GWA [1], using the all-electron mixed basis approach [11–13]. The GWA is based on the time-dependent many-body perturbation theory, and can be constructed from the Hartree–Fock approximation by replacing the bare Coulomb interaction in the Fock exchange term with the dynamically screened Coulomb interaction W evaluated within the RPA [2]. However, we use it ﬁrst iteratively starting from the LDA for the reason explained in Chap. 6. Here we focus on the DEA spectra, which are interpreted as the energy required for adding two electrons to the neutral system [20]. In contrast to the single electron aﬃnity, the DEA reﬂects strongly the electron–electron repulsive interaction. Therefore, they are generally not given by the simple addition of two one-particle energies which may be obtained within the GWA. Here, we take into account of the electron–electron repulsive interaction eﬀectively by using the T -matrix theory to calculate the DEA spectra of alkali-metal dimers (Li2 , Na2 , and K2 clusters). 7.3.2 Eﬀect of the Coulomb Interaction in the DEA Spectra Figure 7.3a, b (dotted lines) shows the results of the DEA spectra obtained by the simple addition of the two GWA quasiparticle energies for K2 and

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Fig. 7.3. DEA spectra of (a) K2 and (b) Li2 . The dotted lines represent the addition of two one-particle energies obtained within the GWA, while the solid lines represent the present result including the eﬀect of electron–electron repulsive interaction approximately. After Noguchi [21]

Li2 clusters, respectively. In these ﬁgures, the energy zero means the vacuum level. The calculated spectra are those constructed only by the combination of 25 single particle levels. Therefore the resulting spectra in high energy side are artiﬁcially truncated. We notice that there are several peaks in the region of negative energies, which might suggest the existence of the bound state. However, this is a 2− wrong result because K2− 2 and Li2 are experimentally unstable. To improve this GWA result, we estimated the correction due to the electron–electron repulsive interaction approximately as 2 eV for K2 and 1.5 eV for Li2 from a rough estimate based on the T -matrix calculation [21]. Then the spectra are constantly shifted to the higher energy side by these amounts. The resulting spectra are depicted by solid lines in Fig. 7.3a, b. Obviously the bound state

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no longer exists in this corrected curves, which indicate that K2− and Li2− 2 2 are really unstable. 7.3.3 Short-Range Repulsive Coulomb Interaction Within the T -Matrix Theory An accurate calculation of the DEA on the basis of the T -matrix theory was performed in [20]. Table 7.1 represents the energy gain for attaching one or two electrons to the neutral Li2 , Na2 , and K2 , i.e., the electron aﬃnity (EA) or the DEA. In the “electron aﬃnity” column, the absolute values of the LUMO level energy obtained by the LDA and the GWA are shown together with the corresponding experimental values. The result of the LDA is larger than the experimental value, since the LDA tends to overestimate the absolute value of the LUMO quasiparticle energy. That the electron aﬃnity is estimated to be positive within the GWA demonstrates the electronic stability of these anions. The resulting electron aﬃnity is in good agreement with the experiments [22,23]. The same level GWA calculation has already been performed by Ishii et al. in [12, 13] (see also Chap. 6). In the present calculation, the spherical cutoﬀ of the Coulomb interaction is also introduced in the LDA calculation. However, the renormalization of the quasiparticle energy ((6.21) in Chap. 6 or (4) in [4]) is not considered in the present GWA. Although, for the LUMO level close to the vacuum level, the oﬀ-diagonal elements of n|Σ GWA − µLDA xc |n might become important, we did not consider such elements in this calculation. The “G1” in the DEA represents twice of the electron aﬃnity obtained by the GWA and all of these values are positive. Since the interaction between two electrons is not considered at all in deriving these values of “G1,” there appears an unphysical result that the dianions of alkali-metal dimers (Li2 , Table 7.1. The calculated electron aﬃnity within the LDA and the GWA, and the DEA within the T -matrix theory (in eV) of Li2 , Na2 , and K2 [20]

Li2 Na2 K2

Electron aﬃnity (energy gain to attach one electron) LDA GWA Experiment 1.84 0.40 0.44a 1.99 0.34 0.54b 1.86 0.50 0.55b

Double electron aﬃnity (energy gain to attach two electrons) G1 T-matrix 0.80 −1.03 0.68 −1.17 1.00 −0.77

Uαααα Coulomb −4.75 −4.49 −3.63

In the column of the double electron aﬃnity, “G1” and “T -matrix” denote, respectively, twice of the electron aﬃnity within the GWA (which does not include the eﬀect of the Coulomb repulsive interaction), and the values obtained in the T -matrix theory. In the last column, Uαααα shows the on-site Coulomb repulsive interaction sandwiched by the LUMO level (α) with negative sign a Reference [22] b Reference [23]

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Na2 , and K2 ) are stable. On the other hand, when the interaction between two electrons is taken into account by solving the BSE for the T -matrix (see the result shown in the “T -matrix” column), all these values are negative. The negative values in the column denoted by “T -matrix” mean that the dianions of alkali-metal dimers (Li2 , Na2 , and K2 ) are electronically unstable. This conclusion within the present T -matrix theory is also physically acceptable. The eﬀective Coulomb interaction between the two electrons, estimated from the diﬀerence between the values given in the columns denoted by “T -matrix” and “G1,” is almost the same for three alkali-metal dimers and amount to be about −1.8 eV for Li2 , −1.9 eV for Na2 , and −1.8 eV for K2 . The matrix elements of the on-site Coulomb interaction, Uαααα , where α is the LUMO level, is presented in the “Coulomb” column. We note that it has much larger value than the diﬀerence between “T -matrix” and “G1.” This Uαααα is the matrix element which carries the largest contribution to the DEA. In the case of Li2 , two attached electrons are more closely populated compared to the cases of Na2 and K2 (due to the shortest bond length among three alkali-metal dimers), and |Uαααα | takes the largest value. Its absolute value tends to be smaller as the atomic number of the constituent atoms increases. This tendency is, however, not seen either in the eﬀective Coulomb interaction discussed above or in the DEA (“T -matrix.”) This fact indicates that the oﬀ-diagonal elements of Uαβνµ are very important and cannot be discarded in the BSE. 7.3.4 Summary To summarize, we have demonstrated that the inclusion of the eﬀect of the strong Coulomb interaction is required in the calculation of the DEA. We have formulated the ﬁrst-principles T -matrix theory by beginning with the stateof-the-art GWA and then including the eﬀect of multiple scattering between two extra electrons due to the repulsive Coulomb interaction. By calculating the DEA of Li2 , Na2 , and K2 , we have conﬁrmed that the dianions of these alkali-metal dimers are all unstable and cannot bound two extra electrons, although these dimers can bound one extra electron. Our present formalization is also applicable to the calculation of the DIE, if the hole–hole two-particle Green’s function is considered instead of the electron–electron two-particle Green’s function. It is highly desired to apply the present formulation to calculate the DIE spectra and the DEA spectra of variety of atoms and molecules.

7.4 Double Ionization Energy Spectra 7.4.1 Introduction In electronic systems, two-particle excitations that are related to the strong electron correlation between two particles are interesting phenomena. In particular, the DIE (DEA), which is deﬁned as the energy required (acquired)

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for removing (adding) two electrons from (to) a neutral system, reﬂects the strong Coulomb interaction between holes (electrons). During the last few decays a lot of theoretical attempts were performed to calculate the twoparticle excitation spectra. For example, the double electron aﬃnities of small-sized molecules were calculated by using the delta self-consistent ﬁeld (∆SCF) and conﬁguration interaction (CI) methods [24–27], while the double ionization energies (DIEs) were evaluated from the potential energy curves of the dications using the multireference conﬁguration interaction (MRCI) method [28, 29]. Another method to calculate two-particle excitation spectra is the twoparticle Green’s function method. A merit of this method is that one can determine the whole spectra all at once in a single calculation in contrast to the MRCI method. Tarantelli et al. formulated a two-particle Green’s function method known as algebraic diagrammatic construction [ADC(2) or ADC(3)] [30–32] which is based on second- or third-order diagrammatic perturbation expansion, calculating the DIEs of various molecules [33–37]. However, the present approach, which is completely diﬀerent from the ADC(n), sums up special kinds of diagrams (e.g., ring or ladder diagrams) up to the inﬁnite order. In the calculation of the DIE spectra, the ladder diagrams up to the inﬁnite order play an important role, because the multiple scattering between two holes due to the short-range repulsive Coulomb interactions dominantly aﬀects the DIE spectra particularly in the small-sized systems. Recently, the ﬁrst-principles Green’s function methods that sum up special kinds of diagrams up to the inﬁnite order were successfully applied to the excited state physics. One-particle ionization energy spectra can be accurately evaluated as the poles of the one-particle Green’s function by the state-of-the-art GW calculations [3, 4]. Moreover, optical absorption spectra can be calculated by starting from the GWA [1] and solving the BSE for the electron–hole Green’s function [5–10]. In these methods, the physics such as the electronic screening and the exciton binding can be treated more transparently. The spectra of DIEs can be calculated from ﬁrst principles as the poles of the two-particle Green’s function in a manner analogous to calculations of the optical absorption spectra using the BSE. However, there has been no such theoretical attempt so far. Only one theoretical calculation was carried out by Springer et al. [17] who used the T -matrix theory for the electron– electron and hole–hole Green’s functions to calculate the imaginary part of the one-particle self-energy. In this section, we show that the spectra of DIEs can be accurately calculated by beginning with the GWA, solving the BSE for the electron–electron or hole–hole Green’s function in the ﬁrst-principles T -matrix theory successfully [38]. The key point is that the DIE strongly reﬂects the eﬀects of the particle– particle interactions in contrast to the ﬁrst ionization potential (IP) or electron aﬃnity (EA). It is inevitable to treat the multiple scattering between two particles within the ladder approximation. The smaller the system, the stronger

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the eﬀect of this interaction. This is because the eﬀect of the electron–electron Coulomb interaction becomes quite signiﬁcant in conﬁned geometries. Here, we show that the present method can well reproduce the experimental values of the DIE spectra of alkali-earth atoms (Be, Mg, and Ca), inert gas atoms (Ne, Ar, and Kr) alkali-metal dimers (Li2 , Na2 , and K2 ), and molecules (CO, and C2 H2 ). 7.4.2 Two-Valence-Electron Systems First of all, we calculate small two-valence-electron systems such as alkaliearth atoms (Be, Mg, Ca) and alkali-metal dimers (Li2 , Na2 , and K2 ). The calculated IPs and DIEs of alkali-earth atoms (Be, Mg, and Ca) and alkalimetal dimers (Li2 , Na2 , and K2 ) are presented in Table 7.2 together with available experimental data [39,40]. To our knowledge, there is no experimental result for DIE of lithium, sodium, and potassium dimers (Li2 , Na2 , and K2 ). In the column “ﬁrst ionization potential,” the IP is evaluated in either the LDA or the GWA. The LDA underestimates the experimental values for all atoms and molecules [39, 40]. On the other hand, the agreement between the GWA and the experiments is good (the values for Li2 , Na2 , and K2 are already given in [12]). In the column “double ionization energy,” “T -matrix” denotes the values evaluated by the ladder diagrams of the bare Coulomb interaction U . The column “GWA×2” represents the DIEs obtained from the evaluation of only the disconnected part of the two-particle Green’s function K, not considering the Coulomb repulsive interactions between two created holes. Table 7.2. The calculated IPs within the GWA and the LDA, and the DIEs within the T -matrix theory (in eV) of alkali-earth atoms (Be, Mg, Ca) and alkali-metal dimers (Li2 , Na2 , and K2 ) [38]

Be Mg Ca Li2 Na2 K2

First ionization potential (energy to remove one electron) LDA GWA Expt. 5.61 9.19 9.32a 4.79 7.62 7.64a 3.87 5.92 6.11a 3.24 5.27 5.14b 3.20 4.90 4.93b 2.74 3.98 4.05b

Double ionization energy (energy to remove two electrons) GWA×2 T -Matrix Experiment 18.39 27.79 27.53a 15.24 22.97 22.67a 11.84 17.82 17.98a 10.54 16.62 – 9.80 15.22 – 7.96 12.37 –

In the column of the “double ionization energy,” the DIEs in the “GWA×2” are twice the IPs within the GWA (which does not include the eﬀect of the Coulomb repulsive interaction); T -matrix denotes the values evaluated by the ladder diagrams of the bare Coulomb interaction U a Reference [39] b Reference [40]

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The diﬀerence between the DIEs in the “GWA×2” and the experiment is about 9.1 eV for Be, 7.4 eV for Mg, and 6.1 eV for Ca. However, these disparities are signiﬁcantly reduced by taking into account the Coulomb interaction between two holes. The diﬀerence between “T -matrix” and the experiment becomes about 0.3 eV for Be, 0.3 eV for Mg, and 0.2 eV for Ca. The largest contribution to the DIE of the two-valence-electron systems is an on-site Coulomb interaction Uαααα in (7.6) where α is the highest occupied molecular orbital (HOMO) state. The evaluated Uαααα is about 9.6 eV for Be, 7.9 eV for Mg, and 6.2 eV for Ca. This value tends to be smaller as the system size increases, because the valence orbital tends to extend further. The same is true for the Uαααα of Li2 , Na2 , and K2 , being about 6.3, 5.6, and 4.6 eV, respectively. 7.4.3 Inert Gas Atoms The calculated IPs of inert gas atoms (Ne, Ar, and Kr) are presented in the Table 7.3. The values of column “LDA” in the Table 7.3 are considerably smaller than the experimental values [39], while the results within the GWA are in good agreement with experiments. Table 7.4, on the other hand, shows the calculated spectra of DIEs of Ne, Ar, and Kr, and the corresponding experiments [41]. In the every case of inert gas atoms, “T -matrix” considerably improves GWA×2. The HOMO state of these atoms is threefold degenerated. The spectra of DIEs split into several states such as Π, ∆, and Σ, due to the symmetry of the holes when two arbitrary holes are created from the p-type valence orbitals. The three states listed in Table 7.4 can be identiﬁed from the information of the eigenfunctions (7.5). However, if we neglect the interaction between two holes, the GWA×2 would not be able to distinguish these states. The on-site Coulomb interaction Uαααα is about 28. eV for Ne, 16.4 eV for Ar, and 14.1 eV for Kr. It decreases as the atomic number increases and exhibits the same tendency as in the case of the two-valence-electron systems. Moreover, we obtain that the eﬀective Coulomb interaction, which can be expressed as T1−GWA×2, is about 24.2 eV for Ne, 13.7 eV for Ar, and 11.6 eV for Kr. The eﬀective Coulomb interaction of the inert gas atoms is diﬀerent by 2.5–4.3 eV from the on-site Coulomb interaction Uαααα , namely showing the importance of the ladder diagrams up to the inﬁnite order. Table 7.3. The calculated IPs within the GWA and the LDA of Ne, Ar, and Kr (in eV) [38]

Ne Ar Kr a

First ionization potential LDA GWA Expt.a 13.54 22.28 21.56 10.40 15.91 15.76 9.42 14.08 14.00

Reference [39]

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Y. Noguchi et al. Table 7.4. The calculated DIE spectra of Ne, Ar, and Kr (in eV) [38] Ne Π 1 ∆ 1 Σ

GWA×2 44.56 44.56 44.56

T -Matrix 68.80 71.65 74.11

Experimenta 62.52 65.73 69.43

Ar Π 1 ∆ 1 Σ

GWA×2 31.82 31.82 31.82

T -Matrix 45.50 47.33 47.79

Experimenta 43.38 45.11 47.50

Kr Π 1 ∆ 1 Σ

GWA×2 28.16 28.16 28.16

T -Matrix 39.78 41.41 42.90

Experimenta 38.36 40.18 42.46

3

3

3

The DIEs in the “GWA×2” are twice the IPs within the GWA (which does not include the eﬀect of the Coulomb repulsive interaction); “T -matrix” denotes the values evaluated by the ladder diagrams of the bare Coulomb interaction U . The assignments in the table are determined by the information of the eigenfunctions in (7.5) a Reference [41]

The present results for Ne are good but not perfect. The reason is not clear and further investigation is required. 7.4.4 CO and C2 H2 Molecules The IP of CO molecule evaluated using the LDA and the GWA is about 9.1 and 13.9 eV, respectively. The calculated IP within the GWA is in good agreement with the experiment [42] (the values for CO are already given in [43]). In Table 7.5, the DIE spectra up to 47 eV of the CO molecule are presented together with the corresponding experimental results. The column of “T -matrix” in the Table 7.5 denotes the values obtained from the evaluation of the ladder diagrams up to the inﬁnite order of the bare Coulomb interaction. The ﬁve DIEs in the table are characterized from lower to higher energies: 13 Πu , 11 Σg+ , 11 Πg , 13 Σu+ , and 21 Σg+ , where 1Π, 1Σ, and 2Σ are the twohole states composed mainly of (1π)(5σ), (5σ)2 , and (4σ)(5σ), respectively. Their characteristics are consistent with the experiment. Because “GWA×2” does not include the eﬀect of the Coulomb interactions between two holes, the values of “GWA×2” are smaller than the corresponding experimental values by about 10 eV. The ladder diagrams up to the inﬁnite order considerably improve the results of “GWA×2.” The present results are in good agreement with the experiment as well as previous calculations using MRCI method [44]. If we neglect the Coulomb interaction between two holes, 11 Σg+ becomes the smallest DIE when two electrons are removed from the HOMO level.

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Table 7.5. The calculated DIE spectra of the CO molecule (in eV) [38, 44] 3

1 Πu 11 Σg+ 11 Πg 13 Σg+ 21 Σg+

GWA×2 30.37 27.82 30.37 34.05 34.05

T -Matrix 40.66 40.79 41.61 42.56 46.83

Uαβγδ 11.26 14.98 11.26 11.02 11.02

Experimenta 41.29 41.70 41.81 43.57 45.48

“T -matrix” denotes the values obtained by evaluating the ladder diagrams up to the inﬁnite order of the bare Coulomb interactions and “Uαβγδ ” denote the matrix element carrying the largest contribution to the DIE a Reference [44]

In reality, however, 13 Πu becomes the smallest DIE when the electrons are individually removed from the HOMO and HOMO-1, because creating two holes in diﬀerent orbitals can reduce the Coulomb interaction (see the column “Uαβγδ ” in the Table 7.5). Therefore, GWA×2 + Uαβγδ results that 1Π (41.6 eV) is smaller than 1Σ (42.8 eV), although 13 Πg and 11 Πg (or 13 Σg+ and 11 Σg+ ) are still same values. To distinguish the spin triplet and singlet state of the created holes, we need to treat the oﬀ-diagonal elements of Hαβνµ in the (7.5). To diagonalize the full Hαβνµ is equal to evaluate the ladder diagrams up to the inﬁnite order. Next, let us discuss the results of C2 H2 molecule. The calculated IP within the LDA and the GWA is about 7.3 and 11.2 eV. The GWA considerably improves the result of the LDA. Five lower DIEs of C2 H2 are presented in Table 7.6 together with the experiments [45–47] for comparison. The experimental values are measured by double charge transfer (DCT) between projectile M+ (M = H or OH) and the target molecule. The DIEs of the molecule are uniquely determined by the conservation of energy and momentum in terms of the formation energy from M+ to M− . The characteristics determined by the calculation are consistent with the experiments, although 1 ∆ and 1 Σ are indistinguishable in the experimental H+ /DCT. The DIEs in the GWA×2 are again identical at 3 Σ, 1 ∆, and 1 Σ because of the twofold degeneracy of the (1π) level of C2 H2 . However, if we include the eﬀect of the Coulomb interaction more accurately, 3 Σ becomes the smallest. In other words, the Coulomb interaction can be further increased when two electrons are removed from the same orbital (HOMO or HOMO-1) rather than being individually removed from two diﬀerent orbitals (HOMO and HOMO-1) (see the Table 7.6). If two electrons are removed from the same orbital, the DIEs split into 3 Σ and 1 Σ or 3 Π and 1 Π according to the symmetry of spin of the created holes. We conﬁrmed that it is easier to remove two electrons in a triplet state 3 Σ, 3 Π than in a singlet state 1 Σ, 1 Π. These results are consistent with the experiments.

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Y. Noguchi et al. Table 7.6. The DIEs of C2 H2 (in eV) [38]

3

Σg ∆g 1 Σg 3 Πu 1 Πu 1

Calculation GWA×2 T -Matrix Uαβγδ 22.31 33.69 12.25 22.31 34.71 12.54 22.31 35.36 12.25 27.81 38.21 11.12 27.81 38.96 11.12

ADC(2)a 31.35 32.47 33.24 36.75 37.64

Experiment OH+ /DCTb H+ /DCT 32.7 ± 0.3 – – 33.6 ± 0.5c , 33.7d – 33.6 ± 0.5c , 33.7d 37.9 ± 0.4 – – 38.5 ± 0.7c , 38.4d

“T -matrix” and “Uαβγδ ” denote the results obtained by evaluating the ladder diagrams of the bare Coulomb interaction and the matrix element carrying the largest contribution to the DIE, respectively. The results of the ADC(2) and two kinds of experimental DCT results, H+ /DCT and OH+ /DCT corresponding to singlet and triplet states, respectively, are given for comparison a Reference [37] b Reference [45] c Reference [46] c Reference [47]

7.4.5 Summary In summary, we formulated the ﬁrst-principles T -matrix theory starting from the GWA and evaluated the particle–particle ladder diagrams up to the inﬁnite order by means of the BSE to calculate the DIE spectra. We ﬁrstly and successfully applied this theory to several atoms and small molecules and showed the importance of the ladder diagrams up to the inﬁnite order. The eﬀect of short-range electron correlation plays an important role in these spectra, particularly for atoms and small molecules. Our results are in excellent agreement with available experimental data as well as the previous calculations. These results show the signiﬁcant eﬀect of the electron correlations on the DIEs, i.e., the DIEs resulting from the present T -matrix theory are quite distinct from the simple sum of two one-particle energies obtained within the GWA. This T -matrix theory has an advantage in that it enables to calculate the spectra and wave functions of the two-particle excited states simultaneously and accurately. Moreover, this approach takes a moderate amount of the CPU time and computer memory. Currently, it is highly desirable to apply the present method to larger molecules to calculate their DIE spectra.

7.5 Two-Electron Distribution Functions and Short-Range Electron Correlations 7.5.1 Introduction Two-electron distribution functions provide signiﬁcant information on the electron correlations between two particles. When the interelectron distance is

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very small, there are two kinds of electron avoidance known as the “exchange hole” for parallel spins due to Pauli’s exclusion principle and the “Coulomb hole” for antiparallel spins due to the short-range repulsive Coulomb interaction. If the Coulomb hole is described from the ﬁrst principles, the short-range correlations between electrons can be treated accurately. However, there has been no such theoretical attempt so far because two-particle wave functions are not the simple product of two one-particle wave functions but require a two-particle picture that can eﬀectively include the electron repulsiveness. In an electron gas, a number of calculations of the radial pair distribution function describing the Coulomb hole have already been performed, and these have indicated the importance of the treatment of short-range electron correlations. For example, the Hartree–Fock approximation fails in evaluating the Coulomb hole, that is, the radial pair distribution function for antiparallel spin electrons is constant over the entire region from the long-range to the shortrange limit because of the lack of electron correlations. The RPA also fails to evaluate the Coulomb hole. The RPA fails at low densities, i.e., the radial pair distribution function for antiparallel spin electrons becomes negative in the short-range limit [48]. However, Carbotte [49] found that the negativeness of the antiparallel spin radial pair distribution function within the RPA can be removed by taking into account the multiple scattering between particles, i.e., ladder diagrams up to the inﬁnite order referred to as the T -matrix; Hede and Canbotte [50] and Yasuhara [19] later succeeded in calculating the Coulomb hole in an electron gas by evaluating the particle–particle ladder diagrams. By calculating the DIE spectra in which the short-range repulsive Coulomb interaction plays an important role, we have already demonstrated in previous sections that the T -matrix correctly describes the strong short-range electron correlations in atoms and molecules. The calculated DIE spactra are in good agreement with the corresponding experimental values. The short-range electron correlations cannot simply be expressed by an expectation value of the bare Coulomb interaction 1/r. Further, the multiple scattering between two electrons or two holes is very important when the interparticle distance is very small; in case of the 1,2,4-trithia-2,4,6-triazapentalenyl (TTTA), the eﬀect of the short-range interaction reduces the bare Coulomb interaction by a few eV [51]; see Chap. 5. To elucidate the eﬀect of the short-range electron correlations between two particles, we explicitly calculate the two-electron distributions of real systems from the two-particle Green’s function in the T -matrix theory that describes the multiple scattering between two particles. That is, we solve the BSE for the T -matrix via eigenvalue problem and compose the two-electron distribution functions from the eigenfunctions. To be explicit, we calculate the two-electron distribution functions of Ar, CO, CO2 , and C2 H2 with the ﬁrst-principles T -matrix theory [52]. Describing the two-electron distribution functions and comparing them with the one-electron distribution functions, we visually conﬁrm the eﬀect of short-range electron correlations as the Coulomb hole. Moreover, we discuss

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the relationship between the Coulomb hole and the short-range electron correlations in detail. 7.5.2 Methodology Our approach solving the eigenvalue problem, (7.5) with (7.6), can be used to calculate the two-electron distribution functions from (7.7). The Bethe– Salpeter amplitude (two-particle wave functions) ΨΩ (r, r ) is in a double space with coordinates r and r that refer to the positions of electrons. If the eigenfunctions Aνµ (Ω) are replaced by δνµ , ΨΩ (r, r ) will not include any electron correlation between the two particles, i.e., we have ΨΩ (r, r ) without the interaction. To discuss ΨΩ (r, r ) with the interaction, we ﬁx the position of one electron at r and consider the distribution of the other electron as a function of r. In this section, the eﬀect of the short-range electron correlation and the Coulomb hole are discussed in detail. 7.5.3 Ar First, we solve (7.5) for the argon atom. Figure 7.4 shows the absolute square of the two-electron distribution functions |ΨΩ (r, r )|2 of the 3p valence level with (solid lines) and without (dotted lines) the interaction. Here, an atomic nucleus is the origin of the horizontal axis and an electron is ﬁxed at the position r = (a) −2, (b) −9, (c) −19, and (d) −44 on the horizontal axis.

Fig. 7.4. Two-electron distribution functions of Ar with (solid lines) and without (dotted lines) the interaction. Here, an atomic nucleus is the origin of the horizontal axis given in a mesh unit (1 mesh is about 0.08 ˚ A) and one electron is assumed to be ﬁxed at r = −2 (a), −9 (b), −19 (c), and −44 (d) on the horizontal axis. After Noguchi [52]

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The dotted lines are identical to the absolute square of the one-particle wave functions |ψν (r)|2 (ν = 3p), which do not include the electron correlations between two particles and have a point symmetry at the origin of the horizontal axis (and no amplitude at the origin). Consequently, the dotted lines do not change according to the position of the ﬁxed electron. On the other hand, the solid lines include the eﬀect of the short-range electron correlations (i.e., multiple scattering) via the T -matrix theory. Because there is a strong repulsive Coulomb interaction from the ﬁxed electron, the shapes of the four solid lines in Fig. 7.4 are diﬀerent from those of the dotted lines. The remarkable common characteristic of the solid lines in Fig. 7.4a–d is the breaking of the point symmetry, and a ﬁnite amplitude at the origin, although this state is nevertheless the 3p level. The repulsive Coulomb interactions have a stronger eﬀect when the ﬁxed electron is closer to the atomic nucleus; therefore, the change in shape from the dotted lines is the largest in Fig. 7.4a and the smallest in Fig. 7.4d. In the case of Fig. 7.4a, one electron is ﬁxed at the position (r = −2) where the dotted lines have the largest amplitude, and as s-type character newly appears to screen the atomic nucleus in addition to the electrons that have distributed around the atomic nucleus. Moreover, there is clearly a change in the electron population over the broad region (not only in the negative region but also the positive region of horizontal axis except at the position of the atomic nucleus). Next, we ﬁx one electron at the outer position (r = −9 on the horizontal axis). The solid line in Fig. 7.4b is closer to the dotted line in contrast to the case of Fig. 7.4a, although there still remains the s-type character and the breaking of the point symmetry. Another remarkable point is that electron depletion appears only in the negative region of the horizontal axis, while the electron in the positive region distributes slightly more than the dotted line. Because the repulsive Coulomb interaction from the ﬁxed electron is suﬃciently screened by the other electron with the same p-type character or the electrons with the s-type character at the origin, there is no decrease in the electron distribution in the positive region. When an electron is ﬁxed further away from the atomic nucleus, the shapes of the solid and dotted lines are quite similar (see Fig. 7.4c, d). In particular, the solid lines in Fig. 7.4d are nearly identical to the dotted lines. Two-electron distribution functions shown in Fig. 7.4a–d are at the state characterized as 1 Σ, which corresponds to a DIE of Ω = 47.8 eV (these values are already given in [38]). Then, the actual Coulomb interaction given by [16] Ω − Eν − Eµ (ν = µ = 3p), which acts between two particles with antiparallel spin in the same 3p level and includes the short-range correlations (i.e., multiple scattering) via the T -matrix theory, is estimated to be about 16.0 eV. This value is slightly diﬀerent as compared to the simple expectation value of the bare Coulomb interaction Uνµνµ (= 16.4 eV). The decrease of about 0.4 eV by the multiple scattering is rather small compared to the other molecules discussed below, by it clearly indicates the eﬀect of the short-range correlations, i.e., the Coulomb hole.

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7.5.4 CO We performed a similar calculation for the CO molecule. Figure 7.5 shows the calculated two-electron distribution functions (a) without and with (b,c) the Coulomb interaction in the HOMO level. Contour lines are plotted on a plane including the molecular axis, and the two black circles represent a carbon atom and an oxygen atom from up to down, respectively. One electron is ﬁxed at both the point marked by the cross (b) near the oxygen atom and that (c) near the carbon atom. In Fig. 7.5b, c, the electron distribution around the cross is considerably decreased as compared to Fig. 7.5a due to the strong short-range Coulomb interaction from the ﬁxed electron, indicating the Coulomb hole. Another remarkable feature in Fig. 7.5b, c is that the Coulomb hole moves according to the position in which one electron is ﬁxed. In particular, a considerable enhancement in the electron distribution at the opposite side of the ﬁxed electron is observed in Fig. 7.5c. To investigate the Coulomb hole of CO molecule in more detail, we plot the contour lines of Fig. 7.5 along the molecular axis and show it in Fig. 7.6. Moreover, the points of ﬁxed electron marked by the crosses in Fig. 7.5 corresponds to the position marked by a solid square and a solid triangle on the horizontal axis; therefore, the dotted line, the solid line with squares and solid line with triangles correspond to Fig. 7.5a, b, and c, respectively. Comparing the dotted line and the solid line with triangles, we ﬁnd a shift in the electron distribution from the carbon atom to the oxygen atom and an enhancement of the electron distribution around the oxygen atom; and identify the Coulomb

Fig. 7.5. Two-electron distribution functions of CO molecule (a) without and (b, c) with the Coulomb interaction. The contour lines are plotted on the molecular plane. The two black circles are a carbon atom and an oxygen atom from up to down. In (b) and (c), one electron is ﬁxed at the point marked by the cross. The contour lines in (b) and (c) clearly indicate the Coulomb hole around the cross. After Noguchi [52]

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Fig. 7.6. Two-electron distribution functions of the CO molecule plotted along the molecular axis. The dotted line, the solid line with squares, and the solid line with triangles correspond to Fig.7.5a, b, and c, respectively. In the cases of the solid lines with squares and those with triangles, one electron is ﬁxed at the position marked by a solid square and a solid triangle on the horizontal axis. After Noguchi [52]

hole around the position marked by a sold triangle on the horizontal axis. Similarly, the solid line with squares has a slight shift in the electron distribution from the oxygen atom to the carbon atom and indicates the Coulomb hole around the position marked by a solid square on the horizontal axis. The electron at the HOMO level of the CO molecule originally has a higher distribution in the carbon atom side than the oxygen atom. Therefore, the shift in electron distribution from the oxygen atom to the carbon atom is not signiﬁcant even if one electron is ﬁxed near oxygen atom. These results are physically reasonable and correctly show the eﬀect of the strong short-range electron correlations. Here, using the T -matrix theory, we estimate the actual Coulomb interaction between electrons to be about 13.0 eV, while the corresponding bare Coulomb interaction Uνµνµ (ν = µ = HOMO) is about 15.0 eV. This diﬀerence of 2.0 eV is due to the Coulomb hole. To discuss the Coulomb hole, the short-range electron correlations in the present study have to be accurately studied. The short-range electron correlations obtained from the evaluation of the ladder diagrams up to the inﬁnite order correctly include the eﬀect of multiple scattering between two particles.

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7.5.5 CO2 We plot the two-electron distribution functions of the CO2 molecule in the HOMO level on the molecular plane as shown in Fig. 7.7. The center circle is a carbon atom and the others are oxygen atoms. The two-electron distribution function without the interaction is shown in Fig. 7.7a. It has two-mirror symmetries an up and down symmetry and a right and left symmetry. Furthermore, the mirror axes are nodes with no amplitude. On the other hand, the two-electron distribution function with the interaction, which has one electron ﬁxed at the point marked by the cross, is represented in Fig. 7.7b. The node between the up and down sides disappears in Fig. 7.7b due to the eﬀect of the ﬁxed electron; in this ﬁgure, the symmetry such as that in Fig. 7.7a is not observed. We see again the remarkable eﬀect of short-range electron correlations, although there is no signiﬁcant Coulomb hole in Fig. 7.7b. However, it is diﬃcult to visualize the Coulomb hole precisely because a three-dimensional analysis is required. The two-electron distribution function discussed here is characterized as 1 ∆. The corresponding DIE Ω is calculated to be about 38.4 eV and is in good agreement with the experimental value (38.5 eV). We also estimate that the actual Coulomb interactions between two particles in the HOMO level is 11.4 eV and the corresponding bare Coulomb interaction is 13.5 eV. Therefore, the multiple scattering between electrons reduces the repulsive Coulomb interaction by 1.9 eV, thus reﬂecting the eﬀect of the Coulomb hole.

Fig. 7.7. Two-electron distribution functions of the CO2 molecule without (a) and with (b) the interaction. The closed circle at the center is a carbon atom while the others are oxygen atoms. In (b), one electron is ﬁxed at the point marked by the cross. After Noguchi [52]

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Table 7.7. The DIEs of CO2 (in eV) [52]

3

Σ ∆ 1 Σ 3 Π 1 Π 1

Calculation GWA×2 T -Matrix 27.37 37.95 27.37 38.44 27.37 38.75 31.89 41.10 31.89 41.52

Uνµνµ 13.41 13.46 13.41 12.78 12.78

Experiment PEPECO 37.35a , 37.65 ± 0.3b 38.52a 39.30a 41.43a 42.30a

“T -matrix” and “Uνµνµ ” denote the results obtained by the T -matrix and the matrix element carrying the largest contribution to the DIE, respectively. The available photoelectron–photoelectron coincidence spectroscopy (PEPECO) experimental data are given for comparison a Reference [53] b Reference [54]

To discuss the short-range electron correlations quantitatively, we present the DIE spectra of the CO2 molecule. Table 7.7 shows the calculated DIEs and the corresponding experiments [53,54]. The DIEs shown in the column of “GWA×2” are −Eν − Eµ obtained from the evaluation of the disconnected part of the two-particle Green’s function Kνµ , namely, twice the ionization potential within the GWA. The ionization potential within the GWA is evaluated to be about 13.7 eV and is in good agreement with that of the experiment [42]. The HOMO level of CO2 is the π orbital and is twofold degenerated. When two electrons are removed from the HOMO level, the DIEs split into three states namely, 3 Σ, 1 ∆, and 1 Σ, according to the two-hole states. Nevertheless,“GWA×2” is identical in Σ and ∆, and a diﬀerence in the spin symmetries of 3 Σ and 1 Σ can be ignored because they do not include the eﬀect of the repulsive Coulomb interaction. In every state given in Table 7.7, the diﬀerences between “GWA×2” and the experiments are more than 10 eV. On the other hand, the “T -matrix,” which denotes the values evaluated by the T -matrix theory, considerably improve “GWA×2” and is in good agreement with that of the experiments. The diﬀerence with the experiments is about 0.8 eV or less for all the states shown in Table 7.7. The column of “Uνµνµ ” denotes the matrix element of the bare Coulomb interaction carrying the largest contribution to the DIEs. On the other hand, the actual Coulomb interaction, the diﬀerence between the T -matrix (Ω) and GWA×2 (−Eν − Eµ ), which includes the eﬀect of the multiple scattering, is deviated from “Uνµνµ ” by 1.5–3.4 eV. The actual Coulomb interaction between antiparallel spin electrons is more than that between parallel spin electrons because parallel spin electrons are unable to approach each other due to Pauli’s exclusion principle. Therefore, the DIEs show a tendency in that two electrons of the triplet state are removed easier than those of the singlet state. (This tendency was also seen in the DIEs of the CO or C2 H2 molecule [38].)

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7.5.6 C2 H2 The two-electron distribution functions with and without the interaction in the 5σ level (i.e., HOMO-1) of the C2 H2 molecule are plotted on a molecular plane as shown in Fig. 7.8. The four black circles denote hydrogen, carbon, carbon, and hydrogen atoms from up to down. To compare the two-electron distribution function with interaction and that without interaction shown in Fig. 7.8a, we ﬁx one electron at the point marked by the cross (near the hydrogen atom) in Fig. 7.8b. The decrease in the electron distribution in the lower CH side in Fig. 7.8b is considerable and clearly indicates the Coulomb hole. To observe the situation in more detail, the two-electron distribution functions without (dotted line) and with (solid line) the interaction plot-

Fig. 7.8. Two-electron distribution functions of C2 H2 molecule (a) without and (b) with the interaction. The contour lines are plotted on the molecular plane. From up to down, the four black circles denote hydrogen, carbon, carbon, and hydrogen atoms, respectively. In (b), one electron is ﬁxed on the point marked by the cross. After Noguchi [52]

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Fig. 7.9. Two-electron distribution functions of C2 H2 molecule without (dotted line) and with (solid line) the interaction plotted along the molecular axis. The dotted and solid lines correspond to Fig. 7.8a, b, respectively. One electron is ﬁxed at the point marked by the cross on the horizontal axis. After Noguchi [52]

ted along the molecular axis are shown in Fig. 7.9. The position marked by the cross in Fig. 7.9 corresponding to the same position of the ﬁxed electron marked by the cross in Fig. 7.8b. A signiﬁcant decrease in electron distribution around the ﬁxed electron is observed. On the other hand, there is an increase in the electron distribution around the right carbon on the horizontal axis (at the opposite side of the ﬁxed electron). We ﬁnally conﬁrm the Coulomb hole around the ﬁxed electron and the shift in the electron distribution from the CH of the ﬁxed electron side to the CH of the opposite side. The change in the electron distribution is considerable and is caused by the short-range electron correlation that is clearly indicated by the diﬀerence with the bare Coulomb interaction Uνµνµ (= 12.8 eV) (with ν = µ = HOMO-1). The estimated short-range electron correlation is about 8.8 eV. The large eﬀect of the short-range interaction, which reduces the bare Coulomb interaction by about 4.0 eV, clearly indicates the existence of the Coulomb hole. 7.5.7 Summary In summary, we calculated the two-electron distribution functions and the DIE spectra of Ar, CO, CO2 , and C2 H2 and discussed the relations between

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them. To account for the short-range electron correlations, we treated the twoparticle Green’s function G2 by using the ﬁrst-principles T -matrix theory. The calculated two-electron distribution functions, which include the eﬀect of multiple scattering via the evaluation of the ladder diagrams up to the inﬁnite order, clearly indicate the Coulomb hole. Moreover, we explained the relationship between the short-range electron correlations (the diﬀerence between Ω −Eν −Eµ and Uνµνµ ) and the Coulomb hole in detail. In the cases of the Ar atom and CO molecule in particular, we found that the electron distribution is aﬀected by the other electron and may in some cases concentrate on the atomic nucleus to screen its positive charge. For the CO2 molecule, we presented new results of the ﬁrst ionization potential and the DIE spectra, and compared them with the corresponding experiments as well as our previous study on other molecules [38]. The calculated ﬁrst ionization potential and DIEs for the CO2 molecule are in good agreement with the experiments. The short-range electron correlations play an important role in the determination of the DIEs. Again, we conﬁrmed that the T -matrix correctly gives the short-range correlations between two particles.

7.6 Summary In summary, we formulated the ﬁrst-principles T -matrix theory beginning with the GWA based on the many-body perturbation theory beyond the framework of the DFT and evaluated the ladder diagrams up to the inﬁnite order to discuss the two-particle excited states of real systems. We calculated the DEAs of alkali-metal dimers (Li2 , Na2 , and K2 ). Although the clusters can bound one extra electron, their dianions are all unstable (namely, the clusters are unable to bound two extra electrons) due to the enormous repulsive Coulomb interactions between extra electrons. On the other hand, we calculated the DIE spectra of atoms alkali-earth atoms (Be, Mg, and Ca), inert gas atoms (Ne, Ar, and Kr), alkali-metal dimers (Li2 , Na2 , and K2 ), and molecules (CO, CO2 , and C2 H2 ). The DIE spectra obtained from the T -matrix theory are in good agreement with the corresponding experiments. The evaluation of the ladder diagrams up to the inﬁnite order (T -matrix) is quite important in calculating the DIE spectra of small systems. The eﬀective (including short-range) Coulomb interactions, which include the eﬀect of the multiple scattering between holes, are smaller by a few eV than the bare Coulomb interactions. To investigate the short-range Coulomb interaction in more detail, we calculated the two-particle wave functions of real systems (Ar, CO, CO2 , and C2 H2 ) using the ﬁrst-principles T -matrix theory and showed the “Coulomb hole” in comparison between the one-particle and the two-particle wave functions. The avoidance of electrons (Coulomb hole) results in the reduction of the short-range Coulomb interactions. The present results are all consistent and in good agreement with experiments. Our method, which solves the eigenvalue problem of the BSE for the

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T -matrix, has a merit that it enables to calculate the DIE or the DEA specra and two-particle excited states simultaneously treat the strongly correlated systems. Currently, it is highly desirable to apply the present method to other strongly correlated systems such as transition metal oxides or X-ray photoemission spectroscopy (XPS) for the core electron spectra. We have recently succeeded in evaluating the Auger spectra of small carbon-hydride molecules. The results will be reported elsewhere [55]. Acknowledgments The authors are grateful to the Information Initiative Center, Hokkaido University, for the use of the HITACHI SR8000 and SR11000 supercomputing facilities. This work has been partly supported by Grant-in-Aids of the Scientiﬁc Research B (no. 17310067) and Scientiﬁc Research on Priority Areas (no. 18036005) from Japan Society for the Promotion of Science and from the Ministry of Education, Culture, Sports, Science, and Technology. Y.N. is supported by the Research Fellowship, Japan Society for the Promotion of Science.

7.7 Appendix 7.7.1 Fourier Transformation of Green’s Function We deﬁne the one-particle Green’s function as follows: G(2, 1) = iΘ(t1 − t2 )N | φ∗ (1)φ(2) | N − iΘ(t2 − t1 )N | φ(2)φ∗ (1) | N , (7.8) where φandφ∗ are ﬁeld operators. Fourier transformation of Green’s function is given by ∞ eiω(t2 −t1 ) G(2, 1)d(t2 − t1 ), (7.9) G(r2 , r1 ; ω) = −∞

G(2, 1) =

1 2π

∞

−∞

e−iω(t2 −t1 ) G(r2 , r1 ; ω)dω.

(7.10)

If we deﬁne τ = t2 − t1 , the particle-Green’s function Gp becomes G (2, 1) = −iΘ(t2 − t1 ) p

emp

N | φν (2) | N + 1N + 1 | φ∗ν (1) | N

ν

= −iΘ(τ )

emp ν

ψν (r2 )ψν∗ (r1 )e−iεν τ ,

(7.11)

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and the hole-Green’s function Gh becomes occ

Gh (2, 1) = iΘ(t1 − t2 )

N | φ∗ν (1) | N − 1N − 1 | φν (2) | N

ν

= iΘ(−τ )

occ

ψν∗ (r1 )ψν (r2 )e−iεν τ ,

(7.12)

ν

using the relationships N | φν (1) | N + 1 = ψν (1) = ψ(r1 )eiεν t1

(7.13)

N − 1 | φν (1) | N = ψν (1) = ψ(r1 )eiεν t1 .

(7.14)

and

Then, (7.11) and (7.12) are substituted for (7.9), G (r2 , r1 ; ω) = −i p

emp ∞ −∞

ν

=

emp ν

Gh (r2 , r1 ; ω) = i =

ψν (r2 )ψν∗ (r1 ) , ω − εν + iη

occ ν

Θ(τ )ψν (r2 )ψν∗ (r1 )ei(ω−εν +iη)τ dτ

∞

−∞

Θ(−τ )ψν∗ (r1 )ψν (r2 )ei(ω−εν −iη)τ dτ

occ ψ ∗ (r1 )ψν (r2 ) ν

ν

(7.15)

ω − εν − iη

.

(7.16)

7.7.2 Fourier Transformation of K-Matrix Next, we consider the Fourier transformation of K-matrix. K is deﬁned by K(12|12 ) = iG(1 , 1)G(2 , 2),

(7.17)

where the statics approximation is used t1 = t2 , t1 = t2 . Using the relationship between (7.9) and (7.10), the Fourier transformation of K becomes ∞ K(r1 r2 |r1 r2 ; ω) = eiω(t1 −t1 ) K(1, 2|1 , 2 )d(t1 − t1 ) −∞ ∞ i = eiω(t1 −t1 ) d(t1 − t1 ) 2 (2π) −∞ ∞ G(r1 , r1 ; ω1 )e−iω1 (t1 −t1 ) dω1 × −∞

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∞

215

G(r2 , r2 ; ω2 )e−iω2 (t1 −t1 ) dω2 −∞ ∞ i = G(r1 , r1 ; ω1 )G(r2 , r2 ; ω2 )δ(ω − ω1 − ω2 )dω1 dω2 2π −∞ ∞ i = G(r1 , r1 ; ω1 )G(r2 , r2 ; ω − ω1 )dω1 . (7.18) 2π −∞ ×

Here, G(r1 , r1 ; ω1 )G(r2 , r2 ; ω − ω1 ) includes next four terms Gp Gp , Gp Gh , Gh Gp , and Gh Gh . However the two terms of them Gp Gh and Gh Gp vanish by taking their poles. Therefore, the Fourier transformation of K is rewritten as follows: ∞ i Gp (r1 , r1 ; ω1 )Gp (r2 , r2 ; ω − ω1 )dω1 K(r1 r2 |r1 r2 ; ω) = 2π −∞ ∞ i + Gh (r1 , r1 ; ω1 )Gh (r2 , r2 ; ω − ω1 )dω1 2π −∞ emp ψν (r1 )ψν∗ (r1 )ψµ (r2 )ψµ∗ (r2 ) i ∞ dω1 = 2π νµ −∞ (ω1 − εν + iη)(ω − ω1 − εµ + iη) occ ψν∗ (r1 )ψν (r1 )ψµ∗ (r2 )ψµ (r2 ) i ∞ dω1 + 2π νµ −∞ (ω1 − εν − iη)(ω − ω1 − εµ − iη) =+

emp νµ

−

ψν (r1 )ψν∗ (r1 )ψµ∗ (r2 )ψµ (r2 ) ω − εν − εµ + iη

occ ψν∗ (r1 )ψν (r1 )ψµ (r2 )ψµ∗ (r2 )

ω − εν − εµ − iη

νµ

.

(7.19)

7.7.3 Fourier Transformation of T -Matrix T -matrix satisﬁes the following BSE: T (1, 2|3, 4) = U (1, 2)δ(1 − 3)δ(2 − 4) +iU (1, 2) d1 d2 K(1, 2|1, 2 )T (1 , 2 |3, 4).

(7.20)

The Fourier transformation is T (r1 r2 |r3 r4 ; ω) =

∞

−∞

eiω(t3 −t1 ) T (12|34)d(t3 − t1 ).

Here, we deﬁne the J as follows: ∞ J(t3 − t1 ) ≡ K(t1 − t1 )T (t3 − t1 )dt1 . −∞

(7.21)

(7.22)

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Similarly, K and T are K(t1 − t1 ) =

1 2π

T (t3 − t1 ) =

1 2π

∞ −∞

∞ −∞

K(ω1 )e−iω1 (t1 −t1 ) dω1 ,

(7.23)

T (ω3 )e−iω3 (t3 −t1 ) dω3 .

(7.24)

If we substitute (7.23) and (7.24) for (7.22), J is ﬁnally rewritten ∞ ∞ 1 2 ∞ J(t3 − t1 ) =

K(ω1 )eiω1 t1 dω1

2π

1 = 2π

∞

−∞

T (ω3 )e−iω3 t3 dω3

−∞ −iω1 (t3 −t1 )

K(ω1 )T (ω1 )e

dω1 ,

ei(ω3 −ω1 )t1 dt1

−∞

(7.25)

−∞

...

J(ω) = K(ω)T (ω).

(7.26)

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

L. Hedin, Phys. Rev. 139, A796 (1965) S.L. Adler, Phys. Rev. 126, 413 (1962) M.S. Hybertsen, S.G. Louie, Phy. Rev. B 34, 5390 (1986) R.W. Godby, M. Schl¨ oter, L.J. Sham, Phys. Rev. B 35, 4170 (1987) G. Onida, L. Reining, R.W. Godby, R. Del Sole, W. Andreoni, Phys. Rev. Lett. 75, 818 (1995) G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74, 601 (2002) S. Albrecht, R. Del Sole, G. Onida, Phys. Rev. Lett. 80, 4510 (1998) M. Rohlﬁng, S.G. Louie, Phys. Rev. Lett. 80, 3320 (1998) M. Rohlﬁng, S.G. Louie, Phys. Rev. B 62, 4927 (2000) L.X. Benedict, E.L. Shirley, R.B. Bohm, Phys. Rev. Lett. 80, 4514 (1998) K. Ohno, F. Mauri, S.G. Louie, Phys. Rev. B 56, 1009 (1997) S. Ishii, K. Ohno, Y. Kawazoe, S.G. Louie, Phys. Rev. B 65, 245109 (2002) S. Ishii, K. Ohno, V. Kumar, Y. Kawazoe, Phys. Rev. B 68, 195412 (2003) F. Herman, S. Skillman, Atomic Structure Calculations (Prentice-Hall, New Jersey, 1963) S. Ishii, K. Ohno, Y. Kawazoe, Mater. Trans. JIM 42, 2150 (2001) J. Kanamori, Prog. Theor. Phys. 30, 275 (1963) M. Springer, F. Aryasetiawan, K. Karlsson, Phys. Rev. Lett. 80, 2389 (1998) U.C. Grossman, M. Rohlﬁng, L. Mitas, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 86, 472 (2001) H. Yasuhara, Physica 78, 420 (1974) Y. Noguchi, S. Ishii, K. Ohno, Mater. Trans. 46(6), 1103 (2005) Y. Noguchi, S. Ishii, Y. Kawazoe, K. Ohno, Sci. Technol. Adv. Mater. 5, 663 (2004)

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8 Green’s Function Formulation of Electronic Transport at Nanoscale A.A. Farajian, O.V. Pupysheva, B.I. Yakobson, and Y. Kawazoe

8.1 Introduction Electronic transport is among the unique physical phenomena whose applications have shaped human civilization as we know it today. From old telegrams and light bulbs to modern televisions, mobile phones, laptops, and supercomputers, all make use of electronic transport. In fact, nowadays we can hardly ﬁnd any signiﬁcant technological product in which electronic transport is not used one way or another. This enormous technological impact is a result of basic scientiﬁc research on electron tunneling and scattering, in diﬀerent environments and including various levels of interactions and correlations. The basic research in the ﬁeld of electronic transport is expected to yield equally unique, and even more important, fruits in future, as the challenges of this vibrant ﬁeld are ever increasing. One of the main areas of interest which has been the focus of numerous scholarly works is electronic transport at nanometer length scales. The reason is that miniaturization of electronic components has caused the device dimensions to reach nanoscale. At nanoscale, the atomistic character of the systems can no longer be treated using rather rough models applicable at micrometer length scales. Therefore, fundamentally new approaches are necessary, in both theory and experiment, to deal with electronic transport at nanoscale. In this chapter we investigate atomistic modeling of electronic transport at nanoscale, and consider typical approaches which are widely used nowadays. The idea is to see how speciﬁc atomic structures of the systems, which result in unique electronic structures, aﬀect electronic tunneling and scattering. The main features of the examples that we consider in this chapter are generally similar. These main features consist in employing Landauer’s formulation of electronic transport combined with Green’s function treatment of scattering properties. Despite diﬀerences in minor details and occasional discussions on alternative treatments, Landauer’s formulation together with Green’s function treatment is commonly used. Basic features of these

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formalisms are mentioned in this chapter on several instances, and the reader is referred to available references for details. Our aim is to introduce atomistic modeling of electronic transport using several typical sample systems, which were the subjects of pioneering calculations in this ﬁeld. We follow a more or less chronological order, to provide an image of the rapid progress of this ﬁeld. The choice of the methods and the systems for which transport characteristics are calculated are meant to show typical modeling approaches, rather than a complete progress account. We hope that the materials of this chapter, together with the cited references, will provide a short review of the subject and emphasize its main features.

8.2 Landauer’s Transport Formalism: The Green’s Function Implementation 8.2.1 Multichannel Landauer’s Formula Landauer’s formalism [1, 2] is the basic formalism that is widely used for calculating transport properties at the nanoscale. Within Landauer’s formalism, the conductance of the system is related to its transmission properties. These can be obtained via diﬀerent approaches available for calculating scattering matrix. The multichannel generalization of the original formulation of Landauer is expressed as follows [3]. Assume that t and r represent the transmission and reﬂection matrices, respectively, with matrix elements tαβ and rαβ between conduction channels α and β. If it is possible to go to a basis in which t and r are both diagonal, then the conductance, Γ , is given by [3] Γ =

e2 |tαα |2 . π¯ h α |rαα |2

(8.1)

As in general it is not possible to diagonalize t and r using the same representation, (8.1) is written as Γ =

e2 1 e2 |t |2 αβ |tαβ |2 = . 2 π¯ h α |rαα | π¯ h 1 − β |tαβ |2 β

(8.2)

αβ

This is the multichannel four-terminal conductance. Similarly, for the multichannel two-terminal conductance we obtain [4, 5] Γ =

e2 e2 T r{tt† }. |tαβ |2 = π¯ h π¯ h

(8.3)

αβ

For further details on Landauer’s formalism, one can consult several extensive reviews which are available on this subject (see [4] and the references therein).

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A powerful procedure for calculating scattering matrix and, hence, transport properties is through Green’s function methodology. This naturally provides us with the probabilities for transmission and reﬂection, when a charge carrier propagates through a medium that may contain irregularities and defects. 8.2.2 Surface Green’s Function Matching Method In this section we discuss the surface Green’s function matching (SGFM) method [6,7], which is a powerful approach for obtaining the Green’s function of a system which consists of several subsystems connected together. It will be shown that the Green’s function of the whole system can be derived from its projection on the interface regions, together with “transfer” matrices of the subsystems. Consider a bulk crystal and take the orientation of the surface eventually to be studied. The surface in bulk crystal (or the interface between two bulk crystals) is taken to be oriented perpendicular to the direction of current ﬂow. This deﬁnes the corresponding atomic layers, which are parallel to the surface. By two-dimensional Fourier transform, with two-dimensional wave vector k, we go from atomic orbitals to k-dependent layer orbitals. If N is the number of diﬀerent atomic states in the basis employed, then in the layer representation the Hamiltonian matrix is a k-dependent N × N matrix. Next we deﬁne a principal layer as a combination of one or more atomic layers, such that each principal layer interacts only with its nearest neighboring principal layers. In other words, each principal layer interacts only with two other principal layers: the immediate one to its left and the immediate one to its right, where left and right are deﬁned along the direction perpendicular to the surface. Here and henceforth the term layer will be used to denote a principal layer. The number of basis orbitals is then increased, according to the size of the layer. The dependence on k will also be understood throughout. Consider now the A/B bicrystal, that is a system consisting of two subsystems (crystals) A and B attached together at an interface. Let M denote one ¯ the other one. The corresponding bulk of the two mediums A and B and M Hamiltonian and Green’s function of medium M are denoted by HM and GM , respectively. In other words, HM and GM are the Hamiltonian and Green’s function of the (inﬁnite) bulk system M, with no interface included. Let PM indicate the projector formed with all the orbitals of the semi-inﬁnite crystal M, and IM that part of PM which belongs to the interface of the bicrystal system A/B. Here and henceforth the calligraphic letters are used to indicate interface objects, i.e., objects that exist in the interface domain. The complete interface projector is I = IA + IB , (8.4) and the Hamiltonian of the (bicrystal) system is HS = PA HS PA + PB HS PB + IA HS IB + IB HS IA .

(8.5)

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The crossterms describe the coupling interaction across the interface. They involve one-terminal layer on each side. Next we consider the Green’s functions GM . Each GM is deﬁned as the propagator of an extended, inﬁnite bulk medium. The Green’s function of the bicrystal system will be denoted by GS . If we choose two arbitrary (principal) layers pM and pM in medium M, then we have pM GS pM = pM GM pM + pM GM IRIGM pM ,

(8.6)

where R is the reﬂection oﬀ the interface. In the tight-binding formulation, for example, pM GS pM is given by a matrix whose elements indicate propagation from sites/orbitals in principal layer pM to those in principal layer pM . Similarly, for two layers pM and pM ¯ on two sides of the interface the propagation is written as pM (8.7) ¯ GS pM = pM ¯ GM ¯ IT IGM pM . Here T is the transmission across the junction. R and T are deﬁned as surface objects which, by deﬁnition, exist only in the space of I, the matching domain. Note that in both cases after scattering – reﬂection or transmission – has taken place, the amplitudes propagate with the corresponding bulk propagator. Now we project (8.6) and (8.7) onto the interface region (matching domain). Therefore we obtain GS = GM + GM RGM , GS = GM ¯ T GM ,

(8.8) (8.9)

whence −1 −1 R = GM (GS − GM )GM , −1 T = GM ¯. ¯ GS GM

(8.10) (8.11)

Feeding these back to (8.6) and (8.7), we obtain −1 −1 pM GS pM = pM GM pM + pM GM GM (GS − GM )GM GM pM , −1 pM ¯ GS pM = pM ¯ GM ¯ GM ¯ GM pM . ¯ GS GM

(8.12)

From these relations, it is seen that propagation between any two arbitrary layers – and eventually, any two arbitrary sites (through the matrix elements of GS ) – of the bicrystal A/B can be derived once the projection of the total Green’s function GS onto the interface region, GS , is known. It remains to relate GS to the physics of the system characterized by the Hamiltonian HS . Consider the deﬁnition of the Green’s function GS : (ω − HS )GS = I,

(8.13)

in which ω is the (complex) energy, and I is the unit matrix of the whole system given by I = PA + PB . Therefore we can write

8 Green’s Function Formulation of Electronic Transport at Nanoscale

(ω − HS )(PA + PB )GS = PA + PB .

223

(8.14)

Projecting this equation onto the interface region and multiplying by GS−1 from right, we obtain GS−1 = I(ω − HS )(PA + PB )GS IGS−1 .

(8.15)

This equation can be written as −1 GS−1 = IA (ω − HA PA GA GA )IA + IB (ω − HB PB GB GB−1 )IB − ∆HS .

Here ∆HS = ∆HA +∆HB +VAB +VBA , with ∆HA and ∆HB being the changes in the Hamiltonian of the interface parts of A and B due to the presence of the junction, and Vs are the couplings at the junction. Further, we have made use of the fact that for any layer orbital
(8.16)

It is now useful to deﬁne transfer matrices T and T¯ such that G21 = T G11 and G01 = T¯G11 . Using these deﬁnitions, (8.16) is ﬁnally written as GS−1 = IA [(ω − HA )11 − HA,10 T¯A ]IA + IB [(ω − HB )11 − HB,12 TB ]IB − ∆HS .

(8.17)

In Sect. 8.2.3, we will use this equation, together with a very powerful algorithm [8,9] for calculating transfer matrices T¯A and TB , to obtain the scattering matrix and transport through junctions in doped nanotubes and atomic wires. 8.2.3 Scattering Matrix and Transport Properties To obtain the conductance and the I–V characteristics of the matched system, we choose the following approach. First, the band structures of crystals A and B are calculated for two inﬁnitely long systems, separately. These eigenstates (Bloch states) of the bulk crystals indicate diﬀerent propagating channels of

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the matched system; electrons start as linear combinations of the eigenstates (channels) of crystal A, move toward the junction, and after scattering oﬀ the potential barrier continue to propagate as a linear combination of eigenstates (channels) of crystal B. Using the bulk crystal band structures, one can determine which channels are conducting in each crystal, at any arbitrary Fermi energy (which is imposed by the external reservoirs). Next, transmission matrix elements are calculated. These matrix elements specify the coeﬃcients of the expansion of conducting eigenstates of crystal A after scattering, in terms of conducting eigenstates of crystal B. Finally, conductance and I−V characteristic of the system are calculated using transmission matrix and Landauer’s formalism. Here we exploit the SGFM method, introduced in Sect. 8.2.2, to calculate transmission matrix elements. This method is capable of handling problems formulated within diﬀerent Hamiltonian formalisms (e.g., tight-binding and ab initio formalisms). Some of the early applications of the tight-binding transport formalism, for example, have been carbon nanotube heterostructures [10, 11]. Following Chico et al. [10], we assume that the principal layers of medium A are successively indexed as . . . , −3, −2, −1 and those of medium B as 1, 2, 3, . . .. Therefore, the interface region consists of two principal layers; −1 of medium A on one hand, and 1 of medium B on the other. To use the SGFM method, we ﬁrst calculate the transfer matrices for medium M (M = A or B), TM , SM , T¯M , S¯M , which are deﬁned as follows: GMn+1,m = TM GMn,m , (n ≥ m), GMn−1,m = T¯M GMn,m , (n ≤ m),

(8.18) (8.19)

GMn,m+1 = GMn,m SM , (m ≥ n), GMn,m−1 = GMn,m S¯M , (m ≤ n).

(8.20) (8.21)

In these relations, GMn,m is the block Green’s function matrix of medium M between principal layers n and m. The dimension of these block matrices is equal to the number of atoms per principal layer times the number of orbitals per atom. We use the algorithm proposed by Lopez Sancho et al. [8, 9] to calculate TM and T¯M , and then calculate SM and S¯M through ¯ SM = G−1 Mn,n TM GMn,n , S¯M = G−1 TM GMn,n , Mn,n

(8.22) (8.23)

where GMn,n is the block Green’s function matrix of the bulk (inﬁnite) crystal M projected onto an arbitrary principal layer n. This matrix is given by (cf. [12]) GMn,n = (E − HMn,n − VMn,n+1 TM − VMn,n−1 T¯M )−1 .

(8.24)

In this equation, E is the (complex) energy, HMn,n is the Hamiltonian matrix of an isolated principal layer of medium M, and VMn,n−1 and VMn,n+1 are hopping matrices of bulk crystal M.

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Within the SGFM method, the Green’s function of the matched system projected onto the interface region is [6, 7] GBB GBA G ≡ IGI ≡ GAB GAA −1 E − HB1,1 − VB1,2 TB −VI1,−1 = , (8.25) −VI−1,1 E − HA−1,−1 − VA−1,−2 T¯A where VI−1,1 and VI1,−1 are hopping matrices at the interface of the matched system. Moreover, calligraphic letters indicate interface objects, i.e., matrices projected onto the interface region. For example, G is the Green’s function of the matched system projected onto the interface region. (I = IA + IB and IM indicate the projection operator onto that part of the interface region which belongs to medium M.) Using these results, we calculate the reﬂected and transmitted amplitudes. The reﬂected amplitude in crystal B and the amplitude transmitted from side A to B are given by [6, 7]

and

Ψnr = GBn,1 GB−1 (GBB − GB )GB−1 GB1,n G−1 Bn ,n ϕBn

(8.26)

−1 Ψnt = GBn,1 GB−1 GBA GA GA−1,n G−1 An ,n ϕAn ,

(8.27)

respectively (cf. (8.12)). In these equations, ϕA and ϕB indicate incident eigenstates in mediums A and B. It is then straightforward to obtain the transmission matrix form medium A to B, n ¯n −1 SBA nn (E, V ) = TB GBA SA GAn ,n ,

(8.28)

which transfers an excitation from principal layer n of medium A to principal layer n in medium B. In the tight-binding basis, SBA nn is an NB × NA matrix, where NB and NA are the number of atoms times the number of orbitals of each principal layer in crystals B and A, respectively. The dependence on V (the external applied potential diﬀerence) results from the fact that the matrices involved in the deﬁnition of SBA nn depend on the on-site energies of crystals A and B, which in turn depend on V . To calculate conductance, we make use of the Landauer’s formula, and write (8.3) as [13] 2e2 vβ 2e2 α 2 T (E, V ) = |ϕβBn |SBA Γ (E, V ) = nn (E, V )|ϕAn | , (8.29) h h vα αβ nn

where T (E, V ) is the transmission coeﬃcient between A and B, and the sum over α and β runs over all allowed eigenstates (channels) of mediums A and B (i.e., those Bloch states whose energy is E), and vα and vβ are the group velocities of those allowed channels. For computational purposes, the sum over

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n and n should include a few principal layers, far from interface, such that the sum is converged. Equation (8.29) is a key result that can be used to calculate the conductance by summing over the contributions of diﬀerent conducting channels. It is possible to restrict the sum over some speciﬁc channels to obtain their contribution to conductance. In other words, the contribution of each input–output pair of channels can be calculated separately. Finally, the current across the junction is obtained using the formula [14] 2e ∞ dE T (E, V )[fB (E) − fA (E)], (8.30) I(V ) = h −∞ where fA and fB are the Fermi distributions of mediums A and B. 8.2.4 Alternative Formulation of the Total Conductance There is another, alternative, formulation for the total conductance which is commonly used in atomistic calculations. In this formulation, one avoids ﬁnding individual conducting channels (Bloch states) of mediums A and B at each carrier energy and summing over them. Instead, a collective sum is carried out to obtain the total conductance. In this approach, the contribution of each input–output pair of channels is indistinguishable, as one obtains the total conductance at each carrier energy. We consider a system with three parts: a semi-inﬁnite part on the left, a ﬁnite (junction) part in the middle, and a semi-inﬁnite part on the right. These three parts are denoted by A, J, and B, respectively. Although in the previous sections it was assumed that the Hamiltonian and Green’s function matrices are described in an orthogonal basis, in this section we assume a general nonorthogonal basis. The surface Green’s functions of the left and right parts, Gs;A and Gs;B , are deﬁned as the projection of the Green’s functions of the left and right parts onto the surface layers of these parts [4, 15]. To match the surface Green’s functions of the left and right contacts to the Green’s function of the middle junction, we deﬁne the self-energies [16] of the left and right contacts as ΣA = (HAJ − zSAJ )† Gs;A (HAJ − zSAJ ) , †

ΣB = (HJB − zSJB )Gs;B (HJB − zSJB ) .

(8.31) (8.32)

Here, HAJ and HJB denote the coupling Hamiltonian matrices connecting the last layer of the left system to the junction, and the junction to the ﬁrst layer of the right system. SAJ and SJB represent the corresponding overlap matrices, and z is the (complex) energy. The total Green’s function of the system projected onto the junction region, Gt;J , is then obtained by matching at the junction [6, 7, 10, 12, 17]: Gt;J = (zSJ − HJ − ΣA − ΣB )−1 .

(8.33)

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In this equation, HJ and SJ represent the junction Hamiltonian and overlap matrices, respectively. It is clear that the SGFM method [6, 7] is a powerful approach for obtaining the Green’s function of a system which generally consists of several subsystems connected together. The Green’s function of the whole system can be derived from its projection on the interface regions, together with the transfer matrices of the subsystems. The formalism that we present is therefore applicable to the general case of several functional junctions attached by metallic contacts. By expressing the elements of the scattering matrix for diﬀerent conducting channels in terms of the Green’s function [13], the transmission probability T (E, V ) can be written as [18]

in which [16, 18–21]

T (E, V ) = Tr[ΓB Gt;J ΓA G†t;J ],

(8.34)

† ΓA,B = i(ΣA,B − ΣA,B ).

(8.35)

8.3 Carbon Nanotube Heterostructures We start with the carbon nanotube heterostructures, which were among the ﬁrst systems for which electronic transport was calculated using atomistic modeling. Since the discovery of carbon nanotubes [22], i.e., cylindrical-shell structures of carbon, with the atoms arranged in rolled-graphene geometry [23], their transport properties have been investigated extensively. An atomistic theoretical investigation of transport properties of carbon nanotubes was presented by Chico et al. in 1996 [10], where the eﬀects of vacancies and pentagon–heptagon defects on conductance were studied. Esfarjani et al. [24] and Farajian et al. [25] studied doped nanotube junctions. They showed that doped nanotube junctions possess some of the essential properties of the components used in electronic circuits, namely rectifying and negative diﬀerential resistance (NDR) eﬀects. These are sample theoretical investigations which make use of the Green’s function formulation of the Landauer’s approach introduced in Sect. 8.2. In this section, we investigate these results in some detail. 8.3.1 Conductance of Nanotubes with Vacancy or Pentagon–Heptagon Defects The systems considered for transport calculations by Chico et al. [10] consist of diﬀerent nanotubes with vacancy or pentagon–heptagon defects. In the former case the chirality of the nanotubes remains unchanged after introducing the vacancy, while in the latter, a chirality change happens along the nanotube upon passing the pentagon–heptagon defects.

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The model that is used in these studies is a one orbital per atom tightbinding model, in which to each carbon atom only one π orbital is assigned. This model is known to produce basic characteristics of the electronic structures of the nanotubes provided that their curvature is not very large. (For more accurate electronic structure, one needs more sophisticated ab initio methods. Alternatively, tight-binding formulations with larger number of orbitals per atom can be used, where σ–π hybridization is included.) Within this one orbital per atom tight-binding model, the Hamiltonian of the system is written as: † ci cj + c.c. (8.36) H = −Vppπ

i,j

The sum is over all nearest neighbors, Vppπ = 2.66 eV [11], and the on-site energies are set to zero. A vacancy is simulated by setting the hoppings to zero around the vacant site and its on-site energy equal to a large value. Figure 8.1 shows the results of conductance calculation for a (4,4) nanotube [26], with and without vacancy. We see a reduction of conductance upon introducing vacancy, at all the carrier energies. This could be expected, as removing one atom from the carbon nanotube lattice results in creating a scattering center which increases the probability of back-scattering. Within the Green’s function formulation, the amount of reduction at each energy can be calculated. The case of a nanotube heterostructure, made up of attaching a (12,0) nanotube to a (6,6) nanotube, is depicted in Fig. 8.2. The two semi-inﬁnite nanotubes (12,0) and (6,6) are attached at a junction via a ring of six pentagon–heptagon pairs around the circumference. Figure 8.2 shows several interesting features. The conductance of the matched system is lower than the conductances of each of the perfect nanotubes (12,0) and (6,6). This is again due to the presence of extra scattering centers, i.e., the pentagon–heptagon pairs at the junction. Here, however, there is a diﬀerence compared to the case of a nanotube with vacancy: In the matched (12,0)–(6,6) system, each semi-inﬁnite part can be considered a perturbation when considering transport through the other semi-inﬁnite part. It is therefore important to include the details of the scattering mechanism between diﬀerent pairs of conducting channels, in the semi-inﬁnite (12,0) and (6,6) nanotubes. One important aspect of this scattering is angular momentum considerations. The rotational symmetries of the nanotubes are reﬂected in the symmetries of their eigenstates. The LDOS curves in Fig. 8.2 show nonzero values around the Fermi energy (taken to be zero), while the conductance is zero. This means that the reason for the suppression of conductance is not lack of conducting states. Indeed, the rotational symmetry considerations explain why the charge carriers cannot tunnel from the (12,0) nanotube to the (6,6) nanotube at energies near the Fermi energy. The arrangement of the pentagon–heptagon pairs at the interface between the two nanotubes satisﬁes

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Fig. 8.1. Conductance characteristics of a perfect (4,4) nanotube (dashed line) show larger transmission as compared to the conductance characteristics of the same nanotube but with a single vacancy (solid line) [10]

the same sixfold rotational symmetry, similar to the perfect (12,0) and (6,6) nanotubes. This means that the rotational symmetry of the tunneling carriers cannot change upon scattering. However, it turns out that the rotational symmetries of the Bloch states near Fermi energy in the (12,0) and (6,6) nanotubes do not match. This results in complete back-scattering of the carriers with energies close to the Fermi energy at the interface, and zero conduction in this energy range. As the energy range changes, states with the same rotational symmetry become available within both the (12,0) and (6,6) nanotubes, and the conductance becomes nonzero. It should be mentioned that when the symmetry of the interface is diﬀerent from the symmetries of the matched nanotubes, tunneling between the states with diﬀerent symmetries on diﬀerent sides of the junction may be possible [10].

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Fig. 8.2. Conductance results for the matched (12,0)–(6,6) nanotube system: (a) conductance of the matched system (solid line) is lower than the conductances of perfect (12,0) nanotube (dashed line) and perfect (6,6) nanotube (dotted line). (b) The local density of states (LDOS) for the interface unit cell of the (12,0) nanotube (solid line) and for the perfect (12,0) nanotube (dotted line). (c) The LDOS for the interface unit cell of the (6,6) nanotube (solid line) and for the perfect (6,6) nanotube (dashed line) [10]

8.3.2 Doped Nanotube Junctions: Rectiﬁcation and Novel Mechanism for Negative Diﬀerential Resistance The conductance characteristics of nanotubes containing pentagon–heptagon defects were considered in the previous section. We saw that the conductance characteristics deviates from that of perfect nanotube owing to a vacant site

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Fig. 8.3. A doped nanotube junction, formed by inserting donor and acceptor dopant atoms inside a (10,0) nanotube [24]

or a change in chirality which is caused by the defect(s). Now we consider another source of changing conductance in nanotubes, namely the eﬀect of doping. A doped nanotube junction can be formed by diﬀerently doping two semiinﬁnite parts of a nanotube. This can be achieved by either inserting diﬀerent dopant atoms inside the nanotube [24], as depicted in Fig. 8.3, or depositing the nanotube on a double crystal substrate. In the former case, the diﬀerence in the charge transfer from donor and acceptor atoms (K and I, for example) results in a diﬀerence in the chemical potential of the two semi-inﬁnite parts of the nanotube (with respect to the density of states (DOS) of the pristine nanotube). The same diﬀerence in the chemical potentials is achieved in the latter case due to a diﬀerence in the work functions of the two crystals constituting the substrate. The conductance characteristics of diﬀerent doped nanotube junctions have been calculated using the multichannel Landauer’s formalism introduced earlier [24,25]. A simple estimate, based on the tunneling probability through a step potential barrier at the junction, reveals the dependence of conductance on the applied bias [24]. A more accurate treatment, based on the Green’s function formalism, can be used to calculate the conductance characteristics and current–voltage (I–V ) curves, with and without self-consistent treatment of charge transfer at the junction [25]. Figure 8.4 shows the I–V characteristics of various doped nanotube junctions, made up of diﬀerent nanotubes. For these calculations, a simple one-orbital tight-binding model has been used [25], similar to the one formulated in (8.36). The initial shifts of the chemical potentials of left and right semi-inﬁnite parts, which arise from asymmetric doping of the two sides of the nanotubes, are chosen to be 0.2 and 0.1 for metallic nanotubes, and 0.3 and 0.0 for semiconducting nanotubes (in units of the hopping energy Vppπ in (8.36)). In Fig. 8.4a, regions of NDR are observed in the I–V characteristics of both (3,3) and (4,4) nanotubes. The NDR feature is also seen in Fig. 8.4b for the zigzag metallic (3,0) nanotube. It should be noticed that the I–V results based on the self-consistent potential drop at the doped nanotube junction (3,0) deviate only slightly as compared to the results of (nonself-consistent) step potential. On the other hand Fig. 8.4b shows regions of zero current for nonzero biases for the semiconducting (5,0) and (7,0) nanotubes, which are

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asymmetric with respect to positive and negative biases. This is the rectifying eﬀect, similar to what is shown by ordinary diodes. The reason for the rectifying eﬀect is basically the diﬀerence in the initial (i.e., zero bias) shifts of the chemical potentials due to diﬀerent dopings of the two sides of the nanotubes [25]. This diﬀerence causes the band structures

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of the two sides of the nanotube to be shifted asymmetrically under positive and negative biases. The calculated transmission probabilities, conductance characteristics, and I–V curves are therefore asymmetric as well. The NDR features of Fig. 8.4 can be explained in terms of the rotational symmetry selection rule [25]. Figure 8.5 shows the band structure of the (4,4) nanotube under Vbias = 0 (a) and Vbias = 1 (hopping energy units) (b). When a potential diﬀerence ∆V = 1 is applied between the two sides of the (4,4) nanotube, the band structures will be shifted with respect to each other as depicted in Fig. 8.5. The labels of the bands in Fig. 8.5 indicate their eigenvalues under rotation, i.e., exp(imπ/2); m = 0, 1, 2, 3. At the energy −0.5 (hopping energy units), for example, Fig. 8.5a shows that there are two bands with the rotational eigenvalue 1 and positive group velocities. Therefore at zero bias there are two conductance units at the carrier energy −0.5. For Vbias = 1, Fig. 8.5b shows that at carrier energy −0.5 there is just one band with the rotational eigenvalue 1 and positive group velocity. Therefore under ∆V = 1 there would be two conduction channels (i.e., bands) available on one side of the nanotube (which is assumed to be under zero bias), but only one conduction channel would be available on the other side (assumed to be subject to Vbias = 1). As a result, one conductance channel is suppressed and the current is decreased. It should be noticed that this mechanism of NDR is caused by the unique properties of the nanotubes, i.e., their transport and rotational symmetry, and is basically diﬀerent from the standard NDR mechanism observed in Esaki diodes and resonant tunneling structures.

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8.3.3 Eﬀects of Random Disorder on Transport of Nanotubes In Sect. 8.3.1 we saw how introducing a single vacancy or pentagon–heptagon defects aﬀects transport characteristics. Although the concentration of defects in the nanotubes used experimentally can be very low [27], cases of predominantly defective nanotubes can also exist [28]. There has been suggestions that transport through defective nanotubes can be used for eﬀective sensor applications [29], as the defects make the nanotubes more reactive and prone to adsorption of gas molecules from the environment. An example of the methods that can be used for generating defects is electron irradiation [30]. Calculating transport through defected carbon nanotubes is therefore interesting, and can be used in designing nanodevices, e.g., nanosensors. The eﬀects of randomly distributed defect centers on transport characteristics of carbon nanotubes are studied by Anantram and Govindan [31]. The model that is used for the nanotubes is a one-orbital tight binding model, similar to the one introduced in Sects. 8.3.1 and 8.3.2. Speciﬁcally, the Hamiltonian is written as † H= ε0i c†i ci − Vppπ ci cj + c.c., (8.37) i

i,j

where ε0i are the on-site energies. The defects are introduced by varying ε0i randomly within a portion of the nanotube. As a particular example, a (10,10) nanotube is considered, and the defects are randomly distributed along a ◦ 1,000 A portion of this nanotube. The scattering region is attached on its left and right to two semi-inﬁnite perfect nanotubes, so that the methodology of Sect. 8.2 can be used. Transmission probability is calculated using (8.34), and I–V characteristics are obtained using (8.30). One important aspect of the results is the dips observed in conductance characteristics of defected nanotubes, at the energies corresponding to the band minima or maxima [31]. At these energies the bands are basically ﬂat. These give rise to van Hove singularities in DOS, and zero carrier velocities (= dE/dk). Such low-velocity carriers are easily scattered even by weak defects. As a result, in presence of defects, the conductance drops more signiﬁcantly at the energies corresponding to the top or bottom of the bands, as compared to the energies close to the middle of the bands. Figure 8.6 depicts the current and diﬀerential conductance characteristics of a defected (10,10) nanotube. There are 10 strong-isolated defects, randomly ◦ distributed along a length of 1,000 A. It is assumed that the potential drops linearly along this scattering region. Although the perfect (10,10) nanotube is metallic, it is observed that the I–V characteristics in Fig. 8.6 resemble that of semiconducting nanotubes. The reason is that the presence of defects indeed results in opening of a transmission gap around Fermi energy [31]. The calculated diﬀerential conductance agrees with experimental results [32]. One should notice, however, that in experiments the low conduction region

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Fig. 8.6. Current and conductance characteristics for a (10,10) nanotube, with 10 ◦ strong-isolated defects distributed randomly along a length of 1,000 A [31]

around zero bias may have been caused by inter-nanotube interactions [33], as nanotube bundles have been used.

8.4 Functional Molecule Between Two Metallic Contacts Up to now, all the systems that we have considered for calculating transport included carbon nanotubes. The methodology presented in Sect. 8.2 is, however, general and can be applied to any system with two semi-inﬁnite contacts and one functional part in between. To show this general applicability, here we consider transport through functional molecules between two metallic contacts. 8.4.1 Transport Through Xylyl-Dithiol Molecule Attached to Two Gold Electrodes In this section we consider a xylyl-dithiol molecule attached to a gold substrate. The molecule is also weakly interacting with a scanning tunneling

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Fig. 8.7. A xylyl-dithiol molecule sandwiched between a gold substrate and scanning tunneling microscope (STM) tip. The substrate and STM tip are used as contacts for applying bias and measuring transport through the molecule [19]

microscope (STM) tip, as shown in Fig. 8.7. Tian et al. have presented both theoretical and experimental results for this system [19]. The xylyl-dithiol molecules generate a self-assembled monolayer (SAM) on the gold substrate and get covalently bonded to it. The connection of the functional molecules to the substrate, which is used as an electrode, is therefore strong. On the other hand, the STM tip which is used to probe the transport properties of the molecules has no covalent bond with the molecules. The STM contact therefore couples weakly with the molecules. Owing to the conical feature of the STM tip, it interacts with a small number of the molecules of the SAM. This way, it is in principle possible to measure transport properties of a single molecule. It should be noticed that this experimental setup is diﬀerent from that of a break junction [34], where it is assumed that the functional molecule forms covalent bonds with both of the contacts. To calculate bias-induced transport in the conﬁguration depicted in Fig. 8.7, one should know the pattern of potential drop along the molecule. The applied bias determines the integration window in (8.30) by shifting the Fermi distributions of the left and right contacts. This shift is relative to the equilibrium Fermi level, which is deﬁned under zero bias conditions. The exact value of the Fermi energy can be calculated by integrating the LDOS of the functional molecule such that the result gives the correct electronic charge, including charge transfer to/from the contacts [19]. This calculation, however, is very sensitive to the energy broadening used in the deﬁnition of the Green’s function. Tian et al. [19] have therefore treated both the equilibrium Fermi energy and the pattern of potential drop as “ﬁtting parameters” to adjust their calculations to experimental results. The diﬀerential conductance characteristics, calculated by taking the derivative of the current values with respect to the applied bias, are shown in Fig. 8.8. This ﬁgure also shows the experimental results. From Fig. 8.8a it is

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Fig. 8.8. Theoretical and experimental diﬀerential conductance characteristics of the xylyl-dithiol system depicted in Fig. 8.7. (a) and (b) correspond to small and large tip-molecule distances, respectively [19]

observed that for the case of small tip-molecule distance, the calculated conductance characteristics are in good agreement with the experimental data when the Fermi energy is assumed to be 0.9 eV above the highest occupied molecular orbital (HOMO). As for the potential drop, it is assumed that it occurs equally at the contact–molecule interfaces. Therefore, half of the drop

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occurs at the left contact, and half of it at the right contact. The potential along the molecule itself is considered constant. Figure 8.8b depicts the conductance results for the case of large tipmolecule distance. For calculating the conductance characteristics, here it is assumed that the equilibrium Fermi energy is 1.4 eV above the HOMO. This larger diﬀerence compared to the case of small tip-molecule distance is justiﬁed by noting that when the tip-molecule distance is small the molecule is under strong electric ﬁeld which reduces the HOMO–LUMO (lowest unoccupied molecular orbital) gap. The reduction of the gap brings the Fermi energy closer to HOMO. It should be mentioned that for the xylyl-dithiol calculations, the selfenergies of the contacts, ΣL and ΣR ((8.31) and (8.32)), are approximated by the s-band LDOS of the gold surface at the Fermi energy [19]. This is in contrast to the conductance calculations of nanotubes presented earlier, where the contact self-energies or surface Green’s functions are calculated self-consistently [10, 25, 31]. 8.4.2 Transport Through Benzene-Dithiol Molecule Attached to Two Gold Electrodes Similar to xylyl-dithiol, benzene-dithiol is another functional molecule which is used in transport measurements and calculations. In this case, a benzene molecule is attached to two gold electrodes via two sulfur clips. Instead of the experimental setup explained in Sect. 8.4.1, however, for measuring transport through benzene-dithiol, break junctions were used [34]. Diﬀerent theoretical calculations are performed to provide better understanding of the transport through benzene-dithiol [35–37]. One of these calculations [37] uses an approach similar to the one explained in Sect. 8.4.1, i.e., the self-energies of the contacts are modeled by the LDOS of the gold surface at the Fermi energy. Here, however, instead of just one s-band LDOS, s-, p- and d-bands are used [37]. After estimating the contact self-energies with the surface LDOS at the Fermi energy, the rest of the calculations proceed as explained in Sect. 8.2.4. The calculation results are depicted in Fig. 8.9. Attachment to two gold surfaces is modeled by connecting each of the sulfur atoms to a single gold atom. There are two optimized structures; with the gold atoms being in the same plane as that of benzene-dithiol (planar structure), and with the gold atoms being out of the plane of benzene-dithiol (nonplanar structure). Both the planar and nonplanar calculation results are shown in Fig. 8.9, together with the experimental results. In the calculations it is assumed that the potential along the molecule is constant, and the potential drop occurs equally at the contacts (similar to the case of the xylyl-dithiol discussed in Sect. 8.4.1). Moreover, when estimating the self-energies of the contacts by the gold surface LDOS, it is assumed that half of the LDOS corresponds to the sharp gold surfaces which provide connection to the functional molecule. Figure 8.9

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Fig. 8.9. Current and diﬀerential conductance results for the planar attachment of benzene-dithiol to two gold atoms (light gray) as well as nonplanar attachment (dotted gray). The experimental results (dark gray) are also shown [37]

shows that the calculation results are in good agreement with the experimental ones, indicating that the Green’s function methodology can be used to analyze transport at the nanoscale.

8.5 Summary In this chapter we discuss a general methodology for calculating electronic transport at nanoscale. The basic formalism is based on calculation of the Green’s function of the system in real space. When the Hamiltonian of the system is expressed in a localized basis, it is shown that the transmission and reﬂection amplitudes can be derived from the Green’s function. The transmission amplitude provides the conductance of the system within Landauer’s formalism. Current–voltage characteristics are calculated by integrating the conductance curves. Several applications of the method are presented. These include transport through carbon nanotube heterostructures and some functional molecules attached to gold electrodes. These show the general applicability of the methods. In particular transport calculations can be performed at the ab initio or semiempirical levels, with or without ﬁtting parameters, e.g., the position of the Fermi energy and the potential drop pattern. The general Green’s function formalism can therefore be used to investigate transport properties of nanoscale systems, to get detailed description of the experimental results and to design systems with desired transport characteristics for speciﬁc applications.

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References 1. R. Landauer, Philos. Mag. 21, 863 (1970) 2. R. Landauer, J. Math. Phys. 37, 5259 (1996) 3. P.W. Anderson, D.J. Thouless, E. Abrahams, D.S. Fisher, Phys. Rev. B 22, 3519 (1980) 4. A.A. Farajian, R.V. Belosludov, O.V. Pupysheva, H. Mizuseki, Y. Kawazoe, in Nanostructures – Fabrication and Analysis, ed by H. Nejo (Springer, Berlin Heidelberg New York, 2007) 5. M. B¨ uttiker, Y. Imry, R. Landauer, S. Pinhas, Phys. Rev. B 31, 6207 (1985) 6. M.C. Munoz, V.R. Velasco, F. Garcia-Moliner, Prog. Surf. Sci. 26, 117 (1987) 7. F. Garcia-Moliner, V.R. Velasco, Theory of Single and Multiple Interfaces (World Scientiﬁc, Singapore, 1992) 8. M.P. L´ opez Sancho, J.M. L´ opez Sancho, J. Rubio, J. Phys. F: Met. Phys. 14, 1205 (1984) 9. M.P. L´ opez Sancho, J.M. L´ opez Sancho, J. Rubio, J. Phys. F: Met. Phys. 15, 851 (1985) 10. L. Chico, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. B 54, 2600 (1996) 11. L. Chico, V.H. Crespi, L.X. Benedict, S.G. Louie, M.L. Cohen, Phys. Rev. Lett. 76, 971 (1996) 12. P.A. Lee, D.S. Fisher, Phys. Rev. Lett. 47, 882 (1981) 13. D.S. Fisher, P.A. Lee, Phys. Rev. B 23, 6851 (1981) 14. C.B. Duke, in Tunneling Phenomena in Solids, ed by E. Burstein, S. Lundqvist (Plenum, New York, 1969) 15. A.A. Farajian, R.V. Belosludov, H. Mizuseki, Y. Kawazoe, Thin Solid Films 499, 269 (2006) 16. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995) 17. A. MacKinnon, Z. Phys. B: Condens. Matter 59, 385 (1985) 18. H.M. Pastawski, Phys. Rev. B 44, 6329 (1991) 19. W. Tian, S. Datta, S. Hong, R. Reifenberger, J.I. Henderson, C.P. Kubiak, J. Chem. Phys. 109, 2874 (1998) 20. M.B. Nardelli, Phys. Rev. B 60, 7828 (1999) 21. A. Rochefort, Ph. Avouris, F. Lesage, D.R. Salahub, Phys. Rev. B 60, 13824 (1999) 22. S. Iijima, Nature 354, 56 (1991) 23. R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998) 24. K. Esfarjani, A.A. Farajian, Y. Hashi, Y. Kawazoe, Appl. Phys. Lett. 74, 79 (1999) 25. A.A. Farajian, K. Esfarjani, Y. Kawazoe, Phys. Rev. Lett. 82, 5084 (1999) 26. Representing diﬀerent nanotubes by a pair of indexes is a common categorization practice which is explained in, e.g., Ref. [23] 27. Y.W. Fan, B.R. Goldsmith, P.G. Collins, Nat. Mater. 4, 906 (2005) 28. H.E. Unalan, M. Chhowalla, Nanotechnology 16, 2153 (2005) 29. L. Valentini, F. Mercuri, I. Armentano, C. Cantalini, S. Picozzi, L. Lozzi, S. Santucci, A. Sgamellotti, J.M. Kenny, Chem. Phys. Lett. 387, 356 (2004) 30. F. Beuneu, C. l’Huillier, J.-P. Salvetat, J.-M. Bonard, L. Forr´o, Phys. Rev. B 59, 5945 (1999)

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31. M.P. Anantram, T.R. Govindan, Phys. Rev. B 58, 4882 (1998) 32. P.G. Collins, A. Zettl, H. Bando, A. Thess, R.E. Smalley, Science 278, 100 (1997) 33. P. Delaney, H.H. Choi, J. Ihm, S.F. Louie, M.L. Cohen, Nature (London) 391, 466 (1988) 34. M.A. Reed, C. Zhou, C.J. Muller, T.P. Burgin, J.M. Tour, Science 278, 252 (1997) 35. M. Di Ventra, S.T. Pantelides, N.D. Lang, Phys. Rev. Lett. 84, 979 (2000) 36. L.E. Hall, J.R. Reimers, N.S. Hush, K. Silverbrook, J. Chem. Phys. 112, 1510 (2000) 37. P.A. Derosa, J.M. Seminario, J. Phys. Chem. B 105, 471 (2001)

9 Self-Assembled Quantum Dot Structure Composed of III–V Compound Semiconductors K. Mukai

9.1 Introduction Self-assembled quantum dot (QD) composed of III–V semiconductors is one of the most promising materials for the devices of next generation optical telecommunication and future single-photon quantum computation [1–8]. The QD structure is grown by molecular beam epitaxy (MBE) or metalorganic chemical vapor deposition (MOCVD), and has been investigated to exploit the delta-function-like state density which arises from the three-dimensional quantum conﬁnement of carriers. Among other methods such as the electrostatical deﬁnition of QDs in a two-dimensional electron gas and the colloidal clustering using solution chemistry, the self-assembled epitaxial growth provides the best means of incorporating QDs into a variety of devices with high quantum eﬃciency. In this decade, there has been great progresses in the development of QD devices for optical telecommunication applications. The target devices are semiconductor laser [4–6], semiconductor optical ampliﬁer (SOA) [9, 10], and photodetector [11]. QD laser was proposed originally in 1982 by Arakawa and Sakaki [12], suggesting that the atom-like state density in a QD causes the improvement of semiconductor laser performance. It was theoretically predicted that semiconductor QD laser will exhibit ultra-low threshold current, temperature-insensitive operation, narrow spectral linewidth, and large modulation bandwidth. All of these features are attributed to the enhanced gain properties, especially the extremely high-diﬀerential gain induced by the discrete state density. Multiwavelength and high-speed pattern-eﬀect-free signal ampliﬁcation and processing of QD SOA was ﬁrst proposed by Sugawara et al. based on the QD’s nonlinear optical properties. With the QD SOA, the ampliﬁcation of low power consumption, high saturation power, and broad gain bandwidth are also expected. These features of QD SOA will provide highperformance ampliﬁers as well as all-optical switches in the next generation photonic networks, by being involved in regenerators, wavelength converters and time-division demultiplexers. Research aiming at infrared detectors

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of semiconductor QD has also been activated. The quantum dot infrared photodetector (QDIP) has attracted a lot of interests because QDIP will be substantially superior than the quantum well infrared photodetector (QWIP). First advantage is that QDIP allows normal incidence. Fabrication of a grating coupler is required in the standard QWIP and causes diﬃculties in realizing a wide and multiple wavelength coverage. Another potential advantage is lower dark currents of QDIP. This is originated by the diﬀerence in mobile carrier density in barrier that is a function of potential barrier height. Also, long-excited electron lifetime in QD will lead to a high responsibility, high operating temperature and high dark current limited detectivity. These technologies of various QD devices are going to be applied to commercial products in the near future [13]. QD has also shown potentiality as a main medium for generating and operating qubits in quantum information devices [5,6,14]. Single-photon pulse generated from QD attracts much attention for its possibilities to improve the performance of ﬁber-based quantum communication system. Self-assembled single QD oﬀers advantages of a sharp luminescence line, optical stability, and controllability of wavelength. For example, Santori et al. [7] reported the observation of polarization correlation of the biexiton and single-electron photons emitted by a single InAs QD in a two photon cascade, intending to detect entangled photon pair [7]. Their attempt did not succeed, but the importance of spin degeneracy due to QD symmetry in the realization of an entangled-photon generation revealed. Miyazawa et al. [14] reported the ﬁrst success in generating single-photon pulses in 1.55 µm band from single-photon generator composed of QD [14]. They also performed to transmit the pulse through 30 km optical ﬁber. This may be a milestone to realize quantum telecommunication using single photon. In this chapter, we review representative recent progress of the research in self-assembled QD, especially material characteristics. The QD is now no longer fully self-assembled, and some properties are controlled intentionally. Since the material was InGaAs/GaAs system in the early stage of the development, the research in the system has much progressed. InGaAs/GaAs QD covers the wavelength from 0.9 to 1.4 µm. To expand application wavelength of QD beyond the region, InGaAs/InP QD has come to be investigated. Aiming at blue emission, InGaN QD has been investigated recently. We review the development of growth technique and characterization technique of these QDs.

9.2 Control of QD Structure by Growth Condition 9.2.1 Control of Growth Parameters To meet the requirements of the device developments, various growth techniques have been researched to acquire high controllability of QD properties [13, 14]. The target properties of the research are size, composition, emission

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Fig. 9.1. Two-dimensional layout of qubit cells which contain ﬁve quantum dot (one large dot and surrounding four small dots) in each cell: (a) schematic and (b) AFM image. The bigger and smaller QDs are 30 and 20 m in diameter, respectively (after Ohshima et al. [15])

eﬃciency, numerical density, uniformity, and positioning. Size and composition determine the energy states of electrons in QD and dominate device functions. Emission eﬃciency, numerical density, and uniformity of QD ensemble are important for the device in which optical gain spectra of QD dominate the device performance. Positioning of QD is a precondition for the device in which the interaction of electron across QDs is operation principle (Fig. 9.1) [15]. Primary investigation of growth parameters succeeded in the control of basic QD properties to some extent. The parameters are growth temperature, amount of source, composition of supplied materials, growth rate, and growth interruption (in situ annealing) [16, 17]. Researches of these parameters are summarized in [18] and [19]. Because the QD formation is dominated by the migration length of adatoms, the QD formation process also depends on substrate orientation. Tested major orientations are (001), (113)A, (113)B, (114)A, (114)B, (110), (111)A, and (111)B [20–23]. Maes et al. [24] presented the substrate orientation dependence of QD formation process using very high magnetic ﬁeld. They reported that an abrupt change from one-dimensional to three-dimensional carrier conﬁnement on (100)GaAs substrate was observed at 1.5 monolayer while the change was replaced on (311) substrate by a slow evolution from wetting layer surface ﬂuctuations and QDs are fully developed from 1.9 monolayer. As for another growth parameter, Kaiander et al. [25] reported that environment friendly alternative precursor tertiarybutylarsine can be used for 1.24-µm emitting InGaAs/GaAs QD laser in spite of highly toxic arsenic hydride. Usage of the alternate source reduced QD size at the same growth temperature. An in situ annealing step led to the evaporation of plastically relaxed defect clusters. Amano et al. [26] proposed MBE growth that uses dimeric arsenic (As2 ) for a GaAs-based 1.3-µm InAs QD. They succeeded to obtain uniform QD having high density and high emission eﬃciency. Using strain reducing layer with the gradient composition technique, they were able to achieve the density of 3.3 × 1011 cm−2 and a full width at half maximum of 23 meV of photoluminescence emission peak at room

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Fig. 9.2. Photoluminescence spectrum of triple-stack QDs with a high dot density of 3.3 × 1011 cm−2 at room temperature (after Amano et al. [26])

temperature (Fig. 9.2). To modify the growth process of surface directly, Matsuura et al. [27] investigated a surfactant eﬀect for self-assembled InGaAs QD. They reported that the introduction of antimony (Sb) into QDs caused a large blueshift of photoluminescence wavelength with a decrease in the full width at half maximum (improvement of uniformity) and an increase in intensity (improvement of emission eﬃciency) in comparison with QDs without Sb. The surfactant expanded the critical thickness of the growth mode change from two-dimensional to three-dimensional. They expected that the incorporation of Sb into the highly strained InGaAs reduces the surface free energy of the epitaxial layer. 9.2.2 Closely Stacked QDs The technique of stacking QD layers closely has been one of the major research subjects for the control of uniformity, positioning in growth direction and eﬀective height for three-dimensional carrier conﬁnement. The eﬀect of size ﬂuctuations on photoluminescence spectral linewidth calculated via the eﬀective mass approximation method suggests that the linewidth is determined much more by height ﬂuctuations than by diameter ﬂuctuations. In 1994, Xie et al. ﬁrst reported that coherent InAs islands separated by GaAs spacer layers exhibited self-assembled growth along the vertical direction [28]. In 1996, Solomon et al. reported that up to ten InAs islands were vertically aligned in columns [29]. Closely stacking resulted in redshift of emission wavelength and reduction of emission linewidth, which was attributed to electronic coupling between islands in the columns (Fig. 9.3) [30]. The electronic coupling weakens quantum conﬁnement eﬀect. Taking into account the eﬀect of vertical coupling, Endoh et al. [31] calculated the relationship between the spectral

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Fig. 9.3. Photoluminescence spectra of a InAs single QD layer, and a three-layer and a ﬁve-layer stacked InAs QD layer grown at 3 nm GaAs intervals as measured at 77 K (after Nakata et al. [30])

linewidth and the spacer layer thickness for the closely stacked QD. As the spacer layer thickness decreases, the spectral linewidth decreased. Unfortunately, emission eﬃciency is much degraded with the decrease. Changes in size and density of QDs with spacer layer thickness were reported in detail by Shiramine et al. [32]. The dependence on spacer layer thickness was explained in terms of the vertical pairing probability of islands, detachment of indium from QD top and surface segregation of indium atoms. Nakata et al. [30] succeeded to stack QDs using even thinner spacer layers, e.g., three monolayers (less than 1 nm), so that the QD would physically contact each other in the perpendicular direction. At ﬁrst, normal Stranski–Krastanov islands were formed on the growth surface as base QDs. Then the thin spacer layer and small QDs were grown repeatedly on the islands. Monitoring of refractive high energy electron diﬀraction (RHEED) pattern intensity told that the surface morphology was repeated between ﬂat and island during stacking. The cyclic supply under precise control enabled the production of columnar-shaped structure as shown in Fig. 9.4 [33]. The columnar-shaped QD demonstrated the photoluminescence spectra with narrow linewidth of 42 meV and emission

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Fig. 9.4. Cross-sectional TEM image of two QDs having columnar shape. Inset shows the plan view of the columnar dots (after Mukai et al. [33])

eﬃciency as high as ordinary Stranski–Krastanov QD at 300 K. The symmetric shape also succeeded to suppress the polarization of photo emission [34]. Further development of stacking technique has been reported. Frigeri et al. [35] investigated the application of atomic layer molecular beam epitaxy (ALMBE) to closely stacked QD. Considering the number of QD layers and spacer layer thickness, they report that the ALMBE-grown QDs are more ordered and have peculiar properties such as large dimension, sharp size distribution, bright photoluminescence at room temperature, long wavelength emission, and narrow emission linewidth, compared with MBE-grown QDs. Lipinski et al. [36] reported that the blueshift of emission wavelength was observed when large QDs was stacked closely while usual small QDs show redshift by stacking (Fig. 9.5). Transmission electron microscopy revealed clearly the increase of dot size in the second layer in both cases. The reason of blueshift was not clear, but they pointed out the inﬂuence of strain driven intermixing on the blueshift of large QD. 9.2.3 QD Buried in Strained Layer Another signiﬁcant growth technique is an overgrowth of QD by strained layer. Since the method enables QD to satisfy the demand of telecommunication wavelength, InGaAs QD buried in strained InGaAs layer has been recent most widely adopted structure. In 1999, Mukai et al. achieved the ﬁrst 1.3µm continuous-wave lasing with the InGaAs-layer overgrowth technique, that was a milestone for the practical application of QD lasers [37,38]. Low growth rate was also a key for the successful growth of high density 1.3-µm emission dots using MBE [39]. Figure 9.6 shows the photoluminescence wavelength

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Fig. 9.5. Photoluminescence spectra of twofold stacks of slowly grown InAs QDs with spacer layers varied from d = 40 to 17 nm. With decreasing spacer layer a second peak on the high energy side evolves, which is attributed to emission from the dots in the upper layer. The energy diﬀerence E of the emission from the upper and lower dot layer vs. the spacer thickness d is plotted as an inset (after Lipinski et al. [36]) 1.4

300 K

1.3

1010 1.2

Density (cm−2)

PL wavelength (µm)

1011

0.5 X 0.5 µm2 1.1 10−3

10−2

10−1

109

Growth rate (ML/sec.) Fig. 9.6. Photoluminescence wavelength and dot sheet density as a function of growth rate. AFM images are indicated as insets (after Mukai et al. [39])

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and dot sheet density as a function of growth rate. Atomic force microscopy images are indicated as insets. By reducing the growth rate, the sheet density decreases and the dot diameter increases with the emission wavelength shift to 1.3 µm. However, the sheet density of 1.3-µm emission dots was small as under 1010 cm−2 . In addition, when the QDs were stacked by the low growth rate, the upper dot layers contained an extremely small number of dots. Also, the emission eﬃciency was degraded much by achieving 1.3 µm emission only with strained layer overgrowth. These were severe obstacles to obtaining high optical gain. Owing to the facts that dot sheet density is rather high and degradation of the density with stacking is weak with the medium growth rate, the combination of low growth rate and strained layer overgrowth was used instead. Consequently, high-density QDs emitting at 1.3 µm with high emission eﬃciency were successfully formed. The characteristic temperature of threshold currents, T0 , of 82 K was achieved by the QD buried in strained InGaAs layer near room temperature. Low temperature study suggested that an inﬁnite T0 with a low threshold current (∼1 mA) is available if the nonradiative recombination process is eliminated. In 2000, the record low threshold current density of 19 A cm−2 was attained by Park et al. [40] using QDs buried in InGaAs. The buried QD structure is also called as dot in well (DWELL) structure. The controllability of QD properties by strained layer overgrowth was theoretically investigated, based on the three-dimensional ﬁnite element method (Fig. 9.7) [41]. The theory suggested that the energy separation between the ground state and the ﬁrst-excited state exhibits a maximum value as a function 1.4

75

1.3

InGaAs -buried

70

GaAs -buried

1.2

65

Wavelength, λgl (µm)

Energy separation, ∆E12 (meV)

80

1.1 60 0.4

(300 K)

0.5

0.6

0.7

0.8

0.9

1.0

QD indium composition Fig. 9.7. Dependence on indium composition of QD on the ground state emission wavelength and the energy separation between the ground and the second state (after Mukai et al. [53])

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of QD composition, enabling us to identify the composition of the QD structure from energy states. It was also shown that the wavefunction symmetry is improved by burying the QD in the InGaAs layer. As mentioned in the introduction of this chapter, symmetry of the wavefunction in a QD state is critical for generating quantum-entangled photon pairs [7]. The Stranski–Krastanov type QD has a quite asymmetric shape. The structural symmetry is remarkably improved by closely stacking. The QD structure buried in a strained layer also improves wavefunction symmetry, because the crystal lattice of the QD is expanded in the growth direction [42] and the strain just above the QD is smaller in the InGaAs layer than the strain in the GaAs layer above a conventional QD. These strain eﬀects reduce the quantum conﬁnement in the growth direction. Mukai et al. [43] ﬁrst pointed out the very distinctive feature of QD buried in highly strained InGaAs layer. The temperature sensitivity of interband emission energy was suppressed signiﬁcantly in 1.3-µm emitting selfassembled InGaAs/GaAs QDs by the overgrowth (Fig. 9.8). Transmission electron microscopy measurements indicated that lattice distortion was enhanced

Fig. 9.8. Emission energy shift of dots buried in a 10-nm thick Inx Ga1−x As (0 < x < 0.3) overgrowth layer as a function of temperature. Energy of bulk GaAs is indicated as a reference (after Mukai et al. [37])

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on dots in the overgrowth layer. Photoluminescence spectra showed that the emission energy shift with increasing temperature was nearly negligible above 150 K when the indium composition of overgrowth layer, x, is greater than 0.25. The shift between 4.2 and 200 K was less than half that of bulk GaAs when x = 0.3. The results revealed the potential of InGaAs-covered dots in realizing temperature-insensitive lasing wavelength. Similar anomalous temperature dependence of photoluminescence peak energy was also reported for InAs QD embedded in GaNAs layer [44] and InAs QD grown on InAlAs quantum wire [45]. The InGaAs-buried QD has shown recent further success in the application to lasers. Temperature-insensitive high-bit-rate modulation of 1.3-µm QD laser was reported by Otsubo et al. (2004) using the InGaAs-buried QD. A 6.5-dB extinction ratio between 20 and 70◦ C and a 10-Gb s−1 direct modulation were demonstrated [46]. In their report, another key technology they adopted was p-type doping into QD structure. This was ﬁrst proposed by Takahashi et al. [47]. Shchekin et al. reported in 2002 that a stacked p-doped quantum dot active region was eﬀective to obtain high T0 of 213 K is obtained between 0 and 81◦ C for continuous-wave operation [48]. A characteristic temperature The p-doping reduces the phonon bottleneck eﬀect in valence band, improving optical gain at the ground state in QD.

9.3 Growth Process of QD Structure Analysis of the growth process of self-assembled QD has been promoted eagerly. Early study by Xie et al. [28] reported that transmission electron microscopy observation of AlGaAs marker layer grown on InAs QD revealed the presence of strain dominated atom migration away from the islands over ◦ dynamically evolving length scales of 100–400 A. Seen in Fig. 9.9, there are the

Fig. 9.9. (a) and (b) are (2 0 0) dark ﬁeld images of sample B showing the tilted nature of AlGaAs marker layers viewed along the [011] and [01¯ 1] azimuths. (c) schematically depicts the GaAs growth front evolution and identiﬁes the distinct regions. (d) shows a high-resolution electron microscope image around an InAs island with the electron beam along [01¯ 1]. The arrows indicate the slope break points for successive growth between marker layers (after Xie et al. [28])

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InAs island, the InAs wetting layer between the islands, the GaAs cap layer with AlGaAs marker layers, and the strained dark region in the substrate below the islands. The atom migration caused depressions in GaAs capping layer above QDs during growth. Three regions can be classiﬁed according to the AlGaAs marker layer images in the ﬁgure (1) a central region where the contrast of the AlGaAs markers is lost, (2) the tilted region which represents the transition from the central region to a ﬂat region, and (3) the ﬂat region away from the InAs islands where the marker layers are approximately parallel to the interface plane. The growth temperature of the sample was 480◦ C. Similar marker growth at 420◦ C on InAs island indicated fairly ﬂat layers, suggesting that the formation of the depression in GaAs was impeded by lowering the substrate temperature. The mass transportation of indium atoms has been believed to form partial wetting layer on the surface of capping layer [32, 49]. Theoretical investigation by Ledentsov et al. [50] suggested the QD stacking process as follows. Seen in Fig. 9.10, after the deposition of GaAs on InAs QD, the gradient of surface chemical potential leads to a locally directional migration of gallium adatoms away from the InAs islands. In other words, the elastic relaxation of the upper InAs layers of the dots makes it an energetically unfavorable site for GaAs atoms, and retards eﬃciently GaAs growth on the top of InAs islands. In the next step, it is energetically favorable for InAs to evaporate from the

Fig. 9.10. Vertical QD alignment process with retardation of GaAs coverage and following indium hole formation during growth interruption (after Mukai et al. [53])

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InAs islands and to cover the free surface of GaAs, forming a new partial wetting layer on top of the GaAs cap. That is, there exists a thermodynamically favored tendency for indium atoms to be detached from the InAs islands and cover the free GaAs surface. Enhanced surface energy will make the planarization of the GaAs surface by directional migration of gallium adatoms energetically more favorable, and then, the rest of the InAs island will be completely conﬁned by GaAs. Second island is formed above the ﬁrst islands since the arrangement is also energetically beneﬁcial. Recently, the surface depression on the overgrowth layer has come to be observed directly in MBE system combined with in situ scanning probe microscope. Joyce et al. [51] reported that surface diﬀusion produced a characteristic valley-ridge structure above the low-growth-rate QDs and the surface was not ◦ planarized even after a cap thickness was over 400 A (Fig. 9.11). The feature highlights the importance of anisotropic gallium diﬀusion on the (2 × 4) surface. The overgrowth behavior of low-growth-rate QDs diﬀered from that of conventional-growth-rate QDs, which was attributed primarily to initial strain state diﬀerence. Heidemeyer et al. [52] also reported that hole and trench structures were observed on the surface of capping layer just above QDs when the growth rate was low (Fig. 9.12). Atomic force microscopy images revealed that, after deposition of a thin spacer layer, rhombus-shaped nanostructures with a trench through their center existed on the GaAs surface. Width and length of the nanostructures were approximately 66 nm in the [110] direction and approximately 150 nm in the [1¯ 10] direction. The rim of the trench had

Fig. 9.11. Change in mean QD or mound height (triangle) and valley depth (square) as a function of GaAs capping layer thickness. The solid line is the expected change in QD height assuming that the GaAs grows only around the QD (after Joyce et al. [51])

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Fig. 9.12. AFM images of the GaAs surface after capping the ﬁrst QD layer with (a) 20 nm, (b) 15 nm, and (c) 10 nm GaAs. The surface was decorated with nanostructures. In (a) and (b) the structure has a trench through the middle whereas in (c) a hole in the center of a rhombus-shaped structure is observed. The scan area is 1 µm×0.75 µm and the height scale was 7 nm. The horizontal direction corresponds to the [1¯ 10] crystal direction (after Heidemeyer et al. [52])

a height of approximately 1.6 nm. As a spacer layer thickness was reduced to 10 nm, the trench gradually changed to a hole in the middle of a rhombusshaped structure. The hole had a depth of approximately 0.7 nm with respect to the surface level of the sample, and was surrounded by a rim of 1 nm height. The second QD was grown on the center of the nanostructures.

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The vertical self-alignment process must be dominated by these surface structure. Calculation by Mukai et al. [53] based on the ﬁnite element method revealed mechanism of QD alignment in growth direction taking into account the eﬀect of indium hole formed in stacked wetting layer just above underlying QD. Assuming that the QD in the stacked layer will be grown at the position where the lattice distortion is minimal, they estimated the critical thickness of spacer layer between QD layers to make QDs align in growth direction. The theoretical values agree well with values estimated by experiments.

9.4 Analysis of QD Structure The exact value of the composition of postgrown QD was left unknown in many published studies. The molecular component of QD may not be InAs even when only indium and arsenic atoms are supplied during QD formation, due to mass transport at the growth surface. The composition of nanosized crystal buried in the matrix should be identiﬁed carefully. It should be noted that most analytical methods, such as energy dispersive X-ray spectroscopy (EDX) or X-ray rocking curve measurements, detect information averaged in the probing direction of the sample. This section deals with two representative methods to analyze detailed information of QD structure. 9.4.1 Grazing Incidence X-Ray Scattering In contrast to the X-ray rocking curve analysis which can detect average information of QD layer [54], the grazing incidence small-angle X-ray scattering (GISAXS) measurements were applied to the determination of the strain ﬁeld, composition distribution, and shape of a single QD. Zhang et al. [55] revealed that detailed structural analysis can be performed using synchrotron radiation as X-ray source. They showed that the dot shape was an octagonal-based truncated pyramid with {111} and {1 0 1} facet families as shown in Fig. 9.13. They suggested that the strain inside QDs was elastically relaxed variously and that the volume distribution of partially strained InAs was peaked at intermediate strain values. Hsu et al. [56] succeeded very detailed analysis of QD structure. The GISAXS measurements were performed at light source of SPring-8 to determine the strain ﬁeld, composition distribution and shape of the dots. The QD shape and strain distribution they investigated are shown in Fig. 9.14. They found that indium composition was less than the nominal composition, 50%, near the dot/substrate interface and exceeded 50% near the top of the dots, and that the relation between the strain and the composition did not obey Vegard’s law. The grazing incidence method also reveals QD ordering. Quantitative GISAXS experiments by Zhang et al. [57] revealed pronounced nonspecular diﬀuse scattering satellite peaks with high-diﬀraction orders, indicating a lateral ordering in the spatial positions of InAs QDs. The mean dot-to-dot

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Fig. 9.13. Logarithmic GISAXS intensities with q along [110] direction. The satellite peaks are symmetrically located with respect to the specular beam at q = 0. The inset (a) is the GISAXS spectrum investigated at an exit angle of 1.3◦ . Addi◦ tional broad facet crystal truncation rods peaks at q = 0.07 A−1 are indicated by thick arrows. The inset (b) depicts the position of the facet crystal truncation rods peaks in the q –qz space. The inset (c) is a sketch of the InAs QD shape suggested on the basis of our GISAXS data (After Zhang et al. [55])

Fig. 9.14. The variation of radii as a function of height along (square) [2¯ 20] and (circle) [220] directions. The marked numbers are the lattice mismatches with respect to GaAs at corresponding height (after Hsu et al. [56])

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distance and correlation lengths of the dot lateral distribution were found to be anisotropic. They observed the sharpest dot distribution in the [110] direction. Ordering in growth direction, i.e., closely stacking eﬀect, was also evaluated by the GISAXS by Gonzalez et al. [58]. They showed that we can access one of the weak (200) X-ray reﬂections to evaluate the degree of vertical alignment of InAs QDs in multilayers. The degree of alignment was assessed by ﬁtting the X-ray diﬀuse scattering proﬁles using a model of Gaussian lateral displacement of QDs, and the average value of the mistake in stacking positions of QDs was determined. Similar grazing incidence X-ray technique was applied to a GaN QD in an AlN matrix by Chamard et al. [59]. The values of the interdot vertical and lateral correlation lengths were determined. 9.4.2 Scanning Tunneling Microscopy In situ cross-sectional scanning tunneling microscopy (XSTM) has been shown to be a eﬀective method to obtain direct structural and chemical information on QDs. After epitaxial growth, the sample is cleaved in situ to expose clean cross-sectional surface for the analysis. The XSTM observation of QD structure was ﬁrst performed by Wu et al. [60]. Evaluations by XSTM have revealed that the indium composition in a QD is not uniform. Liu et al. [61] reported XSTM measurement indicating the shape and composition distribution of InGaAs QDs formed after strained layer overgrowth. The composition appeared highly nonuniform with an indium-rich core having an inverted triangle shape as shown in Fig. 9.15. Bruls et al. [62] also investigated the shape and indium distribution of InAs QDs grown with low growth rate of 0.01 monolayer per second by XSTM observation. From outward relaxation (Fig. 9.16)

Fig. 9.15. (a) 250 nm × 80 nm STM image of InGaAs QDs grown by heterogeneous droplet epitaxy and (b) its zoom-in view (after Liu et al. [67])

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Fig. 9.16. Measured and calculated line proﬁles across the center of the dot. All indium gradients and the measured proﬁle are plotted from bottom to top (after Bruls et al. [62])

and lattice constant proﬁles, they concluded that their QDs consisted of an InGaAs alloy and that the indium concentration increased linearly in the growth direction. Thus, the proﬁle of the composition in the literature is revealed to vary due to the variation of growth conditions. Various type of QD structures have been investigated by XSTM. Lenz et al. [63] reported that InGaAs QD buried by an InGaAs layer had reversed truncated cone composition distribution. They have shown that the structure and composition distribution strongly depends on the size of buried QD [64]. Closely stacked QDs have been also analyzed [65,66]. Bruls et al. [66] reported that the strain in the GaAs matrix around the dots was strongly aﬀected by the stacking process, from comparison of the lattice constant proﬁles of stacked and unstacked dots. They showed that an increasing deformation of the dots and a reduced growth rate of the GaAs spacer layers caused the formation of terraces on the growth surface on which new dots form. In addition to above, XSTM was used to determine the composition distribution of droplet-epitaxygrown QDs [67] and GaSb/GaAs QDs [68].

9.5 Summary and Perspective In this chapter, we review the recent progress of epitaxial growth technique and characterization of self-assembled QD composed of III–V compound semiconductor. We have shown that the control of self-assembled QD’s properties has much progressed. Undeveloped subject is a control of in-plane positioning with any size and high-emission eﬃciency. Due to the lack of information on the control technique, for example, the prospect of planar circuit for quantum computing is still far from certain. We have also shown that the development of evaluation technique enables us to analyze atomic scale QD properties. QD

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technology will have a great future with a good feedback of the evaluation to the growth technique. Now, the commercial products of semiconductor QD device are going out into the market, and self-assembled QD is increasingly expected as an essential material for the future ubiquitous world.

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10 Potential-Tailored Quantum Wells for High-Performance Optical Modulators/Switches T. Arakawa and K. Tada

10.1 Introduction Semiconductor quantum well (QW) and related structures have been studied intensively since the concept of the superlattice was proposed [1]. The fabrication of QWs became possible owing to remarkable progress in semiconductor crystal growth techniques such as molecular beam epitaxy (MBE) and metalorganic vapor phase epitaxy (MOVPE). The QW structures are utilized for various kinds of photonic devices and electron devices. These days quantum wires (QWRs) and quantum dots (QDs) [2, 3] also attract much attention as advanced low-dimensional conﬁnement structures for higher performance devices. The low-dimensional structures such as the QWRs and the QDs are expected to exhibit excellent optical properties and are promising for highperformance photonic devices, however, commercial devices using the QWRs and QDs have not yet been in the market. This is because fabrication techniques for the low-dimensional structures with high-uniformity and highdensity have not matured and established. Therefore QWs still play the main part in the electronics and optoelectronics ﬁelds. For photonic devices such as semiconductor lasers, modulators, and switches, rectangular QW (RQW) structures, which have simple rectangular potential proﬁles, are usually used. However, if a sophisticated QW structure with a nonrectangular potential proﬁle is utilized, it is possible to obtain new and excellent optical characteristics and improved device performance. Such QWs are called “potential-tailored quantum well” or “potential-modiﬁed quantum well.” The quantum-conﬁned Stark eﬀect (QCSE) [4–6] is one of the most important physical properties for a QW. This shows a signiﬁcant amount of change of the absorption coeﬃcient with an applied electric ﬁeld because of the

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enhanced exciton binding energy in a two dimensional structure of a QW [7]. The energy shift due to the QCSE in a rectangular QW is expressed as Cm∗ e2 L4 F 2 , (10.1) h2 ¯ where m∗ is the eﬀective mass of the particle, L is the width of a QW, F is the applied ﬁeld, e the electron charge, and C is the constant. Various kind of photonic devices such as electroabsorption modulators (EAMs) [8], self-electrooptic eﬀect devices (SEEDs) [9] have been demonstrated and developed. A rectangular potential proﬁle is usually utilized for the QCSE for its simplicity of fabrication, however, the potential-tailored QW is expected to exhibit new physical properties based on the QCSE that cannot be obtained in RQWs. In this chapter, we review the potential-tailored QW structures and their application to photonic devices. As the photonic devices, we focus on optical modulators and switches that are key devices for next generation optical networks because the advanced QW structures are very eﬀective for improving performances of such devices. ∆E = −

10.2 Parabolic Potential Quantum Well A quantum well with a parabolic band proﬁle, the so-called “the parabolic potential quantum well” has been proposed and studied theoretically and experimentally. Miller et al. [10] demonstrated a GaAs/AlGaAs parabolic potential QW grown by MBE and showed that it can lead to forbidden transitions with strengths greater than that of nearby allowed transitions. Chu-liang and Qing [11] calculated the energy levels, wavefunctions of electrons and holes, and exciton binding energies in a parabolic QW. They showed that the parabolic potential QW exhibits stronger conﬁnement than in a single RQW, and harmonic-oscillator-like behavior even under an electric ﬁeld. Chuang and Ahn [12] studied interband and intersubband optical transitions in a parabolic potential QW with an applied electric ﬁeld and evaluated the dipole moment matrices, the absorption coeﬃcients, and the changes in the refractive indices with the intrasubband relaxations taken into account. Exciton eﬀects, however, were neglected. Ishikawa et al. [13] analyzed the Stark shift, exciton binding energies, absorption coeﬃcient change and refractive index change considering exciton eﬀect in parabolic potential QWs, and demonstrated their polarizationindependent characteristics for transverse electric (TE) and transverse magnetic (TM) mode light. Figure 10.1a shows the potential proﬁle of the parabolic potential QW. When an electric ﬁeld F is applied, the energy shift of the ground state of an electron ∆Ee is expressed as ∆Ee = −

e2 L2p F 2 , 16Vpc

(10.2)

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Electron energy Conduction band

Vpc

e1

z hh1 Vpv

lh1 Valence band

Lp

(a) Conduction band

GaAs

Al0.3Ga0.7As

Valence band

(b) Fig. 10.1. Potential proﬁles of (a) parabolic potential quantum well and (b) equivalent parabolic potential quantum well

where Lp is the width of parabolic potential quantum well and Vpc is the potential depth for the conduction band. For the energy shift of the ground state of a heavy hole and a light hole, ∆Ehh = ∆Ehh = −

e2 L2p F 2 , 16Vpv

(10.3)

where Vpc is the potential depth of the valence band. The above equations show that the energy shift is independent of a particle mass. The Stark shift ∆E for a heavy hole-electron transition is identical to that for a light holeelectron transition and is expressed as

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light hole

Al0.3Ga0.7As

Llh Lhh GaAs

Fig. 10.2. Potential proﬁle of a quantum well with mass-dependent width

e2 L2p F 2 ∆E = − 16

1 1 . + Vpc Vpv

(10.4)

The TE mode light mainly interacts with heavy holes, and the TM mode light interacts only with light holes. Therefore, polarization independence can be realized more easily in the parabolic potential QW than in the conventional rectangular QW. This interesting property was experimentally demonstrated using an equivalent parabolic potential QW [13] as shown in Fig. 10.1b, and was successfully applied to a polarization-independent EAM [14]. As a polarization-independent QW, simpler and improved structure was also proposed by Yamaguchi et al. [15] as shown in Fig. 10.2. This QW structure has eﬀective width dependent on the eﬀective mass of a heavy hole and a light hole. It is a rectangular QW with thin potential barriers near the both edges. The light hole can more easily tunnel through the thin barriers than the light hole, and the eﬀective well width for the light hole is larger than that for the heavy hole. The energy shift due to the QCSE in a rectangular QW is expressed as (10.1), therefore, by adjusting the each well width as m∗hh L4hh = m∗lh L4lh ,

(10.5)

the energy shift for the heavy hole and that for the light hole can be equalized, and this QW structure shows the polarization-independent properties. By adding tensile strain in QW layers and adjusting band edge for TE and TM light, polarization-independent characteristics can be further improved [16].

10.3 Graded-Gap Quantum Well A quantum well with an asymmetric potential proﬁle is also an interesting structure because it can enhance the Stark shift, oscillator strengths leading to large absorption coeﬃcient change and refractive index change.

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F Conduction band

InAlAs

electron

Conduction band

InGaAs

light hole Valence band

Valence band

heavy hole

Lz

(a)

(b)

Fig. 10.3. Potential proﬁles of (a) graded-gap quantum well and (b) prebiased quantum well with thin barrier layer

Hiroshima and Nishi [17] calculated exciton states in a graded-gap QW (GGQW, Fig. 10.3a) with an electric ﬁeld. The GGQW structure produces “prebias state” for an electron and a hole, leading to the modiﬁed QCSE. They showed that for the ground state heavy hole exciton in this QW, the dependence of Stark shift and the oscillator strengths on the electric ﬁeld is quite diﬀerent from the case of an RQW, that is, the Stark shift and oscillator strength of the heavy hole exciton are enhanced in the GGQW. Kato and Nakano [18] demonstrated polarization-insensitive EAM with InGaAs/InAlAs tensile-strained prebiased QW for 1.55 µm. It is an RQW with a thin barrier layer inserted near one edge of the well as shown in Fig. 10.3b. This structure equivalently realizes GGQW with simpler structure. In this prebias state, large amount of the Stark shift for electrons is obtained, whereas the Stark shifts of the heavy hole and the light hole are quite small. Therefore, the shift of the absorption peak is dominated by the Stark shift of electrons. That is, it makes it possible to achieve almost the same small amount of Stark shift for the heavy hole and the light hole excitons, leading to polarization independence. Miyazaki et al. [19] investigated improvement of the QCSE for high-speed EAM with a tensile-strained asymmetric InGaAsP QW grown by MOVPE. In the QW structure, a low-barrier layer is inserted in the well layer as shown in Fig. 10.4 to modify the wavefunctions of holes. The wavefunctions of the heavy and light holes are localized more strongly in the lower potential region (left region in Fig. 10.4) compared to an RQW and spatial separation of an electron and a hole is enhanced, leading to fast quenching of an absorption peak of the exciton. They also successfully demonstrated 40-Gbps modulation with a 75-µm-long EAM.

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Valence band

Fig. 10.4. Potential proﬁles of asymmetric quantum well with low-barrier region

10.4 Asymmetric Coupled Quantum Well As described in the previous sections, the EA eﬀect based on the QCSE is usually used for semiconductor modulators. On the other hand, the MachZehnder (MZ) multiple QW (MQW) modulator using the electrorefractive (ER) eﬀect is a promising candidate as well because of its low chirp and adjustable chirp modulation characteristics [20,21]. For such modulators based on phase modulation, a large electrorefractive index change ∆n with a small absorption loss is required. In an RQW, however, the region of the large ∆n is usually at or very close to the absorption edge of the e1(ground state of an electron)–hh1(ground state of a heavy hole) transition (e1–hh1). Hence, in that region ∆n cannot be used because the absorption loss is too large. In the longer operation wavelength region, ∆n of the RQW decreases steeply, because the positive and negative absorption coeﬃcient changes cancel each other in the Kramers–Kronig integral relation, ∆α(λ) ∆n(λ0 ) ∝ dλ, (10.6) λ20 − λ2 where ∆α is the absorption coeﬃcient change, λ is the wavelength of light, and λ0 is the operation wavelength. Figure 10.5a shows the schematic spectral shapes of ∆α that lead to smaller values of ∆n in the longer wavelength region away from the absorption edge. The smaller ∆n is caused by the cancelation of positive and negative components in the integral in (10.6). On the other hand, spectral shapes as shown in Fig. 10.5b give us much larger values of ∆n (positive ∆n on the left-hand side and negative ∆n on the right-hand side). The ∆α in an RQW has the shape shown in Fig. 10.1a, and therefore its ∆n is very small at longer wavelength region. To obtain the spectra such as Fig. 10.5b for a large ∆n, asymmetric coupled QW structures were proposed.

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∆α +

+

+ −

−

+

−

−

λ0 Wavelength positive ∆n

negative ∆n

positive ∆n

(a)

negative ∆n

(b)

Fig. 10.5. Spectra of absorption coeﬃcient change ∆α leading to (a) small values of refractive index change ∆n and (b) large values of n, in the long wavelength region away from the absorption edge (after [24]) F

AlAs QW1

QW2

Al0.3Ga0.7As

GaAs 16

3 4 3

12 ML

Fig. 10.6. Potential proﬁles of GaAs/AlGaAs ﬁve-layer asymmetric couple quantum well

Susa [22, 23] proposed and investigated theoretically a three-layer asymmetric coupled QW and predicted enhancement of absorption coeﬃcient change and refractive index change. In the QW, the oscillator strength of the heavy hole in the ground state can be reduced by more than one order of magnitude when an electric ﬁeld is applied. A group that included one of the authors previously proposed a ﬁve-layer asymmetric coupled quantum well (FACQW) [24] as a novel potential-tailored quantum well for giant ∆n. The structure of the GaAs/AlGaAs FACQW is depicted in Fig. 10.6. The positive direction of an applied electric ﬁeld F is deﬁned as shown in Fig. 10.6. It consists of a 16-monolayer (ML) quantum well (QW1) and a modiﬁed 16 (= 4 + 12)-ML QW (QW2) with a three-ML Al0.3 Ga0.7 As barrier layer inserted for potential modiﬁcation. Here one ML is approximately 0.283 nm. The thin AlGaAs barrier layer makes the structure

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asymmetric and QW2 is eﬀectively shallower than QW1. The two QWs couple with each other through a three-ML AlAs barrier layer at the center. The AlAs barrier layer increases the conﬁnement of electrons and holes in QW1 and QW2. The mechanism for obtaining the ∆α spectrum shown in Fig. 10.5b can be explained as follows: At zero bias (F = 0), the wavefunctions of hh1 and hh2 (the ﬁrst excited state of a heavy hole) are dominantly in QW1 and QW2, respectively, due to the asymmetry of the structure and isolation by the AlAs high barrier. The wavefunctions e1 and e2 (the ﬁrst excited state of an electron) are uniformly distributed in QW1 and QW2. At the negative bias state (F < 0), the wavefunction of e1 is concentrated to the QW2 and that of e2 is concentrated to QW1, while those of hh1 and hh2 maintain their initial positions. Through potential modiﬁcation due to the three-ML Al0.3 Ga0.7 As intervening barrier in QW2, the transition energies of e1–hh2 and e1–lh2 become closer to those of e2–hh1 and e2–lh1, respectively, under negative bias. Hence, the exciton absorption regions of e1–hh2 and e2–hh1, e1–lh2, and e2–lh1 come to overlap each other, respectively. This leads to the two strong combined absorption peaks that are important for obtaining giant ∆n. From the above discussion, it is clear that the asymmetric-mode transitions such as e1–hh2 and e2–hh1 have similar transition energies, large wavefunction integral values and exciton binding energies under the negative bias state. As a result, strong exciton absorption change without redshift will occur near the same wavelength region as each other, leading to the absorption coeﬃcient change spectra as shown in Fig. 10.5b for a large electrorefractive index change ∆n. This behavior is quite diﬀerent from that of the QCSE in conventional RQWs. Figure 10.7a shows the calculated dependence of ∆n of the FACQW on applied electric ﬁeld at a wavelength of 830 nm, which is 40 nm from the absorption edge. The ∆n of a 10-nm (35-ML)-thick GaAs/Al0.3 Ga0.7 As RQW at a wavelength of 870 nm is also shown for comparison. For the RQW, the same calculation program as that for the FACQW is employed. The wavelength for the RQW is also approximately 40 nm from the absorption edge. As shown in the ﬁgure, the FACQW produces almost linear and giant ER sensitivity |dn/dF| (∼2 × 10−4 cm kV−1 ) under a positive and negative F , respectively, though the RQW shows a quadratic dependence on F . The |dn/dF| of the FACQW is 10–100 times as large as that of the RQW. Figure 10.7b shows the calculated values of ∆n of the FACQW and the RQW as functions of wavelength at an applied electric ﬁeld F = 25 kV cm−1 . The wavelength range for giant ∆n is over 80 nm. These properties are highly useful for ultra-wideband, ultrafast, and low-voltage optical modulation devices. It was demonstrated that a Mach-Zehnder interferometer traveling-wave optical modulator with multiple FACQW structures could achieve a modulation bandwidth of over 50 GHz [25]. The inﬂuence of layer thickness ﬂuctuations of the FACQW on ∆n [26], a modiﬁed FACQW for an electrorefractive index change with a “negative” sign [24,27], InGaAs/InAlAs FACQW

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Wavelength (RQW) (nm) 840 0.01

FACQW (@ 830 nm)

∆n/∆F ~ 2 x 10−4 cm / kV

0.006

0.004 0.002

0 −0.002

RQW (@ 870 nm)

−40

−20

0

20

40

Refractive Index Change ∆n

Refractive Index Change ∆n

0.008

860

880

900

920

940

960

F = −25 kV/cm

FACQW

0.005

0

RQW −0.005

−0.01 780

Applied Electric Field F (kV/cm)

(a)

800

820

840

860

880

900

Wavelength (FACQW) (nm)

(b)

Fig. 10.7. Electrorefractive index change ∆n for TE-mode light of the FACQW and GaAs (10 nm)/Al0.3 Ga0.7 As RQW as functions of (a) applied electric ﬁeld at a transparent wavelength and (b) wavelength in the transparent wavelength region at −25 kV cm−1

for 1.55 µm wavelength region [28,29] were also studied. A giant ER sensitivity |dn/dF | of TE-mode light as large as 1.7×10−4 cm kV−1 was observed in the GaAs/AlGaAs FACQWs using the Fabry–Perot resonance method [30].

10.5 Intermixing Quantum Well Nonsquare QW structures fabricated with intermixing (interdiﬀusing) techniques as shown in Fig. 10.8 [31] have received much attention for the fabrication of monolithic photonic integrated circuits. These techniques are called quantum well intermixing (QWI). As the QWI techniques, we have impurity-induced layer disordering (IILD) [32], photoabsorption-induced disordering (PAID) [33], and impurity-free vacancy disordering (IFVD) [34], etc. The QWI can control or trim the bandgap of III–V semiconductor after epitaxial growth. In addition, the intermixed QW itself can realize the improved electroabsorption, low chirp characteristics, etc. The potential proﬁle is determined by the interdiﬀusion of constituent atoms across the interface between a well layer and a barrier one due to postgrowth thermal process. Li et al. [35] theoretically analyzed physical properties of a GaAs/AlGaAs intermixed QW. They found that the subband energy in the intermixed QW increases for small diﬀusion length and then decrease for larger diﬀusion length, and the absorption peak for the TM light has a larger magnitude than does the corresponding peak of the TE light. They also investigated electrooptic and electroabsorptive eﬀects [36] caused by the modiﬁed potential, and their applications to Fabry–Perot modulators [37], photodetectors [38].

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Ld = 4nm Ld = 1nm

Ld = 0nm

10nm

Fig. 10.8. Potential proﬁle for an electron and a heavy hole in an InGaAs/GaAs intermixed QW with various diﬀusion length of In and Ga atoms Ld (after [31])

Kleckner et al. [39] reported the large modulation of optical properties of a GaAs/AlAs intermixed superlattice. They obtained large blueshift of band edge (approximately 170 meV) and large refractive index shift (∼9×10−2). This means that the intermixed superlattice is promising for direct fabrication of integrated optical waveguides, splitters, and interferometers. On the interdiﬀused QWs, various interesting researches have been reported: a GaAs/AlGaAs multiwavelength laser using QWI [40], QWI for an InGaAsN laser [41], the interdiﬀusion eﬀect on SiGe/Si QW [42], etc.

10.6 Summary Various types of potential-tailored QW structures are overviewed. These QWs are expected to produce excellent optical characteristics for high-performance photonic devices such as modulators and switches, and have advantages in fabricating layered structures with high-uniformity compared to QWRs and QDs. In addition, nonlinear optical eﬀects such as four-wave mixing (FWM), second harmonic generation (SHG), and quasiphase matching (QPM) can be realized in QW structures, though they are not introduced in this chapter. These eﬀects are expected to be enhanced in potential-tailored QWs.

References 1. L. Esaki, R. Tsu, IBM J. Res. Develop. 14, 61 (1970) 2. H. Sakaki, Jpn. J. Appl. Phys. 19, L735 (1980)

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11 Thermodynamic Properties of Materials Using Lattice-Gas Models with Renormalized Potentials R. Sahara, H. Mizuseki, K. Ohno, and Y. Kawazoe

11.1 Introduction To study thermodynamic properties of materials, lattice model simulation such as lattice Monte Carlo (MC) simulation is one of the simple and fast method. One advantage of the method is that it can treat larger systems both in time scale and in spatial size compared with atomic-scale molecular dynamics (MD) simulations so that it can treat thermodynamic equilibrium or diﬀusion phase transition phenomena. However, it has limitation in the description of disordered or liquid phases because displacement of atoms from regular lattice points that may be important at high temperatures could not be considered. That is, lattice models neglect the vibration entropy as well as the elastic energy. The shortcomings lead to overestimation of the phase transition temperatures and underestimation of the width of single-phase ﬁelds. To overcome the shortcomings of the model, one of us proposed the potential renormalization technique [1, 2]. This technique maps interatomic potentials for example, classical MD potentials as well as potentials obtained by ﬁrst-principles calculations onto lattice models. In other words, this bridges between oﬀ-lattice models and lattice models. The fundamental idea of the potential renormalization is to determine a new potential function without changing the partition function when we discretize the conﬁguration space at a given temperature. A two-step renormalization scheme using A, B sublattices dividing whole space was proposed and formulated in a tractable way. It was applied to the following problems including (1) the solid–liquid phase transition in Si using a body-centered-cubic (BCC) lattice model [3, 4] with a three body potential proposed by Tersoﬀ [5] and (2) the order–disorder phase transition of Cux Au1−x using a face-centered-cubic (FCC) lattice model [6, 7] with a Finnis–Sinclair-type potential [8] by Ackland et al. [9, 10] and slightly reﬁned later by Deng et al. [11]. (3) The thermodynamic properties such as the thermal expansion coeﬃcients in seven transition and noble metals are also analyzed [12] with second moment approximation (SMA) tightbinding (TB) potentials [13, 14] proposed by Rosato et al. [15] as well as by

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Cleri et al. [16]. Finally, (4) an attempt to apply the potential renormalization technique combined with ﬁrst-principles calculations will be shown to study the order–disorder phase transition in L10 FePt alloy [17]. The rest of the chapter is organized as follows. In Sect. 11.2, the two-step potential renormalization scheme is explained. In this scheme, short-range interatomic potentials can be mapped appropriately onto the generalized lattice-gas models, where the particle number at each lattice point may exceed unity. In Sect. 11.3, it is demonstrated brieﬂy that the present scheme is in principle applicable to the four models mentioned earlier.

11.2 Scheme of the Potential Renormalization The fundamental idea of potential renormalization is, as is mentioned already, to make a new potential function for discretized space without changing the value of the partition function for continuous space at a given temperature. Consider that N -atom system in the continuous space with the original potential U (x1 , x2 , · · · xN ) is decomposed into M lattice sites. The conﬁgurational term of partition function is expressed as follows; U (x1 , x2 , · · · xN ) 1 dx1 dx2 · · · dxN exp − N! kB T N MΩ F (i1 , i2 , · · · iN ) , (11.1) = ··· exp − N! kB T i i i 1

2

N

where Ω is the volume of one lattice site, ik runs over all the M lattice sites, and F (i1 , i2 , . . . , iN ) is the desired renormalized potential, which is determined by a knowledge of the original MD potential. Here, how to use this technique in the case of FCC metals is explained as an example. Since four sublattices are needed for BCC and FCC lattices so as to exclude A–A type nearest neighbors, one might use a “four-sublattice, four-step renormalization” technique for complete study. However, to reduce a large amount of the calculations, a two-sublattice, two-step renormalization technique is used here. Consider a case where lth atom is surrounded by n atoms located in adjacent and diﬀerent Wigner–Seitz (WS)-cells numbered as m+ 1, m+ 2, . . . , m+ n. Figure 11.1 illustrates a two-dimensional square lattice in which the WS cell is given by a unit square. WS cell is decomposed into two sublattices, and we call white-colored WS cell as A cell and gray-colored WS cell as B cell. In this situation, only gray-colored B cells are neighbors of an whitecolored A cell and vice versa. When the cutoﬀ radius of the original potential is short, one may assume that the cutoﬀ length of the renormalized potential to be the same order or shorter than the lattice constant so that one atom in the A WS cell is aﬀected only by the ones in the nearest neighbor B cells.

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Fig. 11.1. Illustration of the potential renormalization scheme in the case of a twodimensional simple square lattice. Division of the conﬁgurational space is shown. Solid lines represent the original MD potential and broken lines represent the desired renormalized potential (on-site free energy)

We also assume that the n coordinated atoms are surrounded further by similar conﬁgurations. Solid lines in Fig. 11.1 represent the original potential and broken lines represent the renormalized potential, which is evaluated as follows: In the ﬁrst step of the renormalization, the trace of the lth A cell is evalu ated. That is, the renormalized potential function f r m+1 , rm+2 , · · · r m+n , which is a function of the atomic positions r m+i (i = 1 ∼ n) in the adjacent B cells, is determined by evaluating the integral of the conﬁgurational exponential (Boltzmann factor) with respect to the atomic position rl inside the lth A cell, and is given as follows: exp −βf r m+1 , r m+2 , · · · r m+n 1 dr l exp −βV rl ; (rm+1 , · · · , r m+n ) , (11.2) = Ωl where V r l ; (r m+1 , · · · , rm+n ) expresses the original potential, β = 1/kB T reciprocal temperature, and Ωl volume of lth WS cell. Here, the left-hand side of (11.2) is decomposed into the product of functions, each of which depends

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only on one position, e.g., rm+i , only. That is, the following parameterization is introduced: f r m+1 , rm+2 , · · · , rm+n = f (n) |rm+1 | + f (n) |rm+2 | + · · · + f (n) |r m+n | . (11.3) To reduce the large amount of calculation, further approximation is introduced. That is, when the dependence on the atomic position in the (m+m )th B cell is estimated, other atomic positions in the (m+m )th B cells (m = m ) are ﬁxed at the center of the cell, R(m+m ) . The second step of the renormalization is to evaluate the trace of the mth B cell. From the ﬁrst step, the renormalized potential function f (n) (r m+i ) has already been estimated. As a result of this step, the desired renormalized potential F (n) for the n coordinated conﬁguration can be determined as follows: n 1 exp −βnF (n) = drm exp −β f (n) (r m ) , (11.4) Ωm i=1 where Ωm is the volume of the mth WS cell, and nF (n) in the left-hand side is the renormalized potential for all the desired n coordinated conﬁgurations. The temperatures assumed in the potential renormalization are the same as those used in Metropolis’s MC simulations [18] with Kawasaki’s spin exchange algorithm [19].

11.3 Application of the Potential Renormalization 11.3.1 Application to Melting Behavior of Si It is of technologically importance to demonstrate atomic-scale MD simulations on Si system using empirical potentials proposed by Stillinger and Weber (SW) [20], Tersoﬀ potential [5, 21, 22], or other realistic three-body potentials to compute local atomic conﬁgurations during crystal growth and melting processes. Such studies have achieved a great success, for example, in the determination of the structure of Si melts [23]. However, since the investigation of crystal growth and phase-transition processes mediated by atomic diﬀusion requires large timescale and large system-size simulations, there is a limitation in the application of the MD technique. We tried to reproduce the solid–liquid phase transition of Si as quantitatively as possible by using MC simulation on the basis of a BCC lattice model to study equilibrium thermodynamics. Figure 11.2 shows primitive 2 × 2 × 2 BCC cells. As is shown in Fig. 11.2, the model can realize a regular diamond structure by a 50% ﬁlling of BCC lattice positions. The solid phase is characterized by the coordination number equal to 4, while the liquid state is

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Fig. 11.2. BCC lattice model with 50% ﬁlling. The regular 50% ﬁlling corresponds to the regular diamond structure of Si

characterized by the coexistence of various coordination numbers including six or even more, as has been observed experimentally. Therefore, instead of deﬁning the liquid state of Si as usual by measuring the pair-distribution function, one may alternatively distinguish solid and liquid phases by checking the distribution of coordination numbers, and also by densities which was discussed by Runnels [24]. In the study, Tersoﬀ potential [5], which has been widely used for Si in MD simulations, is introduced. Here, we note that Cook et al.’s previous MD simulations [25] say that the Tersoﬀ potential reproduces a melting temperature of around 3,000 K, which is twice as high as the experimental data (1,700 K), and the potential also gives a 2% volume expansion upon solidiﬁcation, whereas 5.4% is measured in actual Si. Here, only some characteristics of the determined renormalized potential (on-site free energy) are summarized. Remember that the technique incorporates not only the internal entropy in the length scale shorter than the lattice constant but also the eﬀect of the nonlinear lattice vibration at ﬁnite temperatures. At ﬁrst, Table 11.1 shows some renormalized potential values in the unit of electronvolt at several temperatures evaluated for typical conﬁgurations, which are indicated with the coordination numbers at nearest neighbors. In the table, a conﬁguration with coordination four corresponds to tetrahedral structure, while conﬁguration with coordination ﬁve (or three) is one more

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Table 11.1. Renormalized potential values (eV) at several temperatures vs. nearest neighbor coordination Coordination 3 4 (Tetrahedral structure) 5

0K

1,000 K

1,500 K

2,000 K

2,500 K

3,000 K

−4.15 −4.56 −4.17

−4.67 −5.01 −4.68

−5.05 −5.38 −5.08

−5.47 −5.80 −5.52

−5.92 −6.25 −5.98

−6.40 −6.72 −6.47

(or less) atom included in the tetrahedral structure. The table shows that the potential depth becomes deeper as the temperature increases. On the other hand, the potential diﬀerence between conﬁgurations with coordination four and three (or between four and ﬁve) becomes smaller as the temperature increases. For example, the energy diﬀerence between the conﬁgurations with coordination four and three decreases from 0.41 eV at 0 K to 0.32 eV at 3,000 K. One may expect that the shallowing of the potential diﬀerence may reduce the melting temperature estimated in discretized models (discussed later). Next, we discuss the thermal expansion of a perfect diamond structure of Si at ﬁnite temperatures by using the lattice constants, which minimize the renormalized potential. Note that the desired renormalized potential of a perfect crystal can be derived by assuming the tetrahedral conﬁguration. Figure 11.3 shows a relationship between the lattice constant and the renormalized potential per atom at several temperatures. For instance, values at 1,500, 2,000, and 2,500 K are shown. The solid marks indicate the lattice constants, which give the minimum of the renormalized potential at each temperature. It is clear that the lattice constant becomes larger as the temperature increases. The coeﬃcient of linear expansion, α, is expressed as α=

1 da a dT

(11.5)

where a is lattice constant. α obtained by the potential renormalization at lower temperature is slightly higher than the coeﬃcients obtained by the MD simulation; for example, the value, around 7.6 × 10−6 at 500 K, is about 10% larger than the coeﬃcient obtained by Cook et al.’s MD simulation. As temperature increases, they become smaller, and, i.e., around 4.2×10−6 at 2,000 K is as same as that of the MD simulation. The reason of the reproduction of the thermal expansion is that the techniques can incorporate unharmonic term of the original potential function. Finally, we demonstrated MC simulation to study the melting transition of Si. It should be noted here that when we apply Tersoﬀ potential directly to the BCC lattice in the tentative study, we found that the observed phase transition temperature is about 11,000 K, which is signiﬁcantly higher than that of either the oﬀ-lattice MD or MC results of 3,000 K.

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Fig. 11.3. Relationship between the lattice constant and renormalized potential per atom at several temperatures. The solid marks are the lattice constants, which gives the minimum value at each temperature. It is clear that the lattice constant becomes larger as temperature increases

The system used in the simulation has 12 × 12 × 600 BCC cells (172,800 lattice points) and 4,320 Si atoms, which form a regular diamond structure on 12 × 12 × 30 BCC cells (see Fig. 11.4). The system has two free surfaces and it can calculate the diﬀerence in the density between the solid and liquid. To avoid the eﬀect of the surface, 12 × 12 × 10 BCC cells are used when we analyze the results. Figure 11.5 shows the density of the crystal as well as liquid phases at zero pressure. We can compare the densities determined by (1) the lattice constants that minimize the renormalized potential of the perfect crystal and (2) the lattice occupancies obtained by the MC simulations after 1,000,000 MC steps. In Fig. 11.5, open circles are the density of (1) perfect crystal, while open squares are ones obtained by the MC simulation. We can see that it decreases up to around 2,000 K, and then it increases up to 3,000 K, at which the density takes a local maximum. This is caused by a sensitive balance between the expanded lattice constant and the coordination number of atoms.

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Fig. 11.4. The simulation cell. 12 × 12 × 600 BCC cells are introduced. For the initial condition, in the lower 120th region in the Z-direction, silicon atoms are put so as to form a regular diamond structure. This corresponds to 4,320 atom system

The behavior reproduces well the results of the MD simulation. It shows that the potential renormalization technique is very eﬀective to treat the melting behavior of Si. 11.3.2 Application to Cu–Au Phase Diagram The Cu–Au system is known as a typical FCC alloy, which exhibits an order– disorder phase transition at relatively low temperature region. The system has often been investigated with lattice-gas models within various approximation of estimating entropy of mixing such as Bragg–Williams approximation [26], Bethe approximation [27], and tetrahedral approximation of cluster variation method of Kikuchi [28, 29]. A large scale MC simulation was also performed by Binder [30]. However, since their models were over simpliﬁed Ising Hamiltonian having nearest neighbor interactions only, the results could not be compared directly with the actual Cu–Au system.

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Fig. 11.5. Temperature dependence of the density at zero pressure. Open circles are the density of a perfect crystal, while open squares are the values obtained after 1,000,000 MC steps starting from the solid state

Horiuchi et al. tried to reproduce the phase diagram using ﬁrst-principles calculations as the interatomic interactions [31]. In their study, they ﬁrst estimated the formation energies for ﬁve structures [32] and then the internal energy was approximated by means of the cluster expansion method [33]. The conﬁgurational entropy was estimated within the tetrahedron approximation of CVM. However, the resultant order–disorder transition temperatures were overestimated by a few hundred degrees and the width of single phase ﬁelds was underestimated when compared with experimental data. In our study, a Finnis–Sinclair-type potential [8] by Ackland et al. [9, 10] and slightly reﬁned later by Deng et al. [11] was introduced (hereafter, we call Ackland potential). MC simulations with the renormalized Ackland potential are carried out to estimate the order–disorder phase transition temperature. At ﬁrst, lattice constants of pure Cu and Au crystals are investigated as a function of temperature by minimizing the renormalized potential, from which one may guess the lattice constants of Cu–Au alloys by interpolating the concentration. Figures 11.6a and b show, respectively, the temperature dependences of Cu and Au lattice constants. The obtained lattice constants increase monotonically with the temperature for both the cases. The linear thermal expansion coeﬃcients, α, and the lattice constants at 0 K, r0 , are estimated and tabulated in Table 11.2. The table shows that the

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Fig. 11.6. Lattice constants that are obtained by the potential renormalization as a function of the temperature; (a) is for pure Cu and (b) is for pure Au Table 11.2. The linear thermal expansion coeﬃcients, α, and the lattice constants at 0 K, r0 Linear thermal expansion coeﬃcient, α (×10−6 ) Cu Au Potential renormalization Experimental values [34] MD results [35, 36]

14.3 18.3 22

17.1 15.4 26

Lattice constant at 0 K ◦ r0 (A) Cu Au 3.64 3.60 3.70

4.12 4.06 3.60

values, α and r0 obtained by the potential renormalization and the experimental values [34] are very close to each other. That is, the diﬀerence in α for Cu and Au between the two methods is around 20 and 10%, respectively. The diﬀerence in r0 between the renormalization and the experiment is about 1%. Although the values of α obtained by the MD simulation [35, 36] are visibly diﬀerent when compared with the present study and the experiment, the MD values are about 30% higher than that of the renormalization, and the diﬀerence in r0 for Cu and Au between the MD simulation and the other studies are within 2 and 15%, respectively.

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By using the renormalized Ackland potential obtained above, MC simulations are carried out to estimate the order–disorder phase transition temperature. In the present study, the phase transition between disordered phase and short period (L12 or L10 ) structures are observed. According to the following expression of the lattice constant, the renormalized potentials are determined for several temperatures and compositions. That is, the lattice constants of alloys, aTCuc Au(1−c) , are assumed as a linear function of the concentration c as follows: aTCuc Au(1−c) = caTCu + (1 − c)aTAu ,

(11.6)

where c means the concentration of Cu. aTCu and aTAu are the lattice constants of the pure metals at temperature T . For the initial condition of the MC simulations, Cu and Au atoms are put so as to form ordered phase. So, in the case of stoichiometric concentrations, the system has a long-range order parameter as unity. However, in the nonstoichiometric region, long-range order parameters do not reach unity. The order–disorder transition temperatures are determined by the point where the long-range order parameter comes down to zero. A phase diagram of the Cu–Au system can be estimated using similar MC conditions except for composition and temperature. Figure 11.7 shows the ﬁnally obtained phase diagram calculation after 100,000 MC steps, which are suﬃcient to obtain thermodynamical equilibrium in the present simulation. Here, open circles indicate the result of the original Ackland potential, while the solid circles indicate the result of the renormalized Ackland potential. Solid triangles indicate the experimentally observed transition temperatures for three stoichiometric compositions [37].

Fig. 11.7. The phase diagram of the Cu–Au system obtained by MC simulations after 100,000 MC steps. Solid and open circles indicate the results obtained by using the renormalized and original potentials, respectively, while, solid triangles indicate the experimental results from [34]

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For all the calculated concentration regions, it is found that the renormalized potential gives lower phase transition temperatures than the original potential. For example, in the case of Cu3 Au, the estimated phase transition temperatures are 810 and 710 K, respectively, for the original and renormalized potentials. The transition temperature obtained by the renormalized potential is about 100 K lower than that obtained by the original potential, and it agrees with the experimental temperature (663 K). Note that the transition temperature obtained by the renormalized potential is slightly above the experimental temperature for compositions close to Cu3 Au, while it is slightly below for compositions close to CuAu3 . In contrast, near the equiatomic composition range, the transition temperature obtained by the renormalized potential is slightly lower than the experimental value. In this case, the original potential case agrees better with experiment. The fact that the transition temperatures obtained by the renormalized potential do not lie consistently above or below the experimental values is probably due to the limited accuracy of the Ackland potential used as input. 11.3.3 Application to Transition and Noble Metals We also applied the potential renormalization technique to seven FCC transition metals and noble metals (Ni, Cu, Rh, Pd, Ag, Ir, and Pt) [12] to analyze the thermodynamic properties such as the thermal expansion coeﬃcient. In the study, potential renormalization of second moment approximation (SMA) tight-binding (TB) potentials [13, 14] was demonstrated. Here, two sets of SMA parameters have been used; The ﬁrst is proposed by Rosato et al. [15] whose cutoﬀ radius is restricted to the ﬁrst neighbors (hereafter, Rosato potential), and the second is proposed by Cleri et al. [16] whose cutoﬀ radius is extended to the ﬁfth neighbors (Cleri potential). The results are compared with the previous – in principle exact – MD simulations [15, 16] and experimental results [34] when available. Since the cutoﬀ radius of Rosato potential is restricted to the ﬁrst neighbors, it is suitable for the potential renormalization scheme. Cleri potential whose cutoﬀ radius is extended to ﬁfth neighbors was introduced for Ni and Cu to check the eﬀect of the cutoﬀ radius of the potential parameters into the potential renormalization scheme. Table 11.3a, b shows the obtained values of α and a0 for seven FCC metals computed with the Rosato potential, and for Ni and Cu with the Cleri potential whose cutoﬀ radius is extended to ﬁfth neighbors, respectively. The corresponding MD results [15, 16] and experimental data are also shown. From Table 11.3a, we can see that the values, α, obtained in the present study are very close to MD results for Ni and Cu. That is, the diﬀerence in α between the two methods is within 10%. Furthermore, for all the metals, the diﬀerence in α between the present simulation and the experimental results is within 30%.

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Table 11.3. Thermal lattice expansion coeﬃcient and the lattice constant at 0 K Linear thermal expansion coeﬃcient, α (×10−6 )

Lattice constant at 0 K, r0

Present MD result Experimental Present work result work

MD result

(a) With Rosato potential Ni Cu Rh Pd Ag Ir Pt

1.6 2.2 0.88 1.8 2.7 0.73 1.1

1.7 2.4

1.53 1.89 0.96 1.35 2.17 0.77 0.97

3.52 3.61 3.80 3.89 4.09 3.84 3.92

3.52 3.61 3.80 3.89 4.09 3.84 3.92

3.6 2.4

1.4 2.1

1.53 1.89

3.52 3.68

3.52 3.62

(b) With Cleri potential Ni Cu

While, from Table 11.3b, it is shown that although α of Cu is in agreement with MD result within 20% that of Ni is as twice as that of MD result. For a0 , there is a large discrepancy between the renormalizations and MD results. This indicates that large-ranged potentials are less suited for the potential renormalization scheme with the currently used approximations. 11.3.4 Order–Disorder Phase Transition of L10 FePt Alloy Using the Renormalized Potential Combined with First-Principles Calculations In this section, we show the thermodynamic properties obtained by the potential renormalization combined with ﬁrst-principles calculations. It was applied to study the L10 -disorder phase transition phenomena of Fe–Pt alloy. It is well known that the ordered L10 FePt alloy as a material for permanent magnets because of their large magnetocrystalline anisotropy, and its order–disorder phase transition temperature is experimentally observed as 1,650 K [38]. It has been demonstrated by Mohri et al. [39] the phase transition of the alloy using CVM combined with the ﬁrst-principles calculations. In their study, thermal vibrational eﬀect was considered within Debye–Gruneisen model [40], and found that it improved the calculated transition temperature. Recently, Misumi et al. tried to investigate L10 -disorder phase transition in FePt alloy using lattice MC simulation with the renormalized potential combined with ﬁrst-principles calculations [17]. When they deduce the manybody interactions to be used on the FCC lattice, they ﬁrst estimated the total energies for ﬁve systems (FCC Fe, L12 Fe3 Pt, L10 FePt, L12 FePt3 ,

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Table 11.4. Renormalized potential values at 1,650 K (electronvolt/tetrahedral structure) for the ﬁve systems System FCC Fe L12 Fe3 Pt L10 FePt L12 FePt3 FCC Pt

Before renormalization

After renormalization

−35.709 −35.185 −33.860 −31.484 −27.948

−35.675 −35.156 −33.842 −31.469 −27.933

and FCC Pt) using the ab initio ultrasoft pseudopotential plane wave method with Vienna ab initio Simulation Program (VASP) [41, 42]. Here, the cutoﬀ energy for the plane wave expansion is taken to be 237.63 eV, and for Brillouin zone (BZ) integrations, 4 × 4 × 4 k-points is introduced. When many-body interactions were estimated, the lattice constants for all the systems are ﬁxed as that of L10 FePt, and only the ﬁrst step of the potential renormalization was introduced to reduce the huge amount of calculations. To calculate the r l integrals in (11.2), a WS cell of the sublattice A was subdivided into several dozen of grid points by considering symmetry of the systems. As an example, Table 11.4 shows the obtained renormalized potentials at 1,650 K for the ﬁve systems. For comparison, the original potentials are also shown. As is discussed in Sect. 11.3.1, one may again expect that the shallowing of the potential diﬀerence may reduce the order–disorder transition temperatures in the lattice MC simulation. Here, it should be noted that on contrary the case of Si in Table 11.1 and in Fig. 11.3, the absolute value of the renormalized potential is smaller than that of the corresponding original potential. This is simply because the kinetic free energy is not added to the renormalized potential in the present case. When MC simulation was demonstrated, 64,000-atom system was introduced. It was shown that the estimated phase transition temperatures after typically 1,000,000 MC steps are 1,900 K and 1,800 K for the direct use of the ﬁrst-principles calculations and combined with the potential renormalization, respectively. Remember that corresponding experimental value is 1,650 K [38]. The study says that even though only the ﬁrst step of the renormalization was applied, the transition temperature obtained by the ﬁrst-principles calculations combined with the renormalized potential is 100 K decreased than that obtained by the direct use of the ﬁrst-principles calculations. All of the results presented above enable us to conclude that the potential renormalization scheme improves the quantitative analysis of thermodynamic properties such as phase transition behavior of materials. Finally, here we note that the scheme of the potential renormalization has already been applied to other study such as statistical analysis of semiﬂexible polymer solutions [43].

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11.4 Summary In the chapter, the potential renormalization scheme is explained in detail. It overcomes the several problems of lattice models owing to the omission of motions of atomic positions from regular lattice points. A two-step renormalization scheme is proposed and formulated in a tractable way. One advantage is that the present scheme can use diﬀerent potential calculations, for example, classical MD potential or ﬁrst-principles descriptions of the systems. The application of the technique to Si melting, order–disorder phase transition of Cu–Au, analysis of thermodynamic properties of transition and noble metals with classical MD potentials are described. Finally the technique is combined with ﬁrst-principles calculations to study order–disorder phase transition of Fe–Pt binary alloy. It is found that, for all the cases, the renormalized potentials give improved thermodynamic properties compared with the original potentials applied directly on the lattice. Acknowledgments The authors are grateful to the Computer Science Group of the IMR, Tohoku University and Computer Center of Tohoku University and for their continuous support of their computing systems.

References 1. K. Ohno, The Sci. Rep. Res. Inst. Tohoku Univ. A 43, 17 (1997) 2. K. Ohno, K. Esfarjani, Y. Kawazoe, Solid-State Sciences, Vol. 188, (Springer, Berlin Heidelberg New York, 1999) 3. R. Sahara, H. Mizuseki, K. Ohno, S. Uda, T. Fukuda, Y. Kawazoe, J. Chem. Phys. 110, 9608 (1999) 4. R. Sahara, H. Mizuseki, K. Ohno, H. Kubo, Y. Kawazoe, J. Cryst. Growth 229, 610 (2001) 5. J. Tersoﬀ, Phys. Rev. B 38, 9902 (1998) 6. H. Ichikawa, R. Sahara, H. Mizuseki, K. Ohno, Y. Kawazoe: Mater. Trans. JIM 40, 914 (1999) 7. R. Sahara, H. Ichikawa, H. Mizuseki, K. Ohno, H. Kubo, Y. Kawazoe, J. Chem. Phys. 120, 9297 (2004) 8. M.W. Finnis, J.E. Sinclair, Phil. Mag. A 50, 45 (1984) 9. G.J. Ackland, V. Vitek, Phys. Rev. B 41, 10324 (1990) 10. G.J. Ackland, G. Tichy, V. Vitek, M.W. Finnis, Philos. Mag. A 56, 735 (1987) 11. Hui-fang Deng, David J. Bacon, Phys. Rev. B 48, 10022 (1993) 12. R. Sahara, H. Mizuseki, K. Ohno, Y. Kawazoe, Mater. Trans. 46, 1127 (2005) 13. F. Ducastelle, F. Cyrot-Lackmann, J. Phys.Chem. Solids. 31, 1295 (1970) 14. D. Tomanek, A.A. Aligia, C.A. Balseiro, Phys. Rev. B 32, 5051 (1985) 15. V. Rosato, M. Guillope, B. Legrand, Phil. Mag. A 59, 321 (1989) 16. F. Cleri, V. Rosato, Phys. Rev. B 48, 22 (1993) 17. Y. Misumi, Graduation thesis of Yokohama National University (2006)

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18. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21, 127 (1953) 19. K. Kawasaki, in Phase Transitions and Critical Phenomena, Vol. 2, Chap. 11, ed. by C. Domb (M.S. Green Academic, London, 1972) 20. F.H. Stillinger, T.A. Weber, Phys. Rev. B 31, 5262 (1985) 21. J. Tersoﬀ, Phys. Rev. Lett. 56, 632 (1986) 22. J. Tersoﬀ, Phys. Rev. B 37, 6991 (1988) 23. H. Ogawa, Y. Waseda, Z. Naturforsch., A: Phys. Sci. 49A, 987 (1994) 24. L.K. Runnels, in Phase Transition and Critical Phenomena, Vol. 2, Chap. 8, ed. by C. Domb (M.S. Green, Academic, London, 1972) 25. S.J. Cook, P. Clancy, Phys. Rev. B 47, 7686 (1993) 26. W. Schockey, J. Chem. Phys. 6, 130 (1938) 27. Y.Y. Li, J. Chem. Phys. 17, 447 (1949) 28. R. Kikuchi, Phys. Rev. 81, 988 (1951) 29. J.M. Sanches, D. de Fontaine, Phys. Rev. B 17, 2926 (1978) 30. K. Binder, Phys. Rev. Lett. 45, 811 (1980) 31. T. Horiuchi, S. Takizawa, T. Suzuki, T. Mohri, Metall. Mater. Trans. 26A, 11 (1995) 32. K. Terakura, T. Oguchi, T. Mohri, K. Watanabe, Phys. Rev. B 35, 2169 (1987) 33. J.W.D. Connolly, A.R. Williams, Phys. Rev. B 27, 5169 (1983) 34. Y.S. Touloukian et al. (eds.), Thermal Expansion: Metallic Elements and Alloys, Thermophysical Properties of Metal, Vol. 12, (IFI/PLENUM, New York, 1972) 35. J.M. Holender, J. Phys. 2, 1291 (1990) 36. J.M. Holender, Phys. Rev. B 41, 8054 (1990) 37. T.B. Massalski (ed.), Binary Alloy Phase Diagrams, (Metals Park, OH), p. 254 (1986) 38. T.B. Massalski (ed.), Binary Alloy Phase Diagrams, (Metals Park, OH), (1986), p. 1093 39. T. Mohri, Y. Chen, Mater. Trans. 43, 2104 (2002) 40. V. Morruzi, J.F. Janak, K. Schwarz, Phys. Rev. B 37, 790 (1988) 41. G. Kresse, J. F¨ urthmuller, Phys. Rev. B 54, 11169 (1996) 42. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990) 43. K. Ohno, Trans. Mater. Res. Soc. Jpn. 29, 3787 (2004)

12 Optically Driven Micromachines for Biochip Application S. Maruo

12.1 Introduction In the last 1980s, microelectromechanical systems (MEMS) such as microgears and microactuators have been developed by using silicon-based micromachining techniques [1]. Most of MEMS utilize electrostatic actuators as a major driving source. The electrostatic actuators have been used in practical microdevices, including digital micromirror device (DMD) and RF MEMS. However, it is diﬃcult to utilize electrostatic force in liquids such as electrolytes. For this reason, electrostatic actuators are not suitable for biological applications such as microﬂuidic devices and micromanipulation tools for cells. On the other hand, radiation pressure from a tightly focused laser beam can be used as optical tweezers to conﬁne, position and transport microparticles in liquid. Ashkin’s group ﬁrst demonstrated this technique in 1986 [2]. Optical tweezers provide unique features such as remote manipulation of micro-/nanoparticles in liquid, noninvasive manipulation of biological samples, precise manipulation in sealed environment and extremely small torque of the order of 10−12 N m. For these reasons, optical tweezers and its related techniques have been widely applied to studies on biological samples such as cells and DNA molecules, and microchemistry with microdroplets and microbeads. The history and previous works on optical tweezers were introduced in some review reports [3–6]. In most applications of optical manipulation techniques, target samples such as microparticles, droplets and cells are merely trapped or transferred to the desired position. By contrast, sophisticated three-dimensional (3D) microstructures produced by microfabrication techniques such as two-photon microstereolithography [7–9] can also be maneuvered according to the desired motions. The optically driven micromachines oﬀer unique remotely controlled microdevices such as micropumps and micromanipulators for biochip applications. In this chapter, optically driven micromachines produced by a two-photon microstereolithography technique are introduced.

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Fig. 12.1. Two-photon-absorbed photopolymerization generated by a focused laser beam

12.1.1 Two-Photon Microstereolithography for Production of 3D Micromachines Fabrication Principle and Apparatus The optically driven micromachines were fabricated by a two-photon microstereolithography technique [7–9]. As shown in Fig. 12.1, in the two-photon process, a liquid photocurable resin absorbs two near infrared photons simultaneously in a single quantum event whose energy corresponds to the ultraviolet (UV) region. The rate of two-photon absorption is proportional to the square of the intensity of light, so that near infrared light is strongly absorbed only at the focal point within the liquid photocurable resin. The quadratic dependence of two-photon absorption assists to conﬁne the solidiﬁcation to a submicron volume. This virtue of the two-photon process enables us to create a 3D microstructure by scanning a focus inside a photocurable resin. Two-photon microstereolithography, also known as two-photon 3D microfabrication, was ﬁrst demonstrated in 1997 [7]. It has been used to make various types of 3D microstructure including microtubes [8], microsprings [9, 10], and photonic crystals [11, 12]. However, almost all the microstructures were ﬁxed on a base. Some types of free microrotators have been demonstrated [13, 14], but they were simple, wire-frame components. By contrast, in the recent years, we reported the fabrication of more sophisticated movable micromachines such as a pair of micromanipulators, microgears and a micropump [15–17]. Figure 12.2 shows our current fabrication system of the two-photon microstereolithography. In this fabrication system, a mode-locked Ti:sapphire laser (Mira-900F, Coherent, Inc., wavelength, 752 nm; repetition, 76 MHz; pulse width, 200 fs) is used to generate the two-photon absorption. The beam from the laser is introduced into the galvano-scanner system (M2 scanners, GSI Lumonics, Inc.) to deﬂect its direction in two dimensions, and then it is focused with an objective lens set at a numerical aperture of 1.35. The beam

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Fig. 12.2. Fabrication system of two-photon microstereolithography

scans laterally in the photocurable resin while the stage (Mark-204, Sigma KK, Japan) that supports the resin is scanned vertically, thereby moving the point of focus in three dimensions. By controlling both the galvanoscanner system and stage with 3D computer-aided design (CAD) data, a 3D microstructure is fabricated. The system has attained a peak resolution of 140 nm, thus exceeding the diﬀraction limit of light. After the 3D fabrication process, the unsolidiﬁed resin is washed out with a rinse (EE04210, Olympus Optical Co., Ltd), leaving only the resultant microstructure. The photocurable resin used was an epoxy resin for general UV stereolithography (SCR-701, Japan Synthetic Rubber Co., Ltd). 12.1.2 Assembly-Free, Single-Step Fabrication Process of Movable Microparts In conventional microstereolithography, although simple raster scanning of a laser beam has been used to make a microobject ﬁxed to a base, it is not suitable for the assembly-free fabrication of freestanding microparts because of shrinkage of the photocurable resin during polymerization. The problem is overcome by optimizing the scanning pattern of a laser beam to reduce the deformation of the solidiﬁed object. For example, circularly scanning of a laser beam is utilized to make a rotation-symmetric movable microstructure such as a microgear [18]. The process of fabricating a microgear is illustrated in Fig. 12.3 [15]. An attached shaft and stopper are ﬁrst fabricated by scanning a laser beam circularly at each layer, while the stage is lowered step-by-step with a constant interval along a central axis. Next, the stage is moved to the position to make a movable gear wheel. The circular part of a gear wheel is fabricated by scanning the laser beam circularly with increasing the radius. Finally, the teeth of the gear wheel are fabricated by circularly scanning the laser beam with

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Fig. 12.3. Assembly-free, single step fabrication process based on laser direct writing. A microgear with a shaft is fabricated without any supporting parts [8]

Fig. 12.4. Sequential optical microscopic images of the process of fabricating a microturbine [8]

increasing the radius, while the scanning speed is changed periodically. In this case, the nonlinear response of two-photon absorption makes possible to solidify the resin only in the region where the scanning speed is low. Figure 12.4 shows sequential optical microscopic images of the process of fabricating a microturbine. These optical microscopic images were observed in real time with a CCD camera. This result demonstrates that circularly scanning of a laser beam is useful for making such rotation-symmetric structure. Actually, through this approach, a rotation-symmetric microstructure

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Fig. 12.5. Optical microscopic images of microwheels fabricated by two methods with diﬀerent starting points. (a) Stating point locates on the contour of inner hole, and (b) starting point locates inside the shape of the wheel

whose inner diameter is smaller than about 20 µm can be fabricated without undesirable deformation. However, a breakthrough is needed to make a movable microstructure whose inner diameter is larger than 10 µm. Therefore, we devised an alternative way to make any movable microstructures. In this method, the starting point of a laser scanning is put not on the contour of a microstructure but inside its shape. Figure 12.5a, b shows optical microscopic images of microwheels fabricated by two methods with diﬀerent starting points. In Fig. 12.5a, since the starting point is located on the contour of the inner hole, the shape of the resultant wheel is deformed owing to the shrinkage of the photocurable resin. On the other hand, in case the starting point is located inside its shape, the deformation is suﬃciently reduced. These results demonstrated that the optimization of the starting point is eﬀective in making a sophisticated movable microstructure in which even large holes are included. Thus, the combination of several types of scanning pattern makes possible to fabricate any movable microstructures with high precision. Not only thin but also thick movable microparts can be fabricated through the direct laser writing method. Figure 12.6a–d shows scanning electron microscope (SEM) images of microgears with diﬀerent thickness [19]. The microgears were fabricated by changing both the numerical aperture of an objective lens and exposure conditions. As the numerical aperture is reduced, a thicker microgear is fabricated. This is because the focal depth is enlarged by reducing the numerical aperture of the objective lens. Using an objective lens of low numerical aperture, the input laser power must be increased to generate two-photon-absorbed photopolymerization. However, if the laser power exceeds the critical laser power, the solidiﬁed microparts are destroyed owing to laser breakdown. For this reason, the thickness of movable microstructure is limited by the critical laser power. To make thicker microparts, layer-by-layer process is introduced into the laser direct writing method. Figure 12.7a–e shows SEM images of microgears fabricated by the layer-by-layer process

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Fig. 12.6. SEM images of microgears fabricated at diﬀerent numerical aperture objective lenses [19]. (a) NA: 1.25, laser power: 100 mW, thickness: 0.59 µm, (b) NA: 1.15, laser power: 150 mW, thickness: 0.97 µm, (c) NA: 1.05, laser power: 200 mW, thickness: 1.25 µm, and (d) NA: 0.95, laser power: 250 mW, thickness: 1.57 µm

without any supporting parts. The layer-by-layer process makes possible to fabricate a microgear 8 µm thick in the experiments.

12.2 Optically Driven Micromachines 12.2.1 Optical Trapping One of the most promising applications of polymeric movable micromachines produced by two-photon microstereolithography is optically driven micromachine for biochip application. The driving force of the optically driven micromachines results from momentum of photon. If the direction of light propagation is changed due to the reﬂection and refraction, a momentum change of photon is generated. According to Newton’s third law, photon pressure is exerted on the interface between two media as a reaction to the momentum change. In electromagnetic theory of light, the photon force is well known as optical radiation pressure. In case of the size of a microparticle is larger than the wavelength of incident light, the radiation pressure exerted on the microparticle can be analyzed by ray optics [20]. As shown in Fig. 12.8, when a laser beam is focused on a transparent microparticle whose refractive index is larger than that of the surrounding liquid, the laser beam is reﬂected and refracted at the surface of the

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Fig. 12.7. SEM images of microgears fabricated by the layer-by-layer process based on direct laser writing. (a) Number of stacking layers: 2, thickness: 1.66 µm, (b) number of stacking layers: 4, thickness: 2.29 µm, (c) number of stacking layers: 6, thickness: 2.71 µm, (d) number of stacking layers: 8, thickness: 3.73 µm, and (e) Number of stacking layers: 10, thickness: 4.83 µm

microparticle. The laser beams A and B incident to the particle are mainly refracted; thereby radiation pressure is exerted at a vertical direction on the top of the microparticle. When the refracted light exits from the microparticle, radiation pressure is also exerted on the bottom of the particle. The net radiation pressure surrounding the surface of the microparticle is directed to the focus of the laser beam. Since a part of the laser beam whose incident angle is large play an important role to levitate the microparticle, an objective lens of high numerical aperture is normally used for the 3D manipulation of

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Fig. 12.8. Optical trapping of a dielectric microparticle with a focused laser beam

microparticles. The force of optical trapping is proportional to the intensity of the trapping laser beam. By using optical trapping, polymeric movable microstructures that are produced by two-photon microstereolithography can be driven and controlled remotely. Figure 12.9a, b illustrates the driving method of a microgear. As shown in Fig. 12.9a, when the laser beam is focused on the center of a gear tooth, the radiation pressure exerted on the gear tooth is balanced both in lateral and in depth. As a result, the arm is stably trapped at the focus. By contrast, as shown in Fig. 12.9b, if the focus is slightly moved to the side of the gear tooth, the net radiation pressure is directed to the focus. As a result, the microgear is attracted and moved to the focus. Therefore, the circular scanning of a single laser beam makes possible to rotate the microgear. In the same way, various motions of micromachines can be generated according to the desired trajectory. 12.2.2 Optical Driving Method of Multiple Micromachines For the simultaneous driving of multiple micromachines using a single laser beam, there are several methods: holographic optical tweezers [21], continuous laser scanning method [22], and time-divided laser scanning method [23,24]. In particular, the time-divided laser scanning method is a simple, feasible way to control multiple micromachines with a low-cost apparatus using galvano scanners or piezoelectric mirrors. We use galvano scanners to scan the laser beam; we can generate any trajectory of the laser beam with a vector scanning method. The trajectory of vector scanning consists of a series of discrete foci separated with a constant distance as shown in Fig. 12.10. In our method, on–oﬀ control of the laser power is not employed for the simpliﬁcation of the optical scanning system.

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(c) Fig. 12.9. Optical driving method of a movable microstructure. (a) Schematic, (b) a gear tooth trapped at a focus, and (c) attractive force exerted on a gear tooth

The drawback owing to the simpliﬁcation is easily canceled by the optimization of the laser scanning conditions such as trajectories, division distances and waiting times of each focus. For example, in case two microgears are rotated simultaneously, two circular trajectories are generated by the timedivided laser scanning. The repetition rate between two circular trajectories is an important parameter for the synchronized rotation of two microgears. Figure 12.11 shows two methods with diﬀerent repetition rates: single-step and multistep methods. In the single-step method, since averaged optical intensity distribution between two circular trajectories is higher than that of each circular trajectory, two microgears cannot be rotated eﬃciently. On the other hand, if the repetition rate is reduced by use of multistep method, the averaged optical intensity along the target trajectories is high enough to rotate two microgears smoothly. The optimization of both target trajectories and repetition rates enables to drive multiple micromachines simultaneously.

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Fig. 12.10. Vector scanning of a laser beam with galvano scanners. The target circular trajectory is divided into discrete foci separated with a constant division distance

Fig. 12.11. Synchronized laser scanning along multiple trajectories with diﬀerent repetition rate

12.2.3 Optimization of Time-Divided Laser Scanning Time-divided laser scanning method for the control of a micromachine is examined using micromanipulators [25, 26]. The micromanipulators used in our experiments were produced by two-photon microstereolithography. The optical system shown in Fig. 12.2 is utilized as an optical manipulation system of micromanipulators. Figure 12.12 show an SEM image of a prototype of micromanipulators (length: 8 µm, thickness: 2 µm). In experiments, a manipulator arm is trapped and rotated by scanning a trapped laser beam circularly with diﬀerent division distances, while the scanning speed is increased. The maximum following velocity at each division distance is measured by changing the waiting time. At the maximum following velocity, optical driving force is

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Fig. 12.12. SEM image of a prototype of optically controlled micromanipulators (length: 8 µm, thickness: 2 µm)

Fig. 12.13. The dependence of the following velocity of the manipulator arm on laser power at diﬀerent division distances (100, 200, 300, 400, 500, 750, 1000 nm)

balanced with viscous drag force exerted on the manipulator arm. Figure 12.13 shows the dependence of the following velocity of the manipulator arm on laser power. As the laser is increased, the following velocity is higher. This is because the optical driving force is proportional to the intensity of light. At the division distance over 500 nm, although the manipulator arm follows the trajectory of the focused laser beam, the following velocity is extremely low. This means that the optical trapping force is insuﬃcient to attract the arm in short waiting time at large division distance. Therefore, the division distance should be smaller than 500 nm to drive the manipulator arm smoothly in our optical manipulation system. This limit of division distance depends on several parameters, including the spot size of the focused laser beam, the shape of

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the manipulator arm, and viscosity of the surrounding liquid. The maximum following velocity was achieved at the division distance of 400 nm when the laser power is 500 mW. In this case, the grip force of the tip of manipulator arm is estimated at 3.5 × 10−13 N. 12.2.4 Cooperative Control of Micromanipulators A pair of micromanipulators was cooperatively controlled by the time-divided laser scanning method. Figure 12.14 shows optical microscopic images of micromanipulators handling a microglass bead (diameter: 5 µm). The manipulator arms were controlled by scanning a laser beam along two fan-shaped trajectories. By the optimization of repetition rate between two trajectories, two manipulator arms were operated to handle a glass microbead. The repetition rate is about 100 Hz in this demonstration. Three-hand micromanipulators as shown in Fig. 12.15 were also driven and controlled by the same way. Figure 12.16 shows the sequential optical

Fig. 12.14. Optical microscopic images of micromanipulators handling a microglass bead (diameter: 5 µm)

Fig. 12.15. SEM image of three-hand micromanipulators

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Fig. 12.16. Handling of a microglass bead (diameter: 5 µm) with three-hand micromanipulators

microscopic images of handling a glass microbead (diameter: 5 µm) with the three-hand micromanipulators. In this case, three arms are driven sequentially. The three-hand manipulators can handle larger object stably compared to the two-hand manipulators shown in Fig. 12.12b. Conventional micromanipulators for biological applications are operated in a petri dish with glass micropipettes, so they cannot be used inside a sealed microspace such as a microchannel and a microreactor. By contrast, our optically controlled micromanipulators are suitable for handling microobjects such as microbeads, cells, and microbes in sealed environments. This is because the micromanipulators can be easily built in a microchannel by using two-photon microstereolithography. The micromanipulators are remotely controlled by light. Therefore, the optically controlled manipulators will open a novel way to manipulate biological samples inside a sealed environment such as microﬂuidic circuits. Figure 12.17 shows the operation of a prototype of a microﬂuidic circuit in which both micromanipulators and a microseparator are integrated into a y-branch microchannel (width: 13 µm, height: 20 µm) [26]. Both the microchannel and the micromachines are fabricated on a cover glass by using two-photon microstereolithography. In this case, the micromanipulators and the microseparator are cooperatively controlled by time-divided laser scanning, so that target biological samples are selected to the upper channel and examined by the micromanipulators. 12.2.5 Optically Driven Micropump In the development of functional biochip, built-in micropumps based on pressure-driven ﬂow are necessary for the further extension of application ﬁelds. Although various kinds of pressure-driven micropumps have been developed using a range of micromachining techniques [27, 28], most of the

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Fig. 12.17. Optically controlled micromanipulation chip with a microseparator [26]

previously developed micropumps are diaphragm types, in which piezoelectric actuators or pneumatic actuation are utilized to deform an elastic membrane that drives ﬂuid transport. To obtain adequate displacement of the membrane, the diameter of the membrane must range from 100 to 500 µm. As a result, the overall dimensions of the micropumps are much larger than the microchannels. This prevents the further miniaturization and integration of microﬂuidic components. In addition, the use of built-in high-precision microactuators makes biochips expensive. For these reasons, most current disposable microﬂuidic chips for chemical synthesis and cell analysis still utilize external pumps such as syringe types rather than built-in micropumps. As a promising method of addressing the above issues, we have developed an optically driven micropump by using two-photon microstereolithography [17,29]. The micropump consist of two lobed rotors that are incorporated into a microchannel. The lobed rotors are individually conﬁned to a microchannel by their own shaft, thus preventing the rotors from moving unless continuously irradiated with light. This allows the micropump to be easily integrated into a mass-produced biochips made from plastics and glass. Figure 12.18a shows a schematic diagram of the optically driven lobed micropump. This micropump is driven by means of radiation pressure generated by focusing a laser beam. When the laser beam is focused on the side of the rotor, the net radiation pressure applied to the rotor points toward the focus, allowing the rotor to be controlled by scanning the laser beam along two circular trajectories alternately. Figure 12.18b shows the time-divided laser scanning method to control the two rotors simultaneously. The black points shown in Fig. 12.18b are trapping points for each status. The laser beam is divided and scanned along two circular trajectories in the opposite direction. In this case, on–oﬀ control of the laser beam is not performed when the laser beam alternates between the two trajectories. While the rotors are

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(b) Fig. 12.18. Optically driven lobed micropump [17]. (a) Schematic diagram; (b) ﬂuid transport by optically rotating two rotors

Fig. 12.19. SEM image of a prototype of the lobed micropump [17]

tightly engaged and rotated, ﬂuid is trapped in the spaces between the rotors and the cage, and carried around in them as shown in Fig. 12.18b. Figure 12.19 shows an SEM image of a prototype of the lobed micropump. The length of the major axis of each rotor is 9 µm. The thickness of the rotor is 2.5 µm. The lobed rotors were incorporated into a microchannel whose inlet and outlet are in 5 µm wide and 7 µm high. The microchannel was also fabricated by using two-photon microstereolithography.

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Fig. 12.20. Sequential images taken while driving the lobed micropump [17]

Fig. 12.21. Concept of an all-optically controlled biochip [25]

We demonstrated that ﬂuid was transported with the optically driven micropump. Figure 12.20a, b shows sequential images of the micropump being driven. A tracer particle was also fabricated in the microchannel using twophoton microstereolithography. The tracer particle was optically trapped to prevent it from moving around while unsolidiﬁed photocurable resin was being washed out. After the washing process, the tracer particle was released, after which the rotors were interlocked using the time-divided laser scanning technique. As a result, the tracer particle was successfully brought into the microchamber by the pumped ﬂow. When the rotation direction was reversed, the tracer particle was pushed to the outlet. Figure 12.21 shows the dependence of the velocity of the tracer particle on the rotational speed of the rotors. It is clear that the velocity of the tracer particle is proportional to the rotational speed of the rotors. The ﬂow rate was estimated at less than 1 pL min−1 . Ultralow ﬂow rates such as this give our mechanism potential for use in micro-/nanoﬂuidic devices.

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12.2.6 Concept of All-Optically Controlled Biochip The optically driven micromachines such as micromanipulators and micropumps realize all-optically controlled biochips. Figure 12.21 shows the concept of all-optically controlled biochip proposed by Maruo [25]. In this chip, micromachines such as microtweezers, microstirrers, and micropumps are cooperatively controlled by scanning a single laser beam. Target biological samples are transported into a chamber by pressure-driven ﬂow, and examined with the microtweezers. To analyze the biological sample, chemical reagents are supplied to it using micropumps and microvalves. The all-optically controlled biochip oﬀers unique advantages as follows: The combination of various types of micromachines can provide custom-made biochip for versatile applications. Since the optically controlled biochip does not need expensive and sophisticated microactuators, it is suitable for disposable use. Two-photon microstereolithography can provide 3D microstructures by a direct laser scanning inside photocurable resin, so both micromachines and a microchannel are easily fabricated together. For these reason, disposable, low-cost biochip made from photocurable resin can be provided. In addition, the micromachines can be also built in a microchannel made from glass and other plastics [30]. The integration of optically controlled micromachines into a mass-produced microchannels can provide low-cost functional biochips.

12.3 Conclusion and Future Prospect We have developed several types of optically driven micromachines, including micromanipulators and a micropump, using two-photon microstereolithography. One of the most promising applications of the optically driven micromachines is microﬂuidic devices such as micrototal analysis systems, lab-on-a-chip devices and BioMEMS. We proposed a concept of all-optically controlled biochip. In the biochip, since built-in micromachines are remotely controlled by light, the biochip is suitable for disposable usage. The disposable biochips suit medical diagnosis and analysis from the standpoint of prevention of biohazard. For this reason, other types of optically driven microﬂuidic devices using colloids have also been developed in recent years [31–33]. Although the conventional optical microscopes and lasers are still used for the operation of the current optically controlled biochip, on-chip lasers or optical ﬁber systems are ideal for practical use in medical diagnosis and palm-top analysis systems. Some pioneer works have been reported in this context. For example, optical ﬁbers have been utilized to manipulate microparticles inside a microchannels [34, 35]. In addition, semiconductor lasers are integrated into a microchannel for the sequential manipulation of microparticles [36]. In the near future, all-optically controlled biochips that are fully integrated not only with light sources but also with detectors could be created by advanced

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micromachining techniques. The fully integrated biochips driven by light are expected to open a path toward the future biotechnology and medical care and diagnosis. Acknowledgment This research was supported in part by research grants from the Japan Society for the Promotion of Science (Grant-in-Aid for Young Scientists (A), Exploratory Research and Scientiﬁc Research in Priority Areas: Systems Cell Engineering by Multi-scale Manipulation) and PRESTO from Japan Science and Technology Agency.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

K.D. Wise, K. Najaﬁ, Science 254, 1335 (1991) A. Ashkin, J.M. Dziedzic, T. Yamane, Nature 330, 769 (1987) A. Ashkin, IEEE J. Sel. Top. Quantum Electron. 6, 841 (2000) D.G. Grier, Nature 424, 810 (2003) K.C. Neuman, S.M. Block, Rev. Scientiﬁc Instrum. 75, 2787 (2004) J.E. Molloy, M.J. Padgett, Contemp. Phys. 43, 241 (2002) S. Maruo, O. Nakamura, S. Kawata, Opt. Lett. 22, 132 (1997) S. Maruo, S. Kawata, J. Microelectromech. Syst. 7, 411 (1998) S. Kawata, H.B. Sun, T. Tanaka, K. Takada, Nature 412, 697 (2001) H.B. Sun, K. Takada, S. Kawata, Appl. Phys. Lett. 79, 3173 (2001) B.H. Cumpston et al., Nature 398, 51 (1999) K.K. Seet, V. Mizeikis, S. Juodkazis, H. Misawa, Appl. Phys. Lett. 88, 221101 (2006) P. Galajda, P. Ormos Appl. Phys. Lett. 78, 249 (2001) P. Galajda, P. Ormos, Appl. Phys. Lett. 80, 4653 (2002) S. Maruo, K. Ikuta, H. Korogi, J. Microelectromech. Syst. 12, 533 (2003) S. Maruo, K. Ikuta, H. Korogi, Appl. Phys. Lett. 82, 133 (2003) S. Maruo, H. Inoue, Appl. Phys. Lett. 89, 144101 (2006) S. Maruo, K. Ikuta, Appl. Phys. Lett. 76, 2656 (2000) H. Inoue, S. Haga, S. Maruo, IEEJ Trans. SM 126, 216 (2006) A. Ashkin J. Biophys. 61, 569 (1992) J. Leach, G. Sinclair, P. Jordan, J. Courtial, M.J. Padgett, J. Cooper, Z.J. Laczik, Opt. Express 12, 220 (2004) K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, Opt. Lett. 16, 1463 (1991) C. Mio, T. Gong, A. Terray, D.W.M. Marr, Rev. Sci. Instrum. 71, 2196 (2000) F. Arai, K. Yoshikawa, T. Sakami, T. Fukuda, Appl. Phys. Lett. 85, 4301 (2004) S. Maruo, Y. Hiratsuka, in Proceedings of Micro Total Analysis Systems 2005, 2005, p. 1206 Y. Hiratsuka, S. Maruo, IEEJ Trans. SM 125, 473 (2005) D.J. Laser, J.G. Santiago, J. Micromech. Microeng. 14, R35 (2004) M.A. Unger, H.P. Chou, T. Thorsen, A. Scherer, S.R. Quake, Science 288, 113 (2000)

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29. S. Maruo, H. Inoue, in Proceedings of Micro Total Analysis Systems 2005, 2005, p. 590 30. S. Maruo, K. Oishi, T. Hasegawa, in Proceedings of Micro Total Analysis Systems 2006, 2006, p. 669 31. A. Terray, J. Oakey, D.W.M. Marr, Science 296, 1841 (2002) 32. J. Leach, H. Mushﬁque, R. Leonardo, M. Padgett, J. Cooper, Lab Chip 6, 735 (2006) 33. J. Gl¨ uckstad, Nat. Mater. 3, 9 (2004) 34. P. Domachuk, M. Cronin-Golomb, B.J. Eggleton, S. Mutzenich, G. Rosengarten, A. Mitchell, Opt. Express 13, 7265 (2005) 35. C. Jensen-McMullin, H.P. Lee, E.R. Lyons, Opt. Express 13, 2634 (2005) 36. S.J. Cran-McGreehin, K. Dholakia, T.F. Krauss, Opt. Express 14, 7723 (2006)

13 Study of Complex Plasmas M. Shindo and O. Ishihara

Micron-sized dust particles in laboratory plasmas have large negative charges (|Q| = 103 ∼ 105 e) as a result of interaction with ambient plasma. They form structures known as Coulomb crystals when the electrostatic energy is much more than the thermal energy of dust particles. In this section, fundamental physics of a complex plasma is described. Numerical simulation shows particle dynamics forming Coulomb clusters and experiments show the change of the charges of dust particles by irradiating the electron beam and Coulomb cluster formation in cryogenic environment.

13.1 Overview of Complex Plasma Research A plasma including dust particles is called “dust plasma” and found everywhere in space, laboratories and factories. Large quantity of nanosized particles are produced in processing plasmas for manufacturing semiconductors using reactive gases such as SF6 , CF4 , SiH4 , and most of them have no net charge. However, nanoparticles grow via coagulation processes into micron in size [1]. Micron-sized particles are also produced via the sputtering of wafer materials [2]. Note that the charged micron-sized particles can levitate at the plasma-sheath boundary, since the electrostatic force on a dust particle is in balance with the gravity in the sheath region. Levitating dust particles in processing plasmas should be removed of course, since they degrade the quality of the products such as thin ﬁlms and electronic devices. One of the interesting features of dust plasma is that the levitating charged particles often form designed structures with macroscopic scale. We call especially the plasma including micron-sized dust particles “complex plasmas” [3, 4], in which micron-sized particles have a large quantity of negative charge and form macroscopic systems via various kinds of interactions with ambient plasma. Experiments revealed the presence of Coulomb crystals by charged dust particles in plasmas [5–7]. Many experimental researches and numerical simulations

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have been made in addition to the theoretical analyses to clarify the formation mechanism of those structures. In this section, the behavior of micron-sized particle in complex plasmas is reviewed to explore the possible complex plasma design. We treat the situation where artiﬁcial spherical particles with 1 µm or more in diameter are put in a plasma. In Sect. 13.2, the charging of dust particles is discussed. The charging of dust particles is estimated in an electron beam plasma device as described in Sect. 13.3. Interactions between charged microparticles and ambient plasma are discussed in Sect. 13.4. A numerical simulation is a powerful tool to study charged dust particle behavior as shown in Sect. 13.5, where designed structures of dust particle clouds are investigated in detail. In Sect. 13.6, some recent experimental results of Coulomb cluster formation in cryogenic environment in our laboratory are presented.

13.2 Charging of a Dust Particle in a Plasma Micron-sized dust particles in laboratory plasmas are usually charged negatively [3, 4], since electrons with larger mobility than ions rapidly attach onto the dust particle surface. The system of charged particles shows a collective behavior characterized by dust plasma frequency ωpd −1/2 2π nd Q 2 = 2π ωpd M d ε0 1/2 5 −3 1/2 a ρd 10 cm 104 ∼ 2.38 [ms] |Q/e| 1 µm 1 g cm−3 nd (13.1) which gives 2π/ωpd ∼ 2 ms when the dust particle radius a = 1 µm, ρd = 1 g cm−3 , nd = 105 cm−3 , and Q = −104 e. This fact indicates that dust particles show very slow collective behavior in comparison with the plasma with the inverse of the electron plasma frequency 2π/ωpe ∼ 3 (ns) when the electron density is ne ∼ 109 cm−3 . As shown in (13.1), the behavior of complex plasma is strongly dominated by the dust particle charge. The charge of a dust particle is determined by dust–plasma interaction, i.e., electrons and ions buildup charges on a dust particle and ions with directed ﬂow drag the charged dust particles [8–11]. The charge of an isolated spherical particle in a plasma with the surface potential φd is expressed by means of capacitance of an isolated sphere with the radius a Q = 4πε0 aφd ,

(13.2)

where ε0 is the dielectric constant in vacuum. The surface potential of a dust particle is determined such that a net current into the dust is zero. In a typical laboratory plasma containing no high-energetic electrons, Q is about −104 e for a = 1 µm and φd ∼ −10 V.

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13.3 Measurements of the Charge of Dust Particles Levitating in Electron Beam Plasma [12] As described in Sect. 13.2, the charge Q accumulated on a dust particle surface characterizes a complex plasma. To estimate Q in dust plasma experiments, (13.2) is generally used where the surface potential is determined from the balance between electron and ion current into a dust particle as described in Sect. 13.2. On the other hand, when plasma includes high-energy electron beam, the magnitude of φd increases with the electron beam energy [13, 14], i.e., we can control the charge of dust particles in electron beam plasma. In this section, we report on an experimental attempt to control the surface charge of dust particles by irradiating the electron beam in a glow discharge plasma. Figure 13.1 shows a schematic diagram of experimental apparatus of an electron beam plasma system developed by Dr. Yoshiharu Nakamura. The cylindrical vacuum chamber was made of pyrex glass with the inner diameter of 15 cm and ﬁlled with Xe gas. The gas pressure was 0.044 Pa. Hot electrons were emitted from the heated tungsten ﬁlament cathode and accelerated toward the anode by applying VG ∼ −110 ∼ −80 [V] to generate a plasma. Electrons were injected into the plasma as an electron beam with the energy of eVG = 110 ∼ 80 [eV]. The electron temperature and density were measured

Fig. 13.1. Experimental apparatus of the electron beam plasma system

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Fig. 13.2. Position from the electrode of levitating particles as the function of the electron beam energy for the Xe gas pressure of 0.044 Pa

with the Langmuir probe as Te ∼ 7 eV and ne = 2.5 × 107 cm−3 , respectively. The gold-coated particles with the diameter of 5 µm were set in a piezobuzzer placed beneath the plate electrode at z = 0. Without plasma, particles had initial velocity v0 at z = 0 and were blown up to z = h0 = 3.5 cm from the electrode by the piezobuzzer, while they levitated higher z = h1 > 10 cm with plasma. The plate electrode potential was kept ﬂoating. The levitated particles formed a thin layer in a plasma and were observed by irradiating He–Ne laser light. The vertical position of particles is plotted in Fig. 13.2 as a function of the electron beam energy. The increase in vertical position of particles with the electron beam energy is due to the increase in the charge of a dust particle, as described in the following analysis. We consider a situation where a particle with the charge Q and the initial velocity v1 at z = 0 is levitating at z = h in the plasma and reaches the position at z = h1 . Since the space potential is φp at z = 0 and φs at z = h, the following equation of energy balance for a dust particle can be obtained: 1 Md gh − Md v12 + Wn = |Q(φs − φp )|, 2

1 Md v12 = Md gh0 , 2

(13.3)

where Md is the mass of a dust particle, g is the gravitational acceleration. Equation (13.3) includes the energy loss term due to the neutral drag Wn = nn vn2 mn πa2 h [15] where nn , vn and mn are the density, velocity, and mass of neutrals. Note that space potential at z = 0, φp , is equal to the plate electrode potential. In our experiment, |eφp | is nearly equal to the electron beam energy and φs is almost constant about 7 V over the levitation space for all the electron beam energy range. We also assume v1 ∼ v0 since dust particles are blown up mechanically. Using Md = 2 × 10−13 kg in (13.3), we obtain the dependence of the charge of a dust particle on the electron beam energy as shown in Fig. 13.3. As expected, the magnitude of charge increases with the electron beam energy. Therefore, the charge of a dust particle was successfully controlled with the electron beam energy. On the other hand, the

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Fig. 13.3. The dependence of estimated charge of dust particles on the electron beam energy for the pressure of 0.044 Pa

surface potential of a dust particle in the levitating region is about −33 V for eVG = 80 eV, which is the ﬂoating potential measured with the Langmuir probe. In this case, Q is also obtained from (13.2) to be |Q/e| ∼ 57,000, twice as large as that estimated from (13.3). In summary, we succeeded in controlling the charge of dust particles by varying the electron beam energy in a glow discharge plasma. The Coulomb cluster was not formed in the present experiment since the gas pressure is too low for particles to lose their kinetic energy. However, the present experiment suggests that we can design the Coulomb clusters of negatively charged particles by controlling the charge of the dust particles.

13.4 Various Approaches to Plasma-Aided Design of Microparticles System in Ion Flow When a few particles are injected in a plasma, they levitate around at the plasma-sheath boundary. Here, the sheath is the ion-rich region in front of the electrode where the electric ﬁeld occurs and then the subsonic or supersonic ion ﬂow exists in the direction toward the electrode [16]. The interactions between the ion ﬂow and dust particles are very important in designing system with microparticles in a plasma. For example, the ion ﬂow forms a wake potential behind a charged dust particle, which arranges dust particles along the ﬂow. On the other hand, when particles line up perpendicularly to the ﬂow, the force perpendicular to the ﬂow occurs due to a release of thermodynamic free energy in charged particles. The most dominant interaction is so-called ion drag force [8–11], which is calculated as a momentum transfer from ions to a charged particle. In this section, our researches on dust–ion ﬂow interactions are summarized.

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13.4.1 Analysis of Ion Trajectories Around a Dust Particle in Ion Flow [17] Ion motion around a negatively charged particle is generally described with an orbital-limited motion theory [18, 19], where the conservation laws of energy and angular momentum of ions are used to provide orbiting ions. Selfconsistent PIC [20] and ﬂuid simulation [21] models succeeded in reproducing the ion distribution around an isolated dust particle in ion ﬂow, as deduced from the wake ﬁeld model [22–24]. In this section, ion trajectories around a negatively charged and ﬁnite-sized dust particle are studied in detail based on equations of motion including a hydrodynamic term representing the behavior of ion ﬂuid bypassing a spherical obstacle. The simple model suggested here will help us to understand the mechanisms for dust particle charging and ion drag force generation. Figure 13.4 shows the schematic diagram of the present system. A dust particle with radius a and negative charge Q is placed at the origin. The uniform ion ﬂow is assumed to be in the z direction, having the velocity V at inﬁnity. The external electric ﬁeld E0 is applied also in z direction producing the dipole moment p in the dust particle. In this system, the equation of motion for an ion ﬂuid element including ions with mass m and charge q is expressed as follows: v = µE + u.

(13.4)

Here, v = (vr , vθ ) is the ion velocity, µ is the ion mobility, and u is the hydrodynamic velocity. And the electric ﬁeld E is given by E = E 0 +E d +E C

Fig. 13.4. System where a dust particle with the radius a and the charge Q is placed at the origin and uniform ion ﬂow exists with the velocity V at inﬁnity

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with the external ﬁeld E 0 , the dipole ﬁled E d , and the Coulomb ﬁeld E C . We assume that u is expressed with Stokes’ stream function ψ [25] for an axial symmetric ﬂow around a particle with radius a by 1 ∂ψ 1 ∂ψ , − , (13.5) (ur , uθ ) = r2 sin θ ∂θ r sin θ ∂r where ψ(r, θ, V ) = −

V 2 2 3 a 1 a 3 r sin θ 1 − + . 2 2 r 2 r

(13.6)

In this model, two parameters, Q/Q0 and V /V0 , are introduced where Q0 = 4πε0 E0 a2 and V0 = µE0 . The charge Q0 represents the threshold charge beyond which the ion trajectory deviates from a straight line. This is apparent from (13.4) since vr = 0 at r = a when Q = Q0 . When V V0 , the trajectories of ions are determined dominantly by the ﬂuid motion described by the Stokes’ stream function rather than the electric ﬁelds. Equation (13.4) allows us to describe the ion trajectories as a series of contour lines expressed by the following equation [26]: Q 2 r 2 sin2 θ − 2 cos θ = K, (13.7) + −2ψ + a r/a Q0 where K is a constant. The typical trajectories are plotted in Fig. 13.5 for |Q/Q0 | = 0.1, 3, 10 when V /V0 = 1 and 10. When |Q| > |Q0 |, some trajectories departing at (|x/a| > 1, z → −∞) are bent and absorbed onto the dust surface in downstream side. The ion inﬂow on the dust surface increases with |Q/Q0 | while decreases with increase in the ion ﬂow velocity. To estimate the impact parameters for ion absorption onto the dust particle surface, we ﬁrst determine K in (13.7) for the outermost trajectory. The critical condition for a trajectory reaching the dust surface is vr |r=a = 0, which is equivalent to cos θc = −(1/3)|Q/Q0|, where θc is a critical angle. We ﬁnd that the critical angle becomes constant, or θc = π for |Q/Q0 | > 3. Then we obtain the impact parameter bc for the outermost trajectory terminated at (r, θ) = (a, θc ). The resultant impact parameters are:

bc a bc a

2 =

(3 + |Q/Q0 |)2 3(1 + V /V0 )

for |Q/Q0 | < 3,

=

4|Q/Q0| 1 + V /V0

for |Q/Q0 | ≥ 3.

2

(13.8)

Equations (13.8) indicate that the cross section for ions absorbed on the dust particle increases with the increase in the charge of the dust particle but decreases with the increase in the ion ﬂow velocity as shown in Fig. 13.5.

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Fig. 13.5. The orbits of ions moving in z direction around a dust particle (solid circles) with negative charge Q, for (a) |Q/Q0 | = 0.1 and V /V0 = 1, (b) |Q/Q0 | = 3 and V /V0 = 1, (c) |Q/Q0 | = 10 and V /V0 = 1, (d) |Q/Q0 | = 0.1 and V /V0 = 10, (e) |Q/Q0 | = 3 and V /V0 = 10, and (f ) |Q/Q0 | = 10 and V /V0 = 10

When an additional second particle is placed along the z-axis in upstream region, our model provides the ion trajectories in Fig. 13.6 when |Q/Q0 | = 10 and the interparticle distance is 5a. This ﬁgure shows that the ions absorbed onto the ﬁrst particle cannot give any charge or any force on the second particle. The similar trajectory is obtained even when the interparticle distance is fairly larger. We can also ﬁnd that the second particle will receive only downward force which will get the second particle to the ﬁrst one, in other words, the attractive force occurs between two particles. 13.4.2 Wake Potential Formation to Bind Dust Particles Aligned Along Ion Flow In the sheath region, a few particles are sometimes observed aligning along the ion ﬂow. On the other hand, from the point of view of collective motion of a plasma, a spatially oscillating potential appears behind a charged particle in ion ﬂow, which is called wake potential [22–24]. The wake potential will

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Fig. 13.6. The ion trajectories around two particles situated along the ion ﬂow when |Q/Q0 | = 10 and the interparticle distance is 5a

trap another particle, in other words, it makes particles attract each other. When the ion ﬂow with the velocity v f exists in −z direction, the oscillating potential φw (r, z) formed in the downstream region of a particle with charge Q is obtained as follows [24]: φw (r, z) Q M2 = 2πε0 (M 2 − 1)3/2

1/λDe

dk⊥ 0

k⊥ kDe

2

J0 (k⊥ r) sin

k⊥ z √ M2 − 1

, (13.9)

where M is the Mach number deﬁned as M = |v f |/Cs with the ion sound velocity Cs and kDe is the inverse of the electron Debye length λDe , kDe = 1/λDe = (ne e2 /ε0 kB Te )1/2 , with temperature Te . The integrand is a function of k⊥ which is the wave number perpendicular to the ion ﬂow and J0 (k⊥ r) is zeroth-order Bessel function. Equation (13.9) represents the wake potential, and φw (x) with M = 1.5 is shown in Fig. 13.7, indicating that particles will be trapped in the potential minima of the wake formed by the leading particle, in another words, particles behave in such a way that they attract each other. Such an attraction between highly charged dust particles overcome the Coulomb repulsion and may explain the observed crystal formation in a plasma. Actually, the formation of wake potential was observed in numerical simulations [27] and laboratory experiment [28]. The theory explains the

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Fig. 13.7. Wake potential behind a dust particle in the ion ﬂow where M = 1.5. A large negative potential at the front, indicating the presence of a dust particle, is followed by a large positive potential and then an oscillatory nature in space [29]

observed characteristic spacing of the order of a Debye length and observed alignment of dust grains along the plasma ﬂow. 13.4.3 Attractive Force Between Dust Particles Aligned Perpendicular to Ion Flow [30] Dust particles are observed to align perpendicular to ion ﬂow. The attractive force between dust particles aligned perpendicular to the ion ﬂow should be also discussed. In this section, the attractive force perpendicular to the ﬂow is shown to result from a release of thermodynamic free energy in charged ﬁne particles. The electrostatic energy in the presence of a supersonic ion ﬂow is given by Z 2 e2 k2 |∇φ(x)|2 3 −ik·(xi −xj ) UE = ε0 d x∼ d , (13.10) 2 )2 e 2 2ε0 V i,j (k 2 + kD e k where we put Q = Zd e. We note that the electrostatic energy is induced by a pair displacement xi − xj , a novel feature of a multiparticle system. Such a displacement necessarily introduces the volume change in the system through the relationship G = F + P V , where G and F are the Gibbs free energy and the Helmholtz free energy, respectively, and P is the pressure. The thermodynamic excess energy (Gibbs energy) thus may be evaluated as η ∂ ∂Zd ∂ E(η ) G= ns + ns dη Uij , ∂ns ∂ns ∂Zd η 0 s=e,i i,j 1 Zd2 e2 Uij = 1 − kDe Rij e−kDe Rij , (13.11) 2πε0 Rij 4 where Rij = |xi − xj | is the interparticle distance and Uij is the electrostatic energy between ith and jth particle. The force acting on a dust pair is then given by

13 Study of Complex Plasmas

Fij = −

∂Uij Zd2 e2 = 2 ∂Rij 2πε0 Rij

321

1 2 2 1 + kDe Rij − kD e−kDe Rij . (13.12) R 4 e ij

In the complex plasma experiments, dust particles characterized by the charge Zd e ﬁnd their positions on an equipotential line in the sheath region. Each dust particle on the same equipotential line perpendicular to the direction of the ion ﬂow is subject to the force given by (13.12). In this case, the interparticle distance is given by Rij = (1+a coth kDe a)(1+ 1 + 2/(1 + kDe a coth kDe a)). In other words, the measurement of interparticle distances may be used to estimate plasma parameters; the interparticle distance perpendicular to the ion ﬂow will provide the ion ﬂow velocity, while the interparticle distance along the ion ﬂow will provide the Debye screening length.

13.5 Simulation Study of Cluster Design of Charged Dust Particles For many-particle system, a numerical simulation study is also a powerful tool to investigate the microparticle designs. A few tens of charged particles injected in a plasma form a 3D Coulomb crystal consisting of a few vertical chains, as shown by means of a particle simulation [31]. In typical laboratory experiments, dust particles form a thin multilayer structure at the edge of a sheath by the balance of the sheath electric force and the gravitational force. To investigate a 3D structure in a plasma, microgravity experiments have been carried out [32]. Three-dimensional multishell structures of dust particles conﬁned in a spherical symmetric potential are extensively investigated partly because of its analogy to other physically interesting situations, e.g., structure of cold ionic systems [33], classical artiﬁcial atoms [34], and onecomponent plasma [35]. In these studies, however, the repulsive force between the charged particles is assumed to be the long-range Coulomb repulsive force without screening eﬀect. The observed Coulomb dust crystal indicates that interparticle distance is comparable to the Debye length, indicating the screening eﬀect to be an essential factor to determine the cluster structuring. In this section, we review the structure change of 3D dust cluster conﬁned in a gravitation-free plasma by changing the screening length [36]. The normalized equation of motion for the ith dust particle with the mass Md and the charge Q may be written as ⎡ ⎤ 2 dX i d Xi (13.13) = −∇ ⎣Ψ˜ (X i ) + Ψ˜sc (X i − X j )⎦ − dT 2 dT j=i

with the conﬁning potential Ψ˜ (X i ) = αRi2 and the screened Coulomb potential |X i − X j | γ exp − . Ψ˜sc (X i − X j ) = |X i − X j | ΛDe

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Here, the position and the Debye length are normalized to the system length l which gives the system volume by V = (2l)3 , and the time T is normalized to the damping frequency of the particle motion ω0 . Here, two parameters are introduced as: α=

a k ω0

and γ =

Q2 /4πε0 l2 , k ω0 l

(13.14)

where a is the conﬁnement strength and k is the friction coeﬃcient of the particles with neutral gas. In our numerical analysis, the conﬁnement strength is controlled by changing the value of α, while the proportional constant γ appeared in the screened Coulomb potential is kept constant to see the eﬀect of the screening length. Dust particles, with initial velocities randomly chosen, are introduced into the system at arbitrary positions to make particles form spherically symmetric structure. The equilibrium structure attained after suﬃciently long time (T ≥ 2×104) is examined in detail for γ = 4.7×10−5 which corresponds to a typical laboratory condition for dusty plasma experiments, or Md = 6.3 × 10−13 kg, Q = −1.6 × 10−15 C, l = 6.0 × 10−3 m, k = 3.8 × 10−11 kg s−1 . To investigate the structure of 3D cluster, we increased the number of injected dust particles N from 1 to 20. Up to eight dust particles, they form polyhedrons. All the particles have the equal distance from the origin for a polyhedron. And we regard the cluster structure as a shell structure. We note the radius of the shell structure increases with the number of the particles. A shell is ﬁlled for N = 8, and a new shell appears at the center when N ≥ 9. The particle number at which a new shell appears changes with α and the Debye length ΛDe . In the weak conﬁnement α = 2.23 × 10−3 , the ﬁnite screening length does aﬀect the conditions of the shell ﬁlling, while the screening eﬀect has no eﬀect on the shell ﬁlling in the strong conﬁnement α = 9.81×10−2. The radius of the shell structure increases with the increase in ΛDe but saturates at around ΛDe ∼ 1. Now, we discuss whether the shell structures obtained above particle simulations are really in steady states, i.e., in minimum energy states for the purpose of material design. A normalized Hamiltonian of our system including N dust particles in a steady state may be written as H(X 1 , X 2 , · · · , X N , N ) =

N i=1

|X i |2 +

i

|X i − X j |−1 . (13.15)

j>i

Here, we set α = γ = 1 for simplicity, i.e., particles are in strong conﬁnement potential. In this case, the screening has no eﬀect on the structure as described above. Therefore, we set ΛDe → ∞ to ignore the Debye shielding eﬀect. Stable conﬁgurations of dust particles are numerically found by the method of Newton optimization using a condition for minimal total energy, i.e., ∇H = 0. Stable and metastable energy states appear in the conﬁgurations. Here the metastable state is deﬁned as the local minimum state, while

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the stable state as the state which has the lowest total energy in metastable states. The obtained stable and metastable conﬁguration is summarized in Table 1 in [37], which clearly shows that a dust cluster with N ≤ 8 forms a unique stable state with a shell structure near the center of the system. For more than eight dust particles, a dust cluster has both stable and metastable states. In the metastable state, dust particles form a shell structure with dust particles near the center. We notice that when the number of dust particles is 13, the energy diﬀerence ∆E (= EN −1,1 − EN,0 ) changes the sign where EN −1,1 and EN,0 are the energy of conﬁgurations with and without a dust particle at the center, respectively.

13.6 Complex Plasma Experiment in Cryogenic Environment [38] In a complex plasma, the coupling constant Γ is often used to know about whether the cluster is in strongly coupled state or not. The Coulomb coupling constant Γ is deﬁned as a ratio of Coulomb energy to thermal energy of dust particles, i.e., Q2 exp(−¯ x/λD ) Γ = , (13.16) 4πε0 x¯kB Td where x ¯ is the mean interparticle distance. Dust particles are strongly coupled when Γ 1. Equation (13.16) indicates that Γ is controlled by the dust temperature. Most of the dust plasma experiments in the laboratory have been carried out in the room temperature (300 K), where plasmas are characterized by electrons and ions with temperature of about 0.1 ∼ a few eV. When neutral gas temperature is quite low, the dust temperature may also be low. Recently, dust structures, super-dense dust structures and a boundary-free worm-like structure were observed in the striations of dc discharges cooled in cryogenic temperature [39, 40]. In this section, we introduce an experiment to generate dust plasmas in cryogenic environment using liquid nitrogen (77 K) and liquid helium (1 ∼ 4 K) [40]. We show two kinds of dust plasma experiments (1) an experiment performed in a plasma produced in a narrow glass tube ﬁlled with cryogenic (∼77 K) He gas, and (2) an experiment performed in a plasma generated in He vapor above liquid helium surface. The experimental apparatuses are shown in Fig. 13.8a, b for the experiments (1) and (2), in which liquid nitrogen and liquid helium are stored in a glass Dewar bottle of 10 cm in inner diameter and 100 cm in length. In the experiment (1), as shown in Fig. 13.8a, an additional narrow glass tube of 1.6 cm in inner diameter with 70 cm in length is set inside the Dewar bottle. The helium gas is ﬁlled in the glass tube and is evacuated from the upper part. The pressure in the narrow glass tube can be controlled externally and monitored at the upper part of the tube. The plasma is produced by rf discharge between a ring electrode and a plate electrode which are mounted

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Fig. 13.8. Apparatus in a cryogenic condition [38]

Fig. 13.9. Self-organized clusters of dust particles. (a) Top view at 300 K. (b) Side view at 77 K

outside of the glass tube surrounded by liquid nitrogen. Acrylic particles of 3 µm in diameter are dropped into the plasma. Dust particles are levitated and observed at a position of about 5 mm from the bottom of the glass tube and are illuminated by the He–Ne laser light. The images are ﬁlmed by two CCD cameras from the top of the glass tube and from the side. Figure 13.9a, b shows the observed dust particle clusters, where the former one is the top view of the structure at the pressure of 66.5 Pa at room temperature and the latter is side view at liquid nitrogen temperature. Although the neutral gas density is equal in (a) and (b), dust particles at liquid nitrogen temperature were observed to move more rapidly than at room temperature. Figure 13.10 shows side-views of dust particles at liquid nitrogen temperature at diﬀerent pressures. The spatial spread of dust particles at liquid nitrogen temperature is observed to become smaller in size as pressure lowers. In the experiment (2), the temperature of liquid helium in a Dewar bottle is cooled down to about 1 K by evaporative cooling. The pressure at the upper part of the Dewar bottle is around 300 Pa of neutrals at 1 K. In Fig. 13.8(b), the electrodes of 5 mm apart are placed about 20 cm above the surface of

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Fig. 13.10. Variation of cluster size for helium gas pressure (a) 66.5 Pa, (b) 39.9 Pa, (c) 26.6 Pa, and (d) 13.3 Pa at surrounding temperature 77 K [38]

liquid helium, while a mesh and a plate of 5 cm in diameter are placed below the electrodes. Plasma was generated by applying 10 kHz ac voltages to the electrode. The electrodes are protected by a cylindrical insulator to suppress the rapid increase in the temperature of both liquid helium and the vapor gas, while the electrodes, the mesh and the plate, are in an acrylic tube of 5 cm in diameter to conﬁne dust particles. The acrylic particles with the diameter of 1.5 µm are dropped from the upper part of the apparatus and reach the region above the liquid helium after going through the discharge region. A very dense dust cloud was observed at about 1 cm above the plate placed just above liquid helium.

13.7 Summary Fundamental physics of a complex plasma is studied theoretically, numerically, and experimentally. Micron-sized dust particles present in a plasma have a large quantity of negative charges depending on a plasma condition. The charge of particles is a key parameter to control the characteristics of a complex plasma. We estimated the dust charge immersed in a laboratory plasma by irradiating the electron beam. The trajectory analysis of ions in

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the ﬂow reveals the presence of turning-back ions onto the dust surface in downstream region. The ion ﬂow produces a wake potential behind a leading dust particle resulting in other particles in the potential minima of the wake, while the attractive force perpendicular to the ﬂow results from a release of thermodynamic free energy. Numerical simulation showed the stable and metastable structures of Coulomb clusters. The experiments in cryogenic environment are introduced to study strongly coupled Coulomb clusters of charged particles. Acknowledgments We acknowledge that the charge measurement experiment was carried out in collaboration with Y. Nakamura. We also acknowledge our students, T. Yamanouchi, C. Kojima, T. Maezawa and M. Kugue in carrying out particle simulation and cryogenic experiments. This work is in part supported by AOARD (Asian Oﬃce of Aerospace Research and Development) under award number FA4869-07-1-4047 and in part by JSPS (Japan Society for the Promotion of Science) Grant-in-Aid for Scientiﬁc Research (B) under Grant No. 19340173.

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17. M. Shindo, O. Ishihara, in Abstracts of International Conference on Plasma Science, Jeju, Korea, 2–5 June 2003 18. J.E. Allen, R.L.F. Boyd, P. Reynolds, Proc. Phys. Soc. B 70, 297 (1957) 19. F.F. Chen, in Plasma Diagnostic Techniques, ed. by R.H. Huddlestone, S.L. Leonard (Academic, New York, 1965) 20. S.A. Maiorov, S.V. Vladimirov, N.F. Cramer, Phys. Rev. E 63, 017401 (2000) 21. F. Melandso, J. Goree, Phys. Rev. E 52, 5312 (1995) 22. M. Nambu, S.V. Vladimirov, P.K. Shukla, Phys. Lett. A 203, 40 (1995) 23. S.V. Vladimirov, O. Ishihara, Phys. Plasmas 3, 444 (1996) 24. O. Ishihara, S.V. Vladimirov, Phys. Plasmas 4, 69 (1997) 25. H. Lamb, Hydrodynamics (The University Press, Cambridge, 1924) 26. F.J.W. Whipple, J.A. Chalmers, Q. J. R. Meteorol. Soc. 70, 103 (1944) 27. D. Winske, W. Daughton, D.S. Lemons, M.S. Murillo, Phys. Plasmas 7, 2320 (2000), M. Lampe, G. Joyce, G. Ganguli, V. Gavrishchaka, Phys. Plasmas 7, 3851 (2000) 28. K. Takahashi, T. Oishi, K. Shimonai, Y. Hayashi, S. Nishino, Phys. Rev. E 58, 7805 (1998) 29. O. Ishihara, JASMA 22, 3 (2005) 30. O. Ishihara, N. Sato, Phys. Plasmas 12, 070705 (2005) 31. T. Kamimura, Y. Suga, O. Ishihara, Phys. Plasmas (in press) 32. G.E. Morﬁll, H.M. Thomas, U. Konopka, H. Rothermel, M. Zuzic, A. Ivlev, J. Goree, Phys. Rev. Lett. 83, 1598 (1999) 33. R. Rafac, J.P. Schiﬀer, J.S. Hangst, D.H.E. Dubin, D.J. Wales, Proc. Natl. Acad. Sci. USA 88, 483 (1991) 34. Y.G. Cornelissens, B. Partoens, F.M. Peeters, Physica E 8, 314 (2000) 35. H. Totsuji, T. Kishimoto, C. Totsuji, K. Tsuruta, Phys. Rev. Lett. 88, 125002 (2002) 36. T. Yamanouchi, M. Shindo, O. Ishihara, T. Kamimura, Thin Solid Films 506– 507, 642 (2006) 37. T. Yamanouchi, Doctoral Dissertation, Yokohama National University, 2006 38. C. Kojima, M. Kugue, T. Maezawa, M. Shindo, Y. Nakamura, O. Ishihara, in Proceedings of 13th International Congress on Plasma Physics, (May 22–26, 2006, Kiev, Ukraine), E134P 39. V.E. Fortov, L.M. Vasilyak, S.P. Vetchinin, V.S. Zimnukhov, A.P. Nefedov, D.N. Polyakov, Dokl. Phys. 47, 21 (2002) 40. S.N. Antipov, E.I. Asinovskii, V.E. Fortov, A.V. Kirillin, V.V. Markovets, O.F. Petrov, AIP Conf. Proc. 799, 125 (2005)

Index

1,3,5-trithia-2,4,6-triazapentalenyl (TTTA), 8, 143 1D antiferromagnetic chain, 146, 167 1×1 structure, 56, 57, 67 7×7 structure, 52, 56 absorption coeﬃcient change, 264, 268 activation energy, 56 active fabrication, 53, 68 adatom, 22, 36, 53, 55, 82–84, 245, 253 adiabatic potential, 74 adsorbate density, 58 adsorption, 26, 34, 78, 85, 86, 88, 89 adsorption site, 55 adsorption site preference, 39, 48 Aharonov–Bohm (AB) eﬀect, 10 AlGaAs, 252 algebraic diagrammatic construction, 197 aligned cluster, 63 alkali-earth atom, 191, 198, 212 alkali-metal cluster, 176, 177, 179, 180, 193 alkali-metal dimer, 191, 193, 196, 198, 212 all-electron mixed basis approach, 157, 162, 163, 175, 176, 187, 191, 193 all-optically controlled biochip, 307 AlN, 258 anatase, 5, 121, 131 angle-resolved ultraviolet photoelectron spectroscopy (ARUPS), 78 angular momentum, 228 anneal under hydrogen stream, 125

anneal under oxygen pressure, 124 annealing rate, 68 antibonding state, 74 antiferromagnetic insulator, 161 antiferromagnetic spin order, 167 antimony, 246 argon, 191, 198–200, 203, 211, 212 arsenic, 245 asymmetric coupled quantum well, 268 atom migration, 253 atomic force microscopy (AFM), 53, 254 atomic monolayer (ML), 53, 62 atomic orbital (AO), 175, 191 atomic scale, 55, 73 Auger electron spectroscopy (AES), 78 Auger process, 73 Auger spectra, 213 back-bond, 32, 35, 47, 53, 60 back-scattering, 228 background-free, 98, 106 barrier, 38 benzene, 183 benzene-dithiol molecule, 238 beryllium, 191, 198, 199, 212 β-carotene, 103, 112, 115 Bethe–Salpeter amplitude, 204 Bethe–Salpeter equation (BSE), 4, 8, 163, 189, 192, 196, 197, 202, 203, 213, 215 birefringence, 101 bistability, 148, 151 blocking electrode, 136

330

Index

Boltzmann’s constant, 59 bonding conﬁguration, 56 Br/Si(001), 60 Bragg–Williams approximation, 282 bunched step, 63 C60 , 11 C2 H2 , 191, 198, 200–203, 209–212 calcium, 191, 198, 199, 212 capacitance, 8 capping layer, 253 carbon nanotube, 105 carbon nanotube heterostructures, 227 carrier conﬁnement, 245 carrier control, 137 carrier density, 135 carrier mobility, 136 CdSe cluster, 4 characteristic temperature, 250 charge conservation law, 185 charge transfer, 53 charging, 312 chemical vapor transport method, 122 chlorophyll, 5 Cl/Si(111), 60, 66 closely stacked QD, 246 cluster, 3–5, 8, 9, 46, 171, 172, 176–180, 193, 212 cluster alignment, 62, 64, 65 cluster variation method, 282, 287 CO, 191, 198, 200, 203, 206, 207, 209, 211, 212 CO2 , 191, 203, 208, 209, 211, 212 coadsorption, 78 colored anatase single crystal, 123 colorless anatase single crystal, 124 columnar-shaped QD, 247 complex plasma, 11, 311 concentric circle, 62 conductance, 9 conduction band, 72 conﬁguration interaction (CI) method, 197 continuous-wave lasing, 248 correlation, 55 correlation length, 258 Coulomb blockade, 9 Coulomb cluster, 311, 312, 315, 326 Coulomb crystal, 311

Coulomb hole, 8, 162, 164, 165, 191, 193, 203–208, 210–212 coupled quantum well, 269 coupling constant, 323 coverage, 32, 36, 60 cryogenic temperature, 323 CT, 155, 156 Cu(001) surface, 77, 78, 82, 85, 86, 88 Cu(001)-c(2×2)N, 77, 79, 82–85, 88 Cu(111) surface, 88 Cu-Au alloy, 283 Cu3 Au, 286 current–voltage (I–V ), 231 d-band, 88 dangling bond, 32, 35, 44, 55 Davydov component, 151 defective nanotubes, 234 delta self-consistent ﬁeld method, 197 delta-function-like state density, 243 demultiplexer, 243 dendrimer, 5 densities, 55 density functional theory (DFT), 161, 171, 189 density of states (DOS), 9, 231 deposition, 52 desorption, 39, 53 desorption energy, 88 desorption induced by electronic transition (DIET), 68 desorption induced by multiple electronic transition (DIMET), 72, 75 desorption yield, 56 Dewar bottle, 323 diamagnetic susceptibility, 167 diamagnetism, 143, 167 diatomic step, 62, 67 diﬀerential conductance, 236 diﬀraction limit, 51 diﬀusion, 62, 84, 86 diﬀusion barrier, 84, 88 diﬀusion rate, 78 dimer, 60 dimer-adatom-stacking fault (DAS), 22, 52, 55, 62 dimerization, 148, 154 dislocation, 65

Index dissociation, 79, 85, 86 dissociation barrier, 86–88 dissociation of H2 O, 7 dissociation probability, 78 domain boundary, 52, 65 dopant atoms, 231 doped nanotube junction, 227, 231 dot in well, 250 double crystal substrate, 231 double electron aﬃnity (DEA), 189, 191, 193–196, 212, 213 double ionization energy (DIE), 189, 191, 193, 196–203, 205, 208, 209, 211–213 drag delivery system (DDS), 15 dressed Green’s function, 173 droplet-epitaxy, 259 dust plasma, 11, 311, 323 dust–ion ﬂow interaction, 315 dust-plasma interaction, 312 dynamically screened Coulomb interaction, 172, 189, 191, 193 Dyson equation, 172, 173, 184, 191 eﬀective mass, 266 EHP, 108 elastic energy, 275 electric conductivity (of anatase), 134 electroabsorption modulator (EAM), 264 electron aﬃnity (EA), 3, 53, 177, 189, 195 electron and hole transfer, 5 electron beam (e-beam), 73, 313 electron gas, 185 electron irradiation, 234 electron paramagnetic resonance (EPR) (of anatase), 132, 138 electron spin resonance (ESR), 146 electron tunneling, 219 electron–electron Green’s function, 193, 196, 197 electron–electron ladder diagrams, 190 electron–hole Green’s function, 189, 197 electron–hole ladder diagrams, 189 electron-impact, 73 electron-stimulated desorption (ESD), 38, 73 electron–hole plasma (EHP), 108

331

electronic excitation, 53, 68 electronic structure, 219 electronic transport, 219 electron spin resonance (ESR), 146, 149, 150 electrorefractive index change, 268 electrorefractive (ER) eﬀect, 268 emission eﬃciency, 250 endohedral fullerene, 12 energy broadening, 236 energy transfer, 97, 112 energy dispersive X-ray spectroscopy (EDX), 256 entangled photon pair, 244 epitaxial growth, 84 equilibrium, 72 equilibrium Fermi level, 236 ER sensitivity, 271 Esaki diodes, 233 ESR, 146, 149, 150, 155 etching, 21, 22, 50, 52, 75 etching energy, 45 exchange-correlation potential, 175, 184 excited states, 171, 189, 213 exciton binding energy, 189, 264 exciton eﬀect, 264 exciton–phonon interaction, 130 extinction ratio, 252 Fabry–Perot resonance method, 271 femtosecond, 72 femtosecond laser spectroscopy, 97 FePt alloy, 287 FePt cluster, 5 ﬁnite element method, 250 Finnis–Sinclair–type potential, 275, 283 ﬁrst Brillouin zone, 182 ﬁrst-order process, 60 ﬁrst-principles calculation, 4, 5, 77, 171, 189, 191 ﬁrst-principles molecular dynamics simulation, 12 ﬁtting parameters, 236 ﬂuctuation, 55 ﬂuence, 72 four-terminal conductance, 220 fullerene, 11 fullerene polymer, 15

332

Index

gallium arsenide (GaAs) cluster, 180 gallium arsenide (GaAs) crystal, 182 GaN, 258 GaNAs, 252 generalized gradient approximation (GGA), 171, 189 generalized plasmon-pole (GPP) model, 175, 177, 178, 191 germanium cluster, 178–180 Gibbs free energy, 320 graded-gap quantum well (GGQW), 266, 267 grazing incidence X-ray diﬀraction (GIXD), 77 grazing incidence X-ray scattering, 256 Green’s function, 189, 219 grid pattern, 52, 78, 82 group velocity dispersion (GVD), 99 growth interruption, 245 growth rate, 245, 250 growth temperature, 245 g-value, 133 GVD, 102, 103, 112, 113 GW approximation (GWA), 3, 171, 172, 174, 176, 178–180, 182, 189, 191–202, 209, 212 Hall coeﬃcient, 135 Hall mobility, 135 halogen, 21 Hamiltonian of the (bicrystal) system, 221 Hartree–Fock approximation, 173, 183 heating rate, 60 heavy-hole (hh) exciton, 107 Helmholtz free energy, 320 Herman–Skillman code, 175 heterointerface, 68 heterostructure ﬁeld eﬀect transistor (HFET), 10 high electron mobility transistor (HEMT), 10 highest occupied molecular orbital (HOMO), 3, 177, 237 hole injection, 75 hole–hole Green’s function, 193, 196, 197 hole–hole ladder diagrams, 190 hydrogen implantation, 127

impurity-induced layer disordering (IILD), 271 InAlAs, 252 inelastic scattering, 75 inert gas atom, 191, 198, 199, 212 InGaAs, 244 InGaN, 244 inter-nanotube interactions, 235 interband, 251 interface layer, 68 interface region, 222 intermixed superlattice, 272 intermixing, 248 intermixing quantum well, 271 internal conversion, 103 interparticle distance, 318 inverse photoemission, 3 ionization potential (IP), 3, 177, 184, 189 isothermal desorption, 58 jellium model, 176 Josephson junction, 10 Kerr eﬃciency, 100–102 Kerr medium, 98–102 Knotek–Feibelman (KF) model, 73 Kohn–Sham eigenvalue, 171, 177 Koopmans theorem, 171, 183 Kramers–Kronig integral relation, 268 krypton, 191, 198–200, 212 Kubo eﬀect, 1 L10 , 276, 285, 287, 288 L12 , 285, 288 ladder diagrams, 197–203, 207, 212 ladder-climbing, 75 Landauer’s formulation, 219, 220 Langmuir probe, 314, 315 laser, 76 lattice distortion, 251 lifetime, 67, 72, 75 light-harvesting property, 5 linear, 32 liquid helium, 323 lithium cluster, 176, 178 lithium dimer, 191, 193, 196, 198, 212 lithography, 1, 51 local charge conservation, 185

Index local density approximation (LDA), 161, 171, 177, 189, 191, 195, 198, 200, 201 local density of states (LDOS), 228 local spin density approximation (LSDA), 161 local vibrational mode, 75 long-range order parameter, 285 low-velocity carriers, 234 lowest unoccupied molecular orbital (LUMO), 3, 177, 238 luminescence up-conversion technique, 98 Mach-Zehnder interferometer, 270 magic angle, 113 magic number cluster, 176 magnesium, 191, 198, 199, 212 magnetic linear/circular dichroism (MLD/MCD), 78 magnetic phase transition, 144, 147, 148 many-body perturbation theory (MBPT), 171, 172, 187, 189, 193, 212 mass transportation, 253 matching domain, 222 melting transition (of silicon), 280 Menzel–Gomer–Readhead (MGR) model, 74, 75 metal cluster, 4, 176, 178, 181 metalorganic chemical vapor deposition (MOCVD), 243 metastable state, 322, 323 microelectromechanical systems (MEMS), 291 microﬂuidic circuit, 303 microgear, 293, 295 micromanipulator, 300, 302 micropump, 304, 306 microstereolithography, 1 microturbine, 294 migration, 254 migration length, 245 modulator, 263 molecular dynamics (MD) simulation, 275 molecular beam epitaxy (MBE), 243, 254, 263

333

monochloride, 22, 28, 45, 55, 57, 60, 70–72 monolayer (ML), 31 Monte Carlo (MC) simulation, 275 Moore’s law, 1 Mott insulator, 8, 143, 160, 161, 165–167 multilayer structure, 321 multiple excitation, 72, 75 multiple scattering, 190, 191, 196, 197, 203, 205, 207–209, 212 multiple step, 62 multiple vibronic excitation, 75 multireference conﬁguration interaction (MRCI) method, 197 nanocluster, 77 nanodot, 78 nanoisland, 88 nanolithography, 1 nanopattern, 51, 52, 69, 77, 79, 88 nanoscale, 219 nanosecond, 72 nanostructure, 77, 78 nanotechnology, 89 nanotube, 11 nanowire, 77 Nd:YAG laser, 70 N´eel temperature, 167 negative diﬀerential resistance (NDR), 227, 231 neon, 191, 198–200, 212 niobium-doped anatase, 123, 127, 135 noble metal, 286 nonadiabatic, 72 nonequilibrium, 68 nonlinear, 72 nonlinear crystal, 98, 99 nonlinear susceptibility, 31 nonorthogonal basis, 226 nonradiative recombination, 250 nucleation, 56, 79, 82 OKG, 98, 109, 111 on-site Coulomb energy U , 8, 161–164, 166, 167 one-particle Green’s function, 172, 189, 192, 197, 213

334

Index

one-particle wave function, 164, 203, 205 optical absorption, 71, 121 optical absorption spectra, 3, 5, 189 optical gain, 250 optical lithography, 50 optical trapping, 296 optical tweezers, 291 optical Kerr gate (OKG) , 98 optimized eﬀective potential (OEP) method, 184 orbital-limited motion theory, 316 order-disorder phase transition, 275, 276, 282, 283, 285, 287, 289 organic radical, 143, 150 orthogonal basis, 226 overgrowth, 248 oxygen defect, 121 parabolic potential quantum well, 264 paramagnetic susceptibility, 146, 148, 167 partition function, 276 passive fabrication, 62 passive process, 53 pattern, 20, 39, 53 pattern formation, 20 patterning, 38, 68 Peierls instability, 158 Peierls insulator, 143 Peierls transition, 143, 158 pentagon–heptagon defects, 227 persistent photoconductance, 137 phase mask, 107 phase transition temperature, 275, 280, 283, 286–288 phase-matching, 98 phonon bottleneck eﬀect, 252 photoabsorption-induced disordering (PAID), 271 photocatalysis, 5 photoconductivity, 137 photocurable resin, 292 photodegradation, 115, 117 photodetector, 243 photoemission, 3 photoexcitation, 69 photoinduced phase transition, 143, 150, 151

photoluminescence, 4, 131, 245, 252 photon beam, 73 photonic crystal, 292 photosynthesis, 5, 97, 112 phtodegradation, 115 π-conjugated dendrimer, 5 pinning, 62, 64 π–π transition, 154–156 pit, 62 planar circuit, 259 plane wave (PW), 175, 191 plasma-sheath boundary, 311, 315 Polanyi–Wigner’s rate equation, 58 polarizability function, 172, 174 polarization, 248 polarization-independent EAM, 266 polarization-independent quantum well, 266 polarization-insensitive EAM, 267 polybromide, 55 potassium cluster, 176, 178 potassium dimer, 191, 193, 196, 198, 212 potential barrier, 32, 59 potential energy, 26 potential renormalization technique, 275, 276, 282, 286 potential-modiﬁed quantum well, 263 potential-tailored quantum well, 263, 264 pre-biased quantum well, 267 preexponential constant, 59 principal layer, 221 projection on the interface regions, 221 p-type doping, 252 pulse laser, 70, 72 QCSE, 266, 267, 270 quantum communication system, 244 quantum computer, 10 quantum computing, 10, 243, 259 quantum conﬁnement, 251 quantum dot, 4, 8 quantum eﬃciency, 243 quantum well (QW), 9, 263 quantum-entangled photon pairs, 251 quasiparticle (QP) energy, 171, 174–180, 182 qubit, 10, 244

Index radiation pressure, 291, 296 random phase approximation (RPA), 173, 174, 189, 191, 193, 203 rate equation, 58 reaction order, 56 real-time pump-probe imaging spectroscopy, 109, 110, 112, 114, 117 reconstruction strain, 55 rectifying, 227 rectifying eﬀect, 232 reﬂected amplitude, 225 refractive index change, 264 refractive high energy electron diﬀraction (RHEED), 247 regenerator, 243 regular alignment, 52 relaxation, 72 renormalized potential, 276, 278–281, 283, 285–287 resistivity, 135 resonant tunneling structures, 233 rest-atom, 22, 36, 53 rest-surface, 22, 71 rhombus-shaped nanostructure, 254 rotational symmetry, 229, 233 Rutherford backscattering spectroscopy (RBS), 77 scanning probe microscopy (SPM), 53 scanning tunneling microscope (STM) tip, 236 scanning tunneling microscopy (STM), 20, 53, 62, 77–79, 85, 258 scanning tunneling spectroscopy (STS), 53 scattering, 219 scattering matrix, 223 SDR Spectra, 26, 55 second harmonic (SH) intensity, 32 second harmonic generation (SHG), 24, 25, 31, 32, 42, 272 second moment approximation (SMA), 286 second-order perturbation, 197 second-order process, 60 self-assembled monolayer (SAM), 236 self-consistent potential drop, 231 self-energy, 172, 189, 191, 197, 226

335

self-energy correction, 176 self-organization, 1, 19, 51, 68, 77 self-organized process, 51 self-trapped exciton, 130 self-alignment, 256 self-assembled quantum dot (QD), 243 self-electro-optic eﬀect device (SEED), 264 semiconductor cluster, 4, 178, 181 semiconductor laser, 243 semiconductor optical ampliﬁer (SOA), 243 semiﬂexible polymer solution, 288 sensor applications, 234 sextuplet, 138 sheath, 311, 315, 318, 321 sheet density, 250 SHG, 24, 25, 31, 42, 272 short-rage correlations, 8 Si(001) surface, 22, 34, 60 Si(111) surface, 53 Si(111)–(7×7) surface, 55, 62 Si/SiO2 interface, 1 silicon, 278 silicon cluster, 178–180 silicon dichloride (SiCl2 ), 55 silicon monochloride (SiCl), 55, 57 silicon polychlorides (SiCln ), 57 silicon trichloride (SiCl3 ), 55 single crystal, 122 single-shot, 110, 117 skin eﬀect, 132 sodium cluster, 176–178 sodium dimer, 191, 193, 196, 198, 212 spacer layer, 246 spatial diﬀusion, 105 spectral shape, 62 spin concentration grating, 107 spin degeneracy, 244 spin–orbit coupling constant, 133 spot proﬁle analyzing low-energy electron diﬀraction (SPA-LEED), 77 SPring-8, 256 stacking, 246 stacking fault, 67 star-shaped stilbenoid phtalocyanine (SSS1Pc), 5 Stark shift, 264, 265, 267

336

Index

steepness parameter, 130 step edge, 52 step retreat, 53, 63, 64 sticking probability, 84 Stillinger-Weber potential, 278 Stokes shift, 4 strain, 46, 51, 62, 78, 79, 82–85, 88, 89 strain reducing layer, 245 Stranski–Krastanov, 247 stretched-exponential function, 137 stripe pattern, 79, 82 structural deterioration, 110 substrate orientation, 245 superconducting circuit, 10 supersonic ion ﬂow, 315 surface density, 60 surface diﬀerential reﬂectivity (SDR), 24, 26, 40, 55 surface free energy, 246 surface Green’s function, 226 surface Green’s function matching (SGFM) method, 221 surface magneto-optical Kerr eﬀect (SMOKE), 78 surface reconstruction, 19, 51, 57 surface segregation, 247 surfactant, 246 switch, 263 symmetry of wavefunction, 251 synchrotron, 76 synchrotron radiation, 256 synthetic metal, 143 T-matrix calculation, 8 tensile-strained, 267 terrace, 53, 63, 65 Tersoﬀ potential, 278 tertiarybutylarsine, 245 thermal desorption, 62 thermal desorption spectroscopy (TDS), 26, 56 thermal expansion, 280 thermodynamic excess energy, 320 third-order perturbation, 197 third-order correlation function, 100, 102, 104, 111 threshold current, 250 tight-binding (TB) potential, 286

time-dependent density functional theory, 4 time-of-ﬂight (TOF) method, 3, 71, 136 TiO2 , 5, 121 Ti:sapphire, 99, 103, 110 titanium deposition on top of silicon surface, 1 titanium dioxide, 5, 121 T -matrix theory and calculation, 162–164, 189–193, 195–200, 202, 203, 205, 209, 212, 213, 215 total Green’s function, 222, 226 transfer matrix, 223 transient grating, 105 transition metal, 286 transmission coeﬃcient, 225 transmission matrix, 224, 225 transmitted amplitude, 225 trichloride, 22, 23, 39, 55 TTTA, 8, 143 tunable, 69 tunable light, 76 two-electron probability distribution, 164 two-particle excited states, 189, 190, 202 two-particle Green’s function, 4, 162, 163, 189, 191, 192, 196–198, 203, 209, 212 two-particle wave function, 162–164, 191, 193, 203, 204, 212 two-step renormalization scheme, 275, 289 two-terminal conductance, 220 two-photon microstereolithography, 292 ubiquitous, 260 ultrafast dynamics, 76 ultraﬁne machining, 20 undressed Green’s function, 173 unit cell, 54, 62, 173 urbach’s tail, 129 vacancy, 228 van Hove singularity, 234 Vegard’s law, 256 vertex correction, 187 vertex function, 172, 185 vibration entropy, 275

Index vibronic excitation, 72 vibronic quantum, 75 vicinal surface, 79, 82, 88

wavelength converter, 243 work function, 231

wake potential, 318, 319, 326 Ward–Takahashi (WT) identity, 175, 185, 187 wavefunction control, 10

X-ray rocking curve, 256 xylyl-dithiol molecule, 235 Z scheme, 7

337