Handbook of Nanophysics Handbook of Nanophysics: Principles and Methods Handbook of Nanophysics: Clusters and Fullerenes Handbook of Nanophysics: Nanoparticles and Quantum Dots Handbook of Nanophysics: Nanotubes and Nanowires Handbook of Nanophysics: Functional Nanomaterials Handbook of Nanophysics: Nanoelectronics and Nanophotonics Handbook of Nanophysics: Nanomedicine and Nanorobotics

Nanoelectronics and Nanophotonics

Edited by

Klaus D. Sattler

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-7550-2 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress CataloginginPublication Data Handbook of nanophysics. Nanoelectronics and nanophotonics / editor, Klaus D. Sattler. p. cm. A CRC title. Includes bibliographical references and index. ISBN 978-1-4200-7550-2 (alk. paper) 1. Nanoelectronics--Handbooks, manuals, etc. 2. Nanophotonics--Handbooks, manuals, etc. I. Sattler, Klaus D. II. Title. TK7874.84.H36 2010 621.381--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 2010001108

ContentsPreface........................................................................................................................................................... ix Acknowledgments ........................................................................................................................................ xi Editor .......................................................................................................................................................... xiii Contributors .................................................................................................................................................xv

Part I Computing and Nanoelectronic Devices

1 2 3 4 5 6 7 8 9 10 11

Quantum Computing inSpin Nanosystems ......................................................................................1-1Gabriel Gonzlez and Michael N. Leuenberger

Nanomemories Using Self-Organized Quantum Dots ..................................................................... 2-1Martin Geller, Andreas Marent, and Dieter Bimberg Vincent Meunier and Bobby G. Sumpter

Carbon Nanotube Memory Elements ................................................................................................ 3-1 Ferromagnetic Islands ........................................................................................................................ 4-1Arndt Remhof, Andreas Westphalen, and Hartmut Zabel Gilles Micolau and Damien Deleruyelle

A Single Nano-Dot Embedded in a Plate Capacitor ......................................................................... 5-1 Nanometer-Sized Ferroelectric Capacitors ....................................................................................... 6-1Nikolay A. Pertsev, Adrian Petraru, and Hermann Kohlstedt Vincent Bouchiat

Superconducting Weak Links Made of Carbon Nanostructures .......................................................7-1 Micromagnetic Modeling of Nanoscale Spin Valves...............................................................................................8-1Bruno Azzerboni, Giancarlo Consolo, and Giovanni Finocchio Gabriel Gonzlez and Michael N. Leuenberger Natalya A. Zimbovskaya

Quantum Spin Tunneling in Molecular Nanomagnets..................................................................... 9-1 Inelastic Electron Transport through Molecular Junctions ........................................................... 10-1 Bridging Biomolecules with Nanoelectronics .................................................................................. 11-1Kien Wen Sun and Chia-Ching Chang

v

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Part II Nanoscale transistors

12 13 14 15 16

Transistor Structures forNanoelectronics ...................................................................................... 12-1Jean-Pierre Colinge and Jim Greer Andr Avelino Pasa

Metal Nanolayer-Base Transistor .................................................................................................... 13-1 ZnO Nanowire Field- ffect Transistors ................................................................................................14-1 EWoong-Ki Hong, Gunho Jo, Sunghoon Song, Jongsun Maeng, and Takhee Lee Akihiro Hashimoto Jos Aumentado

C 60 Field Effect Transistors .............................................................................................................. 15-1 The Cooper-Pair Transistor ............................................................................................................. 16-1

Part III

Nanolithography

17 18 19 20

Multispacer Patterning: ATechnology for the Nano Era................................................................. 17-1Gianfranco Cerofolini, Elisabetta Romano, and Paolo Amato Zhijun Hu and Alain M. Jonas

Patterning and Ordering with Nanoimprint Lithography .............................................................. 18-1 Nanoelectronics Lithography .......................................................................................................... 19-1Stephen Knight, Vivek M. Prabhu, John H. Burnett, James Alexander Liddle, Christopher L. Soles, andAlainC.Diebold Obert R. Wood II

Extreme Ultraviolet Lithography ..................................................................................................... 20-1

Part IV Optics of Nanomaterials

21 22 23 24 25 26 27 28

Cathodoluminescence of Nanomaterials ..........................................................................................21-1Naoki Yamamoto

Optical Spectroscopy of Nanomaterials .......................................................................................... 22-1Yoshihiko Kanemitsu

Nanoscale Excitons and Semiconductor Quantum Dots ................................................................ 23-1Vanessa M. Huxter, Jun He, and Gregory D. Scholes

Optical Properties of Metal Clusters and Nanoparticles ................................................................ 24-1Emmanuel Cottancin, Michel Broyer, Jean Lerm, and Michel Pellarin Amir Saar

Photoluminescence from Silicon Nanostructures........................................................................... 25-1 Polarization-Sensitive Nanowireand Nanorod Optics ................................................................... 26-1Harry E. Ruda and Alexander Shik

Nonlinear Optics with Clusters ........................................................................................................27-1Sabyasachi Sen and Swapan Chakrabarti

Second-Harmonic Generation in Metal Nanostructures ................................................................ 28-1Marco Finazzi, Giulio Cerullo, and Lamberto Du

Contents

vii

29 30 31

Nonlinear Optics in Semiconductor Nanostructures ..................................................................... 29-1Mikhail Erementchouk and Michael N. Leuenberger Vladimir G. Bordo Andrew R. Parker

Light Scattering from Nanofibers .................................................................................................... 30-1 Biomimetics: Photonic Nanostructures............................................................................................ 31-1

Part V Nanophotonic Devices

32 33 34 35 36 37 38

Photon Localization at the Nanoscale ............................................................................................. 32-1Kiyoshi Kobayashi

Operations in Nanophotonics .......................................................................................................... 33-1Suguru Sangu and Kiyoshi Kobayashi Makoto Naruse

System Architectures for Nanophotonics ........................................................................................ 34-1 Nanophotonics for Device Operation and Fabrication ................................................................... 35-1Tadashi Kawazoe and Motoichi Ohtsu Takashi Yatsui and Wataru Nomura

Nanophotonic Device Materials ...................................................................................................... 36-1 Waveguides for Nanophotonics ........................................................................................................37-1Jan Valenta, Tom Ostatnick, and Ivan Pelant Grigory E. Adamov and Evgeny P. Grebennikov

Biomolecular Neuronet Devices ...................................................................................................... 38-1

Part VI

Nanoscale Lasers

39 40 41

Nanolasers ........................................................................................................................................ 39-1Marek S. Wartak Frank Jahnke

Quantum Dot Laser ......................................................................................................................... 40-1 Mode-Locked Quantum-Dot Lasers .................................................................................................41-1Maria A. Cataluna and Edik U. Rafailov

Index .................................................................................................................................................... Index-1

PrefaceThe Handbook of Nanophysics is the first comprehensive reference to consider both fundamental and applied aspects of nanophysics. As a unique feature of this work, we requested contributions to be submitted in a tutorial style, which means that state-of-the-art scientific content is enriched with fundamental equations and illustrations in order to facilitate wider access to the material. In this way, the handbook should be of value to a broad readership, from scientifically interested general readers to students and professionals in materials science, solid-state physics, electrical engineering, mechanical engineering, computer science, chemistry, pharmaceutical science, biotechnology, molecular biology, biomedicine, metallurgy, and environmental engineering. interdisciplinary projects and incorporate the theory and methodology of other fields into their work. It is intended for readers from diverse backgrounds, from math and physics to chemistry, biology, and engineering. The introduction to each chapter should be comprehensible to general readers. However, further reading may require familiarity with basic classical, atomic, and quantum physics. For students, there is no getting around the mathematical background necessary to learn nanophysics. You should know calculus, how to solve ordinary and partial differential equations, and have some exposure to matrices/linear algebra, complex variables, and vectors.

What Is Nanophysics?Modern physical methods whose fundamentals are developed in physics laboratories have become critically important in nanoscience. Nanophysics brings together multiple disciplines, using theoretical and experimental methods to determine the physical properties of materials in the nanoscale size range (measured by millionths of a millimeter). Interesting properties include the structural, electronic, optical, and thermal behavior of nanomaterials; electrical and thermal conductivity; the forces between nanoscale objects; and the transition between classical and quantum behavior. Nanophysics has now become an independent branch of physics, simultaneously expanding into many new areas and playing a vital role in fields that were once the domain of engineering, chemical, or life sciences. This handbook was initiated based on the idea that breakthroughs in nanotechnology require a firm grounding in the principles of nanophysics. It is intended to fulfill a dual purpose. On the one hand, it is designed to give an introduction to established fundamentals in the field of nanophysics. On the other hand, it leads the reader to the most significant recent developments in research. It provides a broad and in-depth coverage of the physics of nanoscale materials and applications. In each chapter, the aim is to offer a didactic treatment of the physics underlying the applications alongside detailed experimental results, rather than focusing on particular applications themselves. The handbook also encourages communication across borders, aiming to connect scientists with disparate interests to begin

External reviewAll chapters were extensively peer reviewed by senior scientists working in nanophysics and related areas of nanoscience. Specialists reviewed the scientific content and nonspecialists ensured that the contributions were at an appropriate technical level. For example, a physicist may have been asked to review a chapter on a biological application and a biochemist to review one on nanoelectronics.

OrganizationThe Handbook of Nanophysics consists of seven books. Chapters in the first four books (Principles and Methods, Clusters and Fullerenes, Nanoparticles and Quantum Dots, and Nanotubes and Nanowires) describe theory and methods as well as the fundamental physics of nanoscale materials and structures. Although some topics may appear somewhat specialized, they have been included given their potential to lead to better technologies. The last three books (Functional Nanomaterials, Nanoelectronics and Nanophotonics, and Nanomedicine and Nanorobotics) deal with the technological applications of nanophysics. The chapters are written by authors from various fields of nanoscience in order to encourage new ideas for future fundamental research. After the first book, which covers the general principles of theory and measurements of nanoscale systems, the organization roughly follows the historical development of nanoscience. Cluster scientists pioneered the field in the 1980s, followed by extensiveix

x

Preface

work on fullerenes, nanoparticles, and quantum dots in the 1990s. Research on nanotubes and nanowires intensified in subsequent years. After much basic research, the interest in applications such as the functions of nanomaterials has grown. Many bottom-up MATLAB is a registered trademark of The MathWorks, Inc. Forproduct information, Please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA, 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: www.mathworks.com

and top-down techniques for nanomaterial and nanostructure generation were developed and made possible the development of nanoelectronics and nanophotonics. In recent years, real applications for nanomedicine and nanorobotics have been discovered.

AcknowledgmentsMany people have contributed to this book. I would like to thank the authors whose research results and ideas are presented here. I am indebted to them for many fruitful and stimulating discussions. I would also like to thank individuals and publishers who have allowed the reproduction of their figures. For their critical reading, suggestions, and constructive criticism, I thank the referees. Many people have shared their expertise and have commented on the manuscript at various stages. I consider myself very fortunate to have been supported by Luna Han, senior editor of the Taylor & Francis Group, in the setup and progress of this work. I am also grateful to Jessica Vakili, Jill Jurgensen, Joette Lynch, and Glenon Butler for their patience and skill with handling technical issues related to publication. Finally, I would like to thank the many unnamed editorial and production staff members of Taylor & Francis for their expert work. Klaus D. Sattler Honolulu, Hawaii

xi

EditorKlaus D. Sattler pursued his undergraduate and masters courses at the University of Karlsruhe in Germany. He received his PhD under the guidance of Professors G. Busch and H.C. Siegmann at the Swiss Federal Institute of Technology (ETH) in Zurich, where he was among the first to study spin-polarized photoelectron emission. In 1976, he began a group for atomic cluster research at the University of Konstanz in Germany, where he built the first source for atomic clusters and led his team to pioneering discoveries such as magic numbers and Coulomb explosion. He was at the University of California, Berkeley, for three years as a Heisenberg Fellow, where he initiated the first studies of atomic clusters on surfaces with a scanning tunneling microscope. Dr. Sattler accepted a position as professor of physics at the University of Hawaii, Honolulu, in 1988. There, he initiated a research group for nanophysics, which, using scanning probe microscopy, obtained the first atomic-scale images of carbon nanotubes directly confirming the graphene network. In 1994, his group produced the first carbon nanocones. He has also studied the formation of polycyclic aromatic hydrocarbons (PAHs) and nanoparticles in hydrocarbon flames in collaboration with ETH Zurich. Other research has involved the nanopatterning of nanoparticle films, charge density waves on rotated graphene sheets, band gap studies of quantum dots, and graphene foldings. His current work focuses on novel nanomaterials and solar photocatalysis with nanoparticles for the purification of water. Among his many accomplishments, Dr. Sattler was awarded the prestigious Walter Schottky Prize from the German Physical Society in 1983. At the University of Hawaii, he teaches courses in general physics, solid-state physics, and quantum mechanics. In his private time, he has worked as a musical director at an avant-garde theater in Zurich, composed music for theatrical plays, and conducted several critically acclaimed musicals. He has also studied the philosophy of Vedanta. He loves to play the piano (classical, rock, and jazz) and enjoys spending time at the ocean, and with his family.

xiii

ContributorsGrigory E. Adamov Open Joint-Stock Company Central Scientific Research Institute ofTechnology Technomash Moscow, Russia Paolo Amato Numonyx and Department of Materials Science University of Milano-Bicocca Milano, Italy Jos Aumentado National Institute of Standards andTechnology Boulder, Colorado Bruno Azzerboni Department of Matter Physics andElectronic Engineering Faculty of Engineering University of Messina Messina, Italy Dieter Bimberg Institut fr Festkrperphysik Technische Universitt Berlin Berlin, Germany Vladimir G. Bordo A.M. Prokhorov General Physics Institute Russian Academy of Sciences Moscow, Russia Vincent Bouchiat Nanosciences Department Centre National de la Recherche Scientifique Nel-Institut Grenoble, France Michel Broyer Laboratoire de Spectromtrie Ionique et Molculaire Centre National de la Recherche Scientifique Universit de Lyon Villeurbanne, France John H. Burnett Atomic Physics Division National Institute of Standards andTechnology Gaithersburg, Maryland Maria A. Cataluna Division of Electronic Engineering andPhysics School of Engineering, Physics andMathematics University of Dundee Dundee, United Kingdom Gianfranco Cerofolini Department of Materials Science University of Milano-Bicocca Milano, Italy Giulio Cerullo Dipartimento di Fisica Politecnico di Milano Milano, Italy Swapan Chakrabarti Department of Chemistry University of Calcutta Kolkata, West Bengal, India Chia-Ching Chang Department of Biological Science andTechnology National Chiao Tung University Hsinchu, Taiwan and Institute of Physics Academia Sinica Taipei, Taiwan Jean-Pierre Colinge Tyndall National Institute University College Cork Cork, Ireland Giancarlo Consolo Department of Matter Physics andElectronic Engineering Faculty of Engineering University of Messina Messina, Italy Emmanuel Cottancin Laboratoire de Spectromtrie Ionique et Molculaire Centre National de la Recherche Scientifique Universit de Lyon Villeurbanne, France Damien Deleruyelle Centre National de la Recherche Scientifique Institut Matriaux Microlectronique Nanosciences de Provence Universits dAix Marseille Marseille, France Alain C. Diebold College of Nanoscale Science andEngineering University at Albany Albany, New Yorkxv

xvi

Contributors

Lamberto Du Dipartimento di Fisica Politecnico di Milano Milano, Italy Mikhail Erementchouk NanoScience Technology Center and Department of Physics University of Central Florida Orlando, Florida Marco Finazzi Dipartimento di Fisica Politecnico di Milano Milano, Italy Giovanni Finocchio Department of Matter Physics andElectronic Engineering Faculty of Engineering University of Messina Messina, Italy Martin Geller Experimental Physics and Center for Nanointegration Duisburg-Essen University of Duisburg-Essen Duisburg, Germany Gabriel Gonzlez NanoScience Technology Center and Department of Physics University of Central Florida Orlando, Florida Evgeny P. Grebennikov Open Joint-Stock Company Central Scientific Research Institute ofTechnology Technomash Moscow, Russia Jim Greer Tyndall National Institute University College Cork Cork, Ireland

Akihiro Hashimoto Department of Electrical and Electronics Engineering Graduate School of Engineering University of Fukui Fukui, Japan Jun He Department of Chemistry Institute for Optical Sciences and Centre for Quantum Information andQuantum Control University of Toronto Toronto, Ontario, Canada Woong-Ki Hong Department of Materials Science andEngineering Gwangju Institute of Science andTechnology Gwangju, Korea Zhijun Hu Center for Soft Matter Physics andInterdisciplinary Research Soochow University Suzhou, China Vanessa M. Huxter Department of Chemistry Institute for Optical Sciences and Centre for Quantum Information andQuantum Control University of Toronto Toronto, Ontario, Canada Frank Jahnke Institute for Theoretical Physics University of Bremen Bremen, Germany Gunho Jo Department of Materials Science andEngineering Gwangju Institute of Science andTechnology Gwangju, Korea

Alain M. Jonas Institute of Condensed Matter andNanosciences Division of Bio- and Soft Matter Catholic University of Louvain Louvain-la-Neuve, Belgium Yoshihiko Kanemitsu Institute for Chemical Research Kyoto University Uji, Kyoto, Japan Tadashi Kawazoe Department of Electrical Engineering and Information Systems School of Engineering The University of Tokyo Tokyo, Japan Stephen Knight Office of Microelectronics Programs National Institute of Standards andTechnology Gaithersburg, Maryland Kiyoshi Kobayashi Department of Electrical Engineering and Information Systems The University of Tokyo Tokyo, Japan and Core Research of Evolutional Science and Technology Japan Science and Technology and Department of Electrical and Electronic Engineering University of Yamanashi Kofu, Japan Hermann Kohlstedt Christian-Albrechts-Universitat Zu Kiel Faculty of Engineering Nanoelectronics Kiel, Germany Takhee Lee Department of Materials Science andEngineering Gwangju Institute of Science andTechnology Gwangju, Korea

Contributors

xvii

Jean Lerm Laboratoire de Spectromtrie Ionique et Molculaire Centre National de la Recherche Scientifique Universit de Lyon Villeurbanne, France Michael N. Leuenberger NanoScience Technology Center and Department of Physics University of Central Florida Orlando, Florida James Alexander Liddle Center for Nanoscale Science andTechnology National Institute of Standards andTechnology Gaithersburg, Maryland Jongsun Maeng Department of Materials Science andEngineering Gwangju Institute of Science andTechnology Gwangju, Korea Andreas Marent Institut fr Festkrperphysik Technische Universitt Berlin Berlin, Germany Vincent Meunier Oak Ridge National Laboratory Oak Ridge, Tennessee Gilles Micolau Institut Matriaux Microlectronique Nanosciences de Provence Centre National de la Recherche Scientifique Universits dAix Marseille Marseille, France Makoto Naruse National Institute of Information andCommunications Technology Koganei, Japan

and Department of Electrical Engineering and Information Systems School of Engineering The University of Tokyo Tokyo, Japan Wataru Nomura School of Engineering The University of Tokyo Tokyo, Japan

Michel Pellarin Laboratoire de Spectromtrie Ionique et Molculaire Centre National de la Recherche Scientifique Universit de Lyon Villeurbanne, France Nikolay A. Pertsev A.F. Ioffe Physico-Technical Institute Russian Academy of Sciences St. Petersburg, Russia Adrian Petraru Christian-Albrechts-Universitat Zu Kiel Faculty of Engineering Nanoelectronics Kiel, Germany Vivek M. Prabhu Polymers Division National Institute of Standards andTechnology Gaithersburg, Maryland Edik U. Rafailov Division of Electronic Engineering andPhysics School of Engineering, Physics andMathematics University of Dundee Dundee, United Kingdom

Motoichi Ohtsu Department of Electrical Engineering and Information Systems School of Engineering The University of Tokyo Tokyo, Japan

Tom Ostatnick Faculty of Mathematics and Physics Department of Chemical Physics and Optics Charles University Prague, Czech Republic

Andrew R. Parker Department of Zoology The Natural History Museum London, United Kingdom and School of Biological Science University of Sydney Sydney, New South Wales, Australia

Andr Avelino Pasa Laboratrio de Filmes Finos e Superfcies Departamento de Fsica Universidade Federal de Santa Catarina Santa Catarina, Brazil

Arndt Remhof Division of Hydrogen and Energy Department of Environment, Energy andMobility Swiss Federal Laboratories for Materials Testing and Research Dbendorf, Switzerland Elisabetta Romano Department of Materials Science University of Milano-Bicocca Milano, Italy Harry E. Ruda Centre for Advanced Nanotechnology University of Toronto Toronto, Ontario, Canada

Ivan Pelant Institute of Physics Academy of Sciences of the Czech Republic Prague, Czech Republic

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Contributors

Amir Saar Racah Institute of Physics and The Harvey M. Kruger Family Center forNanoscience and Nanotechnology The Hebrew University of Jerusalem Jerusalem, Israel Suguru Sangu Device and Module Technology Development Center Ricoh Company, Ltd. Yokohama, Japan Gregory D. Scholes Department of Chemistry Institute for Optical Sciences and Centre for Quantum Information andQuantum Control University of Toronto Toronto, Ontario, Canada Sabyasachi Sen Department of Chemistry JIS College of Engineering Kolkata, West Bengal, India Alexander Shik Centre for Advanced Nanotechnology University of Toronto Toronto, Ontario, Canada

Christopher L. Soles Polymers Division National Institute of Standards andTechnology Gaithersburg, Maryland Sunghoon Song Department of Materials Science andEngineering Gwangju Institute of Science andTechnology Gwangju, Korea Bobby G. Sumpter Oak Ridge National Laboratory Oak Ridge, Tennessee Kien Wen Sun Department of Applied Chemistry National Chiao Tung University Hsinchu, Taiwan Jan Valenta Faculty of Mathematics and Physics Department of Chemical Physics andOptics Charles University Prague, Czech Republic Marek S. Wartak Department of Physics and Computer Science Wilfrid Laurier University Waterloo, Ontario, Canada

Andreas Westphalen Department of Physics and Astronomy Institute for Condensed Matter Physics Ruhr-Universitt Bochum Bochum, Germany Obert R. Wood II GLOBALFOUNDRIES Albany, New York Naoki Yamamoto Department of Physics Tokyo Institute of Technology Tokyo, Japan Takashi Yatsui School of Engineering The University of Tokyo Tokyo, Japan Hartmut Zabel Department of Physics and Astronomy Institute for Condensed Matter Physics Ruhr-Universitt Bochum Bochum, Germany Natalya A. Zimbovskaya Department of Physics and Electronics University of Puerto Rico Humacao, Puerto Rico and Institute for Functional Nanomaterials University of Puerto Rico San Juan, Puerto Rico

Computing and Nanoelectronic Devices1 Quantum Computing inSpin NanosystemsIntroduction Qubits and Quantum Logic Gates Conditions for thePhysical Implementation ofQuantum Computing Zeeman Effects AtomLight Interaction LossDiVincenzo Proposal Quantum Computing withMolecular Magnets Semiconductor Quantum Dots Single-Photon Faraday Rotation Concluding Remarks Acknowledgments References

IMartin Geller, Andreas Marent, and Dieter Bimberg .......2-1

Gabriel Gonzlez and Michael N. Leuenberger ............................... 1-1

2 Nanomemories Using Self-Organized Quantum Dots

Introduction Conventional Semiconductor Memories Nonconventional Semiconductor Memories Semiconductor Nanomemories A Nanomemory Based on IIIV Semiconductor Quantum Dots CapacitanceSpectroscopy Charge Carrier Storage in Quantum Dots Write Times in Quantum Dot Memories Summary andOutlook Acknowledgments References Introduction CNFET-Based Memory Elements NEMS-Based Memory Electromigration CNT-Based Data Storage General Conclusions Acknowledgments References Introduction Background State of the Art Summary and Discussion References

3 Carbon Nanotube Memory Elements Vincent Meunier and Bobby G. Sumpter .........................................................3-1 4 Ferromagnetic Islands Arndt Remhof, Andreas Westphalen, and Hartmut Zabel ......................................................4-1

5 A Single Nano-Dot Embedded in a Plate Capacitor Gilles Micolau and Damien Deleruyelle..................................5-1Introduction Studied Configuration: Geometry and Notations Metallic Dot SemiconductorDot:Theoretical Approach Finite Element Modeling ofaSilicon Quantum Dot Conclusion Appendix A: A Numerical Validation of the Semi-Analytical Approach References

6 Nanometer-Sized Ferroelectric Capacitors

Introduction Fundamentals of Ferroelectricity Deposition and Patterning Characterization of Ferroelectric Films and Capacitors Physical Phenomena in Ferroelectric Capacitors Future Perspective Summary andOutlook References

Nikolay A. Pertsev, Adrian Petraru, and Hermann Kohlstedt ............6-1

7 Superconducting Weak Links Made of Carbon Nanostructures

Introduction Superconducting Transport through a Weak Link Superconducting Transport inaCarbon Nanotube Weak Link Nanotube-Based Superconducting Quantum Interferometers Graphene-Based Superconducting Weak Links Fullerene-Based Superconducting Weak Links Concluding Remarks Acknowledgment References

Vincent Bouchiat ..............................................7-1

8 Micromagnetic Modeling of Nanoscale Spin Valves Bruno Azzerboni, Giancarlo Consolo, andGiovanniFinocchio ........................................................................................................................................... 8-1Introduction Background Computational Micromagnetics of Nanoscale Spin-Valves: State of the Art Conclusionsand Future Perspective References

9 Quantum Spin Tunneling in Molecular Nanomagnets

Introduction Spin Tunneling in Molecular Nanomagnets Phonon-Assisted Spin Tunneling in Mn12 Acetate Interference between Spin Tunneling Paths in Molecular Nanomagnets Incoherent Zener Tunneling inFe8 Coherent Nel Vector Tunneling in Antiferromagnetic Molecular Wheels Berry-Phase Blockade inSingle-Molecule Magnet Transistors Concluding Remarks Acknowledgment References

Gabriel Gonzlez and Michael N. Leuenberger .............. 9-1

I-1

I-2

ComputingandNanoelectronicDevices

10 Inelastic Electron Transport through Molecular Junctions

Introduction Coherent Transport Buttiker Model forInelastic Transport Vibration-Induced Inelastic Effects Dissipative Transport Polaron Effects: Hysteresis, Switching, and Negative Differential Resistance Molecular Junction Conductance and Long-Range Electron-Transfer Reactions Concluding Remarks References

Natalya A. Zimbovskaya ....................................... 10-1

11 Bridging Biomolecules with Nanoelectronics

Introduction and Background Preparation of Molecular Magnets Nanostructured Semiconductor Templates: Nanofabrication and Patterning Self-AssemblingGrowth of Molecules on the Patterned Templates Magnetic Properties of Molecular Nanostructures Conclusion and Future Perspectives References

Kien Wen Sun and Chia-Ching Chang.........................................11-1

1Quantum Computing inSpin Nanosystems1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Introduction ............................................................................................................................. 1-1 Qubits and Quantum Logic Gates ........................................................................................ 1-2 Conditions for the Physical Implementation of Quantum Computing .......................... 1-3 Zeeman Effects ......................................................................................................................... 1-3Weak External Field Strong External Field JaynesCummings Model RKKY Interaction

AtomLight Interaction ..........................................................................................................1-4 LossDiVincenzo Proposal .................................................................................................... 1-7 Quantum Computing with Molecular Magnets................................................................. 1-9 Semiconductor Quantum Dots ........................................................................................... 1-12Classical Faraday Effect Quantum Faraday Effect

Single-Photon Faraday Rotation ......................................................................................... 1-14Quantifying the EPR Entanglement Single Photon Faraday Effect and GHZ Quantum Teleportation Single-Photon Faraday Effect and Quantum Computing

Gabriel GonzlezUniversityofCentralFlorida

Michael N. LeuenbergerUniversityofCentralFlorida

1.10 Concluding Remarks............................................................................................................. 1-20 Acknowledgments ............................................................................................................................. 1-20 References ........................................................................................................................................... 1-20 computing devices have been developed based on atomic, molecular, optical, and semiconductor physics and technologies. Before contemplating the physical realization of a quantum computer, it is necessary to decide how information is going to be stored within the system and how the system will process that information during a desired computation. In classical computers, the information is typically carried in microelectronic circuits that store information using the charge properties of electrons. Information processing is carried out by manipulating electrical fields within semiconductor materials in such a way as to perform useful computational tasks. Presently it seems that the most promising physical model for quantum computation is based on the electrons spin. A strong research effort toward the implementation of the electron spin as a new information carrier has been the subject of a new form of electronics based on spin called spintronics. Experiments that have been conducted on quantum spin dynamics in semiconductor materials demonstrate that electron spins have several characteristics that are promising for quantum computing applications. Electron spin states possess the following advantages: very long relaxation time in the absence of external fields, fairly long decoherence time d 1 s, and the possibility of easy spin manipulation by an external magnetic field. These characteristics are very promising1-1

1.1 IntroductionThe history of quantum computers begins with the articles of Richard Feynman who, in 1982, speculated that quantum systems might be able to perform certain tasks more efficiently than would be possible in classical systems (Feynman, 1982). Feynman was the first to propose a direct application of the laws of quantum mechanics to a realization of quantum algorithms. The fundamentals of quantum computing were introduced and developed by several authors after Feynmans idea. A model and a description of a quantum computer as a quantum Turing machine was developed by Deutsch (1985). In 1994, Shor introduced the quantum algorithm for the integer-number factorization and in 1997, Grover proposed the fast quantum search algorithm (Grover, 1997). Later on, Wooters and Zurek proved the noncloning theorem, which puts definite limits on the quantum computations, but Shors work challenges all that with the quantum error correction code (Shor, 1995). In the last years, the development of quantum computing has grown to enormous practical importance as an interdisciplinary field, which links the elements of physics, mathematics, and computer science. Currently, various physical models of quantum computers are under intensive study. Several types of elementary quantum

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HandbookofNanophysics:NanoelectronicsandNanophotonics

since longer decoherence times relax constraints on the switching speeds of quantum gates necessary for reliable error correction. Typically quantum gates are required to switch 104 times faster than the loss of qubit coherence. Spin coherent transport over lengths as large as 100 m have been reported in semiconductors. This makes electron spin a perfect candidate as an information carrier in semiconductors (Adamowski et al., 2005). The spin of particles exhibiting quantum behavior is specially suitable for the construction of quantum computers. The electron spin states can be used to construct qubits and logic operations in different ways. They can be constructed either directly with the application of a magnetic field or indirectly with the application of symmetry properties of the many-electron wave function (namely by the resulting singlet or triplet spin states). Traditionally, in nanoelectronic devices, the charge of the electrons has been used to carry and transform information, which makes the interaction of spintronic devices with charge-based devices important for compatibility with existing classical computing schemes. Overall, we can say that spin nanosystems are good candidates for the physical implementation of quantum computation.

| = c0 | 00 + c1 | 00 + c2 | 10 + c3 | 11,

(1.2)

where the normalization condition takes the form |c0|2 + |c1|2 + |c2|2 + |c3|2 = 1. Another basic characteristic for quantum computing is the so-called entanglement. The two quantum two-level systems can become entangled by interacting with each other. This means that we cannot fully describe one system independently of the other. For example, suppose that the state (| 01 | 10)/ 2 gives a complete description of the whole system. Then a measurement over the first subsystem forces the second subsystem into one of the two states |0 or |1. This means that one measurement over one subsystem influences the other, even though it may be arbitrarily far away. Qubits can be transformed by unitary transformations (observables) U that play the role of quantum logic gates and which transforms the initial qubit into a final qubit according to| f = U | i .

(1.3)

Depending on the type of qubit on which they operate, we deal with either a 2 2 or a 4 4 matrix. For example, the quantum NOT gate is defined as 0 UNOT = 1 1 . 0 (1.4)

1.2 Qubits and Quantum Logic GatesWe know that the information stored in a classical computer can take one of the two values, i.e., 0 or 1, with probability 0 or 1 each. Quantum bits or qubits are the quantum mechanical analogue of classical bits. In contrast, the qubit can be defined as a quantum state vector in a two-dimensional Hilbert space. Suppose |0, |1 forms a basis for the Hilbert space, then the qubit can be expressed as the superposition of the two states as | = c0 | 0 + c1 | 1, (1.1)

The effect of the quantum logic gate (4) in the one-qubit state given in Equation 1.1 is to exchange the probability amplitudes i.e., c0 0 UNOT = c1 1 1 c0 c1 = . 0 c1 c0 (1.5)

where c0 and c1 are complex numbers and the modulus squared of each complex number represents the probability to obtain the qubit |0 or |1, respectively. Additionally, they must satisfy the normalization condition |c0|2 + |c1|2 = 1. Contrary to the classical bit, the quantum bit takes on a continuum of values, which are determined by the probability amplitudes given by c0 and c1. If we perform a measurement on qubit |, we obtain either outcome |0 with probability |c0|2 or outcome |1 with probability |c1|2. However, if the qubit is prepared to be exactly equal to one of the states of the computational basis, i.e., | = |0 or | = |1, then we can predict the exact result of the measurement with probability 1. This nondeterministic characteristic between the general state of the qubit and the precise result of the measurement in the basis state plays an essential role in quantum computations. To carry out a quantum computation, we require at least a two-qubit state, i.e., the states of a two-particle quantum system. The two-qubit states can be constructed as tensor products of the basis states |0, |1. The two-qubit basis consists of the states |00, |01, |10, |11, where in the shorthanded notation |0 |1 |01, etc. implied. An arbitrary two-qubit state has the form

An example of an important quantum logic gate that operates on a two-qubit state is the so-called controlled-NOT gate UCNOT, for which the first qubit is the control qubit and the second qubit is the target qubit. The controlled-NOT gate transforms the twoqubit basis states as follows: UCNOT | 00 = | 00, UCNOT | 01 = | 01, UCNOT | 10 = | 11, and UCNOT | 11 = | 10, (1.6)

which means that the CNOT (conditionalNOT) gate changes the second qubit if and only if the first qubit is in state |1. The matrix representation of the CNOT gate is 1 0 = 0 0 0 1 0 0 0 0 0 1 0 0 . 1 0

UCNOT

(1.7)

It has been shown that the set of logic operations, which consists of all the one-qubit gates and the single two-qubit gate UCNOT is universal in the sense that all unitary transformations on

QuantumComputinginSpinNanosystems

1-3

N-qubit states can be expressed by different compositions of the set of universal gates (DiVincenzo, 1995). Using the concepts of superposition and entanglement, we can describe another important characteristic of quantum computation, which is known as quantum parallelism. Quantum parallelism is based on the fact that a single unitary transformation can simultaneously operate on all the qubits in the system. In a sense, it can perform several calculations in a single step. In fact, it can be proved that the computing power of a quantum computer scales exponentially with the number of qubits, whereas a classical computer the scale is only linear.

| = c0 | 0 + e ic1 | 1,

(1.8)

1.3 Conditions for thePhysical Implementation ofQuantum ComputingThe successful implementation of a quantum computer has to satisfy some basic requirements. These are known as the DiVincenzo criteria and can be summarized in the following way (DiVincenzo, 2001). 1. Physical realizability of the qubits. We need to find some quantum property of a scalable physical system in which to encode our information so that it lives long enough to enable us to perform computations. 2. Initial state preparation. It should be possible to precisely prepare the initial qubit state. 3. Isolation. We need a controlled evolution of the qubit; this will require enough isolation of the qubit from the environment to reduce the effects of decoherence. 4. Gate implementation. We need to be able to manipulate the states of individual qubits with reasonable precision, as well as to induce interaction between them in a controlled way, so that the implementation of gates is possible. Also, the gate operation time s has to be much shorter than the decoherence time d, so that /d r, where r is the maximum tolerable error rate for quantum error correction schemes to be effective. 5. Readout. We must be able to accurately measure the final qubit state. The conditions listed above put certain limitations on the quantum computing technology. For example, the complete isolation of the qubit with respect to the environment disables the read/write operations. Therefore, some slight interaction of the quantum system and the environment is necessary. On the other hand, this interaction leads to decay and decoherence processes, which reduce the performance of the quantum computer. In the decay process, the quantum system jumps in a very short time to a new state, releasing part of its energy to the environment. The decay is characterized by the relaxation time, which for the spin states can be very long. Decoherence is a more subtle process because the energy is conserved but the relative phase of the computational basis is changed. As a result of the decoherence the qubit changes as follows:

where the real number denotes the relative phase. The appearance of the nonzero relative phase results due to the coupling between the quantum system and the environment and can lead to significant changes in the measurement process. The ratio of the decoherence time to the elementary operation time s, i.e., R = d/s, is an approximate measure of the number of computation steps performed before the coupling with the environment destroys the qubit. This ratio changes abruptly for different quantum computing schemes. For example, R = 103 for the electron states in quantum dots, R = 107 for nuclear spin sates, and R = 1013 for trapped ions. An important factor that should always be kept in mind when constructing quantum computers is the scalability of the device. We should be able to enlarge the physical device to contain many qubits and still fulfill the DiVincenzo requirements described above.

1.4 Zeeman EffectsAn external magnetic field superimposed on an atom perturbs its state in a definite way. The Hamiltonian for such an atom may be divided into the operator H0 for the unperturbed atom and the perturbation operator H due to the magnetic field. The external magnetic field H causes the vectors L and S to precess about its direction. We shall examine the two cases where the magnetic field is weak and is strong. Let us first write down the perturbation operator explicitly. Consider an electron in the simplest possible atom (hydrogen) rotating around the nucleus. The electron orbit can be regarded as a current loop. The current is the charge per unit time past any fixed point on the orbit, therefore j= e( p /me ) e ev = = , T 2r 2r (1.9)

where p and m e are the momentum and mass of the electron, respectively. The magnitude of the magnetic moment of the loop is the current times the enclosed area, i.e., orb = r 2j, therefore orb = e r p = B L, 2me (1.10)

where we have introduced the Bohr magneton, B = e/2me. We do not have a good semiclassical picture for spin and therefore we can only conclude that the spin magnetic moment is sp = g s B S, (1.11)

where gs is called the gyromagnetic factor and its value is approximately equal to 2. Therefore, the total magnetic moment is

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HandbookofNanophysics:NanoelectronicsandNanophotonics

=

B

(L + 2S ).

(1.12) and using

H =

BH J z S J 1 + 2 , J

(1.16)

The perturbation energy caused by the magnetic field is given then by H = H = B ( J + S ), (1.13)

S J j( j + 1) + s(s + 1) l(l + 1) = =g 2 j( j + 1) J2 we get

(1.17)

where the plus sign resulted because the charge of the electron is e.

H = B Hgm j

for m j = j, j + 1,, j.

(1.18)

1.4.1 Weak External FieldIn a hydrogen atom, the electron circles around the proton; but in a system fixed to an electron, the proton circles around the electron and generates an inner magnetic field at the position of the electron given by H in = 0e L. 4mpr 3 (1.14)

Thus, the energy levels with a given J split into as many levels are as there are different projections of J on the magnetic field, i.e., 2j + 1. The factor g is called the Land factor and takes a given value for a given L and S and each corresponding value of J. The Zeeman effect in a weak magnetic field is called anomalous.

1.4.2 Strong External FieldWe shall now consider the opposite extreme case, when the external field is strong compared the internal field, so that with the coupling between the vectors L , S , J is disrupted. This can be explained by the fact that S precesses twice as fast L. Then, from the classical analogy, each of the vectors S and L precesses independently about the magnetic field. For this case, the correction to the energy is given by H = B eH (Lz + 2Sz ). (1.19)

Equation 1.14 gives the following energy for the electrons spin Hso = sp H in = e2 S L, 4 0r 3m2c 2 (1.15)

which is called the spinorbit interaction. We interpret the spin orbit interaction as an internal Zeeman effect because it splits the energy levels without an external magnetic field. Let the external field be weak compared with the effective internal field. Since the Larmor frequency is proportional to the magnetic field, the tri angle L S J in Figure 1.1rotates about J considerably faster than theprecession around H. Therefore, the coupling of the vectors L , S , and J in the triangle is not disrupted. Let us now find the correction to the energy due to the exter nal magnetic field. Equation 1.13 involves the vectors J and S , if we consider that the z-axis coincides with the direction of the external magnetic field, then, the only projections of these two vectors that contribute to the energy are the ones along the z-axis. Thus, the mean value of H is equal toH

In Equation 1.19, Lz and Sz are the projections of the orbital angular momentum and spin along the z-axis, respectively, and are given by Lz = ml 1 for ml = l ,, l , Sz = ms , for ms = . 2 (1.20)

The projections Sz and Lz are changed by unity, therefore all the levels in Equation 1.19 are equidistant. Of course, certain values of H may be repeated several times if the sum Lz + 2Sz assumes the same value. This Zeeman effect in a strong magnetic field is called the normal Zeeman effect.

1.5 atomLight InteractionS

J L

FIGURE 1.1

Schematic of the spinorbit coupling.

Though an atom has infinitely many energy levels, we can have under certain assumptions a two-level atom. These assumptions are (1) the difference of the energy levels approximately matches the energy of the incident photon, (2) the selection rules allow transition of the electrons between the two levels, and (3) all other energy levels are sufficiently detuned in frequency separation with respect to the incoming frequency such that there is no transition to these levels.

QuantumComputinginSpinNanosystems

1-5

Consider that we apply the two-level approximation to a simple atom where the interaction with visible light involves a single electron. The corresponding wave function of the system is | = c0 | 0 + c1 | 1, (1.21)

dc1(t ) i * = E0 1 | d | 0ei(10 + )t + E0 1 | d | 0ei(10 )t c0 (t ), 2 dt (1.28) where 10 = (1 0)/ and 0|d |1 = 1|d |0* = 0|er |1. For the case when 10, we can drop the terms containing exp[(10 + )t] in Equation 1.28 since they oscillate rapidly in time and can be neglected with respect to the near resonant terms, i.e., the terms of the form exp[(10 )t]. This approximation is called the rotating wave approximation (RWA), and the coupled differential equations become dc0 (t ) i* it = R e c1(t ), dt 2 dc1(t ) iR it = e c0 (t ), dt 2 where R = E0 . 1|d |0/ is known as the Rabi frequency = (10 ) is the so-called detuning The general solution for Equation 1.29 for strictly monochromatic fields, i.e., R = |R|ei = constant, is given by c0 (t ) = 1 (1ei2t 2ei1t ),

where |0 (|1) corresponds to the nonexcited (excited) state and the coefficients c0(c1) represent the probability amplitude to find the system in either state, respectively. The probability amplitudes satisfy the normalization condition |c0|2 + |c1|2 = 1. The wavelength of visible light, typical for atomic transitions, is about a few thousand times the diameter of an atom. Therefore, there is no significant spatial variation of the electric field across an atom, and E(r , t) E(t) can be taken as independent of the position (dipole approximation). Consistent with this long-wavelength approximation is that the magnetic field H is approximately zero, i.e., H 0, so that the interaction energy between the field and the electron of the atom is given by H = er E, hence the total Hamiltonian is H = H0 + H, (1.22)

(1.29)

where H0 represents the unperturbed atom system, i.e., H0 | 0 = 0 | 0 and H0 | 1 = 1 | 1 , where 0 and 1 are the energy values for the nonexcited and excited states, respectively. Seeking a solution to the Schrdinger equation H | (t ) = i in the following form| (t ) = c0 (t )e i0t / | 0 + c1(t )e i1t / | 1,

| (t ) , t

(1.23)

i t c1(t ) = R eit / 2 sin , 2 where 1,2 = 2 + 2 , R 22 2 R

(1.30)

(1.24)

we get i cl (t ) ck (t )e i( k l )t / l | H | k , for l = 0, 1. = dt k = 0 ,1

(1.31)

(1.25)

= 1 2 = + . Equation 1.31 gives the transition probabilities from the nonexcited to the excited state t |c0 (t )|2 = R sin2 , |c1(t )|2 = 1 |c0 (t )|2 , 2 2

Representing the light with frequency by means of the complex electric field vector E 0 in the form 1 * E(t ) = (E0e it + E0 eit ), 2 (1.26)

(1.32)

and using the fact that the diagonal elements of the interacting Hamiltonian are zero due to the parity of the eigenfunction, i.e., k | H | k = 0, we end up with the following coupled differential equations dc0 (t ) i ) * E0 0 | d | 1e i(10 + )t + E0 0 | d | 1e i(10 )t c1(t ), = dt 2 (1.27)

which oscillate with frequency (Rabi flopping frequency) between levels 0 and 1. Any two-level system can be represented by a 2 2 matrix Hamiltonian and hence can be expressed in terms of Pauli matrices. For example, choosing the energy zero to be half way between the excited and nonexcited state |0 and |1, we have the following matrix Hamiltonian for this system H0 = 1 2 0 0 = z , 1 2 (1.33)

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HandbookofNanophysics:NanoelectronicsandNanophotonics

where = 1 0 z is a Pauli matrix Therefore, we can write the total Hamiltonian of the atomlight interaction in the RWA asH= 2 *e it R R eit z (R eit + + *e it ), = R 2 2

dependent amplitude that varies harmonically with time, i.e., . 2 2 A k = ikAk, where k = kx + k 2 + kz = | k | c . Assuming that y the field is periodic in space, and the lengths of the periods in three perpendicular directions are equal to the dimensions ofthe cubic box, then kj = 2nj/L for j = x, y, z and nj are integers of any sign. Using Equation 1.38 we get E(r , t ) = and H (r , t ) = i0V

i0

V

k k k

k

* * Ak eik r + k Ak e ik r ,

(1.41)

(1.34) where 0 + = 0 1 0 and = 0 1 0 . 0 (1.35)

(k k 0

* * )Ak eir r (k k )Ak e ir r .

(1.42)

1.5.1 JaynesCummings ModelSo far, we have used a semiclassical description of the interaction between a two-level atom and an electric field. A quantum description of the interaction would require a quantization of the radiation field. This description is known as the JaynesCummings model. Consider the sourceless electromagnetic field in a cubic cavity of volume V = L L L. Working with the Coulomb gauge for which the vector potential satisfies the requirement A = 0, (1.36)

The classical Hamiltonian for the field is given by HE M = 2

V

(| E | +c | H | ) dx dy dz.2 2 2

(1.43)

Substituting Equations 1.41 and 1.42 into Equation 1.43 one gets HE M = If we introduce the variable Ak = Pk 1 Qk + i , 2 k (1.45)

A A* .2 k k k k

(1.44)

and, since there are no sources, it satisfies the homogeneous wave equation 2 A 1 2 A = 0. c 2 t 2 (1.37)

then Equation 1.44 becomes HE M = 1 2

Then the fields are fully specified by the vector potential in the form E= A and B = A. t (1.38)

(Q 2 k k

2 k

+ Pk2 ).

(1.46)

We can expand the vector potential in the form A(r , t ) = 10V

Equation 1.46 corresponds to the sum of independent harmonic oscillators. This suggests that each mode of the field is dynamically equivalent to a mechanical harmonic oscillator. The canonical quantization of the field consists of the substitution of the variables Qk and Pk for operators, which fulfill the commutation relation [Q k , Pk ] = i k k , this means that [ Ak , Ak ] =

A (t )ek k

ik r

* + Ak (t )e ik r .

(1.39)

(1.47)

* Equation 1.36 implies that Ak and Ak are perpendicular to the vector of propagation k , therefore we can rewrite Equation 1.39 in the following form: A(r , t ) = 10V

k

k k .

(1.48)

k

k

* * Ak (t )eik r + k Ak (t )e ik r ,

(1.40)

Introducing the creation and annihilation operators ak = k Ak and ak =

where k = (e1, e2) is called the polarization vector, which is perpendicular to the direction of propagation and Ak(t) is a time

k

Ak ,

(1.49)

QuantumComputinginSpinNanosystems

1-7

we can write the electromagnetic field Hamiltonian in the form HEM =

H=

k

k ak ak +

1 . 2 k

10 z + ak ak + g k ( + ak + ak ). 2

(1.57)

(1.50)

Equation 1.57 is the JaynesCummings model.

The total Hamiltonian that describes the interaction of an atom with the quantized electromagnetic field can be written in the form H=

1.6 LossDiVincenzo ProposalThe first proposals for quantum computing made use of cavity quantum electrodynamics (QED), trapped ions, and nuclear magnetic resonance (NMR). All of these proposals benefit from long decoherence times due to a very weak coupling of the qubits to their environment. The long decoherence times have led to big successes in achieving experimental realizations. A conditional phase (CPHASE) gate was demonstrated early on in cavity QED systems. The two-qubit controlled-NOT gate has been realized in single-ion and two-ion versions. The most remarkable realization of the power of quantum computing to date is the implementation of Shors algorithm to factor the number 15 in a liquid-state NMR quantum computer to yield the known result 5 and 3. However, these proposals may not be scalable and therefore do not meet the DiVincenzo criteria. The requirement for scalability motivated the LossDiVincenzo proposal for a solid state quantum computer based on electron spin qubits (Loss and DiVincenzo, 1998). The spin of an electron in a quantum dot can point up or down with respect to an external magnetic field; these eigenstates, | and |, correspond to the two basis states of the qubit. The electron trapped in a quantum dot, which is basically a small electrically defined box that can be filled with electrons, can be defined by metal gate electrodes on top of a semiconductor (GaAs/AlGaAs) heterostructure. At the interface between GaAs and AlGaAs conduction band, electrons accumulate and can only move in the lateral direction. Applying negative voltages to the gates locally depletes this two-dimensional electron gas underneath. The resulting gated quantum dots are very controllable and versatile systems, which can be manipulated and probed electrically. When the size of the dot is comparable to the wavelength of the electrons that occupy it, the system exhibits a discrete energy spectrum, resembling that of an atom. Initialization of the quantum computer can be achieved by allowing all spins to reach their equilibrium thermodynamic ground state at a low temperature T in an applied magnetic field, so that all the spins will be aligned if the condition |gBH| kBT is satisfied (where kB is Boltzmann constant). To perform single-qubit operations, we can apply a microwave magnetic field on resonance with the Zeeman splitting, i.e., with a frequency f = EZ/h (h is Plancks constant). The oscillating magnetic component perpendicular to the static magnetic field H results in a spin nutation. By applying the oscillating field for a fixed duration, a superposition of | and | can be created. This magnetic technique is known as electron spin resonance (ESR). In the LossDivincenzo proposal, two-qubit operations can be carried out purely electrically by varying the gate voltages between neighboring dots. When the barrier is high, the spins are decoupled. When the interdot barrier is pulsed low,

| ii | + i i k

k ak ak

i | d | j | i j | E,i, j

(1.51)

where i is the energy corresponding to the state |i of the atom and we have dropped the constant terms corresponding to the zero point energy. The electric field operator is given by E=i

k

k

k ik r a e a k e ik r . k 0V .

(1.52)

Making the dipole approximation, i.e., eik r 1, and assuming that we are dealing with a two-level atom, i.e., i, j = 0, 1, then we can write Equation 1.51 as H= 10 z + 2

k

k ak ak +

( g *k k

+

g k )(ak ak ), (1.53)

where g k = i k/ 0V d01 k . The scalar product between the dipole moment and the polarization vector yields a complex number that can be written as d01 k = | d01 k | ei. This allows * one to choose the phase in such a way that g k = g k . If the atom interacts only with one mode of the electromagnetic field, then we can drop the sum over the vector of propagation to get H= 10 z + ak ak + g k ( + )(ak ak ) = H0 + H. (1.54) 2

In the interaction picture, i.e., HI = e i H0t / H ei H0t / , the Hamiltonian of interaction will have the form HI = g k +ak e i(10 )t + ak ei(10 )t +ak ei(10 + )t

(

ak e i(10 + )t ,

)

(1.55)

using the RWA, i.e., dropping the rapidly oscillating terms containing e i(10 + )t , we end up with HI = 10 z + ak ak + g k ( + ak e it + ak eit ), 2 (1.56)

where = 10 is the detuning. The Hamiltonian of Equation 1.56 corresponds in the Schrdinger picture to

1-8H Hac

HandbookofNanophysics:NanoelectronicsandNanophotonics

e

e

e

e

2DEG

Magnetized layer

Back gate

FIGURE 1.2 Theoretical proposal by LossDiVincenzo for quantum computing using quantum dots and electric gates.

an appreciable overlap develops between the two electron wave functions, resulting in a nonzero Heisenberg exchange couplingJ (see Figure 1.2). The Hamiltonian describing this time dependent process is given by H(t ) = J (t )Sn Sn +1. (1.58)

called indirect interaction. The basis of this interaction lies in an exchange interaction, which was proposed by Ruderman and Kittel (1954), and extended by Kasuya (1956) and Yosida (1957), and now known as the RKKY interaction. This interaction refers to an exchange energy written as Hexch = J (r )

Equation 1.58 is sometimes referred to as the direct interaction. The evolution of the quantum state is described by the propagator given by U (t ) = T exp[i H(t )dt/ ] , where T is the time ordering operator. If the exchange is pulsed on for a time s such that J (t )dt / = J 0 s / = , the states of the two spins will be exchanged. This is the SWAP operation. Pulsing the exchange for a shorter time s/2 generates the square root of SWAP operation, which can be used in conjunction with single-qubit operator to generate the controlled-NOT gate. A last crucial ingredient requires a method to read out the state of the spin qubit. This implies measuring the spin orientation of a single electron. Therefore, an indirect spin measurement is proposed. First, the spin orientation of the electron is correlated with its position, via spin to charge conversion. Then an electrometer is used to measure the position of the charge, thereby revealing its spin. In this way, the problem of measuring the spin orientation has been replaced by the much easier measurement of charge. The LossDiVincenzo ideas have influenced an enormous research effort aimed at implementing the different parts of the proposal and has been quickly followed by a series of alternative solid state realizations for trapped atoms in optical lattices that may also be scalable. It should also be stressed that the efforts to create a spin qubit are not purely application driven. If we have the ability to control and read out a single electron spin, we are in a unique position to study the interaction of the spin with its environment. This may lead to a better understanding of decoherence and will also allow us to study the semiconductor environment using the spin as a probe.

s S,i i

(1.59)

where The exchange parameter J(r) falls off rapidly with distance r between the center of a localized magnetic ion and elec tron spin s i is the spin state of a conduction electron S is the spin of a localized magnetic ion The minus sign in Equation 1.59 is related to the Pauli exclusion principle (lowest energy of electron occupancy). Below some critical magnetic ordering temperature, the itinerant or localized spins may condense into an ordered array, i.e., ferromagnetic or antiferromagnetic. As in the case of atomic scattering, any disorder within this array will cause additional electron scattering. This scattering may be elastic, in which case there is no change in energy or spin-flip, or it may be inelastic, in which case the spin state of the conduction electrons changes. Usually RKKY interaction is of significance only in compounds with a high concentration of the magnetic atom. Thus, localized ions may start to interact indirectly via the conduction electrons. Now, consider the case in which there exist two localized spins at lattice points Rn and Rm. By the interaction between spin Smz localized at Rm and the spin density of conduction electrons polarized by spin nz localized at Rn, the following interaction S between the spins S n and S m is found: Hexch = 9 where F (x) = x cos( x ) + sin(x ) . x4 (1.61) J 2 Ne F 2kF | Rn Rm | Smz Snz , F N 2

(

)

(1.60)

1.6.1 rKKY InteractionThe RKKY interaction is a long-range magnetic interaction that involves nearest-neighbor ions as well as magnetic atoms that are further apart; this interaction is sometimes

QuantumComputinginSpinNanosystems

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Recently, the optical RKKY interaction between two spins was introduced as a means to produce an effective exchange interaction (Piermarocchi et al., 2002). Similar to the LossDiVincenzo scheme, in this scheme, the two qubits are defined by the excess electrons of semiconductor quantum dots. Instead of using the direct exchange interaction between the two electrons, the exchange interaction is indirectly mediated by the itinerant electrons of virtual excitons that are optically excited in the host material, which can be made of bulk, quantum well, or quantum wire structures. This scheme has the advantage that twoqubit gates can be performed on the femtosecond timescale due to the possibility of using ultrafast laser optics. The Coulomb interaction between the photoexcited itinerant electrons and the localized electrons in the two quantum dots contains direct and indirect terms. While the direct terms give rise to state renormalization of the localized electrons, the exchange terms lead to an effective Heisenberg interaction of the form HORKKY = J12S1 S2 P122 2 HC H X P12 , 3

HX =

1 V

, , k , k

J (k , k )Si s , ck , ck , .

(1.64)

The predicted exchange interaction J12(R) (see Figure 1.2) can be of the order of 1 meV, which is of the same order as the Heisenberg interaction in the LossDiVincenzo scheme.

1.7 Quantum Computing withMolecular MagnetsShor and Grover demonstrated that a quantum computer can outperform any classical computer in factoring numbers (Shor, 1997) and in searching a database (Grover, 1997) by exploiting the parallelism of quantum mechanics. Recently, the latter has been successfully implemented (Ahn et al., 2000) using Rydberg atoms. Leuenberger and Loss (2001) proposed an implementation of Grovers algorithm using molecular magnets (Friedman et al., 1996; Thomas et al., 1996; Sangregorio et al., 1997; Thiaville and Miltat, 1999; Wernsdorfer et al., 2000); their spin eigenstates make them natural candidates for single-particle systems. It was shown theoretically that molecular magnets can be used to build dense and efficient memory devices based on the Grover algorithm. In particular, one single crystal can serve as a storage unit of a dynamic random access memory device. Fast ESR pulses can be used to decode and read out stored numbers of up to 105, with access times as short as 1010 s. This proposal should be feasible using the molecular magnets Fe8 and Mn12. Suppose we want to find a phone number in a phone book consisting of N = 2 n entries. Usually it takes N/2 queries on average to be successful. Even if the N entries were encoded binary, a classical computer would need approximately log2 N queries to find the desired phone number (Grover, 1997). But the computational parallelism provided by the superposition and interference of quantum states enables the Grover algorithm to reduce the search to one single query (Grover, 1997). This query can be implemented in terms of a unitary transformation applied to the single spin of a molecular magnet. Such molecular magnets, forming identical and largely independent units, are embedded in a single crystal so that the ensemble nature of such a crystal provides a natural amplification of the magnetic moment of a single spin. However, for the Grover algorithm to succeed, it is necessary to find ways to generate arbitrary superpositions of spin eigenstates. For spins larger than , this turns out to be a highly nontrivial task as spin excitations induced by magnetic dipole transitions in conventional ESR can change the magnetic quantum number m by only 1. To circumvent such physical limitations, it was proposed to use multifrequency coherent magnetic radiation that allows the controlled generation of arbitrary spin superpositions. In particular, it was shown that by means of advanced ESR techniques, it is possible to coherently populate and manipulate many spin states simultaneously by applying one single pulse of a magnetic a.c. field containing an appropriate number of matched frequencies. This a.c. field creates a nonlinear response of the magnet via multiphoton

(1.62)

which is calculated in fourth-order perturbation theory using the diagram depicted in Figure 1.3. P12 is the projection operator on the two-spin Hilbert space of the two localized spins. The control Hamiltonian HC =

k ,

k , (t ) iP e ck , h k , + h.c. 2

(1.63)

describes the creation of the virtual excitons by means of an external laser field that is detuned by the energy from the continuum states of the host material, or more precisely from the exciton 1s level. ck, and hk, are electron and hole creation operators, respectively. The RKKY interaction between a localized electron in the quantum dot and the itinerant electrons in the host material is given in second quantization byP , +

, ke, 2 J2

1

P , kh, 2

3

, k, c J11

P , +

, ke, 2

FIGURE 1.3 Effective spinspin interaction for the localized electrons in the dots 1 and 2 (indicated by dotted lines) induced by a photoexcited electronhole pair (the solid and dashed lines, respectively). The indices and denote the spin states of the electrons localized in the dots. The photon propagator is depicted by a wavy line. (From Piermarocchi, C. etal., Phys. Rev. Lett., 89, 167402, 2002.)

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HandbookofNanophysics:NanoelectronicsandNanophotonicsEnergy

absorption processes involving particular sequences of and photons, which allows the encoding and, similarly, the decoding of states. Finally, the subsequent read-out of the decoded quantum state can be achieved by means of pulsed ESR techniques. These exploit the nonequidistance of energy levels, which is typical of molecular magnets. Molecular magnets have the important advantage that they can be grown naturally as single crystals of up to 10100 m length containing about 10121015 (largely) independent units so that only minimal sample preparation is required. The molecular magnets are described by a single-spin Hamiltonian of the form H spin = Ha + V + Hsp + HT (Leuenberger and Loss, 1999, 2 4 2000a,b), where Ha = ASz BSz represents the magnetic anisotropy (A B > 0). The Zeeman term V = gBH . S describes the coupling between the external magnetic field H and the spin S of length s. The calculational states are given by the 2s + 1 eigenstates of Ha + g B H z Sz with eigenenergies m = Am2 Bm4 + gBHzm, s m s. The corresponding classical anisotropy potential energy E() = As cos2 Bs cos4 + gBHzs cos is obtained by the substitution Sz = s cos , where is the polar spherical angle. We have introduced the notation m, m = m m. By applying a bias field Hz such that gBHz > Emm, tunneling can be completely suppressed and thus HT can be neglected (Leuenberger and Loss, 1999, 2000a,b). For temperatures below 1 K, transitions due to spinphonon interactions (Hsp) can also be neglected. In this regime, the level lifetime in Fe8 and Mn12 is estimated to be about d = 107 s, limited mainly by hyperfine and/or dipolar interactions (Leuenberger and Loss, 2001). Since the Grover algorithm requires that all the transition probabilites are almost the same, Leuenberger and Loss (2001) and Leuenberger et al. (2003) propose that all the transition amplitudes between the states |s and |m, m = 1, 2, , s 1, are of the same order in perturbation V. This allows us to use perturbation theory. A different approach uses the magnetic field amplitudes to adjust the appropriate transition amplitudes (Leuenberger et al., 2002). Both methods work only if the energy levels are not equidistant, which is typically the case in molecular magnets owing to anisotropies. In general, if we choose to work with the states m = m 0, m 0 + 1, , s 1, where m 0 = 1,2,,s 1, we have to go up to nth order in perturbation, where n = s m 0 is the number of computational states used for the Grover search algorithm (see below) to obtain the first nonvanishing contribution. Figure 1.4 shows the transitions for s = 10 and m 0 = 5. The nth-order transitions correspond to the nonlinear response of the spin system to strong magnetic fields. Thus, a coherent magnetic pulse of duration T is needed with a discrete frequency spectrum {m}, say, for Mn12 between 20 and 300 GHz and a single low-frequency 0 around 100 MHz. The low-frequency field Hz(t) = H0(t) cos(0t)ez, applied along the easy-axis, couples to the spin of the molecular magnet through the Hamiltonian Vlow = g B H 0 (t )cos( 0t )Sz , (1.65)

|5

5 |6

6 7 |8 |7

8

|9

0

9 |10(5 FIGURE 1.4 Feynman diagrams F that contribute to Sm,)s for s = 10 and m 0 = 5 describing transitions (of fifth order in V) in the left well of the spin system (see Figure 1.5). The solid and dotted arrows indicate and transitions governed by Equations 1.66 and 1.65, respectively. ( ( (n We note that Smj,)s = 0 for j < n, and Smj,)s n.

where 0 n. Using rectangular pulse shapes, Hk(t) = Hk, if T/2 < t < T/2, and 0 otherwise, for k = 0 and k m0, one obtains (m m0)

of width 1/T, ensuring overall energy conservation for T > 1. > The duration T of the magnetic pulses must be shorter than the lifetimes d of the states |m (see Figure 1.5). In general, the magnetic field amplitudes Hk must be chosen in such a way that perturbation theory is still valid and the transition probabilities are almost equal, which is required by the Grover algorithm. According to Leuenberger and Loss (2001), the amplitudes Hk do not differ too much between each other due to the partial destructive interference of the different transition diagrams shown in Figure 1.5. Leuenberger et al. (2002) show that the transition probabilities can be increased by increasing both the magnetic field amplitudes and the detuning energies under the condition that the magnetic field amplitudes remain smaller than the detuning energies. In this way, both high-multiphoton Rabi oscillation frequencies and small quantum computation times can be attained. This makes both methods (Leuenberger

+T /2

T / 2

eit dt = sin(T/2)/ is the delta-function

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HandbookofNanophysics:NanoelectronicsandNanophotonics

and Loss, 2001; Leuenberger et al., 2002) very robust against decoherence sources. In order to perform the Grover algorithm, one needs the rela(n tive phases m between the transition amplitudes Sm,)s , which is determined by m = m=+s11 k + m, where m are the relative k phases between the magnetic fields Hm(t). In this way, it is possible to read-in and decode the desired phases m for each state |m. The read-out is performed by standard spectroscopy with pulsed ESR, where the circularly polarized radiation can now be incoherent because only the absorption intensity of only one pulse is needed. We emphasize that the entire Grover algorithm (read-in, decoding, read-out) requires three subsequent pulses each of duration T with d > T > 0 1 > m1 > m1, m 1. This gives a clock-speed of about 10 GHz for Mn12, that is, the entire process of read-in, decoding, and read-out can be performed within about 1010 s. The proposal for implementing Grovers algorithm works not only for molecular magnets but for any electron or nuclear spin system with nonequidistant energy levels, as is shown by Leuenberger et al. (2002) for nuclear spins in GaAs semiconductors. Instead of storing information in the phases of the eigenstates |m (Leuenberger and Loss, 2001), Leuenberger et al. (2002) use the eigenenergies of |m in the generalized rotating frame for encoding information. The decoding is performed by bringing the delocalized state (1/ n ) m | m into resonance with |m in the generalized rotating frame. Although such spin systems cannot be scaled arbitrarily, large spin s (the larger a spin becomes, the faster it decoheres and the more classical its behavior will be) systems of given s can be used to great advantage in building dense and highly efficient memory devices. For a first test of the nonlinear response, one can irradiate the molecular magnet with an a.c. field of frequency s2,s /2, which gives rise to a two-photon absorption and thus to a Rabi oscillation between the states |s and |s 2. For stronger magnetic fields, it is in principle possible to generate

superpositions of Rabi oscillations between the states |s and |s 1, |s and |s 2, |s and |s 3, and so on (see also Leuenberger et al., 2002).

1.8 Semiconductor Quantum DotsIn the following, we mainly focus on quantum dots made of IIIV semiconductor compounds with zincblende structure, like GaAs or InAs. The electronic bandstructure of a three-dimensional semiconductor with zincblende structure is illustrated in Figure 1.6. The bands are parabolic close to their extrema, which are all located at the point. The conduction (c) states have orbital s symmetry and are spin degenerate. The valence (v) band consists of three subbands: the heavy-hole (hh), the light-hole (lh), and the split-off (so) band. The v-band states have orbital p symmetry. The bottom of the c band and the top of the v band are split by the band-gap energy Egap. The v-band 3 states with different j ( j = 1 for the so-band, j = 2 for the hh and 2 lh band) are split by so in energy due to spinorbit interaction. The hh states have the angular momentum projections 3 J z = 2 and the lh states J z = 1 . For finite electron wavevectors 2 k 0, and the hh and lh subbands split into two branches according to the different curvatures of the energy dispersion, which implies different effective masses of heavy and light holes. The v-band states with spin can be written in terms of the orbital angular momentum basis by using the ClebschGordon coefficients which gives us 3 , 3 = 1,1 2 2 Heavy hole 3 3 2 , 2 = 1, 1 | 3 , 1 = 2 2 Light hole 3 1 | 2 , 2 = E1 3 2 3

,

(1.69)

| 1,1 | +

| 1, 0 | , 1 | 1, 0 | + 3 | 1, 1 | 2 3

(1.70)

E c

c

Egap K||so hhlh

Egap K||

(a)

so

lh

hh

(b)

so

lh

hh

FIGURE 1.6 Electronic band structure in the vicinity of the point for (a) a three-dimensional crystal and (b) a quantum well. The conduction and valence bands are shown as a function of the wavevector.

QuantumComputinginSpinNanosystems

1-13

| 1 , 1 = 2 2 Split off 1 1 | 2 , 2 =

1 3 2 3

| 1, 0 | +

| 1,1 | . 1 | 1, 1 | + 3 | 1, 0 | 2 3

(1.71)H

Quantum confinement along the crystal axis quantizes the wavevector component, consequently the hh and lh states of the lowest subband are split by an energy hhlh at the point. Uniaxial strain in the semiconductor crystal can also lift the degeneracy of the heavy and light holes, and thus define the spin quantization axis. If we have a spherically symmetric quantum dot known as colloidal quantum dots, then we can have degeneracy between the heavy and light hole band, i.e., hhlh = 0. Via photon absorption, an electron in a v-band state can be excited to a c-band state. Such interband transitions are determined by optical selection rules. The source or the optical transition rules are due to spinorbit interaction. The electronhole pair created with an interband transition is called an exciton. The electron and hole of an exciton form a bound state due to the Coulomb interaction, similar to that of a hydrogen atom. We refer to the system of two bound excitons as a biexciton.

d E

FIGURE 1.7 The figure shows how the polarization of a linearly polarized beam of light rotates when it goes through a material exposed to an external magnetic field.

1.8.1 Classical Faraday EffectMichael Faraday first observed the effect in 1845 when studying the influence of a magnetic field on plane-polarized light waves. Light waves vibrate in two planes at right angles to one another, and passing ordinary light through certain substances eliminates the vibration in one plane. He discovered that the plane of vibration is rotated when the light path and the direction of the applied magnetic field are parallel. In particular, a linearly polarized wave can be decomposed into right and left circularly polarized waves where each wave propagates with different speeds. The waves can be considered to recombine upon emergence from the medium; however, owing to the difference in propagation speed, they do so with a net phase offset, resulting in a rotation of the angle of linear polarization. The Faraday effect occurs in many solids, liquids, and gases. The magnitude of the rotation depends upon the strength of the magnetic field, the nature of the transmitting substance, and Verdets constant, which is a property of the transmitting substance, its temperature, and the frequency of light. The relation between the angle of rotation of the polarization and the magnetic field in a diamagnetic material is = VHd, (1.72)

A positive Verdet constant corresponds to an anticlockwise rotation when the direction of propagation is parallel to the magnetic field and to a clockwise rotation when the direction of propagation is antiparallel. The Faraday effect is used in spintronics research to study the polarization of electron spins in semiconductors.

1.8.2 Quantum Faraday EffectThe Faraday effect is expected to emerge in low dimensional systems such as semiconductor quantum dots in which the spin states of the electron in the conduction band and the light and heavy hole in the valence band provide a system where different circular polarizations of light couple differently during the process of virtual absorption. The quantum analogue of the Faraday effect does not require an external magnetic field because it is created by selection rules (one circular polarization interacts with the heavy hole band while the other circular polarization interacts with the light hole band) and by the Pauli exclusion principle (the absorption of a right polarized wave is excluded because the allowed transition state between bands have the same spin) (Leuenberger et al., 2005b). In particular, we will be interested in the quantum Faraday effect in a semiconductor colloidal twolevel quantum dot system. We can have a two-level system in a colloidal quantum dot where the heavy hole and the light hole bands are degenerate at the point. This two-level system is achieved by the valence and the conduction band under certain assumptions: (1) the split-off band can be ignored since typical split-off energies are around 102 meV, thus bringing the energy level out of resonance with the single photon; (2) under the appropriate doping and thermal conditions, it can be assumed that the top of the valence band is filled with four electrons, while there is one excess electron in the conduction band; and (3) the energy of the electromagnetic wave is taken to be slightly below the effective band-gap energy, so that the transition from the top of the valence band to the bottom of the conduction band is the strongest transition by far (Leuenberger, 2006).

where is the angle of rotation V is the Verdet constant for the material H is the magnitude of the applied magnetic field d is the length of the path where the light and magnetic field interact (see Figure 1.7)

1-141, 1 2 2 1,+1 2 2

HandbookofNanophysics:NanoelectronicsandNanophotonics1, 1 2 2 1, + 1 2 2

+

+ . e iSo

hh

+lh

+ . e iSo

hh

. e iSo

. e iSo

lh

(a)

3, 3 2 2

3, 1 2 2

3,+ 1 2 2

3,+ 3 2 2

(b)

3, 3 2 2

3, 1 2 2

3, + 1 2 2

3, + 3 2 2

FIGURE 1.8 The solid circles represent filled states and dashed circles represent empty states. The electron in the conduction band interacts with the right or left circularly polarized electromagnetic wave allowing virtual transitions between the conduction and valence band states. After the interaction, the RCP and LCP electromagnetic wave acquire different phases, which produce the rotation of the incident wave.

Interestingly, the idea of the conditional single-photon Faraday rotation first developed by Leuenberger et al. (2005b) and already patented by the authors has been copied (Hu et al., 2008), which demonstrates the importance of this scheme. We now turn our focus to the o