Mesoscopic Systems

Yoshimasa Murayama

Mesoscopic Systems Fundamentals and Applications

@WI LEY-VCH Weinheim New York . Chichester . Brisbane Singapore .Toronto

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Yoshimasa Murayama

Mesoscopic Systems

@IWILEY-VCH

Yoshimasa Murayama

Mesoscopic Systems Fundamentals and Applications

@WI LEY-VCH Weinheim New York . Chichester . Brisbane Singapore .Toronto

Prof. Yoshimasa Murayama Dept. Materials Sci. & Tech. Faculty of Engineering Niigata University Ikarashi-2-no-cho 8050 Niigata-shi, Niigata 9502181 Japan

This book was carefully produced. Nevertheless, author and publisher do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Die Deutsche Bibliothek - CIP-Cataloguing-in-PublicationData A catalogue record for this publication is available from Die Deutsche Bibliothek ISBN 3-527-29376-0 Cover picture: This photograph was supplied by courtesy of Dr. T. Hashizume, Advanced Research Laboratory, Hiachi, Ltd. It is a part of the low-temperature (96 K) STM image of a dangling bond quantum wire fabricated on a passivated Si (100)-2 x 1-H surface (7 nm x 4 nm) by removing hydrogen atoms. The arrows denote the center of the Si dimer rows (after Hitosugi, et al.: Phys. Rev. Lctt. vol. 82 (1999) 4034).

0 WILEY-VCH Verlag Berlin GmbH, D-69469 Weinheim (Federal Republic of Germany), 2001

Printed on acid-free paper All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing: betz-druck GmbH, D-64291 Darmstadt Bookbindung: Wilh. Osswald & Co., D-67433 Neuatadt/Weinstr. Printed in the Federal Republic of Germany

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To my family

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Preface

Yoshimasa Murayama Niigata, Japan, May, 2001 In 1991 a project t o publish a bulky series of an “Encyclopedia of Applied Physics” was begun under the editorship of Professor Trigg. This was undertaken by Publishers VCH (now, Wiley-VCH) under the sponsorship of the American Institute of Physics, Deutsche Physikalische Gesellschaft, The Japan Society of Applied Physics, and The Physical Society of Japan. All editing and publishing tasks were completed in 2000 with 23 volumes and a few Update volumes. For this project I was nominated to be an editorial consultant by the Physical Society of Japan. The first-stage of the project was to list items (that were comprehensive enough t o require as many as 30 pages to interpret), and one of the suggested subjects was “Mesoscopic Systems”. As a consultant, I proposed that I myself could elaborate on this subject and write the article and the Editorial Board accepted this proposition. The result was published as a part of Vol. 10 in 1994. According to my understanding, mesoscopic systems cover a wide range of condensedmatter physics, since they occur between macro- and microscopic worlds, the former of which is non-quantum mechanical, whereas the latter is purely quantum mechanical. As will be described in the text, the quantum mechanical world is gradually being degraded into a non-quantum mechanical one, through the degradation of coherence, that is, “decoherence” or “dephasing”. In the universe as well as specifically in condensed matter the origins of decoherence are plentiful and, hence, in actual situations it could be claimed that almost every system is mesoscopic. In bulk systems, however, such decoherence-related phenomena are not usually noticeable, whereas in tiny systems they are sometimes clearly observable. Tiny systems are often closely related to low-dimensional systems. Thus, in thin films, thin wires and in tiny particles (clusters) it is fairly easy t o discover mesoscopicity-related phenomena. In particular, in such low-dimensional systems the so-called “inelastic (scattering) mean-free-path” specifying decoherence may be comparable t o their linear dimensions. The terminology “mesoscopic system” is very often used to mean “nanoscale” or “nanostructured” devices. More generally they manifest themselves in the “nano-world” . This textbook deals with both up-to-date and somewhat older topics. In a Japanese proverb it is said that one should “Recognize the new through the old”. This is always

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Preface

true and I believe that many mesoscopic phenomena have already been revealed in well-established old-fashioned physical phenomena and devices. In that sense I have chosen only typical phenomena, new or old, which I think are sufficient to elucidate what mesoscopic systems are, to what phenomena they are related with and also how decoherence intrigues into the systems of concern. Today various new phenomena are being found and further developments are being performed in this rather new field. In particular, matured skills to deal with tiny samples in semiconductor technology are enthusiastically introduced into scientific researches on mesoscopic devices, which prove to be more and more successful. Those topics should be treated in other books than this in near future. Many textbooks dealing with related topics have already been published. Authors of such books are apt to describe interesting phenomena often utilizing the Green function techniques or similar sophisticated means, which appear very difficult to undergraduate students. The present textbook is aimed at such less-trained students, i.e., 3rd and 4th grades of undergraduate courses as well as graduates on masters’ course. They should at least be able t o understand preliminary quantum mechanics and condensed matter physics. I have included all of the rather difficult mathematical manipulations in Appendices. In the main body of the text I have tried to describe only clear physical insights and interpretations. For many years throughout my career at the Central/Advanced Research Laboratory, Hitachi, Ltd. and Niigata University I have studied various physically interesting phenomena, most of which would be called “mesoscopic phenomena” under modern terminology. Some of them have actually been interpreted in this textbook. This book was initiated by cooperation with the Editorial Board of the Encyclopedia of Applied Physics. Above all, I would like to thank G. L. Trigg, E. H. Immergut and M. Tanaka, the last of whom nominated me to an editorial consultant on behalf of the Physical Society of Japan. Many professors and doctors are thanked for helping complete this book: N. Nakajima, M. Namiki, the late R. Kubo, N. Saito, Y. Sugita, H. Fujiwara, A. Tonomura, T. Ando, Y. Katayama, N. Kotera, the late K.F. Komatsubara, M. Hirao, T. Uda, K. Yamaguchi, T. Ichiguchi, J. Kasai, T. Mishima, Y . Shiraki, T . Shimada, the late S. Saito, E. Yamada, Y. Kamigaki, K. Koike, T. Furukawa, T. Tanoue, V. A , Ivanov, K. Nakazato, K. Yano, S. Watanabe, Y.A. Ono, M. Goda, Y. Ishino and Y. Ishiduki. Particular thanks are due t o Dr. T . Onogi, who kindly read through this book carefully and made valuable comments, and to Dr. T. Hashizume who kindly provided me a photograph for the book cover. In addition, I also thank Dr. H. Takayanagi, the discussions with whom were valuable just at the last stage of completion of this book. In particular thanks are due t o Prof. Goda. According t o his endeavor it happened on me to have a five-year position in Niigata University, that made me possible to finalize the long-lasting project to write this book. Thanks are also due to staff at VCH: The late M. Poulson, R. Wegenmayr, M. Baer, Vera Dederichs, and C. Reinemuth.

Contents

1 Introduction 1.1 Mesoscopic Systems . . . . . . . . . . . . . . . . . . 1.2 Nanoscale Structures . . . . . . . . . . . . . . . . . . 1.3 Electronics . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

.......... .......... .......... ..........

2 Quantum versus Classical Physics 2.1 QuantumEffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quantum Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Particle-Wave Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Measurement of an Ensemble of Particles . . . . . . . . . . . . . . . . 2.5 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Coherence versus Incoherence . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 6

7 11 11 12 13 14 16 19 24 26

3 Quantization 27 3.1 Schrodinger Equation and Discrete Energies . . . . . . . . . . . . . . . 27 3.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Dimensionality 4.1 DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dimensionality of a Landau Electron . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 36 38

5 Junctions 5.1 Metal-Metal Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Homogeneous Semiconductor Junction . . . . . . . . . . . . . . . . . . 5.3 Heterogeneous Semiconductor Junction . . . . . . . . . . . . . . . . . 5.4 Schottky Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Metal-Oxide-Semiconductor (MOS) . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 42 43 45 48 49

2

Contents

6 3D Quantum Systems References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 56

Quantum Systems Single Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Quantum Wells . . . . . . . . . . . . . . . . . . . . . . . . . . Superlattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Landau Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogenic State of Impurity . . . . . . . . . . . . . . . . . . . . . . . Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 61 61 63 63 64 65 68

7 2D 7.1 7.2 7.3 7.4 7.5 7.6 7.7

8 1D Quantum Systems References . . . . . 9 OD Quantum Systems References . . . . . .

. . . . . . . . .. . . . . .

......................

71 72 73 76

77 10 Transport Properties 77 . . . . . . . . . . . . . . . . . . . . . 10.1 Transport Perpendicular to QW 81 10.2 Transport Parallel to QW . . . . . . . . . . . . . . . . . . . . . . . . . 86 10.3 Magnetic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 10.4 Concepts of Electric Conductivity . . . . . . . . . . . . . . . . . . . . . 96 10.5 Universal Conductance Fluctuation . . . . . . . . . . . . . . . . . . . . 98 10.6 Quantized Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Integral Quantized Hall Effect (IQHE) . . . . . . . . . . . . . . . . . . 102 10.8 Fractional Quantized Hall Effect (FQHE) . . . . . . . . . . . . . . . . 106 108 10.9 Ballistic Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.1OCoulomb Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 10.11Atomic Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11 Optical Properties 121 11.1 Single/Multiple Quantum Wells . . . . . . . . . . . . . . . . . . . . . . 129 . . . . . . . . 11.2 Exciton Absorption in Various Dimensional Geometries 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Magnetic Properties 12.1 Fine Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Magnetic Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 140 155

3

13 Properties of Macroscopic Quantum States 13.1 Kosterlitz-Thouless Mode in High-T, SC . 13.2 Superconducting Thin Wires . . . . . . . . 13.3 Superconducting Tunnel Junction . . . . . . 13.4 Transport Properties of High-T, Cuprates . 13.5 Proximity Effect . . . . . . . . . . . . . . . 13.6 Andreev Reflection . . . . , . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

.., . .. . . .. . . .. . . .... .... .. . .

... . ... , .... . . . . . . . . .... .. ..

... ... . . . ... ... ... . ..

... . ... . ... . ..., .. .. .... ....

157 158 161 163 168 171 176 178

14 Future Prospects

181

A FDM Solution of Schrodinger Equations

183

B Effective-Mass Approximated Equation

187

C Boundary Conditions for an Interface

191

D Hydrogenic Envelope Function in 3D and 2D

197

E Qansition Probability of Optical Processes

207

F Eigenvalue Problem for a Linear Electric Potential

211

G Calculation of Conductivity Based on the Kubo Formula

213

H Calculation of Conductivity Tensor in a Magnetic Field

217

I

223

Landau State in ,+Representation

J Micromagnetism of Stripe Domain

225

K Physics Underlying Josephson Junctions

231

Index

235

Author Index

243

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1

Introduction

1.1 Mesoscopic Systems Today we very often hear such terminologies as “Mesoscopic Systems”, “Mesoscopic Physics”, or “Mesoscopic World”. As is known “meso-” comes from the Greek word pwoc meaning “middle” or “intermediate”. Meson, an elementary particle, has been known for a long time, since the original work of Yukawa and, in meteorology, mesosphere is often used. The terminology Mesoscopic first appeared in paper by van Kampen (1976). After its introduction, it has been gradually recognized that mesoscopic system should mean a physical world, which is realized between classical and quantum physics. Historically, classical physics started with the Greek philosophers, mainly represented by Aristotle, and then established through the ingenious work of Galileo, Newton and others. Classical physics was successful enough to interpret almost every phenomenon which can be conceived and sensed by us, at least up till the mid-19th century. The so-called Macroscopic world is interpreted by classical physics. After classical physics matured, various unfamiliar phenomena appeared, which will never been understood by means of macroscopic physics. Quantum physics must be developed to understand those unfamiliar facts fully. Quantum physics interpreted the Microscopic world. These specific facts often appear in very tiny systems, such as molecules, atoms, and radioactively decayed particles. Initially, these “particles” seemed to be the limiting entity, essentially describable in terms of classical physics. However, this was not the case. When we refer to the microscopic world, it seems to imply a very tiny world in geometric terms. In fact a microscope is a means of observing tiny objects which are unseen by the naked eye. This poses the question: Can a system unseen by our eyes always be interpreted by quantum physics? The answer is not so simple. The means of discriminating a classical from a quantum world is not necessarily size. More discussion on this point will be given later. Let us assume that typical tiny systems require quantum mechanics to be understood, then it is easy to conceive that a macroscopically large entity should exist in the classical world, while a microscopic one exists in the quantum world. Nowadays, most physicists consider that quantum mechanics provides the most basic physical rules and classical physics gives only approximate rules which are valid only in large-scaled systems. In this sense, it may be safe to say that quantum mechanics involves classical physics as a subsystem.

1 Introduction

6

Then a serious question arises. What conditions enable a system to be validated by classical physics, or, equivalently, by quantum mechanics? What distinguishes both types of physics? Although this question is not easily answered, it is sure that the halfway, intermediate world bridging the quantum and classical worlds is the Mesoscopic world.

1.2

Nanoscale Structures

Most tiny devices are known to show peculiar features specific to mesoscopic systems. So, the terminology mesoscopic system is very often used for the same systems as nanoscale stuructures or nanoscale devices. Here, “nano-” means a size on the order of several nanometers, namely, lo-’ m. In other words, the systems correspond to a scale of less than one micrometer, 1 pm=10-6 m. Therefore, they are sometimes called alternatively “submicron systems” .

1.3

Electronics

It is known that the development of solid-state devices was initiated by consideration of a solidified vacuum tube during World War I1 at the Bell Telephone Laboratories. The development was enthusiastically driven by Shockley, Bardeen and Brattain. This was the beginning of Electronics. Even before that time, electrons had also been utilized inside vacuum tubes, where they were emitted from a heated cathode, eventually reaching an anode, and in-between they were controlled by an electric field. An electron itself is an elementary particle specified by quantum mechanics. However, the vacuum tube with the emitted electrons was not an electronic device. The name electronics was given to diode, triode, and pentode vacuum tubes when they were replaced by solid-state devices. Specifically, they became diodes and transistors. In vacuum tubes, electrons were classical corpuscular objects. They were not quantum particle-waves, but pure classical particles. On the other hand, in condensedmatter electrons behave as quantum entities, that is, electrons have the property of particles and, at the same time, of waves. In electronic devices the main role is played by these electrons. Essentially, the performance of electronic devices is achieved by quantum mechanical electrons. The readers may be dubious about this statement. In fact, various efforts have been made to treat quantum mechanical phenomena by means of classical terminology, because quantum mechanics is very often unfamiliar, especially to engineers. For instance, let us take one concept: effective mass. As is known, electrons in the solid state behave not as electrons in vacuum, but as an effective entity, dressing all the specific effects occurring in the solid state. An effective mass is a mass that describes the mass of the electron in the solid state and the concept is very specific to quantum mechanics. The concept of the effective mass, as will be described in more detail in this book, is just what discriminates itself from the mass of a classical particle. The classical

1.3 Electronics

7

mass is the mass that a particle exhibits in a vacuum.’ In contrast, an effective mass is not definite but varies depending on what state the electron of concern is in. In this book, we are interested in the contrast between the classical and quantum mechanical interpretation of various phenomena. A physical constant such as electronic mass is one of these examples. To summarize, the contrast is termed the name of quantum effect. Sometimes, quantum mechanics refers to this effect as quantization. In quantum mechanics many quantities of interest should be quantized eventually, resulting in specific quantum effects. Recently, several good reviews of mesoscopic physical phenomena have appeared. So the readers should refer to these textbooks for more specialized details. Several textbooks and conference proceedings on these topics are listed in References. In this book, specific fundamental aspects of quantum mechanics which are required to understand mesoscopic systems will be described. This provides only a basic understanding, which is necessary for further study specialized topics. In particular, the point about what is the difference between quantum and classical mechanical understanding of the same physical phenomena is stressed. Also, only a small number of important experiments are referred to in this book, which are milestones in the vast area of this rapidly developing field. In the References many books are cited which are more advanced or treat similar topics in a more general, sometimes mathematically high-brow, way. The most expected promise for this field is to enable us to develop new types of devices which could replace the matured and deadlocked large scale integration of silicon technology in the near future. Although nobody can definitely predict the possibility of replacement, such updated silicon devices which are SO much reduced in size, must be brought into the so-called mesoscopic domain and the newcomer will necessarily replace conventional silicon devices just with mesoscopic effects.

References Adriaco Res. Conf. (Ed.) (1991), Proc. of the Adriaco Research Conf. o n Quantum Fluctuations in Mesoscopic and Macroscopic Systems, W e s t e , 3-6 July, 1990, Singapore: World Sci. Al’tshuler, B.L., Lee, P. A., Webb, RA. (Eds.) (1991), Mesoscopic Phenomena in Solids, Amsterdam: North-Holland. Akkermans, E., Montambaux, G., Pichard, J.-L. (Eds.) (1996), Physique Quantique Mesoscopique (Les Houches Summer School Proc., Vo1.61), Amsterdam: NorthHolland. Ando, T., Arakawa, Y .(Eds.) (1998), Mesoscopic Physics and Electronics (Nanoscience and Technology) , Heidelberg: Springer. ‘The electron in a vacuum also behaves as a complex, quantum mechanical entity from the “renormalization” point of view in the field theory of elementary particles. “Renormalized mass” is a concept similar t o the effective mass in the solid state, in the sense that all possible interactions of a bare electron with vacuum (in other words, Dirac’s Fermi sea) contributes significantly to the bare mass eventually t o result in an observed mass in the actual world.

8

1 Introduction

Arai, T., Mihama, K., Yamamoto, K., Sugano, S. (Eds.) (1999), Mesoscopic Materials and Clusters: Questions Emerging from Mesoscopic Cosmos (Fundamental Theories of Physics, Vol. 87) , Heidelberg: Springer. Bariakhtar, V. G., Wigen, P. E. (Eds.) (1998), Frontiers in Magnetism of Reduced Dimension Systems (NATO Asi Series. Partnership Sub-series 3, High Technology, Vo1.49), London: Kluwer Academic. Beenakker, C. W. J., van Houten, H. (1991), Quantum Transport in Semiconductor Nanostructures, in: H. Ehrenreich, D. Turnbull (Eds.), Solid State Physics, Vol. 44, New York: Academic Press. Capasso, F. (Ed.) (1990), Physics of Quantum Electronic Devices, New York: Springer. Cerdeira, H. A., Kramer, B., Schon, G. (Eds.) (1995), Quantum Dynamics of Submicron Strucutres (NATO AS1 Series. Ser. El Applied Sciences, Vol. 291), London: Kluwer Academic. Chamberlain, J. M. et al. (Eds.) (1990), Electronic Properties of Multilayers and LowDimensional Semiconductor Structures, New York: Plenum. Chow, T. S. (2000), Mesoscopic Physics of Complex Materials (Graduate Texts in Contemporary Physics) , Heidelberg: Springer. Datta, S. (1997), Electronic Transport in Mesoscopic Systems, (Cambridge Studies in Semiconductor Physics and Microelectronic Engineering, No. 3), Cambridge: Cambridge University Press. Dinardo, N. J . (1994), Nanoscale Characterization of Surfaces and Interfaces, Berlin: VCH. Ezawa, H., Murayama, Y. (Eds.) (1993), Quantum Control and Measurement, Amsterdam: North-Holland. Ferry, D. K., Goodnick, S. M. (1997), Transport in Nanostructures, Cambridge: Cambridge University Press. Ferry, D. K., Grubin, H. L., Jacoboni, C. (Eds.) (1997), Quantum 'Pransport in Ultrasmall Devices, New York: Plenum. Fujikawa, K., Ono, Y. (Eds.) (1996), Quantum Coherence and Decoherence, Amsterdam: North-Holland. Fukuyama, H., Ando, T. (Eds.) (1993), Transport Phenomena in Mesoscopic Systems: Proc. of the 14th Taniguchi Symposium, Shima, Japan, Nov. 10-14, 1991, Heidelberg: Springer. Garcia, N., Nieto-Vesperinas, M. (Eds.) (1998), Nanoscale Science and Technology (NATO Asi series, Series El Applied Sciences, No. 348), London: Kluwer Academic.

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Giovannella, C., Lambert, C. J . (Eds.) (1998), Lectures on Superconductivity in Networks and Mesoscopic Systems: Pontignano, Italy, Sep. 1997 (AIP Conf. Proc., Vol. 427), Heidelberg: Springer. Gravert, H., Devoret, M. H. (Eds.) (1992), Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostructures, Nato ASI, Series B: Physics, Vol. 294, New York: Plenum. Grinstein, G., Mazenko, G. (Eds.) (1986), Directions in Condensed Matter Physics, Singapore: World Scientific. Imry, Y. (1997), Introduction to Mesoscopic Physics, New York: Oxford University Press. James, T. W. (Ed.) (1985), Characterization and Behavior of Materials with Submicron Dimension, Singapore: World Scientific. Jauho, A.-P., Bezaneva, E. V. (Eds.) (1997), Frontiers in Nanoscale Science of Micron/Submicron Devices (NATO Asi Series El Applied Physics, No. 328), London: Kluwer Academic. Kamefuchi, S., Ezawa, H., Murayama, Y., Namiki, M., Nomura, S., Ohnuki, Y. (Eds.) (1990), Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan. van Kampen, N. G. (1976), in: L. PB1, P. Szkpfalus (Eds.), Statistical Physics, Proc. of Int. Conf., 25-29 August 1975, Budapest, Amsterdam: North-Holland. Kobayashi, S., Ezawa, H., Murayama, Y., Nomura, S. (Eds.) (1990), Proc. 3rd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan. Koch, H., Lubbig, H. (Eds.) (1992), Single Electron Tunneling and Mesoscopic Devices: Proc. of the 14th Int. Conf. SQUID '91 (Session on SET and Mesoscopic Devices), Heidelberg: Springer. Kramer, B. (Ed.) (1991), Quantum Coherence in Mesoscopic Systems, New York: Plenum. Kulik, I. O., Ellialtioglu, R. (Eds) (2000), Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics (NATO Science Series: C Mathematical and Physical Sciences Vol. 559), London: Kluver Academic. De Martino, S., De Siena, S., De Nicola, S. (Eds.) (1997), New Perspectives in the Mesoscopic Systems, Singapore: World Sci. Matel, 0. C. (1999), Mesoscopic Charge Density Wave Wires, Coronet Books. Namba, S., Hamaguchi, C. (Eds.) (1989), Proc. Symp. on New Phenomena in Mesoscopic Structures, Dec. 1989.

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Namiki, M., Ohnuki, Y., Murayama, Y., Nomura, S.(Eds.) (1987), Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan. Ono, Y., Fujikawa, K. (Eds.) (1996), Quantum Coherence and Decoherence, Amsterdam: North-Holland. Ovchinnikov, A. A., Ukrainskii, I. I. (Eds.) (1991), Electron Correlation Eflects in Low-Dimensional Conductors and Superconductors, Berlin: Springer. Reed, M. A., Kirk, W. P. (Eds.) (1989), Proc. Int. Symp. Nanostructure Physics and Fabrication, New York: Academic Press. Reguera, D., Platero, G., Bonilla, L. L., Rubi, J. M. (Eds.) (2000), Statistical and Dynamical Aspects of Mesoscopic Systems: Proc. of the Sitges Conf. on Statistical Mechanics, Sitges, Barcelona, Heidelberg: Springer. Richter, K., Kuhn, J., Muller, T., Steiner, F. (Eds.) (2000), Semiclassical Theory of Mesoscopic Quantum Systems (Springer Tracts in Modern Physics, 161), Heidelberg: Springer. Serikaya, M., Wickramasinghe, H. K. (Eds.) (1994), Determining Nanoscale Physical Properties of Materials by Microscopy and Spectroscopy: Symp. Nov. 29-Dec. 3, 1993, Boston, M A , London: Kluwer Academic. Sohn, L. L., Kouwenhoven, L. P., Schon, G. (Eds.) (1997), Mesoscopic Electron Transport (NATO Asi Series, Series E l Applied Sciences, No. 345), London: Kluwer Academic. Thouless, D. J. (1978), Les Houches, Ecole dJEte' de Physique The'orique, Session X X X I ,Amsterdam: North-Holland. Trigg, G. L. (Ed.) (1991-99), Encyclopedia of Applied Physics, Vol. 1-23,New York: VCH. Tsukada, M., Kobayashi, S., Kurihara, S., Nomura, S. (Eds.) (1993), Proc. 4th Int. Symp. Foundations of Quantum Mechanics, Tokyo: Publ. Office Jpn. J. Appl. Phys. Yamamoto, Y., Imamoglu A. (Eds.) (1999), Mesoscopic Quantum Optics, New York: John- Wiley.

2

Quantum versus Classical Physics

2.1

Quantum Effects

Those effects specific to quantum mechanics are called quantum effects and discerned from classical physical ones. The most striking effect results from the fact that a particle such as an electron has the nature of a wave along with that of a particle. This results in quantum features that are peculiar to the wave nature, such as interference and diffraction of the electron. The concept of the effective mass of an electron in a solid-state as described in Chapter 1 is one example of these quantum effects. Another example is diamagnetism. As shown long before by van Leeuwen (1921), any ensemble of classical electrons does not show diamagnetism in principle. In contrast , according t o Landau, a quantum mechanical electron obeys the so-called Landau quantization under an applied magnetic field, which causes the appearance of diamagnetism. Diamagnetism is one type of orbital magnetism and is another example of the quantum effect (Peierls, 1979). Besides orbital magnetism, the most important magnetic behavior of electrons comes from their spin magnetic moments. An ensemble of electrons which occupy all states below the Fermi level and obey the Fermi-Dirac statistics is known to show Pauli paramagnetism. This is also another example of the quantum effects caused by spins. Spin angular momentum is quite a typical quantum effect, since the angular momentum of a spinning electron should be vanishingly small, when the radius of the electron goes to zero. Nevertheless, in quantum mechanics, it is known that the spin angular momentum of an electron is allowed to have only k i h , where fi is the Planck J s. constant divided by 2 ~ namely, , 1.0545x This Planck constant itself is a quantity that describes quantum effects. In other words, if there is no quantum effect revealed, we do not need the Planck constant. As another example, electric conductivity o (the inverse of the electric resistivity) is known to be given by

ne2r m* where n is the density of the carriers in three dimensions, e the electronic charge, m* the effective mass, and T the relaxation time describing an average of various scattering times which cause dissipation of the electrons’ kinetic energy. This formula does not include f i , but a microscopic theory that derives this is based on h, which incidentally disappears in the final expression. In this sense, electric conductivity is also a quantum g=-

12

2 Quantum versus Classical Physics

effect. In particular, in two dimensions, o includes A explicitly, as will be discussed later. Thus, all phenomena in modern physics are, explicitly or implicitly] related to quantum effects.

2.2

Quantum Fluctuation

Quantum fluctuation is obviously a typical example of quantum effects. In quantum mechanics an electron behaves as a wave, and hence we may confine an electron within an infinitesimally small region, only at the expense of an infinite increase in energy. The electron seems to be forced to have a high energy, since its wavelength X becomes infinitesimally small. A quantum mechanical wave accompanying no mass has an energy corresponding to its wavelength given by

where u is the frequency, w the angular frequency, v a phase velocity of the wave and k = 2~ f X the wavevector. An example of such waves is light wave, or photons in quantum mechanical version. For simplicity we will discuss the one-dimensional state. In such a state, a fluctuation of the position of the wave, in other words, the variance of the position defined by = ((x - (Z))?

(2.3)

is not at zero around a mean value of the position (x), but is specifically related to a variance of the momentum defined by

(W2= ((P - (P)I2)

(2.4)

In both equations we used (. . .) for an average in some sense. According to Heisenberg, h AX * Ap 2 2 and the minimum value A/2 is realized only when the wave is described by a wavepacket profiled with a Gaussian envelope function as V(Z, t ) = N P 1 l 2

AioXk(Z)e-iEkttl* k

x

(:)

1 /4

[-

(Alc)1/2exp

.( - hkot/m)2 AX)^

I

eikox

where A? = exp[-(k - k 0 ) ~ / 4 A l c is ~ ] the Gaussian envelope function, and X k = L-lI2eikx is the wavefunction extending over the region with a span L. N is given , = 1. Any wavepackets are known by the normalization condition: J-", I ~ ( z t)12da: to diffuse (i,e., quantum diffusion) with time, so that the above formula is valid only

13

2.3 Particle- Wave Duality

when (Ak)2ht/m << 1. This wavepacket is a quantum extension of an electron in the classical version and obeys a time-dependent Schrodinger equation

d h2 82 ih--cp(x, t ) = -- --cp(x, at 2m d x 2

t)

of a free electron with mass m , and expresses the dynamic features of the electron as a function of the spatial coordinates. For such a free wavepacket, Ap = hk and Ek = h2k2/2m.Necessarily, Ax = 1/(2Ak). Let us discuss a linear oscillator in terms of classical terminology where it oscillates with a monochromatic angular frequency w . As discussed above, the minimum energy EO of the oscillator can be evaluated for p = hAk and x = Ax. Since the energy of the oscillator is

Eo

=

1 1 - ( A P ) ~+ -mw2(Ax)2 2m 2

-

- [(Ap-

1

2m

WAX)^+ ~ ~ L J A ~ A x ]

1 2

2 -hw the minimum energy, the zero-point energy in quantum mechanics, is i h w . This zeropoint energy is characteristic of quantum mechanics and reveals a quantum fluctuation phenomenon. Thus, a linear oscillator in quantum mechanics has an energy nhw in addition to i h w . This n is called the quantum number and a specific quantization procedure is needed to obtain such a discrete energy spectrum for problems of interest. The basis on which we may derive this n will be discussed later.

2.3

Particle-Wave Duality

As described in the previous section, a wavepacket is conveniently used to express a particle-wave state. There Ax, the extent of the wavepacket, specifies how widely it extends, that is, if Ax -+ 0, it corrresponds to a particle state without any spatial extent. At the other limit, Ax -+ 00, it is a pure monochromatic wave composed only of a single wavelength, or equivalently a single wavevector. Here we should stress that a wavepacket is defined by summing over an appropriate spectrum of wavevectors, all with a fixed phase relationship. The criterion is that this type of summation is related to the concept of coherence that will be further discussed in detail in the next section. Let us add some comments about the state of an electron that is considered to be a wave and a particle simultaneously. In quantum mechanics a state always corresponds to a wavefunction. A wavefunction has the meaning of a probability amplitude of finding the particle distributed over a particular spatial range. On the other hand, a particle is always detected as

14

2 Quantum versus Classical Physics

a single entity, as a whole. This is because no matter ever disappears. Thus, for a wavefunction $(z, t) and a probability P ( z ,t)dz

1, 00

P ( x ,t)dz = l$(z, t)I2dz,

l$(z,t)12dz = 1

(2.9)

always hold irrespective of t. An electron exists dispersed over the extent of the spatial range but is detected just like a single particle within an infinitesimally small region. This curious nature is called duality. The duality always appears at the center of the so-called theory of measurement. The theory concerns how a particle-wave is measured, that is, a particle-wave is detected by a detector as a single particle and never divided into a multiple of fractions. Even if a wave is specifically extended, it only means that the particle can exist either a t one site or another, the probability of the existence of which is given by the squared wavefunction. If the particle is detected, the detected entity is never a wave but a definite particle, with at most a size as large as a classical electron. When an electron hits a fluorescence plate, fluorescence occurs minimally t o the extent of one atom. So, do these two ways of interpretation provide a contradiction? The answer is “No”. In the next section we will interpret how these two are compromised.

2.4

Measurement of an Ensemble of Particles

Figure 2.1 shows how an ensemble of particles are detected on a fluorescence plate inside an electron microscope (Tonomura, 1989). A single beam from a coherent field emitter source of electrons is decomposed by means of a tiny solenoid carrying a current and condensed by an electrostatic potential, both of which work as a biprism for the electron beam, as suggested in Fig.2.1 on the left-hand side. After being decomposed once, two beams are superposed on the detector plate. Then a single electron forms a single spot on the plate and an ensemble eventually builds up a sinusoidally varying interference fringe of dense and lean stripes, as seen in this figure. Each spot corresponds to each electron, and the interference fringe corresponds t o an ensemble of these spots. Each spot is randomly formed, but all spots as a whole are never random. They form a fringe just parallel to the two waves interfering on the surface of a pond, that is, the region with denser spots meets the region with a higher probability amplitude, in other words, a larger wavefunction. On obtaining this kind of interference pattern, it makes no sense to ask where one spot drops. However, quantum mechanics tells us that an ensemble makes a predetermined pattern in a definite way. This is a very curious thing, but it is the case since quantum mechanics can only predict the probability distribution, or a wavefunction. Moreover, it is established deterministically once appropriate initial and boundary conditions are set. In classical optics the same type of interference is well known, which forms a smoothly sinusoidal variation of bright and dark stripes. However, in the optics case too, if precisely observed, a similar pile-up of numerous events can be seen to form smooth fringes (Tsuchiya et al., 1985).

2.4 Measurement of a n Ensemble of Particles

Fig. 2.1: Experimental apparatus of electron interference on the left-hand side. An electron is expressed by a plane wave. On the right-hand side, photographs record each electron detection process. In them about 100 (a), 3000 (b), 20 000 (c), and 70 000 (d) events are recorded. Each event is controlled so that only a single electron flies inside the electron microscope and it is considered that each event occurs independently but in a correlated manner. Such correlation is said to occur when the electron beam has “coherence”. (By courtesy of Dr. A. Tonomura et al., 1989).

15

16

2.5

2 Quantum versus Classical Physics

Coherence

A quantum mechanical electron must be a wave, so that it may interfere just like waves on the surface of water. This fact was first conjectured by de Broglie and was given the name matter-wave. A classical electron has a momentum mv as a particle. Hence, according to de Broglie, this momentum was identified with that of a wave hk = h/X, where h appeared again. This relationship is as important as the relationship between energy and wavelength in Eq. (2.2), which assumes a massless matter-wave, On the other hand, according to Einstein’s theory of relativity, the energy of a massive particle is E = cp = m c d m M mc2 + mv2/2 for v << c. This relation tells that mc2 is also a (mass-)energy. A classical electron cannot interfere, as has been intuitively conceived. The reason is that it is a particle. A particle has only an infinitesimally small size; therefore, it is formidably difficult t o superpose two particles. Here we used a terminology “to superpose”. This term is different from “to mix” or “to add”. In quantum mechanics it is usual t o use “to superpose’’ to mean the addition of two amplitudes of waves. As is known, two classical waves are constructive when they have the same phase and destructive when they are out of phase with each other. This situation is the same in quantum mechanics. In this case the amplitude of the wave is expressed in terms of wavefunction. Recently, in addition to electrons and neutrons, heavier and more complicated matter such as atoms and ions are proved to interfere in actual experiments, although at very low temperatures. Superposed waves induce such phenomena as interference and diffraction of matterwaves in quantum mechanics. Just as an X-ray is diffracted by a crystal, electron beams are also diffracted. This is the electron diffraction phenomenon. If two beams have a constructive optical path length, they acquire a stronger amplitude. This type of condition of diffraction was proposed by Bragg and Bragg (father and his son), and is called the Bragg condition. Let us consider a crystal with an array of atoms separated with a lattice constant d l between neighboring layers and dll between the atoms within the same layer, as shown in Fig.2.2. Each atom can scatter an electron. Each beam has a wavevector parallel to its path, so that it is easy to calculate geometrically the difference between the path lengths of a couple of scattered electron beams as suggested in the figure. There are two cases. Let us call the first one (a) the Bragg case and the second (b) the Laue case. The Bragg condition for the Bragg case (the same as in Newton ring) is

2 d l sin8 = nX

(2.10)

where n is an integer and X the wavelength of the electron beam used in the experiment. Meanwhile for the Laue case (the same as in a grating), (2.11) 2dll)sin8 - sin@’(= nX where 8 and 8’ are, respectively, the incidence and diffraction angle. When one of these conditions holds, we refer to the diffraction as being the n-th. In the Bragg geometry, when we scan the incidence angle 8, several peaks of diffracted beams appear depending on n, all of which meet the above condition with

2.5 Coherence

17

(a) Bragg Case

nA=2d I s i n 8 - s i n 8 '

II

II

I

(b) Laue Case

Fig.2.2: Interpretation of the reason why diffraction occurs. From the geometrical difference of the path lengths of the two beams, two cases are differentiated. One is for the Bragg case (a) and the other for the Laue case (b). The Bragg conditions for both cases are given by Eqs. (2.10) and (2.11), respectively. with fixed d l and A. For an angle which satisfies the Bragg condition the diffracted beam is enhanced much more than the simple specularly reflected beam in the same direction. In contrast to the Bragg case, the Laue diffraction is easily detected when we scan 8' for fixed 8, A, and dil, that is, around the 0-th order specular reflection several spots appear for which the beam satisfies the above-mentioned condition. The Bragg condition plays a key role in the electron diffraction experiment as well as solid-state physics in general. Any textbook of quantum mechanics will explain that for a couple of wavefunctions $1 (2,t ) and $2(x, t ) there exists a superposed wavefunction $(x, t ) such as

$(Tt ) = $1

(2,t ) f $ 2 ( 2 ,

t)

(2.12)

In this equation the +-sign is only a mathematical symbol. To make sense physically, there should exist some actual physical device to make both waves superposed, say a superposer. Even if two waves happen t o encounter each other in a vacuum, nothing occurs (Murayama, 1990).

2 Quantum versus Classical Physics

18

se shifter

lodetectors

'H

Fig. 2.3: Illustration of a silicon interferometer for a neutron beam. Three atomically parallel plates are carved out of a single crystalline silicon block on the order of macroscopic size. The first plate acts as a decomposer and the second one as a mirror reflector, and the last one as a superposer of two incident beams. For example, in neutron interferometry, a silicon interferometer is usually used to superpose neutron beams. Figure 2.3 shows this interferometer (Rauch et al., 1974). Todays thanks to highly developed silicon technology, we can obtain a large single crystalline silicon block on the order of several tens of centimeter in the cross section. If carved out of the bock as shown in the figure, three plates of Si have their arrangement atomically regulated. Thus, as suggested in Fig. 2.3, a neutron beam satisfying the Bragg condition on one plate always satisfies the same condition on the other two plates. This is necessary for the neutron beams to be superposed. In the figure, decomposition of a single beam is also shown. On the first plate it is decomposed into two and on the second reflected specularly and on the final stage two beams are superposed. All of these physical processes must satisfy the same Bragg condition] otherwise, either division or superposition does not occur. In Fig.2.4 is shown a numerically simulated neutron beam superposition (Murayama, 1990al 1990b, 1990cl 1990d). Figure 2.4 shows simulated results of a wavepacket flying through a periodic potential region which causes the wavepacket to be decomposed when the Bragg condition is fulfilled. The time-dependent Schrodinger equation was solved by the method described in Appendix A. On the later half figure are shown that both decomposed wavepackets are again superposed in a superposer which has the same Bragg condition

19

2.6 Visibility

as the decomposer. The potential region mimicking a Si lattice has a one-dimensional array of 33 quasi-Gaussian potentials. The initial wavepacket is assumed to exist in a potential-free region and then to start to proceed to the potential region (denoted as 33 x 3 A). The time elapsed is denoted in units of ps. As already simulated by Schiff (1955), it is apparent that a part of the wavepacket is trapped within the potential region and reappears outside the region with a time delay. The fact that a pair of wavepackets do not cross each other without being superposed but are actually superposed is proved when we observe that the outgoing intensities of wavepackets are certainly dependent on the ad hoc given phase shift between the both. The second example of a superposer is the fluorescence plate for an electron beam. When the electron beam hits the fluorescence plate and excites a fluorescent atom from its ground to a higher state and then the excited electron is de-excited, fluorescence results. During this process two beams are superposed and simultaneously detected as a single electron. Here superposition and detection occur as a combined single event. The third example is the cavity wall for a light (electromagnetic) wave. When the cavity wall reflects light waves, it imposes a boundary condition on the wave field. If two light waves drop on the same position on the wall and reflected at the same time, the wall works as a reflector as well as a superposer, when an appropriate condition is satisfied for the two waves. For instance, a forth-propagating wave eik(x-ct)and a backward reflected wave e-ik(x+ct)are superposed to give a standing wave expressed by 2ieVikct sin kx which vanishes at the wall x = 0. The same situation holds for an electron too. This point will be interpreted in detail in the next chapter. The other vitally important condition for the beam to be superposed is whether two beams are coherent or not. If coherent, the two can be superposed, otherwise, i.e., incoherent, they can not. Now it must be clear that some purely physical process or device is needed for a matter-wave to be superposed. Once this condition is fulfilled, superposition is physically expressed with a +-sign as in Eq. (2.12). In reverse, if two beams are superposable, they are coherent; if they are not superposable, they are incoherent. $2, it makes physical Although it is always possible to mathematically denote as $1 sense only when the two wavefunctions are coherent. Otherwise, the notation makes no sense.

+

2.6

Visibility

When performing an experiment using the silicon interferometer as shown in Fig. 2.3, it is usual to insert a phase shifter on one branch of the separate beams between two plates. A phase shifter gives the beam an additional phase equal to $ I I = $re i 2 a ( 1 - n ) D / A (2.13)

20

2 Quantum versus Classical Physics

system size : 1000 x 0.75 8,

Bragg c o n d i t i o n :

( A/2) s i n (W4) =3 A No. o f Gaussian p o t e n t i a l s :

33 w i d t h o f wavepacket: 25 A w i d t h o f Gaussian p r o f i l e :

0.4 A p o t e n t i a I he i ght : V, k i n e t i c energy: E,, V,/E

kO

0.1

2.6 Visibility

Fig. 2.4: Numerically simulated dynamics results of a wavepacket as being representative of a neutron beam. When a stack of silicon atomic planes produces a periodic potential t o the neutron beam and the Bragg condition is satisfied, a single beam is decomposed into two and the two are reflected on the middle plate (in the same manner as the decomposer performs), and eventually superposed. Whether two beams are physically superposed is known from whether the superposed beam intensity varies or not when the phase of one of the beams is modulated.

21

2 Quantum versus Classical Physics

22

where D is the thickness of the phase shifter made of an appropriate material transparent to neutrons and n is the index of refraction of the material. If we rotate the shifter around the axis perpendicular t o the beam path, then the path length through the shifter varies. Thus by letting the phase vary, the outlet of the neutron beam intensity superposed at the final plate is measured. Figure 2.5 shows how the neutron beam intensity in the direction 0 (in the same direction as the incident beam) varies as a function of the added phase shift in contrast to that in H (in the direction out of the incident beam). The sum of both intensities is almost constant, but the intensity in one direction oscillates to give an interference fringe as a function of the phase shift. In such an experiment, we can define “visibility”, which is the ratio of half the to the average intensity, that is, maximum I,,, minus minimum Imin Visibility =

Imax

- Imin

Imax

+ Imin

(2.14)

As is obvious, when there is no contrast between maximum and minimum, visibility is 0, whereas when the minimum value is a zero, it is 100%. As seen in Fig. 2.5, there is ordinarily a constant background in the interference fringe pattern. This background comes from the incoherent part of the neutron interference experiment. In an ideal case, visibility must be loo%, but in fact any neutron source has a fraction of incoherent neutrons, which has a very short Ax in the wavepacket representation. There are usually some mechanisms to cause coherent neutrons t o degrade to incoherent ones, when they pass through the silicon plates as well as the phase shifter. This degradation is caused by some type of scattering inside as well as outside the silicon. These scatterers may be imperfections in the silicon single crystal, impurities, phonons, and other quasiparticle excitations. This point will be described in more detail in later chapters. In the double-slit Young electron interferometer, the interference fringe is ordinarily expressed for a couple of waves $1 = $,geikrl and $2 = q!J,geikrz as

which behaves as proportional to cos(kyd/D) in terms of the separation of the double slits d and the distance between the slit and screen D (Fig. 2.6). In realistic cases, a single beam is not decomposed into a couple of beams of equal intensities, so that the above expression must be modified to

Visibility = 2141 I . 1421

l41I2+ l42I2

We should notice that the visibility becomes very small when either 1411 >>

1411 << 1421.

(2.17) or

2.6 Visibility

23

v)

0 v)

5

a

3000

0 0

3

\

2000

cn + c

3

8c

1000

0 L

0

200

400

600

800

AD / pm Fig. 2.5: Plots of neutron beam intensities in the 0- and H-directions. The horizontal axis gives a difference in path lengths 6D given by rotating the phase shifter. Phase shift is proportional to A D (after Rauch, 1984).

Fig. 2.6: Illustration of the Young-type double-slit interferometer for light waves. For D >> y, d, the interference fringe on the screen is proportional to cos(kdy/D).

2 Quantum versus Classical Physics

24

Figure 2.6 is an optical interferometer after Young and an equivalent of Fig. 2.1 for electron beams and Fig. 2.3 for neutron beams. The probability for photons to drop on a screen after passing through a couple of slits is given by the squared superposed wavefunction of the photon field. That is, each photon field propagates from the corresponding slit and both are superposed, as suggested in Eq. (2.16). This is a probabilistic process and is the case if and only if we do not know “which path” each photon took. Otherwise, that is, if we detected somehow and knew the path each photon passed (the detector is assumed to be transparent), then the photon field spreads only out of the single slit which the photon deterministically took. In the latter case, about half of photons, for instance, pass the upper slit, and the remainder photons pass the lower slit. Thus, each family of photons individually produces a single peak distribution on the screen around the slit they have passed and there occurs no longer interference fringes. When one photon may pass either one slit or the other probabilistically and we do not know the path it took deterministically, the probability must be given in such a manner that the probability amplitude, or wavefunction, must be superposed (added) t o result in a total amplitude. This superposition may be allowed when both beams are coherent, in other words, when both beams have intrinsically the same feature in their energies, whatever phase difference they have. Their phases are determined by their entire path lengths. Next, let us consider the detecting process in more detail. There should be two different cases.

(A) A process which makes a coherent state incoherent. In this case some inelastic process must be included in the detector. The two outgoing fields are no longer equienergetic and, accordingly, the plus-sign between the upper and lower fields loses its physical meaning. When the degree of inelasticity is merely slight, coherence may partially remain (“partial coherence”).

(B) A process t o determine the path causes incoherence. This may be expressed as follows.

This process resides always in the central part of the theory of measurement. Here the +-sign is not physically defined, but only means an acausal and irreversible process. No Schrodinger equation can describe this process at all. The meaning is that the prediction with some probability became an actually occurred event, or history, by detecting which path one beam traced. Such process is no longer probabilistic, so that we can say nothing about the probability of interference.

2.7

Coherence versus Incoherence

Finally we reach the stage of how to understand the mesoscopic world from the viewpoint of a quantum mechanical concept.

2.7 Coherence versus Incoherence

25

We have previously noted that visibility varies from 100% to 0 and the case with visibility 0 corresponds to the classical physical world, since there particles are not waves so that they do not interfere. Thus, visibility is one of the keywords t o differentiate the macroscopic from the microscopic world. However, visibility itself gives only the necessary conditions but this is not sufficient for the system to be microscopic, quantum mechanical. Even if visibility is vanishing, we must be careful about saying that the system is incoherently classical, because if all the microscopic events occur randomly, then visibility become zero incidentally. For instance in the experiment shown in Fig. 2.1, this type of interference fringe can be obtained only for a coherent source of electrons; in other words, if the electrons arriving are in randomly varying states, then each event gives one spot in randomly varying fringes, which causes the fringe to be washed out. In such a case, measurement will give us misleading information that the process is incoherent, although each event itself is a microscopically pure quantum mechanical, coherent one. Thus, the electron source is required to be such that any electrons emitted from the source must be stable and coherent, that is, they must be in the same (or very similar to each other at least) well-defined quantum condition. For the electron source having this condition, people in the electron microscope research field say that the electron source is coherent. The scientists in optics field say also that the laser (light amplification by stimulated emission of radiation) is a coherent light source. It is known that, if all photons in an optical system have the same energy and their phase are the same according to the stimulated emission process, then they behave as a macroscopically quantum coherent system. This fact is enabled because photons are bosonic particles and can occupy the unique state with the same energy and phase. Macroscopic quantum coherence will be discussed in more detail in another chapter. Next, we ask what can bring coherence into incoherence? When using a silicon interferometer, we can observe coherent superposition only if both beams have the same energy, since the superposer must fulfill the same Bragg condition for both waves. This means that, once the energy of one of the two beams changes t o another, then the Bragg condition does not hold, even if the other beam fulfills the condition. At least for this superposer it seems necessary t o require both beams to have the same energy in order to make superposition possible. In other words, if an inelastic scattering occurs inside the phase shifter, the incoherent part in the visibility will increase. Here, inelastic scattering means such scattering as the energy of the scattered particle changes between before and after scattering in contrast to elastic scattering where the energy is conserved. Let us simply assume that an equienergy condition is necessary for coherence to be kept.l In summary, any particles in classical physics are incoherent and any matter-waves in quantum mechanics, if incoherent, behaves almost as particles. In contrast, in quantum mechanics, if any states are coherent, they are superposable. If there occurs 'This point needs more investigation, since we cannot say anything about interference between beams with different energies. We can only say that at this moment we do not have such a superposer that makes neutron beams with different energies interfere.

26

2 Quantum versus Classical Physics

any inelastic scattering, the state looses coherence and a couple of waves once decomposed can never be superposed again, since both have different energies. This type of degradation process in coherence is called decoherence or dephasing. Thus it is safe to say that transition from quantum mechanics to classical physics is described in terms of decohering processes. The world where decoherence enters into a microscopic system and degrades it to an incoherent macroscopic world is called mesoscopic.

References Leeuwen, van J. H. (1921), J. de Physique 2, 361. Murayama, Y. (1990), Study of Quantum Measurement Processes by Simulation, Dissertation: Waseda University. Murayama, Y. (1990a), Phys. Lett. A140, 469. Murayama, Y. (1990b), Phys. Lett. A140, 329. Murayama, Y. (199Oc), Phys. Lett. A140, 334. Murayama, Y. (1990d), Found. Phys. Lett. 3,103. Peierls, R. (1979), Surprises in Theoretical Physics, Princeton, N J : Princeton University Press. Rauch, H. (1984), in: S. Kamefuchi, H. Ezawa, Y. Murayama, M. Namiki, S. Nomura, Y. Ohnuki (Eds.), Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan. Rauch, H., Treimer, W., Bonse, U. (1974), Physics Lett. 47A, 369. Schiff, L. I. (1955), Quantum Mechanics, McGraw-Hill, New York, Chapter 5. Tonomua, A . , Endo, J., Matsuda, T., Kawasaki, T. (1989), Am. J . Phys. 57, 117. Tsuchiya, Y., Inuzuka, E., Kurono, T., Hosoda, M. (1985), in: B.L.Morgan (Ed.), Advances in Electronics and Electron Physics, Vol. 64A, New York: Academic Press, p.21.

3

Quantization

We have been accustomed to classical physics, at least until the advent of quantum mechanics. Classical physics was based on Newton’s laws of motion and mathematically expressed by differential equations which track, for instance, the motion of an electron. The electron was an infinitesimally small particle in this case. Since it is a particle, its motion is specified only by the initial conditions, not by the boundary conditions. In the spatial domain it may not touch a boundary, even if it were to exist near one boundary.

3.1

Schrodinger Equation and Discrete Energies

In quantum mechanics any equation must be modified to meet quantum mechanical requirements, that is, an electron becomes a wave (a particle-wave), so that the state must be specified by appropriate boundary conditions. Thus newly developed equation is the Schrodinger equation. The Schrodinger equation itself is, for example in one dimension, represented as

where V ( x )is a potential that the wave exists in; H is called the Hamiltonian of this problem and $(x) is the wavefunction or the state of the particle-wave. So far, the energy E has not been determined, i.e., it may be any quantity. As was described above, the wave must be subject to boundary conditions, which determine E . Such an equation is called an eigenequation, where “eigen” means “specific to” or “its own” in German. Thus, the solved wavefunction is an eigenstate and correspondingly the solved energy is eigenenergy. In matrix terminology, the eigenvector and eagenvalue are often used corresponding to, respectively, eigenstate and eigenenergy. Let us elucidate how the boundary conditions determine eigenenergies for the onedimensional Schrodinger equation. A differential operator may be approximated by a difference, this method being called the FDM (finite difference method). For example, a continuous variable x is discretized to be a series of equally spaced xi such that xi = z ~ + i A(i = 0,1,2, ...,N ; A is a constant), then

3 Quantztation

28

Then, it is easy to derive a set of simultaneous equations defined by xi:

a($N

- 21CIN-1

+ ‘$N-2) + V N - l $ N - 1

(3.3) = E$N-I

where Vi = V ( z i )$i, = $(xi) and a = - A 2 / 2 r n A 2 . Boundary conditions are important here. Let us assume the wavefunction vanishes at the furthermost left as well as furthermost right coordinate, i.e., $0 = ‘$N = 0. Then, these equations are neatly represented in a matrix form as -2a

+ VI a 0 0 0

0 a

a

-2a

+ VZ

a 0 0

-2a

+ V3

a 0

0 ... 0 ... a 0 ... -2 a + V4 a ... 0 0

..

0

0 0 0 0

.. 0 ... - 2 a + V N - l

This FDM matrix equation is very similar to Heisenberg’s matrix dynamics. As is known, the Schrodinger equation has been mathematically proved to be equivalent to the matrix eigenvalue problem formulated by Heisenberg. We are often interested in other boundary conditions. Please refer to Appendix A for further details. In Heisenberg matrix dynamics, the eigenvalue problem reads ordinarily as

(Hij)($j) = W $ i ) (3.5) where the matrix elements are Hij = ($ilHl$j) $*(z)H$(x)dx and ( H i j ) is a matrix whose elements are given by Hij; ( + i ) is a columnar vector with components $i. In this method, a problem occurs such that, even if we want to calculate matrix elements, we do not know the eigenvectors $i in advance. Accordingly we temporarily utilize any orthonormal sets of eigenvectors defined in a Hilbert space. The closer the orthonormal sets are the eigenstates to be solved, the simpler the matrix becomes, i.e., more elements are zeros. The transition from a classical to a quantum equation is sometimes called “quantization”. Once an equation is quantized, it can treat a particle-wave. If we are treating a matrix eigenequation, then it is fairly easy to understand that the eigenvalues should be given discretely, in particular when the dimensions of the matrix are finite and small. The eigenenergies thus obtained should be discrete, not continuous.

szy

29

3.2 Bloch Theorem

If we want to describe an electron correctly, we must solve the Schrodinger equation under appropriate boundary conditions. What state the electron is in, is certainly determined by these boundary conditions and, hence, the conditions are vitally import ant. For example, the energy of the linear oscillator satisfying appropriate boundary conditions, that is, the wavefunction vanishes at extreme positive and negative endpoints, is given by an integral multiple of some frequency w (plus the zero-point energy). Discrete energies result from such boundary conditions. Whenever we question what the state of an electron is, we must necessarily be concerned about what the boundary conditions are. In the next chapter, we ask what dimensions the electron is in, or whether it is in an infinitely extended space or in a confined geometry. In mesoscopic systems, mostly confined within a small region, some boundary conditions make electronic states very different from those of Bloch electrons extending, limitlessly but periodically, over three dimensions. In reality the wavefunction is subject to appropriate boundary conditions. Sometimes the boundary conditions force the wavefunction to be localized within a small spatial region, and sometimes they let the wavefunction be extended widely.

3.2 Bloch Theorem The Bloch theorem has the central importance in condensed matter physics. So that we list in References a t the end several standard books covering the theorem as well as various basic topics related to this theorem in general. An electron residing in a completely regular periodic lattice was investigated by Bloch, and is known as the Bloch electron or Bloch state. The solution in one dimension is known to be

(3.6)

$k(2) = eik’suk(2)

where uk(2)is a periodic function with a period a equal to the lattice spacing. From now on we will restrict ourselves to a 1D case, for simplicity. The Bloch condition may be specified in another way. Using u(x + T) = u(x), where T is any translation operator, i.e., a shift of the coordinate by an integral multiple of a, it is easy to show that

$(x

+ T)

+

= eik(z+T)uk(x T) -

eikx i k T

e uk(x) - eikT?+!l,k(x)

(3.7)

Although x in the function is a continuous coordinate variable, this function was originally defined only on each lattice point with its periodicity and extended into the region between the neighboring lattice points. The extension is justified if we assume that the state in between also has the same periodicity. hk is the “crystal momentum” and k is a wavevector that is a “good quantum number”, when the lattice is completely regular. k = 27r/X runs only from -7r/a to r / a , since half wavelengths shorter than

3 Quantization

30

a is meaningless; that is, to some extent, equivalent to a discretized version of the wavefunction as in the FDM previously described. In a slightly irregular periodic lattice, e.g., a lattice with a single impurity, the eigenstate @(x)is a superposed state of Bloch functions, which is given approximately by the following equation (Appendix B): {E(-iV)

+ AV(z)} @(x)= E @ ( z )

(3.8)

where AV(x) is the potential difference between the lattice with an impurity and the regular lattice. E(-iV) is an operator E k of the form with k replaced by -iV. This Schrodinger-like equation is called effective-mass equation in the effective-mass approximation, since ordinarily E k has the effective form h2k2/2m*in the lowest order (Luttinger et al., 1955). This @(x)is a type of wavepacket composed of Bloch functions. In this respect it is often called the envelope function t o Bloch functions. The concept of eflective mass will be described in the following in more detail.

3.3

Effective Mass

According to the Bloch theorem, the eigenenergy in a regular lattice is given by E k as a function of the wavevector k , since k is a good quantum number. This energy is called the band energy. Since a collection of electrons, one electron per atom and N atoms as a whole, constitutes a total of N - ( k , Ek) points, they appear as a continuous spectrum for N >> 1. As is obvious, the minimum length scale in the lattice is a , so that no half wavelengths smaller than a have any meaning. In terms of the wavevector, k is defined as modulo K = 2n/a =(the reciprocal lattice vector). k - k' = K is known to satisfy the Bragg condition. In other words, - r / a is equivalent to r / a in one dimension. Accordingly, the basic k space is constructed only between - r / a and r / a , which is called the first Brillouin zone with K / 2 = f k o 3 f r l a . This fact means that the Bloch wave is reflected a t the Brillouin zone boundary. For K / 2 = f k o the two equivalent waves must be added to obtain a superposed wave (the origin of the x-coordinate is taken at an atomic position):

From symmetry considerations, both waves are such that they have nodes and antincorresponds to the occuodes just a t the atomic positions, respectively. Since /$~(z)1~ pation probability of an electron at x, $+ oc cos(kox) has antinodes (with maximum amplitude) at the atomic positions, whereas $- c( sin(k0z) has nodes (with no amplitude) at the same positions also. Because between an electron and a positive ion (each atomic position is assumed to have a positive ion) there is effectively a negative Coulomb energy, obviously the $+ is the state with a lower energy and the $--state must have a higher energy with a finite difference from the former energy. This energy difference is called band gap.

3.3 Effective Mass

31

Rk 0

xh

n -

0

T/2

T

3T/2

2T

5T/2

Fig.3.1: Calculated t-dependences of (a) hk, (b) E k , (c) v g , (d) m a , and (e) the coordinate in real space z, based on the model Ek-spectrum given by Eq. (3.10). Group velocity 2rg is defined by dEk/hdk and z = S,’v,[k(t’)]dt’. T is the period of the Bloch oscillation. Finally, we may model the band energy to be given by (3.10) where W is the band width. Here we define an “effective mas” m* by

1 - 1 d2Ek - - _-

m*

?i2 dk2

(3.11)

then for k 2 0, m*(O)= 2h2/Wa2and m * ( k o )= -2h2/Wa2. A positive effective mass describes an electron, whereas a negative effective mass describes a hole. Thus, the electron as an elementary particle behaves sometimes as an electron with a different mass or sometimes as a positive hole in a regular lattice, which is also an example of quantum effects. Of particular interest is that at both inflection points, i.e., when k = k o / 2 in this model, the effective mass becomes infinite. Let an electron start from k = 0 in the direction of positive k. The closer it approaches the inflection point, the larger and larger the effective mass and suddenly it becomes a hole with an infinite negative

32

3 Quantization

mass. Then the hole decreases its mass, which reaches a finite value at the Brillouin zone boundary. At the same time the hole disappears a t k = ko and reappears at the opposite k = -ko, and then the hole mass increases its magnitude to eventually reach -m. Again at the inflection point, the hole is converted into an electron with a positive infinite effective mass and continues to decrease its mass down to the initial value again. The process repeats. This type of oscillation in the Brillouin zone occurs, theoretically at least, when the lattice contains only a single Bloch electron and is accelerated by an electric field E. This is the so-called Bloch oscillation (Fig. 3.1) with a period T in time. It should be noted here that the acceleration is always constant under a constant electric field, following e E = hk, so that k increases unidirectionally with time t , but in real space the electron behaves oscillatorily as shown in Fig.3.1, since a negative velocity appears when it is converted into a hole. In this respect, the electron oscillates in real space. In an actual solid state, electrons very often suffer inelastic scatterings before they are accelerated sufficiently by the electric field and reach the Brillouin zone boundary. This decoherence mechanism suppresses the Bloch oscillation and at the same time decoherence brings electrons into the nature of a classical particle, not a particle-wave state. In the following list of references, several standard textbooks on condensed-matter physics are included, although they are not referred to in the text.

References Cohen, M. L., Chelikowsky, J. R. (1989), Electronic Structure and Optical Properties of Semiconductors, 2nd ed., New York: Springer. Falikov, L. M. (1973), Electrons in Crystalline Solids, Vienna: IAEA. Harrison, W. A. (1980), Electronic Structure and the Properties of Solids, New York: Freeman. Luttinger, J. M., Kohn, W. (1955), Phys. Rev. 97, 869. Seeger, K. (1989), Semiconductor Physics: an Introduction, 4th ed. , Heidelberg: Springe Seitz, F., Turnbull, D., Ehrenreich, H., Saepen, F. (Eds.) (1955-2000), V01.l-55, New York: Academic Press. Ziman, J. M. (1972), Principles of the Theory of Solids, Cambridge: Cambridge University Press.

4

Dimensionality

Dimensionality is the dimensions of a device. The most commonly known is a threedimensional bulk system. Very thin layers obey the two-dimensional rule and thin wires must demonstrate one-dimensional world. Obviously zero-dimensional is an ultra small particle or an atomic cluster. In order for the system of concern to have a specific dimensionality, it must be subject to appropriate boundary conditions. The conditions eventually cause a specific quantization scheme. Thus, how coherently the device behaves is strongly related to its quantized states, and, hence, its dimensionality. Corresponding to quantized states, there is an important quantity called the “density of states” (DOS). DOS is also closely connected with the dimensionality. In this sense, we should first consider in what form DOS is given in each dimension.

4.1 DOS The DOS is the number of discrete eigenenergy levels within a unit energy width. Ordinarily when a larger volume is considered, more energy levels are included in the volume, so that this number is an extensive quantity. To let the DOS be sizeindependent, the DOS per unit volume must be concerned. The unit of DOS is hence defined per volume as in (states/eV/m3). As was previously described, the wavevector k is an enumerative number in regular lattices, which plays the role of a quantum number. It is called very often the good quantum number. We may define the DOS p ( E ) by

when the quantized energy is given by Ek. 6 ( E ) is the Dirac’s delta function. The factor 2 in front of the summation is according to the spin degeneracy. For mag) be differentiated, corresponding to the spin netic materials, p t ( E ) and ~ J ( Emust orientation (t,4). If we wish to know some physical quantity, say, an operator 0 of electronic origin based on quantum mechanical and statistical theory, it is sufficient to calculate

4

34

Dimensionality

where f(&) is the Fermi-Dirac statistical function for the electron with an eigenenergy Ek. Now, let us rewrite this summation in a more tractable form. For this purpose we need to reason in the following way. According to the Bloch theorem, each electronic eigenstate corresponds to an inis fulfilled for the maximum teger n = -[N/2] 1, ...,[N/2]for which k = 2k,,,n/N = n / u . N u is the size of the lattice of conBrillouin zone boundary value k,, cern. In an actual calculation n is rearranged so that it runs from 0 to N and then An = (Nu/2n)dk means that An states exist within the width of dk weighted with a factor Nu/2n. Ordinarily N u is taken to be the size L in one dimension (1D). Thus, obviously x k can be replaced by an integral: (L/2n)Jdk. In non-magnetic cases, this integral should be multiplied by 2. Finally, in a three-dimensional non-magnetic case, the above equation is rewritten as

+

(4.3) Here V = L3 in three dimensions (3D). A DOS again appeared and it is easy to show that the above equation really gives Eq. (4.2) when the definition of DOS in Eq. (4.1) is employed. After simple mathematics, it is shown that 71

1

(4.4) Generally speaking, it is not always possible to take the right-hand side as a function only of El since it still depends on the direction of the k-vector. If this is not the case, the DOS is given in the more sophisticated form, (4.5) with the areal integral over an equienergy surface E = E k . We can calculate explicit forms of p(E) assuming that E k is dependent only on the magnitude k (cubic symmetry). Then

, V =volume p2D(E) = ;k

A

(d3-1 -

, A=area

are obtained. For the parabolic energy spectrum of a quasi-free particle E k = h2k2 f 2m*,

(4.10) (4.11)

35

4.1 DOS

The most peculiar features seen in these DOS values are that the DOS is constant in two dimensions (2D), and proportional t o the fiin three dimensions and t o inverse in one dimension. From these facts, it is reasoned that the states around the of bottom of the band play a more important role in one dimension than in two and three dimensions. In three dimensions, the higher energy states predominantly contribute to determining the averaged physical quantity. Obviously the DOS of a zero-dimensional system (OD) is straightforwardly written as

a

(4.12) n

where En is the discrete eigenenergy of the system, similar to the line spectrum observed in atoms. Needless to say we live in three dimensions, except for the fourth-temporal dimension according to Einstein, so that, if we discuss a two-dimensional system, the other dimension is ordinarily the “confined” degree of freedom. The details will be discussed in the next chapter, but here we will briefly interpret how the three dimensions are broken, for instance, into two and one. In order to actualize a two-dimensional system, we often utilize a solid-state thin plate with a vacuum (or at least an atmosphere) on both sides. This means the potential energy for electrons is high outside the plate and low inside. The potential may confine electrons firmly inside the system that is two-dimensional. Electrons can nearly move almost freely within the plane, but not across the thickness of the plane, on both sides of which a new boundary condition is imposed. This boundary condition causes the energy to be ‘‘size quantized” and confines electrons into the so-called “confined states”. For example, a plate that is thin in the z-direction has discrete energy levels En, specified by a quantum number nz due to confinement in the z-direction and Bloch energies in the zy-plane. Similarly, a thin wire will have discrete energy levels, such as En,,ny,corresponding t o the confined degrees of freedom in the x- and y-direction, resulting in quantum numbers nz and nu, respectively. Thus, the DOS in 2D and 1D should be written more generally as

(4.13) (4.14) where O(E - E,) is the step function, which has a value 1 only for E 2 Ei and 0 for E < E,. It is useful t o derive the expressions for the DOS values in terms of the Fermi energy and density of electrons in the free electron gas model. Since the total number of electrons N is given by 2 C k -+ 1 at zero temperature, it is easy to establish that n 3 = ~ k:/3n2, n 2 = ~ kf2/21r, and nlD = kf/n for three, two, and one dimension, respectively. Here kf is the Fermi wavevector defined by Ef = A2kf2/2m*. From

4

36

Dimensionality

Eqs. (4.6-4.8), we may write (4.15) (4.16) (4.17) where we interestingly see that the factor decreases from 312 for three dimensions to 112 for one dimension as 312 + 1 + 112. Schematic representations of the DOS values in three, two and one dimension are given in Fig. 4.1.

4.2

Dimensionality of a Landau Electron

We will next consider an electron under a magnetic field B . In this geometry, a vector potential A is given by (-By, 0,O) because rotA = B . Such an electron is called the Landau electron, since Lev Landau first studied how an electronic state is quantized when it is in a magnetic field. Let a magnetic field be applied in the z-direction: then the effective mass equation for the Landau electron reads. h2 d 1 , d2 - - - - $ ( T ) = E$(r) (4.18) 2m* [(- i-dx - -y) l2 dy2 dz2 d2

1

for the simplest parabolic energy spectrum. 1 = is the magnetic length, or, classically, the cyclotron orbit radius. According to the Bloch theorem t , and t , are still good quantum numbers as is obvious from this Hamiltonian. Thus, the wavefunction should have the form: ei(kzr+kzz)X(y).Consequently, the equation to solve is

ENXN(Y), Y = l2kZ

(4.19)

This is the same as that for the one-dimensional linear oscillator with its origin at y = Y, whose eigenvalues and eigenstates are well known and given by the following: 1 + -), 2

E(Y, k,, N ) = - EN 2m* (w, = eB/m*:cyclotron angular frequency) and

EN = b , ( N

+

(4.20)

in terms of the Hermite polynomial H N . ~(Erdklyi et al., 1953-55; Moriguchi et al., 1956-60; Landau et al., 1958). This eigenstate is "Landau quantized", whose quantum 'The definition of the polynomial is as follows: Ifn(<) = (-)net2d"(e-t2)/d
4.2 Dimensionality of a Landau Electron

37

numbers are k , = Y/12, k , and N . In particular N is called the Landau quantum number. The first point to note is that the motion of Bloch electrons is not influenced by the magnetic field in the same z-direction as the magnetic field. The second feature is that, for Bloch states with no magnetic field, the eigenenergy depends on lc,. However, for Landau electrons the eigenenergy is degenerate for all k , = Y/12with Y standing for the center position of the linear oscillator. This degeneracy appears to be reasonable, since the eigenenergy does not depend on the position of the oscillator and the eigenstate is determined only by the relative coordinate y - Y with an arbitrary value of Y. Thus, applying a magnetic field suppresses the degrees of freedom of the Bloch electrons in the plane perpendicular to the field. The only remaining degree of freedom is in the direction parallel to the field, just as in a thin wire. Likewise, Bloch electrons in two dimensions are Landau quantized (with a magnetic field perpendicular to the plane) t o give zero-dimensional bound states. For one-dimensional Bloch electrons, Landau quantization hardly occurs, since cyclotron motion does not occur within the limited small space. If the size of the thin wire is larger than the magnetic length I , quantization may be complete. Otherwise, quantization must be incomplete. In summary, the DOS values under a magnetic field are:

(4.22)

Ck,

1 = (L/27r) dY/12 = L2/27r12. The DOS values Here we utilized the relation in three and two dimensions under a magnetic field look, respectively, like that in oneand zero dimensions without the field (see Fig. 4.1).

energy Fig. 4.1: Schematic illustrations of (Eq. (4.14)) as a function of E .

P3D

(Eq. (4.9)), p2D (Eq. (4.13)), and

p1D

38

4 Dimensionality

References Erdklyi, A., et al., (1954-55), Higher Transcendental Functions, I, 11, 111, New York: McGraw-Hill. Landau, L. D., Lifshitz, E. M. (1958), Quantum Mechanics, London: Pergamon Press, Mathematical Appendices Moriguchi, S., Udagawa, K., Hitotsumatsu, S. (1956-60), Mathematical Formulas I, 11, 111, Tokyo: Iwanami.

5

Junctions

The history of development of solid-state devices can be traced as that from bulk to composite materials. In mesoscopic devices some specific artificial boundary conditions are often imposed so that quantization, such as geometrical confinement, is applied and the electronic energies are made discretized. In order to impose such boundary conditions various technologies that are used t o fabricate junctions are very often utilized, thanks to recent advancement in semiconductor process engineering. In this chapter we will review what occurs at the interfaces between different materials. The first example of such a junction was the semiconductor p-n junction. In addition to the p-n junction, other types of junctions have been studied and actually utilized. Here, we will discuss what happens at the interfaces of, for example, met a1-metal, met a1-semiconductor, metal -insulator, semiconductor -semiconductor, and so on. As a special case, a normal metal-superconductor junction is also of interest and importance, as will be described in Chapter 13. The interface at junctions impose a new boundary condition to the electrons thereabout. Let us start with the simplest metal-metal interface.

5.1

Metal-Metal Junction

Obviously something specific occurs in a metal-metal junction only if both metals are different (DuMond et al., 1935; Chang et al., 1985; Dhez et al., 1987; Shinjo et al., 1988). Even for the same metals this may also be the case when they have different crystallographic orientations. However, ordinary metals are polycrystalline, so that this type of junction is not important realistically. The electronic states of a metal are characterized by several physical quantities: the work function, the Fermi level, and the band width (Fig. 5.1). The work function q5 is the minimum energy needed to excite an electron inside the metal and for it be emitted outside the metal. This quantity is measured in photoemission spectroscopy experiments. The Fermi level, needless to say, is the highest energy below which all energy states are occupied by electrons at zero temperature. The Fermi energy Efis here defined as the energy at the Fermi level measured from the band bottom. An important concept is the vacuum potential level ,,$, relative to the band bottom. The work function q5, accordingly, is equal to the difference between q5vac and Ef.The band width W is the maximum energy spread of the band of concern. In the following, the

5 Junctions

40

Metal B

Metal A

EfB fA

(a)

*

dipole layer

Fig. 5.1: Schematic presentations of the energy scheme of metal-metal interface. (a) After contact of metal A and B for 4~ > 4~ and WA < WB. The vacuum potential level and the band bottom are offset between both metals so that the Fermi level is leveled, caused by the movement of certain number of electrons from B to A. (b) Energy scheme of metals before contact. metal is assumed to have a single energy band, for simplicity. Let metals A and B constitute a composite. This is the case with 4~ > $B and W A< W,. As shown in Fig. 5.1, when metal A and B are brought together to form an interface, the requirements are as follows. (1) Both of the Fermi levels must be leveled at the furthermost positions off the interface, at least at the instance when both were made to be in contact. (2) The vacuum potential levels must be uniquely defined for both metals and be continuous. These two requirements are satisfied only with movement of a small number of electrons from B to A in the situation depicted here. Thus, metal B has a thin layer with less electrons than those neutralizing the metal, whereas metal A accommodates more electrons. These electrons constitute an electric dipole layer at the interface. Since an electric dipole layer produces a polarization field, the vacuum potential level is forced to be offset but it connects smoothly between both metals. In metals, polarization charges and, hence, a polarization field are readily screened out. However, they survive within a macroscopic scale, which is characterized by k: = (e2/co)p3D(Er) = (e2/~0)(3njg/2Ef), i.e., the Thomas-Fermi screening parameter k,, which is calculated based on Eq. (4.15) for the free electron gas model (Kittel, 1963; , J8iT2€&/3n3~e2. Ziman, 1972). The screening length X is specified by X 2 i ~ / k = N

5.1 Metal-Metal Junction

4

c. b.boo tm,

41

I

n-type

++ (a)

depletion layer e

I

T ee e e e

I

A7

I

(b) (b) forward forward bias bias

7h7

h h h

7797

(c) reverse bias

7797

Fig.5.2: (a) Illustration of the energy scheme in p-n junction. A dipole layer is constituted between positively ionized donors in the n-region and negatively ionized acceptors in the p-region. e and h stand for electrons and holes, respectively. c.b. and v.b. mean, respectively, the conduction and valence band. (b) For a forward bias condition both electrons and holes are lured to their opposite regions, causing a current to flow easily. (c) For a reverse bias condition, both electrons and holes are repelled from each other making the depletion layer wider and causing hardly any current to flow.

5 Junctions

42

This length is roughly estimated to be 2.5 m-3.

5.2

-

10

A

for Ef

-

5 eV and

n 3 ~ N

Homogeneous Semiconductor Junction

The polarity of a semiconductor is known to be controlled by doping impurities, which substitute the mother atoms, for example Si. (See, e.g., Sze, 1983). When Si is doped with group I11 elements such as B or Gal then these impurities behave as acceptors which are composites of a negatively ionized acceptor (A-) and a hole. When doped with group V elements such as P or As, each impurity is equal to a positively ionized donor ( D f ) plus an electron. The semiconductor is still neutral even after doping. At room temperature most donors and acceptors are ionized by thermal excitation and the number of electrons and holes are, respectively, the same as the number of donors and acceptors in an ideal case. The Fermi level is situated between the conduction band bottom and the donor levels in an n-type semiconductor, whereas it is between the valence band top and the acceptor level in a p-type. In the n-region the conduction band bottom above the Fermi level is occupied by electrons (negative charges) and the donor levels below it are left with positively ionized donors, since the energy difference between the donor levels and the Fermi level is small enough to excite electrons from the donors at a high enough temperature. At room temperature it is usual that the concentration of electrons a t the band bottom is roughly the same as the concentration of doped donors, i.e, ND x ND+ = n. This temperature range is called the saturation range. This fact seems curious because the Fermi level is between the conduction band bottom and the donor levels, and their separation is so small that the number of ionized donors ND+ are likely t o be equal to ~ N atD the donor levels. However, this is not the case in the saturation region. The readers should study the reason in Sze (1981), Chapter 1. In non-ideal cases, when the semiconductor has unequal numbers of acceptors and donors in the same region, some degree of compensation occurs. In these cases too, charge neutrality holds among all charges: ionized acceptors, ionized donors, holes and electrons. That is, ND+ p = NA- n holds. Let us assume ND > N A . The Fermi level in this case may be situated just below the donor levels. Consequently, electrons trapped in the donors prefer being de-excited down to the acceptor levels to being excited to the conduction band. In other words, although all acceptors become ionized, there appear few holes in the valence band, and the number of electrons n is made equal to ND+ - NA- and NA- = N A . This is the result of compensation. As far as ND > N A , this region remains to be n-type. First we will interpret a homogeneous p-n junction, which is a composite of nand p-type of the same semiconductor. In this case physical quantities of concern are considerably different from those in the metal-metal interface. As shown in Fig. 5.2, the energy difference between the vacuum potential level and the conduction band bottom is usually taken to be x, the electron affinity. There is an energy gap Eg in semiconductors between the conduction and valence band, which is approximately equal to the difference in the Fermi levels in n-type and p-type

+

+

43

5.3 Heterogeneous Semiconductor Junction

semiconductors, since the donor and acceptor levels are very close, respectively, to the conduction band bottom and valence band top. In a semiconductor-semiconductor p-n junction, too, an appropriate number of electrons move from the n-region leaving the donors net positively ionized: D+ . In the p-type semiconductor excess charges appear as negatively ionized acceptors: A- . Thus, an electric dipole layer is produced just parallel to the metal junction, although a dipole layer consists of immobile charges. When considering Fig. 5.2, be careful about the fact that the energy levels far above the Fermi level are always occupied by positive charges (positively ionized donors) or holes, whereas those states far below the Fermi level are occupied with negative charges (negatively charged acceptors) or electrons. That is, the unoccupied states in the conduction band may look equal to the states occupied by holes. This is not the case for levels very close to the Fermi level. For example, the conduction band bottom and donor levels may accommodate, respectively, appreciable electrons (not holes) and positively (not negatively) ionized donors. The dipole region is called the depletion layer, since mobile charges are depleted there. In the semiconductor junction, the Fermi energy Ef for electrons in the conduction band is very small or slightly negative (below the band bottom) depending on the degree of doping, so that the screening effect due to mobile charges is much less than that in metals. In such a case, classical Boltzmann statistics rather than quantum statistics must be employed to estimate the screening effect. Now, not the Thomas- Fermi screening but the Debye screening is concerned, the screening length of which is given by

where n, is the (relative) static dielectric constant of the semiconductor and N is the density of electron or holes; T is the ambient temperature and kg is the Boltzmann /m3 and ns 12, X D ~120 A, ~ and ~ it ~ is constant. For Si doped with N known that the depletion layer thickness is about 10 times the XDebye. It should be instructive to interpret how the energy scheme varies when a bias voltage is applied to this junction. As is known, a p-n junction works as a rectifier. Under a forward bias with a negative potential on the n-region a non-equilibrium condition develops and the Fermi level is forced to separate for electrons and holes, as shown in Fig. 5.2(b). Then, electrons easily drift toward the p-region affected by the potential. Meanwhile, for a reverse bias with positive potential in the n-region and negative potential in the p-region, electrons and holes are evacuated, respectively, out of the n-region and p-region causing a wider depletion layer. Concomitantly holes go away from electrons (Fig. 5.2(c)). This is a reverse bias condition and current hardly flows. N

5.3

-

-

Heterogeneous Semiconductor Junction

This type of junction is not always a p-n junction and is often called a heterojunction (Esaki et al., 1970; Ando et al., 1982; Capasso et al., 1987; Bastard, 1988). Here we

5 Junctions

44

%aAs

b

e e e e

!e

'AlGaAs e

c.b.bottom

I n-type AIGaAs ++

n-type GaAs

dipole layer Fig. 5.3: Band energy scheme for GaAs-AlGaAs heterojunction. This is realized because X G ~ A> ~XA]GaAs.

are concerned with an interface constituted between semiconductors with the same type of polarity but of different materials. The most popular and frequently utilized is the junction between GaAs and A1,Gal-,As (0 5 x 5 l), the latter of which is heteroepitaxially grown over the former because the mismatching of the lattice constants is very small. The same n-type GaAs and A1,Gal-,As constitute a specific interface, since both materials have different electron affinities X , as shown in Fig. 5.3. In this junction, a dipole layer is formed between positively ionized donors in n-type AlGaAs and the excess of electrons in n-type GaAs. The latter electrons are mobile parallel to the interface, in contrast t o the depletion layer in the p-n junction. The conductive 2D sheet is called the channel. Two types of mobile carriers exist in GaAs: one is those ordinarily excited from donors in the bulk and the other is those spilled out of AlGaAs. If GaAs is originally undoped, only the latter type of carriers exists and they may be controlled to perform the expected characteristics of the device. The interface is formed because of different affinities as X G ~ A>~XAlGaAs and both conduction band bottoms are parallel to the vacuum level with an offset A E , 320 meV for x M 0.3. A heterojunction is often applied t o confining electrons around the interface, as will be discussed later under the name of a quantum well. When a GaAs layer is sandwiched by two AlGaAs layers with a higher barrier potential, carriers generated

-

5.4 Schottky Junction

45

in AlGaAs are displaced to and accommodated in the thin GaAs layer, constructing a confinement geometry. It has been recognized that this type of geometry has a couple of remarkable merits when applied to actual devices. One is that the structure generates a twodimensional energy scheme which makes a merit of a constant DOS stemming from two-dimensional discrete levels. Energy level separation is very often larger than the ambient temperature and causes definite quantum effects. The other is its transport property parallel to the interface. In bulk semiconductors, carriers are generated from dopants in the bulk so that the carriers are always forced to be affected, i.e., scattered, by the dopants; that is, the Coulomb scattering of carriers by the dopants is unavoidable. On the other hand in a heterojunction, a conductive channel may be constructed separate from the dopants in the AlGaAs layers. This causes a very large mobility of carriers in the channel. The region with dopants may be taken far away from the interface beyond a spacer layer, so that this technique is called modulation doping (Dingle et al., 1978).

5.4

Schottky Junction

A so called Schottky diode is formed at the metal-semiconductor interface. (See, e.g., Sze, 1983). As shown in Fig.5.4(a), a metal-n-type semiconductor interface has a rectifying characteristic when &, > x. (1) For a current from the semiconductor to the metal, i.e., a forward current, electrons in the n-type semiconductor can easily flow toward the metal over the depleted region (Fig. 5.4(b)). (2) For the current in the reverse direction, i.e., an electronic flow from the metal toward the semiconductor, a barrier exists with immobile positively charged donors in between (depletion layer). All electrons below the Fermi level in the metal may not be allowed to flow into the valence band in the semiconductor, because all the states are completely occupied and have no vacancies (Fig. 5.4(c)). Figure 5.4(d) shows a Schottky junction made of a metal-p-type semiconductor, when &, < x + Eg.In order for electrons in the metal to flow into the p-region with holes, the curvature of the valence band top is lessened as shown in Fig. 5.4(e) and, accordingly, they flow more easily than in the zero-bias case shown in Fig. 5.4(d). For a reverse bias, a depletion layer with immobile negatively charged acceptors widens, as suggested in Fig. 5.4(f), which suppresses the reverse current. Both junctions made of a metal-n-type semiconductor with x > 4m and of a metal-p-type semiconductor with 4m > x + Eg behave rather Ohmic than rectifying.

5 Junctions

46

vacuum level \ n-type semiconductor

6-n

I'

Metal

c.b.bottom Fermi level

+++++ \L

v. b. (a)

++ depletion layer

(b) forward bias

A

(c) reverse bias

77h

5.4 Schottky Junction

47

/\

,fp-type semiconductor

A

Metal

X

@In

(d) depletion layer b .e

e

e

(e) forward bias

(f) reverse bias Fig. 5.4: Energy scheme for (a) Schottky junction made of a metal-n-type semiconductor, and (d) a metal-p-type semiconductor. Electrons likely to flow from the metal toward the n-type semiconductor feel a potential barrier against the depletion layer with immobile ionized donors (c), whereas for the opposite direction they can easily flow due to an applied potential (b). The conditions for (a) and (d) to hold are, respectively, &, > x and &, < x Eg. A current flows easily under a positive potential applied to the p-region (forward bias) which repels holes toward the metal (e). Under a reverse bias a negative potential lures holes off the depletion layer causing it to widen. Because of a wide depletion layer with immobile ionized acceptors, current hardly flows (f).

+

48

5 Junctions

Fermi level

Fig. 5.5: Schematic presentation of the energy scheme in an MOS (metal-SiOz -p-type Si). An n-type inversion layer is formed between the insulator and depletion layer, under which p-type Si exists. The necessary condition for this type of MOS to be realized is xs,oZ Ezioa > xsi EgSi > &,.

+

5.5

+

Metal- Oxide- Semiconductor (MOS)

The most useful and powerful device available at the present moment is the MOS, which stands for a metal-oxide-semiconductor structure, a type of field-effect transistor. Here an explanation is given on an n-MOS made of SiOz and p-Si. An nMOS is an MOS whose active layer is an inverted n-type channel that appears at the interface between oxide (SiOa) and p-type Si (Fig.5.5). An n-MOS transistor has essentially the same geometry as a Schottky diode, with an additional thin oxide layer between the metal and semiconductor. Despite the similarity between both structures, their usage is very different. Schottky diodes (metal-p-semiconductor; Fig.5.4(d)) are used with a current perpendicular to the junction, whereas an MOS is used with a current parallel to the interface in the channel formed between the insulator and the p-type semiconductor. Ordinarily xsioz E$'02 >> xsi Ezi > &,. This structure is a metal-SiOz-n-inversion layer-depletion layer-p-Si. In this device A$ = xsi Ezi - &, plus an appropriate &bstrate (a bias potential between the substrate and gate) is applied over a very thin SiOa plus a depletion layer so that a strong electric field may form an inverted n-type layer between SiOz and the depletion layer. Accordingly, the conduction band of Si is highly bent to reach below the Fermi level and form an inversion layer. Current flow is exclusively within the channel when being used, since Ohmic contact is only in the channel at the drain and source electrodes.

+

+

+

5.5 Metal- Oxide-Semiconductor (MOS)

49

References Ando, T., Fowler, A.B., Stern, F. (1982), Rev. Mod. Phys. 54, 437. Bastard, G. (1988) , Wave Mechanics Applied to Semiconductor Heterostructures, Les Ulis: Les Editions de Physique. Capasso, F., Margaritondo, G. (1987), Heterojunction Band Discontinuities. Physics and Device Applications, Amsterdam: North-Holland. Chang, L. L., Giessen, B. (Eds.) (1985), Synthetic Modulated Structures, New York: Academic. Dhez, P., Weisbuch, C. (Eds.) (1987), Physics, Fabrications, and Applications of Multilayered Structures, NATO Advanced Study Institute Ser. B, Physics, No.182, New York: Plenum. Dingle, R. C., Stormer, H., Gossard, A. C., Wiegmann, W. (1978), Appl. Phys. Lett. 33, 665. DuMond, J. W. M., Youtz, J. P. (1935), Phys. Rev. 48, 703. Esaki, L., Tsu, R. (1970), IBM J. Res. Develop. 14, 61. Kittel, C. (1963), Quantum Theory of Solids, New York: Wiley. Shinjo, T., Takada, T. (Eds.) (1987), Metallic Superstructures: Artificially Structured Materials, New York: Elsevier. Sze, S. M. (1981), Physics of Semiconductor Devices, New York: Wiley. Ziman, J. M. (1972), Principles of the Theory of Solids, Cambridge: Cambridge University Press.

This page is intentionally left blank

6

3D Quantum Systems

In an ideal three-dimensional system, all Bloch waves are extended throughout the whole system. This means that it is still very similar to a microscopic system, however large it is, as long as there are no decoherence mechanisms. At first glance a regular lattice looks like as if it would cause some electric resistance to the Bloch waves because of scattering by the lattice points. However, this is not the case, because Bloch waves are eigenstates of the lattice. This means that if an electron enters from the leftmost electrode, it can instantaneously reach the rightmost electrode since as the electron is a Bloch wave it completely covers the whole lattice, so that there is no resistance or friction to bridge the two electrodes. Then what causes a resistance? The answer is any deviations of the lattice points from the regular positions. For example, impurities substituted for the mother atoms have a potential different from that of the regular lattice. The most common is spatial deviations of atoms from the fixed regular positions due t o thermal excitation. Such a mode of irregularity induces lattice vibrations and, hence, a certain electric resistance. If a specific atom happens to deviate from the pre-determined regular position, the deviation also causes the neighboring atoms to deviate with a slight time delay, so that this deviation propagates as a wave. The reason why neighboring atoms couple is that all atoms are coupled to each other according to the forces which cause cohesion, i.e., metallic, covalent, ionic and hydrogen bonds, and van der Waals force. Such a wave is also characterized by a specific wavevector k (= 27rlX). This wave, when quantized, is called a "phonon" using the terminology of quantum mechanics. At high temperatures, many phonons occur. In such a situation most atom positions are blurred. To describe such a situation we often use the so-called Debye-Waller factor for atomic positions. Let us express the wavy displacement of the lattice points by xj = xjo 6 x j ( z j o : regular lattice points at a site j ) , then

+

Here ( . . . ) a v , ~means an average over the direction of 6 x j relative to the fixed k as well as the thermostatistical average over their magnitudes at ambient temperature. In an

52

6 3 0 Quantum Systems

ideal X-ray diffraction experiment infinitesimally minute Laue spots appear for the Xray satisfying the Bragg condition on regular lattice points. The second exponential in the above expression, known as the Debye- Waller factor, however, makes these spots blur to finite sizes. The higher the temperature is, the more becomes the ((SZ~)~)Tterm. Phonons, acoustic and optic, are known to exert an inelastic scattering on Bloch electrons. In addition to phonons, photons, plasmons, and so on behave similarly in solids. Electron-electron interactions also cause inelastic scattering on a specific electron, although the overall energy and momentum are conserved from before and after the electron-electron scattering. In solids the most frequent scattering is elastic scattering as a result of any disorders: for example, impurities, grain boundaries, dislocations. Elastic scattering does not change the energy of the scattered electrons, so that it does not decohere Bloch states but only changes the phase of the waves. Historically it has been presumed a priori that even in very small devices it would not be possible to observe coherent phenomena directly, since there are lots of elastic scatterers in solids and the product rarely exceeds unity (lei, the mean-free path under elastic scatterings; kf,the Fermi wavevector). However, this presumption has proved not to be the case. In contrast to elastic scattering, inelastic scatterings change the energy from the initial value, which reduces the coherence, that is, decoherence or dephasing. After physicists realized this fact, many of the phenomena that had been investigated so far in small electronic devices were recognized as showing whatever small fraction of coherence remained, especially at lower temperatures where only a few inelastic scattering events are expected. Thus, the above-mentioned criterion should be replaced by kfli, 2 1 although kflel << 1 for coherence to be maintained, where li, means the mean-free path under inelastic scatterings, which ordinarily equals the phase-breaking (dephasing) mean free path 16. In the time domain, the scattering time for inelastic scatterings, qn or 76, is normally used, the latter of which is called the phase-breaking (dephasing) time. Understanding this type of decoherence is very important from the viewpoint of quantum measurement theory. It is often assumed a priori that in order to acquire information from a detection the detection must be an inelastic process. This is also wrong. If we can somehow measure a phase shift using a detector, then this is a sort of measurement. To detect a phase change, e.g., in the two separate beams in neutron interferometry, we need lots of similar events, not a single event, because of the uncertainty between the phase and the number of events of concern. In fact, we can not deduce anything from a single event on the neutron detector shown in Fig. 2.1, because a small amount of data can never make an interference pattern, and, consequently, no information about the phase can be obtained from a pattern with such sparsely distributed data, as is obvious in Fig. 2.3. Elastic scattering is actually a limiting case of inelastic scattering with an infinitesimal energy transfer. A well-known difficulty in quantum measurement theory is how a microscopic system evolves into a macroscopic one at the moment of detection. Thus, unless an inelastic process exists, the detection itself is elastic and the change caused by the

53 process is merely a phase shift, and hence the whole system remains in a quantum mechanical microscopic state, as was discussed by Wigner (1963). Even in this case, we can detect the change in phase by means of an ensemble average of similar events (Machida et al., 1980; Namiki et al., 1993; Murayama, 1990). In actual solid-state devices, a phase of Bloch electrons will frequently suffer from some elastic scatterings but ordinarily we need not know how much of a phase shift occurred in each scattering. We are only concerned with how much phase shift occurred between the inlet and outlet Bloch state and how the original Bloch wave is converted into another one with coherence preserved. Figure 6.1 shows the concept of a localized state in solids, after Bergmann (1983, 1984). This type of localization is known as the Anderson localization, after the first paper by Anderson (1958) (see also Nagaoka et al., 1982; Kramer et al., 1984; Lee et al., 1985; Ando et al., 1988). In solids with disorders, an electron is scattered elastically many times by these disorders and eventually happens to return to the original position. When a wave is superposed with itself after an itinerary, it must have a peak intensity at a specific site on the path if the phase relationship between itself and the adjoint after the itinerary is constructive. This resembles a neutron interference experiment in a silicon interferometer (Fig. 2.3). This state is well-defined in the sense that it is stable and long lasting, unless it obtains enough energy to be activated from a localized to a de-localized state. Such activation is only possible through inelastic scattering, for example, by optic phonons, photons, plasmons and the like. This type of Anderson localized state in solids is as important a concept as the extended state described by the Bloch theorem. If a crystal is a perfectly regular lattice, then the eigenstate in that lattice extends over the whole crystal and the wave shows no resistance unless there is some deviation from regularity. On the other hand, if the crystal had an appropriate amount of disorder, the Bloch wave shrinks to a lozalized wave around a specific position, which can be transported only when a sufficient amount of energy is added through, e.g., an inelastic scattering. This means that a localized state can be constructed only under coherent multiple elastic scatterings. However, on inelastic scattering the coherence is destroyed, as is the localized state, resulting in delocalization. The conductance of Bloch electrons is infinite and any elastic and/or inelastic scattering causes the conductance to reduce to a finite value. Localized states under elastic scatterings do not convey electric current, but inelastic scattering sometimes makes the conductance recover again to a larger value. Now, it is worth mentioning that localized states due to disorders in three dimensions behave similarly to mesoscopic systems. Take a hypercube of d dimensions with a volume L d . As shown in Fig. 6.1, localization is expected due to disorders. Disorders tend to decrease the extension of electron waves from delocalized t o localized states. This decrease occurs due to the change in the boundary conditions or, in other words, because the translational symmetry is broken. According to Thouless (1977, 1984), each localized state has a characteristic length scale, i.e., coherence length 6, over which the state extends. If L 5 <,the system seems ideally microscopic, where all states are extended over the whole hypercube, otherwise

54

6 3D Quantum Systems

Fig. 6.1: Illustration to explain the Anderson localized state. Each node represents a crystalline defect, causing an elastic scattering which eventually allows the wave return to itself with an appropriate phase shift after multiple scatterings. At some specific position, the phase shift almost vanishes and the scattered and original waves are constructively superposed so that a peak occurs in its occupation probability.

dlng dlnL

1 -------------n

"I

lng

Fig.6.2: Schematic diagram of d(lng)/d(lnL) vs lng, where g = ( r h / e 2 ) G , for d=l, 2, and 3 (after Abraham et al., 1979).

55 the state would appear to be localized. Thus we may take the conductivity of such an electron to be proportional to the extent of the state eVL/C. The following discussion makes use of the scaling theory by AALR (Abraham et al., 1979). The size dependence of conductance is schematically illustrated in Fig. 6.2. When we discuss the conductance G in this system, it is natural to assume that the normalized conductance is given by g(L) = (nh/e2)aLd-2 = (nh/e2)G

(6.2) This fact is easily understood in the following consideration. In d dimensions, the cross section of the current is scaled to Ld-', whereas the length parallel to the current is scaled to L. Accordingly, as is know in the relationship of resistivity p to resistance R, R = pL/Ld-l = pL2-d. In particular in two dimensions (d = 2), R = p, or G = u (a = l/p: conductivity). As will be discussed in Chapter 7, in two dimensions, it is know that o = G m e2/xA, and, hence, we may normalize conductivity by xA/e2. The g given above is, accordingly, dimensionless. Now, let this g be proportional to e-L/C; i.e., g(L) = gae-L/C,or d(lng)/d(ln L) ln(g/ga) when the state is well localized and, hence, g is small. Meanwhile, when g is large, g(L) must be expressed in a power series in g and is asymptotically given by the solution of the equation:

-

d(lng)/d(ln L)

N

d - 2 - a/g

+ ...

(6.3)

Thus, the asymptotes for d = l , 2, and 3 are d(lng)/d(ln L) = -1,O, and 1, respectively. If we plot d(lng)/d(ln L) vs lng, ln(g/ga) is negative for small value of g/ga, i.e., L >> <. In three dimensions, G(L) = aL , which increases with the size of the system L and d - 2 = 1 is approached from below when L o( g + 00 ( a > 0). d(lng)/d(ln L) This fact means that in one and two dimensions, the electrons are always localized for L + 00, if they are disordered. The case for two dimensions is typical. As is easily integrated from Eq. (6.3), g = - a l n L. On the other hand, from a microscopic theory in the so-called weakly -(e2/nh)ln L or g -In L. localized regime, g is known to be given by G = u Thus, cy = 1 was derived. In two dimensions, d(lng)/d(lnL) is always negative and g(L) + -0 for L 00. g(L) shows a cross-over from logarithmic (for large g: weakly localized regime) t o exponential localization (for small g: strongly localized regime). In the weakly localized regime, g(L) -In L, where L is very often replaced by 14 or (D: diffusion constant). Since ~4 is proportional to T-P (T: temperature, p : a number of order unity) according to the perturbational calculation of the electron-phonon and electron-electron interaction, it is seen that conductance has a logarithmic dependence on T . This reflects the fact that localized states can diffuse when inelastic scattering supplies them with the necessary amount of energy to delocalize. The strongly localized regime is for 14 >> 5. In the opposite case, i.e., 14 5 5, the conduction process is classified as: (I) for L > 14, classical hopping conduction occurs; (11) for L 2 14, electron waves are extended and coherent.

-

-

-

-

fi

6 3D Quantum Systems

56

In order to discuss conductance in this system, Thouless (1977) introduced a di, w(L) is mensionless number called the Thouless number, g ( L ) = v ( L ) / w ( L )where a typical energy, which may be expressed in terms of h / r ~(TD: the specific time for diffusion, defined by D = L 2 / m ) ,and w ( L ) is the characteristic separation of levels at the Fermi energy, given by l/[pd(Ef)Ld] for a density of states p d ( E f ) in d dimensions. The equivalence of this Thouless number g ( L ) to the above-mentioned g(L) = (7rh/e2)oLd-2can be easily proved from the generalized Einstein relation (T = e2Dpd(Ef).l It is interesting to note that localized states can convey an electric current via hopping through the most probable percolation path. Hopping itself is possible by the assistance of, e.g., phonon absorption. Following Mott (1968, 1974; Ziman, 1979), DC conductivity is given by [T exp-(To/T)l/(d+l), where d is the number of dimensions in the system of concern (variable-range hopping).

-

References Abraham, E., Anderson, P. W., Licciardello, D. C., Ramakrishnan, T. V. (1979), Phys. Re? 42,673. Anderson, P. W. (1958), Phys. Rev. 109,1492. Fukuyama, H. (Eds.) (1985), Anderson Localization, Berlin: Springer. Ando, T., Bergmenn, G. (1983), Phys. Rev. B28,2914 Bergmann, G.(1984), Phys. Rep. 107,1. Kramer, B.,Begmann, G., Bruynseraede, T. V. (Eds.) (1985), Localization, Interaction, and Transport Phenomena, Berlin: Springer. Lee, P. A., Ramakrishnan, T. V. (1985), Rev. Mod. Phys. 57,287. Machida, S.,Namiki, M. (1980), Prog. Theor. Phys. 63,1457, 1833. Mott, N. F. (1968), Rev. Mod. Phys. 40,677. Mott, N.F. (1974), Metal-Insulator Transitions, London: Taylor & Francis. Murayama, Y. (1990), Foud. Phys. Lett. 3,103. Nagaoka, Y., Fukuyama, H. (Eds.) (1982), Anderson Localization, Berlin: Springer. Namiki, M., Pascazio, S. (1993), Phys. Rep. 232,301. Thouless, D.J. (1977), Phys. Rev. Lett 39,1167. Thouless, D. J. (1984), J . Phys. C: Solid State Phys. 13,93. Wigner, E. P. (1963), A m . J. Phys. 31,6. Ziman, J. M. (1979), Models of Disorder, Cambridge: Cambridge University Press.

-

' 0 = n e p = n e 2 D / A E = p d e 2 D , where p is the mobility, and Pd n / A E is the &dimensional DOS within an energy width A E . Classically A E = k B T and a generalized Einstein relation D = ( A E / e ) p was utilized.

7

2D Quantum Systems

Mesoscopic systems are closely related to low-dimensional systems. In ideal three-dimensional systems, a Bloch state is a well-defined quantum state, so that it is easy to argue that the state is coherent. However, it should be asked how the state may be degraded by sufferinginelastic scattering. So long as a system remains three-dimensional, it may be hard to know whether a particular Bloch state has lost its coherence, since we need some specific configuration in which to recognize its coherent nature, allowing an electron to interfere with itself. This type of configuration is ordinarily devised in lower dimensional set-ups. We have described a special case, the Anderson localized state, which reveals coherence in a three-dimensional configuration (Chapter 6). The maximum number of dimensions is three in condensed matter. It is usual t o impose “size quantization” to reduce one of the dimensions, and normally a wavefunction is geometrically “confined” between high potential barriers on both sides. Two-dimensional quantum systems are those where three dimensions are reduced to two dimensions by confining electrons within a thin slab, thus eliminating the dimension perpendicular to the slab. The degree of freedom in that direction is thus lost, whereas the other two degrees of freedom remain unconfined.

7.1

Single Quantum Well

Let us take, for example, a well-like potential with infinite barrier heights to confine the electrons. This geometry is frequently called a “quantum well” (QW). The solution of a Schrodinger-type equation with a potential of this type is well known and is usually described in most textbooks. To be precise, this equation is an effective mass approximated (EMA) equation, which has been interpreted in Chapter 4 and in Appendix B. The solution to this equation gives an envelope function to Bloch states within the well potential. The simplest conceivable two-dimensional system is as follows. Take a non-doped semiconductor, Si for example, with a thickness small enough to confine electrons between both surfaces. Assume the outside surfaces to be in a vacuum, then the vacuum potential relative to the conduction band bottom (assumed to be the origin of the potential) is so high that the confinement is considered to be perfect. This means that the electronic wavefunction completely vanishes outsides. Let us take the z-axis in the confining direction. The motion of the electrons in the z- and y-direction are still free. Note that the mass appearing in this equation is

7 2D Quantum S y s t e m

58

an effective mass caused by a periodic lattice potential, not a bare mass in a vacuum. Then the EMA equation reads, d E(-i-)@(z) = En@(z) (7.1) dz where the eigenenergy of the Bloch electron E (k)= (h2/2m*)(ki+ kt ki) is taken for simplicity. The resultant eigenenergy is E n k , k , = En (h2/2m*)(k% ki). Now a size quantization occurs in the z-direction under the boundary conditions where the envelope function vanishes at z = f d / 2 for a Si thickness of d. En is well known, that is,

+

+ +

(7),

h2 En = 7

2

n = 1,2,3,... 2m Thus, the de Broglie half wavelength must equal to d/n, which means that a type of resonance occurs between a half de Broglie wavelength and the confinement geometry. This produces standing waves sandwiched between both surfaces. It is instructive to estimate the magnitude of the quantized energy difference when it equals the thermal energy at T . From simple arithmetic, we obtain approximately

where T is the ambient temperature in K. For example, for n = 1, m* = O.Olmo and T = 300 K, the thickness is approximately 700 A. Actually, the single quantum well

(SQW) is fabricated with a thickness of less than 100 A. It is recognized, also classically, that a standing wave can be constructed from coherent interference between forward- and backward-moving waves. So the confined, size-quantized wavefunction may not be constructed without coherence of the state. A classical particle, analogous to an incoherent quantum wavepacket, can only go and return, bounced back by the barriers, and does not constitute a standing wave. Thus it is obvious and also important that without coherence quantized eigenstates may not be well defined. In particular, if there are some mechanisms to destroy coherence in QW, then quantization only occurs incompletely. In this respect , mesoscopic systems, where partial coherence is assured, are closely related to two-dimensional systems as a typical example. The allowable width to assure coherence should be less than the minimum of the mean-free path lengths for existing inelastic scatterings. In actual devices the simplest and almost ideal QW can be prepared by growing two heterojunctions made of a conduction band of n-type GaAs sandwiched by that of n-type Gal-,A1,As. Both semiconductors have a band offset equal to 240 meV for x = 0.3, as shown in Fig. 5.3, according to the difference in the electron affinity. When one wishes to solve an EMA equation, there is an annoying problem concerned with the boundary condition at the interface. A specific boundary condition problem is discussed in Appendix B, following Ando et al. (1982) and Ando (1984). Because the true wavefunction is a product of a Bloch function and an envelope function, the product itself should satisfy the continuity boundary conditions at the interface.

7.1 Single Quantum Well

59

00

A =3U2 A =L

x=o

x=L

Fig. 7.1: Size-quantized eigenstates in a QW confined by infinite large potential barriers on both sides. Therefore, there is no reason to require that the Bloch and envelope function must be individually continuous there. We emphasize, however, that the problem is known t o be simplified for GaAs-GaAlAs interface. In this case both the envelope function and its derivative must simply satisfy the continuity conditions: i.e., d d @A(()) = @"B(O), (m>)-'-@A(O) dz = (m;)-'-@B(o) dz (7.4) should be satisfied between materials A and B. A numerically solved envelope function for the QW has been obtained and is shown in Fig. 7.2. The ground state envelope function is weII confined, whereas the first excited state energy is almost near the bottom of the conduction band of the barrier material. This latter fact means that the excited state is apt to spill out of the QW.

60

7 2 0 Quantum Systems

El=-2.lmeV

mB=0.5mo

B I W l B m,=0.0665mo

Fig. 7.2: Numerically solved squared envelope function for a GaAs-GaAlAs QW. The parameters employed are: m i = 0.5m0,m;Y = 0.0665m0, LW = 50 A,where B and W are, respectively, barrier and well. The quantized energy obtained is: El = -1.99.5 meV and E2 = -2.1 meV, the origin of which is taken at the bottom of the conduction band of GaAlAs. Note that the n = 2 level comes very close to the bottom of the GaAlAs conduction band, which means that the state is almost resonating to the ground state of the barrier material.

00

00

00

Fig.7.3: Eigenstates within a coupled double QW. and $, denote, respectively, symmetric and antisymmetric eigenstates, each of which has an eigenenergy E, and E,. $J,

61

7.2 Multiple Quantum Wells

7.2

Multiple Quantum Wells

Figure 7.2 shows eigenstates within a single quantum well (SQW). Next we will consider a pair of SQWs. Let us assume that each QW has a single eigenstate with an equal eigenenergy Eo. When lowering the barrier in between, coupling between the QWs is introduced. Then eventually we have two eigenstates, each of which has an energy other than that without coupling, i.e., E, = Eo - A and E, = Eo A. A is a measure of the degree of the coupling, and s and a correspond, respectively, to symmetric and antisymmetric states, denoted as & and (Fig. 7.3). Thus, it is obvious that the eigenenergies within the multiple QWs (MQW) have n different values when the MQW consists of n SQWs. In general, these N eigenenergies disperse between the maximum and minimum value, the entirety of which is called the energy “band”. The original energy band stems from a series of N-atoms ( N >> l), each of which has a Coulomb potential according to the positive charge of the inner core. This is the reason why we obtain energy bands in periodic lattices, not restricted t o in one or two dimensions but in three dimensions also. Each atom, if it is similar t o a hydrogen atom, has discrete energies such as Is, 2s, 2p, 3s, 3p, 3d, ... and when they are mutually coupled they make, respectively, Is-, 2s-, 2p-, 3s-, 3p-, 3d-energy bands (Fig.7.4). A single energy band consists of many states, which are specified by a good quantum number k, or wavevector, equal to 27r/X (A: wavelength). The maximum energy is characterized by a wavelength X = a/2 ( a : lattice constant), and the minimum value by X = N a x 00 ( N >> 1). In this original band case, the barrier potential between atoms is so thin that tunneling of electronic states occurs easily, i.e., the coupling between atomic eigenstates is strong.

+

7.3

Superlattice

Just parallel to the ordinary atomic lattice, we can define a superlattice (SL) following the idea of Esaki (Esaki et al., 1969, 1979). An SL is a modification of the regular lattice with a long period, e.g., d = dw dg = nu, where a series of A atoms with dw in thickness are accompanied by another series of B atoms with d B . The system has new periodicity of length d. In general, materials A and B have different potentials, so that if material A constitutes a well, B makes a potential barrier. Thus the original series of atoms with period a is modulated with a longer period d. Although this system has double periods, the property of the SL is ordinarily described by an EMA equation, if d >> a , which gives an envelope function to the original Bloch states. For this new periodicity, an ad hoc energy band appears, called a miniband. The position where the original energy band is broken corresponds t o k which is equal to r / d , as shown schematically in Fig. 7.5. As a matter of course a minigap appears at the same wavevector. This is another example of size quantization. The readers should be careful in that SL properties also manifest themselves for energies above the barrier height. Because the additional periodicity modifies the original Bloch states whatever the energy is, the states above the barrier appear as virtual bound states, or virtual states for short. In contrast, the states within the well

+

62

7 2 0 Quantum Systems

nucleus

Coulomb potential

energy 't'

wFig. 7.4: Illustration of a series of hydrogen-like atomic orbitals producing energy bands when coupling is introduced. All s-, p-states and so on are coupled through the so-called quantum tunneling process.

k

QW

superlattice Fig. 7.5: Illustration of the relationship between the ordinary band energy in a regular lattice with period a, miniband energy and minigaps in a n SL as a function of wavevector k. The position of the minigaps is at k = r / d , whereas the original band gap is at k = r / a . Ordinary SLs have d ( a new period) equal to 5a to several tens of a, i.e., typically 10 100 A.

-

are called real bound states. In order to fabricate an MQW or SL, it is usual to utilize the potential barrier produced by the difference in the electronic affinity x for an n-type well or the sum of x and the band gap Eg for a p-type well. In general the offset between different materials produces a certain amount of a potential barrier at the heterojunction (Fig. 5.3). A variety of combinations of semiconductor materials have been known to produce various types of potentiaI barriers. FoIIowing Esaki's classification (19901, they are named Type I and 11, as follows.

7.4 DOS

63

Type I: XA XA

> XB for the conduction band offset g for ~ the valence band offset

+ E g A < XB + E

Type 11: XA

XA

> XB for the conduction band offset g for ~ the valence band offset

+ E g A > XB + E

All of these are illustrated in Fig. 5.6. As already listed, A GaAs-A1As heterojunction is classified as Type I. An example of a Type I1 heterojunction is the combination of InAs-GaSb. In Type I1 heterojunction there appears an electric dipole layer formed of electrons and holes adjoining to each other (Fig. 7.6(b)).

7.4

DOS

The most remarkable feature of a two-dimensional electronic system appears in its DOS. Since material properties are those most strongly determined by DOS, it is expected that the property varies stepwise reflecting that the DOS is a constant, independent of energy, for energies above some specific discrete energy, as described in Chapter 4 (Eq. (4.13)). In a QW, when several quantized bound states exist, the DOS adds a constant whenever a quantized energy value is reached. so In three-dimensional system, in contrast the DOS grows proportionally to that any material property varies only gradually as a function of energy. In two-dimensional systems, we will later explain several interesting phenomena which are caused by this stepwise nature of the DOS. That is, those phenomena result only from a quantum effect according to size quantization in two dimensions.

7.5

Landau Electrons

As has already been described in Chapter 4, applying a magnetic field reduces the degrees of freedom in the plane perpendicular to the field. So, if a magnetic field exists perpendicular to a two-dimensional sheet, the DOS is something like that in an atom or a cluster in zero dimensions, i.e., a series of discrete quantum levels. A typical and interesting example of such a system is found in the Hall effect as observed in a two-dimensional electronic sheet. Such a phenomenon is known as the integral quantum Hull eflect (IQHE) in a MOSFET and the fractional quantum Hall Eflect (FQHE) in a heterojunction, as will be discussed later.

64

7 2 0 Quantum Systems

n-GaAs

n-A I GaAs

p-GaAs

p-A I GaAs

Fig. 7.6: Illustrations of various heterojunctions after Esaki (1990). (a) Type I: both band offsets between the conduction bands and the valence bands have the opposite signs for the combination of materials A and B. (b) Type IT: both band offsets have the same signs. A Type I1 heterojunction produces an electric dipole layer of electrons and holes just neighboring to each other.

7.6

Hydrogenic State of Impurity

If a non-doped semiconductor has an impurity, its state may be described by an EMA equation, so long as it is an impurity with a small binding energy (“shallow” donor or acceptor). Shallowness means that the ionization energy from the impurity potential

65

7.7 Surfaces

is small. For a small ionization energy the wavefunction is considered to extend to a large extent. Then the effective mass approximation is valid. Let us consider an impurity in a two-dimensional system. The calculation is explained in Appendix D. In three dimensions, the hydrogenic radial wavefunction reads (e.g., Schiff, 1955; Landau et al., 1977)

@ ( r , eVJ) , 0; X m ( 6 ,V ) R n l ( r )

(7.5) where X m is a spherical harmonic function and normalization is ignored. The radial part is given by

with the Bohr radius a; = 4 7 ~ 0 t c l i ~ / r n *Since e ~ . this is a solution of an EMA equation, all values for mass should be those for an effective mass and simultaneously a; an effective Bohr radius. tc is a relative dielectric constant of the semiconductor material. The eigenenergy is given by En = -Ry*/n2, where Rydberg Ry' = ( e 2 / 4 ~ ~ ~ n ) 2 m * / 2isAa2 unit of energy in terms of the effective mass and dielectric constant of its environment. Meanwhile in a two-dimensional system (Stern et al., 1967), (7.7) where m is the magnetic quantum number and LE is the generalized Laguerre polynomials. p = 2 r / ( n 1/2)a;.

+

En = -

RY *

(n

+ 1/2)2 ' n = 0,1,2,...

Thus the extent of the ground state wavefunction in a hydrogen-like impurity potential is half that in three dimensions, whereas the ionization energy is four times as large as that in three dimensions. This in turn means that the electron trapped in an impurity center is comparatively well localized and is rather difficult t o ionize in two-dimensional systems. This comparison between two and three dimensions holds true for exciton binding energy. The energy is 4 times larger than that in three dimensions, so that it is said that the lifetime of an exciton in two-dimensional system is longer. Even if an exciton easily decays at ambient temperature in three dimensions, the same in two dimensions may be hardly de-excited.

7.7

Surfaces

It is recognized that the Bloch theorem formaHy holds in a perfect crystal. However, for the Bloch theorem to work, it is implicitly assumed that the crystalline system assures that the Bloch states are coherent throughout the system. To have coherent Bloch states, an infinite long-range crystalline order is needed and no decohering mechanisms are allowed to exist.

66

7 2 0 Quantum Systems

Regarding the long-range order of atoms on a surface, the situation is the same. Coherence must exist for all the surface Bloch states. On surfaces a long-range order of atoms are very often rearranged/reconstructed so that they have a lower cohesion energy. This is because the energy minimum state intrinsic to the surface is not the same as that in the bulk, and, in addition, crystalline imperfections very often occur such as adsorbed atoms/molecules or voids. Such imperfections produce surface states. The other variation to occur on surfaces is accumulation or depletion of electrons at the surface, corresponding to the difference in the Fermi level and vacuum potential. Because surface atoms have no bonding partners in the outward direction, whereas they have several inward, they form dangling bonds which are responsible for the surface reconstruction of atoms. In ordinary bulk crystalline systems, electronic states have the same periodicity as the crystal. The surface electronic states are, in contrast, modulated by means of the envelope wavefunctions subject to the newly imposed boundary conditions specific to the reconstructed surface. Sometimes a charge density wave (CDW) appears, whose wavelength is ordinarily incommensurate with that of the bulk crystal, although the CDW itself is constituted by the Bloch states. In scanning tunneling microscopy (STM) studies of surfaces Slough et al. (1986) succeeded in observing CDW-modulated images of surface atoms on TaSz samples in their very early stage of STM investigation. Another marvelous observation by STM was done by Hess et al. (1989) on the internal and nearby structures of the DOS of quasiparticles in a single fluxon in the mixed state of superconducting NbSea. An astonishing direct observation of surface properties was made by IBM scientists in 1993 (Crommie et al., 1993a; Fig. 7.7). They used an STM to scan a surface with a few atomic steps (terraces) and obtained a straightforward image of the electronic standing waves. The surface monoatomic step creates a potential step for the surface electrons. The potential step imposes new boundary conditions on the Bloch electrons and modulates the wavefunction so that the distribution probability is affected by these boundary conditions; that is, a CDW is created perpendicular to the potential step and the electronic charge is distributed in the STM image as a standing ripple wave. The ripple is obviously caused by the interference between the waves going back and forth. The IBM scientists later proceeded to create do-it-yourself boundary conditions by utilizing an STM-related technique. Their man-made corral was made of Fe atoms that encircled a certain Cu surface area and the generated CDW was confined within the corral. Although not displayed here, the co-centered circular CDW ripple is well worth seeing (Crommie et al., 1993b). Several documents refer to these observed images as a “wavefunction”. Aharonov et al. (1993) proposed that it could be possible to observe a wavefunction itself, if a “protective measurement” is applied to it. However, it is commonly recognized that it is difficult to observe a wavefunction itself by using an ordinary measurement process. Thus, it is safe to say that what we observe is only a squared wavefunction that represents the probability distribution of electrons over various states. To establish this ripple, it is necessary to have coherence in two senses. The first is the coherence for a single electron. This assures that wavefunctions are coherent in their incident

7.7 Surfaces

67

w2

Fig. 7.7: A 500 x 500 constant-current image of a C u ( l l 1 ) surface. Three monoatomic steps and a few tens of point defects are visible. Spatial oscilla15 A are evident. The small circles surrounded by tions with a period of circular ripples are point defects on the surface, whose origin are not known (after Crommie et al., 1993a).

-

Fig. 7.8: Schematic representation of Takayanagi’s DAS model (after Takayanagi et al., 1985). (1) Four Si atoms at the corners of the lowest layer 7 x 7 rhombus are missing. The small open circles neighboring the small solid circles combine in pairs t o form dimers. (2) The next stacking-fault-layer atoms are denoted with larger open circles. Their configurations are faulted between the right and left half rhombus. (3) The dark large circles are for adatoms on the topmost layer. and reflected directions. The other means that an ensemble of electrons must be coherent, in the same sense as the people in the electron microscope field say that an electron source is “coherent”. This meaning ensures that all the electrons involved in the CDW should be the same as one another under the same or very similar quantum conditions. Otherwise, all waves composing a CDW must be added randomly and the ripple would be smeared out.

7 2 0 Quantum Systems

68

All these features represent the local density of states (LDOS) defined by

P ( E ,T ) =

C lQi(r)I2s(E- Ei)

(7.9)

a

where 6(z) is the Dirac’s delta function, and & ( T ) is the wavefunction of the i-state with its eigenenergy Ei. The atomic reconstructed surface is another interesting example of a 2D system. Under reconstruction, atoms are displaced from the bulk positions to construct a new periodicity and the surface Bloch electrons follow the same period. A marvelous example is related to the complicated reconstruction of Si atoms on the Si(ll1) surface, as Binnig and Rohrer and others observed by STM (1983). It has long been known that a surface constitutes a 7 x 7 structure, and this was later modeled by Takayanagi et al. (1985) as a DAS model (Dimer-Adatom-Stacking fault model) based on low-energy electron diffraction (LEED) observation. The schematic structure is shown in Fig. 7.8. This 7 x 7 structure was later re-investigated precisely by STM, and the structure proposed by Takayanagi was confirmed. Such surface reconstruction is realized as a very long-range order, which tells us that the coherence of the surface Bloch states is also long.

References Aharonov, Y., Anandan, J., Vaidman, L. (1993), Phys. Rev. A47, 4616. Ando, T. (1984), Physics of Semiconductor Superlattices and their Applications (in Japanese), Tokyo: Baifukan, p.21. Ando, T., Mori, S. (1982), Surf. Sci. 113, 124 Binnig, G., Rohrer, H., Gerber, C., Weibel, E. (1983), Phys. Rev. Lett. 5 0 , 120. Crommie, M. F., Lutz, C. P., Eigler, D. M. (1993a), Nature 363, 524. Crommie, M. F., Lutz, C. P., Eigler, D. M. (1993b), Science 262, 218. Esaki, L., Tsu, R. (1969), IBM Research Note RC-2418. Esaki, L., Tsu, R. (1970), IBM Research J. Res. Develop. 14, 61. Esaki, L. (1990), in: S. Kobayashi, H. Ezawa, Y. Murayama, S. Nomura (Eds.), Proc. 3rd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, p.369. Hess, H. F., Robinson, R. CB., Dynes, R. C., Valles, J.M., Wasczak, J . V . (1989), Phys. Rev. Lett. 62, 214. Landau, L. D., Lifshitz, E. M. (1977), Quantum Mechanics, 3rd ed., Oxford: Pergamon Press. Schiff, L. (1955), Quantum Mechanics, 2nd ed., New York: McGraw-Hill.

7.7 Surfaces

69

Slough, C. G., McNairy, W. W., Coleman, R. V. (1986), Phys. Rev. B34,994. Stern, F., Howard, W. E. (1967), Phys. Rev. 163, 816. Takayanagi, K., Tanishiro, Y., Takahashi, M., Takahashi, S. (1985), J. Vac. Sci. Technol. A3, 1502.

This page is intentionally left blank

8

1D Quantum Systems

A one-dimensional quantum system is a system where two-dimensional degrees of freedom are frozen, whereas the other degree is still active. Namely, it is a wire-like world and the unique degree of freedom runs along the wire. Such a system is often called the quantum wire. The readers will accept that the Bloch theorem holds along the wire, e.g., specified by a good quantum number k,. However, if the system obeys boundary conditions that an infinite potential surrounds the surface of the wire, geometrical confinement quantizes the state perpendicular to the wire. Let us assume the cross section is rectangular with length .(I/ x) and b(ll y). Then, it is easy to derive its quantized energy scheme around the band bottom to be

where mz, m;, and m: are, respectively, the effective mass in the x-, y- and zdirections. For another boundary condition, namely, a circular cross section of the wire with radius R , the effective mass equation in the cylindrical coordinates reads

where m is the magnetic quantum number and the eigenstate was assumed to have cylindrical symmetry, i.e., qnrn(r,8) = $nm(r) exp(im8). Fkom the well-known property of the Bessel function, it is easy to obtain quantized energies

in terms of the n-th zeros 2 1 ~ 1of, the ~ Iml-th order Bessel function. The boundary condition requires that qnm(r,8) = Jim/( ~ l ~ l , ~ r / Rshould ) e ~ ~vanish ' at T = R. Several zeros are listed below (Moriguchi et al., 1956-60).

8 1D Quantum Systems

72

k 1 2 3 4 5

z0,k

zl,k

2.4048 5.5201 8.6537 11.7915 14.9309

3.8317 7.0156 10.1735 13.3237 16.4706

z2,k

z3,k

z4,k

5.1356 6.3802 7.5883 8.4172 9.7610 11.0647 11.6198 13.0152 14.3725 14.7959 16.2235 17.6160 17.9598 19.4094 20.8269

In addition, in such a system, a thinness of less than N 100 A is required to obtain a considerably large energy level separation under an ambient temperature such as a few 100 K, even if the effective mass is small enough (cf. Eq. (7.3)). The DOS peculiar to a one-dimensional system was roughly sketched in Fig.4.1. There each point of steps corresponds t o En, for a rectangular cross section (Eq. (8.1)) or En, for a circular one (Eq. (8.3)). In order to fabricate such quantum wires, one method available is a lithographic technique to obtain thin lateral widths starting from a flat material surface. For rather thicker wires an ultraviolet (UV) beam or X-rays may be utilized. However, for thinner ones, only electron beam lithography can attain such thinness. It may be said that a thin wire of less than 1000 A is very hard to obtain in a stably reproducible manner even with the latest technology. The other method is to apply the so-called scanning probe microscope (SPM) technology. The most easily available SPM is the scanning tunneling microscope (STM). The method can move individual atoms from one region to another or remove them from their existing region by applying a high voltage between the microprobe and the sample surface. By this method it is easy to obtain a wire as thin as 10 to 100 A,but difficult to obtain rather thick ones of around 1,000 A. It is often said that a thin wire may be fabricated by a “self-organization” scheme. For example, the depassivated quantum wire on a Si surface shown on the cover of this book may accept Ga atoms just onto the wire; such wire, however, can only be fabricated following a prefabricated structure in a “self’-organized manner. In particular, it is extremely difficult to obtain self-standing quantum wires by whatever methods. Ordinarily, thin wires are fabricated on some substrate. Hence, there is obviously a certain electronic interaction between the wire and substrate. This means that the wire does not really perform as a self-standing quantum wire in most cases. We will further discuss one example of such a quantum wire in Chapter 10.

References Moriguchi, S., Udagawa, K., Hitotsumatsu, S. (1956-60), Mathematical Formulas I, 11, 111, Tokyo: Iwanami.

OD Quantum Systems

9

An zero-dimensional system produces an atom- or cluster-like state. It has no energy bands, but a series of discrete spectra. It is sometimes called the quantum dot or quantum box. Let us consider a tiny sphere of radius R. It must resemble a spherically symmetric atom. An EMA equation to consider is similar to that of a hydrogen atom except for the Coulomb potential, that is,

where Rnlmis proportional to the spherical harmonic function for its angular dependence K m ( 9 ,cp), as was the case in a hydrogen atom and discussed in Appendix D. By treating this equation in a dimensionless form, the reduced equation reads

This equation is known t o have the spherical Bessel functions j l ( p ) = a J l + 1 , 2 ( p ) as solutions, the boundary conditional requirement of which is that they should have no singularity at the origin. J1+1/2is the (cylindrical) (1+1/2)-th order Bessel function (Moriguchi et al. 1956-60). From the boundary conditions that j l ( p ) vanishes for p = RJ=/A, eigenen/ R )obtained. ~ Here Cln is the n-th zero of j l ( p ) . lli ergies Enlm = ( F ~ ~ / 2 m * ) ( ( l , are and mli are, respectively, the orbital and magnetic angular momentum. Energies are degenerate for different m. Now, p = r(ln/R. We will list several specific lowest-order spherical Bessel functions. jo(p) = jl

(p)

p-l sinp

= pP2(sin p - p cos p )

j2(p)

= pP3[(3- p 2 ) sinp - 3pcospl

j,(p)

= ~-~[(1 56p2)sinp - p(15 - p 2 ) cosp] = p-5 [(105 - 45p2 15p4)sin p - p( 105 - lop2)cos p]

j4( p )

+

(9.3) (9.4) (9.5) (9.6) (9.7)

... It seems easy to obtain zeros of these arithmetically fairly simply defined functions. For example, (0, = nr, n = 1,2,3,... is obvious.

74

9

OD Quantum Systems

In the case of a tiny disk (microdisk) with radius R and thickness d, if surrounded by an infinite potential barrier, a similar EMA equation gives the eigenenergy as

with ~ 1 ~ the 1 ,n-th ~ zero of the Iml-th order Bessel function. (See Table I in Chapter 8). Such a microdisk is becoming easier to fabricate using the advanced semiconductor miniaturization techniques. There is another way to obtain mesoscopic quasi-zero-dimensional systems, or microdisks. They are actualized in a thin layer surrounded by a Schottky barrier. The nature of a Schottky barrier was interpreted in Chapter 5 and the geometry is depicted in Fig. 9.1. When an n-type uniformly doped disk is depleted by applying a negative potential on a gate surrounding it, the potential is given by the solution of the Poisson equation, i.e., a quadratic function of the radial coordinate. First we will solve the eigenstates in such a system. In the Hamiltonian (9.9) we separate the angle from the radial coordinate, the former of which belongs to an angular momentum mA, when assuming cylindrical symmetry, as follows.

$ ( r , 0) = R(r)eime

(9.10)

Now the radial part of the equation reads

[:$

m2 ( r i ) - r2 -

k]

2m* h2

R ( r ) = ---ER(r)

E

E - -R( r).

4

(9.11)

1; = A/m*wo is the characteristic length for the linear oscillator. Let us change r into p = r 2 , and take x ( p ) = p n F ( p ) . If we put n = lml/2,

pF”

P + ( I m l t 1)F’- -F 41:

If we take F = eaP@ and

(Y

E

= - - F.

(9.12)

4

= -1/21;, the final equation is

( p=p/li) (9.13)

and it is known that this equation is satisfied by the generalized Laguerre polynomials (Appendix D; Moriguchi et al., 1956-60). As v = &/4, the eigenfunction thus obtained is (9.14) with E,,, = 2v(h2/m*l,”j = 2vhw0. From the angular property of the wavefunction m must be &(integers) and the Laguerre function demands v - (Iml + 1)/2 to be a

75 positive integer. Consequently the above solution is rewritten, as follows (Fock, 1928; Darwin, 1930).

En,

= (2n

+ Iml + l)tZwo,

m = O , f l , f 2 , ..., n = 0 , 1 , 2 , ....

(9.15)

= 2hw0, Thus, the ground state energy is EOO= hwo, the first excited state is and the second excited state is EO*2 = Elo = 3hw0, and so on. Such a microdisk as an artificial atom was prepared using heterojunctions and measured by Tarucha et al. (1996). In their sample a quasi-harmonic potential in the radial direction was formed by the Schottky junction, and the confinement perpendicular to the disk was affected by a three-barrier heterostructure, i.e., a pair of quantum dots.

I

I

H

Ri{,yot /

side gate

I

n-GaAs

3

Fig. 9.1: Actualization of a cylindrically symmetric microdisk having a harmonic potential as a function of the radial coordinate.

According t o their investigation, the number of accommodated electrons N showed a shell model-type structure. As we calculated above, a series of closed shell structures were observed at N = 2, 2+4, 2+4+6, ... (including spin degeneracy by 2 ) . In addition they observed that spin degeneracy occurs with a considerable amount of correlation energy; that is, even if the quasi-atom is stabilized for N = 2, the state occupied by the first electron differs in energy from the state for the second electron by the correlation energy. This is just the same effect as the Coulomb blockade, which will be explained in Chapter 10. Nowadays the most pronounced zero-dimensional quantum system must be clusters called C60, or more generally fullerenes (Kroto et al., 1985; Dresselhaus et al., 1996).

76

9 OD Quantum System

They are clusters made of pure carbon atoms. A Cs0cluster made of 60 carbon atoms is spherically symmetric, but is empty inside. One of the most remarkable derivatives in fullerene science is the carbon nanotubes (Iijima, 1991; Iijima et al., 1992; Dresselhaus et al., 1996). They are formed t o be basket-like tubes with both ends open or terminated, and made of networked carbon atoms of a few nm in diameter. Carbon nanotubes may be a one-dimensional rather than a zero-dimensional system.

References Darwin, C. G. (1930), Proc. Cambridge Philos. SOC.27, 86. Dresselhaus, M. S., Dresselhaus, G., Eklund, P. C. (1996), Science of Fullerenes and Carbon Nanotubes. New York: Academic Press. Fock, V. (1928), 2. Phys. 47, 446. Iijima, S. (1991), Nature 354, 56. Iijima, S., Ichihashi, T., Ando, Y. (1992), Nature 356, 776. Kroto, H. W., Heath, J. R., O’Brien, S. C., Curl, R. F., Smalley, R. E. (1985), Nature 318,162. Moriguchi, S., Udagawa, K., Hitotsumatsu, S. (1956-60), Mathematical Formulas I, 11, 111, Tokyo: Iwanami. Tarucha, S., Austing, D. G., Honda, T. (1996), Phys. Rev. Lett. 77, 3613.

10

Transport Properties

10.1 Transport Perpendicular to QW Resonant Tunneling Devices Esaki and the group at IBM first expected nonlinear conductance in a superlattice (SL), which carries a current perpendicular to the two-dimensional planes of SL (Esaki et al., 1969, 1970). However, they observed only a slight decrease in the current as a function of applied voltage, contrary t o their expectations. They later developed an idea called the “resonant tunneling” effect (Tsu et al., 1973; Chang et all 1974; Sollner et al., 1983; Mendez et al., 1996; for a monogram, Mizuta et al., 1995). The basic construction of a resonant tunneling device (RTD) is a QW sandwiched by a pair of barrier layers, i.e., double-barrier RTD, as shown in Fig. 10.1. For a more pronounced performance, multiple-barrier RTDs, i.e., those with multiple QWs every two of which are separated by a barrier, are now being proposed and studied. In a double-barrier RTD, there is a source of carriers on the left-hand side (an emitter) and a drain on the right-hand side (a collector). Between the sides are a pair of barriers and also a QW. The electronic states in the QW are known t o be those in two dimensions, that is (see Eq. (7.1) and below), (10.1) if we take (kz,ky) in the two-dimensional layer. The degree of freedom of the electrons in the k, direction is confined by the two barrier layers. In the figure, En is shown by a horizontal line in the QW. According to the well-known quantum tunneling effect, those carriers incident from the left-hand-side electrode can tunnel into the QW, when the potential barrier is thin and of sufficiently low height. When tunneling, the conservation law of energy E n k , k , is obeyed. Under an electric potential difference applied between the emitter and collector, En goes down relative to the energy in the emitter, since the barrier is usually a non-doped semiconductor with a high resistivity, so that a considerable potential drop occurs within this barrier layer. Similarly, another potential drop occurs within the other barrier. Let us assume the emitter to be an n-type semiconductor with considerable dopants and all regions to have the same effective mass, for simplicity. If we take the origin of the energy to stand at the conduction band bottom, the energy conservation law

10 Ransport Properties

78

reads

(10.2) for the nearly resonant state Eo. An electron in the Fermi level is assumed t o have a wavevector kf. Since the transverse component kI of the momentum across the QW conserves, i.e., kf, = kk and kf, = kh, the above equation requires that kf, =

JziE!%/h. As is known in the treatment of tunneling phenomena, a forward and a reverse current across a barrier cancels out exactly, when there is no difference between energy states on both sides. This is because tunneling current flows only between equal energy states. It is also known that the tunneling probability depends only on the energy. Let us assume that a finite energy difference equal to eV exists due to an applied electric potential, obviously the forward current is proportional to f(E)[l- f(E e V ) ]and the reverse one to f(E+ e V ) [ l- f ( E ) ] both , of which cause a net current proportional to [ f ( E )- f(E e V ) ] .That is,

+

+

=

2e

1

- dE,dk:[f(E)

(243fi

-f

( E + eV)]T(E,)

(10.3)

where E, = li2k,2/2m*.The transmission probability T ( E )for a single barrier is given by the following formula (e.g., Mizuta et al., 1995):

(10.4) with IE, = J2m*(Vo - E,)/li. VOis the barrier height and d is the thickness of the barrier. Now, we interpret a sketchy I - V characteristic of an RTD. For very low electric potentials hardly any current flows. When approaching the potential satisfying the energy conservation requirement for the lowest resonant state energy Eo, the current increases, but above this potential it again decreases (offresonance condition). For a higher voltage, another resonance condition may be satisfied, e.g., for E l , where we must have another peak of the current. Thus, there is a differential negative resistance (DNR) characteristic between EOand El. Figure 10.2 is the I - V characteristic data obtained by Tanoue et al. on an InAlAs-InCaAs double-well (triple-barrier) RTD at 127 K. Here we must emphasize that no DNR devices show a net negative resistance. If they did, then electric power might be stored inside them, i.e., they would act as an endothermal phenomenon, which contradicts the second law of thermodynamics. Thus they are only allowed to have a local, differentially-negative I - V characteristic. This tunneling device reminds us the Esaki tunnel diode (Esaki, 1958, 1974, 1976). In this case, a similar I - V characteristic is obtained when a heavily doped p n junction diode is utilized (Fig. 10.3). In the forward bias condition, there is not much current for the low potentials, but it increases when the potential difference is large

10.1 Transport Perpendicular to Q W

79

GaAs AlGaAs GaAs AlGaAs GaAs

potent i a I d i fference

potent i a I difference Fig. 10.1: Presentation of the configuration of an RTD and suggested operating principle.

Fig. 10.2: Observed DC current vs DC voltage showing a differential negative resistance on an InAIGa-InGaAs double-well (triple-barrier) RTD with a mesa area of 6 pm x 6 pm measured a t 127 K. (After Tanoue et al., 1988).

80

10 Transport Properties

V

V

Fig. 10.3: Presentation of the configuration of an Esaki tunnel diode and its operational principle. (a) Reverse bias; (b) zero bias; (c) forward bias with a peak current; (d) forward bias with a minimum (valley) current; (e) forward bias with an ordinary thermal, field-driven current.

10.2 iknsport Parallel to QW

81

enough to make the depletion layer between n- and p-region sufficiently thin and a tunneling current component dominant. For larger voltages the energy states of electrons in the n-region fall equal to such values as those that do not exist in the valence band but in the forbidden energy gap. The current decreases again, since tunneling probability is suppressed. Above such biases current increases following the ordinary forward-bias diode characteristic, where electrons in the n-region flow into the conduction band in the p-region. The Esaki diode requires no pair of barriers as in RTD, since electrons tunnel into hole states, which current disappears under above an appropriate potential difference. Meanwhile in RTD, electrons tunnel into electronic states across the first barrier. Such electronic states may appear not only once but numerous times for some larger voltages.

10.2

Transport Parallel to QW

Crystalline Anisotropy-Related Nonlinear Transport in MOS As already described in Section 5.5, an n-MOS has a built-in electric field, say, F , between the oxide- semiconductor interface and the inner semiconductor. This field tilts the bottom of the conduction band and confines electrons in a narrow region called the “n-channel”. In the case where the interface is made of an Si-[OOl] surface, size-quantization occurs in the direction perpendicular t o the interface (z-direction from now on) and electrons behave freely in the channel in the zcy-plane. The first Brillouin zone of diamond-structured Si is a truncated octahedron, like that shown in Fig. 10.4(a) (Brillouin, 1963; Ziman, 1964; Chelikowsky et al., 1976). The shape has a high rotational symmetry. On the other hand, the conduction band of n-type Si has a well-known anisotropic feature, as shown in Fig. 10.4(b). The band minima exist around Ic, = 27r/a[0.85,0,0] and its equivalents, where a is the lattice constant. These equivalent energy minima are called “valleys”, so that Si is structured by six-valley conduction bands. From now on we will briefly refer t o each as a [loo]-, [010]-, [001]-valley and so on. The equienergetic surface of electrons in a [loo]-valley is an ellipsoid with a long axis in the x-direction and two short axes in the other directions. This means that the effective mass mi in the Ic,-direction is larger than the transverse one mt in this valley. Let us consider a quantization scheme caused by an electric field applied in the z-direction. Then the size quantization occurs with mi in the [001]- and [00T]-valleys, whereas the same occurs with m,+in the [loo]-, [TOO]-, [010]-, and [OiOI-valleys. So, as suggested in Fig. 10.4(b), the quantized energy in the [001]-valley is (see Eqs. (F.lO) and (F.12) in Appendix F) (10.5)

82

10 Transport Properties

ENERGY

e

Fig. 10.4: (a) Illustration of the first Brillouin zone for diamond structured silicon. The shape is a truncated octahedron (After Chelikowsky et al., 1976). (b) Illustration of the conduction band of Si, when an electric field is applied in the [OOl]-direction. According to the effective mass anisotropy, the lowest energy band appears in the [001]- and [OOI]-valley for n = 0 as Era'] with an effective mass mt in the zy-plane. The next lowest band may be Erol] or EIool (or its equivalents) depending on the magnitudes of F , m; and mt. (After Murayama et al., 1972).

10.2 Transport Parallel to QW

83

6

5 4

3

?)2 \

h

l

P a,

-;n

co

a,

-I -2 -3 -A

L

ctiii

r

CIOOI x

L

CIIII

r

CIOOI

x

wavevecto r

wavevector

3h

300 K

(b)

in

GaAs

I

1

1

2 E / (MV / m)

I

I

3

Fig. 10.5: (a) Energy band structures of Si and GaAs, where E, is the energy bandgap. (+) and (-) indicate, respectively, holes in the valence bands and electrons in the conduction bands. (After Chelikowsky et al., 1976). (b) Measured data of velocity vs electric field in GaAs (after Ruch et al., 1967).

84

10 Transport Properties

dipole layer

I

0

L

high f i e l d domain

b X

(d)

(el

(c) Assumed fluctuation in the density of carriers to form a tiny dipole layer (d) Plausible field distribution corresponding t o the carrier distribution (e) . whereas the quantized energy in another direction is

(10.6) and ELoo1] <

loF&'

= Ek1'1

An introduction to the interpretation of the relationship between anisotropy and the quantized energy scheme was so far given. When both doping level and temperature are low enough, electrons occupy only the lowest quantized state, say, E,[ O O l I . When the number 3f electrons increases according t o an applied positive gate voltage, they rise t o occupy higher quantized states as well. The ambient temperature has the same effect. For a higher temperature, electrons begin to occupy higher states. Such

10.2 Transport Parallel to QW

85

variations in the occupation mode of electrons should cause the transport properties to change. Historically Gunn (Gunn, 1962, 1963) was the first to observe a differential negative resistance (DNR) effect in a bulk GaAs single crystal. GaAs has an energy band scheme as shown in Fig. 10.5(a). The lowest band for electrons is known to situate at the so-called r-point, i.e., the center of the Brillouin zone. (See the Brillouin zone of Si shown in Fig. 10.4(a), which is essentially the same as that of GaAs.) On the other hand the next higher conduction band minimum is around the L-point. The energy difference between the r- and L-bands is 0.31 eV. Under a high electric field, carriers ordinarily have a high velocity. However, in GaAs, under a high field, electrons are sufficiently accelerated and become “hot”, i.e., a type of “hot electron effect” occurs (Ridley et al., 1961; Hilsum, 1962; Kroemer, 1964). An elevated electron temperature brings a considerable fraction of the carriers into a higher energy band, where they flow slower than in the lowest band, thus revealing a DNR phenomenon. A Gunn diode is made of bulk GaAs, which generates an oscillating current under a constant DC voltage according to this DNR effect. The mechanism is as follows. First consider Fig. 10.5(b), where data for velocity vs electric field curve are plotted. The velocity of carriers v is given by p ( F ) F where the mobility p is a function of the electric field F . Let us assume that a tiny fluctuation in carrier density occurs by chance (Fig. 10.5(c)). It produces a high voltage region between the regions with, respectively, increased and decreased carriers as in in Fig. 10.5(d). The electric field is connected with accumulated or depleted carriers through a Maxwellian equation: c,dF/dx = eAn, where F increases for positive An whereas it decreases for negative values as shown in Fig. 10.5(e). Thus the dipole layer is called the “high field domain”. For an elevated electric field, their velocity may decrease. This leads to the carriers at the front of the domain being decelerated and those at the rear being accelerated. Thus the dipolar distribution of the carriers is enhanced and the initial instability is stabilized by forming a domain. On the other hand, a reversed dipolar distribution, i.e., accumulated carriers at the rear and depleted ones at the front would not be enhanced by the same DNR effect. In GaAs a high field domain propagates as it grows between the two ends causing a depression of the current. On disappearance of the domain the current increases. Thus, an oscillating current is generated. Next it should be asked whether such a domain can be formed or not (Hobson, 1974; Sze, 1981). A necessary condition is that the device is long enough to grow the domain while it is also running through it. The time needed to grow the domain is TR determined by the CR time constant. For a domain that is S wide and 1 long, C = v;cOS/l and R = l / a S = l/nepS, accordingly T R = v;co/nep. Here, p is a certain typical value of mobility and 6 is the relative static dielectric constant. Thus, L > TRV must be fulfilled, for a device of length L, of typical velocity v inside the device and carrier density n. This condition can be rewritten as n L > rccov/ep. For GaAs it is 10l2 cm-2. If nL < rccov/ep, a stable field known that the right-hand side equals distribution is established and the device does not oscillate. In an n-MOS, Katayama et al. (Katayama et al., 1972) observed similar DNR

-

10 Transport Properties

86

phenomenon at sufficiently low temperatures. The reason why the mobility of an nMOS decreases with higher electric fields may be attributed to the same mechanism as in the Gunn diode. However, the fact that they observed no oscillatory current in the n-MOS sample must be questioned. The estimated nL product in this case was found in the region where there was a stable field distribution differing from the long Gunn diode, where an oscillation actually occurs.

10.3 Magnetic Response Classical Response Let us discuss the response of a magnetic field in condensed matter. It is known that a magnetic field does not perform any mechanical work in the direction parallel to the field. Accordingly, it is sufficient to consider a two-dimensional system to check the effects performed by a magnetic field. In other words, three-dimensional systems respond to the field in the same manner as two-dimensional systems do. First we will start with a classical response. A Hamiltonian of N electrons in a magnetic field in a vacuum is given by 1 H = -(pi - eAi)2 (10.7) 2m i

C

As is known, an orbital magnetic moment of i-th electron is combined with a circular current through e mi = -Ti x vi (10.8) 2 where ri and wi are, respectively, the coordinate and velocity of i-th electron. For a field B in the z-direction the magnetic response mi, parallel to the field is (e/2)(zitji yiii). The vector potential is A = B(-y/2, z/2). In classical Boltzmann statistics,

where Z is the partition function of this system. Now considering that xi = aH/apxi and tji = dH/dpyi, it is easy to show that the integral is proportional to

which must vanish because it is obvious that the Boltzmann factor goes to an infinitesimal for p + f m . These statistics prove that no magnetic moment exists as a result of the orbital magnetic response so long as classical statistics are employed. This theorem is known as the van Leeuwen theorem (van Leeuwen, 1921; Peierls, 1979).

10.3 Magnetic Response

Fig. 10.6: Illustration of classical Ramour orbitals eventually producing a large anticlockwise orbital enclosing all of them, since each inner part is canceled by the inner part of the other neighboring one. If the sample has a definite boundary, then the above-mentioned large orbital is also canceled by a series of clockwise, incomplete Ramour orbitals engendered by the existing boundary.

Fig. 10.7: (a) Illustration of the original Aharonov-Bohm configuration. (b) Simplified version of Fig. 6.1 explaining the Anderson localized orbital. A wavefunction is first divided at the 0-site and eventually superposed at the A-site after a series of elastic scatterings, the amplitude of which is given in the figure. 0 = C , O i ( r Z+ ) k L is the total phase acquired after traveling along the path and x = 2.ir@/@O is the Aharonov-Bohm phase.

87

10 D-ansport Properties

88

The physics existing in the theorem is depicted conceptually in Fig. 10.6. In the figure all magnetic moments are assumed to be generated by equal circular orbits. At the interface between two circles the current is effectively canceled to leave an anticlockwise circuit enclosing all four orbits and a small clockwise inner circuit. The latter inner circuits go to a zero moment if all the circles are taken as being infinitesimally small. In addition to the large encompassing circuit there is another almost linear downward flow along the boundary of the sample. This flow eventually encloses the whole sample clockwise. Thus a diamagnetic response caused by an inner anticlockwise circuit is completely canceled by an outer clockwise flow along the boundary. This fact demonstrates that some sort of quantization is necessary to explain the magnetic response actually observed, that is, we need quantum mechanics as well as quantum statistics. We have already interpreted the Landau quantization scheme in quantum mechanics in Chapter 4. Now we will proceed to the phase effect of an electron under a magnetic field.

Aharonov- Bohm Phase and Magnetic Flux The EMA equation of an electron under a magnetic field is

[

&(p

I

- eA)2+ V ( r )

$ ( r )= E 9 ( r )

(10.11)

and under the gauge transformation, that is, 9 ( r )= $ ( T ) exp(ie/h) J A . d r , it is easy to obtain a gauge transformed equation (10.12) Apparently the magnetic field disappeared from the equation, but the eigenfunction has an additional phase factor according to the field. It is obvious that the electron feels the magnetic field exclusively through the phase of the wavefunction, when it passes through the vector potential field. If it flies along a path, the overall phase acquired should be expressed as ( e l t i ) A . ds, where integration must be taken along the path. Let us consider the configuration shown in Fig. 10.7(a), which can be interpreted as an interference experiment. Originally Aharonov and Bohm (1949) assumed that the tiny solenoid was infinitely long that produces a magnetic field perpendicular t o the plane for the electron to pass (Ruijsenaars, 1983). The electron incident at 0 is divided by the solenoid and passes through a clockwise or anticlockwise semicircle to be superposed at A. They claimed that the electron interference pattern obtained at site A should be a function of the field, even though the electron is never in touch with the field. This point has long been controversial but a t the present moment the issue is being settled. Although no magnetic field exists, a vector potential actually exists in the region, where the electron passes through. It is just the vector potential that causes an additional phase for the electron passing through the field.

89

10.3 Magnetic Response

The phase difference given to the electron at site A is expressed as (e/h)(J& - Jc,)A. ds, which is converted to (e/h) S,(rotA),dS = (e/h) J, B,dS according to the Stokes theorem, and eventually t o 27r@/@0.n is the normal of the surface to integrate on and is the areal integral. @ is the magnetic flux passing inside the circuit. $0 = h/e is the flux quantum in the normal state. (In superconducting states the same expression becomes h/e* = h/2e, as will be discussed in Chapter 13.) The readers should note that the phase given to the electron beam at site A is simply the difference between those phases acquired on both paths, not the phase itself; accordingly it can be converted into a magnetic flux. The absolute value of the phase is never observable. In interference experiments that usually occur easily in any condensed matter environment, an applied magnetic field affects the interference pattern through the phase = cos 27r@/@o;that is, for a magnetic flux @ = B,S difference factor R(exp i27~@/@0) equal to an integral multiple of @o, visibility becomes bright (“constructive superposition”), whereas that equal to half-an-odd-integral multiple, becomes dark (“destructive superposition”). We will proceed further to discuss a variety of Aharonov-Bohm type experiments. In Chapter 6, we discussed the fact that Anderson localization is interpreted as an interference phenomenon of an electron on which multiple elastic scatterings are exerted. A schematic representation of the phenomenon was shown in Fig. 6.1. Now we will consider a magnetic field applied to the circuit causing localization, as shown in Fig. 10.7(b). The figure was highly simplified just to give the essence of the interference. Along the circuit there exist lots of scatterers which may change the phase of the electron. For simplicity the phase change is assumed to occur homogeneously over the whole circuit. Then an electron circumventing the whole circuit suffers a phase 0 = CiO,(Ti) + IcL. The first term represents the phase given by scattering and the second term assumes free flying along the path L. L is the perimeter and Ic is the wavevector. If the circuit is divided into fractions c and 1-c, then an additional phase factor eiC(@+X) is given to the wave passing through the c part of the circumference. Correspondingly, the Aharonov-Bohm phase is c27r@/@o. The other wave passing through the 1 - c part is given a factor ei(l-c)(O-x). For the Aharonov-Bohm phase the sense is opposite to that for the pass c and 1- c, which thus have opposite signs. The superposed wave at site A is, hence,

ss

q ( A ) = $(c) + $ ( I - c )

(10.13)

- ~ ~ ~ i c x ~ i ( @ -ei~0e-i(@-~)/2 x)/2 + complex conjugate

(

)

(10.14)

In particular we are interested in the c = 1/2, c = 0 and c = 1 cases. They are (10.15) (10.16) (10.17)

10 Bansport Properties

90

and when sites 0 and A coincide, i,e., c = 0 or c = 1, *(A; c = 0) and *(A; c = 1) waves must be added, that is, the interfered electron intensity is )*(c=O)+P(c=

1))2 =2)~rJ)2(1+cos2x+2cosxcos0)

(10.18)

This final result means that in this configuration the electron interference intensity varies not only with a period B,S = a0 = h/e but also with a period B,S = @0/2 = h/2e. S is the area inside the circuit.

Aharonov- Bohm Effect - Experiments Figures 6.1 and 10.7(b) showed a conceptual description of Anderson localization, the point of which was that after multiple elastic scatterings the wave comes back to itself, and the original and the returned waves can be superposed either constructively at a certain site or destructively at other sites. When the superposition is constructive, the wave can have a larger amplitude than the average. In 1981, Sharvin and Sharvin (1981) (also Aronov et all 1987) prepared a quartz fiber with a diameter of N 2 pm covered with a vacuum-evaporated thin Mg film, and measured the transport properties of the sample with two leads separated from each other in the direction of the fiber under a parallel magnetic field (Fig. 10.8). This experiment was suggested by a previous theory of Al’tshuler-Aronov-Spivak (1981). The evaporated Mg must be disordered and show a sort of localization. This sample simulates the geometry shown in Fig. 10.7(b) with c = 0 and 1. There, scattering occurs against the crystalline defects, not within an analytically connected but within a disconnected region (thin Mg film only covers the outside of the quartz fiber). In their experiments shown in Fig. 10.8, the input and output leads are considered as being situated above and below, on the same line, along the fiber (Sharvin et al., 1981). In other words, point 0 coincides with point A. Hence, as discussed above, it is expected that interference intensity, i.e., visibility, varies with a period A B = &/2S = hf2eS or AB = @o/S = h/eS, depending on the degree of contribution determined by the weight of cos0. They actually observed the former period as a function of magnetic field B, by measuring the magentoresistance. Another conventional Aharonov-Bohm effect in the solid state was observed using a small metallic ring, by an IBM group, Webb et al. (1985a, 1985b; Washburn, 1985) (Fig. 10.9). In this case, the effect is understood to be analogous t o the electron beam interference experiment explained in Fig. 2.1. The period is A B = @o/Sin contrast to the Sharvins’ experiment. The difference comes from whether the propagation of the waves occurs along the fiber axis or circumvents a circle turning around in one, when the orbital is localized within the plane perpendicular to the direction of B. After the first observation of periodic variation of conductance through a small ring, other experiments improved the periodic behavior by using samples with a higher aspect ratio (the diameter versus the width of the ring). This improvement results from the fact that the magnetic field can penetrate the arm of the ring itself, when the arm is very wide. An enhanced aspect ratio gives fewer paths to be averaged over. If there are many paths inside the ring arm, the Aharonov-Bohm effect occurs for each path and the conductance must eventually be averaged over all the paths.

91

AR/R

.

,AR/Q

- .1 0 1 2 3 4 5 BImT Fig. 10.8: Observed Aharonov-Bohm effect on a disordered thin Mg hollow cylinder under a magnetic field. The period A B agrees roughly with +0/2S = h/2eS with S being the area penetrated by the magnetic field. 29.6

I

29.5

G ..

\

29.4

29.3 0

2

6

4

a

HIT

(bj 2 1

; i l

loOD Ih/e/

large ring T = 0.06 K

1.Opm

(a)

I

100

200

AH-’/ T - l Fig. 10.9: (a) Photo of a small Au ring sample. (b) Measured variation in resistance vs magnetic field B and its Fourier-transformed spectrum vs 1/B (after Washburn et al., 1985).

300

10 Contents

92

The Aharonov-Bohm effect with a period @o/Sis sensitive to the ensemble average. This fact is evident from the strong dependence on the aspect ratio of the sample. On the contrary, the previous Sharvin-Sharvin’s effect, with the period of @0/2S, is not sensitive to the ensemble average.

Concepts of Electric Conductivity

10.4

Kubo Formula It is known that any linear electric circuit has minimal thermal noise, or Johnson noise, and it is proportional to the inherent resistance and the temperature. Ordinarily noise is expressed in terms of fluctuations in some specific electric quantity. Some time ago Johnson (1927) discovered that fluctuations in electromotive force occurred between both ends of a resistive element and Nyquist (1928) formulated this fluctuation, i.e., noise, as follows 00

[

V ( t ) V (+t s)eiwsds= ~ R ~ B T

(10.19)

J-00

where R, T and k g are, respectively, the resistance, temperature and Boltzmann constant. The overline means that an average over the time s is taken. If ergodicity is assumed, time average is equated with the ensemble average. This kind of thermal noise (Lee,its power is proportional t o temperature) is called Johnson noise or Nyquist noise. Equation (10.19) is called the Nyquist theorem. It has a remarkable significance in that fluctuation is related to resistance causing dissipation, although it may be difficult to intuitively imagine this type of relationship. Whether it is difficult or not, such a relationship is often referred to as the “fluctuation-dissipation” theorem, which holds universally between various types of fluctuation and dissipation phenomena. We can slightly rewrite the above equation using the Ohm’s law, V ( t )= R J ( t ) , 00

J ( t ) J ( t + s)eiwsds= 2kgT/R

(10.20)

which seems to be a reasonable relationship for defining conductance G = 1/R in this circuit. In a quantum statistical version, Kubo (1957) was the first t o define electric conductivity per unit volume in a similar context (10.21)

, = xeiwip (vi is the velocity operator in terms of a one-particle current operator j of i-th electron) and j,(-ihA) eX3Ljue-XH.R is the volume of the system and R is the unperturbed Hamiltonian.’ For a sufficiently high temperature in comparison

=

’The term “unperturbed” means in this context that the Hamiltonian does not include the electric potential term, which brings the system t o a non-equilibrium state. T h e Hamiltonian may include any scattering terms which are effective t o causing electric resistance.

10.4 Concepts of Electric Conductivity

93

with the energy eigenvalues of concern, i.e., e*AR x 1, the integral over X is easily obtained and gives 1 / k B T with jv(-i7iX) x jv(0),which corresponds to the classical formula Eq.(10.19). The ensemble average of any correlated operator product A and B ( t ) is given in quantum statistics by ( A B ( t ) )= TrpAB(t) with the density matrix p = e-H/kBT, which should be equal to the time average 1

rt’

(10.22) under the ergodicity assumption. It is said that the Kubo formula defines conductivity in the most general and exact way so far as the electric response is approximated up where it is linear; that is, it gives the Ohmic conductivity correctly. However, it is rather difficult to actually calculate. To calculate the correlation we are usually obliged to rely on the Green function and/or diagram techniques, but a simple, semi-phenomenological example of the formula is given in Appendix G for readers’ understanding purpose. (See, e.g., Abrisokov et al., 1963; Zubarev, 1971; Landau et al., 1969-80; Toda et al., 1991; Kubo et al., 1991). Kubo also proved sum rules in an exact way :lm9?{uPP(w)}dw

=

ne2

(10.23)

m*

(10.24) for a system of electrons. Here up,, is the diagonal part, such as the (2, x), (y, y) or ( z , z ) components of the conductivity. These two relations suggest that uPP(w) must have an explicit form such as DC(W) uPP

ne2 (-i) m* w - i / r ’

= -~

w+o

(10.25)

It is easy to show that this up,, actually satisfies Eqs. (10.23) and (10.24) as well as the historically well-known DC conductivity (10.26) This equation may be written in another form as

e2 OFF= -n(r) m*

E

r(E)p(E)f(E)dE

(10.27)

where p ( E ) and f ( E )are, respectively, the DOS and Fermi-Dirac statistical function. In particular in two dimensions, since & D ( E ) = (m*/rh2) B(E - Ej) with the step function B(E - Ej), the expression finally reached reads

ci

(10.28) In the above expression the prefactor e 2 / r A is a typical quantity in two dimensions. In fact, in the integral quantum Hall effect (IQHE) to be explained later (von Klitzing

10 Contents

94

et al., 1980), the observed Hall resistance showed remarkable plateaus at h/ie2 with the integer i as a function of the gate voltage in an MOS. After the discoverer of the effect, the 2rh/e2 = h/e2 is called the von Klitzing constant and h/e2=25.812807 k n (e.g., Taylor, 1992). Let us discuss the integral J T ( E ) d E / hfurther. This corresponds to the area in the phase space, i.e., the energy-time product, divided by the quantum action h. In other words, this quantity stands for the number of degrees of freedom visited by a single electron during the average relaxation time. The number may be an integer or a fraction, depending how specifically the system is quantized and further equated to the number of conductive channels in the two-dimensional system. The T is the scattering time or relaxation time which specifies that electrons are scattered within an average time T repeatedly relaxing to an equilibrium value after acceleration due to an electric field. The dominant scatterers causing electric conductivity to be finite are phonons, magnons, and other quasiparticles, as well as impurities, and crystalline defects.

Landauer- Buttiker Formula The other useful, but intuitively defined formula of conductance which is specifically applicable to mesoscopic systems is called the Landauer formula (1957), after the inventor, or the Landauer-Buttiker formula (Buttiker, 1985), attaching the name of the researcher who extended the formula to a more general case. Let us follow Imry’s discussion (1986). As shown in Fig. 10.10, the electron incident .ij from the left channel i tunnels into the right channel j with a probability f ~ C j T i j T is the tunneling or transmission probability with the Fermi-Dirac statistical function f~ representing the existence of an electron in the left-hand side lead. Similarly, fR is the Fermi distribution function in the right lead. Both leads are now treated as reservoirs of electrons. In an ordinary case there are numerous channels between both

region L

region R

.L

f

~

li

v

......

I...

CfRR,l

*

Fig. 10.10: Definition of channel i and its probability of occupation f i around region R and L with a tiny barrier in between. T and R are transmission and reflection, respectively. leads. The electron incident from the right channel i may be reflected into the right

10.4 Concepts of Electric Conductivity

95

channel j with a probability f R C j R i j . (Rij is the reflectivity probability.) The sum of these probabilities minus the equilibrium distribution gives the current transported on channel i from the left-hand to right-hand side, i.e.,

(10.29)

(10.30) where the density of states in one dimension is piD(E) = l / r A v i and C j Rij+Cj Tij = 1 is utilized. p~ and p~ are the chemical potentials, respectively, in regions L and R, the difference between which causes an electric current, i.e., eAV = ( p -~p ~ ) Since . the conductance G is given by C i I i l A V , it is easy to obtain the Landauer formula

G = (e2/rh)CijTij

(10.31)

Very often CijTij is written as tr(ttt) in terms of the transmission probability amplitude t i j as Tij = tijt;,. This formula holds when electrons are fed directly from both of the classical reservoirs. “Classical” means that electrons are completely incoherent within the reservoir due to frequent inelastic scattering. On the other hand, with two “ideal” leads between the sample and reservoirs, the formula is given by Landauer’s original expression:

G = ( e 2 / r h ) T / ( 1- T )

(10.32)

where T is the transmission per channel, as will be discussed below. Note that conductivity and conductance coincide in two dimensions. Thus, conductance is given in terms of the inverse quantum resistance e 2 / r h times the overall transmission T . The transmission is sometimes restated as the number of conductive channels, since each probability of unity, i.e., C j Tij = 1 corresponds t o the channel connecting the i-state with all possible j-states. Let us return to the conductance of the sample itsey, i.e., when there is no difference between the leads and the parts of the sample, e.g., for ideal leads. The density of carriers on the left- and right-hand sides is described, as follows:

(10.33) (10.34)

+

Here a factor of 1 / 2 was used so that the sum of nL and n R gives nL nR = dE p:”( f L f ~ ) Ti . and Ri are, respectively, the transmission and reflection Tij and Ri = Rij. These formulae using such an for i-th channel, i.e., Ti = approximation as is adopted above, give

xi

nL

+

- nR M p L2 r- hP R / d E ( - g )

xj

1 C(I+R~-T,)i

Vi

(10.35)

10 Contents

96

From Eqs. (10.30) and (10.35),the current expression is in another form such as

(10.36) or, using the thermal equilibrium density of carriers n L - n R x n A - n B x -J d E 7rA

(-g)

1

(PA - PB)wi

(10.37)

the conductivity of the sample itself (i.e., not between both of the reservoir leads): Gsample = I / ( P A- P B ) is eventually given by

or, again using Ti = 1 - Ri,

which resembles the formula G = (e2/7rA)T/R= ( e 2 / n h ) T / ( 1- T ) initially given by Landauer. The denominator 1 - T reads to mean R or CiRi in his formula. In these formulae, T = Ti may be larger than unity because Ti ( 5 1) per single channel must be collected to give a total of T . The fact that conductivity is given in terms of a universal constant in two dimensions is elucidated in another way. Conductivity is given by Eq. (10.26):

xi

(10.40) where r is the scattering (relaxation) time, and r = 1/vf and of = Ef/Akf (vf:the Fermi velocity; kf: the Fermi wavevector; Ef: the Fermi energy) were utilized. Since the density of carriers n in two dimensions is expressed as (27r)-1(2m*Ef/A2),it is easy to reach

eL

u = -(kfl)

( 10.41)

7rA

This equation gives a universal constant when kfl = 1. The last condition is called the Mott-Ioffe-Regel limit beyond which conduction is called to be coherent metallic. Obviously an actually observed conductivity may surpass e2/7rA, but one should rigorously discern the cases where kfl itself surpasses unity from the cases where n results from an aggregation of kfl less than unity per channel.

10.5

Universal Conductance Fluctuation

In Figs. 6.1 and 10.7(a,b) it was shown that a localized state is defined on an idealized quasi-one-dimensional ring with a distributed phase on the perimeter. It is important that the state thus defined is localized around the site a , i.e., is very sensitive to parameters such as magnetic field B , chemical potential p and others: q l a ( Bp, , ...).

10.5 Universal Conductance Fluctuation

1.5

-8 5

I

97

I

1

1

Q 1

>

Q

.5

0‘

I

1 (L/

2

Fig. 10.11: Normalized variance in voltage measured on short thin Au quantum wires. For a vanishing length a UCF appears, where voltage fluctuation does not tend to zero. The inset explains schematically a theoretical variation of V vs the sample size L. The solid symbols represent symmetric contributions against the exchange of current and voltage leads (after Webb et al., 1987). Thus, the manner in which the state is localized appears in its dependence on B , p and so on. This phenomenon was called a magneto-fingerprint by Stone (1985, 1987), since the system always behaves specifically and reproducibly as a function of these parameters, e.g., B. Actually fluctuations observed in the data shown in Fig. 10.11 always manifest themselves in a similar manner, although they look like simple noise. The system of concern intrinsically has disorders. It is characterized by a mode in which such disorders are distributed. In mesoscopic systems, these disorder-induced fluctuations appear in, e.g., conductance and are never averaged out. This is called universal conductance fluctuation (UCF) specific to the system of concern, since it occurs universally, not accidentally (Lee et al., 1985; Stone, 1985; Al’tshuler, 1985; Al’tshuler et al., 1985). When the system is made larger and larger, the fluctuation disappears, as can easily be envisaged, and approaches a macroscopic limit. On the contrary, when the size is reduced, the fluctuation is enhanced. Let us

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consider the conductance G as defined by Kubo. G is proportional to

which is the sum over all possible pairs between ( a ,p); w is the velocity operator. The deviation in G from the mean value as a function of B , p , and so on, should be given by a higher-order term: (SG(B,p , ...)6G(B A B , p A p , ...)) which is the correlation between four localized states. As suggested above, this fluctuation approaches a UCF given by

+

0.729

AG = x

( ;:xi )

for

( ii )

+

(10.42)

after Lee et al. (1987). These three fractional numbers are only numerically calculated. Experimentally, Benoit et al. (1987a, 1987b) (Fig. 10.11) measured the voltage fluctuation as a function of the length of thin gold wires. As shown in the inset, the variance decreases in proportion to L1I2 and approaches a constant value of UCF. For a sufficiently small system L << [ (as discussed in Chapter 6, [ is the coherence length), the voltage fluctuation should disappear as suggested in the inset because coherent conduction means “lossless” conduction, since hardly any localization phenomenon is found.

10.6

Quantized Conductance

Quantization of Conductance We will discuss quasi-one-dimensional transport observed in the so-called quantum point contact (QPC), following van Wees, et al. (1988, 1989). The sample is illustrated in Fig. 10.12(a), which forms a narrow channel (orifice or constriction) utilizing a heterojunction between AlGaAs and GaAs. In the figure the a r e a surrounded by the broken lines just under the electrodes deplete carriers so that the darkened sheet, i.e., a 2DEG (two-dimensional electron gas), has the harmonic potential in the x-direction in between. Van Wees modeled the channel to be quasione-dimensional and gave its energy scheme, as shown in Fig. 10.12(b). (10.43) where n = 0,1,2! 3, ... is the quantum number to specify the eigenenergies of a harmonic oscillator, wg is the angular frequency of the oscillator, whose confining potential is given by m*wix2/2, and eVo is a specific potential corresponding to the gate voltage VG controlling the number of carriers in the 2DEG sheet. Figure 10.12(b) depicts schematically the model eigenenergy in this system. p~ and p~ are, respectively, the chemical potentials in the emitter and collector, between which a potential V is applied.

10.6 Quantized Conductance

99

-Tk+ Eff

'

/

I

--

Fig. 10.12: Illustration of a QPC formed in a 2DEG along with potential barriers to squeeze conducting channel (a). (b) Schematic diagram to model one-dimensional eigenenergy scheme. (c) Illustration of the circuit within which coherent edge current channels are controlled by applying a voltage t o the gates under a magnetic field. With both gates A and B switched] closed-current channels are formed (d), and for either A or B switched] an orifice is formed (not shown here) (after van Wees et al., 1989). Figures 10.12(c) and (d) show another device with a couple of QPCs between which there is space causing edge currents to interfere, in particular] under a magnetic field. In this device, electrons are emitted through orifices formed by potential barriers on upper and lower gate electrodes and flow back and forth through through-channels.

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When there is a single QPC, conductance is known t o show a series of quantized values with plateaus on each step, as shown in Fig. 10.13. This fact is understood from the following basis. Since in the expression of conductance (10.44) the DOS p ( E ) = (l/7r)dky/dE and the velocity v Y ( E ) = dE,(lc,)/dhk, ally in

G= (T)nc

result eventu(10.45)

where nc is the maximum quantum number for the harmonic oscillator. As is obvious in the Landauer formula, each channel carries a current determined by a unit of conductance e2/7rh = 2e2/h. Let the orifice be sufficiently wide and, consequently, its potential well low enough, then most channels run through the orifice without being blocked by the barrier potential, otherwise, some may be blocked. One can count the number of channels running through the orifice. Since each channel carries a conductance of e2/7rh, the total conductance carried by an aggregate of such channels may manifest a pile-up of integral multiples of e2/7rh, the number of which increases from unity t o some integer, when the voltage increases. This is because the applied voltage lowers and/or widens the potential at the orifice, so that a higher number of channels can be accommodated.

-2

-1 .a

-1.6

-1.4

gate voltage VG

-1.2

-1

/ V

Fig. 10.13: Conductance quantized in units of e2/.rrh. = 2e2/h observed on an orifice, which allows a number of conduction channels determined by the gate voltage to pass (after van Wees et al., 1988).

101

10.6 Quantized Conductance

A t the limit where no potential channel remains wide or shallow enough to accommodate a single one-dimensional channel, current is only transported by thermal excitation over or quantum tunneling through the potential barrier at the orifice. Thus, conductance is quantized so that it has plateaus at every integral multiple of e2 f r h .

Interference between Edge Currents Figure 10.14 shows the observed conductance as a function of magnetic field, when the barrier potential B (Fig. 10.14(a))does not exist and, hence, the circuit appears to have a single QPC at A (Fig. 10.12(c)). Figure 10.14(b) is for a similar case with a single QPC at B. If both barrier potentials at A and B are switched on, a circular quantum dot is formed by cooperative performance of the QPCs as well as the circumferences of the barriers. Figure 10.14(c) depicts the observed oscillation of conductance with a period in B given by A B = @'o/S,where S is the area of the circle shown in Fig. 10.12(c).

..........................

3-h

- A

I

......

calculated 2 1

. . . . . . . . I. . . . . , . . . . I . . . . , . . . . 2.5 2.6 2.7

2.4

magnetic field / T Fig. 10.14: Observed conductance as a function of magnetic field, when a single orifice at A (a) or at B (b) is formed and (c) when a closed-channel circuit is formed. (d) A simulation result (after ven Wees et al., 1989).

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The readers should note here that the conductance shown in Figs. 10.12(a)-(c) varies from 3 to 2 times e2/h, not 2e2/h as before. This is because it is observed under a magnetic field and, hence, every electron has either up or down spin without degeneracy because of the Zeeman effect. Strictly speaking, the eigenenergies of this system are not easy to solve, since they should be coupled eigenmodes between harmonic oscillation and cyclotron motions. Aharonov-Bohm Gauge vs Phase Acquired by Cyclotron Motion It would probably be instructive to compare classical and quantum conditions for cyclotron resonance of electrons under a magnetic field. It is known that, by application of the correspondence principle, a cyclotron motion is specified by the condition (10.46) If n >> 1 is assumed and the zero-point value 1/2 is discarded (because a classical case is assumed here), the classical cyclotron orbit radius is known t o be equivalent to a f i l according to the corresponding principle. quantum one where T , = This 1 is sometimes called the magnetic length. On the other hand, quantum mechanics tells us that the phase of an electron circulating under a magnetic field with a radius rc is 27r(7rr;B)/@ol which also results in rC= 6 1 , if the acquired phase equals 27rn. Thus, the motion of a classical particle under a magnetic field can be reinterpreted exactly in terms of quantum mechanics as a change in the Aharonov-Bohm phase, according to the vector potential generating the field.

d

w

Integral Quantized Hall Effect (IQHE)

10.7

As stated before, conductance in two dimensions may be independent of the system geometry as G 0; Lo (Chapter 6) and, accordingly, conductance is equal to conductivity. This means that G is given by universal constants such as e, h, and m independent of the sample geometry, and, in addition, the scattering time T in general. This condition actually occurs in the Hall effect measurement in a 2DEG (two-dimensional electron gas) in a MOS under a magnetic field. Let the width and length of a two-dimensional channel be W and L in the y- and z-directions, respectively. Then (Appendix H) E x

=

jxpx, +jypyx

EY

=

jXPXY + j V P Y Y

(10.47) (10.48)

give electric fields for given current densities. In terms of actually measured quantities Vx and Vy under a constant current I x , the above equations become (10.49) (10.50)

10.7 Integral Quantized Hall Effect (IQHE)

103

In the Hall measurement, the Hall voltage VHis measured in an open condition, i.e, with Iy= 0. Then, it is easy to obtain (10.51) This is the definition of the Hall coefficient RH in a 2DEG with a perpendicular magnetic field and means that, for an observed voltage VHunder a current Ix,the ratio is given by a quantity that includes only the universal constants and, at least, a scattering time T . Thus, it is well established that the diagonal and off-diagonal components of the resistivity tensor are given by pxx

= Po

Pry

= Po@

(10.52) (10.53)

where p is the mobility and po = 1 / 0 0 = l / n e p is the resistivity with no magnetic field, when the system is a degenerate electron gas with a single relaxation time T . Otherwise, in general, we must consider the following formulae: Pxx

=

Pyy

=

0xx

(10.54)

a l x + “2,

(10.55) The density of Landau states in two dimensions is sketchily shown in Fig. 10.15(a) as a function of energy (Ando et al., 1974; Ando, 1974a, 1974b, 1974c; Murayama et al., 1987). Without scattering, there must exist narrow levels with a degeneracy of In fact every Landau 1/27d2. Here, 1 is the magnetic length defined by I = subband seems to be broadened according to the scattering effects. If the Landau subbands are filled up to an integer i , then the density of electrons n = i / 2 d 2 gives BRH = p r y = pB/oo = B/ne = -h/ie2. (We denote an electronic charge with e = -lei throughout this text.) Thus, the observed ratio VH/G equals a universal constant. This is why the effect is called the integral QHE, because the Hall constant times B is quantized to a universal constant divided by an integer. To determine the integer i experimentally, we need at least a plateau around the n = i / 2 d 2 in the observed curves, as will be explained below. Every experiment is performed as a function of the gate voltage VG, or the density of the carriers n. When the DOS is re-plotted as a function of n , the variation should read as in Fig. 10.15(b). The unshaded portions in the figure mean “delocalized states”, whereas the shaded (darkened) mean “localized states”. Let us assume that wCr = pB >> 1 for this experiment, as usual. Then, two relations, pxy = l/crxy = ne and pxy N B2uXr/(ne)’otDOS/n, are obtained. As shown in Fig. 10.15(c), pxx is known to hit a zero only at the instance when n equals (i + $)/27r1’ as a function of n.

w.

104

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li w,/2

3 li w,/2

Fig. 10.15: Conceptual sketch of two-dimensional Landau electrons (a) as a function of energy and (b) as a function of carrier density n 0: Vc. When localization occurs on both sides of the two-dimensional delocalized Landau levels, C J , ~is quantized as shown in (d), where o,, 0: p,, vanishes (c). The thin curves in (c) depict p,, in an unrealistic sample where all states are assumed delocalized.

10.7 Integral Quantized Hall Effect (IQHE)

105

Fig. 10.16: Observed Hall voltage V, between the Hall probes (see the inset) and voltage V, between the source and drain under a constant applied current as functions of gate voltage VGon an n-channel MOS Hall bar sample. VH shows a number of plateaus, i.e., integral QHE (after von Klitzing et al., 1980).

In actual devices, it is known that electrons are very easily localized in particular in two dimensions. Thus, each Landau subband has localized states around the central delocalized narrow band. In Fig. 10.11(c) the thin curves suggest pxr without localization, whereas the thick curves are for that with localization. In the localized regions, carriers are immobile and, hence, oar, accordingly prr disappears, although the DOS has a finite value. In accordance with the p,,, pxy is expected to behave as shown in Fig. 10.11(d). In the figure you will notice plateaus clearly. Thus, unless there is localization, the Hall effect experiment would never have plateaus in the Hall coefficient as a function of n or VG,and accordingly it would not be possible to determine h / e 2 precisely in up to 6 or 7 digits (Yamanouchi et al., 1981; Yoshihiro et al., 1982; Bliek et al., 1983; Kawaji, 1984; Kinoshita et al., 1984). The existence of localization was never previously considered in experiments, but was only deduced from research on the integral quantum Hall effect.

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10.8 Fractional Quantized Hall Effect (FQHE) In the integral QHE, plateaus appeared for the parameters n that allowed the Fermi level to remain between neighboring Landau subbands, where all the subbands below the quantum number N are completely occupied whereas those above are completely unoccupied. Historically the integral QHE was discovered in a 2DEG at the Si-SiO2 interface, which seems to have a considerably low electron mobility. On the other hand, it is well established that the heterostructure consisting of GaAs and AlGaAs gives an excellent quality 2DEG. It shows a very high mobility up to 1000 m2/V s and a mean-free path of several pm. This type of sample often shows very flat plateaus in magnetic fields (Fig. 10.17: Stormer et al., 1983; Tsui et al., 1982), for which the carrier densities are such that a Landau subband is fractionally occupied to v = p / q (v: filling factor) of the total electrons of 1/27r12 per subband and unoccupied beyond. p ( # q ) is an integer and q = 3, 5 , 7, ..., odd numbers because of the Fermi statistical requirement. This effect is called “fractional” QHE in contrast to the integral QHE that occurs for the carrier densities equal to an integral multiples of 1/27r12. Experimentally, an integral QHE is known to occur in dirty samples, which seem to ease localization. On the other hand, a fractional QHE is only observed in clean samples with a high mobility. A fractional QHE shows plateaus within the same Landau subband as in integral QHE. To simplify the explanation, let us limit ourselves to the case with p = 1 and q = 3. To interpret the effect, Laughlin (1983) introduced the possibility of correlated three-particle states. Laughlin (1983) started with the eigenstate of a Landau electron (Appendix I). N

where n and m are, respectively, the Landau and the angular momentum quantum number, and R is the normalization factor. The eigenenergy is ( n 1/2)Rw, in this The eigenenrepresentation and 1 is the magnetic length ( h / e B ) ’ / 2= (T~/m*w,)’/~. ergy is m-fold degenerate. We explicitly write the lowest Landau subband, i.e., n = 0 band as

+

( 10.57) where z = x + i y and E = x -iy. This state is an eigenstate of angular momentum with eigenvalue m. It is known that a single Landau subband includes N electrons subject to the DOS per unit area equal to 1/27r12. If the maximum angular momentum of this system is denoted by M , then m = 0,1,2, ...,M - 1. Thus, m N = M is obtained. For simplicity we take m = 3. A single Landau subband is only partially filled up to one-third (i.e., l / m = 1/3 = N / M ) with all the available states belonging t o the total angular momentum M .

10.8 Fkactional Quantized Hall Effect (FQHE)

5134 2 3 1

I

I I I

3

107

filling factor 4 7 3 5 3 5 I

1

v 1 3

2

5

1

h

N

a,

\

c v \

/

> X

Q

-11

312

1OkR

/

1 1I2 113

(,,,,,,

0 X

X

Q

0

0

5

10

15

2c

magnetic field B I T Fig. 10.17: Observed pzy and pzz as a function of magnetic field B. A number of plateaus appear for fractional filling factors shown on the upper axis, i.e., fractional QHE (after Stormer et al., 1983). Then, Laughlin extended this single-particle state into a many-body correlated state, as follows:

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108

which is known as the Jastrow function and has often been studied to describe the socalled one-component plasma. He tried to minimize energy by varying the many-body state. f is an appropriate function to describe the correlation, which was taken to be f(Zi

- " j ) = (Zi - 4 3 .

The two-dimensional Landau electron gas is degenerate, as has already been mentioned, with a degeneracy of 1/2.rr12,which comes from the fact that the eigenenergy is not explicitly dependent on the quantum number, the center coordinate of cyclotron motion. However, because of the correlation energy between Landau electrons, they are mutually repulsive and are likely t o form, e.g., a triangular lattice, just as in a Wigner crystal or in an Abrikosov lattice in superconductors. The ground state threebody state has three-fold symmetry due to its correlation energy and its occupation is v = 113 - 6 (6: infinitesimal). The Laughlin state is a trial function for a variation to be performed on, and is constructed so as to maximize mutual repulsion. More accurate calculations show that the excited state with an occupation of 1/3+6 has a tiny Coulomb gap over the ground state energy with an occupation of 113 - 6. The gap energy was estimated to be on the order of 0.02e2/47rnc,J (n: the relative dielectric constant of the medium). This Coulomb gap plays the same role as the gap between Landau subbands in the integral QHE case. Thus plateaus should be observed a t around 113 occupation as a function of B . The plateau at Y = 213 is easily understood from the electron-hole symmetry within a single Landau subband.

10.9

Ballistic Transport

Through a sufficiently short path, e.g., in a GaAs-AlGaAs heterostructure with a high mobility, electrons can be transported without being scattered. In old vacuum tubes this is the case since there is ordinarily no scattering in a vacuum. In this case electrons are accelerated up to a velocity limited by the accumulated space charges, as given by Richardson's formula. In solid states, a similar situation can occur, especially in high quality interfaces such as in GaAs-AlGaAs. Figure 10.18 shows the variation in resistance as a function of magnetic field, when a current flows from lead k t o lead I, whereas the voltage is measured between leads m and n. The measured resistance is denoted R k l , m n . Even if the current path is bent, there is no resistance due t o the bend, so long as the generated channels consist of eigenmodes (i.e., complete streamlines) of the potential problem. However, the path might become narrow around the bends and, consequently, there may be considerable blocking of the channels, that is, if the path become narrow, geometrically confined energy levels are raised and, accordingly, the number of levels below the Fermi level decreases. In other words, this means that the number of channels decreases. This effect may cause a resistance on the electrons curving around the bends (Timp et al., 1988; Takagaki et al., 1988; Baranger et al., 1989; Beenakker et al., 1989). This additional resistance may be called the bend resistance. In the absence of a magnetic field, a straight path gives the lowest resistance, whereas under a magnetic field paths are more natural with bends and give a lower resistance, as observed in

10.10 Coulomb Blockade

I

0

-.5

I

109

I

I

I

I

I

0 magnetic field / T

I

I

I

.5

Fig. 10.18: Observed magnetoresistance R k l , m n on a sample with a current fed between leads k and 1 and a voltage measured between leads m and n. For a straight current path, the zero field resistance is minimum. For a higher magnetic field, the resistance is almost independent of the path, since the current path tends to be curved (after Takagaki et al., 1989). both of the regions off the central part in Fig. 10.11. Therefore, the difference in resistance is reduced there or the resistances appear to be in reversed in contrast to the central field region.

10.10

Coulomb Blockade

In the band theory of condensed matter physics it is known that we can neglect most of the correlation energy between electrons, so long as their orbitals are sufficiently extended. The correlation energy is simply Coulomb repulsion energy exerting between the two electrons with opposite spins in the same orbital. However, if electron orbitals are localized, this is not the case, because they feel mutual repulsion. Throughout this Section, we will follow the argument described by Averin et al. (1991). In quantum dots, orbitals are limited by the boundary, so that Coulomb repulsion is not negligible in these dots. Let us consider a tunnel junction (Fig. 10.19(a)). A tunnel junction is an analogy of a classical condenser, although the insulator between

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the electrodes must be thin enough for electrons to be able to tunnel in contrast. Take a tunnel junction, each electrode of which has either charge Q = q or Q = - q . Let one electron tunnel from one electrode to the other. Then the net charge becomes q - (el on one electrode and q lel on the other, and a certain current flows on the tunneling instance. This tunneling may occur as a type of quantum fluctuation. One question arises whether the energy increases or decreases after this tunneling. The main point of the underlying physics is that electrons can be transferred via tunneling only under the correlation effect. The correlation energy is efficient for ultrasmall tunnel junctions. In larger tunnel junctions, the electron wavefunction spreads and then the correlation effect is diminished in the same manner as that which makes it almost insignificant in bulk metals. We treat ultrasmall junctions here. Figure 10.19(b) shows an answer to this question. An amount of charge Q is defined by C V , where V is the applied electric potential and C is the capacity of the tunnel junction. Tunneling across a barrier is the same as the electron-hole pair-annihilation process, that is, one electron in the electrode with a charge -Q and one hole in the other with a charge Q disappear at the same time to result in charges, respectively, [el and Q - lei, or, in other words, the capacitor discharges by one elementary -Q charge (el. We treat e as a positive value to avoid complication from now on. If a charge q between -e/2 and e/2 is annihilated by the unit charge e via tunneling through the junction, the electrostatic energy, i.e., correlation, changes from q2/2C to ( q e)2/2C. It increases as shown in Fig.lO.lS(b) on the parabolic curve. Otherwise for the tunneling of either one electron or a hole from the state with a charge either larger than e/2 or smaller than -e/2, the energy decreases according to the Coulomb repulsion, as was first suggested by Likharev et al. (1985). See also, Ben-Jacob et al. (1985); Fulton et al. (1987); Delsing et al. (1990); Geerligs et al. (1990). The latter energy decreasing process actually occurs, whereas the former process can hardly occur. In other words, tunneling conductance is suppressed. This phenomenon is known as the Coulomb blockade. Figure 10.20(a) illustrates a series circuit of an conventional gate capacitor with a capacity CG and a tunnel junction with CJ. The tunnel junction behaves as if it were a capacitor with a leak current, so that when a voltage V is applied the capacitor begins to be charged according to a charging curve such as Q = CGV[l - exp(-t/r)] in terms of the RC time constant r. When the charge reaches e/2, tunneling occurs to reduce the charge in the tunnel junction. Charging continues in the gate capacitor, whereas the tunnel junction repeats a series of charging and tunneling (i.e., leaking). Thus, the electrostatic energy of the junction is given by EQ = (Q - n e ) 2 / 2 C ~after n-times tunneling, as shown in Fig. 10.20(b). The charge in the junction will behave as shown by the curve in Fig.10.20(c). At the instance when tunneling occurs, the charge may be suddenly reduced. Since the current I is given by dQ/dt, I may trace the curve shown in Fig. 10.12(d), where the dips on the curve are only schematically shown (ideally, they are the delta-functions). This oscillatory phenomenon is called the single electron tunneling oscillation (SET oscillation).

+

+

10.10 Coulomb Blockade

E,=Q2

c

111

/2 C

I

J e/2C

V

I \ (b)

-e/2

e/2

Q

(c)

Fig. 10.19: (a) Diagram of the Coulomb blockade circuit consisting of a tunnel junction with a capacity of C and a resistance of R,. A current I flows under an applied voltage V . (b) Schematic interpretation of how energy changes under the electron tunneling of the unit charge e . (c) Expected I-V characteristic according to the Coulomb blockade effect. Since current is suppressed between the voltages equal to - e / 2 C and e/2C, the phenomenon is called the blockade. From a device-application point of view, the Coulomb blockade effect has been applied to only a few actual devices. This is mainly because the electrostatic energy is T meV at room temperature). If kBT too far below the ambient thermal energy ~ B (25 is equated to e2/2C, the capacity C must be as small as 3 aF(= 3 x lo-'* F). This value corresponds to a 3 x 3 nm2 capacitor with a 3-nm gap and a dielectric constant E = 4 ~ 0 (~0: the dielectric constant of vacuum). Such a small capacitor cannot be fabricated with conventional photolithographic techniques. For this reason, plausible devices could initially only be operated at low temperatures (Nakazato et al., 1993).

112

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2

EQ=(Q-ne)/2CJ

I

Q-ne

= ! t i

WG9

-Q+ne

5e

(d)

I

.4t

.8t

1.2t

t

Fig. 10.20: (a) Series circuit consisting of a conventional gate capacitor Cc: and a single-electron tunnel junction CJ. (b) Electrostatic potential on the tunnel junction as a function of charge Q on the gate capacitor. n is the number of tunneling processes. (c) Charges Q and QJ appearing, respectively, on the gate capacitor and the tunnel junction, as a function of charging time t. (d) Charging current flowing in the circuit defined by I = dQ/dt. The dips on the curve correspond to the delta-functions resulting from differentiating the sudden decreases in QJ. An SET oscillation and the period is I l e .

113

10.10 Coulomb Blockade

However, Yano et al. (1994) first observed room-temperature operation of the Coulomb blockade phenomena within granular polycrystalline ultrathin Si films. The device structure they examined was similar to the conventional electrically alterable MOS (EAMOS), or a floating gate non-volatile memory device, where the thickness of the granular Si film was so thin that the potential produced by the gate voltage randomly varied from grain to grain. When the gate voltage was high enough, percolation channels were formed. Otherwise, the channels were broken into pieces of quantum dots. In the latter case, every dot would have acted as an island for the Coulomb blockade capacitor. The size of these islands seems to be sufficiently small compared with the size of the above-mentioned 3 aF capacitor. Figure 10.21(a) shows a model for an ultrasmall MOS with a floating island embedded inside the gate insulator. In this device, an increase in the gate voltage VG induces “write-in” into the floating island (the floating gate) by charging. As suggested in Fig. 10.21(b),the floating charge shifts the effective gate voltage by a specific amount according to the Coulomb blockade effect. This shift in VGis represented by the relationship

before writing channel F~oaiingDo! /

-

+-. e

u .-c

in

threshold shift by a single electron

U erase > JOOK

gate voltage

(b)

(c)

Fig. 10.21: (a) Structure model of a MOSFET with a floating-island gate embedded inside the gate insulator. (b) Illustration of the MOSFET with and without the floating gate charged. If charged, the side potential causes the channel to conduct a less current. (c) Two VGvs I D Scharacteristics are contrasted. One is with the floating gate uncharged and the other is with VG shifted according to a charged floating gate. q means e (after Yano et al., 1994, 0 1 9 9 4 IEEE).

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114

gate voltage / V Fig. 10.22: Measured VGvs. I D . Each group of data corresponds to a single floating-gate condition. Lightly drawn continuous lines trace the characteristics for different numbers of charges inside the islands (after Yano et al., 1994, 01994 IEEE).

AVG

N

e9 -

( 10.59)

cgc

where Cg, is the capacitor specific to the circuit. Thus, the original VGvs I D characteristic where I D is the drain current becomes a shifted (VG+ A V c ) vs ID characteristic as shown in Fig. 10.21(c). The most significant part of this phenomenon is that the shift in VGis quantized. Certainly, a single floating charge seems to block a single conducting channel. As described in Section 10.5, the Landauer formula gives conductance in terms of the number of channels. With more channels, the device becomes more conductive. A further increase in VGmay reduce the number of channels one by one, depending on whether the islands are charged in a floating manner. The observed data are shown in Fig. 10.22. The device introduced here has a great advantage in that it works at room temperature. However, how the gate voltage can be precisely controlled is another formidable problem. In the experiment, the ultrathin Si film unexpectedly produced a network of floating islands. Therefore, the gate voltage itself varied a great deal from sample t o sample. In contrast, the shift itself in the gate voltage was definite due to the Coulomb blockade effect with an integral number of floating charges, which is the biggest advantage of these devices.

10.11 Atomic Wires

10.11

115

Atomic Wires

An atom is very stable and keeps its coherence, as long as it does not interact with any other degrees of freedom. For example, a hydrogen atom in its ground state remains in that state almost forever, unless it absorbs photons or collides with any other high energy particles, atoms or molecules. If one goes from an atom to a linear chain of atoms, what happens? A long linear molecule-like chain of atoms may be still in a coherent state and stable. However, a problem occurs if this system is lying on a substrate in that a variety of interactions are likely to occur between the chain of atoms and its substrate. One-dimensional atomic (quantum) wires have been investigated so far, mostly theoretically, because the fabrication technique is far beyond the present attainable level (Aono et al., 1993; Serena et al., 1997; Joachim et al., 1997). In studies on practical quantum wires, the effect of the interaction with the substrate is very important. Quantum wires are most easily fabricated by removing H atoms in one dimension from a fully passivated Si surface. This can be done by applying a high voltage between the STM tip and the passivated surface. This sort of atomic wire is sometimes called a dangling-bond (DB) wire or a de-passivated (DP) wire. Watanabe et al. (1995) tried a pseudo-potential, density-functional approximation calculation for a DB wire on a Si(ll1) surface and showed that the wire becomes truely conductive (Fig. 10.23). Experimentally Hitosugi et al. (1999a, 1999b) investigated finite-length danglingbond quantum wires fabricated on a hydrogen-passivated Si(100)-(2 x 1) surface using scanning tunneling microscopy. They found that these samples show the so-called “odd-even problem”, where their properties depend on the length (the number of the dangling bonds in one dimension) and show an edge effect. Whether a sample is in a more stable state depending on the number of dangling bonds is caused by their interaction with the Si substrate through the redistribution of charges on the surface. The other example of quantum wires is Ga-bars which are self-organized with adsorbed Ga atoms on the same hydrogen-passivated Si(lO0)-(2 x 1) surface (Hashizume et al., 1996). At last we will mention one important aspect of controlling the polarity of quantum wires by doping. That is, how to obtain a p-n junction utilizing these quantum wires, something not yet achieved, is an important question from a practical point of view. It is also very interesting from a device-physics aspect. In a study on what dopant concentrations are needed, to realize n- or p-type one-dimensional, atomicscale conductors, the present author reasoned as follows (Murayama, 1995). Doping is effective in conventional bulk semiconductors when its level is something like ( a / a i ) 3, where a is the lattice constant and a: is the appropriate effective Bohr radius of a hydrogen-like dopant atom, which is on the order of several tens of angstroms. This means that 1 ppm or a slightly larger dopant concentration works fairly well in the bulk case. However, in one-dimensional wires, since a / a B is approximately 1 %, a large quantity of dopant is needed for the wire to be significantly conductive. This large concentration may lead to a severe stability problem in such heavily doped semiconducting quantum wires.

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(a)

8

0 Y

8

t

0 6 Q

0

7 1

Fig. 10.23: An atomic wire on the H-terminated Si(ll1) surface. (a) Top view. A horizontal array of open circles through No.4 is expected to form a quantum wire of depassivated Si atoms. A rhombus is the 2 x 4 supercell to be calculated. (b) Side view along the A-B line in (a). Small solid and larger circles represent, respectively, H and Si atoms. A dotted circle shows the de-passivated H site. (c), (d) and (e) show the charge density contour maps of the midgap state at the symmetry r-point: (c) along line A-B, (d) along line C-D, and (e) along line E-F (after Watanabe et al., 1995).

References Abrikosov, A. A., Gotkov, L. P., Dzyaloshinski, I. E. (1963), Methods of Quantum Field Theory in Statistical Physics, New York: Dover Publ. Aharonov, Y., Bohm, D. (1949), Proc. Phys. SOC.B62,8. Al'tshuler, B. L., Aronov, A. G., Spivak, B. Z. (1981), JETP Lett. 33,94. Al'tshuler, B. L. (1985), JETP Lett. 41,648. Al'tshuler, B. L., Khmel'nitskii, D. E. (1985), JETP Lett. 42, 359. Ando, T., Uemura, Y. (1974), J . Phys. SOC.Jpn. 36,959. Ando, T.(1974a), J . Phys. SOC.Jpn. 36,1521; (1974b), 37,622; (1974c), 37,1233.

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Aono, M., Kobayashi, A., Grey, F., Uchida, H., Huang, D.-H. (1993), Jpn. J. Appl. Phys. A47, 4616. Aronov, A. G., Sharvin, Yu. V. (1987), Rev. Mod. Phys. 59, 531. Averin, D. V., Likharev, K. K. (1991), in: B. L. Alt’shuler, P.A. Lee, R. A. Webb (Eds.), Mesoscopic Phenomena in Solids, Amsterdam: North-Holland, p.173. Baranger, H. U., Stone, A. D. (1989), Phys. Rev. Lett. 63, 414. Beenakker, C. W. J., van Houten, H. (1989), Phys. Rev. Lett. 63, 1857. Ben-Jacob, E., Gefen, Y. (1985), Phys. Lett. 108A, 289. Benoit, A. D., Washburn, S., Umbach, C. P., Laibowitz, R. B., Webb, R. A. (1987a), Phys. Rev. Lett. 57, 1765. Benoit, A. D., Umbach, C. P., Laibowitz, R. B., Webb, R. A. (1987b), Phys. Rev. Lett. 58, 2343. Bliek, L., Braun, E., Engelmann, H. J., Leontiew, H., Melchert, F., Schlapp, W., Stahl, B., Warnecke, P., Weimann, G. (1983), PTB-Mitteilungen 93, 21. Brillouin, L. (1963), Wave Propagation in Periodic Structures, 2nd ed., New York: Dover. Biittiker, M., Imry, Y., Landauer, R., Pinhas, S. (1985), Phys. Rev. B31, 6207. Chang, L. L., Esaki, L., Tsu, R. (1974), Appl. Phys. Lett. 24, 593. Chelikowsky, J. R., Cohen, M. L. (1976), Phys. Rev. B14, 556. Delsing, P., Likharev, K. K., Kuzmin, L. S., Claeson, T. (1989), Phys. Rev. Lett. 63, 1861. Esaki, L. (1958), Phys. Rev. 109, 603. Esaki, L., Tsu, R. (1969), IBM Research Note RC-2418; (1970), IBM J. Res. Dev. 14, 61. Esaki, L. (1974), Proc. IEEE, 62, 825; (1976), IEEE Trans. Electron Devices ED-23, 644. Fulton, T. A., Dolan, G. J. (1987), Phys. Rev. Lett. 59, 109. Geerligs, L. J., Andregg, V. F., Holweg, P. A., Mooij, J. E., Pothier, J. E., Esteve, D., Urbina, C., Devoret, M. H. (1990), Phys. Rev. Lett. 64, 2691. Gunn, J . B. (1963), Solid State Commun. 1, 88. Gunn, J . B. (1964), IBM J . Res. Dev. 8 , 141.

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Hashizume, T., Heike, S., Lutwyche, M.I., Watanabe, S., Nakajima, K., Nishi, T., Wada, Y. (1996), Jpn. J. Appl. Phys. 35, L1085. Hilsum, C. (1962), Proc. IRE 50, 185. Hitosugi, T., Heike, S., Onogi, T., Hashizume, T., Watanabe, S., et al. (1999a), Phys. Rev. Lett. 82, 4034. Hitosugi, T., Suwa, Y., Matsuura, S., Heike, s., Onogi, T., Watanabe, S., et al. (1999b), Phys. Rev. Lett. 83, 4116. Hobson, G. S. (1974), The Gunn Eflect, Oxford: Clarendon. Imry, Y. (1986), in: G. Grinstein, G. Mazenko (Eds.), Directions in Condensed Matter Physics, Singapore: World Sci., p.101. Joachim, C., Roth, S. (Eds.) (1997), Nanowires, Dordrecht: Kluwer Academic. Johnson, J. B. (1928), Phys. Rev. 32, 97. Katayama, Y., Yoshida, I., Kotera, N., Komatsubara, K. F. (1972), Appl. Phys. Lett. 20, 31. von Klitzing, K., Dorda, G., Pepper, M. (1980), Phys. Rev. Lett. 45, 494. Kawaji, S. (1984) in: S. Kamefuchi, H. Ezawa, Y. Murayama, M. Namiki, S. Nomura, Y. Ohnuki (Eds.) (1990), Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, p.339. Kinoshita, J., Inagaki, K., Yamanouchi, C., Yoshihiro, K., Endo, T., Murayama, Y., Koyanagi, M., Moriyama, J., Wakabayashi, J., Kawaji, S. (1984), in: ibid. p.327. Kroemer, H. (1964), Proc. IEEE 52, 1736. Kubo, R. (1957), J. Phys. SOC.Jpn. 12, 570. Kubo, R., Toda, M., Hashitsume, N. (1991), Statisitical Physics II. Nonequilibrium Statisitical Mechanics, Berlin: Springer. Laughlin, R. B. (1983), Phys. Rev. Lett. 50, 1395. Landau, L. D., Lifshitz, E. M. (1969-80), Statistical Physics, Oxford: Pergamon. Landauer, R. (1957), IBM J. Res. Dew. 1, 223. Lee, P. A., Stone, A. D. (1985), Phys. Rev. Lett. 55, 1622. Lee, P. A., Stone, A. D., Fukuyama, H. (1987), Phys. Rev. B35,1039. Leeuwen, van J. H. (1921), J. de Physique 2, 361. Likharev, K. K., Zorin, A. B. (1985), J. Low Temp. Phys. 59, 347.

10.11 Atomic Wares

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Mendez, E. E., Esaki, L. (1996), in: G. L. Trigg (Ed.) Encyclopedia of Appl. Phys. 16, 437. Mizuta, H., Tanoue, T . (1995), The Physics and Applications of Resonant Tunneling Diodes, Cambridge: Cambridge University Press. Murayama, Y. et al. (1995), US and Japan patents, pending. Murayama, Y., Kamigaki, Y., Yamada, E. (1972), in: Proc. 3rd Conf. on Solid State Devices, Tokyo, 1971 [Suppl. Oyo Butsuri, 41,133.1 Murayama, Y., Ando, T . (1987), Phys. Rev. B35,2252. Nyquist, H. (1928), Phys. Rev. 32,110. Nakazato, K., Blaikie, R. J., Cleaver, J. R. A., Ahmed, H. (1993), Electronics Lett. 29, 384. Peierls, R. (1979), Surprises in Theoretical Physics, Princeton, NJ: Princeton University Press. Ridley, B. K., Watkins, T. B. (1961), Proc. Phys. SOC.London, 78, 293. Ruch, J . G., Kino, G. S. (1967), Appl. Phys. Lett. 10, 40. Ruijsenaars, S. N. M. (1983), Ann. Phys. (N. Y.) 146,1. Serena, P.A., Garcia, N. (Eds.) (1997), Atomic and Molecular Wires, Dordrecht: Kluwer Academic. Sharvin, D.Yu., Sharvin, Yu.V. (1981), JETP Lett. 34,272. Sollner, T. C. L. G., Goodhue, W. D., Tannenwald, P. E., Parker, C. D., Peck, D. D. (1983), Appl. Phys. Lett. 43,588. Stone, A. D. (1985), Phys. Rev. Lett. 54,2692. Stone, A. D. (1987), in: M. Namiki, Y. Ohnuki, Y. Murayama, S. Nomura, S. (Eds.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, p.207. Stormer, H. L., Chang, A., Tsui, D. C., Hwang, J. C. M., Gossard, A. C., Wiegmeann, W. (1983), Phys. Rev. Lett. 50,1953. Sze, S. M. (1981), Physics pof Semiconductor Devices, 2nd ed., New York: Wiley. Takagaki, Y., Gamo, K., Namba, S. (1988), Solid State Commun. 12, 1051. Takagaki, Y., Wataka, F., Takaoka, S., Gamo, K., Namba, S. (1989), Jpn. J. Appl. Phys. 28,2188.

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Tanoue, T . Mizuta, H., Takahashi, S. (1988), IEEE Electron Device Lett. EDL-9, 365. Taylor, B. N. (1992), in: G. L. Trigg (Ed.) Encyclopedia of Appl. Phys. 4, 243. Timp, G., Baranger, H. U., de Vegvar, P., Cunningham, J. E., Howard, R. E., Behringer, R., Mankiewich, P. M. (1988), Phys. Rev. Lett. 60, 2081. Toda, M., Kubo, R., Saito, N. (1991), Statisitical Physics I. Equilibrium Statisitical Mechanic,, Berlin: Springer. Tsu, R., Esaki, L. (1973), Apl. Phys. Lett. 22,562. Tsui, D. C., Stormer, H. L., Gossard, A. C. (1982), Phys. Rev. lett. 48, 1559. Washburn, S., Umbach, C. P., Laibowitz, R. B., Webb, R. A. (1985), Phys. Rev. B32, 4789. Watanabe, S., Ono, Y. A., Hashizume, T., Wada, Y., Yamauchi, J., Tsukada, M. (1995), Phys. Rev. B52,10768. Webb, R. A., Washburn, S., Umbach, C. P., Laibowitz, R. B. (1985a), Phys. Rev. Lett. 54, 2696. Webb, R. A . , Washburn, S., Umbach, C. P., Laibowitz, R. B. (1985b), in: B. Kramer, G. Begmann, T . V. Bruynseraede (Eds.), Localization, Interaction, and Transport Phenomena, Berlin: Springer. Webb, R. A. (1987), in: M. Namiki, Y. Ohnuki, Y. Murayama, S. Nomura (Eds.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, p.193. van Wees, B. J., van Houten, H., Beenakker, C. W. J., Williamson, J . G., Kouwenhoven, L. P., van der Marel, D., Foxon, C. T. (1988), Phys. Rev. Lett. 60,48. van Wees, B. J., J. G., Kouwenhoven, L. P., Harmans, C. J. P. M., williamson, J. G., Timmering, C. E. (1989), Phys. Rev. Lett. 62,2523. Yamanouchi, C., Yoshihiro, K., Kinoshita, J., Inagaki, J., Moriyama, J., Baba, S., Kawaji, S., Murakami, K., Igarashi, T., Endo, T., Koyanagi, M., Nakamura, A. (1981), Precision Measurement and Fundamental Constants 11, B. N. Taylor, W. D. Phillips (Eds.), NBS USA Special Publication No. 617. Yoshihiro, K., Kinoshita, J., Inagaki, K., Yamanouchi, C., Moriyama, J., Kawaji, S. (1982), J. PhysSoc. Jpn. 51, 5. Yano, K., Ishii, T., Hashimoto, T., Kobayashi, T., Murai, F., Seki, K. (1994), IEEE Trans. Electron Devices, 41, 1628. Ziman, J. M. (1964), Principles of the Theory of Solids, Cambridge: Cambridge University Press. Zubarev, D. N. (1971), Nonequilibrium Statistical Mechanics, Moskow: Nauka.

11

Optical Properties

11.1

Single/Multiple Quantum Wells

Excitons an Q Ws Within semiconductors, shallow donors and acceptors as well as excitons, can be described by hydrogenic wavefunctions. In order to discuss excitons, the mass should be taken t o be a reduced value p* of the effective masses of an electron and a hole, mE-'. Hence, the exciton binding energy En is -Ry*/n2, i.e., p*-l = m,*-' where n = 1, 2, ..., and Ry* = m*e4/2n2@i2 is the effective Rydberg energy, 1 3 . 6 x ( p * / m o ) / ~eV. ~ €0 and n are, respectively, the static dielectric constant in vacuum and the relative dielectric constant in the semiconductor of concern. Here it should be noted that the effective Rydberg energy is much smaller than that in a vacuum, since, e.g., m* of an electron in GaAs is as small as 0.067mo and K = 13.1. Accordingly, Ry*=5.31 meV. This is still the case within a QW, if the width is large enough, i.e., the system is three-dimension-like. For a narrow QW, one degree of freedom of the spherical orbital is confined. An extremely thin QW will look like a two-dimensional hydrogen atom. The binding energy in this limit is known t o be given by En = -Ry*/(n + 1/2)2, n = 0, 1, 2, 3, ... (Appendix D). If both the lowest binding energies in three and two dimensions are compared, it is obvious that 4E1(3D) = Eo(2D). Correspondingly, the radial extent of the s-orbital of the two-dimensional exciton is half of that in three dimensions. The fact that the exciton binding energy in a QW is larger than that in three dimensions has an important meaning in its applications. For example, when a QW is formed as an active layer in a semiconductor laser, the emission efficiency is improved, partly because the binding energy can be still larger than the thermal excitation energy at room temperature and the exciton is only slightly excited into a free electron and a hole pair that can more easily recombine without emitting luminescence. The other reason is that the DOS in two dimensions is constant as was stated before and, consequently, the emissive transition probability is higher than in three dimensions, where the DOS starts from 0 and is in proportion to Figure 11.1 shows the dependence of the exciton binding energy against the well width in GaAs-Alo,4Gao,6As.For narrow widths, the energy first increases and then decreases. The reason is that for a diminishing well width the binding energy level comes very close to the conduction band bottom of the barrier material and, conse-

+

a.

11 Optical Properties

122

quently, the wavefunction is no longer confined within the well, just as was shown for the second excited level El in Fig. 7.2. This is also the case for holes. The excitonic state consisting of these extended orbitals is never confined within the well and, therefore, the exciton binding energy approaches again the value for three dimensions, which is lower than the value for two dimensions. The photon energy in photoluminescence (PL) corresponds t o the difference between the energies for the electron and the hole minus the exciton binding energy. The binding energy is given in this figure. The limiting case for a vanishing well width in this figure is equal to the energy gap EG minus the exciton binding energy in the three-dimensional bulk.

30

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l

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25

20

>

!i

\

g 15 w

t n

10

5

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1

2

3

4

5 6 7 8 well width / nm

9

F i g l l . l : Exciton binding energy within QWs vs well width. The labels lh and hh designate the exciton binding energy formed between, an electron and a light hole (with a lighter effective mass), and a heavy hole (with a heavier effective mass), respectively. Since the exciton belongs t o a bound state within the QW, the energy of which is inversely proportional t o the effective mass, an lh exciton has a larger binding energy (after Nelson et al., 1987).

11.1 Sangle/Multiple Quantum Wells

123

2

1.5

8-1

-a u

E

1

0. 5 0

5

10 well width / nm

15

Fig. 11.2: Calculated variation of the power index m in PL given by the equation I p p ~cc I g , plotted with the date obtained by Mishima et al. (1986a, 198613). m = 2 corresponds to a low efficiency of PL because of the infrequent trapping of carriers for the resonance conditions, whereas m = 2 corresponds to a high efficiency because of tightly trapped carriers for the off-resonance conditions.

Trapping of Carriers i n t o Q Ws Another interesting phenomenon is seen in the efficiency of PL (Murayama, 1986). When MQW samples are fabricated with barriers too thick to couple with each other, it is easy t o observe a number of PL spectra from quantum wells with various widths under a single shot excitation. From a well of a certain thickness, PL of a specific wavelength is emitted and its intensity is proportional to the m-th power of the excitation intensity, I p L 0: Ig with m being between 1 and 2 . This power index m depends on the well width, as is shown in Fig. 11.2. These measurements were carried out by Mishima et al. (1986a, 1986b). As already stated, when the well width W increases from a very small value, the lowest quantized tate energy level begins to decrease from the conduction band bottom of the barrier material. Around W M 4.7 nm for Alo.sGao.7As-GaAs MQW, the second quantized energy level appears. This

124

11 Optical Properties

must appear as shown in Fig.7.2. Note that PL always comes from the recombination of an exciton composed of the lowest electron and the highest hole level, each of which, respectively, resides at the bottom of the lowest conduction subband and at the top of the highest valence subband in the QW. When an electron level comes near the barrier conduction band, i.e., the resonance condition holds, photoexcited electrons can easily be freed from within the well. On the other hand, when there is a considerable energy depth between the conduction band bottom of the barrier and the highest level within the well, i.e., the 08resonance condition holds, photoexcited carriers may be trapped tightly within the well and contribute significantly t o PL. This situation reflects on the PL efficiency, which depends on the well width in the manner shown in Fig.ll.2. The present author (Murayama, 1986) has discussed that the efficiency of PL is calculated based on a rate equation considering a relaxation time dependent on the energy difference between the initial trapping state and the band minimum (or maximum, for holes) in the QW, and showed that this very trapping efficiency should determine the PL efficiency. The author called this phenomenon the QWIDDLE (quantum well-width dependent photoluminescence efficiency) effect. m 2 with a low efficiency occurs for the resonance conditions, whereas m 1 with a high efficiency occurs for the off-resonance conditions. A periodic variation in m is caused by the periodic appearance of a highest subband within the QW near the bottom of the barrier potential and the period was accurately interpreted based on the argument above.

-

-

Determination of Tunneling Mass It is said that the tunneling effect is a pure quantum effect. An electron is a quantum entity in quantum mechanics which is never corpuscular but wavy. It is also said that the concept of the effective mass in condensed matter is nothing but a quantum outcome. Both of these quantum concepts must be connected with each other. In the theory of elementary particles, an electron and a positron have the same mass mo. A positron (hole) is said to be a mirror image of an electron. However, such a positron (hole) is within the Fermi sea of electrons, whereas an electron is always in a higher energy state than a hole by a gap energy equal to 2moc2. Then, what about their masses in the energy gap? In the tunneling phenomenon, an electron tunnels through a barrier with a certain mass. Ordinarily we assume that the mass inside the barrier is the same in order to calculate the probability of the effect. However, when the effective masses on both sides are different, then which mass should we take for the intermediate barrier state? Here we will try to obtain the magnitude of the effective mass in a band gap (Murayama et al., 1990). Let us take a single quantum well. Within the QW, we will know how high the confined energy levels are established, by measuring, e.g., a photoluminescence spectrum. The PL data from the recombination of an exciton were obtained by Kasai (Murayama and Kasai, 1992) for GaAs-Alo.3GQ.7As SQW for thicknesses between 1.5 and 15 nm. If an effective mass equation is utilized in a valid manner to calculate the confined energy levels within the QW, it is easy to obtain the difference between the energy

11.1 Sangle/Multiple Quantum Wells

I

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.3

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15

Lw / nm Fig. 11.3: Calculated well-width dependent band-gap mass for the electron in the barrier material with taking the effective masses of an electron and a hole t o be, respectively, mwe = 0.0665mo and m W h h = 0.34mo. The data of the exciton binding energy ( B E ) e xwere taken from Nelson et al. (1987). The open circles are calculated points and the solid lines show a gradual decrease in the mass within the energy gap vs the energy below the conduction band bottom on the horizontal axis. For example, Lw = 10 nm corresponds to E = -215 meV. level for the electron and that for the hole. In that calculation we must assume an effective mass for an electron within the barrier (i.e., the semiconductor band gap). Our calculation took m h e = 0.0665mo and m h h h = 0.34m0, denoting, respectively, the effective mass of the electron and that of the heavy hole within the well. In the calculation the effective mass of an electron within the barrier, mB is a parameter to be determined so that the calculated energy separation meets the observed spectrum. Thus determined effective mass mg is plotted in Fig. 11.3. The heavy hole mass is much larger than the electron mass, so that the variation in the effective mass for a hole within the barrier may be assumed to be insensitive to the energy above the valence band top of the barrier material. We are only concerned with the variation of the effective mass of the electron below the conduction band bottom of the barrier.

126

11 Optical Properties

E

f

Fig. 11.4: Schematic display of the calculation scheme of the effective mass as studied by Ando et al. (1989). Accordingly, in Fig,ll.3, the horizontal axis denotes the thickness of the QW, which dominates the determination of the energy separation between the conduction band bottom of the barrier and the confined energy levels for the electron within the QW. For a diminishing thickness the energy level of interest coincides with that at the conduction band bottom of barrier material AlGaAs. For the thicker QWs, the energy levels decrease more, the closer to the bottom of the conduction band of well material GaAs, i.e., the deeper energy for a tunneling electron below the bottom of the conduction band of AlGaAs. Ando and Akera (1989) tried to calculate the mass within the gap based on the tight-binding method. Their assumption was quite similar to the inversion of the E - lc curve for a negative energy value that causes an imaginary k with an appropriate effective mass. Their calculation gives, at least qualitatively, the same behavior for the tunneling mass as that obtained by the present author as a function of energy from PL measurement.

127

11.1 SingLe/MultipLe Quantum Wells

Thus it was established that the effective mass for an electron within the band gap decreases when the energy concerned gets more when going down more below the conduction band bottom of the barrier. On the other hand, with the case for the effective mass of a hole, it may decrease as the energy goes up above the top of the valence band. This fact suggests that the mass may disappear around the midpoint within the bad gap. The situation is schematically illustrated in Fig. 11.4, where l/m* = 7T2d2E(K)/dK2.Ando et al. (1989) utilized a formula l / m * = h-’dE(K)/KdK for an imaginary wavevector K in their calculation. In reality, the m* seems to vanish around the midpoint. This point should be contrasted to the energy-dependent effective mass within a single energy band. The effective mass of an electron goes to +oo at the inflection point and reappears from -m as the effective mass of a hole, as depicted schematically in Fig. 3.1.

E

- --. -.

c

7 A

-.

-

-.

-.

-

_. -.

-

na= 2

(4 Fig. 11.5: (a) Minibands in an SL. (b) An electron state that has the largest amplitude at the center of the SL, when an electric field is applied. (c) Suggested optical excitation of an electron, in the valence band at the center of the SL, t o several electron states. The largest transition probability occurs between the electron and hole within the same well (after Nakayama et al., 1991).

11 Optical Properties

128

Superlattice

In an SL there are minibands and minigaps, as shown in Fig. 11.5(a). When an electric field is applied perpendicularly to this device, the potential of each QW varies as a function of the well position, as shown schematically in Fig. 11.5(b). So we can expect a series of optical transitions between the level for a hole, e.g., in the SL center, and electron levels with energy offsets, according to the Stark effect. Figure 11.6 plots the photocurrent spectrum and the variation in reflectance spectrum under an electric field against the photon energy for various fields. The strongest signal is obtained when both the electron and the hole are within the same well, whereas weaker ones are for the transitions to the electron levels in neighboring wells. As is obvious, for a diminishing electric field, only a single sharp spectrum should exist between the energy levels for the electron and the hole, since all the spectra shown in Fig. 11.5 should converge to the same one.

.3

v, =

1

- 400 mV (140 kV/c+

'

- 450 rnV

'

X2

S

3

11.2 Exciton Absorption an Various Dimensional Geometries

129

Selection Rules Today Si is the best material for manufacturing inexpensive transport devices in mass fabrication. However, it is never used in light emitting devices, because it is known to perform only indirect optical transitions and, hence, the emitting efficiency is extremely low. Using SLs with an overlaid periodicity of n times the original lattice constant, it is recognized that the Brillouin zone (BZ) in the bulk state can be folded into l / n of the original reciprocal lattice vector in one dimension. For example, by fabricating an SL in the (100)-direction in k-space by alternately growing Si and the alloy SiGe, the conduction band minimum (the so-called valley) of Si around X = (27r/a)(0,010.85) may come close t o the r-point in the BZ. The components of the wavevector in the other two directions remain on the k,-axis. Does such Si with a folded BZ show a higher light emitting efficiency? This is not necessarily so, because the transition matrix element between orbital states remains the same after folding, although the k-selection rule can change from an indirect to a direct transition. Regarding the optical properties of Si, recent photoluminescence studies in porous Si are worth noting from the point of view of mesoscopic systems’ applications. When Si wafers are immersed in HF acid and anodized, the surface is mesoscopically processed and optical transition is no longer indirect as in the bulk. So far, cluster and quantum wire models have been proposed t o interpret a rather high PL efficiency from porous Si, although Si dangling bonds on the surface of the mesoscopic systems must usually be terminated by H and/or 0 atoms.

11.2

Exciton Absorption in Various Dimensional Geometries

Let us start with bulk three-dimensional exciton absorption. We are interested in how the absorption coefficient should be modified in two- and one-dimensional geometry. In the late 1950s and 1960s, there were several theoretical works published concerned with hypothetical two- and one-dimensional hydrogen atoms (for example, Elliott, 1957; Elliott et al., 1960; Loudon, 1959; Shinada et al., 1956). , Recently, it has become not hypothetical, but realistic to consider such squeezed hydrogenic states as are actualized in, e.g., quantum wells or quantum wires. Here we will review dimension-dependent optical responses. We are interested in various types of optical absorption. They are typically classified into four types: bound exciton and unbound exciton absorption both of which are further divided into allowed and forbidden transition cases. The nomenclature “bound” and “unbound ”exciton may sound curious, since “bound exciton” is sometimes used for an exciton that is bound to a crystalline defect or the like. In addition, the concept of an exciton is always concerned with its bound nature between an electron and a hole. However, this usage of terminology has long been used, so that we use the same terminology with ‘Lboundexciton” referring to the ordinary bound state with a binding energy below the band edge, and with “unbound exciton” referring to

11 Optical Properties

130

the exciton, the energy of which is above the band edge and depends on a wavevector k. Both bound and unbound excitons are affected by the Coulomb attractive force. Later we will use “free state” to refer to the direct band-to-band absorption unaffected by the Coulomb interaction between the electron and the hole. Three-Dimensional Optical Response The absorption coefficient is, as discussed in Appendix D, given by

(11.1) where p 3 ~ ( h w )is the three-dimensional joint density of states between the i- and fstates, and wp is the plasma frequency wp = with N being the number of carriers and V is the volume of the present system. q’+i$’ is the complex refractive index (e.g., Ziman, 1972). From this formula it is easy to write the absorption coefficients for bound excitons, unbound excitons as well as for the direct band-to-band transitions without the Coulomb attractive interaction working between an electron and a hole (that is, a “free” state). We will now discuss allowed transitions (denoted ‘a’) as opposed to forbidden transitions (denoted ‘f’).

d

m

(3D-la) Absorption coefficient for direct band-to-band; an allowed transition: (11.2)

(3D-lf) Absorption coefficient for direct band-to-band; a forbidden transition: (11.3) Here we assumed the energy spectrum to be parabolic for simplicity and the average of the squared directional cosine Ik . PI2was taken to be 1/3. K. is the relative dielectric constant of the medium of concern. For an allowed transition of a bound exciton, we use IQnlm(0)I2and the corresponding density of the states defined by P3D = 2VldEn/dnl-’ (2 is the spin degeneracy factor and En = -Ry*/n2). The reason why the value of the envelope function is taken at T = 0 is because an exciton is well localized around the center-of-gravity coordinate. (3D-2a) Absorption coefficient for bound excitons; an allowed process: (11.4)

11.2 Exciton Absorption in Various Dimensional Geometries

131

For a forbidden transition, we must only consider such non-vanishing states as

1 g r a d anlm(0)I. As shown in Appendix D, the following expression is obtained, since = (10n2 - 1)/97r(nai)5.' they are I grad@,om(0)12+ I grad@P,1m(0)(2 (3D-2f) Absorption coefficient for bound excitons; a forbidden process. Here the average of the squared directional cosine between grad and P was again taken to be 1/3.

(0 5 EG - h~ 5 Ry*)

(11.5)

(3D-3a) Absorption coefficient for unbound excitons; an allowed process: (11.6) where CY = l / a b k with wavevector k . For forbidden transition processes, non-vanishing 1 g r a d @(0)l2must be summed to2 (11.7) Thus, the following is reached. (3D-3f) Absorption coefficient for unbound excitons; a forbidden process: (11.8) Let US discuss the Sommerfeld factor, which is defined as the ratio of the excitonic absorption coefficient to that of the free states. From the above-derived equations we will list characteristics of optical responses in three dimensions. These items are to be compared with those in one and two dimensions, which will be described later. (1) The absorption coefficient for bound excitons is insensitive to the quantum number n. (2) The direct band-to-band absorption coefficient is proportional to (Aw - E G ) ~ / ' (allowed transition) and ( h w-EG)3/2 (forbidden transition), respectively, according to . absorption the three-dimensional DOS and k2 x P3D stemming from I g r a d @ ( r ) I 2The coefficient for bound excitonic states is almost constant for both allowed and forbidden transitions around EG M hw. (3) When k approaches 0 and 00, the Sommerfeld factor behaves as 27rlaI;k and approaches 1, respectively, for allowed transitions. On the other hand, the same factor for forbidden transitions behaves as 20n/9(a;l k)3 and approaches 1/9. 'The first term was unreasonably neglected by Elliott. The term is, however, really non-vanishing. 2The same as in the case in footnote 1.

132

11 Optical Properties

10

!lLL--

02 allowed 0

0.2

0.4 0.6 ka;I

0.8

1

Fig. 11.7: Sommerfeld factors of unbound excitons in three dimensions for allowed and forbidden transitions plotted as a function of a-' = ka;. Two-Dimensional Optical Response Now we proceed to two-dimensional cases. In this case

p2D

= m*/7rh2.

(2D-la) Absorption coefficient for direct band-to-band allowed transition:

(11.9) (2D-lf) Absorption coefficient of direct band-to-band forbidden transition: Here the average of the squared directional cosine between grad and P was taken to be 112. (11.10)

For an allowed process of a bound exciton, I@nm(0)12 = l/7rag2(n+ following is obtained.

i)3,hence, the

(2D-2a) Absorption coefficient for bound excitons; an allowed process: (11.11)

For a forbidden transition, we must only consider non-vanishing states I grad Qnrn*l (0) Since they are3

3The first term below was unreasonably neglected by Shinada and Sugano (1956). T h e y must be included.

11.2 Exciton Absorption an Various Dimensional Geometries

133

the following expression is obtained. (2D-2f) Absorption coefficient for bound excitons; a forbidden process:

(0 5 EG - hw 5 4Ry*)

(11.13)

where a factor 1/2 was again included resulting from averaging the squared directional cosine between k and P . o again stands for (ka;)-I. (2D-3a) Absorption coefficient for unbound excitons; an allowed process: (11.14) For forbidden transition processes, non-vanishing I grad 'P(0)l2 must be summed to give

(11.15) Hence, the following is reached. (2D-3f) Absorption coefficient for unbound excitons; a forbidden process: Here the average of the squared directional cosine between grad and P was taken t o be 1/2. (11.16) With the same Sommerfeld factor as discussed for three-dimensional cases, we again list characteristics of optical responses in two dimensions. (1) The absorption coefficient for bound excitons is insensitive to the quantum number n. (2) The direct band-to-band absorption coefficient is constant (allowed transition) and (Aw - E G ) (forbidden transition), respectively, according to the two-dimensional DOS and k2 x p 2 D . The absorption coefficient for bound excitonic states is almost constant for both allowed and forbidden transitions around EG M hw. (3) Figure 11.8 shows the Sommerfeld factors for unbound excitonic states for both allowed and forbidden transitions as a function of a i k . When k approaches 0 and 00, the Sommerfeld factors approach 2 and 1, respectively, for allowed transitions. On the other hand, the same factor for forbidden transitions behaves as 12/(a;k)' and approaches 1/2.

134

11 Optical Properties

20

(2D)

Q,

2

15Y c

n C

3

10-

O

n

-

C

2

Y

I

0

0

allowed 0.5

I

1

1.5

1 2

Fig. 11.8: Sommerfeld factors of unbound excitons in two dimensions for allowed and forbidden transitions plotted as a function of a-1 = Ica;J.

One-Dimensional Optical Response After Elliott's paper (1959), we can easily write down the free electron band-to-band absorption coefficients as follows. All these formulae are approximate, since they are obtainable only for a finite cut-off zo in the Coulomb interaction, i.e., -e2/r;co(lzl+zo). Accordingly, they are valid for 2zo/6,ak << 1 (n: quantum number for the quantized energy, En = -Ry*/62, 6, -+ n, n = 0,1,2, ... under the above approximation). (1D-la) Absorption coefficient for direct band-to-band allowed transition: (11.17)

(1D-lf) Absorption coefficient for direct band-to-band forbidden transition: (11.18)

As for bound exciton absorption, we can obtain the next two formulae; u is the quantum number. (1D-2a) Absorption coefficient for bound excitons; an allowed process:

(11.19)

11.2 Exciton Absorption an Various Dimensional Geometries

135

(1D-2f) Absorption coefficient for bound excitons; a forbidden process:

j

EG - Ry’fS,” 5 AW 5

EG

(11.20)

For the positive, continuous energy state, following Ogawa et al. (1991a, 1991b), the Sommerfeld factors are obtained for both allowed and forbidden processes, using, respectively, 1@Lg’(zo)1and I grad @?’(zo)I. (1D-3a) Absorption coefficient of unbound excitons; an allowed process: (11.21)

(1D-3f) Absorption coefficient for unbound excitons; a forbidden process: (11.22)

Let us summarize the optical responses in two dimensions. (1)The absorption coefficient due to direct band-to-band transitions is proportional / ~ allowed processes and (tuJ - E G ) ’ / ~for t o the DOS in lD, i.e., (hw - E G ) - ~ for forbidden processes. (2) The absorption coefficient of bound excitons is proportional to inverse of the quantum number square. (4)The Sommerfeld factor of unbound excitons is less than unity for allowed processes, i.e., the absorption probability is suppressed below that of the interband free electron process, whereas for forbidden processes the absorption probability is enhanced. (After Ogawa et al., 1991b).

Let us summarize the various dimensional cases in Tables 1-3.

136

1 1 Optical Properties

" 0

2

4

6

0

10

(E - Eg) / R i

0

c

0

2

4

6

0

10

(E - Eg) / R i Fig. 11.9: (a) Calculated Sommerfeld factors for the allowed transition with varying cut-off lengths 2 0 , i.e., Z O / U ; =2, 1, 0.5, 0.2 and 0.05 from top to bottom. (b) The same factors but for the forbidden transition (after Ogawa et al., 1991b).

11.2 Exciton Absorption in Various Dimensional Geometries

137

Table 11.1: Energies of Bound Excitons in Various Dimensions 3D Bound exciton energy with quantum number n (n=l, 2, 3, ... )

2D

-Ry*/6;

-Ry*/(n

1D

+ 1/2)2

-Ry* /u2,

(6,

-+ n - 1 for 20 -+

0)

Table 11.2: Absorption Coefficients of Free Excitons in Various Dimensions

11 Allowed Kfree(hw)

1

3D

("&?)

1/2

2D

1D

constant

Table 11.3: Sommerfeld Factors for Unbound Excitons in Various Dimensions ( a = l/kak)

allowed

I a >> 1

I-

207ra3/9

1 1 12a2

1

138

11 Optical Properties

References Ando, T., Akera, H. (1989), Phys. Rev. B40, 11619. Elliott, R. J. (1957), Phys. Rev. 108, 1384. Elliott, R. J., Loudon, R. (1959), J. Phys. Chem. Solids 8, 382. Elliott, R. J., Loudon, R. (1960), J. Phys. Chem. Solids 15, 196. Loudon, R.(1959), Am. J. phys. 27,649. Mishima, T., Kasai, J., Morioka, M., Sawada, Y., Murayama, Y., Katayama, Y., Shiraki, Y. (1986a), in: Proc. Int. Symp. GaAs snd Related Compounds, (Inst. Phys. Conf. Ser., No.79, Adam Hilger Ltd.), p.445. Mishima, T., Kasai, J., Morioka, M., Sawada, Y., Murayama, Y., Katayama, Y., Shiraki, Y. (1986b), in: Proc. Int. Symp. Modulated Semiconductor Structures (Yamada conf.), [Surf. Sci. 170,311.1 Murayama, Y. (1986), Phys. Rev. B34, 2500. Murayama, Y., Kasai, J. (1992), in: Proc. Yamada Conf. X X X : Electronic Properties of Two-dimensional Systems, M. Saito (Ed.) [Surf. Sci. 263, 604.1 Nakayama, M., Tanaka, I., Nishimura, H., Kawashima, K., Fujiwara, K. (1991), Phys. Rev. B44, 5935. Nelson, D. F., Miller, R. C., TLI,C. W., Sputz, S. K. (1987), Phys. Rev. B36, 8063. Ogawa, T., Takagahara, T. (1991a), Phys. Rev. B43, 14325. Ogawa, T., Takagahara, T. (1991b), Phys. Rev. B44, 8138. Shinada, M., Sugano, S. (1966), J . Phys. SOC.Jpn. 21, 1956. Ziman, J. M. (1972), Principles of the Theory of Solids, Cambridge: Cambridge University Press.

12

Magnetic Properties

12.1

Fine Particles

Among mesoscopic systems, zero-dimensional systems such as quantum dots and clusters are difficult t o deal with, since it is not easy to measure their DC transport properties and only detectable properties are AC responses to high frequencies including light beams. However, apart from optical responses, magnetic properties are fairly easy to detect. General reviews about magnetic materials and properties are found in, e.g., Chikazumi (1964); Cullity (1972); White (1970); Vonsovskii (1974). In 1962, Kubo (1962) discussed the thermodynamic properties, such as heat capacity and paramagnetic susceptibility, of metallic fine particles In fine particles, the typical energy difference between neighboring levels may be larger than the ambient thermal energy, so the thermal excitation is sometimes suppressed at an ambient temperature much more than in the bulk, continuously excitable system. This reduces the heat capacity and increases the paramagnetic susceptibility. According to Kubo’s theory, the number of spins in the particles strongly affects the magnetic properties, depending on whether it is even or odd. When it is odd, the particles must behave like magnetic particles with uncompensated spin. Almost a t the same time, Nkel (1964) considered a similar problem. Suppose that fine particles are composed of an odd number of antiferromagnetic sublattices. The magnetic moment on the surface sublattice will rotate to be parallel with the external magnetic field and the moments inside will be sequentially affected by the neighboring spin orientation. In this manner antiferromagnetic particles show higher magnetic susceptibility because they behave like a spiral spin state with the remnants of a magnetic moment. This particle state was termed superantiferromagnetism. When particles have an uncompensated magnetic moment , e.g., in single-domain ferromagnetic particles, they look like particles with giant magnetic moments. Since these particles may not be in an ordered magnetic state in a non-magnetic medium, the whole system need not ordinarily show ferromagnetism. If the particles are small enough, -100 &, they must behave as a paramagnetic substance under thermal agitation at the room temperature. This state is called superparamagnetism. Superparamagnetism can reverse its whole magnetization when a high potential barrier is overcome at a fairly high ambient temperature. If the temperature is low enough, this type of thermal activation over a high barrier will never occur. A magnetic particle usually has a certain amount of crystalline anisotropy energy. This acts as a barrier, since, if the particle is a quasi-sphere with its easy axis along the (0001)-

140

12 Magnetic Properties

direction of a hexagonal crystal and both M and - M have the same energy, any other directions usually make hard axes. In other words, a bundle of spins forming M performs a reversal only by tunneling through the potential barrier which equals the anisotropy energy. This is an example of a macroscopic quantum phenomenon. Any macroscopic quantum phenomenon seems to occur easily on mesoscopically tiny samples, i.e., wires and particles. This topic has been studied by, e g , Tatara et al. (1997); Hong et a1.(1998); Ruediger, et al. (1998), and so on. It is well known that a y-ray is emitted without recoil in bulk radioactive materials, called the Miissbauer effect. However, it is not the case in a mesoscopically small-sized particle. The recoil energy must be compensated for by the motion of the center of gravity, i.e., the particle as a whole. So, when it is sufficiently small and behaves without coupling to its environment, the recoil energy must be appreciable and be eventually manifested as a shift of the emitted y-ray. This fact has been pointed out by the present author (1966a).

12.2 Magnetic Thin Films For an infinitesimal thickness of magnetic thin film, it is easily imagined that the system behaves as in two dimensions. There, it is known that the Nkel type of magnetic domains appear. The Nkel domains are such that magnetization rotates in the plane, without rising from the film plane, from one direction to the other. When increasing the thickness, magnetization rises up to form three-dimensional domain structures when a perpendicular anisotropy is energetically favored, one typical example of which is the Bloch domains. This thickness-dependent phenomenon is a remarkable dimensional crossover from two to three. We will discuss how a two-dimensional system is converted to a three-dimensional one depending on its thickness and/or various physical constants. The first paradigm of such dimensional crossover is the stripe domain in permalloy thin films. The domain structure is shown schematically in Fig. 12.1, and was discovered independently by Spain (1963) and Saito et al. (1964). Theoretically Kaczkr et al. (1963) proposed the simplest theoretical model where the spin tilting from the plane varies sinusoidally as a function of the coordinate perpendicular to the mean direction of spins and parallel with the plane. The present author (196613) showed that there is a micromagnetic state (Brown, 1963) with a lower energy, as shown in Fig. 12.1, where spins form a circulating Landau-type closure domain in the cross section of the film perpendicular to the resultant spins. In this domain structure, the thickness and stripe width are on a comparable scale of a few 1000 A. Since it would appear to be instructive to show a fairly detailed treatment of the spin structure from a micromagnetic point of view, Appendix J is dedicated to describing the theoretical aspects of the structure. In this Section we are only interested in the critical thickness where the stripe domain structure appears. The critical thickness is such that over which the stripe domain structure appears three-dimensionally and under which all spins lie in the film

12.2 Magnetic Thin Films

141

-! Fig. 12.1: A cross section of a magnetic structure perpendicular t o the mean magnetization direction, i.e., the y-direction. The rising angle q5 also depends on x and z, and the deviation angle 6' from the same direction depends on x and z . The z-axis is taken to be perpendicular t o the film, whereas the x- and y-axes t o be in the plane. The y-direction meets the mean magnetization. X is the full wavelength in the x-direction, and the thickness is 2d.

L

m

fn

.05

film thickness 2d / (100 nm)

Fig. 12.2: Calculated saturation fields vs thickness. The saturation field Hkll corresponds to that under which the stripe domains disappear. H k l = K l / I s is the anisotropy field. Experimental data were communicated by Sugita et al. (1966).

12 Magnetic Properties

142

3

.

.

--L

0 0 3

3

W

.5

. 2

5 10 20 50 film thickness 2d / (100 f-~m) Calculated period X of the in-plane variation 4 (and 8) vs film 1

Fig. 12.3: thickness d. The experimental data were taken from Saito et al. (1964). p = 0.7 was estimated from the data by Fujiwara (1966). A/(1,2/2pO) = 7 x m2 was used in the calculation. plane and form a single domain two-dimensionally. Three cases have been investigated. (Model I) : 4 (Model 11) : 4

4(x),

8 = constant = 0 = $ ( z , z ) , 8 = constant = 0 (Model 111) : 4 = 4 ( x ,z), 8 = 8(x,Z) =

(12.1) (12.2) (12.3)

Model I11 corresponds to Landau’s closure domain structure. As will be intuitively recognized, the larger the wavelength X is, the higher the magnetostatic energy is, which causes the critical thickness 2d t o increase. The reason is that the larger the aspect ratio (i.e., X/2d) of a single domain is, the larger is the demagnetizing factor N and, accordingly, the magnetostatic energy density N 1 ; / 2 p o . As a matter of course, the larger the (perpendicular) anisotropy energy K l is, the smaller the critical thickness is. An applied external magnetic field H I Icauses the thickness t o increase. As typical experimental data, the present author compared his calculation with the data taken by Saito et al.(1964). Figure 12.2 illustrates 1 - H K J I / H Kvs~ the critical thickness 2d. H K is~ the anisotropy field defined by K l = I s H ~ (l K l : the perpendicular anisotropy energy). In the figure, I, 11, and I11 denote that the calculations are performed on Models I, 11, and 111, respectively.

143

12.2 Magnetic Thin Films

The next comparison was done on the wavelength A, the period of sinusoidal variation of q5 in the 2-direction. Figure 12.3 also compares Model I, 11, and I11 calculations with experimental data by Fujiwara et al. (1966). The comparison assures that all calculations are justified, and Model I11 best fits the observations.

Superlattice of Magnetic Thin Films Magnetic thin films sometimes show unusual effects when they are formed in a superlattice (SL) (Himpsel et al., 1998; Levy, 1994). The first data for a Fe-Cr-Fe sandwich were reported by Grunberg et al. (1986), which showed an antiferromagnetic coupling between the two outer Fe layers. Later it was elucidated that this antiferromagnetic coupling between ferromagnetic substances causes a large magnetoresistance, which is usually called giant magnetoresistance (GMR). This large magnetoresistance was reported on a type of SL, i.e., (Fe-Cr), with n = 30 - 60 by Baibich et al. (1988) (Fig. 12.4).

~~

I

I

-2

I

R I R(B=O)

1

-4

I

0

I

2

, 4

magnetic field B I T Fig. 12.4: Observed giant magnetoresistance (GMR) on three superlattice samples with Fe and Cr at 4.2 K (after Baibich et al., 1988).

144

12 Magnetic Properties

Fe

(a) Ferromagnetic alignment ( T

'!)

(b) Ferromagnetic alignment ( I

.1 )

Fe .1 Cr t

Fe t Cr L

Fe 1

(c) Antif err omagn etic aI ignment Fig. 12.5: Conceptual energy band schemes based on the spin-split-band model (SSBM). R? and RL are the resistances owned by up-spin and down-spin electrons. (a),(b) For the F-alignment, where Ry and RJ are combined t o give a total for the parallel resistance circuit, RF = RtRc/(R? R L ) .(c) For the AF-alignment which is half the average resistance between RT and R J , i.e., RAF= (Rt R J ) / ~ .

+

+

Ordinary bulk material has a magnetoresistance of at most a few percent, whereas in this SL it amounts to several tens of percent. Initiated by these studies, enthusiasm

12.2 Magnetic Than Films

145

arose for investigating magnetic multilayers, since they may be applied to reading heads for magnetic file memories such as hard disks and floppy disks. First of all, we will try to give an intuitive explanation as to how the variation in resistance is attained in the SL systems. In Fig. 12.5 conceptual band energy schemes of ferromagnetic and non-magnetic materials are shown (Hathaway et al., 1985). The easiest way to understand the ferromagnetic state is by means of the spin-split-band model (SSBM). A splitting is caused by the exchange-correlation energy difference between majority- and minorityspin electrons. For an up-spin state the resistance is denoted by Rt, whereas a downspin state by RJ. Here we are not concerned what the origin of the electric resistance in this system is. We will apply the so-called two-fluid model (Edwards et al., 1991a, 1991b, 1991c) to explain that the antiferromagnetically coupled system always has a higher resistance than the ferromagnetically coupled one. The two-fluid model means that all current carriers are divided into two groups: carriers with up-spins and those with downspins. In addition, we assume that both carriers are never intermixed upon scattering; in other words, we neglect spin-flipping scattering. Then for ferromagnetic alignment (F) a parallel resistance circuit has a compound resistance RF = RrRJ/(Rr Rk). Similarly in antiferromagnetic alignment (AF) a couple of average resistances R’ = (Rt R J ) / ~flow parallel, which causes a total of RAF= R’/2 = (Rr R J ) / ~Even. tually it is easy to show that

+

+

+

(12.4) Although the terminology “AF” is used here, the reader must be careful that no antiferromagnets appear in this problem. Magnetic substances are always ferromagnetic materials with spins either up (f) or down ($). Antiferromagnetic alignment is only used for antiparallel spin configurations between ferromagnetic substances on both sides. Let us assume that the effective exchange coupling Jeffis negative between the spins on i-th and (if1)-th ferromagnetic layers: i.e., -J,sSi.Si+l, Jeff < 0. Actually, when an external magnetic field is applied in the plane of a multilayer and the resistance is measured within the plane, for example, perpendicularly to the field, the magnetic states are antiferromagnetically aligned for small fields, whereas for a sufficiently large field they are forced to change into ferromagnetic alignment, in the case of which resistance decreases. The field at which the alignment becomes ferromagnetic is called the saturation field B,. The next point is that the effective exchange coupling depends on the thickness of non-magnetic material, say, Cr. It varies from AF alignment for ultimately thin layers to F alignment for thicker layers, and further to AF, and so on. The sign of the effective coupling changes periodically. For thicknesses with F alignment, there is no appreciable change in magnetic alignment and, hence, no significant magnetoresistance. Thus, whether the outside layers are coupled F-like or AF-like is vitally important to cause a GMR.

12 Magnetic Properties

146

Cr thickness

(A)

Cr thickness

(A)

Fig. 12.6: (a) Transverse saturation magnetoresistance a t 4.2 K and (b) saturation field vs Cr layer thickness for three series of structures of the form Si(111)/(100 A)Cr/[(20 A ) F e / t c , C r ] ~ / ( 5 0W)Cr, deposited at temperature 40°C ( N = 30) (A, 0 ) and 125°C (0; N = 20) (after Parkin et al., 1990).

12.2 Magnetic Thin Films

147

The data in Fig. 12.6 were taken by Parkin et al. (1990). In Fe-Cr multilayers oscillatory variations in ARIR and B, are shown as a function of the thickness of Cr sandwiched by the Fe layers.

Ruderman-Kittel-Kasuya- Yoshida (RKKY ) Interaction The easiest concept to understand this oscillatory behavior of the effective magnetic coupling may be the RKKY interaction (Ruderman et al., 1954; Kasuya, 1956; Yoshida, 1957). Historically, this interaction has been elaborated based on indirect Ii Ij-coupling between nuclear magnetic moments mediated by conduction electrons (s-electrons) through the hyperfine interaction. The hyperfine interaction works between the magnetic moment of s-electrons and the nuclear magnetic moment (“contact term”). Such a theory was proposed by Rudermann and Kittel, Kasuya, and Yoshida. As is known as the F’riedel oscillation, a charged center in metals is screened by the surrounding free electrons. Sometimes overscreening occurs in some parts or underscreening in others, depending on the distance from the center; that is, a charge density wave (CDW) is generated around the center as a spherically oscillating wave. In a similar manner a localized spin may be over- or under-depolarized depending on the distance from the localized spin. Such an oscillatory behavior of the spin polarization is also called the spin density wave (SDW). In terms of an effective exchange energy, the interaction changes its sign depending on the distance from the localized spin. If this theory is applied to decide a one-dimensional effective exchange energy between ferromagnetic layers of Fe separated by a Cr layer with thickness d, the interaction will be of the form like cos (2kfd) (12.5) J ( z )c( d2 where kf is the Fermi wavevector (Baltenberger et al., 1990: Slonczewsky, 1993). Ordinarily this k p is something like q r / a with a being the lattice constant and q a fraction less than unity, so that the expected oscillation period may be several A. Experimentally, periods on the order of 10 to 20 A are observed in metals. It was first doubted because of this discrepancy whether the RKKY interaction works in metallic superlattices or not. Many theories have been proposed to reconcile this gap between observations and calculations. Regarding the effective coupling energy and the period of the variation, first-principle calculation of energy is performed (Schilfgaarde et al., 1993; Mirbt et al., 1996; Stiles, 1993, 1996). They obtained, however, too large energy difference between ferro- and antiferromagnetically aligned states. One of the powerful propositions will be discussed below based on the so-called “quantum well model” (QW model: Edwards et al., 1991a, 1991b, 1991c; Ortega et al., 1993).

Quantum Well Model In parallel with semiconductors, a metallic sandwich with different species of metals may constitute a quantum well. For the present metallic multilayer configuration we will also study the quantum well model.

12 Magnetic Properties

148

Let us calculate how the total energy varies from ferromagnetic alignment (F) to antiferromagnetic alignment (AF) (Murayama et al., 1999). Since each layer is so thin and the unit of three layers periodically repeat to construct a superlattice system, we may assume envelope functions specific to the Bloch states, which must be the solutions of the system with a periodic potential. The minimum unit is, e.g., an Fe-Cr-Fe sandwich. From now on, we will refer to the nonmagnetic mediating layer as “M”, and a magnetic layer with spin up as “t” and that with spin down as “$”.

vacuum potential

m

Fig. 12.7: (a) Illustration of a sandwich: f -M- 4,whose work functions ~ q 5~ q 5 ~and EF and the Fermi energies are specified, respectively, by q 5 EP - E i . (b) Band diagram of a f -M- T-sandwich with an approximate rounded-off interfaces due to screening of the dipole layer. (c) The same as in (b) for a -1 -M- $-sandwich. (d) The same as in (b) for a -1 -M- ?-sandwich.

149

12.2 Magnetic Thin Films

As shown in Figs. 12.5 and 12.7(a), we will consider an up-spin band separated from a down-spin band. The work functions of f-, J--,and M-material are denoted, respectively, by q5t, q 5 ~ ,and q 5 ~ . Correspondingly the Fermi energies are defined, respectively, by E,f, E / , and Ef" measured from the band bottom. We are concerned with the total energies of each spin configuration at zero temperature. For F alignment the energy is calculated summing energies for the f -M- f and J- -M- J- -systems, whereas for AF alignment the energy is just that of the f -M- J- - and J.. -M- f -systems. The most important assumption employed in this calculation is the so-called spinsplit-band-model (SSBM), which claims that the energy band with down-spins is shifted upward in comparison with that with up-spins by the exchange-correlation ~ q 5 ~- U . energy, say, a constant U . This means that E i = E i - U along with q 5 = The widths of M- and the magnetic layer are, respectively, 2d and 2w. As is well known in a semiconductor junction, once a pair of metals with different work functions are in contact and attain a common Fermi level, charges are transferred from the layer with the smaller work function to that with the larger one. The potentials eventually become smoothly connected, having the same derivatives across the interface. At that time the quantity t o specify the dipole layer region must be the F The DOS of the concerning Thomas-Fermi screening constant ~ T = dimension p ( E ) was given in Chapter 3. For a constant net excess (or deficiency) of charge, the Poisson equation dV(x)/dx2= n/co (n: the charge density) gives a quadratic variation of the potential eV(z) as a function of x. In metals, this is not the case. However, for a fairly thick dipole layer compared with the thicknesses of the non-magnetic and magnetic layers, an approximation with the quadratic function may be accepted to simulate a Hartree potential. We employed this approximation assuming that the Thomas-Fermi screening length XTF = 2 7 r / k ~is~ larger than 2d: X ~ ~ 2 2 dRegarding . Fig.12.7(b) (F: ff-case), the potential is given by

d-.'

-d

-2d

5 x 5 0,

5 x 5 -d,

-d-w5~<-2d,

d

5 x 5 2 d : (w 2 d)

2 d < x < d + w : (w > d )

V(x) = -E,f -d - w 5 x 5 -d, d5x

(12.6)

5 d + w : (w 5 d)

d 5 x 5 2 d : (w

> d)

'The Fourier transform of the Coulomb interaction is written, using ~ T F , as e 2 / c o ( k 2 which does not diverge for a vanishing wavevector k + 0.

+ k&),

150

12 Magnetic Properties

and for F: $$-case (Fig. 12.7(c)) the same expressions are written with all the T's replaced by 4's. Similarly for the AF case (Fig. 12.7(d)), if the magnetic layer on the left-hand side has r's, then that on the right-hand side has $'s. Once the potential has been given, it is easy t o solve the Schrodinger equation by the FDM method (Appendix A) based on the periodic boundary conditions. For simplicity, we took the origin of the energy to be the Fermi level in the AF alignment case. After obtaining the eigenenergies for this problem, we calculate the total energy of the F case, where a formula

=

C $[E: - (Ef)2]

(12.7)

i= 1

is utilized. For down-spins, in a similar manner,

(12.8) and for A F alignment, the total energy reads

ntL

= - C , ( pE i t.lI 2

(12.9)

i=l . n t ~are, . respectively, the maximum quantum number over which Here n t t , ~ J Jand the eigenenergies Ef', E!', and Ef' exceed the respective Fermi level. The total energy for F alignment is the sum: EF = EFT' EFJJ,whereas that for AF is twice E A F ~ J . Throughout this calculation a constant DOS p specific to two dimensions and a Fermi level Ef common to T t and $4 were assumed. In addition, the number of electrons must be equal for F and AF alignments, the condition of which is described by

+

(12.10) i=l

i=l

12.2 Magnetic T h i n Films

151

This equation determines Ef. We then reach the final expression for the sought difference in the total energies per electron between the F and AF alignments:

A E = (EFTT + E F J J- ~ E A F T ~ ) / ~ N A F

(12.11)

The calculated results are shown in Fig.12.8(a) for 4~ = 4.5 eV, 4~ = 2.0 eV, 41 = 4.0 eV; Ef" = 0.5 eV, Eft = 2.5 eV, E i = 0.5 eV. The energy difference A E is plotted as a function of the thickness 2d with magnetic material thickness 2w fixed to 30 A. Be careful that the horizontal axis is measured in units of eV. That is, the difference in total energy was derived only after canceling four to five digits, since the total energy is on the order of a few tens eV. As already stated referring to the experimental data shown in Fig.12.6, the positive and negative differences mean that an AF and an F alignment is, respectively, favored. Thus, magnetic alignment changes starting from AF near a zero thickness t o F and then again to AF and so forth quasi-periodically. Since the energy difference may be equated with 2 p ~ g B=~A E

(12.12)

( p ~ the : Bohr magneton), the curve is replotted as a saturation field B, vs thickness as in Fig. 12.8(b). The calculated results should be compared with the observation shown in Fig. 12.6. The most meritorious point in this model calculation is that it can give a reasonable order of the saturation field as well as the oscillatory behavior simulating the observation. The present calculation assumes a Fermi energy of the non-magnetic metal equal to 0.5 eV which is small enough to result in a period of about 7.5 A. To check the simplest eigenenergy case, we may assume a constant eigenenergy separation just as in the harmonic potential case. Then the calculated energy difference is negative irrespective of thickness; that is, magnetic alignment should be ferromagnetic for all thicknesses 2d, i.e., no GMR would occur. Finally, we must discuss the period of the energy difference concerned. At first the observed period was unreasonably large and numerous investigations have been published to explain why so a large period is observed. It must be stated that this point has not yet been concluded. However, the explanation described below seems to be one of the convincing models. The larger the Fermi energy, the smaller is the period, since the Fermi wavevector relates to the period by kf = n/(period). Actually for a few eV of Ef, the calculated period becomes something like a few A, which is a small value. However, it is known that the crystalline lattice is composed of a discrete lattice constant, which is not a continuous value. In consequence, we must only choose points of discrete thicknesses on a rapidly oscillating curve. This produces fairly slowly oscillating phenomena in parallel with the Moir6 pattern observed on a couple of periodic lattice patterns. However, this consideration does not yet conclude a very long period as observed. Regarding the temperature dependence of the saturation field, we must take into account the temperature dependence of U , which is proportional to the square of magnetization and may follow, e.g., the MS2-vs-T curve. Actually for a vanishing magnetization, U , and accordingly A E , also vanishes.

152

12 Magnetic Properties

thickness of non-magnetic layer / 8,

10 20 30 40 50 thickness of non-magnetic layer / A Fig. 12.8: (a) Calculated difference in total energies for F and AF alignment as a function of thickness 2d. A positive energy difference means that magnetic state is antiferromagnetically aligned, whereas a negative energy is for ferromagnetic alignment. (b) Plot of saturation magnetic field as a function of thickness, using Eq. (12.12) and the energy differences shown in (a). Both results are for plausible parameters obtained from Hathaway (1985): & = 4.5 eV, 4~ = 2.0 eV, $4 = 4.0 eV; Ef" = 0.5 eV, E: = 2.5 eV, E: = 0.5 eV.

Surface Magnetism A considerably different but similar phenomenon was observed by Koike et al. (1994a, 1994b, 1995). According to them gold thin surface layers deposited onto a polarized

12.2 Magnetic Thin Films

153

4

1

I

6

8 \

0

r 4 a N

0 .w

-2

2

-2

0

-4

0

Q

0

I

I

I

2

4

6

8

Au thickness / nm Fig. 12.9: Oscillating component of polarization as a function of Au layer thickness obtained by subtracting the smoothly decaying background signal from the total output signal of spin-SEM. The solid curves fit the variation of observed vertical bars to produce a period of 15 A for both experiments. E p stands for the primary electron energy (after Koike et al., 1994a). Fe surface showed magnetic polarization which oscillates as a function of Au thickness (Fig. 12.9). Measurement was done utilizing a so-called Spin-SEM (the same as SEMPA, an abbreviation of Scanning Electron Microscope with Polarization Analysis), where the polarization of secondary electrons emitted from the sample surface is analyzed and determined. This phenomenon again suggests an RKKY-like oscillation, i.e., electrons in Au-layer are alternately antipolarized or polarized being affected by the surface polarization of Fe.

Spin-Dependent Tunneling

In the QW model of an AF-aligned t -M- -1 sandwich, the electronic states in the two FM films are offset by the exchange-correlation energy according to the SSBM. For this problem we considered size-quantized scheme of energy. On the other hand, a simple energy offset between a couple of FM materials may be concerned with in a tunnel junction, i.e., an FM-I-FM structured device (Julliere,

12 Magnetic Properties

154

1975; also Slonczewski, 1989). For this device FM materials are not necessarily thin films, but may be bulky. Moodera et al. (1996, 1996a) observed such an effect in a CoFe/A1203/Co tunnel junction by measuring tunnel conductance as a function of magnetic field parallel to the junction plane. The magnetic coercive force of Co (Hc(Co) 5 mT) differs from that of CoFe (Hc(CoFe) 20 mT), and accordingly for a field between both Hcs the spins in the FM materials are different. This means that electrons in one electrode are not easy to conduct t o the counter electrode with the opposite spins. That is, in a magnetic field region between Hc (CoFe) and Hc (Co) MR (magnetoresistance) increases, whereas outside the region MR decreases to an ordinary magnitude for tunneling between the same spin states. Figure 12.10 shows that MR measures up to about 10 7% in contrast to the MR for each electrode material itself up to 0.5 % at the largest. It should be noted that researches on such spin-dependent tunneling phenomenon were initiated by pioneering works done by Meservey et al. (1970), Tedrow et al. (1971) and Tedrow et al. (1971a) in superconducting A1-AI203 -Ag and superconducting Al-A1203 -ferromagnetic Ni junctions, respectively.

-

N

I

1

- 0.50

I 0.0

CoFe/A1203/Co junction

7.5 -

5.02.5-

-60

-40

-20

u

0 20 B/mT

40

60

Fig. 12.10: MR measurements of a tunnel junction FM-I-FM and two magnetic electrodes FM materials themselves as a function of magnetic field applied parallel to the junction plane at room temperature. The arrows indicate the spin orientations in the FM electrodes (after Moodera et al., 1996).

12.2 Magnetic Thin Films

155

References Baibich, M. N., Broto, J . M., Fert, A., Nguyen Van Dau, F., Petroff, F., Eitenne, P., Creuzet, G., Friedrich, A., Chazelas, J. (1988), Phys. Rev. Lett. 61, 2472. Baltenberger, W., Helman, J . S. (1990), Appl. Phys. Lett. 57, 2954. Brown, W. F. Jr. (1963), Micromagnetics, New York: Interscience. Chikazumi, S. (1964), Physics of Magentism, New York: Wiley. Cullity, B. D. (1972) , Introduction to Magnetic Materials, Reading, MA: AddisonWesley. David, J. (1991), Introduction to Magnetism and Magnetic Materials, London: Chapman and Hall. Edwards, D. M., Mathon, J., Muniz, R. B. (1991a), IEEE Trans. Magnetics 27, 3548. Edwards, D. M., Mathon, J. (1991b), J. Magn. Magn. Mater. 93, 85. Edwards, D. M., Mathon, J., Muniz, R. B., Phan, M. S. ( 1 9 9 1 ~ )J.~ Phys.: Cond.Matter 3, 4941. Fujiwara, H., et al. (1966), private communications. Grunberg, P., Schreiber, R., Pang, Y. (1986), Phys. Rev. Lett. 57, 2442. Hathaway, K. B., Jansen, H. J. F., Freeman, A. J. (1985), Phys. Rev. B31, 7603. Himpsel, F. J., Ortega, J. E., Mankey, G. J., Willis, R. F. (1998), Adv. Phys. 47, 511. Hong, K., Giordano, N. (1998), J . Phys. C o d - M a t t e r 10, L401. Jiles, D. (1991), Introduction to Magnetism and Magnetic Materials, London: Chapman and Hall. Julliere, M. (1975), Phys. Lett. 54A, 225. KaczBr, J., Zeleni, M., Siida, P. (1963), Czech.J . Phys. 13, 579. Kasuya, T. (1956), Progr. Theor. Phys. 16,45. Koike, K., Furukawa, T., Murayama, Y. (1994a), Phys. Rev. B50, 4816; (1995), Phys. Rev. B51, 10260. Koike, K., Furukawa, Cameron, G. P., T., Murayama, Y. (1994b), Jpn. J. Appl. Phys. 33, L769. Kubo, R. (1962), J. Phys. SOC.Jpn. 17, 975. Levy, P. M. (1994), in: H. Ehrenreich, D. Turnbull (Eds.) Solid State Physics47, p.367.

156

12 Magnetic Properties

Meservey, R., Tedrow, P. M., Fulde, P. (1970), Phys. Rev. Lett. 25,1270. Mirbt, S.,Niklasson, A.M. N., Johansson, B. (1996), Phys. Rev. B54,6382. Moodera, J. S., Kinder, L. R. (1996), J . Appl. Phys. 79,4724. Moodera, J . S., Kinder, L. R., Nowak, J., LeClair, P., Meservey, R. (1996a), Appl. Phys. Lt 69,708. 79,4724. Murayama, Y. (1966a), Phys. Lett. 23,332. Murayama, Y. (1966b), J. Phys. Soc. Jpn. 11, 2253. Murayama, Y., Ishino, Y., Ishiduki, Y. (19991, unpublished. NCel, L. (1964), Comptes Rendus, SCance du 26 Juin, 4075; SCance du 3 Juillet, 9; SCance du 10 Juillet , 203. Ortega, J. E., Himpsel, F. J., Mankey, G. J., Willis, R. F. (1993), Phys. Rev. B47, 1540. Parkin, S . S.P., More, N., Roche, K. P. (1990), Phys. Rev. Lett. 64,2304. Ruediger, U.,Yu, J., Zhang, S., Kent, A. D., Parkin, S. S. P. (1998), Phys. Rev. Lett. 80, 5639. Ruderman, M. A., Kittel, C. (1954), Phys. Rev. 96,99. Saito, N.,Fujiwara, H., Sugita, Y. (1964), J. Phys. SOC.Jpn. 19,421. van Schilfgaarde, M., Herman, F. (1993), Phys. Rev. Lett. 71,1923. Slonczewski, J. C. (1989), Phys. Rev. B39,6995. Slonczewski, J. C. (1993) , J . Magn. Magn. Muter. 126,374. Spain, R.J. (1993), Appl. Phys. Lett. 3,208. Stiles, M. D. (1993), Phys. Rev. B54,14679. Stiles, M. D. (1996), Phys. Rev. B48,7238. Sugita, Y., et al. (1966), private communications. Tatara, G., Fukuyama, H. (1997), Phys. Rev. Lett. 78,3773. Tedrow, P. M., Meservey, R. (1971), Phys. Rev. Lett. 26,192. Tedrow, P. M., Meservey, R. (1971a), Phys. Rev. Lett. 27,919. Vonsovskii, S.V. (1974), Magnetism, Vol. 1 & 2, New York: Wiley. White, R. M. (1970), Quantum Theory of Magnetism, New York: MaGraw-Hill. Yoshida, K. (1957), Phys. Rev. 106,893.

Properties of Macroscopic Quantum States 13

Four typical examples of macroscopic quantum states are lasers, Bose-Einstein condensates of atoms, superfluids, and superconductors. Here we are most interested in superconductivity (SC) (Schrieffer, 1964a; de Gennes, 1966; Tinkham, 1975). It is well established that the SC state can be described in terms of a macrowave, which is a coherent wave extending from one edge of the sample t o the other. This is almost always the case even if the sample is macroscopically large, so long as it is superconductive. This means that the large sample behaves just like a tiny microscopic system from the point of view of its physical properties, in particular, the property of coherence. The mesoscopic systems we are discussing are concerned with the systems where the physical properties are those such as between the micro- and macroscopic scales. In mesoscopically tiny SC samples, do they not degrade their coherence but behave perfectly as in microscopic systems? Usually the larger size or larger dimensions the system has, the more degrees of freedom are incorporated into the system, which may degrade the coherence of electrons therein. So far we discussed systems tiny enough to maintain coherence so that they behave partially similar to a microscopic system, although they are far beyond a typical microscopic entity such as an isolated atom. The macroscopic coherence is an outcome of the fact that all electrons concerned in, e.g., an SC have lost their identity as Fermi particles because they all fall into a single ground state by means of forming Bose-particle-like Cooper pairs. A wave may be identified by either its wavelength or frequency ( the both are correlated through a dispersion relation, which eventually specifies its energy) and an additional phase. If all electrons incidentally have the same ground state energy and there is only an infinitesimally small fluctuation in phase of these electrons, they are no longer individually differentiated from another. The electrons in such a state happen to have macroscopic coherence which extends over other electrons. However, in SC, too, some sort of degraded coherence can occur, e.g., when the system is thin or tiny enough. An ideal SC is made from bulk samples, where there is no decay in the current between two leads or in a persistent ring current. This type of current can flow persistently only when there is no damping. However, there is always a slight damping mechanism even in the SC state. Nowadays it is well established that high-T, cuprates are modeled t o have CuOz sheets carrying a supercurrent with weak Josephson-like

158

13 Properties of Macroscopic Quantum States

coupling between them. Accordingly, their superconductivity is characterized by two-dimensional features. Typical examples of high-T, cuprates are: Laz-,Sr,Cu04 (LSCO), YBa2Cu307-6 (YBCO, Y123), Bi2Sr2Ca,-1C~,02~+4+6 (BiSCCO; n = 2,3), TlBa2Ca,-1Cu,O2,+3+a (TlBCCO; n = 2,3), T12Ba2Can-1Cu,02,+4+a (TlBCCO; n = 1,2,3), and HgBa2Can-1Cu,O2,+2+6 (HgBCCO; n = 1,2,3) ( e g , Ginsberg, 1989, 1990, 1992, 1994; Kamimura et al., 1989; Bedell, 1990; Cyrot et al., 1992; Seahan, 1994; Anderson, 1997; Chu, 1997). Let us assume magnetic flux quanta ( “fluxons”) exist penetrating the two-dimensional sheets above the lower critical field BC1.It is known that thermally excited fluxonantifluxon pairs (pairs of upward and downward oriented magnetic flux lines) interact with the current that is fed to measure the voltage drop through the sample, particularly in two-dimensional systems. Thus, the I - V characteristic is subject to a power law, according t o the Kosterlitz-Thouless (KT) mode of excitation that is intrinsic to two-dimensional systems, as will be described next. This means that a voltage drop of exactly zero is never expected.

13.1

Kosterlitz-Thouless Mode in High-T, SC

In the original paper by Kosterlitz and Thouless (1973; also Kosterlitz, 1974), they interpreted a new type of phase transition in a two-dimensional system based on a dilute plasma model. Let us consider a logarithmic interaction between charges. ~ ( r-ir j )

-e2 log(lri - rjl/ro) - 2p, = 0, =

~ ri rjl IT2

-‘jl

> TO < To

(13.1)

where 7-0 is a cut-off of the separation between a pair of charges and L,L is the chemical ) satisfy the equation: V 2 @ ( r )= - p ( r ) potential per one charge. For a potential @ ( r to ( p : the charge density), mathematics teaches us that the solution to this equation is easily given by means of the so-called Green function. As is well known the Green function in three dimensions is proportional to r-l , i.e., the Coulomb potential, whereas in two dimensions it is a logarithmic function of the radial coordinate of the form that is assumed in Eq. (13.1). It is easy to show that at sufficiently low temperatures charges form pairs, because isolated charges are not really created, according to an insignificant entropy term. In a low temperature region a finite topological long range order actually occurs. However, beyond a critical temperature, those pairs are apt to become dissociated to create iosolated charges. This type of phase transition can be described by divergent mean square separation between the charges within the pairs. This is so-called the Kosterlitz-Thouless (K -T) transition. In two dimensions, it is easy to have an image of a fluxon as described above, since a vortex cut by the 2D sheet is nothing but a point defect. Here a pair of positive and negative charges corresponds to a fluxon-antifluxon pair. When a magnetic field is imposed on an SC, the field penetrates into the SC in a quantized form of flux as Wb (e* = 2e), for a field larger than B,1 (the lower = h/e* = 2.07 x large as critical field) and smaller than Bc2 (upper critical field), in the case that coherence

159

13.1 Kosterlitz- Thovless Mode in High-Tc SC

length (5London penetration length X or it is an SC of Type 11. This quantized form of magnetic flux is called a fluxon or flux quantum. Thus, in a high-T, SC, superconducting two-dimensional CuO2 sheets are penetrated by fluxon-antifluxon pairs at low temperatures and they become apt to move around as isolated fluxons at higher temperatures beyond a critical point. This is a typical K-T mode of excitation in high-T, SCs, and, following the authors, the effective interaction energy reads (13.2) where ~ ( ris) an appropriate dielectric constant which describes a screening effect of the Coulomb interaction due to other existing pairs. When considering the Boltzmann factor, it is easy to conclude that any thermo-statistically averaged physical quantities follow a power law, as shown, e.g., in a susceptibility,

(13.3) Thus, most physical quantities are shown to obey some power law. Using a BiSCCO sample, Ichiguchi et al. (1989) (Onogi et al., 1989; Ban et al., 1989; Martin et al, 1989) measured I-V characteristics in the SC state under magnetic fields larger than BC1but less than Bc2and obtained typical power law behaviors such ) (c) V 0: Bm(Tl').This as shown in Figs. 13.1 (a) and (b). There V 0: I n ( T > Band

1

2

3

4

5

10

I / 10mV Fig. 13.1: (a) log1 vs logV under various magnetic fields and (b) the same as in (a) with various temperatures under a magnetic field of 0.8 T. (c) log B vs log V biased by various transport currents. All data were taken at T = 77 K (after Ichiguchi et al., 1990).

13 Properties of Macroscopic Quantum States

160

10

1

0.1

0.01 10

1

100

I I mA

.1

.2

.3

.4

B/T

.5

1

13.2 Superconducting Thin Wares

161

observation suggests that high-T, cuprates, especially BiSCCO, have evident twodimensional features. Similar characteristics were observed in YBCO, too (Stamp et al., 1988). These characteristics observed in BSSCO are confirmed by simulation based on Josephson junction-coupled Cu02-layer model after Lawrence and Doniac (1971) by Sugano et al. (1992) and (1993), Ryu et al. (1992). The validity of this model will be also discussed in Section 13.4. High-T, cuprates are known to show peculiar B - T phase diagram significantly different from that in the conventional superconductors. To elucidate these behaviors many works by simulation are being done so far (Nelson et al., 1992; Tachiki et al., 1994; Bulaevskii et al., 1995; Koshelev et al., 1996; Sugano et al., 1998)

13.2

Superconducting Thin Wires

Besides the damping due to the KT mechanism, superconducting diamagnetism caused by a Meissner (persistent) current may change as a result of thermal excitation beyond barriers, which separate a potential minimum with N-fluxons from neighboring minima with Nkl fluxons (Fig.13.2). The case with a persistent current through a ring circuit is depicted schematically in the figure. This quantized state of fluxons is known as London quantization. This type of quantization is simply the case for when the Aharonov-Bohm phase is equal to an integer n times 2x, i.e., 2 ~ @ / @= 6 2n7r. This can occur when the macrowave is perpetuated by returning to itself with no phase shift (mod 2 ~ ) . Let us consider a one-dimensional wire system. Figure 13.3 shows the variation in the phase of the macrowave along the wire when a bias current I is applied, according to the fact that the energy E ( 4 ) is given by E J cos 4 - I@. q5 and are related to each other by the gauge relation 4 = 2n@/@;. E J is known as the energy stored in a Josephson junction. The horizontal axis of the figure is phase 4 and incidentally corresponds t o the coordinate along the wire, since the phase is continuously rounded through the wire in one sense when the state of the sample is homogeneous. When one fluxon jumps from one potential minimum to the neighboring minimum with a lower energy, thermally or via tunneling, the total winding number through the wire decreases. This means that a fluxon crosses the wire perpendicularly. Thus, in wires so thin that fluxons can cross anywhere (where the width WLAL, the London penetration length), the wires always show a voltage drop or, in other words, are resistive. Ordinarily, the resistance depends on the temperature but, at low enough temperatures, a thermally activated phase change can no longer occur so that only tunneling takes place. The cross-over from quantum tunneling to thermal excitation is at a temperature specific to the sample (Fig. 13.3: Giordano, 1988). The theoretical justification for the cross-over was given by Saito et al. (1989).

13 Properties of Macroscopic Quantum States

162

#=0 J=O

QO

J1

2#0 < Je

3#,0 J3

Fig. 13.2: Illustration of the potential felt by an integral number of fluxons in an SC ring circuit. An N-fluxon state is separated from ( N & 1)-fluxon state by a sufficiently high barrier, which may be surpassed by thermal excitation or tunneled through.

E

tunnel.

0

2%

4

Fig. 13.3: “Washboard” potential of a current-biased SC single wire vs phase of the macrowave. Phase slippage occurs via thermal excitation and quantum tunneling. It is also interesting to observe that the same phenonemon occurs in a superconductor as in an SET (single electron transistor) in its normal state. This time, it is not the tunneling of single normal electrons, but quasiparticles in the superconductor. Cooper pairs, which are known to be the substances to carry supercurrent, may show a similar phenomenon, when the Josephson junction has a small enough area; see the next Section. Such phenomenon may be named single Cooper pair transistor.

13.3 Superconducting Tunnel Junction

-0.8

-0.6

163

-0.4 -0.2 ( T T, ) / K

-

0

Fig. 13.4: Temperature vs DC resistance of thin SC wires with diameters 410 A ( e ) , 505 A (+), and 720 A ( o ) (after Giordano, 1988).

13.3 Superconducting Tunnel Junction Quasiparticles ’ Tunneling The nomenclature quasiparticles specific to SC is ordinarily used to mean normallike electrons excited from the ground state condensate of Cooper pairs. In order to describe the behavior of quasiparticles the so-called semiconductor model is often used (e.g., Tinkham, 1975). Quasiparticles are composed of electron-like and hole-like particles both of which are excited from the Fermi level situated at the center of the superconducting energy gap 2A. The Fermi level works as a reservoir for quasiparticles, although they exist as Cooper pairs (i.e., two-particle states), not as single particles, and carry supercurrents so long as they stay at the Fermi level. Thus, the Cooper pairs do not reveal themselves in the semiconductor model. Hence, the semiconductor model is utilized to exclusively describe the behaviors of quasiparticles. The semiconducting property is essentially a one-particle phenomenon, whereas a supercurrent in superconductors is carried by two-particle states. The essential difference between semiconductors and superconductors, both of which have a specific energy gap, is in that there are no carriers at the Fermi level in semiconductors,

164

13 Properties of Macroscopic Quantum States

whereas there are actually superelectrons in superconductors. In actual intrinsic semiconductors the Fermi level is an imaginary level, where no electron exists. In a semiconductor model of an SC, an electron and a hole may be excited from the very Fermi level into the ‘‘Conduction” and “valence” band, respectively. The energy required to destroy a Cooper pair is 2 A ( T ) in genuine terminology, while in the semiconductor model the same energy produces a hole and an electron. Just as in parallel t o the tunneling phenomenon between semiconductors with an insulator barrier in between, there may be a non-vanishing tunneling current between SCs, unless the “valence band” top or “conduction band” bottom energy resides within the energy gap €3, = 2 A ( T ) , as suggested in Fig. 13.5. This is the tunneling phenomenon of quaszparticles, which are called Giaever’s tunneling (Giaever , 1960), after the discoverer. If a bias voltage is given to shift, for example, the valence band top to a position corresponding to the conduction band bottom, it is easy to imagine that a finite current may be produced between both bands. The only difference in this phenomenon from that in semiconductor tunnel junction appears in their DOSs, as follows.

E

electrons

A1

A2

A1

A2 -

Fermi level

holes

Fig. 13.5: Illustration of the band scheme for an SC-I-SC tunnel junction. Beyond the voltage V = 2 A ( T ) / e current can flow between the valence and conduction band. For voltages below the value any current is blocked since there are no energy levels to feed-in.

165

13.3 Superconducting Tunnel Junction

d

m

'

with &k The energy spectrum of a quasiparticle is given by E k = corresponding to the normal spectrum of a Bloch electron relative to the Fermi level. Therefore, k

= P N ( ~ ) ~ k / t / Ek 2A ~ ( T )

(13.4)

This DOS has a peculiar feature in that it diverges at both ends: the valence band top and the conduction band bottom. Thus the expected tunneling current of

W)

I"

IMI2PS(E)PS(V - E ) d E

(13.5)

is sketched in Fig. 13.7 (Schrieffer, 196413). ]MI2 is the squared transition matrix element and is ordinarily taken constant for tunneling. At the threshold voltage 2A(T)/e tunneling current itself is finite but its derivative diverges. This is the case for a tunnel junction made of the same SCs on both sides. For a normal-metal-SC tunnel junction, the rise-up of the current occurs at V = A/e.

Cooper Pairs' Funneling According t o Josephson's prediction (1962, 1974), Cooper pairs are known t o be able to tunnel through a barrier between SCs. A summary of the physics of Josephson junctions (JJs) is given in Appendix K. The most peculiar and surprising characteristic of a Josephson tunneling current is a finite current under a zero bias across the barrier. As is obvious from the preceding discussion, the Josephson tunneling current only flows between the same ground state (the Fermi level) with no offset. The coupling energy of Josephson junction is

E ( 4 ) = EJ C O S ~ ,4 = A 4 - 2 ~ @ / @ : ,

(13.6)

when SCs on both sides of the tunnel barrier are described by respective macrowaves and have a phase difference of A 4 in between. A magnetic flux inside the barrier also generates a phase shift just as in the Aharonov-Bohm effect. When a JJ is short-circuited by an SC wire, which configuration is called the R F SQUID (radio-frequency superconducting quantum interference device), a DC current may flow for a zero bias voltage with an integral number of flux quanta trapped inside the loop circuit. The circuit shown in Fig. 13.8(a) represents an RF SQUID, when the shunt-resistance R is replaced by an SC wire. A more interesting device is a DC SQUID, which connects two JJs in parallel with SC wires. In this case magnetic flux can enter and escape through both JJ, although

166

13 Properties of Macroscopic Quantum States

the number of the fluxons inside the SQUID must also be an integer. However, this possibility suggests that a certain dynamic effect may occur for a finite voltage bias across the JJs. As was explained in Appendix K, a Josephson current oscillates under a finite voltage, known as the AC Josephson effect (Fig. 13.6). For a zero bias, let us assume that a Josephson current can flow between both SCs. If the circuit is open, it may be obvious that the superelectrons are accumulated to one side, if the current is a DC. This means that the current must back flow at the other end of the insulating layer resulting in no net current as a whole. Or, there may occur an alternately forward- and backward-flowing distributed current pattern, as shown in Fig. 13.6. In this case the maximum current density is limited by Jmax and the distribution is formed so that a minor loop current is quantized to give a flux This distribution makes a wave of phase in the y-direction; an oscillatory quantum variation of phase is the same as that of magnetic flux. If both SC electrodes of a Josephson junction are short-circuited by an external normal metal wire, the JJ may be supplied with a finite voltage. If the voltage is Vo, the above-mentioned wave of phase runs in the y-direction which causes an alternate current with an angular frequency of WJ = e*Vo/hunder a constant voltage. This is the AC Josephson effect.

+;.

i

L

Fig. 13.6: Configuration of a Josephson junction with alternately reversed Josephson currents corresponding to an oscillatory phase as a wave in the y-direction. For a finite voltage this wave propagates along the barrier producing an AC current across the junction.

13.3 Superconducting Tunnel Junction

167

' It

dc JoseDhson current

"

I ac Jc

2A/e

0

V

Fig. 13.7: Sketch of I - V characteristic of an SC-I-SC tunnel junction with a Josephson current at V = 0 and a Ciaever's tunnel current beyond V 2 A(T)/e. The voltage region between V=O and A(T)/e is known to generate an AC Josephson current which emits a microwave with an angular frequency w = e*V/fi.

JJ

0

V

Fig. 13.8: (a) A parallel circuit with a JJ (denoted by a union of V and A) and a shunt resistance (a load resistance as an equivalent circuit element). This device can switch for a feed-in current of 10, if the shunt resistance has an appropriate value as a load. (b) Conceptual representation of how work point A switches to point B on another branch, and returns to A by decreasing 10.

168

13 Properties of Macroscopic Quantum States

Let us consider a DC SQUID. When a single fluxon exists inside, its sense can be either upward or downward, which can be utilized to simulate a ‘0’ or ‘1’-bit state. This is in fact a memory cell in binary code. We will now discuss a switching device utilizing a JJ. The I - V characteristic is shown in Fig. 13.7. Both the Josephson and Giaever tunneling currents apparently give a negative resistance characteristic, so that this can work as a switching device, if utilized with an appropriate load resistance. Consider a parallel circuit with a linear shunt resistance R as shown in Fig. 13.8(a). For a feed-in current of 10 such as I0 < IJ (IJis the maximum current, not density, through a J J ) , all current flows through the JJ. However, for a slightly larger current IJ > 10,the J J can no longer tolerate bearing the current and eventually switches t o the Giaever tunneling current branch, as suggested in Fig.13.8(b). For this voltage the work point is B, which is determined by the crossing of IJ = I0 - V / Rand the Giaever branch. By making the current less than IJ,the device returns to a zero-voltage state with no current through the shunt resistance. There is the other switching mode. If an appropriate magnitude of magnetic field is applied t o the junction by means of feeding a current to the SC circuit lying by the junction, the maximum JJ current decreases following Eq. (K.l). This field-producing input current also makes a switching operation to a specific work point on the Giaever branch. This type of switching can be applied to construct logic circuits. In the 1960s, there were actually many efforts to develop JJ computers, but they were in vain being defeated in competition with semiconductor devices. However, recently some efforts are again being made in the US and Japan. The remarkable merit of JJs is their low loss of power, but the crucial point is that they act only at a low temperature. Their speed of performance is now almost caught up by semiconductor logic ICs.

13.4 Transport Properties of High-Tc Cuprates It is known that a supercurrent flows in CuOz planes in high-T, cuprates. Their transport properties are in fact two-dimensional. The existence of a supercurrent is simply recognized through the quantized magnetic flux it produces and the conduction without potential drop. The paper by BCS as well as textbooks on superconductivity (e.g., Tinkham, 1975) describe how an optical conductivity O ( W ) is formulated based on the BCS states. It is contributed by the pairs created from the ground state condensate by absorbing a photon Aw (or, equivalently, quasielectron and quasihole pair creation in the semiconductor model), which is easily obtained from an equation similar to Eq. (13.5). .(w)

0:

/

M

lWE,w)I2ps(E

+ hAJ)ps(E)[f(E)- f ( E + Aw)ldE

(13.7)

-a

+

where IM(E,u)I2must be substituted for by the so-called “coherence factor” f ( 1 A 2 / E E ’ )for pair creation processes ( E = d m ) . The coherence factor comes

13.4 Transport Properties of High-T, Cuprates

169

from the squared matrix element of scattering (i.e., pair creation) through the probability amplitude of an existent quasiparticle.

-0

2

4

6

8

10

hw / 2 A Fig. 13.9: Plots of d ( w ) and d ’ ( w ) (Eqs. (13.7) and (13.8)) as functions of w . Now, it is easy to reach an optical (complex) conductivity in ideal SC samples d ( w ) - id’(w) at T = 0 which is related with a complex dielectric constant d ( w ) id’(w) so that E ( W ) = iu(w)/w (Mattis et al., 1958):

+

(13.8) (13.9) where K ( k ) and E ( k ) are, respectively, the first and second kind of the complete elliptic integrals with k = l2A - liwl/(2A + liw) and k’ = d m . The 0’’behaves roughly as w-l for hw << 2A (unsimilar to the Drude’s law w-2 for coherent conductive metals) and approaches 0 for w + m. Readers must be careful that these expressions do not include the conductivity contributed by quasiparticles and it is assumed a priori that there is an infinite reservoir of superelectrons in the ground state. The two functions are plotted in Fig. 13.9. As is known, 0’vanishes for liw 5 2A because no excitation occurs for photon energies less than the SC gap energy. However, since the fact violates the oscillator-strength sum rule: o’(w)dw = 7rn:e2/2m* (n;: the density of superelectrons; cf., Eq.(10.23)) when an SC gap opens up, we must include an ad hoc term A 6 ( w ) in the expression of ( ~ ’ ( w ) . Now, let us consider empirical expressions taking into account the contribution from quasiparticles at T # 0 as follows after Mare1 et al. (1991). (13.10)

170

13 Properties of Macroscopic Quantum States

(13.11)

The last terms in the both equations above are due to the quasiparticles excited at finite temperatures and their relaxation time is defined as T = y-’. wps is the plasma (angular) frequency of superelectrons and wPo is that of normal electrons (quasiparticles). They also stand for, respectively, the oscillator strengths of super- and normal electrons. The first term in a’ is a delta-function, which restores the “missing area” under the d ( w ) curve t o satisfy the sum rule. Experimentally, the optical conductivity was measured in Sr-doped LSCO: LaZ-=Srz Cu04 with the electric field parallel to CuO2-planes by Uchida et al. (1991) and Romero et al. (1992). Rough tendencies observed in this field configuration meet the above empirical expressions well except for several spectral structures according to phonon excitations. The latter paper actually observes the depression in d ( w ) for T less than T,. The conduction across the C u 0 2 planes is another agendum. Let us discuss such a case with a current perpendicular to the CuO2-planes in LaSrCuO compounds by measuring optical conductivity in the following paragraph. Transport Perpendicular to the Cu02 -Planes Figure 13.10 shows the reflectivity data from superconducting LSCO samples with an electric field perpendicular to the CuO2-planes, which are known to carry supercurrents (Tamasaku et al., 1992). According t o the data, reflectivity spectra have sharp dip structures corresponding to the plasma frequency ups. Since reflectivity R for perpendicular incidence of light beams is expressed in such a manner as n

€(W)/€O

w;S

(13.12)

=K -W2

where

K

is the relative dielectric constant of the medium, and

+ k 2 E’((w) = n2 - k2 + 2ink ( 13.13) (n + 1)2 + k2 ’ it is easy t o obtain the conditions where the dips occur to be n = 1, accordingly, w = w p s / a (because k 0 and c 2 ( w ) n2 = 1). Typically these dips appear in the region around a very small value of frequency v = 50 cm-’ and its temperature dependence is known to closely follow the BCS’s A(T)-vs-T curve (Tamasaku et al., 1992). From this fact Anderson (1995) found out that this plasma frequency is exactly S equal t o the Josephson plasma frequency. That is, the inductance L = @ ~ / J J(S: the area of CuO2-plane; J J : the Josephson maximum current; = h/e*) and the capacitance C = moS/d (d: the distance between the CuO2-planes) result in the resonance frequency R=

(n - 1)2

N

N

( 13.14)

13.5 Proximity Effect

171

frequency / cm-’ Fig. 13.10: Infrared reflectivity spectra of La2--zSrzCu04 with electric field polarization perpendicular to the Cu02 planes above and below T, for (a) z=O.lO (T,=27 K), (b) x=0.13 (Tc=32 K ) , and (c) z=0.16 (T,=34 K) (after Tamasaku et al., 1992). (Appendix K). The Josephson maximum current (density) also appears in the condensate energy of the Josephosn junction per unit area: A E = (2AJ~le’)cos 4 (4: the phase difference between the SC layers on both sides) which is equal to ;psA2 ( p s : the density of states in the SC state). Thus, a physical image of the stack of CuOz-planes eventually obtained is such that all planes are coupled to each other via Josephson junctions.

13.5

Proximity Effect

Take an SC-normal metal (S-N) junction. The normal metal is specified by a vanishing energy gap around the Fermi level, whereas it is not the case in an SC. The SC gap at the interface which is sometimes called the “pair potential” continues into the normal metal with decreasing its magnitude. The specific length of attenuation of the gap may be also named the “coherence length” & in the normal metal. The specific boundary condition at the SC-N interface was discussed by Blonder et al. (1982) from general aspects. See also Hekking et al. (1994).

172

13 Properties of Macroscopic Quantum States

Let us take an SC-N-SC junction with the normal metal of length L in between. According to Seto et al. (1972) (also van Duzer et al., 1981) the pair potential in the normal region may have the magnitude engendered by the proximity effect as follows.

AN = A,

+

d2(1 cosx) cosh2[(2x - L)/2&]

+ 2(1 - cosx) sinh2[(2a:

-

L)/2&]

2 cOSh(l/2<~) ( 13.15)

where x is the phase difference between the SCs on both sides. If x=O, the pair potentials on both sides of the normal metal-SC interface is considered to be the same; otherwise, there is an offset in general. A rough sketch of this situation is depicted in Fig. 13.11. In the figure occupancy probabilities of quasielectrons and quasiholes in the semiconductor model are also shown, which are given by the Fermi-Dirac statistical

electron energy

4

pair

normal metal hole energy

Fig. 13.11: Illustration of the band-gap scheme of a SC-normal metal-SC junction realized according to the proximity effect. The dashed line is for the Fermi level. The darkened areas represent the occupancy probabilities for quasielectrons and quasiholes for A/k*T x 1.5 on the right, 1 in the middle, and 0 on the left.

13.5 Proximity Eflect

173

I

,

,

I

I

1

1

3

I

t I

1 1

2O~~V/div

Fig. 13.12: Measured I - V characteristics in an Nb-pInAs-Nb tor for a couple of gate voltages (after Takayanagi, 1987).

SC transis-

function with energy spectrum d m relative to a zero chemical potential. The SC gap is assumed to be 1.5 ~ B atT the interfaces in this plot. The next problem is what is the coherence length in the normal metal. De Gennes (1964) (also de Gennes, 1969) gives the magnitude calculated in the dirty limit (i.e., the mean-free path is less than the coherence length): (13.16) where D is the diffusion constant in the dirty sample. Takayanagi et al. (1985) (also Takayanagi, 1987) utilized a junction formed by Nb-pInAs semiconductor-Nb system, the surface inversion layer of which works as a normal metal. Isolated from the inversion layer by an SiO thin film is overlaid a gate electrode. On applying a positive bias potential to the gate electrode, the free carriers in the inversion layer increase. Since the mean-free path 1 = u f r = vfm*p/e ( p : the mobility) and the Fermi velocity uf = tikf/m* = ( 2 ~ n ) ' / ~ ( h / m * )(n: the density of carriers) give the diffusion constant D = q1/2 for a two-dimensional electron gas, the above expression is eventually obtained as (13.17) Thus, the coherence length is elongated as n1I2,when an applied gate voltage enhances the density of carriers. Figure 13.12 depicts two I - V-characteristics for different gate voltages in a SC transistor (Takayanagi, 1987). The curves show clearly the same characteristics as observed in the so-called "weak-link" type of Josephson junctions. Weak-link type Josephson junctions are commonly realized on point-contacts between SCs, SC constrictions and SC microbridges. Weak-link type characteristics are those which do

174

13 Properties of Macroscopic Quantum States

not show differential negative resistance features as shown in Fig. 13.7, but a nonlinear conductance composed of a zero-voltage current and a voltage-insensitive rise in current (Beenakker et al., 1991; Beenakker, 1991). The zero-voltage current is easily reduced by applying a magnetic field similar to the SC-I-SC tunnel junction, so that this type of SC transistor may be switched from the zero-voltage state to a non-zerovoltage one. The weak-link type SC transistors are another type of useful devices in mesoscopic electronics area.

Circuit with a Superconducting Reservoir If a normal current circuit is connected to an additional superconductor performing as a supercurrent reservoir, an interesting effect may be expected to occur. The reason is that, even if the superconductor is connected through a “dangling” arm and, accordingly, no net current flows to or from the superconductor, we should notice that an implicit supercurrent may exist under a zero bias voltage. In this context “dangling” means that no bias voltage is imposed on to this superconducting electrode. Such a specific situation was first considered by Volkov (1995) (also Volkov, 1995a) assuming a circuit with two SC-I-N junctions in an SC,-I-N-I-SCb structure. Here we will follow a variation given by Shaikhaidarov et al. (2000) in a more transparent form. In Fig. 13.13 is shown a schematic circuit composed with mesoscopically sized normal wires connecting a normal electrode N with a superconducting SC electrode, at the midpoint of which a normal wire is branched t o reach another superconducting electrode SC’ forming a dangling arm. Let us consider the simplest case to elucidate the peculiar situation more clearly; the two normal wires are assumed equal in length, for simplicity, t o have the same resistance R1 = R2 = R. As is denoted in the figure, a finite voltage VN - Vsc is applied between electrodes SC and N, and no voltage between SC’ and N. A simple circuitry consideration will prove that

11 = Is + I q p l 12 = I, - I q p 2 = 0

( 13.18) (13.19)

where I q p 1 , 2 stand for currents carried by quasiparticles through branch 1 and 2, respectively, and Iqpl= Iqp2 = Iqpholds in the present symmetric case. I, may flow because the electrodes SC and SC’ are connected by a normal wire which is short enough to have a supercurrent according to the proximity effect. The midpoint of the circuit is assumed to have a potential Vo and the SC a potential Vsc (SC’ is earthed). Then we have (13.20) (13.21) (13.22) $J is the phase, the superconductor SC has relative to SC’. Let us consider an applied voltage to be a DC. Then,

(13.23)

175

13.5 Proximity Effect

which means that I, = Iqpand VO = RI, = RI,,,

sin 4 = RIqp

(13.24)

with I = 21qp. For this condition, electric potential increases from 0 at SC' to a positive VOat the midpoint and then decreases to 0 at SC (Le., VSC = 0). From these equations we reach VN

=

(

I = 2I,,,

~

+h $) 21,

sin 4

(13.25)

sin4

(13.26)

An applied voltage VN determines a phase difference 4 between SC and SC' which in turn determines a current I through the horizontal normal branch. First of all, for a zero voltage VN = VSC = 0, 4 and I = 0. For a finite, but sufficiently small voltage, 4 M VN/~I,,,(R~ R/2). On the other hand if VN is high enough, the superconducting reservoir SC' may not work and VN-VSC = I(Rh+R) will be reached, where a finite voltage appears between SC and SC' causing a nonlinear current I, like that shown in Fig. 13.12. In this case too, the nonlinear current is compensated for by Iqpin branch 2 . When VSCbegins to take a non-zero magnitude, VN reaches its maximum called the critical voltage, where I,, = 21max(Vcr)sin4(Vcr).

+

branch 1 "N

branch

W

Fig. 13.13: Normal conductor circuit with an SC dangling arm branched from the midpoint of the circuit. Through branch 2 a zero-voltage Josephson current implicitly flows to compensate for a normal current carried by quasiparticles. SC and SC' are connected superconductively by the proximity effect for small VN'S.

176

13 Properties of Macroscopic Quantum States

In summary, for a voltage VN 5 v,, a Josephson zero-voltage current I = VN/(2&+ R) flows between SC and SC’. For higher voltages VN - Vsc > V,, a negative VSC appears. This new effect and its variation in mesoscopic superconductors have been experimentally confirmed by Shaikhaidarov et al. (2000) and Takayanagi (2001) (see also Hekking et al., 1994).

13.6

Andreev Reflection

Superconductivity is known to result from pairs of electrons which behave like bosons. According to the BCS theory, the most probable pair is composed of an electron with momentum Ak and another electron with h(-k). In Fig.13.14(a) we will consider the case where an electron with momentum Akf at the Fermi level is incident to the normal-meta-SC interface from the normal metal side. In an SC, except for quasiparticle states beyond the gap, only two-particle levels exist degenerate at the Fermi level within the SC gap. Since we are concerned with an incident single electron, we need another electron to let the both enter into the Fermi level in the SC. The partner electron should have a momentum -hkf. Since the creation of an electron with momentum -Rkf is equivalent t o the annihilation of a hole with momentum Akf, we will have a hole with hkf eventually emitted from the interface. This is an interpretation of the Andreev reflection after Andreev (1964). In the reflection momentum is thus conserved. At non-zero temperatures, electrons with an energy 6 higher or lower than the Fermi level may be Andreev reflected. E is measured relative to the Fermi level and 161 5 A. It is obvious that energy is also conserved in the meaning that the incident electron has an energy 6 and the reflected hole the energy - E to result in a zero total energy equal to that of the Cooper pair. At the instance of reflection the variation in phase from the incident electron to the reflected hole must be equal to the macrophase of the SC, e.g., 8. In addition it is known that the electron and hole, respectively, carry the following phases, (13.27)

-12 { 8 - c0s-l

(2)}

(13.28)

for hole

(Kummel, 1969; Griffin et al., 1971; Beenakker et al., 1991). Obviously the phase difference between the electron and hole is equal to 8, i.e., the phase of the SC. Since the energy of concern is equal to that within the gap, or, in other words, the incident electron is forbidden to enter a state in the gap, it must be reflected [Fig. 13.14(a)]. Note that the generated hole has a momentum of -Ak, and, hence, traces the same path as the electron but in the opposite direction. This is shown in Fig. 13.13(b), in comparison with the ordinary normal reflector [Fig. 13.13(c)]. In an experiment on a mesoscopic scale Nishino et al. (1990) observed a geometrical resonance between a Nb injector and a Nb reflector through Si. If the distance d between them is such that ,/A2 ( n h v ~ / 2 d ) ~ eV (V: applied voltage) with an

+

-

177

13.6 Andreev Reflection

integer n, a resistance minimum appears. These data are plotted in Fig. 13.14(d), which clearly shows lst, 2nd, and 3rd order geometrical resonance.

Energy Pair potential

A

Incident electron EF

1.02

I

X

1st 2nd 3rd

1.01

1 .oo

1.o

(4

2.0

v

3.0

4.0

(mv)

Fig. 13.14: (a)Conceptual explanation of the Andreev reflection. Expected paths on reflection for specular scattering (b) in the normal state and (c) for Andreev reflection in a SC. (d) Observed differential resistance vs voltage showing a few geometrical resonance (after Nishino et al., 1990).

178

13 Properties of Macroscopic Quantum States

Rreferences Ambegaokar, V., Baratoff, A. (1963), Phys. Rev. Lett. 10, 486; erratum, (1963) ibid., 11, 104. Anderson, P. W. (1995), Science 268, 1154. Anderson, P. W. (1997), The Theory of Superconductivity in the High-Tc Cuprates, Princeton: Princeton University Press. Andreev, A. F. (1964), JETP 46, 1823. Ban, M., Ichiguchi, T., Onogi, T. (1989), Phys. Rev. B40, 4419. Beenakker, C. W. J. (1991), Phys. Rev. Lett. 67, 3836. Beenakker, C. W. J., van Houten, H. (1991), Phys. Rev. Lett. 66, 3056. Bedell, K. (Ed.) (1990), High Temperature Superconductivity: Proceedings, Redwood City, CA:Addison-Wesley. Blonder, G. E., Tinkham, M., Klapwijk, T. M. (1982), Phys. Rev. B25, 4515. Bulaevskii, L. N. (1995) Phys. Rev. lett. 74, 801. Chu, C. W. (1997), in: Encyclopedia of Appl. Phys. G. L. Trigg (Ed.) 20, 213. Cyrot. M., Paruma, D. (1992), Introduction to Superconductivity and High T, Materials, Singapore: World Sci. van Duzer, T., Turner, C. W. (1981), Principles of Superconductive Devices and Circuits, Amsterdam: Elsevier. de Gennes, P. G. (1964), Rev. Mod. Phys. 36, 225. de Gennes, P. G. (1966), Superconductivity of Metals and Alloys, New York: Benjamin. Giaever, I. (1960), Phys. Rev. Lett. 5, 147, 464. Ginsberg, D. M. (Ed.) (1989), Physical Properties of High Temperature Superconductors I, Singapore: World Sci. Ginsberg, D. M. (Ed.) (1990), Physical Properties of High Temperature Superconductors 11, Singapore: World Sci. Ginsberg, D. M. (Ed.) (1992), Physical Properties of High Temperature Superconductors 111, Singapore: World Sci. Ginsberg, D. M. (Ed.) (1994), Physical Properties of High Temperature Superconductors IV, Singapore: World Sci. Giordano, N. (1988), Phys. Rev. Lett. 61, 2137.

13.6 Andreev Reflection

179

Griffin, A., Demers, J . (1971), Phys. Rev. B4,2202. Hekking, F., Schon, G., Averin, D. (Eds.) (1994), Mesoscopic Superconductivity, in Physica 203B. Ichiguchi, T, Onogi, T., Murayama, Y. (1990), in: S. Kobayashi, H. Ezawa, Y. Murayama, S. Nomura (Eds.), Proc. 3rd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, p.391. Josephson, B.D. (1962), Phys. Lett. 1, 251. Josephson, B.D. (1974), Rev. Mod. Phys. 46, 251. Kamimura, H., Oshiyama, J . (Eds.) (1989), Mechanisms of High Temperature Superconductivity, Heidelberg: Springer. Koshelev, A. E. (1996), Phys. Rev. Lett. 77, 3901. Kosterlitz, J. M., Thouless, D. J. (1973), J. Phys. C: Solid State Phys. 6, 1181. Kosterlitz, J. M. (1974), J. Phys. C: Solid State Phys. 7, 1046. Kiimmel, R. (1969), 2eits.f. Physik 218, 472. Lawrence, W. E., Doniac, S. (1971) in: E. Kanda (Ed), Proc. of Low Temp. Phys. LT-12, Kyoto, 1970, Tokyo: Keigaku, p.361. van der Marel, D., Habermeier, H.-U., Heitmann, D., Konig, W., Wittlin, A. (1991), Physica 176C, 1. Martin, S., Fiory, A. T., Fleming, R. M., Espinosa, G. P., Cooper, A. S. (1989), Phys. Rev. Lett. 62, 677. Mattis, D. C., Bardeen, J . (1958), Phys. Rev. 111,412. Nelson, D. R., Vinokur, V. M. (1992) Phys. Rev. Lett. 68, 2398. Nishino, T., Hatano, M., Hasegawa, H., Murai, F., Kure, T., Yamada, E., Kawabe, U.(1990), in: S. Kobayashi, H. Ezawa, Y. Murayama, S. Nomura (Eds.), Proc. 3rd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, pp. 263. Onogi, T., Ichiguchi, T., Aida, T . (1989), Solid State Commun. 69, 991. Romero, D. B., Porter, C. D., Tanner, D. B., Forro, L., Mandrus, D., Mihaly, L., Carr, G. L., Williams, G. P. (1992) Phys. Rev. Lett., 1590. Ryu, S. Doniac, S., Deutscher, G., Kapitulnik, A. (1992), Phys. Rev. lett. 68, 710. Saito, S., Murayama, Y. (1989), Phys. Lett. A139, 85.

180

13 Properties of Macroscopic Quantum States

Seahan, T. P. (1994), Introduction to High Temperature Superconductivity, New York: Plenum. Seto, J., van Duzer, T. (1972), Proc. of Low Temp. Phys. LT-19 New York: Plenum, vol. 3,pp.328. Schrieffer, J. R. (1964a), Theory of Superconductivity, Redwood City, CA: AddisonWesley. Schrieffer, J. R. (1964b), Rev. Mod. Phys. 36,200. Shaikhaidarov, R., Volkov, A. F.,Takayanagi, H., Petrashov, V. T., Delsing, P. (2000), Phys. Rev. B62,R14649. Stamp, P. C. E., Farro, L., Ayache, C. (1988), Phys. Rev. B38,2847. Sugano, R., Onogi, T., Murayama, Y. (1992), Phys. Rev. B45,10789; (1993), Phys. Rev. B48,13784. Sugano, R., Onogi, T., Hirata, K., Tachiki, M. (1998), Phys. Rev. Lett. 80, 2925. Tachiki, M., Koyama, T., Takahashi, S. (1994) Phys. Rev. B50,7065. Takayanagi, H.,Kawakami, T. (1985), Phys. Rev. Lett. 54,2449. Takayanagi, H.(1987), in: S. Kobayashi, H. Ezawa, Y. Murayama, S. Nomura (Eds.), Proc. 3rd Int. Symp. Foundations of Quantum Mechanics, Tokyo: The Physical Society of Japan, pp. 241. Takayanagi, H. (2001), private communications. Tamasaku, K., Ito, T., Takagi, H., Uchida, S. (1992) Phys. Rev. Lett. 69,1455. Tinkham, M.(1975) , Introduction to Superconductivity, New York: McGraw-Hill. Uchida, S., Ito, T., Takagi, H., Arima, T., Tokura, Y., Tajima, S. (1991), Phys. Rev. B43,7942. Volkov, A. F. (1995), Phys. Rev. Lett. 74, 4730. Volkov, A. F. (1995a), JETP Lett. 61, 565.

14

Future Prospects

As should already be clear to the reader, phenomena where perfect coherence degrades into partial coherence through inelastic scatterings are revealed in every actual event in condensed matter. For that reason mesoscopicity is said to be a widely prevailing phenomenon. In the near future the prospects are that all devices will be more and more minituarized in scale. When the dimensions are thus tiny scaled, mesoscopicity appears more and more clearly in contrast with the performance of bulk devices. This is in particular so in low-dimensional systems such as in two, one, and zero dimensions. In low-dimensional systems ad hoc size quantization occurs which shows specific effects when combined with decoherence. Let us discuss for instance the future prospects of semiconductor devices. It is said that the number of memory cells in a single LSI chip increases four times every three years. Because of yield requirements, the increase in chip size must be kept less than the number of cells, which means that the minimum linear scale must decrease. At the present moment it is on the order of 0.13 pm at the development stage. Take a single cell of 0.13 x 1 pm2 as a typical area, then it sounds incredible that the cell works with as small a number of electrons as roughly 1,300 at a snapshot. This number is really countable, far less than the Avogadro number of loz3. The electrons in the n-channel exert many types of scatterings, the most dominant of which are inelastic scatterings with phonons, other electrons and scatterings to excite impurities to a higher energy level, and elastic scatterings with the roughness of the interface and impurities. The energies of n-channel electrons are quantized according to a large electric field between the interface and the inside of the p t y p e semiconductor substrate. They make a twodimensional system composed of a series of step-wise DOSs. For a fairly large number of electrons the effect due to this quantization may not be clearly revealed, but for a less number it must appear more and more readily, since the lowest quantized levels may be more featured clearly with wide separations of the levels. Moreover, it may happen that we could abandon the conventional down-sizing technology for LSIs and proceed to developing a completely different technology such as single-electron transistors for actual devices. This approach imposes on US far more serious difficulties to overcome at the development as well as the application stage, since it needs very exotic processing and application technologies, e.g., when implemented into actual circuitry in particular. Regarding magnetic devices, along with the increasing capacity of magnetic hard disks a tinier one-bit is pursued, which requires much smaller magnetic reading/writing head and higher bit density. This requirement forces technology to go into thinner

-

182

14 Future Prospects

magnetic layers/wires. This technology area is called Magnetoelectronics [cf., e.g., Physics Today, April (1995) pp. 24-63]. In nanoscale devices the number of spins may again be countable. Then behavior of individual spins must be of concern. In Japan the terminology Spinics or Spintronics is utilized to mean spin-dependent electronics. A typical example is the GMR effect described in Chapter 12. Nowadays it is known that spin significantly affects the conducting properties in ordinary condensed matter. This fact suggests that if a pair of electrons with different spins, i.e., those emitted from a ‘1’- and ‘O’lstate in magnetic memory cells are injected into a logic circuit, they may perform a specific logical operation under a spin-dependent physical effect. This is really a futuristic proposition. A promising new technology in nanoelectronics resides in utilizing an STM (scanning tunneling microscope) t o manipulate a single or a small number of atoms such as in a cluster. These atomic-order-sized electronic devices are often called atomics; however, more plausibly, they should be called atomic electronics or atom-electronics, since the main role in the phenomena of interest is still played by electrons, not by atoms. An STM can also be used to manipulate atoms one by one (or at least, as very tiny atomic clusters), for example by extracting, moving, or depositing them. Atoms on a sample surface can be extracted from this surface, and similarly atoms on the surface of the tip can be deposited onto the surface of a sample by applying a high electric voltage between the tip and sample. This manipulator is often applied to fabricate atomically small terraces (2-dimensional quantum systems), atomic arrays (1dimensional quantum systems), and atomically small clusters (0-dimensional quantum systems). In the early years of this new century, many nanoscale devices will be proposed (cf., Science 290, 24 Nov., pp. 1520-1545). However, studies will have t o continue mainly from a purely scientific view for a while, rather than from actual application-oriented interests. Nevertheless, many breakthroughs are still expected, because today’s prevailing Si devices are so tough and seem likely to survive longer than once expected. The realization of practical nanoscale devices may be considerably later than had once been hoped. This atomic-scale device technology, however, has a promising future and an important role to play in human technological development. The ultimate goal is to mimic the human brain with a network of nanoscale devices on the order of synapse size! In addition, although nobody knows whether these devices will work coherently or not, coherent operation could make the quantum computer attainable.

Appendix A FDM Solution of Schrodinger Equations

Time-Independent Schrodinger Equation Various potential problems that can be solved analytically are usually introduced in most textbook. However, such problems are so limited in number that we are obliged to solve some problems with particular potentials only numerically. The easiest and most tractable method is to discretize the continuous variable, e.g., x into a series of points with a finite difference A in such a manner that xi = xo+iA, i = 0,1,2,3,..., N . This method is called the FDM (finite difference method). So long as the problem is one-dimensional, it is always possible to solve it numerically by the FDM. For two or three-dimensional problems we need a more specific manipulation, even when the same method is utilized. In this Appendix we refer to the other boundary condition needed to solve the same Schrodinger equation than that discussed in the text. In addition how to improve an accuracy of the solution is discussed.

Periodic Boundary Condition In the periodic boundary condition which often appears for the wave equation in crystals, i.e., $0 = $ N , the first and the last two equations in Eq. (3.3) must be replaced, respectively, by ( a = -h2/2mA2) a($2 - 2741

+ $ N ) + VI$I

= E$I

... a($N

- 2$N-l

-k $ N - 2 )

+ VN-l$N-l

+ '$N-1)

a('$] - 2$N

-k v N $ N

= E$N-~ = E$N

Accordingly the matrix equation reads

-2a+V1 a 0 0

\

a

a -2a

0 a

+ V2

a 0 0

-2a

0 0 a

+ V,

a

0

-2a

.,. 0 ... 0

... 0

+ V4 ... ..

0

. . ..

0

a 0 0 0

... a - 2 a + V ~

184

Appendix A FDM Solution of Schrodinger Equations

Solutions with a Higher Approximation The lowest order FDM has been described so far. To improve the degree of approximation we define the 2nd-order derivative with, e.g., 5-point data (5-point FDM) or 7-point data (7-point FDM), and so on. Let us first explain how we can reach the 5-point FDM. Since it is easy to derive

If we use the same a as before, the 5-point FDM N x N matrix for the kinetic energy term in the Schrodinger equation reads under the periodic boundary conditions

185 $0

= $ N , as follows. 5a -$a

'a

o

Ea O

:a --a4 &a

-I24 3a %a 24 -?a

-a1 -'24 3a 5 2a

4 - 3a

1 =a

0

0

24 -3a 1

... ...

=a 1

-fa

...

o o

12

...

0

0

... -?a4

o

5 Ta

7-point FDM Similarly we can easily calculate the 2nd-order derivative in the 7-point FDM, as follows. $kt~

=

+ 24$k-2

$k-3

- 465$k+1 f 24$k+2

- 465$k-1

+ $k+3

360A2

Solution of Time-Dependent Schrodinger Equation To solve a time-dependent dynamic problem, it is usual to rely on a wavepacket, which is localized in space. For example in the problem where a wavepacket is scattered by a spatially localized poential, either elastic or inelastic, it is necessary to first define an unscattered (free) wavepacket to start with. Then it is allowed to proceed into the scattering region. For this purpose, we must derive in what superposed states the wavepacket should be given under the scattering. For a free Gaussian wavepacket, any textbook describes that it is given in the following form:

(A.lO) k

where A: is the Gaussian envelope function with its central velocity given by hp/m* i.e., (A.ll) with the normalization factor N2 = f / m / 2 f i A k and e = 2 T ~ ( A k ) ~ / mk* .is the wavevector to represent a free plane wave such as Xk(5j) = ( k l j ) = exp(ikzj) defined only at discretized points xj. For a Hamiltonian H which is not diagonal in the k-representation the wavepacket is given in the form

$ ( x , t )=

1 A;(klj)(jle-'Htlhlj)

(A.12)

k

since xk(xj) is interpreted as a transformation matrix from the discretized-site representation ( j in FDM) t o the wavevector ( k ) representation. If the Hamiltonian is diagonalized as V-lHV = E I using the diagonalizing matrix V, then e-iHt/h - e-iVV-l HVV-'t/h - Ve-iV-'HVt/h V-l (A.13)

Appendix A FDM Solution of Schrodinger Equations

186

Accordingly, (A.14) is obtained where the eigenvalues En should be calculated in the FDM method. The diagonalizing matrix V is composed of the eigenvectors as

v-'

v

= =

((W) ((jln))

(A.15)

where (jln) is the j-th component of the columnar vector corresponding t o the n-th eigenvalue. This type of newly defined wavepacket has been utilized in a variety of dynamic problems by Murayama (1990a, 1990b, 199Oc, 1990d) (refs. in Chapter 2). Once an eigenvalue equation is set, eigenvalues and eigenvectors are easily obtained by utilizing a specific library software supplied by the computer system.

Appendix B Effective-Mass Approximated Equation

Starting from the following Schrodinger equation:

h2 [-%v2

+ VO(T) + A V ( T ) ]X ( T ) = EX(T)

(B.1)

where VO is the potential of the regular lattice and AV is a deviation from VO of the actual potential of this problem. Now, let us expand x into a series of Bloch states, such that

where $k is the Bloch function, R the coordinate of atom positions, and W ( T - R) is the Wannier function which is localized around R . It is obvious that the Wannier function is defined by

and

As Bloch states constitute an orthonormal set with respect to k, it is easy t o show that Wannier functions do the same with respect to R. After a simple manipulation, we obtain

Here E ( k ) appeared as the eigenenergy of the following equation:

188

Appendix B Effective-Mass Approximated Equation

Let us multiply both sides of Eq. (B.5) by

$;(T)

and integrate over

T,

then we reach

k

Again let us multiply both sides of Eq. (B.7) by eik.. and sum over Ic, then the first term on the left hand side may be rewritten as

and the second term

may be modified into

(B.ll) k

where we assumed that Uk is insensitive to k so that Uk N Uk' N UO. 0 is the volume in real space and, hence, this is eventually equal to AV(T)@(T),when uo N The expression finally reached is [E(-iV)

+ AV(r)]@(r)= E @ ( T )

(B.12)

Take a parabolic band with an effective mass m* in a solid for instance. Then the ), we started with first term of the above equation gives - ( h 2 / 2 m * ) V 2 @ ( ralthough Eq. (B.l) having the bare electronic mass m therein. This is the reason why we refer to such an equation as an effective mass equation or effective mass approximation (EMA). When considering Eq. (B.4), we find that @ ( T ) is equal to C R ~ ( T- R ) , i.e., a series of the delta functions defined only at lattice sites T = R. If we are allowed to connect the peak values at each R smoothly, then the finally obtained form of @ ( T ) is not such highly oscillatory function as in the Bloch state, but a gradually varying function extending over several lattice sites. Such function is a type of envelope function which modifies the original Bloch wavefunction. Actually the effective mass equation is known t o give a good approximation for problems such as an impurity state, an excitonic state in semiconductor and so on. Equation (B.12) is given in the text as Eq. (3.8). In the next we discuss moreover a few actual cases.

CR

States in Quantum Wells

189

Quantum well will be introduced in Chapter 7. Here we may define it by only saying that it is a thin semiconductor layer (“well”) sandwiched with other semiconductor materials with a larger potential on both sides. As is well known, if the potentials outside the well are sufficiently high as in vacuum, the system behaves as in two dimensions. Let the band energy of the well-semiconductor be given by E(k) = h2k2/2m* around k = 0. For a larger k-value, it is known that slight non-parabolicity comes in. That is, E ( k ) = h2k2/2m* a k 4 will be a more plausible band energy. In this case the effective mass equation must read

+

+

1

a@(x)= E @ ( z ) 2m*dx2 dx4 d4 if we take the potential in the well to be zero.

(B.13)

Donor States In n-type semiconductor one Si atom is substituted for a donor atom such as B or Ga. In this case an add hoc potential for an electron around this donor may be intuitively given by -e2/4mOtcr ( K : the relative dielectric constant of the medium). The kinetic energy term in the effective mass equation results from -fL2k2/2m*by replacing k with -id/d. Here we assumed that the system is spherically symmetric. Although it is not evident to replace -h2k2/2m*with -(h2/2m*~2)(d/d~)(r2d/d~), this replacement will bring us an effective mass equation @(r= ) E@(r)

(B.14)

which is known t o give data precise enough to interpret various measurements. T in this problem is the radial coordinate referred to the donor site. Since this equation is the same as that for an electron in hydrogen atom, the envelope function is also named Is, 2s, and so on. The ground state envelope function is in the 1s state.

Excitons For excitons the kinetic energy term should be replaced with that for a relative motion between an electron and a hole, whereas the potential term should be replaced with the Coulomb attraction, as follows. @ ( r= ) E@(T)

(B.15)

where p* is the reduced effective mass and T is the relative coordinate between the electron and hole. The system was again assumed to be spherically symmetric.

190

Appendix B Effective-Mass Approximated Equation

Bloch States to Constitute an Envelope Function Let us return to Eq. (B.7) and consider a state for k = 0. As was described above, the potential term in the effective mass equation for a donor is given in the form -ve-aR/R ( a + 0; v: a constant). If the potential term in Eq. (B.7) has such the off-diagonal term is Fourier transformed to give dependence on k’ as -./(a2 -v(e-aR/R)cR, since this is a convolution and, accordingly, its Fourier transform is a product of the Fourier transforms of (Av)ok, and c k ’ . Thus, the envelope function of the electron bound by a donor is obviously a linear combination of the Bloch k’-states weighted with -v/k’2 around k = 0 (i.e., the bottom of the conduction band). This k’-dependence is also justifiable from the perturbational point of view.

Appendix C Interface

Boundary Conditions for an

Let us describe in this Appendix how energy eigenvalues are obtained based on the tight-binding approximation (TBA) in the former half, and how the boundary conditions should be given at the interface between different materials if formulated using the same method in the latter half.

Tight-Binding Approximation The Bloch function as described in Chapter 3 is the most general expression of an eigenfunction of a crystal so long as it is specified by a completely periodic potential. However, it is still incredibly difficult to determine the periodic part of the Bloch , it has to be subjected to a potential determined self-consistently function u ~ ( T )since with that which will be affected by those functions of the other existing electrons; that is, the problem becomes a nonlinear equation of a many-body problem. The easiest way to give the periodic part is as follows. Let us take the Bloch function as j

where Ck(j)= N-l12 exp(ik . r j ) and cp(r - r j ) is the atomic orbital function, which is determined by a spherically symmetric Coulomb core potential and well-localized around the lattice site r j . These wavefunctions are called the linear combination of atomic orbitals (LCAO). For a sodium atom, for example, the nucleus has a charge of 2 = + l l e , but the inner closed shell (i.e., core) with 10 electrons may be assumed to screen +10e out, thus leaving as a result only +le or the equivalent of a hydrogen atom nucleus. Thus defined Bloch function is easily proved to obey the necessary and sufficient conditions ) any translational operation T for the Bloch state: +k(r + T) = exp(ik. T ) + ~ ( Tfor from one lattice point to another; that is, the periodic part of the Bloch wavefunction is equated to

Appendix C Boundary Conditions for an Interface

192

Effective Mass Equation in TBA-s-p Coupled Band Let us solve the boundary condition problem at the interface between semiconductor materials A and B, following the theory developed by Ando and Mori (1982) (ref. in Chapter 7). Although this problem has not yet been fully developed, the following discussion will give us an important hint about what boundary conditions should be imposed a t a semiconductor interface. The situation for the composite material A and B is depicted in Figs.C.l and C.2. Both material A and B are assumed to be compound semiconductors, each composite atom of which has either s-like or p-like orbitals. Since atomic orbitals are well localized, only the transfer integral between the nearest neighbors are taken into account, as shown below.

potential V(x) Fig.C.1: Assumed configuration of s- and p-orbital atoms in the direction perpendicular to the interface.

material B

material A

TTTT=ryT=T xn -1

I

'n

Xn-

a'4

xn +1

Xn +2

X

interface

Fig. C.2: Illustration to explain that the transfer integrals must have opposite signs. One transfer integral is between the p- and s-orbital of the neighboring atom on the left, and the other is between the p- and the same s-orbital on the right.

193

For a T B wavefunction $ k ( z , T I ) [TI = (y, z ) ] and aHamiltonian H = -(A2/2m)V2+ V ( T )with an effective Coulomb potential V ( T ) ,we choose an orbital cpS(x- z, -

+

a/4, T I ) around an s-orbital atom at site z, a/4 and calculate the matrix elements of the Hamiltonian with respect to the T B wavefunctions, eventually obtaining

ECsk(n) = x -t

csk(n)

J p S ( x - x, - 1 " /4 ,rl) Cpk(n)

-t Cpk(n

J

I

A2 --v2 + V ( T ) (ps(z- z,

[

2m

pS(x - 2, - a/4, r i ) V ( r ) ~ p p ( zz n

+ 1)/ C p s ( z

- z,

- a/4,ri)d3r

+ a/4, v i ) d 3 r

- a/4, rl)v(T)Cpp(z- %+I

(C.3)

+ a/4, r i ) d 3 ~

Here, a one-dimensional case is treated for simplicity, by assuming that the atomic orbital functions are well localized in the plane perpendicular to the z-direction so that all transfer integrals with interatomic distances other than that of the nearest neighbor are negligible. In a similar way choosing a p-orbital q p ( z- z, + 1 " / 4 , r ~ ) around a p-orbital atom at site x, - a/4,

+

Csk(n)J p p ( x - 5,

+

csk(72

- 1)

/

Cpp(x

+ a/4, T L ) V ( T ) -~ ~ (-~1"/4,r l ) d 3 r 2 ,

(C.4)

- 5 , -k a/4, T i ) V ( T ) ( p , ( z- Z,-1 - a/4, r i ) d 3 r

is obtained. The first integrals in (C.3) and (C.4) are, respectively, & s c s k ( n ) and If we take the transfer integral s p s ( z - 2 , - I " / ~ , T ~ ) V ( T- )z,$ ~ ( ~ a / 4 , r l ) d 3 r to be a constant -t, then the second integrals in (C.3) and (C.4) become, respectively, -tCpk(n) and -tCsk(n), as suggested in the figure. Similarly we calculate the last integrals and finally the following equations are reached

+

&pCpk(n).

ECsk(n) = ECpk(n) =

& s c s k ( n ) - tCpk(n)

&pCpk(n)- tcsk(n)

+ tCpk(n -k 1)

+ tCpk(n - 1)

Now let us introduce explicit forms into csk(n)

=

Cpk(n) =

In general, as

Csk,

and

eikz(zn+a14) csk, eik,(z,-a14) cpk,

Cpk,

Csk(71)

(C.5) (C.6)

and Cpk(n). (C.7) (C.8)

are not constant. Then the above Eqs. (c.5) and (C.6) read

(C.10)

Appendix C Boundary Conditions for an Interface

194

which are solved assuming kxa << 1 t o give Es - E p

E +E E=”Pf2

2

J1

4t2(k,a)2

+ (E,

+ Ep)2

((2.11)

, the effective mass m* = h2Eg/2t2a2are introduced If the band gap Eg = . c p - ~and and 2tkxa/Eg <( 1 is assumed, it is easy to calculate approximate eigenenergies to be E , - h2kq/2m* and E,, h2kq /2m*. Now we move on to the effective mass equation. Since itak, cpk, = -(C.12) Cskz E - E~ itak, (C.13) csk, = E - cSCpk,

+

+-

hold, E M

E,

is taken around the valence band top, then Eq. ((2.9) becomes (C.14)

which may be transformed into an effective mass equation by setting ((2.15) kz

as in Appendix B. Originally this @,(R) is defined only on discrete lattice sites R = x n l n = 1,2,3, N , but we smooth it over a specific region covering several lattice points, so that it may satisfy the following equation. The obtained effective mass equation reads (C.16) Here @,(R)was redefined for a continuum variable R, or for an x in this onedimensional case. This procedure may be justified for a function extending widely over at least several multiples of the lattice constant; in other words, if it is an envelope function to the Bloch wavefunction. Similarly, for a p-orbital,

h2 d2 @ (x) c p a p ( x )= E@p(x) 2m*dx2 is the corresponding effective mass equation.

+

((2.17)

Boundary Conditions at the Interface Now let us define the interface between the I-th and (1 + 1)-th cell site and modify Eq. ((2.5) and (C.6) so that they meet the interface condition. In Eq. (C.6) n is replaced by 1. For the last term tCp,(Z) we may continue from material A to B if we define tABCZ(1)

tACA(1).

(C.18)

195 Likewise in Eq. (C.9),

+ 1)

tABC,$(l

tBCpB((1

+ 1).

(C.19)

These conditions are necessary to ensure that Eq. (C.5) and (C.6) are valid across the interface. , is transformed t o give @(z) and in Around the valence band top, i.e., E M E ~ Csk, a similar manner cpk, to give -(ta/Eg)V@(x).Hence, the boundary condition (C.18) is rewritten as tAaA(X1)

(C.20)

tABaB(xl)

or, approximately, (C.21) where we took the interface to be at x. Likewise, from Eq. (C.19)

(C.22) Thus finally

(C.23) with

(C.24) Ando et al. called TBAthe interface matrix. It is now easy t o derive the corresponding matrix for the state near the conduction band bottom:

(C.25)

s-s Coupled

Band

Mori and Ando gave the other case; that is, when coupling occurs between s- and s-states at the interface. Then, a similar TBAis given, as follows:

TBA=

(

$(tA/tAB tA/tAB

+ tAB/tB) - tAB/tB

1

-(tA/tAB a(tA/tAB

- tAB/tB)

+ tAB/tB)

(C.26)

The page is intensily left blank

Appendix D Hydrogenic Envelope Function in 3D and 2D

Hydrogenic State in Three Dimensions An EMA equation describing an electron trapped by an impurity potential is given by

analogous to that of the hydrogen atom, where K is the static dielectric constant of the medium concerned and it is known that the solution is @ n l m ( ~O:) Xm(e,V)Rnl(r)

(D.2)

where Ylmis the spherical harmonic function, the detail of which will be given at the end of this Appendix. The magnetic quantum number m is an integer satisfying -1 5 m 5 1, and Rnl is given by the solution of the equation

as

1 = 0 , 1 , 2,...,n - 1 and

is the generalized Laguerre polynomials. Readers should be careful about the definition of the generalized Laguerre polynomials. For example, in Moriguchi et al. (1960) (ref. in Chapter 4), they are defined by Li?,"_!,= (-)"L;/n!. Refer also to Landau and Lifshitz (1977) (ref. in Chapter 4), Mathematical Appendices. The eigenenergy is En = -Ry*/n2 with the effective Rydberg energy Ry* = ( e 2 / 4 m 0 ~ )x2 (m*/2h2).The effective Bohr radius is aI; = 47r€oKh2/e2m*.In order to solve an impurity state in semiconductors, m* is an effective electron mass m:

Appendix D Hydmgenic Envelope f i n c t i o n in 3D and 2 0

198

trapped in a donor state, whereas m* is an effective hole mass m i trapped in an acceptor state. To treat an excitonic state, m* should be a reduced effective mass defined by p* = m,*mt/(m,* mi). For a positive energy, the exciton is unbound. Assuming the energy to have a parabolic spectrum E = h2k2/2m* (k: wavevector) and a = l/kuG, Eq. (D.1) is rewritten in a dimensionless form as

+

d2R

p~

dP

+ [2(1+ 1) - p]-dR - (ia + 1 + l ) R = 0, dP

p = 2ikr

the explicit solution of which reads, if we follow the k-scale normalization (cf. Shinada et al., 1966, ref. in Chapter l l ) ,

f-+ [ k

=

7r

(21

+

1

' ~ n i = ~ {aj2~}

l)!

sinhm

(-i)le7ra/2 e - i k r (2kr)l 1'2

xF(1 + 1 + ia, 2(1+ 1);2ikr) Km(f3,4)

6

where F is the confluent hypergeometric function defined (Moriguchi, et al., 1960, ref. in Chapter 4) with

+

with ( p ) n p(p - 1)(p- 2) . . . ( p 72 - 1). The normalization factor to the hypergeometric function with a continuous eigenspectrum is a fairly tedious problem. Refer to Shinada et al. (1966) (ref. in Chapter l l ) , Appendix, for this matter. The above normalized envelope function is obtained so that it satisfies the condition

In order t o reach the normalized function with dimension V-1/2 as usual we must ~ instead ~ ofl @ k l~m , the relation of which is given by calculate @ (D.lO) where N k is the number of orbitals within a sphere of radius k in the k-space and V is the three-dimensional volume. Let us start with several definitions. Following, e.g., Ziman (1972) (ref. in Chapter l l ) ,consider the oscillator strength. From the interaction between quantized photon field and electron, it is usual to define a dimensionless oscillator strength of the j-th electron's transition from i- to f-state

199 by the formula

(D.ll) where mo is the bare electron mass. The first line indicates that the transition is caused by the electric dipole moment (unless the integral is divergent) and the second and third lines are converted in terms of momentum pj. ex stands for a specific polarization direction X of the interacting photon field. This conversion is made with p = i y ( H r - r H ) for the Hamiltonian H of concern. The last line includes a photon energy hv which is implicitly equated with Ef - E;. Thus, the absorption coefficient is defined in terms of fj with '

(D.12) where ~ 3 ~ ( h w is )the three-dimensional joint density of states between the i- and fstates, and wp is the plasma frequency, wp = JNe2/Vmoc0,with N being the number of carriers. q' + iq" is the complex refractive index. This definition of the absorption coefficient is rather complicated to derive, but a rough introduction will be outlined in Appendix E. For exciton absorption, it is well known that the transition matrix should be multiplied by the envelope function @ ( r )(c: conduction band; v: valence band) as

(D.13) (flex . Pli) = (clex . PI4 x @(TI for allowed transitions. On the other hand, for forbidden transitions we rely on the k.pperturbation with the perturbation Hamiltonian tik . p/mo. The perturbed envelope function is given by (flex .pli) = (clex .PIv)@(T)

x (-i g r a d ) @ ( r ) (clex .plv)@(r)- i(c[Pxlv) . grad@(r)

(D.14)

Since @(O) = 0 holds in the case of forbidden transitions, we obtain

(D.15) /(flex .PI91 = l(clPxlv)l x Igrad@(r)l This formula is derived from the fact that transition occurs between band c and v only through an intermediate state i coupled through the k , p perturbation term. This time wavevector k operates on the envelope function as k@)nlm(r)= -igrad@,lm(r).

Appendix D Hydrogenic Envelope Function in 3 0 and 2D

200

The envelope function is @nlm(r,8,$) = Rnl(r)Xm(8).Both @ and grad@in the transition matrix may be estimated at T=O, since the exciton is in a fairly localized state. From now we will replace the dependence on the polar angle 8 and azimuthal angle 4 by 1/& after averaging over them. Depending on whether an exciton is bound or unbound, we denote the envelope function as @nlm or @klm, assuming that the unbound exciton energy is expressed with a positive energy E k , whereas a bound exciton energy is E n = -Ry*/n2. We can list explicitly several important formulae which appear in the absorption coefficient. (1) For a bound exciton:

(D.16) (D.17) (D.18) (2) For an unbound exciton: I@ivsOm(0)12 =

aeXa sinh7rar

(D.19)

~

(D.20) (D.21)

Hydrogenic State in Two Dimensions An EMA equation with a Coulomb attractive potential in two dimensions is written using cylindrical coordinates as A21d 2m’rdr

-.__-_

[

d dr

T-R(T)

]

-

e2 fi2 m2 R ( r ) -- R ( r ) = E R ( r ) 47rcorcr 2m’ r 2

+

~

(D.22)

with the magnetic quantum number m. We again follow the paper by Shinada et al. (1966) (ref. in Chapter 11). To solve this equation we first put R ( r ) = r ” F ( r ) , then the equation reads

TF” + (1 + 2v)F’ +

v(v - 1)

+ v - m2F + - 2m*Er F +

T

fi2

2m*e2 ,F = 0 (D.23) 47rE0K h

~

Here let us take v = Iml. The next step is to take F ( r ) = e-P/’L(p) with p = r/ahX = -4E/Ry*). Then the finally reached equation is (D.24)

201 which is satisfied by the generalized Laguerre polynomials Li\y’,ml-1,2(p). The Laguerre polynomial has the highest order in p equal to 2X - Iml - 112, which must be non-negative. If we introduce a quantum number n equal to 2X - 1/2, then n 2 Iml is necessary. Thus normalized wavefunction and eigenenergies obtained are

En,

Iml

RY* ( n 1/2)2 5 n, n = 0 , 1 , 2 , 3 ,...

=

-

(D.26)

+

It is seen that the binding energy is 4 times larger than that in three dimensions, and the radius of the state is half that in three dimensions for the ground state ( n = 0). In parallel to the procedure to reach Eq. (D.5), Eq. (D.l) is modified for a positive energy to be 1 d2R dR (D.27) (2lml+ 1 - p)- (icr ImJ -)R = 0 dP dP 2 in terms of X = -ia/2 and cr = l/u&k. Now the solution of the original equation reads p~

+

+

1

+

eim9

(D.28) x F (l m (+ - + ia; 2(rnl+ 1;2ikr)2 G For an unbound state, as in the three-dimensional case, we should transform the normalization condition for a continuous eigenspectrum from J d2r@&krn(T)@N;rnt (T) = 6(Nk - N#,,, to J d 2 ~ @ l m ( ~ ) @ p m = f6(k ( ~-) k’)drnrn(,where a dimensionless N is the number of orbitals within a circle of radius k in k-space. Thus, the value of the envelope function itself should be taken to be = , / m @ k m . Taking Idk/dNkl = 2x/Ak, we eventually obtained the above normalized eigenfunction. A is the area of the system and Nk = Axk’/(27~)~. In @ k m = R k m ( r ) e i m 9 / f i , the azimuthal angle dependence was replaced by l/G. (3) For a bound exciton:

(D.29) (D.30) (D.31)

Appendax D Hydrogenic Envelope Function an 3 0 and 2D

202

(4) For an unbound exciton: (D.32) (D.33) (D.34) Hydrogenic State in One Dimension

Loudon (1959) (ref. in Chapter 11) and Elliott et al. (1959, 1960) (ref. in Chapter 11) first studied excitons in a magnetic field, which necessarily provoked the problem of an excitonic state in one dimension; that is, an hydrogenic state in the problem of concern has the only unique degree of freedom in the direction parallel to the field. Let us take for simplicity the Coulomb potential t o be given by -e2/4rq,tc(lzl +zo), where it is known that for a vanishing zo the ground state energy becomes a negative infinity. To avoid divergence we must include zo as a cut-off parameter. Taking z = 2(z zO)/aa;, ( z 2 0) or z = -2(-z z o ) / a u l ; , ( z < O), i.e., z = f 2 ( l z l z o ) / a u l ; , the Schrodinger equation is that for the well-studied Whittaker’s function (Moriguchi et al., 1957, ref. in Chapter 4):

+

+

+

(D.35) and the binding energy E, = -Ry*/cy2, where cy = u, u = 0 , 1 , 2 , ..., for zo + 0. Let us introduce a quantum defect 6,, with which we take the eigenvalue a = u 6,. It is known that 6, x 2zo/al; for odd states ( u : odd) and 6, x -[log(2z0/uu~)]-’ for even states ( u : even), when 2zo/aak is very small. In particular for the ground state, do should be taken to be the solution to a transcendental equation: log(2zo/60al;) 1/260 = 0. The approximate wavefunctions for a M u are

+

+

with the associated Laguerre polynomial Lb(<). Thus, the wavefunction is non-vanishing only for even states at z = 0, i.e., z = zo = 2zo/uul;, as given by

(D.37) In parallel, its derivative is non-vanishing only for odd states at x = zo as given by

203 For an unbound exciton, we follow Ogawa et al. (1991) (ref. in Chapter 11) and take x = 2ik(lzl Z O ) , cy = l/kah and E = h2k2/2rn’, then the Schrodinger equation becomes

+

(D.39) the solution of which is given by a linear combination of W-ia,l/2(x) (1) G r(l +ia)xe-z/2 [F(1+ i a , 2 ; z ) + G ( l + i a , 2 ; x ) ]

(D.40)

r(1- ia)xe-”/2[F(1 + icy12;x)- G ( 1 + ia,2;x)]

(D.41)

I W-icl,l12(z) (2)

with the above-defined confluent hypergeometric function F (Eq. (D.8)). G is newly defined by z

G ( l +icy,2;x) = -(e-2HL1 - l){[lnz 27r1

+1-27

-$(1 +ia)]F(l+ i a , 2 ; x )

where the digamma function $(x) = d In r ( x ) / d z was utilized. y is Euler’s constant equal t o -$(l) = 0.57721 .... Since the potential is symmetric about z = 0, two envelope functions (an even and odd function with respect to z ) are solutions. After a tedious calculation, normalized envelope functions are obtained as follows: (D.43) (D.44) where the superscripts g and u denote, respectively, “gerade” (even) and “ungerade” (odd) functions, and

W;” WJ’)’

G

W(j)(2ikzo),

(D.45)

dW(3)(z)/dzlz=2ikzo, ( j = 1,2)

(D.46)

+

W ( l ) ( z ) and W ( 2 ) ( zare ) the functions of x = 2ik(lzl zo) given by Eqs. (D.40) and (D.41) with omission of the subscript (-icy, 1/2). As was so in the previously described two-dimensional case, the optical spectra due to allowed and forbidden transitions are, respectively, determined by (2ikz0)1’ and

/@F)

I@p”(2ikzo)J2.In addition, as was so in two and three dimensions, the k-scale normalized envelope function should be transformed into @ $ z ) ( z )using the relationship that @(g’u) Nk = with Nk = (L/27r)2k. Finally, we must remember that the DOS of the 1D system is

(l/JdNk/dkl)@?’U) m*

PlD(Jq =

-O(E 7rh2k

- Eg)

(D.47)

204

Appendix D Hydrogenic Envelope Function an 3 0 and 2D

where O ( E ) is the step function. All the results are briefly re-described in Chapter 11. Spherical

Harmonic Functions

K,(e, 4) is the eigenfunction satisfying L 2 K m ( e4) , = i(i+ 1)h2Km(e,4)

(D.48)

with the (squared) angular momentum operator ~2

-A'

(-882d2 + cote-ae8 + cosec28-a42)

(D.49)

82

They are explicitly given in the following forms.

(D.50) where PLm(e,4) are the generalized Legendre polynomials defined by p1ywW)= (1 - ,2)lml/2-

dim1 dwlml

9 (w)

(D.51)

for the Legendre polynomials 9 ( w )

I d 1 9 ( w ) = --(20 2l1! dwl

2

- 1)1

(D.52)

Thus defined spherical harmonics obey the following orthogonality relations

(D.53) Several spherical harmonics for the typical orbitals are: s-orbital yo0

=

&

(D.54)

porbitals

(D.55) (D.56) (D.57)

205 or

(D.58) (D.59) (D.60) &orbitals

(D.61) (D.62) I-

Y20

= d & ( 3 c o s 2 8 - 1)

(D.63) (D.64) (D.65)

or

(D.66) (D.67) (D.68) (D.69)

(D.70)

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Appendix E Transition Probability of Optical Processes

Quantization of Radiation Field Optical transition processes occur through the Hamiltonian term HI = ( e / m o ) p .A , which appears as a deviation from p2/2mo in the equation

where A is the vector potential of radiation. Here we rely on the Coulomb gauge, i.e., p . A = -ihV. A = 0 and neglect the small A2 term. Let us decompose A ( r , t ) into X = (k,v)components with an amplitude qx(t), where k is a wavevector and u = It stands for the polarization:

ex is the unit vector representing polarization u. To quantize a radiation field, it is usual to represent radiation by means of harmonic oscillators, which have the form:

nx = alax with [ a x , a i , ] = 6x,xf (E.4) t are, respectively, an annihilation and a creation operator of one photon of ax and ax A. A photon is a Bosonic particle, and, hence, nx-state is considered to be equivalent to the state which is occupied with nx photons. The correspondence relationship between classical amplitudes of radiation and quantum operators are

Finally we obtain the perturbation Hamiltonian

Appendix E Tkaansition Probability of Optical Processes

208

B o r n Approximation

Let us consider a transition rate of an electron in li)-state to If)-state by absorbing or emitting a photon using the first-order Born approximation. 1 1 7=

F;

-

+

where nx is connected with eik."3 expressing an absorption process, whereas n A 1 with e-ik,r3 an emission process. In this expression nx is the number of photons equal to the radiation intensity. The sum should be performed over Xi and Xf ensuring that the momentum conservation rule q = I f q1 = l l c f - kiI holds. As usual, e*'".' in the matrix element is approximated as being 1 from now on. Henceforth, we simply denote the squared matrix element by IC(Xi;Xf)I2 and are concerned only with the absorption process. When we assume that both the initial and final state are, respectively, the valence and conduction electron, and that both are parabolic, so that we take kf = k + 412 and ki = k - 412, the argument of the delta function becomes iiwx - Eg- E with E = .$h2k2(l/m,' l/mL). Here we have neglected a small contribution to the argument of the delta function: i i i 2 q . k ( l / m $ - l/m;). Then the double sums are exerted considering that 2 V / ( 2 7 ~ )d3k ~ ... = d E p 3 ~ ( E..., ) as follows:

+

s

fj

s

is the dimensionless j-th oscillator strength defined as (E.lO)

The p3D in this equation is neither the DOS of the initial state (valence band) or that in the final state (conduction band), but the joint DOS defined in terms of E = hw, - E,, which is proportional to for E 2 Eg in three dimensions. Absorption Coeficient

In order to relate the transition rate with the absorption coefficient, we need to know the complex dielectric response in a form that includes a relaxation rate rq. Let us start with a Boltzmann equation which is physically transparent and intuitively easy to understand (see, e.g., Ziman, 1972, ref. in Chapter 11). (E.ll)

209 From the Newtonian equations of motion for a Bloch electron (F: an electric field): hk=eF,

Vk

=+

(E.12)

the deviation of f from the equilibrium fo, say gk, satisfies

(E.13)

(E.14) which is introduced into

(E.15)

-

where Sd3kevkfo = 0 was utilized. When we neglect a small term r k V k in the denominator of aq (Eq. (E.14)) and an approximation -afo/dE = 6 ( E - Ef) is used, the optical conductivity is simply

(E.16) k Here we utilized the DOS in 2 V ( 2 ~ ) - ~ d ~p(E)dE with (E.17) and I V k E k l = hvk assuming a parabolic band for the electron. The integration is performed on an equi-energy surface. Thus, 1 (E.18) .(W) = .(O) 1 - iW(Tk) where ( r k ) is a certain average of ‘rk on the equi-energy surface E = E f . On the other hand, from the Maxwell equations j = [-iwc(w)

+o(w)]F

(E.19)

is obtained. The u-term comes from the conductivity of conduction electrons. A complex refractive index q/+iq” is now introduced and the complex dielectric response is defined by -iwc(wq)

+ O ( W ) F -iw(q/ + iq/1)2

(E.20)

or

q / 2 - q / / 2 + 2i“‘q” where a’ f processes.

icy“

(E.21)

is the complex polarizability resulting from the interband transition

210

Appendix E Ransition Probability of Optical Processes

It is just the complex polarizability that corresponds to the optical absorption and emission processes, in particular, in semiconductors. This is given by the following expression:

(E.22)

+

with 6 + 0. As usual in this limit the well-known formula l/(z - k)+ P / z i.rrb(z) should be applied. The P stands for taking the principal part on integration. The imaginary part expresses both of the optical processes, since

where the first and second terms denote, respectively, the absorption and emission process. In the equations

(E.24) (E.25) the term including o(0) is much smaller than the first term in semiconductors. In metals, the order is reversed. Since optical absorption coefficient K(w) is given by (2w/c)77”, Eq. (D.12) was obtained.

Appendix F Eigenvalue Problem for a Linear Electric Potential

In the n-channel in an MOS-FET (metal-oxide-semiconductor field-effect transistor), a strong electric field exists between the oxide-semiconductor interface and the semiconductor inside. For simplicity, let us take the field potential to be linearly dependent on the coordinate as e F z , where z is the direction toward the semiconductor with its origin at the interface. This potential is often called the triangular potential. In actual cases, the electrons already occupying low-lying levels cause an ad hoc electric potential onto high-lying electrons, thus forcing the problem to always be solved in terms of the Hartree potential (see Stern, 1967, ref. in Chapter 7). Here we are concerned with a non-Hartree problem, i.e., an empty potential problem for the linear one, so that it is easy to solve quantized energies according to the above-mentioned field. The EMA equation is A2 d2 2m' dz2 If we reduce this equation to a dimensionless form taking z = ( F ~ ~ / 2 r n * e F )and ~/~C E = ( A 2 / Z m ' ) ' / 3 ( e F ) 2 / 3 Then, ~.

(-0: + 011,= E11, is obtained. Let us consider a function defined by the following integral: f (c)

/m

(F.2)

cos[t3 - a ( b - C)t]dt

0

is easily proved, since we are allowed to take limn+m cos R = 0. Let us assume 11, C( f . Then, from

(-0; + C)f(C)

= Sm[(at)' 0

+ C]cos[t3 - a(b - C)t]dt

Appendix F Eigenvalue Problem for a Linear Electric Potential

212

and Eq.(F.2), it is required that 1 - a3/3 = 0 and a3b/3 = E , i.e., b = E must be fulfilled. In the definition of the Bessel functions and one special case named the Airy function Ai(x) (Moriguchi et al., 1960, ref. in Chapter 4), roo

Ai(x) = /o

cos(t3 - xt)dt

;fi [J; (w) J-;(w)] 2x3/2

+

2x3/2

is what we need in this problem with x = 3 1 / 3 (-~ C). In order t o obtain quantized energies, we take the boundary condition that $(O) = 0. Then,

is the eigenfunction. If we consider the n-th zero of the Airy function, j,, be given), we finally reach the eigenenergy formula

( u will

soon

or

(F.lO) Zeros of the Airy Function Using the definition of the Bessel J, and Neumann function Nu (Moriguchi et al., 1960, ref. in Chapter 4),

(

J,+ J-, = I +

cotvn cosecvn

) J, (cot2un +

~

- cosec2un cosecvn

(F.ll)

On the other hand, it is known that cos(a) J, -sin(a)N, has n-th zero of j U n ( a )which , is explicitly given as follows. Since u = 113, we obtain (Y = n / 6 from the equation J1/3 J-1/3= (1 cos $)J113 -sin $N1/3 0: cosaJ1/3 - ~ i n a N ~ / ~ . Now, for a = n/6, the zeros are

+

+

(F.12)

+

where C = 4u2, ,B = ( n / 2 ) ( u 2n - 1/2) - a , j i n ( n / 6 ) n ( n - 1/4), n = 1,2,3,.... N

.... Thus, for the lowest approximation,

Appendix G Calculation of Conductivity Based on the Kubo Formula

It is probably instructive for readers to know how the Kubo formula of electric conductivity should be developed to reach a familiar expression, when based on a simple model. Let us start with Eq. (10.18) explicitly represented in terms of the li)-state of a free electron to diagonalize the unperturbed Hamiltonian 31.

where R is the volume of the system, p = e-D?i/Z is the density matrix with the partition function Z and spin degeneracy is considered to double the integral; p = l / k B T . After integration over t and A, we obtain

where an adiabatic switching-on of an electromagnetic field at t = --oo was assumed, i.e., we took e-iWtfEt/hunder the integral and a limit of E + 0 is taken after integration. Since oPy(w) is analytic in the complex w-plane and it is assured through the fact that w has a tiny imaginary part -if, we can estimate o P V ( w )at w - i / r , which denotes some type of relaxation of the electron after it has interacted with a scatterer such as an electromagnetic field quantum or a phonon. At this moment the relaxation time r is phenomenologically introduced. To treat the Kubo formula correctly, we must calculate T within the same frame of the theory as referring to 12)- and Ij)-states. We are interested only in DC conductivity from now on (w -+ 0). Thus,

Appendix G Calculation of Conductivity Based on the Kubo Formula

214

Here we replaced Z-le-oEi by the Fermi-Dirac statistical function fi. v is the velocity operator ( J p = ewp). Let us calculate the above equation for the free electron energy 2 Ej = ( h 2 / 2 m * ) ( k - ~ / 2 )and 2 an elastic process spectra Ei = ( h 2 / 2 m * ) ( k + ~ / 2 )and such as Ei = Ej. Then

holds for a temperature low enough t o make electrons degenerate. The Fermi energy Efequals ( h 2 / 2 m * ) k ; . First of all, we calculate the matrix elements for ozz:

Since for a spherically symmetric K

it is easy t o reach czz =

2e2 /d$,d(cosB,)d(cos8,)c2-d(n)dn 2T2 (2TlS x

K2

] d$k { ( k cos k2 l/r(k)

x--6

Ok)'

[ h2

-(k2 2m*

-

( K cos(2, K))2}d(cos0,)

+ kKcosBK + ~

~ - /kf2 4

Here the polar angle of K is referred to k and that of k is referred t o the z-axis. l / r ( k ) is assumed t o depend on k in general. Finally we reach Iszz

ne2(r) =-

m*

since

dk

= lEfdEp(E)r(E) = n(r)

(G.9)

with a density of carriers n. In quantum mechanics a relaxation (scattering) time r always appears in the form l / ~in,particular, in perturbational calculation. Consequently, it seems rather curious to find that conductivity is proportional to r , even if the Kubo formula is said to be the most basic one to define electric conductivity. For instance, if there are several scattering mechanisms and they are independent of each other, the so-called Matthiessen pa, each rule holds. According to the rule, electric resistivity is additive, i.e., p =

c,

215 component of which corresponds to one species of scattering mechanism. This rule is easy to understand. Then how should we define the composite DC conductivity under several scattering mechanisms? The answer is as follows. 2e2 (-i)m*(v2) a(0)= 6(hw), w + 0 (G.lO) Rm* w - iCa(l/Ta) has the Matthiessen rule in the denominator. To be exact,

(7)

= 1/ ca(1/7&).

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Appendix H Calculation of Conductivity Tensor in a Magnetic Field

In this Appendix a conductivity tensor in two dimensions, i.e., in the plane perpendicular to a magnetic field, is calculated. Take a magnetic field in the z-direction, and allow electrons to flow in the perpendicular zy-plane. First we rely on the Boltzmann equation to derive conductivity tensor, where it is fairly easy to understand the underlying physics intuitively (cf. Uemura et al., 1966, ref. in Chapter 10). Afterwards we.give more general formulas based on the Kubo formalism. Let us write down the Boltzmann equation to start with, as in Appendix El although the Lorentz force should be included in this case.

where f( k ) and fo ( k ) are, respectively, the non-equilibrium and equilibrium distribution functions. Note that e should take a negative value for an electron. The right hand side collision term was approximated by a scattering time. This scattering time will be specifically proved to be an averaged, as described later. Here the spatial dependence of f(k) is neglected. Now we take f (k)= fo(k) - (v P ) d f O / d ewith a generalized momentum P . In the equation %

= - h( m * w - P ) - afo m* ae may look to be a good approximation, where the relationship w = &/dhk = hk/m* was utilized assuming a parabolic energy spectrum. Thus,

w.P -

-

7

-

e -((E

m*

+ w x B ) . (m*w- P ) 1

03.3)

where E . P term was neglected, since it gives only a higher order in E . Eventually we reach

P + ( p B ) x P = eTE

(H.4)

218

Appendix H Calculation of Conductivity Tensor in a Magnetic Field

in terms of the mobility p = er/m*. p B is also reinterpreted as w c r with the cyclotron frequency wc = eB/m*. This equation is solved t o give

If we define current density with

then

are easily obtained. Since the integration of the term with vxvy vanishes in an isotropic k-space, finally we obtain

(H.lO) (H.ll) with uxx

= uyy= -

uxy

= -uyx =

“J (2.p

(H.12) ‘$‘(pB) dfod3k. 1 (pB)2 da

+

(H.13)

In these formulae r , and thus, p may in general depend on k or v. In the case where the electron gas is degenerate, and, hence, d folds M --6(a - af) (af = Fermi energy) and v: = v2/2 in two dimensions is utilized, then the integrals Eqs. (H.12) and (H.13) can be estimated only at E = af, i.e.,

(H.14) (H.15) with p = p(af). Ordinarily the Hall effect is measured for a vanishing jy, i.e., an open-circuit electric field equal t o Ey vs the current density along the z-axis j , , namely,

(H.16)

219 The thus defined RH is the Hall coefficient and is known t o be l/e&:rp(cf). In the degenerate two dimensional system, the DOS p is constant and, hence, Efp = p J e6(&&f)de= pJE(-8afo/d&)de = pJafo(&)d&= n holds, that is, 1

RH= (H.17) ne results. Here an explanation has been given for two-dimensional system, but it is also the case for a three-dimensional electron gas. In three dimensions, vz in the above equations must be replaced with $v2 and the relationship ( 2 / 3 ) ~ f p ( ~=f )n is utilized (see Eq. (4.15)). The inverse relationships are sometimes used between E and j , that is,

+j Y P Y X

Ex Ey

=

pxx

= Pyy =

jxPxx

= jypyx+jypYy with pxx = pyy and pyx = -pxy. It is easy t o calculate the resistivity tensor as ffxx

ff3x Pxy

=

-pyx

=

+ “2, ffXY

2 ffxx

+ “2,

(H.18) (H.19)

(H.20) (H.21)

In a degenerate electron gas, we calculate from Eqs. (H.14), (H.15) and (H.16) as (H.22)

(H.23) with uo = nep = ne2r/m*. Finally, we note that the most remarkable and very often utilized fact that uxy >> usx holds for pB = w,r >> 1, and consequently pxx

ffxx =-

(H.24)

ff2y Pxy

=

1 ~

(H.25)

ffXY

where the former equation is really curious, since p x x is proportional to uxx.The condition p B = w,r >> 1 means that the cyclotron motion occurs many times before scattering destroys the circular motions. These p x x and pxy have formulae that are easier to understand than uzx and uxy. In fact, von Klitzing et al. (1980) (ref. in Chapterlo) successfully observed the integral quantum Hall effect by measuring pxx and pxy under a constant current configuration, i.e., j , = 0 and j , = constant. They observed Ex and Ey,and accordingly p x z and pxv as a function of n, but not j, or jy,and accordingly, uxx nor uxyunder a constant EX.

We will next discuss the Kubo formula for the same conductivity. Basesd on the Kubo’s formulation of an exact conductivity (1957) (ref. in Chapter lo), he and others

A p p e n d 8 H Calculation of Conductivity Tensor in a Magnetic Field

220

(Kubo, Hasegawa and Hashitsume, 1959, ref. in Chapter 10) developed similar exact conductivity formulae for an electron under a magnetic field. They first separated the coordinates as follows: x

= c+x,

k=(+X

(H.26) (H.27)

y = Q+Y, ~ = ? j + Y where (X, Y) and (<,7 ) represent, respectively, the center coordinates of the cyclotron motion and the coordinates relative t o the center. Now take a Hamiltonian with a scattering potential V ( T )such as

1 V ( r ) , n = p - eA, B = rotA (H.28) 2m* and, then, from [7rx,7ry] = -h2/i12 = -ehB/i (if the particle of concern is an electron, l 2 = h/leJB = -h/eB), an equivalent commutator

3c = E ( n ) + V ( T )= -n2

+

(H.29)

<

is obtained, when we assume = -127ry/h and 7 = l 2 n x / h . Now using X = x - and Y = y - q ,

<

h. (H.30) ieB and [<,XI= [<,Y]= [ q , X ]= [q,Y] = 0 are easily calculated. For a new gauge of A = [-By/2, B2/2,0], the Hamiltonian and wavefunction are obtained by the canonical transformation, respectively, from Eq. (4.18) (Hamiltonian) and Eq. (4.19) (wavefunction) which are defined for a gauge A = [-By, O,O]. The transformed Hamiltonian is

[X,Y] =

12

1

=-

j j = e-ixy/212yeizy/2P

(H.31)

and the wavefunction is

4 = e-ixy/212 " h , k , , N ( T )

(H.32)

Now Y = l2kX. Since the system is quantized within a length LX in the x-direction, the number of the states Nx is given by (H.33) Accordingly, (H.34) (H.35) are easily obtained and from these relationships AXAY = 2 d 2 results. If we take canonical variables to be (X, Y) which satisfy Eq. (H.30), then an uncertainty relation AXAY = 2 d 2 must be obeyed. This fact means at the same time

221 that there is a A/2n12-fold degeneracy of states for a single Landau quantum number N (A: the area of the system), since there is a single state per area AX6Y. Eventually we arrive at

(H.36)

(H.37)

(H.38)

(H.39) except for the motion in the t-direction parallel to the magnetic field that is never affected by the field. Although their derivation is not reproduced here, Kubo and coworkers obtained the following formulae for uzz and uzy: P

dt

uzz = ozy

=

+

e2

dA(X(-iAX)X(t)) 00

dt

l

(H.40)

P

dX(Y(-ifiX)X(t))dt

(H.41)

with p = l / k ~ T and , the volume of the system R , just parallel t o Eq. (10.18). These formulae are valid so long as the relative coordinates (5,q ) are bounded. Obviously the first term on the right hand side of ozy gives the Hall effect independent of scattering. The center coordinates of the cyclotron motion are given by ( X ,Y ) ,which are the constants of motion and, hence, specify a quantum number in addition to the Landau quantum number, unless there is scattering. If some scattering occurs, it is easy to imagine that the center jumps from one site t o another. This type of jump is described in terms of ( X ,Y ) and the conductivity under a magnetic field is known to occur only according to such jumps.

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Appendix I Landau State in x-Representation

An electron in a magnetic field B is usually represented by taking a vector potential to be A = (-By,O) in two dimensions, the eigenstate of which was given in Chapter 4. There is another possible way of expressing the vector potential, i.e., a symmetric gauge with respect t o x and y. Let us take A = (-By/2,Z?x/2), where the effective mass equation of concern reads

= E+ (1.1) Now, we represent (x, y) in the Gaussian space as z = x + iy and 2 = x - iy. It is known that 2. is not an analytical function of z , but a Z / d z = (1/2)(dZ/dx-idz/dy) = 0 is easily proved. Using z and Z the above equation is re-expressed as

= q!J

with c = 2rn*E/h2. To solve the equation take = e z z / 4 1 2 Then, ~. 1 (1.3) xzz -ZXr + -4 I;?+€ x = o 212 where x z and x Z z are, respectively, the derivative of x with respect to 2 and the derivative of xz with respect to z . To find a solution of this differential equation let us try to take x as follows and proceed to calculate

Y1

+

+

)

with any integers m and n. Putting the

P = -1/212 and €

= (272

+ 1)/P

x

into the above equation, we find that (1.5)

Appendix I Landau State in z-Representation

224

or

Thus, the integer n is known to be the Landau quantum number and the m plays a role in the angular momentum. The eigenenergy is degenerate with respect t o the quantum number m. This m is another equivalent of the center coordinate of the cyclotron motion, which was the case in Chapter 4 (and Appendix H) where we found that the energy was degenerate for this center coordinate. Finally it is easy to return to the (2,y)-representation using

i a aZ = -2 (-ax -);i a

d

,

l

=

a

5 (% +);i

This representation of a Landau electron was the starting point for Laughlin (1983) (ref. in Chapter 10) when considering the fractional quantized Hall effect, as interpreted in Section 10.8.

Appendix J Domain

Micromagnetism of Stripe

It would be of interest to readers to describe in detail an example of magnetic structure from a micromagnetic point of view (Murayama, 1966). The simplified and intuitive description of magentic structure is such that it is constructed from magnetic domains and domain walls. In such a description, the total magnetic energy is the sum of the magnetostatic energy attributed to the domains and the energy of the walls that are separate. However, it is obvious that the variation in the magnetization does not change discontinuously from inside the domain to the wall. All geometries must vary gradually and continuously. From such a point of view, we may construct all the necessary equations to fully describe the detailed structure. This scheme is called micromagnetism. Our present model is the so-called stripe domain seen in magnetic thin films such as permalloy films. In the 1 9 6 0 ~magnetic ~ thin films were actively studied because it was thought that they would be usefully applied to a magnetic memory device in the form of magnetic wire. Soon the thin film memory devices were replaced by magnetic cores and eventually by semiconductor memories. Let us start with the assumed structure shown in Fig. 12.1. The film is thick in the z-direction and varies sinusoidally in the x-direction in the film plane. We assume an infinite geometry in the y-direction parallel to the mean magnetization in the zy-plane, and consequently there is no variation. The simplest model is what was discussed by Spain (1963, ref. in Chapter 13), where the rising angle of magnetization 4 depends only on z, not on z (Model I). In the next more complicated Model 11, 4 depends on z and z but the angle of magnetization deviating from the y-direction is taken to be 0, i.e., spin does not deviate from the yz-plane. In addition, the most general and complicated model is such as that described in terms of

The next step is to express the energy of the structure. The magnetostatic energy E , , the exchange energy eex and the magnetic field energy E H are reduced by Ij/2po (Is: saturation magnetization), c,,,,~., the anisotropy energy perpendicular to the film

Appendii J Macromagnetism of Stripe Domain

226

and are given by the following equations:

{

cex =

+

($)2

(g)2+ 4 [(g)2+ (g)2]}

ca

=

-p Jsin24dv

EH

=

-T

ems.

=

s

/

cos2

dv

(5.2)

(5.3)

cos 4 C O S ( ~- 6 ) d ~

(5.4) (5.5)

padv

where q = ( A / d 2 ) / ( I , " / 2 p 0p) ,= K L / ( ~ ? / ~ P oT )=, H I S / ( I ~ / 2 p and o ) [T = p - r / 2 . I,, A , K L and H are, respectively, the saturation magnetization, the exchange stiffness, the perpendicular anisotropy energy and the external magnetic field. The thickness of a film is 2d and the length is measured in units of d . The magnetic pole density p and the potential 9 are defined in the following way:

M H Isp

= I, grad @ = I,(cos

4 sin 6 , cos 4 cos 6 , sin 4)

= H{sin@,cos@,O) - -divM

and

An external magnetic field is applied in the direction deviated by 0 from the y-direction in the xy-plane. Later, we simplify the problem by taking 0 = 0.

Euler 's Equations Applying the ordinary variation method to the total energy, it is easy t o reach Euler's equations on this problem. Both are due to variation with respect to, respectively, 4 and 6 :

+ 4zz+ sin 4 cos 4(63 + 62)]+ p sin 4 cos 4 - -r2 sin 4 cos(0 - 6 ) 1

q[qL

=

-az sin 4 sin 6 + 'P,

cos 4 (J.10)

1 q[(6zcos24)z+(6Z~os24)2] + -rcos4sin(O-6) = ~ , c o s ~ c 0 s 6 (5.11) 2 Here the subscript (...)z and (...)zI represent differentiation by 2 once and twice, respectively. The other subscripts are the corresponding differentiation by z .

227 After rewriting the total energy utilizing the above Euler's equations, we obtain €tot

= q

[/

dx4, tan $JfZ?:

+

s

dxe, tan 4Jtz+:

I (5.12)

Now we are interested in the case with very small tilt angles. In particular, when we discuss the critical condition where stripe domains appear, this assumption may be justified. sin 4 M 4 = X(z)sin6x1 X ( z )<< 1 sine M 0 = [ ( Z ) C O S ~ X , [(z)<< 1

(5.13) (5.14)

In Eq. (J.9), 9 can be solved if we utilize the two-dimensional Green function G(z, 2) = (-1/2) ln(x2 z 2 ) to obtain

+

+

1 9 ( x , z ) = -sin6z 2

dC[E(C) X(C)sgn(z - C)]e-

6lz-c'l

(5.15)

Thus, the integro-differential equations finally reached from which to determine X ( z ) and ((2) are obtained as follows:

(J.16) q["

1 2

- (qh2 + - T ) [ -

[X(C)sgn(z - C)

+ [ ( C ) ] e-61z-CldC

= 0 (J.17)

and the final expression of the total energy is €tot

+ T = qX'X]O+'

(J.18)

From the symmetry consideration, it is obvious that X ( 0 ) = 0 at the mid point of the film. Later, the total energy will be found to have a maximum at a &value, where

a as

-X'(l)X(l)

=0

(J.19)

holds, so that the critical condition should be given by X'(l)X(l) = 0 along with Eq.(J.19). Here the origin of the energy ctot there is no spatial variation in 4 and 0.

(J .20)

+ T is equated to 0, where

228

Appendix J Macromagnetism of Stripe Domain

The above integro-differential equations are easily solved by setting 3

X ( Z ) = CAicoshRiz

(5.21)

i= 1 3

(5.22) i= 1

and we can obtain the following solutions: 3

C A i

1

0

(5.23)

and (5.26)

All solutions for R are, for example, approximate for a small qb2, as given by qn;

23

l-a+

qR;

R

1 -r+ 2

qn;

s

0

-q?x

+

(1 - 0 ) 2 - (0 i r ( 2 - 0)) d2 (1- a)(l - 0 -

+ (2 -

0)kT

ir(1-

0

(5.27)

i.)

-(fry

-

(5.28)

f.)

+

u(l fr) qd2 kr(1- 0)

(5.29)

Since 0: is negative and it is known that the solved X and [ are dominated by tan r , we may assume the critical conditions to take the maximum value asymptotically: r = ~ / 2 Next, . we are interested in the limit: p + kr + 0, i.e., 0 -+ 0.

(A) With no external magnetic field (T = 0) and p << 1. For

T

+ 0, the critical conditions read

407;

1 = 406; x ;P

(5.30)

They are simplified further to give 70

= So

40

=

M

K/2

P/X2

(B) With an external magnetic field with (1/2)r + p

(5.31) (5.32)

- 0 and 0 = p - (1/2)r --t 0.

229

At this limit we may solve the above equations in a different way, i.e., the solutions read yso"

(T

%

-

2

(5.33) (J.34)

From the second equation above and the critical condition with r = 0, i.e., gohi M p/4, we solve the critical thickness as a function of the external magnetic field; that is, M

(1 + P )U2F

Yo

(5.35)

or more explicitly

(5.36) This is the result for Model 111. Here the magnetic field was assumed to be applied in the plane and denoted by H ~ l l H . K is~the perpendicular anisotropy field of the thin film. Similarly in Model 11, we obtain the same condition in the form (5.37) and in Model I (5.38) These relationship indicates that when an external magnetic field HKll is applied in the plane, the critical thickness d increases beyond the critical thickness do with no field applied. A comparison between these calculations and the experimental data observed by Sugita et al. (1964, ref. in Chapter 12) is shown in Fig. 12.2. Similarly, the calculated width of a stripe domain, A, is compared with the experimental data by Fujiwara et al. (1966, ref. in Chapter 12). In Model 111, (5.39) In Model I1 (5.40) In Model I

x

2A

2 Again experimental data are compared with calculations in Fig. 12.3.

(5.41)

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Appendix K Physics Underlying Josephson Junctions

Josephson junctions (JJs) have peculiar physics therein, which are very different from those which describe normal states. Therefore, we will summarize here those peculiar, but important, points of physics underlying them. Several ideas in this Appendix follow Tinkham (1975, Chapter 13). Figure 13.6 illustrates a JJ having a thin insulating layer between two SC layers on both sides. The I - V characteristic is known to have features such as are shown in Fig. 13.7. In the discussion below we use the density N*, the mass m*, and the charge e* of supercurrent carriers, which are, respectively, equal to N/2, 2m, and 2e, since each carrier is known to be composed of a Cooper pair (a pair of electrons). For an SC a flux quantum must be correspondingly equal to = h/e* = h/2e.

(1) The current flowing for V=O is the DC Josephson current which is very sensitive to the effect of a magnetic field B penetrating the insulating layer with an area S, obeying a Fraunhofer-type interferometric pattern:

( 2 ) Assume that a constant magnetic field B is applied inside the insulator region as well as the regions neighboring the insulator as far as the London's penetration length X on both sides. For the effective region with a magnetic flux penetrated we will utilize a notation such as d* z 2X + d from now on.

Then a vector potential is given by B = rotA with A = (-By, 0,O). If we apply the Stokes' theorem to the square ( L x x) shown in Fig. (13.6),

@z(x,~) = l y d y l z d x B = BXY

(K.2)

gives a magnetic flux, which exists in the z-direction inside a square of L X(maximum size d* in the x-direction). a current from the left to the right side of For a macrowave Q = IQlei('#'-2T@z/*z)1 the JJ is given by

Appendix K Physics Underlying Josephson Junctions

232 e* h

m*

dx

@:

dx

This expression indicates that the current is proportional to the phase difference between two SCs on both sides of the JJ differentiated with respect to x. However, to meet a physically realistic situation, we should preferably replace this expression by a sine of the phase difference in terms of modulo 27r as was actually discussed by Josephson. Thus,

j ( d * , L ) = JJSin’y e*h - ---1912 sin(A4 m*d* is the final result. JJ is the maximum Josephson current density, which is known to be equal to

according to the study by Ambegaokar and Baratoff (1964, ref. in Chapter 13). R, is the the tunneling resistance per unit area of the junction and A(T) is the superconducting energy gap at T following Bardeen, Cooper and Schrieffer (BCS).

(3) In an SC, the number of Cooper pairs N* is so large that the quantum fluctuation of N* is also so much larger than unity, i.e., A N * N N*1/2>> 1. This means that the phase, a variable conjugate to N * , is quite definitely defined although they must obey a Heisenberg’s uncertainty relation A N * A 4 1; that is, A 4 N 1/N*1/2 << 1. Accordingly, we may treat N* and 4 as if they were semiclassical variables. From an analogy between the Hamiltonians for translation and for rotation, the following correspondences are allowed: N

2

m . 2- p -x 2 2m L2 = I ’2 -

H(translationa1 motion)

=

H(rotating motion)

54

-2i

where mx = p and I & = L are, respectively, the linear and angular momentum ( I : the moment of inertia). If the angular momentum is conserved and gives a constant L = h N * , then, from Poisson’s brackets of concern,

h4

=

dH

dN*

(x =

g)

are obtained. Since j , = e * N * , the average coupling energy of a JJ is obtained by integrating Eq. (K.4) with respect to the phase. Thus, A JJ

E = EJCOSY,EJ = --

e*

(K.lO)

233

(4)The DC Josephson current flows for a vanishing voltage across the junction, which means that if a current flows, finite amounts of superelectrons are accumulated in the feed-in layer, if the junction is an open circuit. This does not seem to occur. Actually, a forward current occurs on one site, whereas a backward current flows on the other site. This is possible only by forming a current loop as shown in Fig. 13.6. The magnetic flux produced by the loop current must be London quantized to give the least amount of @: (the flux quantum) and, accordingly, a series of upward and downward flux quanta appear alternately. (5) If a finite voltage Vo is applied across the junction, the above-mentioned series of alternating flux quanta move in the y-direction following the equation. (K.ll)

cdm,

where XJ = ( + : / 2 i ~ p J ~ d * ) ' / ~v,, = K is the relative dielectric constant of the insulator, p is the permittivity of the insulator, c is the velocity of light. The XJ is the Josephson penetration depth, which specifies how deep the phase difference penetrates from the edge of the junction. The above equation indicates that the phase including the spatially varying magnetic flux propagates as a wave in the insulating layer in the y-direction. If the wave is specified by a wavevector k,, then WJ

=

k,

=

e*Vo/ii = k,v 27rd*B

(K.12) (K.13)

a(;

This angular frequency is known to be what makes the current across the junction oscillate as an AC current under a constant voltage Vo, i.e., the AC Josephson effect. If Eq. ( K . l l ) is solved by linearizing as in y = yo y1 with 70 >> 71, the dispersion relation of the wave equation is obtained.

+

);(

w2 =

2

cos -yo

+ w;

(K.14)

with a plasma frequency of the JJ, w i = "'/A; = e*dJ~/27rn%h.This frequency is simply 1/m, a resonance (angular) frequency for a circuit with C = neo/d and L = @(;/JJ per unit area.

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Index

B - T diagram, 161 CR time constant, 85, 110 I - V characteristic, 78, 111, 113-114, 159, 168, 173, 231 k-scale normalization, 198 z-representation, 223 Abrikosov lattice, 107 Absorption coefficient, 131-134, 199-200, 207, 210 AC Josephson current, 168 effect, 166, 233 Acceptor, 43, 47, 121, 198 Accumulation, 66, 85 Aharonov-Bohm configuration, 87 effect, 89-91, 161, 165 gauge, 101 Airy function, 212 Al, 154 Alo.3Gao.7AS, 123 A1203, 154 AlGaAs, 105, 126 Andreev reflection, 176, 177 Anisotropy, 84 energy crystalline, 139 field perpendicular, 229 Anti ferromagnetic, 143 Atom, 16, 115-116 Atom-electronics, 182 Atomic wire, 115 Atomics, 182 Au, 90, 152-153 Avogadro number, 180 Ballistic transport, 107 Band bottom, 39, 40, 42-44, 57, 59-60, 71, 77, 121, 123, 125-126, 165 gap, 30, 62, 83, 126

offset, 58, 63-64 top, 39, 42-43, 57, 125, 165 width, 39 Bend resistance, 107 Bessel function, 71, 212 spherical, 73 Bias forward, 42 reverse, 42-43, 45, 47 Binding energy, 121-122, 125 Bloch electron, 29, 32, 68, 165, 209 function, 191 oscillation, 31 state, 29, 51, 57, 61, 66, 148, 188-190 theorem, 29, 36, 53, 65, 71 Bohr magneton, 151 radius, 65, 115 Boltzmann equation, 217 statistics, 43, 86 Born approximation, 207 Bose-Einstein condensate, 157 Bound state, 63, 122 Boundary condition, 14, 19, 27-29, 35, 39, 58, 66, 71, 73, 191-192, 194-195, 212 periodic, 150, 183, 184 B r a g condition, 16-18, 21, 52 Brillouin zone, 81-82, 85, 129 boundary, 30, 32, 34 C60r 75 Capacitance, 170 Carbon atom, 76 Carbon nanotube, 76 Channel, 48, 81, 93-95, 97-99, 113 Charge density wave, 66-67, 147 Cluster, 73, 75, 129, 139, 182 Co, 154 Coercive force, 154

236 CoFe, 154 Coherence, 13, 15, 19, 24-25, 52-53, 55, 58, 65-68, 98, 115 Coherence factor, 168 length, 97, 171, 173 Coherent metallic conduction, 95 Cohesion, 51 Collector, 77 Compensation, 42 Compound semiconductor, 192 Conductivity, 11, 55-56, 92, 95 tensor in a magnetic field, 217, 221 Confinement, 35, 45, 75 Constructive, 16, 54, 89-90 Cooper pair, 157, 162-163, 165, 176 Correlation, 75, 110 energy, 109 Coulomb attraction, 189 blockade, 75, 109, 111, 113-114 gap, 107 gauge, 207 interaction, 130, 134, 149 potential, 73, 158, 191, 202 repulsion, 109, 110 scattering, 45 Critical field lower, 158 upper, 158 Critical thickness, 140, 142, 227-229 Crystal momentum, 29 Cu, 66 CUOZ,161, 157, 170-171 Cyclotron frequency, 218-219, 221 motion, 37, 101, 107, 220 orbit, 101 Dangling arm, 174 bond, 115 DAS model, 67-68 DC conductivity, 213, 215 Josephson current, 231, 233 SQUID, 165, 168 De-passivated, 72, 115-116 Debye-Waller factor, 51 Decoherence, 26, 32, 51-52, 65, 180 Decomposer, 14, 18 Delocalization, 53 Delocalized state, 102, 104 Demagnetizing factor, 142 Density matrix, 213

Index

Density of states, 63, 149, 165, 180, 203, 209 local, 68 Density-functional approximation, 115 Dephasing, 26, 32, 52 Depletion, 66, 85 layer, 42-43, 45, 47-48 Destructive, 16, 89-90 Detector, 14, 52 Deterministic, 14, 24 Diagram technique, 92 Diamagnetic response, 88 Diamagnetism, 11, 161 Dielectric response complex, 207 Differential negative resistance, 78-79, 85, 174 Diffraction, 11, 16-17, 52 Diffusion, 56 Dimension d , 55 one, 12, 19, 30, 33-37, 71-72, 76, 9798, 100, 115, 129, 134, 147, 161, 182, 202 three, 11, 29, 33-37, 51, 53, 57, 61, 63, 65, 86, 121-122, 129-130, 133, 140, 182, 198-199, 201, 207, 219 two, 33-37, 45, 57-58, 63, 65, 71, 77, 86, 92, 95, 101, 10-104, 107, 121-122, 129, 132-133, 150, 158159, 168, 182, 200, 203, 217219, 223, 227 zero, 34, 73-75, 139, 182 Dimensionality, 33 Dimer, 67 Diode, 6 Dipole layer, 40, 43-44, 63-64, 84, 149 moment, 199 Disorder, 52 Donor, 43, 47, 121, 189-190, 198 DOS, 33-37, 45, 56 Double-well, 79 Down-sizing, 180 Duality, 14 Edge current, 100 Effective mass, 6, 11, 30-32, 58, 71-72, 77, 81-82, 121-122, 124-127, 194, 197 approximation, 65, 88

Index equation, 30, 36, 57-58, 61, 64, 71, 7374, 124, 187-190, 194 Eigenenergy, 73, 101, 151, 201, 224 Eigenstate, 27, 36, 51, 58, 74 Eigenvalue, 27, 202, 211 Eigenvector, 28 Einstein relation, 56 Electric conductivity, 214 Electron affinity, 44, 58, 62 Electron diffraction low-energy, 68 Electron microscope, 14-15, 25, 67 Electron-phonon, 55 Electronics, 6 Emitter, 77 Endothermal, 78 Energy conservation, 77-78 gap, 81, 163, 171, 232 Envelope function, 57-59, 61, 66, 188-189, 198-201 Ergodicity, 91 Esaki tunnel diode, 78, 80-81 Euler’s equation, 226 Exchange coupling effective, 145 stiffness, 226 Exchange-correlation energy, 145, 153 Exciton, 65, 121, 189 bound, 129-137, 200 unbound, 129-137, 200, 202-203 FDM, 27, 30, 150, 183-186 Fe, 66, 153 Fermi energy, 148 level, 11, 39-40, 42-43, 48, 78, 105, 107, 150, 163, 165, 171, 176 sea, 7 Fermi-Dirac statistics, 11, 92, 172, 214 Ferromagnetic, 143 FET, 48, 113 First-principle calculation, 147 Fluctuation-dissipation theorem, 91 Fluorescence, 14, 19 Flux quantum, 89 Fluxon-anti fluxon pair, 158-159 Folding zone, 129 Forward bias, 81 Friedel oscillation, 147 Fringe, 14, 24 Fullerene, 75

237 Ga, 72 Ga-bar, 115 GaAs, 44, 58, 83, 85, 97, 105, 121, 123 GaAs-AlGaAs, 44, 60, 107 Gap energy, 124 Gauge transformation, 88 Gaussian envelope function, 12 potential, 19 Geometrical resonance, 177 Green function, 92, 158, 227 Ground state condensate, 168 Gunn diode, 85, 86 Hall coefficient, 102, 104, 219 effect, 63, 102, 218-219 effect fractional quantum, 63, 105-106, 224 effect integral quantum, 63, 92, 102, 104 Harmonic oscillator, 207 potential, 75, 97, 151 Hartree potential, 211 Heat capacity, 139 Hermite polynomial, 36 High field domain, 85 Hole heavy, 125 Hopping conduction, 55 Hot electron effect, 85 Hydrogen atom, 61-62, 115, 121, 129, 189, 191, 197 Hydrogenic state, 202 Hyperfine interaction, 147 Hypergeometric function confluent, 198, 203 IC, 168 Impurity, 52, 64 potential, 64 InAlAs-InGaAs, 78 Incoherence, 19, 22, 24-26, 53, 58, 94 Index of refraction, 22 Inductance, 170 Inflection point, 31, 32 InGaAs-InGaAs, 79 Interface, 194 Interference, 11, 14-16, 24-25, 52, 58, 66, 88-90 fringe, 22 Interferometer, 19, 24

238 silicon, 18, 53 Young double-slit, 22-23 Inversion layer, 48, 173 Ion, 16 Ionization potential, 64 Irreversible, 24

Index Anderson, 53-54, 57, 87, 89-90 Localized regime strongly, 55 regime weakly, 55 Localized state, 97, 102 London penetration length, 159, 161 Long-range order, 68 LSI, 7, 180

Jastrow function, 107

JJ computer, 168 Josephson current, 175 penetration depth, 233 Junction, 39-48 hetero, 39, 43-44, 58, 62-64, 75, 97, 107 Josephson, 157, 161, 165-166, 168, 171, 173, 231 metal-metal, 42 metal-metal, 39-40 metal-semiconductor, 39, 45, 47-48 normal metal-SC, 39, 165 p n , 39, 42-43 SC-I-normal metal, 174 SC-I-SC tunnel, 165, 168 SC-N-SC, 172 SC-normal metal-SC, 172 Schottky, 45, 47-48 tunnel, 109-111, 113, 153-154, 163 von Klitzing constant, 93 Kosterlitz-Thouless transition, 158-159 Kubo formula, 91-92, 213-214, 217, 219 Kubo’s theory, 139 Laguerre polynomial, 65, 74, 197, 201-202 Landau electron, 103, 107, 224 level, 103 quantization, 36, 37 quantum number, 37 state, 223 subband, 102, 104-105, 107 Landauer formula, 93-94, 99 Landauer-Buttiker formula, 93 Laue diffraction, 17 LCAO, 191 van Leeuwen theorem, 86 Legendre polynomial, 204 Lithography electron beam, 72 UV, 72 X-ray, 72 Localization, 97, 104

Macroscopic, 5, 18, 25-26, 40, 52 quantum coherence, 25, 157 quantum phenomenon, 140 quantum state, 157 Macrowave, 157, 161-162, 165 Magnetic alignment antiferromagnetic, 143, 145, 147-153 3 alignment ferromagnetic, 143, 145, 147, 149-152 anisotropy energy, 142, 225-226 anisotropy perpendicular, 140 device, 180, 225 Magnetic domain Bloch, 140 closure, 140, 142 N6e1, 140 stripe, 140, 142, 225, 227 Magnetic fine particle, 139 length, 36, 101 susceptibility, 139 thin film, 140, 143, 182, 225 wire, 182, 225 Magneto-fingerprint, 96 Magnetoelectronics, 182 Magnetoresistance, 90, 109, 143, 146, 154 giant, 143, 145, 147 Magnetostatic energy, 225 Magnon, 93 Many-body state, 107 Matrix dynamics, 28 Matter-wave, 16, 25 Mat thiessen rule, 2 14-215 Maximum Josephson current, 166, 168, 170, 232 Maxwell equation, 209 Measurement theory, 14, 52 Meson, 5 Mesosphere, 5 Mg, 90 Microdisk, 74, 75 Micromagnetism, 140, 225 Microscopic, 5, 25-26, 52, 53

Index

coherence, 157 Miniband, 61-62, 127 Minigap, 61-62 Modulation doping, 45 Moire pattern, 151 Momentum conservation, 207 Monoatomic step, 66 MOS, 48, 81, 85, 93, 104, 113 MOS FET, 211 Mossbauer effect, 140 Mott-Ioffe-Regel limit, 95 Multilayer metallic, 147 Nanoscale device, 6 structure, 6 Nb-InAs-Nb SC transistor, 173 NbSez, 66 Neumann function, 212 Neutron, 16, 18, 21-22, 24, 53 Newton ring, 16 Ni, 154 Noise Johnson, 91 Nyquist, 91 thermal, 91 Non-volatile memory, 113 Normal metal-SC interface, 172, 176 Off resonance, 123-124 Ohmic, 91-92 Operator creation/annihilation, 207 Optical, 14 conductivity, 169, 209 Oscillator strength, 169, 198, 207 Pair potential, 171 annihilation, 110 Particle-wave, 6, 13-14, 27-28, 32 Partition function, 213 Passivated, 115 Pauli paramagnetism, 11 Percolation, 56, 113 Persistent current, 157, 161 Phase shifter, 19, 22-23 slippage, 162 Phonon, 51-53, 93, 180 Photocurrent, 128 Photoemission, 39 Photolithography, 72, 111 Photoluminescence, 122-124, 126, 129

239 Photon, 24, 51-53 Planck constant, 11 Plasma frequency, 170 Josephson, 233 Plasmon, 51-53 Plateau, 104 Point contact, 173 Poisson equation, 74 Polarizability complex, 210 Positron, 124 Power index, 123 Probabilistic, 14 Probability, 14, 30, 54, 66, 78, 93-94 amplitude, 13 Protective measurement, 66 Proximity effect, 171-172, 175 Quantization, 7, 11, 13, 28, 58, 84, 99, 123 conductance, 97 Landau, 88 London, 233 radiation field, 207 size, 35, 57, 59, 61, 81, 153, 180 Quantized, 58, 102, 180, 220 energy, 60, 63, 71, 134 magnetic flux, 168 Quantum box, 73 computer, 182 diffusion, 12 dot, 73-75, 113, 139 effect, 7, 11-12, 45, 63 fluctuation, 12-13, 232 number, 13, 33, 35, 97, 99, 131, 133135, 150, 201, 221, 224 number good, 29-30, 61, 71 number Landau, 221, 224 number magnetic, 65, 197, 200 point contact, 97, 99-100 resistance, 94 well, 44, 57, 60, 77, 124, 126, 128-129, 189 well model, 147 well multiple, 61-62, 121, 123 well single, 61, 121 wire, 71-72, 115-116, 129 Quasiparticle, 22, 93, 162-163, 165, 175 QWIDDLE, 124 Ramour orbital, 87

240 Reconstruction, 68 Rectifier, 43 Reflection, 93-94 Reflectivity, 128, 171 Reflector, 19 Refractive index complex, 130, 209 Relaxation time, 11, 102, 124, 170, 207, 213 Renormalization, 7 Resistivity, 11 Resonance, 123-124 Resonant tunneling device, 77-79, 81 RF SQUID, 165 Richardson’s formula, 107 RKKY-like oscillation, 153 RKKY interaction, 147 Rydberg energy, 121 energy effective, 197 Saturation field, 151 range, 42 SC-normal metal interface, 171 Scanning probe microscope, 72 Scattering, 22, 102 elastic, 24-25, 51-54, 57-58, 89, 90, 180 electron-electron, 52, 55 inelastic, 24-25, 51-53, 55, 57-58, 94, 180 Schottky barrier, 74 Screening Debye, 40 over-, 147 Thomas-Fermi, 40, 149 under-, 147 Selection rule, 129 Self-organization, 72, 115 Semiconductor memory, 225 model, 163 SEMPA, 153 Shell model, 75 Si, 68, 81-83, 114, 116, 129, 182 device, 7 porous, 129 technology, 7 Si-SiOz, 105 SiGe, 129 Single Cooper pair transistor, 162 electron transistor, 162 electron tunneling oscillation, 110, 113 Solenoid, 14, 88

Index Solid state device, 6, 53 Sommerfeld factor, 131-136 Spacer, 45 Spherical harmonic function, 65, 73, 204 Spin density wave, 147 Spin-SEM, 153 Spin-split-band model, 143, 145, 149, 153 Spinics, 182 Spintronics, 182 SQUID, 165 Standing wave, 66 Stark effect, 128 STM, 66, 182 Stokes theorem, 89 Sum rule, 169 Superantiferromagnetism, 139 Superconducting ring circuit, 162 thin wire, 161 transistor, 173 Superconductor, 154, 157 LSCO, 158, 170-171 BiSCCO, 158, 161 HgBCCO, 158 high-T,, 159 TlBCCO, 158 YBCO, 158 Supercurrent, 157, 163, 168, 174, 231 Superelectron, 169, 233 Superfluid, 157 Superlattice, 61-62, 77, 127-129, 143 metallic, 147 Superparamagnetism, 139 Superposer, 18-19 Superposition, 14, 17, 21, 54, 90 Surface, 65, 71-72 magnetism, 152 state, 66 TaS2, 66 Thouless number, 56 Tight-binding approximation, 191 method, 126 wavefunction, 193 Time-dependent Schrodinger equation, 185 Topological long-range order, 158 Transfer integral, 192 Transistor, 6 Transition allowed, 130-137, 199

Index band-to-band, 130-133, 135 forbidden, 130-137, 199 indirect, 129 probability, 121 Transmission, 93, 94 probability, 78 Trapping, 123, 124 Triangular potential, 211 Triode, 6 Tunneling, 61-62, 78, 93, 100, 124, 140, 154, 161-162, 165, 232 Giaever, 165, 168 Josephson, 165, 168 Two-dimensional electron gas, 97, 105, 173 Two-fluid model. 145 Uncertainty number-phase, 52 relation, 232 Universal conductance fluctuation, 95-96 Vacuum level, 44

241 potential, 39, 40, 57, 66 tube, 6 Valley, 81-82, 129 Variable-range hopping, 56 Vector potential, 88, 223 Velocity operator, 214 Virtual state, 61 Visibility, 19, 25 Wannier function, 187 Washboard potential, 162 Wavepacket, 12-13, 18-19, 21, 58, 185 Weak-link, 173-174 Which path, 24 Whittaker's function, 202 Wigner crystal, 107 Winding number, 161 Work function, 39, 148 Zeeman effect, 101 Zero-point energy, 13, 29 Zero-voltage current, 174, 176

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Index

Author Index

Abraham, E., 55 Abrikosov, A.A., 93 Aharonov, Y . , 66, 88 Al’tshuler, B.L., 90, 97 Ambegaokar, V., 232 Anderson, P.W., 158, 170 Ando, T., 43, 53, 58, 103, 127, 192 Andreev, A.F., 176 Aono, M., 115 Aronov, A.G., 90 Averin, D.V., 109

Chikazumi, S., 139 Chu, C.W., 158 Crommie, M.F., 66, 67 Cullity, B.D., 139 Cyrot, M., 158 Darwin, C.G., 75 David, J., 139 Delsing, P., 110 Dhez, P., 39 Dingle, R.C., 45 Dresselhaus, M.S., 75, 76 DuMond, J.W.M., 39 Duzer, van T., 172

Biittiker, M., 93 Baibich, M.N., 143 Baltenberger, W., 147 Ban, M., 159 Baranger, H.U., 108 Bardeen, J., 6, 232 Bastard, G., 43 Bedell, K., 158 Beenakker, C.W.J., 108, 174, 176 Ben-Jacob, E., 110 Benoit, A.D., 98 Bliek, L., 105 Bloch, F., 29 Blonder, G.E., 171 Bragg, W.L., 16 Brattain, W.H., 6 Brillouin, L., 81 Brown, W.F.Jr., 140 Bulaevskii, L.N., 161 Biittiker, M., 94

Geerligs, L.J., 110 Gennes, de P.G., 157, 173 Giaever, I., 164 Ginsberg, D.M., 158 Giordano, N., 161, 163 Griinberg, P., 143 Griffin, A., 176 Gunn, J.B., 85

Capasso, F., 43 Chang, L.L., 39, 77 Chelikowsky, J.R., 81, 82, 83

Hashizume, T., 115 Hathaway, K.B., 145, 152 Heisenberg, W., 28

Edwards, D.M., 145, 147 Elliott, R.J., 129, 131, 134, 2( ErdBlyi, A., 36 Esaki, L., 43, 61, 62, 64, 77, 7 Fock, V., 75 Fujiwara, H., 142, 143, 229 Fulton, T.A., 110

Author Index

244 Hekking, F., 171, 176 Hess, H.F., 66 Hilsum, C., 85 Himpsel, F.J., 143 Hitosugi, T., 115 Hobson, G.S., 85 Hong, K., 140 Ichiguchi, T., 159 Iijima, S., 76 Imry, Y., 94 Joachim, C., 115 Johnson, J.B., 92 Josephson, B.D., 165 Julliere, M., 154 Kiimmel, R., 176 KaczBr, J., 140 Kamimura, H., 158 Kampen, N.G.van, 5 Kasai, J., 124 Kasuya, T., 147 Katayama, Y., 85 Kawaji, S., 105 Kinoshita, J., 105 Kittel, C., 40 Klitzing, von K., 94, 105, 219 Koike, K., 152 Koshelev, A.E., 161 Kosterlitz, J.M., 158 Kramer. B., 53 Kroemer, H., 85 Kroto, H.W., 75 Kubo, R., 92, 93, 139, 219, 220 Landau, L.D., 36, 66, 93, 197 Landauer, R., 94 Laughlin, R.B., 106, 224 Lawrence, W .E., 161 Lee, P.A., 53, 96, 97 Leeuwen, virn J.H., 11, 86 Levy, P.M., 143 Likharev, K.K., 110 Loudon, R., 129, 202

Lut tinger , J .M., 30 Machida, S., 53 Marel, van der D., 169 Martin, S., 159 Mattis, D.C., 169 Mendez, E.E., 77 Meservey, R., 154 Mirbt, S., 147 Mishima, T., 123 Mizuta, H., 77 Moodera, J.S., 154 Moriguchi, S., 36, 71, 73, 74, 197, 212 Mott, N.F., 56 Murayama, Y., 17, 18, 53, 82, 103, 123, 124, 140, 148, 186, 225 NBel, L., 139 Nagaoka, Y., 53 Nakayama, M., 127, 128 Nakazato, K., 111 Namiki, M., 53 Nelson, D.R., 122, 161 Nishino, T., 176, 177 Nyquist, H., 92 Ogawa, T., 135, 136, 203 Onogi, T., 159 Ortega, J.E., 147 Parkin, S.S.P., 146, 147 Peierls, R., 11, 86 Rauch, H., 23 Ridley, B.K., 85 Romero, D.B., 170 Ruch, J.G., 83 Ruderman, M.A., 147 Ruediger, U., 140 Ruijsenaars, S.N.M., 88 Ryu, S., 161 Saito, N., 140, 142 Saito, S., 161 Schiff, L.I., 19, 66 Schilfgaarde, van M., 147

245 Schockley, W., 6 Schrieffer, J.R., 157, 165 Seahan, T.P., 158 Serena, P.A., 115 Seto, J., 172 Shaikhaidarov, R., 174, 176 Sharvin, D.Yu., 90 Shinada, M., 129, 132, 198, 200 Shinjo, T., 39 Slonczewski, J.C., 147, 154 Slough C.G., 66 Sollner, T.C.L.G., 77 Spain, R.J., 140, 225 Stormer, H.L., 106, 107 Stamp, P.C.E., 161 Stern, F., 65, 211 Stiles, M.D., 147 Stone, A.D., 97 Sugano, R., 161 Sugita, Y., 141, 229 Sze, S.M., 42, 45, 85 Tachiki, M., 161 Takagaki, Y., 108, 109 Takayanagi, H., 173, 176 Takayanagi, K., 67 Tamasaku, K., 170 Tanoue, T., 79 Tarucha, S., 75 Tatara, G., 140

Taylor, B.N., 94 Tedrow, P.M., 154 Thouless, D.J., 53, 56 Timp, G., 108 Tinkham, M., 157, 163, 168, 231 Toda, M., 93 Tonomura, A., 14 Tsu, R., 77 Tsuchiya, Y., 14 Tsui, D.C., 105 Uchida, S., 170 Uemura, Y., 217 Volkov, A.F., 174 Vonsovskii, S.V., 139 Washburn, S., 90, 91 Watanabe, S., 115, 116 Webb, R.A., 90, 97 Wees, van B.J., 98, 99, 100, 101 White, R.M., 139 Yamanouchi, C., 105 Yano, K., 113, 114 Yoshida, K., 147 Yoshihiro, K., 105 Ziman, J.M., 40, 56, 81, 130, 198, 208 Zubarev, D.N., 93