Mesoscopic Electronics in Solid State Nanostructures
Thomas Heinzel
Mesoscopic Electronics in Solid State Nanostructures
Second, Completely Revised and Enlarged Edition
WILEY-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank Thomas Heinzel
Mesoscopic Electronics in Solid State Nanostructures
Second, Completely Revised and Enlarged Edition
WILEY-VCH Verlag GmbH & Co. KGaA 1807–2007 Knowledge for Generations
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The Author All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained Prof. Dr. Thomas Heinzel in these books, including this book, to be free of Dept. of Solid State Physics errors. Readers are advised to keep in mind that Heinrich-Heine-Universität Düsseldorf statements, data, illustrations, procedural details or [email protected] other items may inadvertently be inaccurate.
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To Carola and Alexander This Page Intentionally Left Blank VII
Contents
Preface XIII
1 Introduction 1 1.1 Preliminary remarks 1 1.2 Mesoscopic transport 2 1.2.1 Ballistic transport 3 1.2.2 The quantum Hall effect and Shubnikov–de Haas oscillations 5 1.2.3 Size quantization 7 1.2.4 Phase coherence 8 1.2.5 Single-electron tunneling and quantum dots 9 1.2.6 Superlattices 10 1.2.7 Spintronics 11 1.2.8 Samples, experimental techniques, and technological relevance 11
2 An update of solid state physics 15 2.1 Crystal structures 16 2.2 Electronic energy bands 18 2.3 Occupation of energy bands 27 2.3.1 The electronic density of states 27 2.3.2 Occupation probability and chemical potential 29 2.3.3 Intrinsic carrier concentration 29 2.3.4 Bloch waves and localized electrons 31 2.4 Envelope wave functions 32 2.5 Doping 36 2.6 Diffusive transport and the Boltzmann equation 40 2.6.1 The Boltzmann equation 41 2.6.2 The conductance predicted by the simplified Boltzmann equation 44 2.6.3 The magneto-resistivity tensor 46 2.6.4 Diffusion currents 47
Mesoscopic Electronics in Solid State Nanostructures. Thomas Heinzel Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40638-8 VIII Contents
2.7 Scattering mechanisms 48 2.8 Screening 50
3 Surfaces, interfaces, and layered devices 57 3.1 Electronic surface states 59 3.1.1 Surface states in one dimension 59 3.1.2 Surfaces of three-dimensional crystals 65 3.1.3 Band bending and Fermi level pinning 67 3.2 Semiconductor–metal interfaces 68 3.2.1 Band alignment and Schottky barriers 69 3.2.1.1 The Schottky model 72 3.2.1.2 The Schottky diode 73 3.2.2 Ohmic contacts 73 3.3 Semiconductor heterointerfaces 74 3.4 Field effect transistors and quantum wells 77 3.4.1 The silicon metal–oxide–semiconductor field effect transistor 77 3.4.1.1 The MOSFET and digital electronics 81 3.4.2 The Ga[Al]As high electron mobility transistor 84 3.4.3 Other types of layered devices 87 3.4.3.1 The AlSb–InAs–AlSb quantum well 87 3.4.3.2 Hole gas in Si–Si1 xGex–Si quantum wells 89 − 3.4.3.3 Organic FETs 89 3.4.4 Quantum confined carriers in comparison to bulk carriers 91
4 Experimental techniques 97 4.1 Sample preparation 97 4.1.1 Single crystal growth 98 4.1.2 Growth of layered structures 100 4.1.2.1 Metal organic chemical vapor deposition (MOCVD) 101 4.1.2.2 Molecular beam epitaxy (MBE) 101 4.1.3 Lateral patterning 107 4.1.3.1 Defining patterns in resists 107 4.1.3.2 Direct writing methods 110 4.1.3.3 Etching 112 4.1.4 Metallization 113 4.1.5 Bonding 115 4.2 Elements of cryogenics 116 4.2.1 Properties of liquid helium 117 4.2.1.1 Some properties of pure 4He 117 4.2.1.2 Some properties of pure 3He 120 4.2.1.3 The 3He/4He mixture 121 4.2.2 Helium cryostats 122 Contents IX
4.2.2.1 4He cryostats 122 4.2.2.2 3He cryostats 125 4.2.2.3 3He/4He dilution refrigerators 125 4.3 Electronic measurements on nanostructures 127 4.3.1 Sample holders 128 4.3.2 Application and detection of electronic signals 128 4.3.2.1 General considerations 128 4.3.2.2 Voltage and current sources 129 4.3.2.3 Signal detectors 130 4.3.2.4 Some important measurement setups 133
5 Important quantities in mesoscopic transport 139 5.1 Fermi wavelength 139 5.2 Elastic scattering times and lengths 139 5.3 Diffusion constant 140 5.4 Dephasing time and phase coherence length 143 5.5 Electron–electron scattering time 144 5.6 Thermal length 144 5.7 Localization length 145 5.8 Interaction parameter (or gas parameter) 145 5.9 Magnetic length and magnetic time 145
6 Magneto-transport properties of quantum films 147 6.1 Landau quantization 148 6.1.1 Two-dimensional electron gases in perpendicular magnetic fields 148 6.1.2 The chemical potential in strong magnetic fields 151 6.2 The quantum Hall effect 154 6.2.1 Phenomenology 154 6.2.2 Toward an explanation of the integer quantum Hall effect 156 6.2.3 The quantum Hall effect and three dimensions 161 6.3 Elementary analysis of Shubnikov–de Haas oscillations 162 6.4 Some examples of magneto-transport experiments 165 6.4.1 Quasi-two-dimensional electron gases 165 6.4.2 Mapping of the probability density 167 6.4.3 Displacement of the quantum Hall plateaux 167 6.5 Parallel magnetic fields 169
7 Quantum wires and quantum point contacts 177 7.1 Diffusive quantum wires 179 7.1.1 Basic properties 179 7.1.2 Boundary scattering 181 X Contents
7.2 Ballistic quantum wires 182 7.2.1 Phenomenology 182 7.2.2 Conductance quantization in QPCs 184 7.2.3 Magnetic field effects 191 7.2.4 The “0.7 structure” 195 7.2.5 Four-probe measurements on ballistic quantum wires 195 7.3 The Landauer–Büttiker formalism 198 7.3.1 Edge states 199 7.3.2 Edge channels 202 7.4 Further examples of quantum wires 204 7.4.1 Conductance quantization in conventional metals 204 7.4.2 Molecular wires 206 7.4.2.1 Carbon nanotubes 206 7.5 Quantum point contact circuits 210 7.5.1 Non-Ohmic behavior of QPCs in series 210 7.5.2 QPCs in parallel 212 7.6 Semiclassical limit: conductance of ballistic 2D systems 214 7.7 Concluding remarks 218
8 Electronic phase coherence 223 8.1 The Aharonov–Bohm effect in mesoscopic conductors 223 8.2 Weak localization 226 8.3 Universal conductance fluctuations 229 8.4 Phase coherence in ballistic 2DEGs 234 8.5 Resonant tunneling and s-matrices 236
9 Single-electron tunneling 247 9.1 The principle of Coulomb blockade 247 9.2 Basic single-electron tunneling circuits 250 9.2.1 Coulomb blockade at the double barrier 252 9.2.2 Current–voltage characteristics: The Coulomb staircase 255 9.2.3 The SET transistor 259 9.3 SET circuits with many islands: The single-electron pump 265
10 Quantum dots 273 10.1 Phenomenology of quantum dots 274 10.2 The constant interaction model 279 10.2.1 Quantum dots in intermediate magnetic fields 283 10.2.2 Quantum rings 285 10.3 Beyond the constant interaction model 287 10.3.1 Hund’s rules in quantum dots 287 10.3.2 Quantum dots in strong magnetic fields 287 Contents XI
10.3.3 The distribution of nearest-neighbor spacings 290 10.4 Shape of conductance resonances and I–V characteristics 294 10.5 Other types of quantum dots 297 10.5.1 Metal grains 298 10.5.2 Molecular quantum dots 299 10.6 Quantum dots and quantum computation 301
11 Mesoscopic superlattices 309 11.1 One-dimensional superlattices 310 11.2 Two-dimensional superlattices 312 11.2.1 Semiclassical effects 312 11.2.2 Quantum effects 318
12 Spintronics 323 12.1 Ferromagnetic sandwich structures 324 12.1.1 Tunneling magneto-resistance (TMR) and giant magneto-resistance (GMR) 324 12.1.2 Spin injection into a non-magnetic conductor 328 12.2 The Datta–Das spin field effect transistor 332 12.2.1 Concept of the Datta–Das transistor 332 12.2.2 Spin injection in semiconductors 333 12.2.2.1 Interface tunnel barriers 333 12.2.2.2 Ferromagnetic semiconductors 335 12.2.3 Gate-induced spin rotation: The Rashba effect 336 12.2.4 Spin relaxation and spin dephasing 339
A SI and cgs units 343
B Correlation and convolution 345 B.1 Fourier transformation 345 B.2 Convolutions 345 B.3 Correlation functions 347
C Capacitance matrix and electrostatic energy 349
D The transfer Hamiltonian 353
E Solutions to selected exercises 355 References 383
Index 393 This Page Intentionally Left Blank XIII
Preface
Exploring new orders of magnitude seems to be a part of human nature. This tendency is reflected even in the language. If a particular “megadeal” is really stunning, we feel more and more obliged to call it a “gigadeal”. It will not take long before the first “terastar” pops up! Each new generation of particle colliders, telescopes, or lasers, in fact of almost any scientific device or tech- nique you can think of, extends the accessible interval of a physical quantity. It is rather the rule than the exception that novel phenomena are discovered in such a process, some of which have been anticipated, others of which have not. Such an evolution seems to gain speed as soon as it gets classified as useful, besides pure scientific interest. It is of course debatable what exactly should be considered as “useful”. In any case, the miniaturization of electronic circuits during the past 50 years has been both scientifically rewarding and useful in our daily life. It is certainly not necessary to support this statement by examples. Scientifi- cally, however, the expression “microstructure” no longer fuels our imagina- tion. Meanwhile, the really exciting electronic circuits are nanostructures.As this name already suggests, nanostructures are objects with structures in the nanometer (nm) regime, which can mean just 1 nm, but also just a little less (in fact, in some cases, even somewhat more) than 1000 nm. The point of nanos- tructure science is that, within the last two decades, tremendous progress has been made in fabricating, controlling and understanding structures in this size regime. This is true for a wide variety of fields, including, for example, gene technology, crystal growth or microchip–excuseme,nanochip–fabrication. The resulting novel possibilities at hand are really breathtaking and get heav- ily explored by a significant fraction of the scientific community. In many cases, having control over the size and shape of an object in the nanometer regime means being able to control its chemical and/or physical properties. For example, the size of a semiconductor nanocluster determines its optical emission spectrum via size quantization; while from introductory solid state physics lectures, we have learned that this property is related to the bandgap, an intrinsic feature of the material. By now, the size reduction has actually
Mesoscopic Electronics in Solid State Nanostructures. Thomas Heinzel Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40638-8 XIV Preface
reached dimensions that are of interest for chemistry and biology, and there is a rapidly growing overlap. For example, you can think of an ionic chan- nel (they reside in cell membranes and control the electrochemical potential of cells by selectively transferring certain types of ions across the membrane) as a molecular transistor. On the other hand, nanostructure physicists have started to use DNA strands as wires and templates for nanocircuit fabrication. One branch of nanoscience deals with the electronic transport properties of solid state nanostructures. This field is often referred to as “mesoscopic transport”, an expression which indicates that the explanations for the ob- served transport phenomena must be sought somewhere in between micro- scopic and macroscopic models. The purpose of the present book is to intro- duce the reader to this topic from an experimental point of view. “The reader” is hereby assumed to be a student of physics or a related field, who has just finished introductory courses, in particular those on solid state physics and quantum mechanics, and plans to study nanoscience more closely. The reader is picked up at the knowledge he/she is likely to have, and a ride is given to ongoing research activities in the field of mesoscopic transport. Along the way, the elementary concepts and nanostructures are introduced. Selecting illustrative experiments for such a purpose is of course a highly subjective matter. The author has tried to pick particularly instructive exam- ples well known to him, which can furthermore be explained within the scope of this book. These examples have thus not necessarily been of high relevance for the evolution of the field, and I apologize for this shortcoming. It should be remarked that, in some of the figures, the original data have been redrawn for better reproduction quality and for a consistent presentation. The text contains a somewhat unusual feature, namely “papers” in the ex- ercises sections. Their purpose is to encourage the student to go through selected, usually quite recent, original publications. They are referred to by [P chapter.number] in the text. The student should be able to summarize the beef of such a paper in a 15 minute talk. The reader is strongly encouraged actually to do this. Besides collecting complementary information and getting exposed to different styles of presentation, the experience of being able to un- derstand the stuff written down not in a textbook but in an original paper can be highly motivating. I hope that after going through the text, the reader will not only be able to join with some confidence an experimental research group working in this field, but also feel well prepared for more advanced theoretical lectures on mesoscopic physics. This book, and with it the author, has enjoyed a lot of encouragement and support from many sides. I would like to thank particularly Hermann Grabert and Wolfgang Häusler, who read through parts of the manuscript and made many valuable comments. I am grateful to my colleagues Klaus Ensslin, Preface XV
Andreas Fuhrer, Miha Furlan, Ryan Held, Thomas Ihn, Silvia Lüscher, Jörg Rychen, and Volkmar Senz for countless fruitful discussions and stimulat- ing ideas. Special thanks go to those who supplied figures for this book, namely Günther Bauer, Mildred Dresselhaus, Andreas Fuhrer, Theo Geisel, Adam Hansen, Roland Ketzmerick, Anupam Madhukar, Andy Sachrajda, Elke Scheer, Jürgen Smet, and Horst Stormer. Furthermore, I thank my stu- dents, whose critical but always constructive comments have shaped and improved the presentation of the material. Last, but not least, I thank my wife Ulrike. With her tremendous energy and selfless support, she managed to supply the refuge I needed to transform my disorganized lecture notes into a book.
Thomas Heinzel, Düsseldorf, 2006 This Page Intentionally Left Blank 1
1 Introduction
1.1 Preliminary remarks
Over the past 30 years, the miniaturization of electronic devices has strongly influenced the technological evolution. Just think of the progress made in communication technology, or of the improvements in personal computers. For the money spent on a pocket calculator (which was barely able to carry out the four basic arithmetical operations) 30 years ago, you can buy today a desktop computer able to solve quite sophisticated numerical tasks, which in the 1970s could be tackled only by supercomputers. Moore’s law states that roughly every three years the number of transistors per microchip doubles. This law has been valid remarkably well in the past three decades, and it is expected to hold for some more years to come, although probably with a slightly reduced rate. This process, however, requires an ongoing reduction of the sizes of features, which up to now has essentially been achieved by using smaller wavelengths for optical lithography (the wavelength determines the resolution limit via diffraction). This is much more challenging than it may sound, for several reasons. First of all, a quick glance in an optics textbook reveals that the index of refraction of all common glasses diverges rapidly as the wavelength gets re- duced to about 200 nm. In addition, metals become transparent at their plasma frequencies, which typically fall in the same range of wavelengths. Hence, constructing both lenses and mirrors for the 100 nm regime is not that easy. Currently, the wavelengths used for lithography are of the order of 250 nm. Alternative lithographic techniques are able to pattern significantly smaller feature sizes. Although electron beam lithography is used in industry for some fabrication steps, it is too expensive for mass production of microchips. Novel patterning schemes, such as self-assembly or lithography with scan- ning probe microscopes, are presently the subject of extensive studies in re- search labs all around the world. It is, however, very unlikely that these tech- niques will replace optical lithography within the foreseeable future. Second, the patterns illuminated in an optical photoresist have to be trans- ferred into a structured device. Processes like developing the photoresist,
Mesoscopic Electronics in Solid State Nanostructures. Thomas Heinzel Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40638-8 2 1 Introduction
semiconductor etching, metal evaporation, alloying or selective doping must be carried out without losing the resolution. Furthermore, the devices must be connected to wires, and the inevitable heating generated during operation must be kept under control. Suppose that nanoscientists find adequate solutions to all these technolog- ical problems – there is in fact little doubt that they will. Then, however, another issue will become more and more important: all the above consid- erations implicitly assume that the components of a microchip can be scaled down arbitrarily without changes in their performance. This is not the case! Conventional transport theory makes presumptions about certain length and energy scales. For example, it is assumed that the electron mean free path is small compared to the feature size of the device, like the gate length of a transistor. The concept of resistivity is based on this assumption. Within the Boltzmann theory of electronic transport, it is assumed that the acceleration of the Fermi sphere by external electric fields gets compensated by many kinds of relaxation processes, which generate friction. In a stationary state, these friction forces balance the effects of the external field, and the resistivity of the sample can be defined. What happens for device sizes comparable to the mean free path, or to other relevant length scales? Well, we then enter the regime of mesoscopic transport. Novel effects occur, which may profoundly change the device performance. Introducing these effects is the major goal of this book. In the following sec- tion, we will look at the specific length and energy scales somewhat more closely and give examples for typical transport properties of samples in the mesoscopic regime.
1.2 Mesoscopic transport
What characterizes the mesoscopic regime? The answer depends on the par- ticular quantity under study. For the above example, the criterion would be that the device size must be comparable to or smaller than the electronic mean free path e. Other length scales are the de Broglie wavelength of the electrons that carry the current, which in all cases studied in this book are those elec- trons at the Fermi edge. Their de Broglie wavelength is the Fermi wavelength λF = h/√2m∗EF,wherem∗ denotes the effective electron mass (see Chap- ter 2 for details), and EF is the Fermi energy. If the feature sizes of the sample are comparable to λF, the wave character of the electrons will become essen- tial, and their kinetic energies will quantize. This fact is often referred to as size quantization, which is nothing but elementary quantum mechanics. If size quantization takes place in one spatial direction only, the electron system is 1.2 Mesoscopic transport 3 confined to two dimensions, and we speak of quantum films, which are the topic of Chapters 3 and 6. Suppose we confine our electrons in a second spa- tial direction. Their motion then becomes one-dimensional, and we have a quantum wire. The basic properties of quantum wires are discussed in Chap- ter 7. Finally, we can confine the electron in all directions, like in an atom. The resulting objects are known as quantum dots or artificial atoms (see Chap- ter 10). Another important length scale is the phase coherence length.Mostof us are aware of the diffraction pattern that electrons produce as they traverse a double slit setup in a vacuum tube. However, we usually do not think of electronic interference effects in solid state devices. Nevertheless, these effects do occur and become particularly important in devices with dimensions of the order of the phase coherence length. Phase coherent electrons are the topic of Chapter 8. Furthermore, it has turned out that the granular character of the electrons, which even in macroscopic samples plays an important role since it is responsible for shot noise, becomes increasingly important in nanostruc- tures. The point here is that the energy needed to charge a small island with a single electron may become significant. The resulting effects are summarized by single-electron tunneling, which is the topic of Chapter 9. Fig. 1.1 gives an overview of the most important mesoscopic regimes, and we continue with a brief survey of the phenomena to be discussed.
1.2.1 Ballistic transport
In order to enter the ballistic regime, the mean free path e, which roughly speaking is the average distance an electron travels before getting scattered,1 must be small compared to the relevant sample length L. At room tempera- ture, a major source of scattering is electron–phonon interaction, with a mean free path of the order of 20 nm. How are we supposed to describe electron transport through a wire with L <