semiconductor nanostructures quantum states and electronic transport feb 2010

SEMICONDUCTOR NANOSTRUCTURES
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Semiconductor Nanostructures
Quantum States and Electronic Transport
Thomas Ihn
Solid State Physics Laboratory, ETH Zurich
1
3
Great Clarendon Street, Oxford ox26dp
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ISBN 978–0–19–953442–5 (Hbk.)
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13579108642
Preface
This book is based on the lecture notes for the courses Semiconduc-
tor Nanostructures and Electronic Transport in Nanostructures that the
author gives regularly at the physics department of ETH Zurich. The
course is aimed at students in the fourth year who have already attended
the introductory lectures Physics I–IV, theoretical lectures in electrody-
namics, classical and quantum mechanics, and a course Introduction to
Solid State Physics. The course is also attended by PhD students within
their PhD programme, or by others working in the field of semiconductor
nanostructures or related scientific areas. Beyond the use of the material
contained in this book as the basis for lectures, it has become a popular
reference for researchers in a number of research groups at ETH work-

ing on related topics. This book is therefore primarily intended to be a
textbook for graduate students, PhD students and postdocs specializing
in this direction.
In order to acquire the knowledge about semiconductor nanostruc-
tures needed to understand current research, it is necessary to look at
a considerable number of aspects and subtopics. For example, we have
to answer questions like: which semiconducting materials are suitable
for creating nanostructures, which ones are actually used, and which
properties do these materials have? In addition, we have to look at
nanostructure processing techniques: how can nanostructures actually
be fabricated? A further topic is the historical development of this mod-
ern research field. We will have to find out how our topic is embedded
in the physical sciences and where we can find links to other branches
of physics. However, at the heart of the book will be the physical ef-
fects that occur in semiconductor nanostructures in general, and more
particularly on electronic transport phenomena.
Using this book as the basis for a course requires selection. It would be
impossible to cover all the presented topics in depth within the fourteen
weeks of a single semester given two hours per week. The author regards
the quantization of conductance, the Aharonov–Bohm effect, quantum
tunneling, the Coulomb blockade, and the quantum Hall effect as the five
fundamental transport phenomena of mesoscopic physics that need to be
covered. As a preparation, Drude transport theory and the Landauer–
B¨uttiker description of transport are essential fundamental concepts. All
this is based on some general knowledge of semiconductor physics, in-
cluding material aspects, fabrication, and elements of band structure.
This selection, leaving out a number of more specialized and involved
vi Preface
topics would be a solid foundation for a course aimed at fourth year
students.

The author has attempted to guide the reader to the forefront of cur-
rent scientific research and also to address some open scientific questions.
The choice and emphasis of certain topics do certainly follow the pref-
erence and scientific interest of the author and, as illustrations, his own
measurements were in some places given preference over those of other
research groups. Nevertheless the author has tried to keep the discus-
sions reasonably objective and to compile a basic survey that should
help the reader to seriously enter this field by doing his or her own
experimental work.
The author wishes to encourage the reader to use other sources of
information and understanding along with this book. Solving the ex-
ercises that are embedded in the chapters and discussing the solutions
with others is certainly helpful to deepen understanding. Research arti-
cles, some of which are referenced in the text, or books by other authors
may be consulted to gain further insight. You can use reference books,
standard textbooks, and the internet for additional information. Why
don’t you just start and type the term ‘semiconductor nanostructures’
into your favorite search engine!
Thomas Ihn,
Zurich, January 2009
Acknowledgements
I want to thank all the people who made their contribution to this book,
in one way or other. I thank my family for giving me the freedom to
work on this book, for their understanding and support. I thank all the
colleagues who contributed with their research to the material presented.
I thank my colleagues at ETH who encouraged me to tackle this project.
Many thanks go to all the students who stimulated the contents of the
book by their questions and comments, who found numerous mistakes,
and who convinced me that it was worth the effort by using my previous
lecture notes intensively.

I wish to acknowledge in particular those present and former col-
leagues at ETH Zurich who contributed unpublished data, drawings,
or other material for this book:
Andreas Baumgartner, Christophe Charpentier, Christoph Ellenberger,
Klaus Ensslin, Andreas Fuhrer, Urszula Gasser, Boris Grbiˇc, Johannes
G¨uttinger, Simon Gustavsson, Renaud Leturcq, Stephan Lindemann,
Johannes Majer, Francoise Molitor, Hansjakob Rusterholz, J¨org Rychen,
Roland Schleser, Silke Sch¨on, Volkmar Senz, Ivan Shorubalko, Martin
Sigrist, Christoph Stampfer, Tobias Vanˇcura.
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Contents
1Introduction 1
1.1 A short survey 1
1.2 What is a semiconductor? 5
1.3 Semiconducting materials 8
Further reading 9
Exercises 9
2 Semiconductor crystals 11
2.1 Crystal structure 11
2.2 Fabrication of crystals and wafers 11
2.2.1 Silicon 11
2.2.2 Germanium 13
2.2.3 Gallium arsenide 15
2.3 Layer by layer growth 15
2.3.1 Molecular beam epitaxy – MBE 15
2.3.2 Other methods 17
Further reading 18
Exercises 18
3 Band structure 19
3.1 Spinless and noninteracting electrons 19

3.2 Electron spin and the Zeeman hamiltonian 27
3.3 Spin–orbit interaction 29
3.4 Band structure of some semiconductors 31
3.5 Band structure near band extrema: k·p-theory 33
3.6 Spin–orbit interaction within k·p-theory 42
3.7 Thermal occupation of states 47
3.8 Measurements of the band structure 49
Further reading 51
Exercises 51
4 Envelope functions and effective mass approximation 53
4.1 Quantum mechanical motion in a parabolic band 53
4.2 Semiclassical equations of motion, electrons and holes 59
Further reading 60
Exercises 61
x Contents
5 Material aspects of heterostructures, doping, surfaces,
and gating 63
5.1 Band engineering 63
5.2 Doping, remote doping 72
5.3 Semiconductor surfaces 76
5.4 Metal electrodes on semiconductor surfaces 77
Further reading 82
Exercises 82
6 Fabrication of semiconductor nanostructures 83
6.1 Growth methods 83
6.2 Lateral patterning 88
Further reading 93
7 Electrostatics of semiconductor nanostructures 95
7.1 The electrostatic problem 95
7.2 Formal solution using Green’s function 96

7.3 Induced charges on gate electrodes 98
7.4 Total electrostatic energy 99
7.5 Simple model of a split-gate structure 100
Further reading 102
Exercises 102
8 Quantum mechanics in semiconductor nanostructures 103
8.1 General hamiltonian 103
8.2 Single-particle approximations for the many-particle
problem 106
Further reading 112
Exercises 113
9 Two-dimensional electron gases in heterostructures 115
9.1 Electrostatics of a GaAs/AlGaAs heterostructure 115
9.2 Electrochemical potentials and applied gate voltage 117
9.3 Capacitance between top gate and electron gas 118
9.4 Fang–Howard variational approach 118
9.5 Spatial potential fluctuations and the theory of screening 122
9.5.1 Spatial potential fluctuations 122
9.5.2 Linear static polarizability of the electron gas 123
9.5.3 Linear screening 125
9.5.4 Screening a single point charge 128
9.5.5 Mean amplitude of potential fluctuations 132
9.5.6 Nonlinear screening 134
9.6 Spin–orbit interaction 135
9.7 Summary of characteristic quantities 138
Further reading 140
Exercises 141
Contents xi
10 Diffusive classical transport in two-dimensional electron
gases 143

10.1 Ohm’s law and current density 143
10.2 Hall effect 145
10.3 Drude model with magnetic field 146
10.4 Sample geometries 150
10.5 Conductivity from Boltzmann’s equation 157
10.6 Scattering mechanisms 161
10.7 Quantum treatment of ionized impurity scattering 165
10.8 Einstein relation: conductivity and diffusion constant 169
10.9 Scattering time and cross-section 170
10.10 Conductivity and field effect in graphene 171
Further reading 173
Exercises 174
11 Ballistic electron transport in quantum point contacts 175
11.1 Experimental observation of conductance quantization 175
11.2 Current and conductance in an ideal quantum wire 177
11.3 Current and transmission: adiabatic approximation 182
11.4 Saddle point model for the quantum point contact 185
11.5 Conductance in the nonadiabatic case 186
11.6 Nonideal quantum point contact conductance 188
11.7 Self-consistent interaction effects 189
11.8 Diffusive limit: recovering the Drude conductivity 189
Further reading 192
Exercises 192
12 Tunneling transport through potential barriers 193
12.1 Tunneling through a single delta-barrier 193
12.2 Perturbative treatment of the tunneling coupling 195
12.3 Tunneling current in a noninteracting system 198
12.4 Transfer hamiltonian 200
Further reading 200
Exercises 200

13 Multiterminal systems 201
13.1 Generalization of conductance: conductance matrix 201
13.2 Conductance and transmission: Landauer–B¨uttiker
approach 202
13.3 Linear response: conductance and transmission 203
13.4 The transmission matrix 204
13.5 S-matrix and T -matrix 205
13.6 Time-reversal invariance and magnetic field 208
13.7 Four-terminal resistance 209
13.8 Ballistic transport experiments in open systems 212
Further reading 223
Exercises 223
xii Contents
14 Interference effects in nanostructures I 225
14.1 Double-slit interference 225
14.2 The Aharonov–Bohm phase 226
14.3 Aharonov–Bohm experiments 229
14.4 Berry’s phase and the adiabatic limit 235
14.5 Aharonov–Casher phase and spin–orbit interaction
induced phase effects 243
14.6 Experiments on spin–orbit interaction induced phase
effects in rings 249
14.7 Decoherence 250
14.7.1 Decoherence by entanglement with the
environment 250
14.7.2 Decoherence by motion in a fluctuating
environment 253
14.8 Conductance fluctuations in mesoscopic samples 256
Further reading 262
Exercises 262

15 Diffusive quantum transport 265
15.1 Weak localization effect 265
15.2 Decoherence in two dimensions at low temperatures 267
15.3 Temperature-dependence of the conductivity 268
15.4 Suppression of weak localization in a magnetic field 269
15.5 Validity range of the Drude–Boltzmann theory 272
15.6 Thouless energy 273
15.7 Scaling theory of localization 275
15.8 Length scales and their significance 279
15.9 Weak antilocalization and spin–orbit interaction 280
Further reading 286
Exercises 286
16 Magnetotransport in two-dimensional systems 287
16.1 Shubnikov–de Haas effect 287
16.1.1 Electron in a perpendicular magnetic field 288
16.1.2 Quantum treatment of E ×B-drift 292
16.1.3 Landau level broadening by scattering 293
16.1.4 Magnetocapacitance measurements 297
16.1.5 Oscillatory magnetoresistance and Hall resistance 298
16.2 Electron localization at high magnetic fields 301
16.3 The integer quantum Hall effect 305
16.3.1 Phenomenology of the quantum Hall effect 306
16.3.2 Bulk models for the quantum Hall effect 309
16.3.3 Models considering the sample edges 310
16.3.4 Landauer–B¨uttiker picture 311
16.3.5 Self-consistent screening in edge channels 318
16.3.6 Quantum Hall effect in graphene 320
16.4 Fractional quantum Hall effect 322
16.4.1 Experimental observation 322
Contents xiii

16.4.2 Laughlin’s theory 324
16.4.3 New quasiparticles: composite fermions 325
16.4.4 Composite fermions in higher Landau levels 327
16.4.5 Even denominator fractional quantum Hall states 328
16.4.6 Edge channel picture 329
16.5 The electronic Mach–Zehnder interferometer 330
Further reading 332
Exercises 333
17 Interaction effects in diffusive two-dimensional electron
transport 335
17.1 Influence of screening on the Drude conductivity 335
17.2 Quantum corrections of the Drude conductivity 338
Further reading 339
Exercises 339
18 Quantum dots 341
18.1 Coulomb-blockade effect in quantum dots 341
18.1.1 Phenomenology 341
18.1.2 Experiments demonstrating the quantization of
charge on the quantum dot 344
18.1.3 Energy scales 345
18.1.4 Qualitative description 349
18.2 Quantum dot states 354
18.2.1 Overview 354
18.2.2 Capacitance model 355
18.2.3 Approximations for the single-particle spectrum 359
18.2.4 Energy level spectroscopy in a perpendicular
magnetic field 360
18.2.5 Spectroscopy of states using gate-induced electric
fields 364
18.2.6 Spectroscopy of spin states in a parallel magnetic

field 365
18.2.7 Two electrons in a parabolic confinement:
quantum dot helium 366
18.2.8 Hartree and Hartree–Fock approximations 372
18.2.9 Constant interaction model 375
18.2.10 Configuration interaction, exact diagonalization 376
18.3 Electronic transport through quantum dots 377
18.3.1 Resonant tunneling 377
18.3.2 Sequential tunneling 387
18.3.3 Higher order tunneling processes: cotunneling 398
18.3.4 Tunneling with spin-flip: the Kondo effect in
quantum dots 403
Further reading 406
Exercises 407
xiv Contents
19 Coupled quantum dots 409
19.1 Capacitance model 410
19.2 Finite tunneling coupling 415
19.3 Spin excitations in two-electron double dots 417
19.3.1 The effect of the tunneling coupling 417
19.3.2 The effect of the hyperfine interaction 418
19.4 Electron transport 420
19.4.1 Two quantum dots connected in parallel 420
19.4.2 Two quantum dots connected in series 420
Further reading 425
Exercises 425
20 Electronic noise in semiconductor nanostructures 427
20.1 Classification of noise 427
20.2 Characterization of noise 428
20.3 Filtering and bandwidth limitation 431

20.4 Thermal noise 434
20.5 Shot noise 436
20.5.1 Shot noise of a vacuum tube 436
20.5.2 Landauer’s wave packet approach 438
20.5.3 Noise of a partially occupied monoenergetic stream
of fermions 440
20.5.4 Zero temperature shot noise with binomial
distribution 441
20.6 General expression for the noise in mesoscopic systems 442
20.7 Experiments on shot noise in mesoscopic systems 445
20.7.1 Shot noise in open mesoscopic systems 445
20.7.2 Shot noise and full counting statistics in quantum
dots 447
Further reading 450
Exercises 451
21 Interference effects in nanostructures II 453
21.1 The Fano effect 453
21.2 Measurements of the transmission phase 458
21.3 Controlled decoherence experiments 461
Further reading 467
Exercises 468
22 Quantum information processing 469
22.1 Classical information theory 470
22.1.1 Uncertainty and information 470
22.1.2 What is a classical bit? 473
22.1.3 Shannon entropy and data compression 475
22.1.4 Information processing: loss of information and
noise 475
22.1.5 Sampling theorem 484
22.1.6 Capacitance of a noisy communication channel 486

Contents xv
22.2 Thermodynamics and information 488
22.2.1 Information entropy and physical entropy 488
22.2.2 Energy dissipation during bit erasure: Landauer’s
principle 492
22.2.3 Boolean logic 493
22.2.4 Reversible logic operations 495
22.3 Brief survey of the theory of quantum information
processing 496
22.3.1 Quantum information theory: the basic idea 496
22.3.2 Qubits 498
22.3.3 Qubit operations 505
22.4 Implementing qubits and qubit operations 506
22.4.1 Free oscillations of a double quantum dot charge
qubit 507
22.4.2 Rabi oscillations of an excitonic qubit 509
22.4.3 Quantum dot spin-qubits 512
Further reading 519
Exercises 520
A Fourier transform and Fourier series 521
A.1 Fourier series of lattice periodic functions 521
A.2 Fourier transform 521
A.3 Fourier transform in two dimensions 521
B Extended Green’s theorem and Green’s function 523
B.1 Derivation of an extended version of Green’s theorem 523
B.2 Proof of the symmetry of Green’s functions 523
C The delta-function 525
References 527
Index 545
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Introduction
1
1.1 A short survey 1
1.2 What is a semiconductor? 5
1.3 Semiconducting materials 8
Further reading 9
Exercises 9
1.1 A short survey
Nanostructures in physics. How is the field of semiconductor nano-
structures embedded within more general topics which the reader may
already know from his or her general physics education? Figure 1.1 is
a graphical representation that may help. Most readers will have at-
tended a course in solid state physics covering its basics and some of its
important branches, such as magnetism, superconductivity, the physics
of organic materials, or metal physics. For this book, the relevant branch
of solid sate physics is semiconductor physics. Particular aspects of this
branch are materials, electrical transport properties of semiconductors
and their optical properties. Other aspects include modern semiconduc-
tor devices, such as diodes, transistors and field-effect transistors.
Miniaturization of electronic devices in industry and research.
We all use modern electronics every day, sometimes without being aware
of it. It has changed life on our planet during the past fifty years enor-
mously. It has formed an industry with remarkable economical success
and a tremendous influence on the world economy. We all take the avail-
ability of computers with year by year increasing computing power for
granted. The reason for this increase in computer power is, among other
things, the miniaturization of the electronic components allowing us to
place a steadily increasing amount of functionality within the same area
of a computer chip. The decreasing size of transistors also leads to de-
creasing switching times and higher clock frequencies. Today’s silicon-

based computer processors host millions of transistors. The smallest
transistors, fabricated nowadays in industrial research laboratories have
a gate length of only 10 nm.
Of course, this trend towards miniaturization of devices has also af-
fected semiconductor research at universities and research institutes all
over the world and has inspired physicists to perform novel experiments.
Solid State Physics
− magnetism
− superconductivity
− soft condensed matter,
organic materials
− semiconductor physics
− physics of metals
−
Semiconductor Physics
− materials
− electrical transport proper-
ties
− optical properties
− semiconductor devices,
miniaturisation
=> NANOSTRUCTURES
Solid State Physics
− magnetism
− superconductivity
− soft condensed matter,
organic materials
− semiconductor physics
− physics of metals
−

Semiconductor Physics
− materials
− electrical transport proper-
ties
− optical properties
− semiconductor devices,
miniaturisation
=> NANOSTRUCTURES
Fig. 1.1 Schematic representation
showing how the field of semiconductor
nanostructures has emerged as a special
topic of solid state physics.
On one hand they benefit from the industrial technological developments
which have established materials of unprecedented quality and innova-
tive processing techniques that can also be used in modern research.
On the other hand, physicists are interested in investigating and under-
standing the physical limits of scalability towards smaller and smaller
devices, and, eventually, to think about novel device concepts beyond
the established ones. Can we realize a transistor that switches with sin-
gle electrons? Are the essentially classical physical concepts that govern
2 Introduction
Fig. 1.2 The physics of semiconductor
nanostructures is related to many other
areas of physics.
Semiconductor
Nanostructures
Quantum mechanics
Quantum statistics
Metal physics
Electronics

Quantum information
processing
Optics
Atom physics
Low temperature
physics
Material research
and a lot more
Semiconductor
Nanostructures
Quantum mechanics
Quantum statistics
Metal physics
Electronics
Quantum information
processing
Optics
Atom physics
Low temperature
physics
Material research
and a lot more
the operation of current transistors still applicable for such novel de-
vices? Do we have to take quantum effects into account in such small
structures? Can we develop new operating principles for semiconductor
devices utilizing quantum effects? Can we use the spin of the electrons
as the basis for spintronic devices?
All these highly interesting questions have been the focus of research
in industry, research institutes, and universities for many years. In the
course of these endeavors the field of semiconductor nanostructures was

born around the mid 1980s. Experiments in this field utilize the techno-
logical achievements and the quality of materials in the field of semicon-
ductors for fabricating structures which are not necessarily smaller than
current transistors but which are designed and investigated under condi-
tions that allow quantum effects to dominate their properties. Necessary
experimental conditions are low temperatures, down to the millikelvin
regime, and magnetic fields up to a few tens of tesla. A number of
fundamental phenomena has been found, such as the quantization of
conductance, the quantum Hall effect, the Aharonov–Bohm effect and
the Coulomb-blockade effect. In contrast, quantum phenomena play
only a minor role in today’s commercial semiconductor devices.
Nanostructure research and other branches of physics. The
physics of semiconductor nanostructures has a lot in common with other
areas of physics. Figure 1.2 is an attempt to illustrate some of these links.
The relations with materials science and electronics have already been
mentioned above. Beyond that, modern semiconductor electronics is an
integrated part of measurement equipment that is being used for the
measurement of the physical phenomena. The physics of low temper-
atures is very important for experimental apparatus such as cryostats
which are necessary to reveal quantum phenomena in semiconductor
nanostructures. Quantum mechanics, electrodynamics and quantum
1.1 A short survey 3
1940
1950
1960 1970 1980 1990 2000 2010
Year
Minimum pattern size
1 mm
1 Mm
1 nm

10 nm
100 nm
10 Mm
100 Mm
10 mm
100 mm
100 pm
atoms
molecules
80386
80486
Pentium
8086
Itanium
first transistor
Xeon
silicon
technology
QHE
QPC
QD
AB
QD
qubit
2DEGs
SdH
Fig. 1.3 Development of the mini-
mum pattern sizes in computer proces-
sor chips over time. Data on Intel
processors were compiled from Intel

publications. The dashed line repre-
sents the prediction of Moore’s law,
i.e., an exponential decrease of pat-
tern size over time. Abbreviations
in the bottom part of the chart indi-
cate milestones in semiconductor nano-
structure research, namely, 2DEGs:
two-dimensional electron gases, SdH:
Shubnikov–de Haas effect, QHE: quan-
tum Hall effect, QPC: quantum point
contact (showing conductance quanti-
zation), QD: quantum dot (Coulomb
blockade), AB: Aharonov–Bohm effect,
QD qubit: quantum dot qubit. This
shows the close correlation between in-
dustrial developments and progress in
research.
statistics together form the theoretical basis for the description of the
observed effects. From metal physics we have inherited models for diffu-
sive electron transport such as the Drude model of electrical conduction.
Analogies with optics can be found, for example, in the description of
conductance quantization in which nanostructures act like waveguides
for electrons. We use the terms ‘modes’, ‘transmission’, and ‘reflection’
which are also used in optics. Some experiments truly involve electron
optics. The field of zero-dimensional structures, also called quantum
dots or artificial atoms, has strong overlap with atom physics. The fact
that transistors are used for classical information processing and the
novel opportunities that nanostructures offer have inspired researchers to
think about new quantum mechanical concepts for information process-
ing. As a result, there is currently a very fruitful competition between

different areas of physics for the realization of certain functional units
such as quantum bits (called qubits) and systems of qubits. The field of
semiconductor nanostructures participates intensely in this competition.
Reading this book you will certainly find many other relations with your
own previous knowledge and with other areas of physics.
History and Moore’s law. Historically, the invention of the transistor
by Shockley, Bardeen, and Brattain, at that time at the Bell laboratories,
was a milestone for the further development of the technological use
of semiconductors. The first pnp transistor was developed in 1949 by
Shockley. In principle it already worked like today’s bipolar transistors.
Since then miniaturization of semiconductor devices has made enormous
progress. The first transistors with a size of several millimeters had
already been scaled down by 1970 to structure sizes of about 10 µm.
Since then, miniaturization has progressed exponentially as predicted by
Moore’s law (see Fig. 1.3). With decreasing structure size the number
4 Introduction
of electrons participating in transistor switching decreases accordingly.
If Moore’s law continues to be valid, industry will reach structure sizes
of the order of the electron’s wavelength within the next decade. There
is no doubt that the importance of quantum effects will tend to increase
in such devices.
The size of semiconductor nanostructures. The world of nano-
structures starts below a characteristic length of about 1 µm and ends
at about 1 nm. Of course, these limits are not strict and not always will
all dimensions of a nanostructure be within this interval. For example, a
ring with a diameter of 5 µm and a thickness of 300 nm would certainly
still be called a nanostructure. The word ‘nano’ is Greek and means
‘dwarf’. Nanostructures are therefore ‘dwarf-structures’. They are fre-
quently also called mesoscopic systems. The word ‘meso’ is again Greek
and means ‘in between’, ‘in the middle’. This expresses the idea that

these structures are situated between the macroscopic and the micro-
scopic world. The special property of structures within this size range
is that typically a few length scales important for the physics of these
systems are of comparable magnitude. In semiconductor nanostructures
this could, for example, be the mean free path for electrons, the structure
size, and the phase-coherence length of the electrons.
Beyond the nanostructures lies the atomic world, starting with macro-
molecules with a size below a few nanometers. Carbon nanotubes,small
tubes of a few nanometers in diameter made of graphene sheets, are at
the boundary between nanostructures and macromolecules. They can
reach lengths of a few micrometers. Certain types of these tubes are
metallic, others semiconducting. Their interesting properties have made
them very popular in nanostructure research of recent years.
Electronic transport in nanostructures. The main focus of this
book is the physics of electron transport in semiconductor nanostruc-
tures including the arising fundamental quantum mechanical effects.
Figure 1.4 shows a few important examples belonging to this theme.
Measuring the electrical resistance, for example, using the four-terminal
measurement depicted schematically at the top left is the basic experi-
mental method. The quantum Hall effect (bottom left) is a phenomenon
that arises in two-dimensional electron gases. It is related to the con-
ductance quantization in a quantum point contact. Another effect that
arises in diffusive three-, two-, and one-dimensional electron gases is the
so-called weak localization effect (top middle). Its physical origin can be
found in the phase-coherent backscattering of electron waves in a spa-
tially fluctuating potential. This effect is related to the Aharonov–Bohm
effect in ring-like nanostructures (top right). A characteristic effect in
zero-dimensional structures, the quantum dots, is the Coulomb-blockade
effect. Its characteristic feature is the sharp resonances in the conduc-
tance as the gate voltage is continuously varied. These resonances are

related to the discrete energy levels and to the quantization of charge in
this many-electron droplet. While the summary of effects shown in Fig.
1.2 What is a semiconductor? 5
0.40.20-0.2
magnetic field (T)
-0.4
10
46
44
42
40
resistance (k
7)
38
36
34
-30 -20 -10 0 10
magnetic field (mT)
20 30
12
R
xx
k7
14
magnetic
field B
sample
16
I
U

H
0
2
4
6
8
0 1 2 3 4 5 6 7 8
0
0.4
0.8
1.2
1.6
magnetic field (T)
R
xx
(k7)
T = 100 mK
R
xy
(k7)
U
-0.6 -0.55 -0.5 -0.45 -0.4
0
0.1
0.2
0.3
0.4
plunger gate voltage (V)
conductance (
e

2
/h)
1 Mm
(a) (b)
(c) (d)
(e)
Fig. 1.4 Summary of important quan-
tum transport phenomena in semicon-
ductor nanostructures. (a) Schematic
drawing of a four-terminal resistance
measurement. (b) Weak localization
effect in a diffusive two-dimensional
electron gas, which is related to the
AharonovBohm effect shown in (c). (c)
Aharonov-Bohm effect in a quantum
ring structure. (d) The longitudinal-
and the Hall-resistivity of a two-
dimensional electron gas in the quan-
tum Hall regime. (e) Conductance of a
quantum dot structure in the Coulomb-
blockade regime.
1.4 cannot be complete, it shows the rich variety of transport phenom-
ena which makes the field of semiconductor nanostructures particularly
attractive.
1.2 What is a semiconductor?
The term ‘semiconductor’ denotes a certain class of solid materials.
It suggests that the electrical conductivity is a criterion for deciding
whether a certain material belongs to this class. We will see, however,
that quantum theory provides us with an adequate description of the
band structure of solids and thereby gives a more robust criterion for

the distinction between semiconductors and other material classes.
Resistivity and conductivity. The electrical conductivity of solid
materials varies over many orders of magnitude. A simple measure-
6 Introduction
ment quantity for the determination of the conductivity is the electrical
resistance R which will be more thoroughly introduced in section 10.1 of
this book. If we consider a block of material with length L and cross-
sectional area A, we expect the resistance to depend on the actual values,
i.e., on the geometry. By defining the (specific) resistivity
ρ = R
A
L
we obtain a geometry-independent quantity which takes on the same
value for samples of different geometries made from the same material.
The resistivity is therefore a suitable quantity for the electrical charac-
terization of the material. The (specific) conductivity σ is the inverse of
the resistivity, i.e.,
σ = ρ
−1
.
Empirically we can say that metals have large conductivities, and insu-
lators small, while semiconductors are somewhere in between. Typical
numbers are shown in Table 1.1.
Table 1.1 Typical resistivities of
materials at room temperature.
Material ρ (Ωcm)
Insulators ∼ 10
14
Macor (ceramic)
SiO

2
(quartz)
Al
2
O
3
(sapphire)
Semiconductors 10
−2
− 10
9
Metals ∼ 2 × 10
−6
Cu 1.7 ×10
−6
Al 2.6 ×10
−6
Au 2.2 × 10
−6
Temperature dependence of the resistance. The temperature de-
pendence of the electrical resistance is a good method for distinguishing
metals, semiconductors and insulators.
The specific resistivity of metals depends weakly and linearly on tem-
perature. When a metal is cooled down from room temperature, electron–
phonon scattering, i.e., the interaction of electrons with lattice vibra-
tions, loses importance and the resistance goes down [see Fig. 1.5(a)].
At very low temperatures T , the so-called Bloch–Gr¨uneisen regime is
reached, where the resistivity shows a T
5
-dependence and goes to a

constant value for T → 0. This value is determined by the purity of,
and number of defects in, the involved material. In some metals this
‘standard’ low-temperature behavior is strongly changed, for example,
by the appearance of superconductivity, or by Kondo-scattering (where
magnetic impurities are present).
In contrast, semiconductors and insulators show an exponential de-
pendence of resistivity on temperature. The resistance of a pure high-
quality semiconductor increases with decreasing temperature and di-
verges for T → 0 [cf., Fig. 1.5(b)]. The exact behavior of the tempera-
ture dependence of resistivity depends, as in metals, on the purity and
on the number of lattice defects.
Band structure and optical properties. A very fundamental prop-
erty that semiconductors share with insulators is their band structure.
In both classes of materials, the valence band is (at zero temperature)
completely filled with electrons whereas the conduction band is com-
pletely empty. A band gap E
g
separates the conduction band from the
valence band [see Fig. 1.6(a)]. The Fermi level E
F
is in the middle of
the band gap.
1.2 What is a semiconductor? 7
resistivity
temperature
resistivity
temperature
(a)
(b)
absorption

energy
absorption
energy
(c)
(d)
E
g
Fig. 1.5 Left: Characteristic tem-
perature dependence of the resistivity
(a) of a metal, (b) of a semiconduc-
tor. Right: Characteristic optical ab-
sorption as a function of photon energy
(c) of a metal, (d) of a semiconductor.
This property distinguishes semiconductors and insulators from met-
als, in which a band gap may exist, but the conduction band is partially
filled with electrons up to the Fermi energy E
F
and the lowest electronic
excitations have an arbitrarily small energy cost [Fig. 1.6(b)].
The presence of a band gap in a material can be probed by optical
transmission, absorption, or reflection measurements. Roughly speak-
ing, semiconductors are transparent for light of energy below the band
gap, and there is very little absorption. As depicted in Fig. 1.5(d), at
the energy of the band gap there is an absorption edge beyond which
the absorption increases dramatically. In contrast, metals show a finite
absorption at arbitrarily small energies due to the free electrons in the
conduction band [Fig. 1.5(c)].
Semiconductors can be distinguished from insulators only by the size
of their band gap. Typical gaps in semiconductors are between zero and
3 eV. However, this range should not be seen as a strict definition of

semiconductors, because, depending on the context, even materials with
larger band gaps are often called semiconductors in the literature. The
band gaps of a selection of semiconductors are tabulated in Table 1.2.
Table 1.2 Band gaps (in eV) of se-
lected semiconductors.
Si Ge GaAs AlAs InAs
1.1 0.7 1.5 2.2 0.4
Doping of semiconductors. A key reason why semiconductors are
technologically so important is the possibility of changing their electronic
properties enormously by incorporating very small amounts of certain
atoms that differ in the number of valence electrons from those found
in the pure crystal. This process is called doping. It can, for example,
lead to an extreme enhancement of the conductivity. Tailored doping
profiles in semiconductors lead to the particular properties utilized in
semiconductor diodes for rectifying currents, or in bipolar transistors
for amplifying and switching.
8 Introduction
Fig. 1.6 Schematic representation of
band structure within the first Bril-
louin zone, i.e., up to wave vector π/a,
with a being the lattice constant. Gray
areas represent energy bands in which
allowed states (dispersion curves) ex-
ist. States are occupied up to the Fermi
level E
F
as indicated by thick disper-
sion curves. (a) In insulators and semi-
conductors, all conduction band states
are unoccupied at zero temperature and

E
F
lies in the energy gap. (b) In metals
E
F
lies in the conduction band and the
conduction band is partially occupied
resulting in finite conductivity.
Energy
Wave vector
valence band
conduction band
E
g
π/a
Energy
Wave vector
valence band
conduction band
E
g
π/a
E
F
E
F
(a) (b)
Energy
Wave vector
valence band

conduction band
E
g
π/a
Energy
Wave vector
valence band
conduction band
E
g
π/a
E
F
E
F
(a) (b)
1.3 Semiconducting materials
Semiconducting materials are numerous and versatile. We distinguish
elementary and compound semiconductors.
Elementary semiconductors. Silicon (Si) and germanium (Ge), phos-
phorous (P), sulfur (S), selenium (Se), and tellurium (Te) are elementary
semiconductors. Silicon is of utmost importance for the semiconductor
industry. Certain modifications of carbon (C
60
, nanotubes, graphene)
can be called semiconductors.
Compound semiconductors. Compound semiconductors are classi-
fied according to the group of their constituents in the periodic table of
elements (see Fig. 1.7). Gallium arsenide (GaAs), aluminium arsenide
(AlAs), indium arsenide (InAs), indium antimonide (InSb), gallium an-

timonide (GaSb), gallium phosphide (GaP), gallium nitride (GaN), alu-
minium antimonide (AlSb), and indium phosphide (InP), for example,
all belong to the so-called III-V semiconductors. In addition, there
are II-VI semiconductors, such as zinc sulfide (ZnS), zinc selenide
(ZnSe) and cadmium telluride (CdTe), III-VI compounds,suchas
gallium sulfide (GaS) and indium selenide (InSe), as well as IV-VI
compounds, such as lead sulfide (PbS), lead telluride (PbTe), lead
selenide (PbSe), germanium telluride (GeTe), tin selenide (SnSe), and
tin telluride (SnTe). Among the more exotic semiconductor materials
there are, for example, the copper oxides CuO and Cu
2
O (cuprite), ZnO
(zinc oxide), and PbS (lead sulfide, galena). Also of interest are organic
semiconductors such as polyacetylene (CH
2
)
n
or anthracene (C
14
H
10
).