The original edition of this book grew out of our somewhat disorganized attempts to teach the physics and electronics of mesoscopic devices over the past decade. Fortunately, these evolved into a more consistent approach, and the book tried to balance experiments and theory in the current, at that time, understanding of mesoscopic physics. Whenever possible, we attempted to first introduce the important experimental results in this field followed by the relevant theoretical approaches. The focus of the book was on electronic transport in nanostructure systems, and therefore by necessity we omitted many important aspects of nanostructures such as their optical properties, or details of nanostructure fabrication. Due to length considerations, many germane topics related to transport itself did not receive full coverage, or were referred to only by reference. Also, due to the enormity of the literature related to this field, we did not include an exhaustive bibliography of nanostructure transport. Rather, we tried to refer the interested reader to comprehensive review articles and book chapters when possible.

The decision to do a second edition of this book was reached only after long and hard consideration and discussion among the authors. While the first edition was very successful, the world has changed significantly since its publication. The second edition would have to be revised extensively and considerable new material added. A decision to go ahead was made only after welcoming Jon Bird to the author's team.

As discussed in the previous chapter, there are two issues that distinguish transport in nanostructure systems from that in bulk systems. One is the granular or discrete nature of electronic charge, which evidences itself in single-electron charging phenomena (see Chapter 6). The second involves the preservation of phase coherence of the electron wave over short dimensions. Artificially confined structures are now routinely realized through advanced epitaxial growth and lithography techniques in which the relevant dimensions are smaller than the phase coherence length of charge carriers. We can distinguish two principal effects on the electronic motion depending on whether the carrier energy is less than or greater than the confining potential energy due to the artificial structure. In the former case, the electrons are generally described as bound in the direction normal to the confining potentials, which gives rise to quantization of the particle momentum and energy as discussed in Section 2.2. For such states, the envelope function of the carriers (within the effective mass approximation) is localized within the space defined by the classical turning points, and then decays away. Such decaying states are referred to as evanescent states and play a role in tunneling as discussed in Chapter 3. The time-dependent solution of the Schrodinger equation corresponds to oscillatory motion within the domain of the confining potential.

Nanostructures are generally regarded as ideal systems for the study of electronic transport. What does this simple statement mean?

First, consider transport in large, macroscopic systems. In bulk materials and devices, transport has been well described via the Boltzmann transport equation or similar kinetic equation approaches. The validity of this approach is based on the following set of assumptions: (i) scattering processes are local and occur at a single point in space; (ii) the scattering is instantaneous (local) in time; (iii) the scattering is very weak and the fields are low, such that these two quantities form separate perturbations on the equilibrium system; (iv) the time scale is such that only events that are slow compared to the mean free time between collisions are of interest. In short, one is dealing with structures in which the potentials vary slowly on both the spatial scale of the electron thermal wavelength (to be defined below) and the temporal scale of the scattering processes.

Since the late 1960s and early 1970s, researchers have observed quantum effects due to confinement of carriers at surfaces and interfaces, for example along the Si/SiO2 interface, or in heterostructure systems formed between lattice-matched semiconductors. In such systems, it is still possible to separate the motion of carriers parallel to the surface or interface, from the quantized motion perpendicular, and describe motion semiclassically in the unconstrained directions.

In the preceding chapters, and indeed in the subsequent chapters, most of the discussion is on semiconductors in which the Bloch theory of extended states prevails. There is another class of semiconductors that has received considerable attention over the past several decades, and that is disordered (or amorphous) semiconductors. Here, in the realm of nanostructures, we really do not want to discuss the entire field of amorphous semiconductors, and would generally ignore strongly disordered materials as well. However, recent experiments have shown the presence of a metal–insulator transition in quasi-two-dimensional systems. Consequently, one needs to understand the difference between localized (disordered) systems, weakly disordered systems, and the normal Bloch band picture of conductance.

Generally, in disordered (or, strongly localized) systems, the Boltzmann equation fails to describe transport adequately except under very special circumstances. Disordered materials can stem from several sources, ranging from amorphous materials to relatively good single crystals with very high doping concentrations. In particular, the latter exhibit a form of impurity-induced disorder when the concentration of the impurity reaches a significant fraction of the atomic concentration of the host lattice. This, in turns connects to weak localization which can also arise from impurity-induced coherence effects even when the concentration is not too high.

The focus of this chapter is a discussion of transport in quantum dots, which are quasi-zero-dimensional nanostructure systems whose electronic states are completely quantized. The confinement of carrier motion in these structures is imposed in all three spatial directions, resulting in a discrete spectrum of energy levels much the same as in an atom or molecule. We can therefore think of quantum dots as artificial atoms, which in principle can be engineered to have a particular energy level spectrum. As in atomic systems, the electronic states in quantum dots are sensitive to the presence of multiple electrons due to the Coulomb interaction between electrons. Rich transport phenomena are therefore observed in these structures, not only because of quantum confinement and the resonant structure associated with this confinement, but also due to the granular nature of electric charge.

In contrast to quantum wells and wires, quantum dots can be sufficiently small that the introduction of even a single electron is sufficient to dramatically change the transport properties due to the charging energy associated with this extra electron. One of the main consequences of this charging energy is to give rise to a Coulomb blockade of transport, where conductance oscillations are observed with the addition or subtraction of a single electron from a quantum dot, which we discuss in detail in Sections 6.1 and 6.2.

The discovery in 1980, by Klaus von Klitzing and his colleagues, of the integer quantum Hall effect (IQHE) may have done more than any other single event to stimulate experimental and theoretical interest in the electrical properties of low-dimensional systems. This phenomenon has now been observed in a variety of different material systems, and is manifest as the appearance of wide and precisely quantized plateaus in the Hall resistance (RH, or Hall resistivity ρxy), which therefore deviates strongly from the linear dependence on magnetic field that is expected classically. It is now understood that this high-magnetic-field phenomenon is associated with the formation of strongly quantized Landau levels in a two-dimensional electron gas (2DEG), under which conditions current flow is carried by ballistic edge states that are the quantum analog of classical skipping orbits (recallSection 2.5). Thus, the quantum Hall effect represents a remarkable manifestation of one-dimensional transport in a macroscopic system.

In this chapter, we begin by discussing the basic phenomenology of the (integer) quantum Hall effect, which, due to the extreme accuracy of its quantization, has now been adopted as an international standard for the definition of the ohm. We present an interpretation of this effect due to Büttiker, which begins from the concepts of the Landauer formula (Section 3.3) and explains the quantization by considering that edge states propagate ballistically, without dissipation, over the entire sample length.

The technological means now exists for approaching the fundamental limiting scales of solid-state electronics in which a single electron can, in principle, represent a single bit in an information flow through a device or circuit. The burgeoning field of single-electron tunneling (SET) effects, although currently operating at very low temperatures, has brought this consideration into the forefront. Indeed, the recent observations of SET effects in poly-Si structures at room temperature by a variety of authors has grabbed the attention of the semiconductor industry. While there remains considerable debate over whether the latter observations are really single-electron effects, the resulting behavior has important implications for future semiconductor electronics, regardless of the final interpretation of the physics involved. Indeed, the semiconductor industry is rapidly carrying out its own advance, with transistor gate lengths in the 20 nm range in production in 2009 (the so-called 35 nm node).

We pointed out in Chapter 1 that the semiconductor industry is following a linear scaling law that is expected to be fairly rigorous. With dimensions approaching 10 nm within another decade, there is a rapid search for possible new technologies that can supplement Si with the offer of improved performance. However, it is clear from a variety of considerations that the devices themselves may well not be the limitation on continued growth in device density within the integrated circuit chip.

In Chapter 2, we introduced the idea of low-dimensional systems arising from quantum confinement. Such confinement may be due to a heterojunction, an oxide–semiconductor interface, or simply a semiconductor–air interface (for example, in an etched quantum wire structure). When we look at transport parallel to such barriers, such as along the channel of an HEMT or MOSFET, or along the axis of a quantum wire, to a large extent we can employ the usual kinetic equation formalisms for transport and ignore the phase information of the particles. Quantum effects enter only through the description of the basis states arising from the confinement, and the quantum mechanical transition rates between these states are due to the scattering potential. This is not to say that quantum interference effects do not play a role in parallel transport. As we will see in the later chapters, several effects manifest themselves in parallel transport studies such as weak localization and universal conductance fluctuations, which at their origin have effects due to the coherent interaction of electrons.

In contrast to transport parallel to barriers, when particles traverse regions in which the medium is changing on length scales comparable to the phase coherence length of the particles, quantum interference is expected to be important. By “quantum interference” we mean the superposition of incident and reflected waves, which, in analogy to the electromagnetic case, leads to constructive and destructive interference.

This chapter treats quantum interference effects in disordered conductors. We have already discussed interference phenomena for the few-channels case in Section 1.6, and in Chapter 2 we have seen that the scattering approach becomes increasingly complicated for many transport channels. This is why we need special methods to treat many-channel diffusive conductors. Some of their properties can be understood if we replace the Hamiltonian or a scattering matrix by a random matrix. In Section 4.1 we discuss random matrices as mathematical objects and review the properties of their eigenvalues and eigenvectors. We then use random matrix theory to describe the properties of energy levels (Section 4.2) and transmission eigenvalues (Section 4.3).

To describe interference effects in a very broad class of conductors, we develop in Section 4.4 the methods to handle interference corrections for a circuit theory, which allows us to take account of all nanostructure details. We evaluate universal conductance fluctuations and the weak localization correction to conductance.

We also find that, in some situations, electrons are localized – confined to small regions of space. Section 4.5 discusses under which conditions this strong localization occurs and briefly outlines transport properties associated with this regime.

Random matrices

In this section, we describe disordered and chaotic systems. Disordered systems contain some defects that scatter electrons; chaotic ones do not have any defects, but scattering at the boundaries induces a very different motion for particles with very close energies. In these systems, the positions of energy levels and transmission eigenvalues are random and vary from sample to sample. We have already encountered such a situation when discussing the transport properties of diffusive conductors in Chapter 2.

This survival kit is intended to provide basic knowledge of superconductivity necessary for understanding the material of the book. We recommend Refs. [41], [56], and [183] to the reader who wishes to acquire a deeper understanding of superconductivity concepts.

Basic facts

• Below a certain temperature Tc (the superconducting transition temperature), the electrical resistance of some metals vanishes. In particular, the most commonly used superconductors are aluminum and niobium; alkali, noble, and magnetic metals never become superconducting. The highest transition temperature found among pure metals is about 9K for niobium; among “usual” superconducting compounds it is 39K for magnesium diboride. There are compounds with even higher transition temperatures, of over 100 K, known as high-temperature superconductors. They possess unusual symmetries, which lead to very uncommon physical properties. We do not consider them in this book.

• Superconductors are ideal diamagnets: weak magnetic fields do not penetrate the bulk of superconductors (Meissner effect). A high magnetic field destroys superconductivity. This critical field, Hc, can vary from 1G (approximately 10-4 T) for tungsten to 1980G for niobium.

• There is a narrow layer at the boundary of the superconductor where an external magnetic field decreases exponentially to zero value in the bulk. The characteristic length of this decay δp, known as the penetration depth, is temperature-dependent and diverges at the transition temperature proportionally to (Tc - T)-12;.

• The transition to the superconducting state is a second-order phase transition in zero magnetic field. It may become a first-order phase transition in finite magnetic fields.

• […]

In Chapter 3, we discussed charging effects in nanostructures: the most important manifestation of electron–electron interaction in quantum transport. In this chapter, we concentrate on another aspect of interactions which we have so far mentioned very briefly. It concerns interaction with slow modes. In most cases these slow modes are electromagnetic excitations in the nanostructure and nearby circuit that form an electromagnetic environment of the nanostructure. The effect of this interaction is threefold. First, it may affect and alter transport properties of the nanostructure. Secondly, it provides energy relaxation: transporting electrons and qubits may exchange their energy with the electromagnetic environment. Thirdly, the environment provides decoherence, inducing time-dependent phase shifts to wave functions of propagating electrons and qubits, thereby destroying the quantum coherence of corresponding states.

The physics discussed in this chapter is sometimes involved and various. It requires effort to see a “common denominator” in all effects mentioned. We choose to present material starting from the ideas of dissipative quantum mechanics: a branch of quantum mechanics developed in the 1970s and 1980s. For several concrete phenomena this presentation manner deviates from that commonly accepted in the literature. Although this may be inconvenient for the reader, we did this for the sake of the “big picture,” which allows us to see links and analogies between formally different phenomena.

The structure of the chapter is as follows. Sections 6.1 and 6.2 are introductory. In Section 6.1, we discuss electromagnetic excitations in linear circuits and the way to treat them quantum-mechanically. In Section 6.2 we review general ideas of dissipative quantum mechanics, which are not specific for quantum transport: the orthogonality catastrophe, shake-up, classification of environments.