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.

It is difficult nowadays to graduate from a department of natural sciences and not hear anything about quantum computing, most likely about the fascinating prospects of it. Quantum computing by its origin is a rather abstract discipline, a branch of math or information science. It has emerged from a persistent search for more efficient ways to process information when quantum mechanics made it to the scope of the search. Being an abstract discipline, quantum computing approximates a physical quantum system with a number of axioms and explores the consequences of these axioms, precisely as conventional math has been. These activities begun in the 1970s, and for a long time it was not obvious why quantum calculational schemes have to be any better than common computer algorithms. This extensive work was rewarded with a break-through in 1994, when Peter Shor discovered a remarkable quantum algorithm for the factorization of large numbers into prime ones. This sounds quite abstract, but many public key cryptosystems will become obsolete if Shor's algorithm is ever implemented in a practical quantum computer. Modern communication security is based on the fact that the factorization is a tough problem for a classical computer: it takes too long to crack a code. The proposed algorithm speeds up the factorization enormously. The discovery by Shor has brought quantum computing to the scientific and even public attention.

This progress has motivated a massive research effort towards the manipulation of individual quantum systems, the practical realization of quantum computing schemes being one of the most attractive goals. This chapter describes this goal, the quantum transport systems suitable for it, and the achievements made so far.

A paradox of solid state physics is that electrons in conductors are almost exclusively regarded as non-interacting particles, even though they do interact. This comes both from physical reasons and from the human need for convenience. The physical reason is that the interacting electrons form a ground state, and charged elementary excitations above the state – quasielectrons – do not interact provided their energies are sufficiently close to the Fermi surface. This makes a model of non-interacting electrons completely adequate for quasielectrons, at least in the low-energy limit. This allows a scattering approach to quantum transport that assumes the absence of interaction. The convenience model is that the physics of non-interacting particles is much easier to understand and apply. Besides, sticking to a convenient picture usually goes unpunished. In fact, in solid state physics there are only a few rather exotic examples where interaction effects really reign and the noninteracting approach produces obviously erroneous results. These cases are notoriously difficult to comprehend and to quantify; some effects revealed almost a century ago (for instance, Mott insulator transition) are still on the front-line of modern research.

In contrast to solid state physics, there is a very common regime in quantum transport where interaction effects are dominant: the Coulomb blockade regime, and here the scattering approach fails. However, in contrast to the situation in solid state systems, Coulomb blockade systems are usually even simpler than those of Chapters 1 and 2 where interaction does not play an important role. Due to this, one can quickly grasp the fascinating features of Coulomb blockade physics, and begin to design and apply Coulomb blockade circuits and devices.

It is an interesting intellectual game to compress an essence of a science, or a given scientific field, to a single sentence. For natural sciences in general, this sentence would probably read: Everything consists of atoms. This idea seems evident to us. We tend to forget that the idea is rather old: it was put forward in Ancient Greece by Leucippus and Democritus, and developed by Epicurus, more than 2000 years ago. For most of this time, the idea remained a theoretical suggestion. It was experimentally confirmed and established as a common point of view only about 150 years ago.

Those 150 years of research in atoms have recently brought about the field of nanoscience, aiming at establishing control and making useful things at the atomic scale. It represents the common effort of researchers with backgrounds in physics, chemistry, biology, material science, and engineering, and contains a significant technological component. It is technology that allows us to work at small spatial scales. The ultimate goal of nanoscience is to find means to build up useful artificial devices – nanostructures – atomby atom. The benefits and great prospects of this goal would be obvious even to Democritus and Epicurus.

This book is devoted to quantum transport, which is a distinct field of science. It is also a part of nanoscience. However, it is a very unusual part. If we try to play the same game of putting the essence of quantum transport into one sentence, it would read: It is not important whether a nanostructure consists of atoms.

We devoted Chapter 1 to a purely quantum-mechanical approach to electron transport: the scattering approach. Electrons were treated as quantum waves that propagate between reservoirs – the contact pads of a nanostructure. The waves experience scattering, and the transport properties are determined by the scattering matrix of these waves. As we have seen, this approach becomes progressively impractical with the increasing number of transport channels, and can rarely be applied for G ≫ GQ, where G is the conductance of the system.

A different starting point is well known from general physics, or, more simply, from general life experience, which is rather classical. In this context, a nanostructure is regarded as an element of an electric circuit, which conducts electric currents. If one makes a more complicated circuit by combining these elements, one does not have to involve quantum mechanics to figure out the result. Rather, one uses Ohm's law or, generally, Kirchhoff rules. The number of parameters required for this description is fewer than in the quantum-mechanical scattering approach. For example, the phase shifts of the scattering matrix do not matter.

In this chapter, we will bridge the gap between these opposite starting points. The first bridge is rather obvious: it is important to understand that these two opposite approaches do not contradict each other. In Section 2.1, we illustrate the difference and the link between the approaches with a comprehensive example of a double-junction nanostructure.

This book provides an introduction to the rapidly developing field of quantum transport. Quantum transport is an essential and intellectually challenging part of nanoscience; it comprises a major research and technological effort aimed at the control of matter and device fabrication at small spatial scales. The book is based on the master course that has been given by the authors at Delft University of Technology since 2002. Most of the material is at master student level (comparable to the first years of graduate studies in the USA). The book can be used as a textbook: it contains exercises and control questions. The program of the course, reading schemes, and education-related practical information can be found at our website www.hbar-transport.org.

We believe that the field is mature enough to have its concepts – the key principles that are equally important for theorists and for experimentalists – taught. We present at a comprehensive level a number of experiments that have laid the foundations of the field, skipping the details of the experimental techniques, however interesting and important they are. To draw an analogy with a modern course in electromagnetism, it will discuss the notions of electric and magnetic field rather than the techniques of coil winding and electric isolation.

We also intended to make the book useful for Ph.D. students and researchers, including experts in the field. We can liken the vast and diverse field of quantum transport to a mountain range with several high peaks, a number of smaller mountains in between, and many hills filling the space around the mountains.