Abstract

This article gives a detailed review of our current theoretical understanding of the crossover from cooperative Cooper pairing to independent bound-state formation and Bose–Einstein condensation in Fermi systems with increasing attractive interactions.

Introduction

There are two well-known paradigms for understanding the phenomena of superconductivity and superfluidity:

1. (I) BCS theory [1], in which the normal state is a degenerate Fermi liquid that undergoes a pairing instability at a temperature Tc ≪ εf, the degeneracy scale. The formation of Cooper pairs and their condensation (macroscopic occupation of a single quantum state) both occur simultaneously at the transition temperature Tc.

2. (II) Bose–Einstein condensation (BEC) of bosons at a Tc of the order of their degeneracy temperature. At a fundamental level these bosons (for example, 4He or excitons) are invariably composite objects made up of an even number of fermions. The composite particles form at some very high temperature scale of the order of their dissociation temperature Tdissoc, and these “pre-formed” bosons then condense at the BEC Tc ≪ Tdissoc.

In most cases of experimental interest the system under consideration clearly falls into one category or the other. For instance, 3He is a Fermi superfluid described by (I) whereas 4He is a Bose superfluid (II). Essentially all cases of superconductivity in metals that we understand reasonably well are much closer to (I) than to (II), as evidenced by the existence of fermionic quasiparticles above Tc.

Abstract

This paper examines the meaning of the “phase” of a Bose condensate, and in particular the degree of validity of the often used analogy with the direction of magnetization of a ferromagnet. It focusses on two specific questions: (i) Under what circumstances is the relative phase of two condensates well defined? (ii) Would it be possible in principle to set up a “standard of phase”? In the most obvious sense, the answer to (ii) is concluded to be no.

As is well known, it is very fashionable nowadays to treat Bose condensation as a special case of the more general idea of spontaneously broken symmetry, which is ubiquitous in condensed-matter physics [1]. The standard account goes something like this: Just as in a magnetic material described by an isotropic Heisenberg model, the Hamiltonian is invariant under simultaneous rotation of all the spins, so in a Bose system described by the standard creation and annihilation operators ψ†(r), ψ(r) it is invariant under the global U(1) gauge transformation ψ(r) → ψ(r)eiϕ, ψ†(r) → ψ†(r)e−iϕ. Thus, at first sight, symmetry forbids either the expectation value < S > of the magnetization of the magnetic material, or the corresponding quantity < ψ > in the Bose system, to take a finite value.

Abstract

We review the theory of coherent pairing of excitons and biexcitons in two-dimensional and three-dimensional semiconductor structures.

Introduction

The coherent pairing of bosons, formally analogous to Cooper pairing of electrons in superconductors, and the coexistence of particle and pair Bose–Einstein condensation (BEC) was first studied in Refs. [1–4]. BEC of excitons in semiconductors was investigated using the approximation of coherent pairing of the electrons and holes [5]. Later on, the idea of coherent pairing of bosons was applied to excitons [6–8]. In Ref. [6], it was assumed that the boson pairs consist of two electron–hole pairs instead of two excitons, and it was suggested that the new collective state is a BEC even though the two-pair bound state does not exist. It was proved independently [7, 8] that in a system of bosons and excitons with an attractive pair interaction sufficient for biexciton formation, the coherent pairing of excitons with momenta k and −k coincides with the BEC of biexcitons with zero translational momentum.

The possibility of particle and pair BEC in a system of four species of excitons with different values of spin projections of the electron and hole was studied in Ref. [9]. It is known [10, 11] that the interaction of an exciton of either type with other exciton species is repulsive on the average, but that pairs of excitons with antiparallel spins of electrons and holes interact attractively, and biexcitons can be formed.

In practical terms, the enormous range of values of resistivity of solids is something which we take for granted. Every day, we happily touch the polymer sleeving or fitments surrounding conductors bearing potentially lethal currents at quite high voltages. Only when one is reminded that the difference of almost 30 orders of magnitude found between the resistivity of the noble metals and some synthetic polymers represents the largest variation of any physical parameter does this apparently mundane phenomenon suddenly appear intriguing. It is tempting to enquire whether the same physical process can be responsible for electrical conduction in all materials. Even if the same process extends over half the range it would be a remarkable achievement. Perhaps we would then appreciate why so much time and effort is devoted to measurements of electrical conductivity.

It is hard to conceive that, prior to the turn of the century, very little was known about the physics of solids. Some ideas on crystal structure had been anticipated from the morphology of natural and synthetic crystals but there existed little understanding of the electrical, thermal or magnetic properties of solids. Solid state physics is a twentieth century branch of science and as such deserves recognition as an important section of ‘modern physics’. As we shall see later, it was not until quantum mechanics was applied to the physics of solids that many real advances were made.

In this chapter we examine the way in which the simple model of an electron in a box can be applied to understand the optical properties of the alkali halides. We then proceed to consider the magnetic properties of electrons which are localized at specific lattice sites. With this discussion of the phenomena of diamagnetism and paramagnetism, we lead up to the next chapter on magnetic order where we see the breakdown of the independent electron approximation giving quite dramatic results.

Point defects in alkali halides

The band gaps of the alkali halides, such as NaCl or KCl, are high and they are found to be excellent insulators. Thus, in the pure and perfect state, the alkali halides are transparent to visible light. However, as a result of either irradiation with X-rays, heating in the vapour of the alkali metal or electrolysis at high temperature, it is possible to create large numbers of negative ion vacancies (Fig. 8.1). These defects correspond to unoccupied lattice sites which would normally contain a halide ion. If the lattice site was simply vacant, the site would appear to be positively charged due to the removal of negative charge on the halide ion. In order to preserve electrical neutrality, it is favourable for an electron to become trapped at this negative ion vacancy. Such an electron trapped at a negative ion vacancy is called an F or colour centre.

The years of the third decade of the present century were heady times. Old social orders were being swept away, monarchies becoming republics and the United States of America was emerging as a world power. In physics too, revolutions were taking place, none more potent than that which resulted in the emergence of quantum mechanics as a model for the behaviour of sub-atomic particles.

It was Newton who first suggested that light was particulate but during the nineteenth century this view had fallen into neglect. Indeed a number of experiments, for example the classic Young's slits experiment, demonstrated quite conclusively that light was a wave motion. How else could interference fringes occur? However, the experiments on the photoelectric effect demonstrated just as conclusively that light energy was carried in packets, or quanta, and that a continuous wave description was not applicable.

This ‘wave or particle’ dilemma was not unique to the behaviour of light. J. J. Thompson showed that cathode rays were charged, had a well defined mass and had all the properties expected of a beam of particles. Nevertheless, Davisson and Germer showed that diffraction of electrons could take place, an effect made visually much more dramatic by G. P. Thompson's transmission electron diffraction patterns. Nowadays, electron diffraction is used as a routine analytical tool in all advanced metallurgical and materials science laboratories (Fig. 2.1). Clearly new axioms were required.

Probably the most spectacular phenomenon associated with the breakdown of the independent electron approximation is that of superconductivity. In the superconducting state, the material loses all resistivity and becomes a perfect conductor. The discovery in 1986 of oxide materials which were superconducting at temperatures above that of the boiling point of nitrogen sparked an unprecedented surge of activity in the field which remains an area of high profile and popular interest.

The discovery of superconductivity

In 1908 Kammerlingh Onnes succeeded in liquefying helium and set about the task of studying the properties of metals at these extremely low temperatures. As we saw in Chapter 1, the resistivity of metals such as platinum fell to a small, non-zero value when extrapolated to T = 0. This residual resistivity fell with improvements in purity and thus Onnes studied mercury, which was the most pure metal available at that time. To the great surprise of Onnes and the whole scientific community, the resistivity fell monotonically until just above the boiling point of helium and then fell abruptly to zero. Fig. 10.1 shows an example of the superconducting phase transition in yttrium barium copper oxide, one of the high temperature superconducting oxides. Onnes was unable to measure precisely the transition width or whether the resistivity was genuinely zero. However, in 1963 File and Mills measured the decay of a persistent current set up in a superconducting ring using nuclear magnetic resonance as the probe.

One of the most remarkable developments of the last decade has been the growth of the semiconductor industry. The exploitation of the properties of semiconducting materials to build large logic arrays on a single piece of crystal has led to a dramatic increase in the processing power and memory capacity of small computers, with a simultaneous fall in price. Development of single chip microprocessors has revolutionized our mode of working in a whole range of fields from the factory floor to the office. In this chapter we will examine the basic physics associated with a number of devices, and as many devices rely on the properties of junctions between n- and p-type semiconductors a fairly detailed discussion of such junctions is given before individual devices are treated. However, because all devices require connection to metallic wires in order to join components together we will first examine the metal–semiconductor junction. As it turns out this proves to be an excellent introduction to the p–n junction as well as providing a glimpse into some of the not-so-obvious pitfalls associated with device manufacture.

Metal–semiconductor junctions

The Schottky barrier

Let us suppose that a piece of metal is brought into contact with a piece of n-type semiconductor. (Of course, in practice, the metal would be evaporated on the semiconductor as a thin film, or attached with a soldered connection in which an alloy is formed, but such a naive picture is useful to fix our ideas.)

Most textbooks on solid state physics begin the exposition from what might be called a ‘structural position’. Space and point groups are discussed, followed by consideration of the Bravais lattice. The reader is thus led on to elementary ideas about crystallography and the use of diffraction techniques for the solution of crystal structures. Having laid the foundation of how atoms and molecules order to form crystalline structures, electron motion in such periodic structures is treated and band theory developed. The free electron model is seen as an approximation of the more general band theory. In many, rather formal, ways this approach is very satisfying. It would seem obvious that in the first instance one must understand the structure of the material on which one is working before attempting to understand its other physical properties. However, in practice, it proves rather hard to teach solid state physics this way and to retain student enthusiasm in the early stages of the teaching of crystallography where one is dealing with rather difficult geometrical concepts and very little physics. There is a very real danger of making the introduction to the subject so unexciting that the inspiration is lost and students come to regard solid state physics as the ‘dull and dirty’ branch of their physics course. However, elementary quantum mechanics, including the one-dimensional solution of the time independent Schrödinger equation, is included quite early in many undergraduate courses and there is much attraction in illustrating at this early stage the important technological context of the apparently abstruse quantum mechanics.

At the beginning of the previous chapter, we reviewed some experimental data which could not be explained using the free electron model. While the Hall effect unequivocally indicates the breakdown of the free electron model, it does not provide direct evidence for the existence of energy bands. There are, however, a number of techniques which do provide a direct measure of the energy gaps and the density of states.

Optical techniques for band structure measurements

Infra-red absorption in semiconductors

The first technique is both the easiest to understand and the easiest to perform experimentally. If one looks at a piece of polished silicon or germanium, it has the appearance of a metal. However, if thinned to below a few micrometres in thickness a piece of silicon is translucent, having a red appearance. A certain amount of light is transmitted in the red end of the spectrum. If one goes further into the infra-red, we find that these semiconductors are transparent.

Monochromatic radiation can be obtained from the continuous spectrum of infra-red radiation emitted by a hot filament by use of a diffraction grating spectrometer. The intensity transmitted through the semiconductor is measured as a function of wavelength, and, as illustrated in Fig. 5.1, a very abrupt drop in transmission is observed at a frequency characteristic of the semiconductor. For germanium and other common semiconductors this ‘absorption edge’, as it is called, occurs at around 1–2 μm wavelength. This is in the near infra-red.