The 1980s were a very exciting time for mesoscopic physics characterized by a fruitful interplay between theory and experiment. What emerged in the process is a conceptual framework for describing current flow on length scales shorter than a mean free path. This conceptual framework is what we have tried to convey in this book. The activity in this field has expanded so much over the last few years that we have inevitably missed many interesting topics, such as persistent currents in normal metal rings, quantum chaos in microstructures, etc.

The development of the field is far from complete. So far both the theoretical and the experimental work has been almost entirely in the area of steady-state transport and many basic concepts remain to be clarified in the area of time-varying current flow as well as current fluctuations. Another emerging direction seems to be the study of mesoscopic conductors involving superconducting components. Finally, as we study current flow in smaller and smaller structures it seems clear that electron–electron interactions will play an increasingly significant role. As a result it will be necessary to go beyond the one-particle picture that is generally used in mesoscopic physics. Single-electron tunneling is a good example of this and it is likely that there will be many more developments involving current flow in strongly correlated systems.

It is well-known that the conductance (G) of a rectangular two-dimensional conductor is directly proportional to its width (W) and inversely proportional to its length (L); that is,

G = σW/L

where the conductivity a is a material property of the sample independent of its dimensions. How small can we make the dimensions (W and/or L) before this ohmic behavior breaks down? This question has intrigued scientists for a long time. During the 1980s it became possible to fabricate small conductors and explore this question experimentally, leading to significant progress in our understanding of the meaning of resistance at the microscopic level. What emerged in the process is a conceptual framework for describing current flow on length scales shorter than a mean free path. We believe that these concepts should be useful to a broad spectrum of scientists and engineers. This book represents an attempt to present these developments in a form accessible to graduate students and to non-specialists.

Small conductors whose dimensions are intermediate between the microscopic and the macroscopic are called mesoscopic. They are much larger than microscopic objects like atoms, but not large enough to be ‘ohmic’. A conductor usually shows ohmic behavior if its dimensions are much larger than each of three characteristic length scales: (1) the de Broglie wavelength, which is related to the kinetic energy of the electrons, (2) the mean free path, which is the distance that an electron travels before its initial momentum is destroyed and (3) the phase-relaxation length, which is the distance that an electron travels before its initial phase is destroyed.

Our purpose in this chapter is to describe an approach (often referred to as the Landauer approach) that has proved to be very useful in describing mesoscopic transport. In this approach, the current through a conductor is expressed in terms of the probability that an electron can transmit through it. The earliest application of current formulas of this type was in the calculation of the current-voltage characteristics of tunneling junctions where the transmission probability is usually much less than unity (see J. Frenkel (1930), Phys. Rev., 36, 1604 or W. Ehrenberg and H. Honl (1931), Z. Phys., 68, 289). Landauer [2.1] related the linear response conductance to the transmission probability and drew attention to the subtle questions that arise when we apply this relation to conductors having transmission probabilities close to unity. For example, if we impress a voltage across two contacts to a ballistic conductor (that is, one having a transmission probability of unity) the current is finite indicating that the resistance is not zero. But can a ballistic conductor have any resistance? If not, where does this resistance come from? These questions were clarified by Imry [2.2], enlarging upon earlier notions due to Engquist and Anderson [2.3]. Büttiker extended the approach to describe multi-terminal measurements in magnetic fields and this formulation (generally referred to as the Landauer–Büttiker formalism) has been widely used in the interpretation of mesoscopic experiments.

In Section 2.6 we briefly studied the effects of electron scattering on resonant tunnelling which are inevitable in a real system operating at room temperature. The phenomenological Breit–Wigner formula was introduced to describe the incoherent aspect of the electron tunnelling which in general results in a broadening of the transmission peak and thus degraded current P/V ratios in RTDs. In this chapter we look in more detail at various scattering processes, both elastic and inelastic, which have been of great interest not only from a quantum transport physics point of view but also because of the possibility of controlling and even engineering these interactions in semiconductor microstructures. The inelastic longitudinal–optical (LO) phonon scattering, introduced in the preceding chapter, is the most influential process, with Г–X-intervalley scattering and impurity scattering also affecting the resonant tunnelling electrons. Section 3.1 describes the dominant electron–LO-phonon interactions. Both theoretical and experimental studies of a postresonant current peak are presented, which provide much information about the electron–phonon interactions in the quantum well. Section 3.2 then discusses the effects of the upper X-valley which become more significant in AlxGa1−xAs/GaAs systems with an Al mole fraction, x, higher than 0.45 since the energy of the X-valley then becomes lower than that of the Г-valley. Finally, in Section 3.3, we study elastic impurity scattering, which may be caused by residual background impurities or those diffused from the heavily doped contact regions.

One of the most significant discoveries of the 1980s is the quantum Hall effect (see K. von Klitzing, G. Dorda and M. Pepper (1980), Phys. Rev. Lett., 45, 494). Normally in solid state experiments, scattering processes introduce enough uncertainty that most results have an ‘error bar’ of plus or minus several per cent. For example, the conductance of a ballistic conductor has been shown (see Fig. 2.1.2) to be quantized in units of (h/2e2). But this is true as long as we are not bothered by deviations of a few per cent, since real conductors are usually not precisely ballistic. On the other hand, at high magnetic fields the Hall resistance has been observed to be quantized in units of (h/2e2) with an accuracy that is specified in parts per million. Indeed the accuracy of the quantum Hall effect is so impressive that the National Institute of Standards and Technology is interested in utilizing it as a resistance standard.

This impressive accuracy arises from the near complete suppression of momentum relaxation processes in the quantum Hall regime resulting in a truly ballistic conductor of incredibly high quality. Mean free paths of several millimeters have been observed. These unusually long mean free paths do not arise from any unusual purity of the samples. They arise because, at high magnetic fields, the electronic states carrying current in one direction are localized on one side of the sample while those carrying current in the other direction are localized on the other side of the sample. Due to the formation of this ‘divided highway’ there is hardly any overlap between the two groups of states and backscattering cannot take place even though impurities are present.

Introduction

The conventional media of magneto-optical data storage, the rare earth-transition metal alloys, are generally produced by radio frequency (rf) sputtering from an alloy target onto a plastic or glass substrate. For protection against the environment as well as for optical and thermal enhancement, the RE–TM alloy films are sandwiched between two dielectric layers (such as SiNx or AINx) and covered with a metallic reflecting and heat-sinking layer, before finally being coated with several microns of protective lacquer. The properties of the MO layer are determined not only by the composition of the alloy, but also by the sputtering environment and by the condition of the substrate's surface. The temperature at which the film grows, the sputtering gas pressure, the substrate bias voltage (self or applied), the rate of deposition, the surface roughness of the substrate, the quality of underlayer and overlayer can all affect the properties of the final product. It is therefore necessary to have accurate characterization tools with which to measure the various properties of the media and to establish their suitability for application as the media of erasable optical data storage.

The most widely used method of characterization for magnetic materials is vibrating sample magnetometry (VSM). With VSM it is possible to measure the component of net magnetization Ms along the direction of the applied field. For MO media, one obtains the hysteresis loop when the field is perpendicular to the plane of the sample.

Introduction

In a digital system, the so-called “user-data” is typically an unconstrained sequence of binary digits 0 and 1. In such devices it is the responsibility of the information storage subsystem to record the data and to reproduce it faithfully and reliably upon request. To achieve high densities and fast data rates, storage systems are usually pushed to the limit at which the strength of the readout signal is of the same order of magnitude as that of the noise. Operating under these circumstances, it should come as no surprise that errors of misinterpretation do indeed occur at the time of retrieval. Also, because individual bits are recorded on microscopic areas, the presence of small media imperfections and defects, dust particles, fingerprints, scratches, and the like, results in imperfect reconstruction of the recorded binary sequences. For these and other reasons to be described below, the stream of user-data typically undergoes an encoding process before it finally arrives on the storage medium. The encoding not only adds some measure of protection against noise and other sources of error, but also introduces certain useful features in the recorded bit-pattern that help in signal processing and future data recovery operations. These features might be designed to allow the generation of a clocking signal from the readout waveform, or to maintain the balance of charge in the electronic circuitry, or to enable more efficient packing of data on the storage medium, or to provide some degree of control over the spectral content of the recorded waveform, and so on.

Introduction

The process of magnetization reversal in thin magnetic films is of considerable importance in erasable optical data storage. The success of thermomagnetic recording and erasure depends on the reliable and repeatable reversal of magnetization in micron-sized areas within the storage medium. A major factor usually encountered in descriptions of the thermomagnetic write and erase processes is the coercivity of the magnetic material. Technically, the coercivity Hc is defined for a hysteresis loop as the value of the applied field at which the net magnetization becomes zero. Coercivity, however, is an ill-defined concept which may be useful in the phenomenology of bulk reversal, but its relevance to the phenomena occuring on the spatial and temporal scales of thermomagnetic recording must be seriously questioned. To begin with, there is the problem of distinguishing the nucleation coercivity from the coercivity of wall motion. Then there is the question of speed and uniformity of motion as the wall expands beyond the site of its origination. Finally one must address issues of stability and erasability, which are intimately related to coercivity, in a framework wide enough to allow the consideration of local instabilities and partial erasure. It is fair to say that the existing theories of coercivity are generally incapable of handling the problems associated with thermomagnetic recording and erasure. In our view, the natural vehicle for conducting theoretical investigations in this area is computer simulation based on the fundamental equations of micromagnetics, the basis for which was laid down in the preceding chapter.

Introduction

Magneto-optical recording is an important mode of optical data storage; it is also the most viable technique for erasable optical recording at the present time. The MO read and write processes are both dependent on the interaction between the laser beam and the magnetic medium. In the preceding chapters we described the readout process with the aid of the dielectric tensor of the storage medium, without paying much attention to the underlying magnetism. Understanding the write and erase processes, on the other hand, requires a certain degree of familiarity with the concepts of magnetism in general, and with the micromagnetics of thin-film media in particular. The purpose of the present chapter is to give an elementary account of the basic magnetic phenomena, and to introduce the reader to certain aspects of the theory of magnetism and magnetic materials that will be encountered throughout the rest of the book.

After defining the various magnetic fields (H, B, and A) in section 12.1, we turn our attention in section 12.2 to small current loops, and show the equivalence between the properties of these loops and those of magnetic dipoles. The magnetization M of magnetic materials is also introduced in this section. Larmor diamagnetism is the subject of section 12.3. In section 12.4 the magnetic ground state of free atoms (ions) is described, and Hund's rules (which apply to this ground state) are presented.

Introduction

The present chapter is devoted to the analysis of the various sources of noise commonly encountered in magneto-optical readout. Although the focus will be on magneto-optics, most of these sources are known to occur in other optical recording systems as well; the arguments of the present discussion, therefore, may be adapted and applied to various other media and systems.

The MO readout scheme considered in this chapter is briefly reviewed below, followed by a description of the standard methods of signal and noise measurement, and the predominant terminology in this area. After these preliminary considerations, the characteristics of thermal noise, which originates in the electronic circuitry of the readout, are described in section 9.1. Section 9.2 considers a fundamental type of noise, shot noise, and gives a detailed account of its statistical properties. Shot noise, which in semiclassical terms arises from the random fluctuations in photon arrival times, is an ever-present noise in optical detection. Since the performance of MO media and systems is approaching the limit imposed by shot noise, it is imperative to have a good grasp of this particular source of noise. In section 9.3 we describe a model for laser noise; here we present measurement results that yield numerical values for the strength of laser power fluctuations. Spatial variations of disk reflectivity and random depolarization phenomena also contribute to the overall level of noise in readout; these and related issues are treated in section 9.4.

Introduction

The diffraction of light plays an important role in optical data storage systems. The rapid divergence of the beam as it emerges from the front facet of the diode laser is due to diffraction, as is the finite size of the focused spot at the focal plane of the objective lens. Diffraction of light from the edges of the grooves on an optical disk surface is used to generate the tracking-error signal. Some of the schemes for readout of the recorded data also utilize diffraction from the edges of pits and magnetic domains. A deep understanding of the theory of diffraction is therefore essential for the design and optimization of optical data storage systems.

Diffraction and related phenomena have been the subject of investigation for many years, and one can find many excellent books and papers on the subject. The purpose of the present chapter is to give an overview of the basic principles of diffraction, and to derive those equations that are of particular interest in optical data storage. We begin by introducing in section 3.1 the concept of the stationary-phase approximation as applied to two-dimensional integrals. Then in section 3.2 we approximate two integrals that appear frequently in diffraction problems using the stationary-phase method.

An arbitrary light amplitude distribution in a plane can be decomposed into a spectrum of plane waves. These plane waves propagate independently in space and, at the final destination, are superimposed to form a diffraction pattern.

This book has grown out of a course that I have taught at the Optical Sciences Center of the University of Arizona over the past five years. The idea has been to introduce graduate students from various backgrounds, either in physics or in one of the engineering disciplines, to the physical principles of magneto-optical (MO) recording. The topics selected for this course were of general interest, since both optics and magnetism are very broad and have many applications in modern science and technology. Each topic was treated in a self-contained and comprehensive manner. The students were first motivated by being told the relevance of a subject to optical data storage, then the subject was developed from basic principles, and examples were given along the way to show its application in quantitative detail. At the end of each chapter homework problems were assigned, so that students could learn certain details that were left out of the lectures or find out new directions in which their newly acquired knowledge could be applied.

The book essentially follows the format of the lectures, albeit with the addition of several chapters whose classroom coverage has usually been inhibited by the finite length of a semester. My primary goal in writing the book is to open the field to graduate students in physical sciences. Most college courses these days are based on single-issue topics that can be covered fairly comprehensively in the course of one or two semesters.

Introduction

Magnetostatics is the study of stationary patterns of magnetization in magnetic media. The central role in these studies is played by domains and domain walls. Domains are regions of uniform magnetization that are large on the atomic scale, but may be small in comparison with the dimensions of the medium under consideration. The walls are narrow regions that separate adjacent domains. In uniform media, the primary source of the spontaneous breakdown of magnetization into domains is the long-range dipole–dipole interaction, commonly referred to as the demagnetizing effect. Breakdown into domains provides a means of lowering the energy of demagnetization at the expense of increased exchange and anisotropy energies. Since within the domains the magnetization is fairly uniform and often aligned with an easy axis, it is at the site of the domain walls that the excess energies of exchange and anisotropy accumulate, as though the walls were endowed with an energy of their own. This chapter is devoted to the study of domains and domain walls and their structure and energy, as well as their magnetostatic interactions.

The basic tenets of magnetism were reviewed in Chapter 12, and the fundamental notions of magnetic field, magnetization, exchange and anisotropy were introduced. In the present chapter we shall confine attention to homogeneous thin-film media, where the magnetization M is treated as a continuous function of position r over the volume of the material (see Fig. 13.1).