Overview and background

Recent progress in crystal growth and microfabrication technologies have allowed us to explore a new field of semiconductor device research. The quantum-mechanical wave-nature of electrons is expected to appear in mesoscopic semiconductor structures with sizes below 100 nm. Instead of conventional devices, such as field effect transistors and bipolar transistors, a variety of novel device concepts have been proposed based on the quantum mechanical features of electrons. The resonant tunnelling diode (RTD), which utilises the electron-wave resonance in multi-barrier heterostructures, emerged as a pioneering device in this field in the mid-1970s. The idea of resonant tunnelling (RT) in finite semiconductor superlattices was first proposed by Tsu and Esaki in 1973 shortly after molecular beam epitaxy (MBE) appeared in the research field of compound semiconductor crystal growth. A unique electron tunnelling phenomenon was predicted for an AlGaAs/GaAs/AlGaAs double-barrier heterostructure, based on electron-wave resonance, analogous to the Fabry–Perot interferometer in optics. In the particle picture, each electron is constrained inside the GaAs quantum well for a certain dwell time before escaping to the collector region. The bias dependence of the tunnelling current through the double-barrier structure shows negative differential conductance (NDC) as a result of RT. Experimental results reported in the early days showed only weak features in current–voltage (I–V) characteristics at low temperatures and did no more than confirm the theoretical prediction of resonant tunnelling.

Tunneling is perhaps the oldest example of mesoscopic transport. Single-barrier tunneling has found widespread applications in both basic and applied research. The latest example is scanning tunneling microscopy which has made it possible to image on an atomic scale. However, our purpose in this chapter is not to discuss single-barrier tunneling; the field is far too large and well-developed. Instead we will focus on what is called a double-barrier structure, consisting of two tunneling barriers in series. Since the pioneering work of Chang, Esaki and Tsu (Appl. Phys. Lett. 24, (1974) 593) much research has been devoted to the study of such structures. Two important paradigms of mesoscopic transport have emerged from this study, namely, resonant tunneling and single-electron tunneling. At the same time, the current–voltage characteristics of these structures exhibit useful features at room temperature and high bias, unlike most other mesoscopic phenomena which are limited to the low temperature linear response regime.

We start in Section 6.1 with a discussion of current flow through a double-barrier structure, assuming that transport is coherent. The current can then be obtained by calculating the coherent transmission through the structure from the Schrödinger equation. In Section 6.2 we discuss how scattering processes inside the well affect the peak current and the valley current.

In Chapter 2 we have tried to establish that there exists a useful quantity called the transmission function in terms of which one can describe the current flow through a conductor. In this chapter we address the question of how the transmission function can be calculated for actual mesoscopic conductors. As we might expect, this chapter is somewhat mathematical and familiarity with matrix algebra is required. It could be skipped on first reading since it is not essential to know how to calculate the transmission function in order to appreciate mesoscopic phenomena, just as it is not necessary to understand the microscopic theory of diffusion or mobility in order to appreciate bulk transport phenomena. However, we will occasionally (especially in Chapter 5) use some of the concepts introduced here.

If the size of the conductor is much smaller than the phase-relaxation length then transport is said to be coherent and one can calculate the transmission function starting from the Schrödinger equation. A large majority of the theoretical work in this field is centered around this coherent transport regime where we can relate the transmission function to the S-matrix as discussed in Section 3.1.

When dealing with a large conductor it is often convenient to divide it conceptually into several sections whose S-matrices are determined individually. We discuss in Section 3.2 how the S-matrices of successive sections can be combined assuming complete coherence, complete incoherence or partial coherence among the sections.

So far in this book we have described the effect of electron–phonon or electron–electron interactions in phenomenological terms, through a phase-relaxation time. In this chapter we will describe the non-equilibrium Green's function (NEGF) formalism which provides a microscopic theory for quantum transport including interactions. We will introduce this formalism using simple kinetic arguments based on a one-particle picture that are only slightly more difficult than those used to derive semiclassical transport theories like the Boltzmann equation. This heuristic description is not intended as a substitute for the rigorous descriptions available in the literature [8.1–8.8]. Our intention is simply to make the formalism accessible to readers unfamiliar with the language of second quantization. We will restrict our discussion to steady-state transport as we have done throughout this book.

The NEGF formalism (sometimes referred to as the Keldysh formalism) requires a number of new concepts like correlation functions which we introduce in Sections 8.1 and 8.2. We then describe the formalism in Sections 8.3–8.6. In Section 8.7 we relate it to the Landauer–Büttiker formalism which, as we have seen, has been very successful in describing mesoscopic phenomena. For non-interacting transport the two are equivalent, and the added conceptual complexity of the NEGF formalism is not necessary. The real power of this formalism lies in providing a general approach for describing quantum transport in the presence of interactions.

In contrast to the preceding chapters, which concentrated mainly on the physics of RTDs, this chapter reviews some applications of RTDs and related three-terminal devices. As briefly described in the first chapter, RTDs have two distinct features over other semiconductor devices from an applications point of view: namely, their potential for very-high-speed operation and their negative differential conductance. The former feature arises from the very small size of the resonant tunnelling structure along the direction of carrier transport; because of the short distance through which carriers must travel, RTDs can be designed to have very high cut-off frequencies. As a result, oscillation in submillimetre wave frequencies has been reported. Besides this highspeed potential, the negative differential conductance makes it possible to operate RTDs as so-called functional devices, which enables circuits to be designed on different principles than conventional devices. For example, signal processing circuits with a significantly reduced number of devices and multiple-valued memory cells using RTDs have been proposed and demonstrated. These functional applications are highly promising since RTDs, with their simple structure and small size, can be easily integrated with conventional devices such as field effect transistors (FETs) and bipolar transistors.

In Section 5.1, high-speed applications, including high-frequency signal generation and high-speed switching, are discussed. Functional applications, such as a one-transistor static random access memory (SRAM) and a multi-valued memory circuit, are described in Section 5.2.

This last chapter is devoted to the study of resonant tunnelling through laterally confined, ultra-small, double-barrier heterostructures. Recent rapid advances in nanofabrication techniques have naturally led to the idea of resonant tunnelling through three-dimensionally confined ‘quantum dot’ structures. Since electrons are confined laterally as well as vertically in these structures, the devices are often called zerodimensional (0D) RTDs and have become of great interest both from the standpoint of the physics of quantum transport through 0D electronic states and also for device miniaturisation towards highly integrated functional resonant tunnelling devices. The 0D RTD is a virtually isolated quantum dot only weakly coupled to its reservoirs and thus is well suited to investigating electron-wave transport properties through 3D quantised energy levels. By designing the structural parameters such as the barrier thickness, the quantum well width and dimensionality of lateral confinement, it is possible to realise a ‘quantum dot’ in which the number of electrons is nearly quantised so the effect of single-chargeassisted transport, or the so-called Coulomb blockade (CB), becomes significant. After Reed et al. reported their pioneering work in 1988 on resonant tunnelling through a quantum pillar which was fabricated by electron beam lithography and dry etching, several theoretical and experimental studies have been reported which investigate the mechanism of the observed fine structures. Transport in the 0D RTD is generally much more complicated than that in the conventional large-area RTDs which we have studied so far in this book: problems such as lateral-mode mixing due to a non-uniform confinement potential, charge quantisation in a quantum well and the interplay between resonant tunnelling and Coulomb blockade single-electron tunnelling have recently been invoked for the 0D RTDs. Such difficulties are still far from being fully resolved.

The tremendous progress of crystal growth and microfabrication technologies over the last two decades has allowed us to explore a new field of semiconductor device research. The quantum mechanical wavenature of electrons, expected to appear in nanometre-scale semiconductor structures, has been used to create novel semiconductor devices. The Resonant Tunnelling Diode (RTD), which utilises the electron-wave resonance occurring in double potential barriers, emerged as a pioneering device in this field in the middle of the 1970s. The idea of resonant tunnelling (RT) was first proposed by Tsu and Esaki in 1973, shortly after Molecular Beam Epitaxy (MBE) appeared in the research field of compound semiconductor crystal growth. Since then, RT has become of great interest and has been investigated both from the standpoint of quantum transport physics and also its application in functional quantum devices. Despite its simple structure, the RTD is indeed a good laboratory for electron-wave experiments, which can investigate various manifestations of quantum transport in semiconductor nanostructures. It has played a significant role in disclosing the fundamental physics of electron-waves in semiconductors, and enables us to proceed to study more complex and advanced quantum mechanical systems.

This book is designed to describe both the theoretical and experimental aspects of this active and growing area of interest in a systematic manner, and so is suitable for postgraduate students beginning their studies or research in the fields of quantum transport physics and device engineering.

We start this chapter with a brief review of some basic concepts. First in Section 1.1 we introduce the gallium arsenide (GaAs)/aluminum gallium arsenide (AlGaAs) material system which provides a very high quality two-dimensional conduction channel and has been widely used in meso-scopic experiments. Section 1.2 summarizes the free electron model that is commonly used to describe conduction electrons in metals and semiconductors. Next we discuss different characteristic lengths like the de Broglie wavelength, mean free path and the phase-relaxation length which determine the length scale at which mesoscopic effects appear (Section 1.3). The variation of resistance in the presence of a magnetic field is widely used to characterize conducting films. Both the low-field properties (Section 1.4) and the high-field properties (Section 1.5) yield valuable information regarding the electron density and mobility.

In Section 1.6 we introduce the concept of transverse modes which plays a prominent role in the theory of mesoscopic conductors and will appear repeatedly in this book. Finally in Section 1.7 we address an important conceptual issue that arises in the description of degenerate conductors, that is, conductors with a Fermi energy that is much greater than kBT. Normally we view the current as being carried by all the conduction electrons which drift along slowly. However, in degenerate conductors it is more appropriate to view the current as being carried by a few electrons close to the Fermi energy which move much faster. One consequence of this is that the conductance of degenerate conductors is determined by the properties of electrons near the Fermi energy rather than the entire sea of electrons.

Following the study in Chapter 3 of the effects of elastic and inelastic scattering on the transmission probability function, this chapter investigates non-equilibrium electron distribution in RTDs. Electron distribution in the triangular potential well in the emitter is studied first (Section 4.1). Then dissipative quantum transport theory is presented based on the Liouville–von-Neumann equation for the statistical density matrix (Section 4.2.1). Numerical calculations are carried out in order to analyse the femtosecond dynamics of the electrons (Section 4.2.2) and the dynamical space charge build-up in the double-barrier structure which gives rise to the intrinsic current bistability in the NDC region (Section 4.3.1). Next experimental studies of the charge build-up phenomenon are presented using magnetoconductance measurements (Section 4.3.2) and photoluminescence measurements (Section 4.3.3). Finally, the effects of magnetic fields on intrinsic current bistability are studied (Section 4.4).

Non-equilibrium electron distribution in RTDs

Let us start with a discussion on electron distribution in the emitter. We have seen in Section 2.4 that the electronic states in the emitter become 2D in the pseudo-triangular potential well formed between the thick spacer layer and the tunnelling barrier (see Fig. 2.16). Sharper current peaks observed for Materials 2 and 3 (Fig. 2.18) have been attributed to the 2D–2D nature of resonant tunnelling. This interpretation is based upon an assumption that the electrons in the triangular well are well thermalised, and that local equilibrium is achieved.

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.