Introduction

In the preceding chapters we have been largely concerned with the simplest coherence effects of optical fields, namely those which depend on the correlation of the field variable at two space-time points (r1, t1) and (r2, t2). As we have seen, these effects include the most elementary coherence phenomena involving interference, diffraction and radiation from fluctuating sources.

In this chapter we present an extension of the theory to cover more complicated situations, which have to be described by correlations of higher order, i.e. by correlations of the field variables at more than two space-time points or the expectation values involving various powers and products of the field variables. Situations of this kind have become of considerable importance since the development of the laser and of nonlinear optics. The basic difference between the statistical properties of thermal light and laser light can, in fact, only be understood by going beyond the elementary second-order correlation theory.

Many of the higher-order coherence phenomena are most clearly manifest in the photoelectric detection process, which can only be adequately described by the quantum theory of detection or by taking into account the quantum features of the field, both of which will be studied in the succeeding chapters. However, because the field is still described classically in the semi-classical theory of light detection, and also because the classical field description provides a natural stepping stone to the quantum description of field correlations, we will now discuss the general description of field correlations of all orders on the basis of the classical theory of the fluctuating wavefield.

This chapter is intended as an introduction to the fundamental physics of resonant tunnelling diodes (RTDs). The idea of global coherent tunnelling is introduced in order to provide an intuitive and clear picture of resonant tunnelling. The theoretical basis of the global coherent tunnelling model is presented in Section 2.2. The Tsu–Esaki formula, based on linear response theory, is adopted and combined with the transfer matrix method to calculate the tunnelling current through double-barrier resonant tunnelling structures (Section 2.2.1). The global coherent tunnelling model is improved by taking Hartree's selfconsistent field (Section 2.2.2) into account. A more analytical transfer Hamiltonian formula is also presented (Section 2.2.3). Section 2.3 introduces the electron dwell time, which is one of the important quantities required to describe the high-frequency performance of RTDs. The effects of quantised electronic states in the emitter are then studied in Section 2.4. Section 2.5 describes resonant tunnelling through double-well structures. Finally, Section 2.6 discusses the idea of incoherent resonant tunnelling induced by phase-coherence breaking scattering. The problem of collision-induced broadening is then discussed in terms of the peak-to-valley (P/V) current ratio of RTDs by using a phenomenological Breit–Wigner formula.

Resonant tunnelling in double-barrier heterostructures

Let us start with a simple discussion of resonant tunnelling through the double-barrier heterostructure depicted in Fig. 2.1 (a). A resonant tunnelling diode (RTD) typically consists of an undoped quantum well layer sandwiched between undoped barrier layers and heavily doped emitter and collector contact regions.

According to Ohm's law, the resistance of an array of scatterers increases linearly with the length of the array. This describes real conductors fairly well if the phase-relaxation length is shorter than the distance between successive scatterers. But at low temperatures in low-mobility samples the phase-relaxation length can be much larger than the mean free path. The conductor can then be viewed as a series of phase-coherent units each of which contains many elastic scatterers. Electronic transport within such a phase-coherent unit belongs to the regime of quantum diffusion which has been studied by many authors since the pioneering work of Anderson (P. W. Anderson (1958), Phys. Rev.109, 1492). In this regime, interference between different scatterers leads to a decrease in the conductance. For a coherent conductor having a overall conductance much greater than ~ (e2/h) or 40 μΩ-1, the decrease in the conductance is approximately (e2/h). Such a conductor is said to be in the regime of weak localization (Section 5.2). This effect is easily destroyed by a small magnetic field (typically less than 100 G), so that it can be identified experimentally by its characteristic magnetoresistance (Section 5.3). This is a very important effect, because unlike most other transport phenomena it is sensitive to phase relaxation and not just to momentum relaxation. Indeed the weak localization effect is often used to measure the phase-relaxation length.

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.