As discussed in the previous chapter, there are two issues that distinguish transport in nanostructure systems from that in bulk systems. One is the granular or discrete nature of electronic charge, which evidences itself in single-electron charging phenomena (see Chapter 6). The second involves the preservation of phase coherence of the electron wave over short dimensions. Artificially confined structures are now routinely realized through advanced epitaxial growth and lithography techniques in which the relevant dimensions are smaller than the phase coherence length of charge carriers. We can distinguish two principal effects on the electronic motion depending on whether the carrier energy is less than or greater than the confining potential energy due to the artificial structure. In the former case, the electrons are generally described as bound in the direction normal to the confining potentials, which gives rise to quantization of the particle momentum and energy as discussed in Section 2.2. For such states, the envelope function of the carriers (within the effective mass approximation) is localized within the space defined by the classical turning points, and then decays away. Such decaying states are referred to as evanescent states and play a role in tunneling as discussed in Chapter 3. The time-dependent solution of the Schrodinger equation corresponds to oscillatory motion within the domain of the confining potential.

Nanostructures are generally regarded as ideal systems for the study of electronic transport. What does this simple statement mean?

First, consider transport in large, macroscopic systems. In bulk materials and devices, transport has been well described via the Boltzmann transport equation or similar kinetic equation approaches. The validity of this approach is based on the following set of assumptions: (i) scattering processes are local and occur at a single point in space; (ii) the scattering is instantaneous (local) in time; (iii) the scattering is very weak and the fields are low, such that these two quantities form separate perturbations on the equilibrium system; (iv) the time scale is such that only events that are slow compared to the mean free time between collisions are of interest. In short, one is dealing with structures in which the potentials vary slowly on both the spatial scale of the electron thermal wavelength (to be defined below) and the temporal scale of the scattering processes.

Since the late 1960s and early 1970s, researchers have observed quantum effects due to confinement of carriers at surfaces and interfaces, for example along the Si/SiO2 interface, or in heterostructure systems formed between lattice-matched semiconductors. In such systems, it is still possible to separate the motion of carriers parallel to the surface or interface, from the quantized motion perpendicular, and describe motion semiclassically in the unconstrained directions.

In the preceding chapters, and indeed in the subsequent chapters, most of the discussion is on semiconductors in which the Bloch theory of extended states prevails. There is another class of semiconductors that has received considerable attention over the past several decades, and that is disordered (or amorphous) semiconductors. Here, in the realm of nanostructures, we really do not want to discuss the entire field of amorphous semiconductors, and would generally ignore strongly disordered materials as well. However, recent experiments have shown the presence of a metal–insulator transition in quasi-two-dimensional systems. Consequently, one needs to understand the difference between localized (disordered) systems, weakly disordered systems, and the normal Bloch band picture of conductance.

Generally, in disordered (or, strongly localized) systems, the Boltzmann equation fails to describe transport adequately except under very special circumstances. Disordered materials can stem from several sources, ranging from amorphous materials to relatively good single crystals with very high doping concentrations. In particular, the latter exhibit a form of impurity-induced disorder when the concentration of the impurity reaches a significant fraction of the atomic concentration of the host lattice. This, in turns connects to weak localization which can also arise from impurity-induced coherence effects even when the concentration is not too high.

The focus of this chapter is a discussion of transport in quantum dots, which are quasi-zero-dimensional nanostructure systems whose electronic states are completely quantized. The confinement of carrier motion in these structures is imposed in all three spatial directions, resulting in a discrete spectrum of energy levels much the same as in an atom or molecule. We can therefore think of quantum dots as artificial atoms, which in principle can be engineered to have a particular energy level spectrum. As in atomic systems, the electronic states in quantum dots are sensitive to the presence of multiple electrons due to the Coulomb interaction between electrons. Rich transport phenomena are therefore observed in these structures, not only because of quantum confinement and the resonant structure associated with this confinement, but also due to the granular nature of electric charge.

In contrast to quantum wells and wires, quantum dots can be sufficiently small that the introduction of even a single electron is sufficient to dramatically change the transport properties due to the charging energy associated with this extra electron. One of the main consequences of this charging energy is to give rise to a Coulomb blockade of transport, where conductance oscillations are observed with the addition or subtraction of a single electron from a quantum dot, which we discuss in detail in Sections 6.1 and 6.2.

The discovery in 1980, by Klaus von Klitzing and his colleagues, of the integer quantum Hall effect (IQHE) may have done more than any other single event to stimulate experimental and theoretical interest in the electrical properties of low-dimensional systems. This phenomenon has now been observed in a variety of different material systems, and is manifest as the appearance of wide and precisely quantized plateaus in the Hall resistance (RH, or Hall resistivity ρxy), which therefore deviates strongly from the linear dependence on magnetic field that is expected classically. It is now understood that this high-magnetic-field phenomenon is associated with the formation of strongly quantized Landau levels in a two-dimensional electron gas (2DEG), under which conditions current flow is carried by ballistic edge states that are the quantum analog of classical skipping orbits (recallSection 2.5). Thus, the quantum Hall effect represents a remarkable manifestation of one-dimensional transport in a macroscopic system.

In this chapter, we begin by discussing the basic phenomenology of the (integer) quantum Hall effect, which, due to the extreme accuracy of its quantization, has now been adopted as an international standard for the definition of the ohm. We present an interpretation of this effect due to Büttiker, which begins from the concepts of the Landauer formula (Section 3.3) and explains the quantization by considering that edge states propagate ballistically, without dissipation, over the entire sample length.

The technological means now exists for approaching the fundamental limiting scales of solid-state electronics in which a single electron can, in principle, represent a single bit in an information flow through a device or circuit. The burgeoning field of single-electron tunneling (SET) effects, although currently operating at very low temperatures, has brought this consideration into the forefront. Indeed, the recent observations of SET effects in poly-Si structures at room temperature by a variety of authors has grabbed the attention of the semiconductor industry. While there remains considerable debate over whether the latter observations are really single-electron effects, the resulting behavior has important implications for future semiconductor electronics, regardless of the final interpretation of the physics involved. Indeed, the semiconductor industry is rapidly carrying out its own advance, with transistor gate lengths in the 20 nm range in production in 2009 (the so-called 35 nm node).

We pointed out in Chapter 1 that the semiconductor industry is following a linear scaling law that is expected to be fairly rigorous. With dimensions approaching 10 nm within another decade, there is a rapid search for possible new technologies that can supplement Si with the offer of improved performance. However, it is clear from a variety of considerations that the devices themselves may well not be the limitation on continued growth in device density within the integrated circuit chip.

In Chapter 2, we introduced the idea of low-dimensional systems arising from quantum confinement. Such confinement may be due to a heterojunction, an oxide–semiconductor interface, or simply a semiconductor–air interface (for example, in an etched quantum wire structure). When we look at transport parallel to such barriers, such as along the channel of an HEMT or MOSFET, or along the axis of a quantum wire, to a large extent we can employ the usual kinetic equation formalisms for transport and ignore the phase information of the particles. Quantum effects enter only through the description of the basis states arising from the confinement, and the quantum mechanical transition rates between these states are due to the scattering potential. This is not to say that quantum interference effects do not play a role in parallel transport. As we will see in the later chapters, several effects manifest themselves in parallel transport studies such as weak localization and universal conductance fluctuations, which at their origin have effects due to the coherent interaction of electrons.

In contrast to transport parallel to barriers, when particles traverse regions in which the medium is changing on length scales comparable to the phase coherence length of the particles, quantum interference is expected to be important. By “quantum interference” we mean the superposition of incident and reflected waves, which, in analogy to the electromagnetic case, leads to constructive and destructive interference.

Introduction

The conventional, single-reference, coupled-cluster method is very effective for electronic states dominated by a single determinant, such as most molecular ground states near their equilibrium geometry. Such states are predominantly closed-shell singlet states, and CC calculations on them produce pure singlet wave functions. But even these states become dominated by more than one determinant when one or more bonds are stretched close to breaking, so that single-reference CC based on RHF orbitals is then not usually appropriate for the calculation of entire potential-energy surfaces. While such problems can be partially treated by using UHF reference functions, which usually separate correctly, the UHF approach makes use of symmetry breaking and is poor in the spin-recoupling region.

Most excited, ionized and electron-attached states are open-shell states, and CC calculations on them using UHF or ROHF orbitals do not usually result in pure-spin wave functions. Furthermore, such states often involve large contributions from more than one determinant and thus do not respond well to conventional single-reference treatments.

One solution to these problems is to resort to multireference methods, such as those described in Chapters 8 and 14, but such treatments are still quite difficult to apply at a high enough level. An effective alternative in many cases is provided by the equation-of-motion coupled-cluster (EOM-CC) method (Emrich 1981, Sekino and Bartlett 1984, Comeau and Bartlett 1993, Stanton and Bartlett 1993a). A closely related approach is the coupled-cluster linear response (CCLR) method (Monkhorst 1977, Dalgaard and Monkhorst 1983, Koch and Jørgensen 1990).

In the present chapter, we discuss the Sutherland model for particles without internal degrees of freedom such as spin. We call this model the singlecomponent Sutherland model. For two particles, the eigenenergies and eigenstates of the Sutherland model (1.3) have been obtained explicitly in the previous chapter. The most striking feature of the Sutherland model is that one can derive not only the energy spectrum but also the dynamics for the many-particle case with an exact account of interaction effects. Thus, the Sutherland model provides an ideal framework to study a one-dimensional quantum liquid in detail.

In Section 2.1, we derive the eigenenergies of eigenstates. In Section 2.2, we present different but equivalent physical pictures for the energy spectrum. Namely, the energy spectrum is naturally regarded as that of interacting bosons or fermions. The same spectrum can also be interpreted as that of free particles obeying nontrivial quantum statistics, i.e., free anyons in one dimension. The exclusion statistics proposed by Haldane will be explained on this occasion. Spectrum and statistics of elementary excitations are derived in Section 2.3. In Section 2.4, we discuss thermodynamic properties, which can be rewritten as those of free anyons. In Section 2.5, we identify the eigenfunctions with Jack symmetric polynomials, and discuss their basic properties. In Section 2.6, we consider dynamical correlation functions such as Green's functions and the density correlation function. These quantities are derived with the use of Jack polynomials, and are naturally interpreted in terms of elementary excitations with fractional charge.

The Sutherland model is the simplest model to realize the Tomonaga–Luttinger liquid.

Bipolar device design can be considered in two parts. The first part deals with designing bipolar transistors in general, independent of their intended application. In this case, the goal is to reduce as much as possible, consistent with the start-of-the-art fabrication technology, all the internal resistance and capacitance components of the transistor. The second part deals with designing a bipolar transistor for a specific circuit application. In this case, the optimal device design point depends on the application. The design of a bipolar transistor in general is covered in this chapter, and the optimization of a transistor for a specific application is discussed in Chapter 8.

Design of the Emitter Region

It was shown in Section 6.2 that the emitter parameters affect only the base current, and have no effect on the collector current. In theory, a device designer can vary the emitter design to vary the base current. In practice, this is rarely done, for two reasons. First, for digital-circuit applications, as long as the current gain is not unusually low or the base current unusually high, the performance of a bipolar transistor is rather insensitive to its base current (Ning et al., 1981). For many analog-circuit applications, once the current gain is adequate, the reproducibility of the base current is more important than its magnitude. Therefore, there is really no particular reason to tune the base current of a bipolar device by tuning the emitter design, once a low and reproducible base current is obtained. Second, as can be seen in Appendix 2, the emitter is formed towards the end of the device fabrication process. Any change to the emitter process to tune the base current could affect the doping profile of the other device regions and hence could affect the other device parameters. As a result, once a bipolar technology is ready for manufacturing, its emitter fabrication process is usually fixed. All that a device designer can do to alter the device and circuit characteristics in this bipolar technology is to change the base and the collector designs, which often can be accomplished independently of the emitter process and hence has no effect on the base current.

The metal–oxide–semiconductor field-effect transistor (MOSFET) is the building block of VLSI circuits in microprocessors and dynamic memories. Because the current in a MOSFET is transported predominantly by carriers of one polarity only (e.g., electrons in an n-channel device), the MOSFET is usually referred to as a unipolar or majority-carrier device. Throughout this chapter, n-channel MOSFETs are used as an example to illustrate device operation and derive drain-current equations. The results can easily be extended to p-channel MOSFETs by exchanging the dopant types and reversing the voltage polarities.

The basic structure of a MOSFET is shown in Fig. 3.1. It is a four-terminal device with the terminals designated as gate (subscript g), source (subscript s), drain (subscript d), and substrate or body (subscript b). An n-channel MOSFET, or nMOSFET, consists of a p-type silicon substrate into which two n+ regions, the source and the drain, are formed (e.g., by ion implantation). The gate electrode is usually made of metal or heavily doped polysilicon and is separated from the substrate by a thin silicon dioxide film, the gate oxide. The gate oxide is usually formed by thermal oxidation of silicon. In VLSI circuits, a MOSFET is surrounded by a thick oxide called the field oxide to isolate it from the adjacent devices. The surface region under the gate oxide between the source and drain is called the channel region and is critical for current conduction in a MOSFET. The basic operation of a MOSFET device can be easily understood from the MOS capacitor discussed in Section 2.3. When there is no voltage applied to the gate or when the gate voltage is zero, the p-type silicon surface is either in accumulation or in depletion and there is no current flow between the source and drain. The MOSFET device acts like two back-to-back p–n junction diodes with only low-level leakage currents present. When a sufficiently large positive voltage is applied to the gate, the silicon surface is inverted to n-type, which forms a conducting channel between the n+ source and drain. If there is a voltage difference between them, an electron current will flow from the source to the drain. A MOSFET device therefore operates like a switch ideally suited for digital circuits. Since the gate electrode is electrically insulated from the substrate, there is effectively no dc gate current, and the channel is capacitively coupled to the gate via the electric field in the oxide (hence the name field-effect transistor).