The two-dimensional electron gas (2DEG) trapped at a doped heterojunction is the most important low-dimensional system for electronic transport. It forms the core of a field-effect transistor, which goes by many acronyms including modulation-doped field-effect transistor (MODFET) and high electron mobility transistor (HEMT). I shall use the first, which emphasizes the close relation to the silicon MOSFET but with MODulation doping. The silicon MOSFET is perhaps the most common electronic device, with electrons or holes trapped in an inversion layer at an interface between Si and SiO2. Many of the ideas in this chapter were originally derived for the MOSFET but it has been almost completely superseded in physics experiments by the MODFET because of the enormous improvement in the mobility of electrons and holes. The highest mobility of electrons in a MOSFET is around 4 m2 V−1 s−1, whereas values over 1000 m2 V−1 s−1 have been achieved in a MODFET. These mobilities are measured at low temperature, where they are limited by scattering from impurities, defects, and interfaces rather than phonons. The almost perfect crystalline quality of III–V heterostructures and the ability to separate carriers from the impurities that provide them by modulation doping mainly account for this huge difference.

First we shall study the electrostatics of modulation-doped layers to get estimates of important quantities such as the density of electrons, and then we shall develop models for the energy level and wave function of a two-dimensional electron gas.

This chapter provides a review of the general properties of heterostructures, semiconductors composed of more than one material. Variations in composition are used to control the motion of electrons and holes through band engineering. Knowledge of the alignment of bands at a heterojunction, where two materials meet, is essential but has proved difficult to determine even for the best-studied junction, GaAs–AlxGa1−xAs. Although effort was initially concentrated on materials of nearly identical lattice constant, current applications require properties that can be met only by mismatched materials, giving strained layers.

A huge variety of devices has been fabricated from heterostructures, for both electronic and optical applications, and we shall survey these before studying them in greater detail in later chapters. Finally, we shall look briefly at the effective-mass approximation. This is a standard simplification that allows us to treat electrons as though they are free, except for an effective mass, rather than using complicated band structure. It means, for example, that an electron in a sandwich of GaAs between Al0.3Ga0.7As can be treated as the elementary problem of a potential well.

We shall neglect the random nature of alloys such as AlxGa1−xAs, where there is assumed to be no ordering of the Ga and Al ions over the cation sites of the lattice. In principle Bloch's theorem does not apply to such materials because they lack translational invariance from cell to cell.

In Chapter 4 we looked at how electrons could be trapped in various examples of potential wells and made to behave as though they were only two-dimensional (or less). In this chapter we shall look at free electrons that encounter barriers or other obstacles as they travel. Again, most of the potential profiles will be one-dimensional and we need only solve the Schrödinger equation in this dimension, although the other dimensions enter into the calculation of the current. We shall use the general tool of T-matrices, which can simply be multiplied together to yield the transmission coefficient for an arbitrary sequence of steps and plateaus. Two particular applications are to resonant tunnelling through a double barrier and to an infinite, regularly spaced sequence of barriers, a superlattice. Two barriers show a narrow peak in the transmission when the energy of the incident electron matches that of a resonant or quasi-bound state between the barriers (Section 5.5). This peak broadens into a band in the superlattice, and Section 5.6 shows how band structure and Bloch's theorem emerge for a specific example.

Many low-dimensional structures cannot simply be factorized into one-dimensional problems but have many leads, each with several propagating modes. These will be treated in Section 5.7 and we shall derive one of the famous results of low-dimensional systems, the quantized conductance.

Fermi's golden rule is one of the most important tools of quantum mechanics. It gives the general formula for transition rates, the rates at which particles are ‘scattered’ from one state to another by a perturbation. ‘Scattered’ is in quotation marks because it is a much more general concept than one might guess. An obvious example is provided by impurities in a crystal, which scatter an electron from one Bloch state to another. They change its momentum but not its energy. Similarly, phonons (vibrations of the lattice) also scatter electrons, but in this case they change the energy of the electrons as well as their momentum. A less obvious example is the absorption of light, which can be viewed as a scattering process in which an electron collides with a photon. The converse process also occurs, where an electron loses energy to a photon, and gives rise to spontaneous and stimulated emission. Thus scattering is a remarkably general concept.

The examples suggest that there are two broad classes of scattering processes that we should treat:

1. (i) potentials that are constant in time, such as impurities in a crystal, which do not change the energy of the particle being scattered;

2. (ii) potentials that vary harmonically in time as cos ωqt, such as phonons and photons, which change the energy of the particle by ±ħωq.

Few low-dimensional systems are periodic (superlattices provide an obvious exception), but they all consist of relatively large scale structures superposed on the structure of a host. This may be a true crystal such as GaAs or a random alloy such as (Al,Ga)As; we shall ignore the complications introduced by the alloy and treat it as a crystal ‘on average’. We must understand the electronic behaviour of the host before treating that of the superposed structure.

This chapter deals first with one-dimensional crystals, followed by three-dimensional materials. The final section is devoted to phonons, lattice waves rather than electron waves, which also have a band structure imposed by the periodic nature of the crystal. Photons are the third kind of wave that we shall encounter, and structures that display band structure for light have recently been demonstrated. Their behaviour can be described with a similar theory but we shall not pursue this.

Band Structure in One Dimension

The potential energy in a real crystal is clearly far more complicated than the systems that we have studied in the previous chapter. In Section 5.6 we shall solve the simple example of a square-wave potential in detail, but the most important results follow from the qualitative feature that the potential is periodic. In one dimension this means that V(x + a) = V(x), where a is the lattice constant, the size of each unit cell of the crystal.

The phase interference between two distinct electron (or hole) waves was treated in the Introduction to this book. Whereas the past few chapters dealt largely with the quasi-ballistic transport of these waves through mesoscopic systems, in this chapter we want to begin to treat systems in which the transport is more diffusive than quasi-ballistic. How do we distinguish between these two regimes? Certainly the existence of scattering is possible in both regimes, but we distinguish the diffusive regime from the quasi-ballistic regime by the level of the scattering processes. In the diffusive regime, we assume that scattering dominates the transport to a level such that there are no “ballistic” trajectories that extend for any significant length within the sample. That is, we assert that l = εF τ ≪ L, where L is any characteristic dimension of the sample. Typically, this means that the material under investigation is characterized by a relatively low mobility, certainly not the mobility of several million that can be obtained in good modulation-doped heterostructures. In a sense, the transport is now considered to be composed of short paths between a relatively large number of impurity scattering centers. Thus, we deal with the smooth diffusion of particles through the mesoscopic system. To be sure, the Landauer formula does not distinguish ballistic from diffusive transport, but its treatment in multimode waveguides is more appropriately considered a ballistic transport. To illustrate the difference, consider the Aharonov-Bohm effect and the presence of weak localization. In the former, the wavefunction (particles) splits into two parts that propagate around opposite sides of a ring “interferometer,” as illustrated in Fig. 1.4.

This book has grown out of our somewhat disorganized attempts to teach the physics and electronics of mesoscopic devices over the past decade. Fortunately, these have evolved into a more consistent approach, and the book tries to balance experiments and theory in the current understanding of mesoscopic physics. Whenever possible, we attempt to first introduce the important experimental results in this field followed by the relevant theoretical approaches. The focus of the book is on electronic transport in nanostructure systems, and therefore by necessity we have omitted many important aspects of nanostructures such as their optical properties, or details of nanostructure fabrication. Due to length considerations, many germane topics related to transport itself have not received full coverage, or have been referred to by reference. Also, due to the enormity of the literature related to this field, we have not included an exhaustive bibliography of nanostructure transport. Rather, we have tried to refer the interested reader to comprehensive review articles and book chapters when possible.

The Introduction of Chapter 1 gives a general overview of the important effects that are observable in small systems that retain a degree of phase coherence. These are also compared to the needs that one forsees in future small electron devices. Chapter 2 provides a general introduction to quantum confined systems, and the nature of quasi-two-, quasione- and quasi-zero-dimensional systems including their dielectric response and behavior in the presence of an external magnetic field. It concludes with an overview of semi-classical transport in quantum wells and quantum wires including the relevant scattering mechanisms in quantum confined systems.

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 4). 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 Schrödinger equation corresponds to oscillatory motion within the domain of the confining potential.

The second type of motion we will be concerned with is that associated with propagating states of the system. Here the carrier energy is such that it lies above that of the confining potentials, or that the potentials are limited sufficiently in extent so that quantum mechanical tunneling through such barriers can occur.

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 Yano et al. [1] 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 to future semiconductor electronics, regardless of the final interpretation of the physics involved.

We pointed out in Chapter 1 that the semiconductor industry is following a linear scaling law that is expected to be fairly rigorous, at least into the first decade of the next century. This relationship will lead to devices with critical dimensions well below 0.1 μm. Research devices have been made with drawn gate lengths down to 20 nm in GaAs and 40 nm in Si MOSFETs. This suggests that such devices can be expected to appear in integrated circuits within a few decades (by 2020 if scaling rules at that time are to be believed). 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 a 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 Chapters 5 and 6, 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. Such a coherent superposition of states is of course what leads to the quantization of momentum and energy in the formation of low-dimensional systems discussed in the previous chapter.

When the temperature is raised above absolute zero, the amplitudes of both the weaklocalization, universal conductance fluctuations and the Aharonov-Bohm oscillations are reduced below the nominal value e2/ħ. In fact, the amplitude of nearly all quantum phase interference phenomena is likewise weakened. There is a variety of reasons for this. One reason, perhaps the simplest to understand, is that the coherence length is reduced, but this can arise as a consequence of either a reduction in the coherence time or a reduction in the diffuson coefficient. In fact, both of these effects occur. In Chapter 2, we discussed the temperature dependence of the mobility in high-mobility modulation-doped GaAs/AlGaAs heterostructures. The decay of the mobility couples to an equivalent decay in the diffuson constant (discussed in Chapters 2 and 5), D = ε2Fτ/d, where d is the dimensionality of the system, through both a small temperature dependence of the Fermi velocity and a much larger temperature dependence of the elastic scattering rate. The temperature dependence of the phase coherence time is less well understood but generally is thought to be limited by electron-electron scattering, particularly at low temperatures. At higher temperatures, of course, phonon scattering can introduce phase breaking.

Another interaction, though, is treated by the introduction of another characteristic length, the thermal diffuson length. The source for this lies in the thermal spreading of the energy levels or, more precisely, in thermal excitation and motion on the part of the carriers. At high temperatures, of course, the lattice interaction becomes important, and energy exchange with the phonon field will damp the phase coherence.

We now turn to transport in nanostructure systems in which the electronic states are completely quantized. In Section 2.3.2 we briefly introduced quantum dots (sometimes referred to as quantum boxes) in which confinement was imposed in all three spatial directions, resulting in a discrete spectrum of energy levels much the same as an atom or molecule. We can therefore think of quantum dots and boxes as artificial atoms, which in principle can be engineered to have a particular energy level spectrum. In Section 4.1, we first consider models for the electronic states of quantum dots and boxes, and then compare these to experimental data. 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. In addition, magnetic fields serve as an experimental probe that one can use to elucidate the energy spectrum of such artificial atoms discussed below.

The primary focus of our attention in this book is on the transport properties in nanostructures; quantum dots provide some of the most interesting experiments in this regard. Transport in quantum dots and boxes implies an external coupling to these structures from which charge may be injected, as discussed in Chapter 3. Rich phenomena are observed 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 quantum wires, quantum dot structures are sufficiently small that even the introduction of a single electron is sufficient to dramatically change the transport properties due to the charging energy associated with this extra electron.