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.

It is often said that nanostructures have become the system of choice for studying transport over the past few years. What does this simple statement mean?

First, consider transport in large, macroscopic systems. Quite simply, for the past fourscore years, emphasis in studies of transport has been on the Boltzmann transport equation and its application to devices of one sort or another. The assumptions that are usually made for studies are the following: (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.

In contrast to the above situation, it has become possible in the last decade or so to make structures (and devices) in which characteristic dimensions are actually smaller than the appropriate mean free paths of interest. In GaAs/GaAlAs semiconductor heterostructures, it is possible at low temperature to reach mobilities of 106 cm2/Vs, which leads to a (mobility) mean free path on the order of 10 μm and an inelastic (or phase-breaking) mean free path even longer. (By “phase-breaking” we mean decay of the energy or phase of the “wave function” representing the carrier.)

Our discussion of lattice-gas models now takes a qualitative turn. We continue to study fluid mixtures as in the previous chapter, but now they will exhibit some surprising behavior—they won't like to mix!

This change in direction also steers us towards the heart of this book: models for complex hydrodynamics. The particular kind of complexity we introduce in this chapter relates to interfaces in immiscible fluids such as one might find in a mixture of oil and water. We are all familiar with the kind of bubbly complexity that that can entail. So it seems all the more remarkable that only a revised set of collision rules are needed to simulate it with lattice gases. Indeed, the models of immiscible fluids that we shall introduce are so close to the models of the previous chapters that we call them immiscible lattice gases.

This chapter, an introduction to immiscible lattice-gas mixtures, is limited to a discussion of two-dimensional models. In the next chapter, we introduce a lattice-Boltzmann method that is the “Boltzmann equivalent” of the immiscible lattice gas. That then sets the stage for our discussion of three-dimensional immiscible lattice gases in Chapter 11.

Color-dependent collisions

In the miscible lattice gases of the previous chapter, the collision rules were independent of color. The diffusive behavior derived instead from the redistribution of color after generic colorblind collisions were performed. Aside from some diffusion, the color simply went with the flow.

In this chapter we give a full derivation of the Navier-Stokes equation for the lattice gas. The first step, the Boltzmann approximation, is an approximation of the exact Liouville dynamics. The Boltzmann approximation should not be confused with the lattice-Boltzmann method of Chapter 6 and Chapter 7, but results that we obtain here for the Boltzmann approximation and the Navier-Stokes equations are also useful for the Boltzmann method. One of these results is the H-theorem for lattice gases. In this chapter we also reopen the tricky issue of the spurious invariants of lattice-gas or Boltzmann dynamics. We specifically discuss non-uniform global linear invariants for which, unlike the general nonlinear invariants, some theoretical results are known. Among them we find the staggered momentum invariants. We discuss their effect on hydrodynamics, which leads us to corrections to the Euler equation of Chapter 2.

General Boolean dynamics

It is useful to express the Boolean dynamics as a sequence of Boolean calculations, as we did in Section 2.6. In this section we shall denote by s or n local configurations. We further define a field of “rate bits”, which are random, Boolean variables, defined independently on each site and denoted by ass′(x, t). They are equal to one with probability 〈ass′〉 = A(s, s′). Thus if the pre-collision configuration is n the post-collision configuration is that single s′ for which ans′ = 1.

Our objectives for this chapter are twofold. First, we review some elementary aspects of fluid mechanics. We include in that discussion a classical derivation of the Navier-Stokes equations from the conservation of mass and momentum in a continuum fluid. We then discuss the analogous conservation relations in a lattice gas. Finally, we briefly describe the derivation of hydrodynamic equations for the lattice gas, but defer our first detailed discussion of this subject to the following chapter.

Molecular dynamics versus continuum mechanics

The study of fluids typically proceeds in either of two ways. Either one begins at the microscopic scale of molecular interactions, or one assumes that at a particular macroscopic scale a fluid may be described as a smoothly varying continuum. The latter approach allows us to write conservation equations in the form of partial-differential equations. Before we do so, however, it is worthwhile to recall the basis of such a point of view.

The macroscopic description of fluids corresponds to our everyday experience of flows. Figure 2.1 shows that a flow may have several characteristic length scales li. These lengths scales may be related either to geometric properties of the flow such as channel width or the diameter of obstacles or to intrinsic properties such as the size of vortical structures. The smallest of these length scales will be called Lhydro.

Non-dissipative, inviscid, compressible flow

We consider a simple fluid such as air or water. This fluid is described by a number of thermodynamical fields, such as pressure, density, etc., as well as a velocity field u. A variable is called specific when it gives a quantity per unit mass. For instance, let E be the internal energy of a finite volume V of fluid. Let M = ρV be the mass of this volume of fluid. Then e = E/M is the specific internal energy. Table C.1 lists all the thermodynamic variables used in this appendix.

In addition to thermodynamic variables there are variables describing the external actions on the fluid. The effect of gravity, for instance may be represented by the acceleration f = g. The heating rate per unit mass q represents sources of heat, for instance from radiation. There may be heat and momentum exchanges inside the fluid, by heat conduction or viscous forces. These are not taken into account in the non-dissipative description.

For a one-component gas, there are only two independent thermodynamic variables. As a special choice, one may choose ρ, T as independent variables and express all quantities such as p, e, h, etc., in terms of ρ and T, but other choices are possible as well. These relations are linked through equations of state. For instance p = p(ρ, T) is the standard form for an equation of state.

We now provide our first detailed derivation of the hydrodynamics of lattice gases. To keep matters from becoming unnecessarily complicated, we mostly restrict the discussion in this chapter to two-dimensional (2D) models. We begin with the simplest possible 2D model on a square lattice. We then repeat the calculation for the hexagonal lattice model. The principal result of this chapter is the derivation of the Euler equation of both models. This equation has the form already indicated in Chapter 2 but here the unknown coefficients are explicitly calculated. For future use we also include a general calculation in arbitrary dimension D and with an arbitrary number of rest particles.

The hydrodynamic behavior that we thus find at the macroscopic scale is a consequence of the existence of a kind of thermodynamic equilibrium. This equilibrium state is described by the Fermi-Dirac distribution of statistical mechanics. How this distribution arises is described in detail in Chapters 14 and 15. In this chapter we give a simpler derivation of some properties of equilibrium, which are sufficient to obtain the Euler equation.

Homogeneous equilibrium distribution on the square lattice

Our first task is to calculate 〈ni〉, the average value of the Boolean variable ni introduced in Section 2.6. Repeated applications of the rules of propagation and collision in the lattice gas cause these average particle populations to quickly reach an equilibrium state regardless of initial conditions.