Pedagogical presentation and analysis of the symmetry aspects of physical systems in terms of group theoretical concepts and methodology has been evolving over the past six or seven decades, since the pioneering textbooks by Weyl and Wigner first appeared. This constantly evolving pedagogy has resulted in over a hundred textbooks on the subject. The impetus behind these efforts has stemmed from the general recognition of the invaluable role that the application of such methodology plays in determining and predicting the properties of a physical system.

Symmetry concepts provide a very useful means for systematizing the description of a physical system in terms of its energy and momentum, and other relevant physical quantities. Furthermore, the incipient methodologies furnish a very efficient framework for classifying its physical states, and a crucial machinery for simplifying the intervening numerical applications of physical laws. By means of the irreducible representations of its symmetry group, one can classify physical states and particles in a logical way and establish selection rules, which predict restrictions on possible transitions between different physical states. The use of symmetry also simplifies numerical calculations, for example, in solving the Schrö equation for condensed matter systems. Moreover, from the symmetry properties of a physical system, one can make conclusions about the values of measurable physical quantities, and, conversely, one can trace a symmetry group of a system from observed regularities in measured quantities. There is also an intimate connection between symmetry, invariance and dynamical laws.

Introduction

Information regarding the electronic structure of a condensed matter system provides the basis for understanding the myriad of its physical properties: optical, mechanical, magnetic, electrical, etc. There are two main hurdles, however, that seriously hamper any attempt to derive the electronic states of condensed matter systems: The first arises from the gigantic difference in the time scales associated with the motions of electrons and nuclei, or ions, which can be in the order of 103−105. The second difficulty concerns the numbers of particles involved, which are at least of the order of Avogadro's number of ∼1024. In order to overcome the first hurdle, we invoke the adiabatic, or Born–Oppenheimer, approximation, which we will discuss in the following chapter. The only impact of this approximation here is that we treat the ions classically and fix all the ionic positions, {R}; we introduce their interactions with the electron system as an external potential, V (r, {R}). For simplicity, we drop {R} from the notation in the remainder of the chapter.

The one-electron approximations and self-consistent-field theories

Similar presentations to the content of this section can be found in references.

The many-body problem

Our objective here is to determine the ground-state properties of an interacting manyelectron system subject to an external potential V (r), representing the interaction with the frozen ions.

Introduction

So far we have mainly considered the symmetries of condensed matter systems with regular periodicity in one-, two-, and three-dimensional space (1D, 2D, and 3D). We demonstrated that manifestations of such periodicities are lattice structures in real Euclidean space, and that when a lattice is decorated with an atomic basis we obtain a crystal. Moreover, we showed that there is a finite number of crystal structures, and that each crystal type has a compatible space-group symmetry, comprising point-group and translation operations. In 2D, we classified the lattices into 5 distinct types, with 17 crystallographic space-groups involving 10 point-groups, while in 3D we were able to identify 14 lattices, 230 monochromatic Fedorov space-groups with 32 point-groups, or 1058 Shubnikov dichromatic spacegroups with 58 point-groups. We exhaustively discussed the effect of the lattice structures and their underlying symmetries on the different physical properties of material systems.

However, over the past three decades we have witnessed the emergence of systems that do not exhibit periodic structures, yet display well-defined long-range order, manifest in well-defined diffraction patterns comprising of sharp Bragg peaks. These systems have been named quasi-periodic (QP). The lack of periodicity in QP systems in 2D and 3D leads to the lifting of related constraints imposed on the regular lattices. For example, it was found that certain QP structures admit the packing of two types of cells, pentagonal and icosahedral. This evidence defies previously accepted notions that long-range order was incompatible with both pentagonal and icosahedral packing, let alone that only one type of cell was admissible in crystal packing.

Groups and their realizations

Abstract group theory defines relationships among a set of abstract elements in terms of binary operations among the elements of the group. The operations are known as group multiplication.

Formally, a group consists of a set of elements with the following properties:

1. (i) The product of any two elements in the set is a member of the set. Thus the set is closed under all group multiplication operations.

2. (ii) If A, B, and C are elements of the group, then A(BC) = (AB)C. The associative law of multiplication holds; the commutative law of multiplication need not hold.

3. (iii) There is a unit element, an identity element, E such that EA = AE = A.

4. (iv) Each element A has a unique inverse A−1 such that AA−1 = A−1A = E.

A typical abstract group multiplication table is given in Table 2.1, for the group we denote by G6, which consists of six elements.

The convention for such tables is that the ij th element in the table is the product of the ith element labeling the rows and the j th element labeling the columns. From the table we see that AB = D, which means that the operation B followed by the operation A is equivalent to the single operation D. Note that AB = D, but BA = F; thus AB ≠ BA in this case.

Abelian groups If XY = YX for all elements of the group, the group is called Abelian. It is clear from the asymmetry about the diagonal of Table 2.1 that the group G6 is not Abelian.

Introduction

The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles.

Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether.

At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant.

Introduction

In previous chapters, fundamental physical processes on the nanoscale, analysis of nanomaterials, and nanofabrication methods were all discussed extensively. The knowledge gained in these previous discussions makes it possible to consider and analyze a variety of different nanostructure devices. In this chapter, we consider electronic, optical, and electromechanical devices. Some of these devices mimic well-known microelectronic devices but with small dimensional scales. This approach facilitates applications to devices with shorter response times and higher operational frequencies that operate at lower working currents, dissipate less power, and exhibit other useful properties and enhanced characteristics. Such examples include the field-effect transistors and bipolar transistors considered in Sections 8.3 and 8.5.

On the other hand, new generations of the devices are based on new physical principles, which can not be realized in microscale devices. Among these novel devices are the resonant-tunneling devices analyzed in Section 8.2, the hot-electron (ballistic) transistors of Section 8.5, single-electron-transfer devices (Section 8.4), nanoelectromechanical devices (Section 8.7), and quantum-dot cellular automata (Section 8.8).

As a whole, the ideas presented in this chapter provide an understanding of the future development of nanoelectronic and optoelectronic devices that may be realized through the wide use of nanotechnology.

Resonant-tunneling diodes

Diodes or, in other words, two-terminal electrical devices, are the simplest active elements of electronic circuits. Some applications of diodes are based on their nonlinear current–voltage characteristics. Another important capability required of diodes is their operational speed.

Introduction

In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values.

Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics.

The Schrödinger wave equation

From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.

Introduction

Now, we begin our analysis of novel developments in electronics that have resulted from the use of nanostructures in modern electronic devices. Importantly, the attributes of nanotechnology make it possible to pursue both devices with smaller dimensional scales and novel types of device. Though the ongoing trend of miniaturization in electronics is extremely important, the unique properties of electrons in nanostructures give rise to novel electrical and optical effects, and open the way to new device concepts. The electric current and voltage in a device are determined by two major factors: the concentration and the transport properties of the charge carriers. In nanostructures, these factors can be controlled over wide ranges. In this and the next chapter we will study nanostructures for which these basic factors that are important for the electronics are engineered, which are being exploited intensively both in research laboratories and in practical nanoelectronics.

To distinguish the nanostructures already having applications from the newly emerging systems, we refer to the former as traditional low-dimensional structures.

Electrons in quantum wells

In this section, we consider a few particular examples of nanostructures with two-dimensional electrons.

As a basis for the further analysis, we will recall and develop several of the previously introduced definitions and properties of an electron gas. In what follows, the effect of the band offset arising at a junction of two semiconducting materials, which was defined in Section 4.5 via electron affinities, is critically important.

Introduction

In previous chapters we studied advances in materials growth and nanostructure fabrication. In the case of electrons, we paid primary attention to the quantization of their energy in nanostructures. In fact, electronics relies upon electric signals, i.e., it deals with measurements of the electric current and voltage. Controlling and processing electric signals are the major functions of electronic devices. Correspondingly, our next task will be the study of transport of charge carriers, which are responsible for electric currents through nanostructures.

The possible transport regimes of the electrons are dependent on many parameters and factors. Some important aspects of these regimes can be elucidated by comparing the time and length scales of the carriers with device dimensions and device temporal phenomena related to operating frequencies. Such an analysis is carried out in Section 6.2. In Sections 6.3 and 6.4 we discuss the role of electron statistics in transport effects. Then, we consider the behavior of the electrons in high electric field, including so-called hot-electron effects. Analyzing very short devices, we describe dissipative transport and the velocity-overshoot effect. Finally, we consider semiclassical ballistic motion of the electrons and present ideas on quantum transport in nanoscale devices in Section 6.5.

Time and length scales of the electrons in solids

We start with an analysis of possible transport regimes of the electrons in nanostructures. Since there is a large number of transport regimes, we introduce their classification in terms of characteristic times and lengths fundamentally inherent to electron motion.