In Appendix A, Section A.2, we considered the oscillations of particles using Newton's laws of classical mechanics. The distinctive feature of Newton's mechanics, when considering interactions between particles (for example, the gravitational interaction), is the instantaneous transmission of the interaction between particles, i.e., transmission occurs with infinite speed. The interaction between charged particles is realized through an electromagnetic field, which possesses energy and momentum and is carried through space with finite speed. Electromagnetic waves can exist without any charges in a space in which there is no substance, i.e., in vacuum. This is substantiated by the fact that the equations of classical electrodynamics allow solutions in the form of electromagnetic waves for such conditions.

The main equations of classical electrodynamics are Maxwell's equations, which were formulated after analyzing numerous experimental data. In this sense they are analogous to Newton's equations of classical mechanics. Maxwell's equations are the basis for electrical and radio engineering, television and radiolocation, integrated and fiber optics, and numerous phenomena and processes that take place in materials placed in an electromagnetic field. Together with Newton's equations, Maxwell's equations are the fundamental equations of classical physics. Just like Newton's equations, Maxwell's equations have their limits of applicability. For example, they do not sufficiently well describe the state of an electromagnetic field in a medium at frequencies higher than 1014–1015 Hz.

The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method, using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics.

Stationary perturbation theory for a system with non-degenerate states

This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.

Three-dimensional quantum-dot superlattices can be considered as nanocrystals. Spherical nanoparticles consisting of a big enough number (from 10 to 1000) of atoms or ions, which are connected with each other and are ordered in a certain fashion, can be considered as the structural units of such nanocrystals. Examples of nanocrystals that are of natural origin are the crystalline modifications of boron and carbon which have as their structural units the molecules B12 and C60. The boron molecule B12 consists of 12 boron atoms, and the carbon molecule C60, which is called fullerene, consists of 60 carbon atoms. The fullerene molecule resembles a soccer ball, i.e., it consists of 12 pentagons and 20 hexagons, with carbon atoms at their corners. These nanoparticles form face-centered superlattices with a period of about 1−10 nm. At these distances between molecules of C60 weak molecular forces, which provide the crystalline state of fullerene, act.

In addition to nanocrystals of natural origin, numerous artificial three-dimensional superlattices consisting of various types of nanoparticles have been fabricated. The variety of nanocrystalline structures as well as of conventional crystals is defined by the differences in the distribution of electrons over the quantum states of atoms. The most significant role in the formation of individual nanoparticles as well as of crystals is played by the electrons in the outer shells of atoms.

In the previous chapter we have analyzed the peculiarities of quantized electron motion in layered structures with one-dimensional potential wells. From the mathematical point of view, the solution of the Schrödinger equation for one-dimensional potential profiles is much simpler. However, many quantum objects, such as atoms, molecules, and quantum dots, are three-dimensional objects. Thus, in order to analyze electron motion in such objects we need to find solutions of the Schrödinger equation for three-dimensional potential profiles. The electron motion in spaces with dimensionality higher than one, especially for rectangular potential profiles with infinite potential barriers, is not so difficult to analyze. At the same time we have to keep in mind that such potential profiles frequently represent some approximation of the more complex, real potential profiles. Depending on the type of structure and on the form of the potential profile, the electron motion may be limited in two directions (two-dimensional quantization) or in three directions (three-dimensional quantization). In this chapter we will show that the existence of discrete energy levels in the electron spectrum is an intrinsic feature of electron motion in potential wells of any form and dimensionality.

An electron in a rectangular potential well (quantum box)

In the previous chapter we studied the electron motion in one-dimensional potential wells. An electron's motion along one direction was confined by the potential profile and the momentum in this direction was quantized.

In Chapters 3 and 4 we have discussed electron behavior in potential wells of various profiles and dimensionalities. We have established that localization of electrons in such potential wells, regardless of their form, leads to the discretization of the electron energy spectrum whereby the distance between energy levels substantially depends on the geometrical size of the potential wells. If this size is macroscopic then the distance between the energy levels is so small that we can consider the energy spectrum to be practically continuous (or quasicontinuous). Electrons in metallic samples of macroscopic sizes have this kind of energy spectrum. Another limiting case is that of small clusters consisting of just a few atoms, where the distance between energy levels is of the order of electron-volts. Gradual decrease of one or several geometrical dimensions of the potential well from macroscopic to about 1 μm practically does not change the form of the electron energy spectrum. Very often macroscopic materials (or macroscopic crystals) are referred to as bulk materials or bulk crystals. Changes happen only when the size of structures is of the order of or less than 100 nm. Such structures are called nanostructures. The change of the electron spectrum from quasicontinuous to discrete implies changes in most of the physical properties of nanostructures compared with those in bulk crystals. In this chapter we will consider the main peculiarities of the electron energy spectrum in nanostructures of various dimensionalities.

The main characteristics of atoms and molecules are their structure and their energy spectrum. Under the term structure of an arbitrary particle we usually understand the size and the distribution of its mass and its charge in space. In quantum physics such a distribution is defined by the square of the modulus of the wavefunction of a particle. The particle itself may consist of a system of other particles that are bound by a certain type of coupling. For example, an atom consists of a nucleus and a system of interacting electrons, a molecule consists of a system of interacting atoms, and so on. If we consider an atom, the nucleus of the atom is assumed to be at rest (the so-called adiabatic approximation). This assumption can be made because the nucleus has a much larger mass than that of an electron (a proton's mass is 1836 times larger than the mass of an electron). Then, the square of the modulus of the wavefunction, ∣ψ(r1, r2, …, rn)∣2, for the system of electrons defines the probability density of finding the j th electron at the point rj. Graphically it is very convenient to depict ∣ψ∣2 in the form of an electron cloud, which can be considered as an averaged distribution of matter added to the mass of the nucleus located at the center of an atom.

Nanoelectronics is a field of fundamental and applied science, which is rapidly progressing as a natural development of microelectronics towards nanoscale electronics. The modern technical possibilities of science have reached such a level that it is possible to manipulate single molecules, atoms, and even electrons. These objects are the building blocks of nanoelectronics, which deals with the processes taking place in regions of size comparable to atomic dimensions. However, the physical laws which govern electron behavior in nanoobjects significantly differ from the laws of classical physics which define the operation of a large number of complex electronic devices, such as, for example, cathoderay tubes and accelerators of charged particles. The laws that govern electron behavior in nanoobjects, being of quantum-mechanical origin, very often seem to be very strange from a common-sense viewpoint. The quantum-mechanical description of electron (or other microparticle) behavior is based on the idea of the wave–particle duality of matter. The wave properties of the electron, which play a significant role in its motion in small regions, require a new approach in the description of the electron's dynamic state on the nanoscale. Quantum mechanics has developed a fundamentally new probabilistic method of description of particle motion taking into account its wave properties. This type of description is based on the notion of a wavefunction, which is not always compatible with the notion of a particle's trajectory. This makes electron behavior harder to understand.

A review of milestones in nanoscience and nanotechnology

It is extremely difficult to write the history of nanotechnology for two reasons. First, because of the vagueness of the term “nanotechnology.” For example, nanotechnology is very often not a technology in the strictest sense of the term. Second, people have always experimented with nanotechnology even without knowing about it. Ironically enough, we can say that the medieval alchemists were the founding fathers of nanoscience and nanotechnology. They were the first researchers who tried to obtain gold from other metals. The ancient Greek philosopher Democritus also can be considered as a father of modern nanotechnology, since he was the first to use the name “atom” to characterize the smallest particle of matter. The red and ruby-red opalescent glasses of ancient Egypt and Rome, and the stained glasses of medieval Europe, can be considered as the first materials obtained using nanotechnology. An exhibition at the British Museum includes the Lycurgus cup made by the ancient Romans. The glass walls of the cup contain nanoparticles of gold and silver, which change the color of the glass from dark red to light gold when the cup is exposed to light. In 1661 the Irish chemist Robert Boyle for the first time stated that everything in the world consists of “corpuscules” – the tiniest particles, which in different combinations form all the varied materials and objects that exist.

Nanotechnology is based on the ability to manipulate individual atoms and molecules in order to assemble them into bigger structures. Such artificial nanoscale structures, usually fabricated using self-assembly phenomena, possess new physical, chemical, and biological properties. The fabrication of various types of nanostructures and study of their properties require new technological means and new principles.

Nanotechnology has initiated a new so-called bottom-up technology. The bottom-up technology is based on the self-assembly phenomenon, i.e., the process of formation of complex ordered structures from simpler ones. The main idea of this technology is in the development of the controlled self-assembly of the atoms, molecules, and molecular chains into nanoscale objects. The bottom-up technology allows the fabrication of nanoobjects, such as quantum dots, quantum wires, and superlattices.

The bottom-up approach is opposite in principle to the traditional approach, which may be called the top-down approach, which is based on the sequential decrease of the object's size by means of mechanical or chemical processing for the fabrication of objects of nanoscale size (nanoobjects). Thus, for example, some of the nanoparticles can be obtained by grinding material consisting of particles of micrometer or larger size in a special grinder. The traditional technologies include laser methods for the processing of semi-conductor surfaces and making masks of various configurations and sizes for photolithography.

In this chapter we will briefly consider the main fabrication and characterization techniques for nanostructures and will give some examples of applications of nanostructures in modern nanoelectronics.