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Solid state physics is a field in modern physics in which one is mainly concerned with the physical properties of and physical processes in various solids. Besides its fundamental significance, a clear understanding of different physical properties of and physical processes in solids and their origin may provide insight for possible practical applications of relevant properties and physical processes. Since the middle of the twentieth century, many achievements in the field have made great contributions to modern science and technology, even resulting in revolutionary developments. We can expect that further achievements in this field will continually bring tremendous benefits to human beings and society.

One-dimensional crystals are the simplest crystals. Historically, much of our current fundamental understanding of the electronic structures of crystals were obtained through the analysis of one-dimensional crystals [–]. Among the most well-known examples are the Kronig-Penney model [], Kramers’ general analysis of the band structure of one-dimensional infinite crystals [], Tamm’s surface states [], and so forth. In order to have a clear understanding of the electronic states in low-dimensional systems and finite crystals, the first step is to have a clear understanding of the electronic states in onedimensional finite crystals. To prepare for this purpose, in this chapter we begin with a more general study on the properties of the solutions of the relevant differential equations the second-order linear homogeneous ordinary differential equations with periodic coefficients. In the theory of boundary value problems for ordinary differential equations, the existence and locations of the zeros of the solutions of such equations are often of central importance. After reviewing some elementary knowledge on the theory of second-order linear ordinary differential equations, we introduce two basic theorems on the zeros of solutions of second-order linear homogeneous ordinary differential equations. In the major part of this chapter, we will learn some more advanced theory on the second-order linear homogeneous differential equations with periodic coefficients and the zeros of their solutions.

A one-dimensional semi-infinite crystal is the simplest periodic system with a boundary. Based on a Kronig-Penney model, Tamm was the first to find that the termination of the periodic potential due to the existence of a barrier at the boundary in a one-dimensional semi-infinite crystal can cause localized surface states to exist in band gaps below the barrier height []. Now after more than 70 years, the investigations of the properties of surface states and relevant physical and chemical processes have become an important field in solid state physics and chemistry [–]. Among the many surface states of different origins, the surface states caused purely by the termination of the crystal periodic potential are not only the simplest but also the most fundamental surface states. In this chapter, we present a general single-electron analysis on the existence and properties of surface states caused purely by the termination of the crystal periodic potential in one-dimensional semi-infinite crystals.

In this chapter, we present a general investigation on the electronic states in ideal one-dimensional crystals of finite length , where is the potential period and is a positive integer. On the basis of the theory of differential equations in Chapter 2, exact and general results on the electronic states in such an ideal finite crystal can be analytically obtained. We will see that in obtaining the results in this chapter, it is the understanding of the of the solutions of a one-dimensional Schrödinger differential equation with a periodic potential that plays a fundamental role.

Starting from this chapter, we extend our investigations in Part II to threedimensional crystals. The major difference between the problems treated in this part and in Part II is that the corresponding Schrödinger equation for the electronic states in a three-dimensional crystal is a differential equation; therefore, now the problem is a more difficult one. This is due to the fact that relative to the solutions of ordinary differential equations, the properties of solutions of partial differential equations are much less understood mathematically [e.g.,1], not to mention solutions of partial differential equations with periodic coefficients []. The variety and complexity of the three-dimensional crystal structures and of the shapes of three-dimensional finite crystals further make the cases more variational and more complicated. Nevertheless, based on the results of extensions of a mathematical theorem in [], we show that in many simple but interesting cases, the properties of electronic states in ideal low-dimensional systems and finite crystals can be understood, how the energies of these electronic states depend on the size and/or the shape of the system can be predicted, and the energies of many electronic states can be directly obtained from the energy band structure of the bulk. Again, the major obstacle due to the lack of translational invariance can be circumvented.

In this chapter, we investigate the electronic states in ideal quantum wires.We are interested in the electronic states in rectangular quantum wires, which can be considered as the electronic states in a quantum film discussed in Chapter 5 further confined in one more direction. In particular, we are interested in those simple cases where the two primitive lattice vectors 1 and 2 in the film plane are perpendicular to each other. By using an approach similar to that used in Chapter 5, we try to understand the further quantum confinement effects in a quantum wire of two-dimensional Bloch waves in (5.28) and in (5.33) in a quantum film that were obtained in Chapter 5. It is found that each type of two-dimensional Bloch waves will produce two different types of one-dimensional Bloch waves in the quantum wire.

The electronic states in an ideal finite crystal or quantum dot can be considered as the electronic states in an ideal quantum wire further confined in one more direction. In this chapter, we are interested in the electronic states in an orthorhombic finite crystal or quantum dot that can be considered as the onedimensional Bloch waves in a rectangular quantum wire discussed in Chapter 6 further confined by two boundary surfaces perpendicularly intersecting the 1 axis at τ11 and (τ1 + N1)1; here, 1 is a positive integer. By using an approach similar to that used in the last two chapters, we can understand that the further quantum confinement of each set of one-dimensional Bloch waves in an ideal quantum wire will produce two different types of electronic states in the ideal finite crystal or quantum dot.

We have presented a single-electron nonspin analytical theory on the electronic states in some simple ideal low-dimensional systems and finite crystals, based on a theory of differential equations approach. By ideal, it is assumed that (i) the potential inside the low-dimensional system or the finite crystal is the same as in a crystal with translational invariance and (ii) the electronic states are completely confined in the limited size of the low-dimensional system or the finite crystal.