Solid State Physics

By Giuseppe Grosso and Giuseppe Pastori Parravicini

Solid State Physics is a textbook for students of physics, material science, chemistry, and engineering. It is the state-of-the-art presentation of the theoretical foundations and application of the quantum structure of matter and materials.

This second edition provides timely coverage of the most important scientific breakthroughs of the last decade (especially in low-dimensional systems and quantum transport). It helps build readers' understanding of the newest advances in condensed matter physics with rigorous yet clear mathematics. Examples are an integral part of the text, carefully designed to apply the fundamental principles illustrated in the text to currently active topics of research.

Basic concepts and recent advances in the field are explained in tutorial style and organized in an intuitive manner. The book is a basic reference work for students, researchers, and lecturers in any area of solid-state physics.

• Features additional material on nanostructures, giving students and lecturers the most significant features of low-dimensional systems, with focus on carbon allotropes
• Offers detailed explanation of dissipative and nondissipative transport, and explains the essential aspects in a field, which is commonly overlooked in textbooks
• Additional material in the classical and quantum Hall effect offers further aspects on magnetotransport, with particular emphasis on the current profiles
• Gives a broad overview of the band structure of solids, as well as presenting the foundations of the electronic band structure. Also features reported with new and revised material, which leads to the latest research

• Language: English
• Publisher: Academic Press
• Release date: Oct 17, 2013
• ISBN: 9780123850317

Giuseppe Grosso

Giuseppe Grosso graduated in Physics at the University of Pisa in 1972 and PhD from the Scuola Normale Superiore in 1977, He is a retired full professor of Solid State Physics at the Physics Department of Pisa. The main research topics addressed concern electronic and optical properties of perfect 3D and nanostructured solids, Green’s function, recursion and renormalization methods, continued fractions coherent transport, Keldysh formalism, conjugated polymers and molecular crystals, silicon and germanium based photonics.

Book preview

Preface to the second edition

These last years have witnessed a continuous progress in the traditional areas of solid state physics, as well as the emergence of new areas of research, rich of results and promises. The countless novelties appeared in the literature with the beginning of this century are one of the main motivations of the second edition of Solid State Physics. Besides the inclusion of new material, the content has been greatly rationalized and the manuscript appears now in a widely renewed dress. Furthermore, this second edition is accompanied and enriched by the presence of numerous Appendices, containing better in-depth treatment of specific topics, or containing significant solved problems, for the readers willing to increase the proper technical abilities.

The chapters of Solid State Physics are in large part self-contained, and can be chosen with great flexibility and split in two semesters to cover an annual course for the degree in physics, or also for the degree in material science and electrical engineering. The volume is also suited for the graduate students, who have not followed similar in-depth courses in their curricula. From a didactical point of view, it is a steady characteristics of this book to enhance gradually the level of difficulty starting from the initial models and leading as near as possible to the frontier science. The purpose is to provide within the reasonable length of a textbook the general cultural baggage needed for all researchers, whose activity is oriented, or is going to be oriented, in the fascinating field of solid state physics.

We wish to express our deepest gratitude to our colleagues for their valuable suggestions, and to our students for their challenging questions; in their own way, they have made the most relevant contribution to improve and inspire this textbook. Many warm thanks are due to Donna de Weerd-Wilson for her encouragement in pursuing this project, and to Paula Callaghan, to Anita Koch, to Sharmila Vadivelan, and to Jessica Vaughan for their assistance and professionalism in the preparation of the manuscript.

Pavia and Pisa, May 2013

Giuseppe Grosso and Giuseppe Pastori Parravicini

Preface to the first edition

This textbook has developed from the experience of the authors in teaching the course of Solid State Physics to students in physics at the Universities of Pavia and Pisa. The book is addressed to students at the graduate and advanced level, both oriented toward theoretical and experimental activity. No particular prerequisite is required, except for the ordinary working knowledge of wave mechanics. The degree of difficulty increases somewhat as the book progresses; however, the contents develop always in very gradual steps.

The material presented in the book has been assembled to make as economical as possible the didactical task of teaching, or learning, the various subjects. The general organization in chapters, and groups of chapters, is summarized in the synoptic table of contents. The first three chapters have a propaedeutic nature to the main entries of the book. Chapter IV starts with the analysis of the electronic structure of crystals, one of the most traditional subjects in solid state physics; Chapters V and VI concern the band theory of solids and a number of specific applications; the concepts of excitons and plasmons are given in Chapter VII. Then, in Chapters VIII and IX, the adiabatic principle and the interdependence of electronic states and lattice dynamics are studied. Having established the electronic and vibrational structure of crystals, the successive chapters from X to XIV describe several investigative techniques of crystalline properties; these include scattering of particles, optical spectroscopy, and transport measurements. Chapters XV, XVI, and XVII concern the electronic magnetism of tendentially delocalized or localized electronic systems, and cooperative magnetic effects. The final chapter is an introduction to the world of superconductivity.

From a didactical point of view, an effort is made to remain as rigorous as possible, while keeping the presentation at an accessible level. In this book, on one hand we aim to give a clear presentation of the basic physical facts, on the other hand we wish to describe them by rigorous theoretical and mathematical tools. The technical side is given the due attention and is never considered optional; in fact, a clear supporting theoretical formalism (without being pedantic) is essential to establish the limits of the physical models, and is basic enough to allow the reader eventually to move on his or her own legs, this being the ultimate purpose of a useful book.

The various chapters are organized in a self-contained way for the contents, appendixes (if any), references, and have their own progressive numeration for tables, figures, and formulae (the chapter number is added when addressing items in chapters differing from the running one). With regard to the references, these are intended only as indicative, since it is impossible to mention, let alone to comment on, all the relevant contributions of the wide literature. Since the chapters are presented in a (reasonably) self-contained way, the lecturers and readers are not compelled to follow the order in which the various subjects are discussed; the chapters, or group of chapters, can be taken up with great flexibility, selecting those topics that best fit personal tastes or needs. We will be very interested and very pleased to receive (either directly or by correspondence) comments and suggestions from lecturers and readers.

The preparation of a textbook, although general in nature, requires also a great deal of specialized information. We consider ourselves fortunate for the generous help of the colleagues and friends at the Physics Departments of the University of Pavia, University of Pisa, and Scuola Normale Superiore of Pisa; they have contributed to making this textbook far better by sharing their expertise with us. Very special thanks are due to Emilio Doni, who survived the task of reading and commenting on the whole preview-manuscript. Several other colleagues helped with their critical reading of specific chapters; we are particularly grateful to Lucio Claudio Andreani, Antonio Barone, Pietro Carretta, Alberto Di Lieto, Giorgio Guizzetti, Franco Marabelli, Liana Martinelli, Attilio Rigamonti. Many heartfelt thanks are due to Saverio Moroni for his right-on-target comments.

Before closing, we wish to thank all contributors and publishers, who gave us permission to reproduce their illustrations; their names and references are indicated adjacent each figure. We wish also to express our gratitude to Gioia Ghezzi for her encouragement, to Serena Bureau, Manjula Goonawardena, Cordelia Sealy, and Bridget Shine for their assistance in the preparation of the manuscript. Last, but always first, we thank our families for their never ending trust that the manuscript would eventually be completed and would be useful to somebody.

Pavia and Pisa, May 1999

Giuseppe Grosso and Giuseppe Pastori Parravicini

Chapter 1

Electrons in One-Dimensional Periodic Potentials

Abstract

Low dimensionality offers a unique opportunity to introduce some relevant concepts of solid state physics, keeping the treatment at a simple level. This chapter begins with the presentation of the Bloch theorem for one-dimensional periodic lattices. A peculiar aspect of the energy spectrum of an electron in a periodic potential is the presence of allowed and forbidden energy regions. One-dimensional approaches are particularly suited to show from different points of view (weak binding, tight-binding, quantum tunneling, continued fractions) the mechanism of formation of energy bands in solids. The focus of this chapter is on one-dimensional periodic potentials, with traditional contributions going back to the early times of the foundation of quantum mechanics; brief digressions are given to evidence the conceptual link between this historical area of solid state and other much younger and very active areas of research such as artificial structures and superlattices, photonic crystals, atomic condensates.

Keywords

Bloch theorem; Quantum wells; Kronig-Penney model; Electron tunneling; Resonant tunneling; Tight-binding model; Nearly free-electron model; Quasi-momentum; Effective mass; Bloch oscillator; Density-of-states; Green’s function Continued fractions

Low dimensionality offers a unique opportunity to introduce some relevant concepts of solid state physics, keeping the treatment at a reasonably simple level. In the early days of solid state physics, just this desire for technical simplicity was the motivation for the studies on one-dimensional periodic potentials. More recently, with the restless developments in the world of nanoscience and nanotechnology, low-dimensional models have had a renaissance for the understanding of a number of realistic situations. At the same time a note of warning is necessary: the features of one-dimensional systems that have any relevance beyond dimensionality must be assessed situation by situation; cavalier extensions of one-dimensional results to actual three-dimensional crystals may be misleading or even completely unreliable.

The material of this chapter is organized and presented so that it can also be embodied in standard courses of Structure of Matter or Quantum Mechanics; in fact no previous knowledge is needed other than elementary ideas about wave mechanics.

This chapter begins with the presentation of the Bloch theorem for one-dimensional periodic lattices. A peculiar aspect of the energy spectrum of an electron in a periodic potential is the presence of allowed and forbidden energy regions. One-dimensional approaches are particularly suited to show from different points of view (weak binding, tight-binding, quantum tunneling, continued fractions) the mechanism of formation of energy bands in solids. The semiclassical dynamics of electrons in energy bands is then considered; together with the Pauli exclusion principle for occupation of states, it gives a qualitative distinction between metals, semiconductors, and insulators.

Before beginning our trip among the one-dimensional models of primary interest in the field of solid state, we wish to notice the frequent trespassings that have occurred, in the course of the years, among areas quite different from the original context. For instance the Kronig-Penney model, originally suggested to justify qualitatively the formation of electronic energy bands in periodic materials, is often encountered as a precious tool in the study of electronic states in artificial superlattices, or of the propagation of electromagnetic waves in photonic crystals. Similarly, the concept of Bloch oscillations of electrons in periodic potentials and external electric fields, experimentally rather elusive even for semiconducting superlattices, has been well verified for atomic condensates in optical lattices, where gravity takes the role played by electric fields in semiconductors. Bloch oscillations of light waves have also been observed in dielectric superlattices, where a refractive index gradient plays the optical analog of the external force. Although the focus of this chapter is on one-dimensional periodic potentials, with traditional contributions going back to the early times of the foundation of quantum mechanics, it is worthwhile to be aware of the cross-fertilization of concepts and techniques between this historical area of solid state and other much younger and very active areas of research, which include for instance artificial structures and superlattices, organic crystals, photonic crystals, atomic condensates.

1.1 The Bloch Theorem for One-Dimensional Periodicity

and the corresponding Schrödinger equation

(1.1)

The solutions of Eq. .

, satisfies the relation

(1.2)

can be expressed in the form

(1.3)

such that

(1.4)

We wish to analyze the implications on the eigenfunctions and eigenvalues of Eq. is periodic, and hence its Fourier spectrum is discrete, according to Eq. (1.3).

Let us start considering Eq. vanishes (empty lattice). In the free-electron case, the wavefunctions are simply plane waves and can be written in the form

(1.5)

. The plane waves (1.5) constitute a complete set of orthonormal functions, that can be conveniently used as an expansion set.

Let us now consider the eigenvalue problem :

, is named first Brillouin zone (or simply Brillouin zone).

, can be expressed as an appropriate linear combination of the type

(1.6)

the function

. Equation (1.6) then takes the form

(1.7)

This expresses the Bloch theorem: any physically acceptable solution of the Schrödinger equation in a periodic potential takes the form of a traveling plane wave modulated on the microscopic scale by an appropriate function with the lattice periodicity.

The Bloch theorem, summarized by Eq. (1.7), can also be written in the equivalent form

(1.8)

is any translation in the direct lattice. It is easy to verify that Eq. .

We can finally notice that, in the case of a generic potential of type includes all plane waves in the form

(1.9)

Nothing specific and general can be inferred from Eq. : in fact, for aperiodic potentials, it is possible to find localized wavefunctions for the whole spectrum, itinerant ones, both types (separated by mobility edges), or even solutions belonging to fractal regions of the spectrum.

The Bloch theorem plays a central role in the physics of periodic systems; not only it characterizes the itinerant form of the wavefunctions summarized by Eqs. (1.7) and (1.8), but also entails the fact that the energy spectrum consists, in general, of allowed energy regions separated by energy gaps , as can be seen from direct inspection of Eq. (1.1) under complex conjugate operation (or in a more formal way by use of group theory analysis of time reversal symmetry of the electronic Hamiltonian).

A feature of one-dimensional periodic potentials is that the allowed energy bands cannot cross there are only two linearly independent solutions of the differential Eq. , where dEin general vanishes. (The non-crossing of one-dimensional bands is confirmed also by group theory analysis; in one dimension the group of symmetry operations (neglecting spin) is too small to imply degeneracy of bands; on the contrary, in two- and three-dimensional crystals crossing of energy bands is possible at high symmetry points or lines in the Brillouin zone.)

So far we have considered Eq. ). The reason to consider a very large macroscopic region, rather than an infinite one, is simply a matter of convenience, mainly for counting states and distributing electrons in the energy bands. In order not to affect the physics by boundary effects we use cyclic or Born-von Karman boundary conditions for the wavefunctions. This consists in the requirement

(1.10a)

are considered as physically equivalent.

; then the boundary condition to the ones that satisfy

(1.10b)

must be thought of as a dense (although discrete) variable; notice that the first Brillouin zone contains a number of uniformly distributed k points, equal to the number N of cells of the lattice.

1.2 Energy Levels of a Single Quantum Well and of a Periodic Array of Quantum Wells

One of the most elementary problems in quantum mechanics is the study of the energy levels of a particle in a single quantum well. Similarly, one of the most elementary applications of the Bloch theorem is the study of the energy bands of a particle moving in a periodic array of quantum wells. The periodically repeated quantum well model was introduced by Kronig and Penney in 1931 to replace the actual crystal potential with a much more manageable piecewise constant potential; in this way, in each well (or in each barrier) the linearly independent solutions of the Schrödinger equation are simple trigonometric (or exponential) functions. Standard boundary conditions of continuity of wavefunctions and currents, combined with Bloch conditions required by periodicity, easily lead to an analytic compatibility equation for the eigenvalues of the crystal Hamiltonian. Successively, the Kronig-Penney model, with appropriate generalizations, has found a revival in the very active field of research of the electronic states of layered structures and superlattices [superlattices are artificial materials, obtained by growing on a substrate a controlled number of layers of two or more chemically similar crystals, in an appropriate periodic sequence]. Applications also include other models of field propagation, such as mechanical waves in phononic crystals and electromagnetic waves in photonic crystals.

Energy Levels of a Single Quantum Well

is indicated in elsewhere. The potential has inversion symmetry around the center of the well; thus the eigenfunctions of Eq. (1.1) must be either even or odd under spatial inversion.

Figure 1.1 . (b) Normalized wavefunctions corresponding to the bound states of the finite quantum well under attention.

) takes the form

(1.11)

are two arbitrary coefficients.

The standard boundary conditions requiring continuity of the wavefunction ) give

We thus obtain the transcendental equation

(1.12a)

which expresses the compatibility relation for even parity states.

) give

(1.12b)

we see that both conditions (1.12a) and (1.12b) are equivalent to the condition

(1.12c)

We will provide a graphical picture of the expressions above, after discussion of the infinite barrier case.

. Equations , and the energy spectrum becomes

(1.13)

where odd integers correspond to even solutions and even integers correspond to odd solutions. Equation (1.13) can also be written in the form

.

In the case of finite height well, we have to consider the transcendental equations (1.12a) and (1.12b) here rewritten respectively in the forms

(1.14)

.

. In the upper part of .

Figure 1.2 Graphical solution of the Eqs. determine the solutions of the compatibility equation. Notice also that at least one bound state always occurs.

, remains finite. From Eq. . It follows:

the binding energy, we have

(1.15)

, as shown also in Problem 1.

Energy Levels of Periodically Repeated Quantum Wells

Consider now the case of two (or more) potential wells; if one decides to proceed with the same technique adopted for the single well enforcing the boundary conditions is the number of potential wells. It becomes soon apparent that such a procedure, from the point of view of achieving analytic results and interpretations, is not particularly encouraging, even if the number of wells is as small as two or three. However, in the case the number of periodically repeated wells is infinite, we can again solve exactly the eigenvalue matching problem, by virtue of the Bloch theorem, as shown below.

Consider an infinite periodic sequence of rectangular wells, as shown in has the form

(1.16)

could be dealt with in a similar way).

Figure 1.3 One-dimensional potential formed by a periodic array of quantum wells.

must be chosen in such a way to satisfy the following four boundary conditions:

(1.17a)

(1.17b)

The two conditions ), as required by the Bloch theorem.

The conditions (1.17) lead to the following linear homogeneous equations for the unknown coefficients A, B, C, D:

vanishes:

minors. With easy calculations and collection of terms we obtain

(1.18)

It is possible to solve graphically the compatibility equation constant; we obtain the simplified compatibility equation

(1.19)

, as in the historical Kronig-Penney model.

The graphical solution of Eq. that are allowed. From Figure 1.4, it can be seen by inspection that the energy levels are grouped into allowed energy bands separated by forbidden energy regions.

Figure 1.4 regions where the compatibility equation is satisfied have been enhanced for convenience.

It is instructive to consider the compatibility equation . The energy levels are

.

, and in appear at the border of the Brillouin zone.

Figure 1.5 Bohr radius).

).

1.3 Transfer Matrix, Resonant Tunneling, and Energy Bands

In this section we consider the electron propagation in one-dimensional crystals from the point of view of the electron optics in solid state, via the determination of the reflected and transmitted components of a wave impinging on a given potential. We present first some general properties of transfer matrices for the description of the elastic propagation through a potential of arbitrary shape. Then the conditions of resonant electron tunneling through a double barrier structure formed by two identical barriers is examined. Finally in the case of a crystal pictured as a periodic structure formed by identical barriers, we illustrate the formation of the energy bands from the point of view of quantum tunneling. In all three subsections, the general concepts are applied to rectangular barriers, where the simple analytic results allow a better insight and understanding of the physical aspects.

1.3.1 Transmission and Reflection of Electrons from an Arbitrary Potential

, and connecting two semi-infinite regions at the same constant potential, taken to be zero for simplicity; the two semi-infinite regions are often addressed as leads and the finite region therein as central device or scatterer (see Figure 1.6). The corresponding Schrödinger equation reads

(1.20)

with

can be different from zero.

Figure 1.6 of finite range and arbitrary shape, connected to two leads at zero potential. Arrows indicate the wavevectors of the propagating plane waves of energy E.

, in the left and right leads, can be written as

(1.21)

, the second order differential Schrödinger equation . One can choose to express any two amplitudes in terms of the other two. A frequently adopted convention expresses the amplitudes on the right (left) lead in terms of those on the left (right) lead by means of the forward (backward) transfer matrixof the incoming and outgoing waves in the left lead, and is defined by

(1.22a)

(the backward transfer matrix is the inverse of the forward one). Equation (1.22a) can be written in extended form

(1.22b)

for simplicity of notations, at times the energy dependence of variables is not indicated explicitly.

Alternatively, and equivalently, it is also frequent the use of the scattering matrix on the scatterer; namely:

(1.23)

the last passage in Eq. (1.23) is a straight elaboration of Eq. (1.22b). With respect to the transfer matrix, the scattering matrix approach may present advantages of numerical stability, in some situations that need large scale numerical calculations or involve strongly space dependent wavefunctions. In view of the rather simple models of our concern, we will continue our elaborations with the forward propagation, summarized by Eqs. (1.22), transferring wavefunctions from the left lead to the right one.

, can be obtained by (numerical or analytic) integration of the Schrödinger equation; before considering specific examples, we discuss the general properties of the transfer matrices. First of all we notice that the complex conjugate of any solution of the Schrödinger equation (1.20) is on its own right a solution at the same energy (this property reflects the time reversal symmetry of the Hamiltonian operator under attention). It follows:

(1.24)

by virtue of time reversal symmetry the independent matrix elements of the transfer matrix are limited to two.

is unimodular, i.e. its determinant equals unity. This property follows from probability current conservation, in conjunction with the assumption that the constant potentials in the left and right leads are equal. (In the case the constant potentials in the left and right leads are different, some of the equations below need simple modifications.) In one dimension the probability current associated to an eigenfunction is given by

(1.25)

Expression . In fact its differentiation gives

where the last equality occurs by virtue of the Schrödinger equation (1.20) and its complex conjugate. Using Eqs. (1.21) and (1.25), it is easily established that

), entails

(1.26)

Using Eqs. (1.22) and (1.24) we have

Comparison with Eq. (1.26) shows that the transfer matrix between two leads at the same potential is unimodular. We now determine the physical meaning of the independent matrix elements of the transfer matrix, and its unimodularity, in terms of reflection and transmission amplitudes and coefficients.

Reflection and Transmission Amplitudes and Coefficients

, and Eqs. (1.22b) read

from these two equations, we can easily obtain the reflection and transmission amplitudes and coefficients as follows.

The reflection amplitude for a particle incoming from the left lead is defined as , and is given by

(1.27a)

the reflection coefficient is

(1.27b)

The transmission amplitude to the right lead for a particle incoming from the left lead is , and is given by

(1.27c)

where the unimodularity of the transfer matrix has been taken into account. The transmission coefficient is

(1.27d)

for a particle incoming from the left lead, the transfer matrix can be written in the form

(1.28)

with a clear physical meaning of its matrix elements, and its unimodularity.

A similar elaboration for the scattering matrix expressed by Eq. (1.23) gives

refer to the same quantities for a particle incoming from the right.

, a particularly interesting situation occurs for those energies (if any) for which the transfer matrix is diagonal. This happens when

because of the unimodularity of the transfer matrix. These energies are called resonance energiesis transparent to the particle flux at these special energies.

. It is seen by straight elaboration of the defining Eqs. (1.21) and (1.22) that

with

(1.29)

.

-axis, is given by the product of transfer matrices

(1.30)

in the indicated ordered sequence. The simplicity of this composition is the key point of the transfer matrix procedure (while the composition of the scattering matrices is somewhat more elaborate, as can be seen by inspection).

As an application of the concepts explained so far, we consider here the case of tunneling through a single rectangular barrier. In the next subsection we examine the case of two barriers of equal shape and focus on the resonant tunneling effects. Finally, we will consider the case of periodically repeated rectangular barriers (i.e. the Kronig-Penney model) and the formation of energy bands within the transfer matrix framework.

Transfer Matrix for a Rectangular Barrier and for a Piecewise Potential

As an application of the concepts explained so far, we determine explicitly the transfer matrix for a rectangular barrier; such a matrix also constitutes the basic ingredient to build up the transfer matrix of any piecewise potential.

, shown in has the form

.

Figure 1.7 connecting two leads at zero potential.

give

in the matrix form

(1.31a)

in the form

(1.31b)

The direct multiplication of the transfer matrices in Eqs. for the rectangular barrier; with straightforward calculations we obtain

(1.32)

Expressions , the appropriate phase factors pointed out in Eq. (1.29) are to be included. The transfer matrix for any arbitrary piecewise potential is obtained simply multiplying in the appropriate order the matrices corresponding to each component barrier.

With the transfer matrix given in Eq. for the transmission amplitude, we obtain

denote the modulus and the phase of the transmission amplitude. The above equality for the phases and for the moduli of the two members implies

(1.33)

and

is then

(1.34a)

we have

(1.34b)

similar calculations can be performed and the transmission coefficient becomes

(1.34c)

The typical behavior of the transmission coefficient , the transmission is close to zero. In such a situation, the unity can be neglected in the denominator of Eq. (1.34a) and, apart from prefactors of the order of unity, Eq. (1.34a) takes the approximate form

(1.34d)

the transmission coefficient decreases exponentially, with an exponent which is linear in the width of the barrier and goes as the square root of its height. The approximate expression of Eq. of the barrier (and keeping the squared modulus).

Figure 1.8  Å (solid line).

is exactly equal to 1 at energies such that the argument of the trigonometric function in Eq. ; this occurs for

.

As summarized by Eq. (1.34d), the transmission coefficient for low energy electrons through a barrier is extremely sensitive to the height and width of the barrier. This high sensitivity is the key working principle of the scanning tunneling microscope [G. Binning and H. Rohrer Scanning tunneling microscopy—from birth to adolescence Rev. Mod. Phys. 59, 615 (1987)]. In the experiments, the sharp tip of the microscope is separated by a small gap of a few angstroms from the conducting surface of the material under investigation. The map of the tunneling current flowing between tip and sample, provides information on the surface landscape with an accuracy that may reach atomic resolution.

1.3.2 Double Barrier and Resonant Tunneling

; however, these resonance energies occur in a region where the medium is basically transparent and the transmission coefficient, apart from oscillations, is approaching to unity. A much more interesting situation occurs in a symmetric double barrier structure where ideal transparency may occur also at energies lower than the barriers height, i.e. in regions where each individual barrier could be quite opaque; this highly selective filter effect has the largest interest in the physics of devices and is called resonant tunneling.

, as indicated in , so that the transfer matrix is obtained applying Eq. (1.29). The total transfer matrix for the symmetric double barrier model is the product of the transfer matrices of the two barrier scatterers, namely:

(1.35)

The upper diagonal element of the transfer matrix (1.35) reads

are the reflection coefficient, the transmission coefficient and the phase of the transmission amplitude of a single barrier provided by Eqs. (1.27). The transmission coefficient of the double barrier is then

(1.36)

); resonance energies of the single barrier are resonance energies for the double barrier, too.

Figure 1.9 .

More importantly, it is seen by inspection that perfect transparency also occurs when

(1.37)

The condition (1.37) can be cast in the form

Using Eq. (1.33) it is seen that the above condition is fully equivalent to the requirement

(1.38a)

, we have that the resonant condition (1.38a) becomes

(1.38b)

The resonant condition . Thus we see that resonant tunneling is sustained by the eigenvalues of the well itself. In Figure 1.10 we show as an example the transmission coefficient of a symmetric double barrier, with illustration of the resonant tunneling effect.

Figure 1.10  Å is also given for comparison (dashed line).

Consider now Eq. and is thus strongly suppressed.

1.3.3 Electron Tunneling through a Periodic Potential

(see in the unit cell, we can express the condition of propagation of Bloch-type wavefunctions throughout the system.

Figure 1.11 within the unit cell.

As indicated in . We specify Eqs. (1.21) to the present geometry and we have

(1.39)

.

We look for solutions of Eqs. , the boundary conditions required by the Bloch theorem

(1.40)

The Eqs. namely:

(1.41)

where the last passage on the right has been done by summing and subtracting the two equations on the left side.

, we obtain from Eqs.

(1.42)

The condition of Bloch propagation and existence of allowed energy bands requires the vanishing of the determinant

Taking into account that the transfer matrix is unimodular, one obtains

(1.43a)

or equivalently

(1.43b)

The above compatibility equation could also be obtained by direct inspection of Eq. .

It is convenient to express the quantities in Eqs. ; we have

The compatibility condition (1.43) for the tunneling of Bloch-type wavefunctions takes the compact and significant form

(1.44)

of the unit cell. Notice that Eq. , we can infer that the compatibility equation (1.44) leads quite generally to a sequence of allowed and forbidden energy regions.

It is instructive to analyze the band structure, produced by the periodic array of barriers of for a single barrier is given by Eq. (1.32). The energy bands are determined by Eq. (1.43b), which reads in the present case

(see Figure 1.3), the above equation becomes

(1.45)

We thus see that the compatibility equation (1.45) for the tunneling of Bloch-type wavefunctions through periodically repeated barriers, exactly coincides with Eq. (1.18), previously obtained using the ordinary boundary conditions procedure.

1.4 The Tight-Binding Model

1.4.1 Expansion in Localized Orbitals

-fold degeneracy is removed and evolves into an energy band. As usual in this chapter we keep our considerations at an introductory level, referring to Chapter 5 for a deeper analysis.

. A schematic picture of the model crystal considered is given in Figure 1.12.

Figure 1.12 leads to the formation of energy bands.

on the atomic orbitals are all equal; similarly the hopping integrals between nearest neighbor orbitals are equal. We have

(1.46)

-like orbitals and attractive atomic potentials).

do not satisfy the Bloch theorem; but this can be remedied considering the linear combinations of atomic orbitals of the type

(1.47)

is the number of unit cells of the crystal. The functions defined by Eq. (1.47) are named Bloch sums have delocalized character, and satisfy the Bloch theorem; in fact

For simplicity we assume orthonormality of orbitals centered on different atoms; in this case, the Bloch sums (1.47) are also orthonormal.

is thus given by

between localized orbitals and appropriate Bloch phase factors, as follows:

has been dropped.

between atomic orbitals are given by Eq. (1.46), the above dispersion relation becomes

(1.48)

(see Figure 1.13).

Figure 1.13 .

We can expand the second member of Eq. around the center of the Brillouin zone and retain terms up to the second order; we have

is defined as

(1.49)

is large, and vice versa. Notice also that the effective mass, at the border of the Brillouin zone, is negative and is the opposite of expression .

1.4.2 Tridiagonal Matrices and Continued Fractions

, with matrix elements given by Eq. has the form

(1.50)

The one-dimensional tight-binding Hamiltonian (1.50), with one orbital per site and nearest neighbor interactions, is the simplest example of problems described by a tridiagonal matrix.

The most general tridiagonal form for the Hamiltonian is

(1.51)

can be brought to real form, without loss of generality, just embodying appropriate phases in the basis functions). The tridiagonal operator (1.51) can be represented graphically in a chain form as indicated in Figure 1.14.

Figure 1.14 .

Numerous physical problems can be described by a tridiagonal Hamiltonian of the type are randomly distributed in an energy interval, while the hopping integrals are constant; in the Anderson model all states are localized, for any degree of disorder. We cannot pursue here the interesting problems posed by localization, and the wide and restless research on Hamiltonians of type (1.51).

Much more generally, the importance of tridiagonal matrices is due to the fact that, by the Lanczos method (see Chapter 5), it is possible to map any quantum mechanical problem into a semi-infinite chain represented by a tridiagonal matrix. This is the reason why tridiagonal operators have a special relevance in quantum mechanics [see for instance D. W. Bullett, R. Haydock, V. Heine and M. J. Kelly, Solid State Physics Vol.35 (Academic Press, New York 1980)].

Tridiagonal matrices have simple and elegant mathematical properties, that allow to focus on physical aspects. From a physical point of view, the most remarkable aspect is that the inversion of tridiagonal matrices is straightforward and thus the Green’s function of tridiagonal operators is easily accessible.

A Note on the Inversion of Tridiagonal Matrices

of the form

(1.52)

.

along the elements of the first row gives

we obtain

we obtain the continued fraction expansion

(1.53)

The continued fraction expansion (as for instance in Figure 1.14), we obtain

(1.54)

we recover Eq. could be expressed, with appropriate elaboration, in terms of continued fractions.

Density-of-States and Green’s Functions

its normalized eigenfunctions and eigenvalues (supposed to be countable, for simplicity). The total density-of-states of the system is defined as

(1.55)

provides the number of states of the system therein.

The density-of-states projected (normalized to unity) is defined as

(1.56)

(also called local density-of-states) gives