Solid-State Physics for Electronics E-Book

Andre Moliton

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 444

Veröffentlichungsjahr: 2013

Beschreibung

Describing the fundamental physical properties of materials used in electronics, the thorough coverage of this book will facilitate an understanding of the technological processes used in the fabrication of electronic and photonic devices. The book opens with an introduction to the basic applied physics of simple electronic states and energy levels. Silicon and copper, the building blocks for many electronic devices, are used as examples. Next, more advanced theories are developed to better account for the electronic and optical behavior of ordered materials, such as diamond, and disordered materials, such as amorphous silicon. Finally, the principal quasi-particles (phonons, polarons, excitons, plasmons, and polaritons) that are fundamental to explaining phenomena such as component aging (phonons) and optical performance in terms of yield (excitons) or communication speed (polarons) are discussed.

Leseprobe

Table of Contents

Foreword

Introduction

Chapter 1: Introduction: Representations of Electron-Lattice Bonds

1.1. Introduction

1.2. Quantum mechanics: some basics

1.3. Bonds in solids: a free electron as the zero order approximation for a weak bond; and strong bonds

1.4. Complementary material: basic evidence for the appearance of bands in solids

Chapter 2: The Free Electron and State Density Functions

2.1. Overview of the free electron

2.2. Study of the stationary regime of small scale (enabling the establishment of nodes at extremities) symmetric wells (1D model)

2.3. Study of the stationary regime for asymmetric wells (1D model) with L ≈ a favoring the establishment of a stationary regime with nodes at extremities

2.4. Solutions that favor propagation: wide potential wells where L ≈ 1 mm, i.e. several orders greater than inter-atomic distances

2.5. State density function represented in energy space for free electrons in a 1D system

2.6. From electrons in a 3D system (potential box)

2.7. Problems

Chapter 3: The Origin of Band Structures within the Weak Band Approximation

3.1. Bloch function

3.2. Mathieu’s equation

3.3. The band structure

3.4. Alternative presentation of the origin of band systems via the perturbation method

3.5. Complementary material: the main equation

3.6. Problems

Chapter 4: Properties of Semi-Free Electrons, Insulators, Semiconductors, Metals and Superlattices

4.1. Effective mass (m*)

4.2. The concept of holes

4.3. Expression for energy states close to the band extremum as a function of the effective mass

4.4. Distinguishing insulators, semiconductors, metals and semi-metals

4.5. Semi-free electrons in the particular case of super lattices

4.6. Problems

Chapter 5: Crystalline Structure, Reciprocal Lattices and Brillouin Zones

5.1. Periodic lattices

5.2. Locating reciprocal planes

5.3. Conditions for maximum diffusion by a crystal (Laue conditions)

5.4. Reciprocal lattice

5.5. Brillouin zones

5.6. Particular properties

5.7. Example determinations of Brillouin zones and reduced zones

5.8. Importance of the reciprocal lattice and electron filling of Brillouin zones by electrons in insulators, semiconductors and metals

5.9. The Fermi surface: construction of surfaces and properties

5.10. Conclusion. Filling Fermi surfaces and the distinctions between insulators, semiconductors and metals

5.11. Problems

Chapter 6: Electronic Properties of Copper and Silicon

6.1. Introduction

6.2. Direct and reciprocal lattices of the fcc structure

6.3. Brillouin zone for the fcc structure

6.4. Copper and alloy formation

6.5. Silicon

6.6. Problems

Chapter 7: Strong Bonds in One Dimension

7.1. Atomic and molecular orbitals

7.2. Form of the wave function in strong bonds: Floquet’s theorem

7.3. Energy of a 1D system

7.4. 1D and distorted AB crystals

7.5. State density function and applications: the Peierls metal-insulator transition

7.6. Practical example of a periodic atomic chain: concrete calculations of wave functions, energy levels, state density functions and band filling

7.7. Conclusion

7.8. Problems

Chapter 8: Strong Bonds in Three Dimensions: Band Structure of Diamond and Silicon

8.1. Extending the permitted band from 1D to 3D for a lattice of atoms associated with single s-orbital nodes (basic cubic system, centered cubic, etc.)

8.2. Structure of diamond: covalent bonds and their hybridization

8.3. Molecular model of a 3D covalent crystal (atoms in sp 3-hybridization states at lattice nodes)

8.4. Complementary in-depth study: determination of the silicon band structure using the strong bond method

8.5. Problems

Chapter 9: Limits to Classical Band Theory: Amorphous Media

9.1. Evolution of the band scheme due to structural defects (vacancies, dangling bonds and chain ends) and localized bands

9.2. Hubbard bands and electronic repulsions. The Mott metal–insulator transition

9.3. Effect of geometric disorder and the Anderson localization

9.4. Conclusion

9.5. Problems

Chapter 10: The Principal Quasi-Particles in Material Physics

10.1. Introduction

10.2. Lattice vibrations: phonons

10.3. Polarons

10.4. Excitons

10.5. Plasmons

10.6. Problems

Bibliography

Index

First published in France in 2007 by Hermes Science/Lavoisier entitled: Physique des matériaux pour l’électronique © LAVOISIER, 2007

First published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd 27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.ukJohn Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com

© ISTE Ltd, 2009

The rights of André Moliton to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Moliton, André.

[Physique des matériaux pour l'électronique. English]

Solid-state physics for electronics / André Moliton.p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-062-2

1. Solid state physics. 2. Electronics--Materials. I. Title.

QC176.M5813 2009

530.4'1--dc22

2009016464

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-062-2

Foreword

A student that has attained a MSc degree in the physics of materials or electronics will have acquired an understanding of basic atomic physics and quantum mechanics. He or she will have a grounding in what is a vast realm: solid state theory and electronic properties of solids in particular. The aim of this book is to enable the step-by-step acquisition of the fundamentals, in particular the origin of the description of electronic energy bands. The reader is thus prepared for studying relaxation of electrons in bands and hence transport properties, or even coupling with radiance and thus optical properties, absorption and emission. The student is also equipped to use by him- or herself the classic works of taught solid state physics, for example, those of Kittel, and Ashcroft and Mermin.

This aim is reached by combining qualitative explanations with a detailed treatment of the mathematical arguments and techniques used. Valuably, in the final part the book looks at structures other than the macroscopic crystal, such as quantum wells, disordered materials, etc., towards more advanced problems including Peierls transition, Anderson localization and polarons. In this, the author’s research specialization of conductors and conjugated polymers is discernable. There is no doubt that students will benefit from this well placed book that will be of continual use in their professional careers.

Michel SCHOTT

Emeritus Research Director (CNRS),Ex-Director of the Groupe de Physique des Solides (GPS),Pierre and Marie Curie University, Paris, France

Introduction

This volume proposes both course work and problems with detailed solutions. It is the result of many years’ experience in teaching at MSc level in applied, materials and electronic physics. It is written with device physics and electronics students in mind. The book describes the fundamental physics of materials used in electronics. This thorough comprehension of the physical properties of materials enables an understanding of the technological processes used in the fabrication of electronic and photonic devices.

The first six chapters are essentially a basic course in the rudiments of solid-state physics and the description of electronic states and energy levels in the simplest of cases. The last four chapters give more advanced theories that have been developed to account for electronic and optical behaviors of ordered and disordered materials.

The book starts with a physical description of weak and strong electronic bonds in a lattice. The appearance of energy bands is then simplified by studying energy levels in rectangular potential wells that move closer to one another. Chapter 2 introduces the theory for free electrons where particular attention is paid to the relation between the nature of the physical solutions to the number of dimensions chosen for the system. Here, the important state density functions are also introduced. Chapter 3, covering semi-free electrons, is essentially given to the description of band theory for weak bonds based on the physical origin of permitted and forbidden bands. In Chapter 4, band theory is applied with respect to the electrical and electronic behaviors of the material in hand, be it insulator, semiconductor or metal. From this, superlattice structures and their application in optoelectronics is described. Chapter 5 focuses on ordered solid-state physics where direct lattices, reciprocal lattices, Brillouin zones and Fermi surfaces are good representations of electronic states and levels in a perfect solid. Chapter 6 applies these representations to metals and semiconductors using the archetypal examples of copper and silicon respectively. An excursion into the preparation of alloys is also proposed.

The last four chapters touch on theories which are rather more complex. Chapter 7 is dedicated to the description of the strong bond in 1D media. Floquet’s theorem, which is a sort of physical analog for the Hückel’s theorem that is so widely used in physical chemistry, is established. These results are extended to 3D media in Chapter 8, along with a simplified presentation of silicon band theory. The huge gap between the discovery of the working transistor (1947) and the rigorous establishment of silicon band theory around 20 years later is highlighted. Chapter 9 is given over to the description of energy levels in real solids where defaults can generate localized levels. Amorphous materials are well covered, for example, amorphous silicon is used in non-negligible applications such as photovoltaics. Finally, Chapter 10 contains a description of the principal quasi-particles in solid state, electronic and optical physics. Phonons are thus covered in detail. Phonons are widely used in thermics; however, the coupling of this with electronic charges is at the origin of phonons in covalent materials. These polarons, which often determine the electronic transport properties of a material, are described in all their possible configurations. Excitons are also described with respect to their degree of extension and their presence in different materials. Finally, the coupling of an electromagnetic wave with electrons or with (vibrating) ions in a diatomic lattice is studied to give a classical description of quasi-particles such as plasmons and polaritons.

Chapter 1

Introduction: Representations of Electron-Lattice Bonds

1.1. Introduction

This book studies the electrical and electronic behavior of semiconductors, insulators and metals with equal consideration. In metals, conduction electrons are naturally more numerous and freer than in a dielectric material, in the sense that they are less localized around a specific atom.

Starting with the dual wave-particle theory, the propagation of a de Broglie wave interacting with the outermost electrons of atoms of a solid is first studied. It is this that confers certain properties on solids, especially in terms of electronic and thermal transport. The most simple potential configuration will be laid out first (Chapter 2). This involves the so-called flat-bottomed well within which free electrons are simply thought of as being imprisoned by potential walls at the extremities of a solid. No account is taken of their interactions with the constituents of the solid. Taking into account the fine interactions of electrons with atoms situated at nodes in a lattice means realizing that the electrons are no more than semi-free, or rather “quasi-free”, within a solid. Their bonding is classed as either “weak” or “strong” depending on the form and the intensity of the interaction of the electrons with the lattice. Using representations of weak and strong bonds in the following chapters, we will deduce the structure of the energy bands on which solid-state electronic physics is based.

1.2. Quantum mechanics: some basics

1.2.1. The wave equation in solids: from Maxwell’s to Schrödinger’s equation via the de Broglie hypothesis

In the theory of wave-particle duality, Louis de Broglie associated the wavelength (λ) with the mass (m) of a body, by making:

[1.1]

For its part, the wave propagation equation for a vacuum (here the solid is thought of as electrons and ions swimming in a vacuum) is written as:

[1.2]

If the wave is monochromatic, as in:

[1.3]

[1.3’]

A particle (an electron for example) with mass denoted m, placed into a time-independent potential energy V(x, y, z), has an energy:

(in common with a wide number of texts on quantum mechanics and solid-state physics, this book will inaccurately call potential the “potential energy” – to be denoted V).

The speed of the particle is thus given by

[1.4]

The de Broglie wave for a frequency can be represented by the function Ψ (which replaces the s in equation [1.2]):

[1.5]

Accepting with Schrödinger that the function ψ (amplitude of Ψ) can be used in an analogous way to that shown in equation [1.3’], we can use equations [1.1] and [1.4] with the wavelength written as:

[1.6]

so that:

[1.7]

This is the Schrödinger equation that can be used with crystals (where V is periodic) to give well defined solutions for the energy of electrons. As we shall see,

these solutions arise as permitted bands, otherwise termed valence and conduction bands, and forbidden bands (or “gaps” in semiconductors) by electronics specialists.

1.2.2. Form of progressive and stationary wave functions for an electron with known energy (E)

In general terms, the form (and a point defined by a vector ) function for an electron of known energy (E) is given by: of a wave

where ψ() is the wave function at amplitudes which are in accordance with Schrödinger’s equation [1.7]:

– if the resultant wave Ψ(r, t) is a stationary wave, then ψ() is real;

– if the resultant wave Ψ(r, t) is progressive, then ψ() takes on the form where f() is a real function, and is the wave vector.

1.2.3. Important properties of linear operators

1.2.3.1. If the two (linear) operators H and T are commutative, the proper functions of one can also be used as the proper functions of the other

For the sake of simplicity, non-degenerate states are used. For a proper function ψ of H corresponding to the proper non-degenerate value (α), we find that:

Multiplying the left-hand side of the equation by T gives:

This equation in fact signifies that ψ is a proper function of T with the proper value being the coefficient of collinearity (t) (QED).

1.2.3.2. If the operator H remains invariant when subject to a transformation using coordinates (T), then this operator H commutes with operator (T) associated with the transformation

Here are the respective initial and final states (with initial on the left and final to the right):

Similarly, the application of the operator T to the quantity Hψ, with H being invariant under T’s effect, gives:

We thus find:

from which:

1.2.3.3. The consequence

If the operator H is invariant to the effect of the operator T, then the proper functions of T can be used as the proper functions of H.

1.3. Bonds in solids: a free electron as the zero order approximation for a weak bond; and strong bonds

1.3.1. The free electron: approximation to the zero order

The electric conduction properties of metals historically could have been derived from the most basic of theories, that of free electrons. This would assume that the conduction (or free) electrons move within a flat-bottomed potential well. In this model, the electrons are simply imprisoned in a potential well with walls that coincide with the limits of the solid. The potential is zero between the infinitely high walls. This problem is studied in detail in Chapter 2 with the introduction of the state density function that is commonly used in solid-state electronics. In three dimensions, the problem is treated as a potential box.

Two approaches can be considered, depending on the nature of the bonds. If the well depth is small (weak bond) then a treatment of the initial problem (free electron) using perturbation theory is possible. If the wells are quite deep, for example as in a covalent crystal with electrons tied to given atoms through strong bonds, then a more global approach is required (using Hückels theories for chemists or Floquet’s theories for physicists).

1.3.2. Weak bonds

This approach involves improving the potential box model. This is done by the electrons interacting with a periodical internal potential generated by a crystal lattice (of Coulumbic potential varying 1/r with respect to the ions placed at nodes of the lattice). In Figure 1.1, we can see atoms periodically spaced a distance “a” apart. Each of the atoms has a radius denoted “R” (Figure 1.1a). A 1D representation of the potential energy of the electrons is given in Figure 1.1b. The condition a < 2R has been imposed.

Depending on the direction defined by the line Ox that joins the nuclei of the atoms, when an electron goes towards the nuclei, the potentials diverge. In fact, the study of the potential strictly in terms of Ox has no physical reality as the electrons here are conduction electrons in the external layers. According to the line (D) that does not traverse the nuclei, the electron-nuclei distance no longer reaches zero and potentials that tend towards finite values join together. In addition, the condition a < 2R decreases the barrier that is midway between adjacent nuclei by giving rise to a strong overlapping of potential curves. This results in a solid with a periodic, slightly fluctuating potential. The first representation of the potential as a flat-bottomed bowl (zero order approximation for the electrons) is now replaced with a periodically varying bowl. As a first approximation, and in one dimension (r ≡ x), the potential can be described as:

The term w0, and the associated perturbation of the crystalline lattice, decrease in size as the relation a < 2R becomes increasingly valid. In practical terms, the smaller “a” is with respect to 2R, then the smaller the perturbation becomes, and the more justifiable the use of the perturbation method to treat the problem becomes. The corresponding approximation (first order approximation with the Hamiltonian perturbation being given by ) is that of a semi-free electron and is an improvement over that for the free electron (which ignores H(1)). The theory that results from this for the weak bond can equally be applied to the metallic bond, where there is an easily delocalized electron in a lattice with a low value of w0 (see Chapter 3).

1.3.3. Strong bonds

The approach used here is more “chemical” in its nature. The properties of the solid are deduced from chemical bonding expressed as a linear combination of atomic orbitals of the constituent atoms. This reasoning is all the more acceptable when the electrons remain localized around a specific atom. This approximation of a strong bond is moreover justified when the condition a ≥ 2R is true (Figure 1.2a), and is generally used for covalent solids where valence electrons remain localized around the two atoms that they are bonding.

Once again, analysis of the potential curve drawn with respect to Ox gives a function which diverges as the distance between the electrons and the nuclei is reduced. With respect to the line D, this discontinuity of the valence electrons can be suppressed in two situations, namely (see also Figure 1.2b):

– If a >> 2R, then very deep potential wells appear, as there is no longer any real overlap between the generated potentials by two adjacent nuclei. In the limiting case, if a chain of N atoms with N valence electrons is so long that we can assume that we have a system of N independent electrons (with N independent deep wells), then the energy levels are degenerate N times. In this case they are indiscernible from one another as they are all the same, and are denoted Eloc in the figure.

Figure 1.2. Strong bonds and: (a) atomic orbitals (s orbitals with radius R) in a periodic lattice (of period denoted a) where a ≥ 2R; (b) in 1D, the resulting potential energy (thick curve) seen by electrons

– If a ≥ 2R, the closeness of neighboring atoms induces a slight overlap of nuclei generated potentials. This means that the potential wells are no longer independent and their degeneration is increased. Electrons from one bond can interact with those of another bond, giving rise to a spread in the band energy levels. It is worth noting that the resulting potential wells are nevertheless considerably deeper than those in weak bonds (where a < 2R), so that the electrons remain more localized around their base atom. Given these well depths, the perturbation method that was used for weak bonds is no longer viable. Instead, in order to treat this system we will have to turn to the Hückel method or use Floquet’s theorem (see Chapter 7).

1.3.4. Choosing between approximations for weak and strong bonds

The electrical behavior of metals is essentially determined by that of the conduction electrons. As detailed in section 1.3.2, these electrons are delocalized throughout the whole lattice and should be treated as weak bonds.

Dielectrics (insulators), however, have electrons which are highly localized around one or two atoms. These materials can therefore only be described using strong-bond theory.

Semiconductors have carriers which are less localized. The external electrons can delocalize over the whole lattice, and can be thought of as semi-free. Thus, it can be more appropriate to use the strong-bond approximation for valence electrons from the internal layers, and the weak-bond approximation for conduction electrons.

1.4. Complementary material: basic evidence for the appearance of bands in solids

This section will be of use to those who have a basic understanding of wave mechanics or more notably experience in dealing with potential wells. For others, it is recommended that they read the complementary sections at the end of Chapters 2 and 3 beforehand.

This section shows how the bringing together of two atoms results in a splitting of the atoms’ energy levels. First, we associate each atom with a straight-walled potential well in which the electrons of each atom are localized. Second, we recall the solutions for the straight-walled potential wells, and then analyze their change as the atoms move closer to one another. It is then possible to imagine without difficulty the effect of moving N potential wells, together representing N atoms making up a solid.

1.4.1. Basic solutions for narrow potential wells

In Figure 1.3, we have W > 0, and this gives potential wells at intervals such that [– a /2, + a/2] where – W < 0.

We can thus state that , and the energy E is the sum of kinetic energy and potential energy.

As the related states are carry electrons then E < 0, and we can therefore write that:

By making α2 (γ2 − k2) > 0, α is real.

Figure 1.3.Straight potential wells of width a

[1.8]

[1.9]

The solutions for equation [1.8] are (with the limiting conditions of ψ(x) being finite when x →±∞):

The solution to equation [1.9] must be stationary because the potential wells are narrow (which forbids propagation solutions). There are two types of solution:

[1.10]

from which it can be deduced that ,

[1.11]

Equations [1.10] and [1.11] can be combined in the form:

[1.12]

In addition, equations [1.10] and [1.11] must be compatible with the equations that define α and k, so that:

[1.13]

Figure 1.4.Solutions for narrow potential wells

[1.14]

The solutions for α and thus E (as ) correspond to the points where 2m the circle of equation [1.14] crosses with the deduced curves from equations [1.10] and [1.11].

If:

Figure 1.5.The first four energy levels along with the appearance of the corresponding wave functions in the narrow potential wells

Figure 1.6.Scheme of the potential energies of two narrow potential wells brought close to one another

1.4.2. Solutions for two neighboring narrow potential wells

Schrödinger’s equation, written for each of the regions denoted 1 to 5 in Figure 1.6 gives the following solutions (which can also be found in the problems later on in the book):

– Symmetric solution:

[1.15]

– Similarly, for the asymmetric solution we obtain:

[1.16]

1.4.2.1. Neighboring potential wells that are well separated

If d is very large, equations [1.15] and [1.16] become:

[1.17]

– if n is even then ;

In effect, we again find the solutions of equations [1.10] and [1.11] for isolated wells, which is quite normal because when d is large the wells are isolated. Here though with a high value of d, the solution is degenerate as there are in effect two identical solutions, i.e. those of the isolated wells.

1.4.2.2. Closely placed neighboring wells

If d is small, we have e–ad << 1 and:

[1.18]

[1.19]

For the single solution (α0) in equation [1.17] (if the wells are infinitely separated) there are now two solutions: one is αs from equation [1.18] and the other is αa from equation [1.19]. For isolated or well separated potential wells, all states (symmetric or asymmetric) are duplicated with two neighboring energy states (as αs and αa are in fact slightly different from α0). The difference in energy between the symmetric and asymmetric states tends towards zero as the two wells are separated (d → ∞). In addition, we can show quite clearly that the symmetric state is lower than the asymmetric state as in Figure 1.7.

Figure 1.7.Evolution of energy levels and electronic states on going from one isolated well to two close wells

The example given shows how bringing together the discrete levels of the isolated atoms results in the creation of energy bands. The levels permitted in these bands are such that:

– two wells induce the formation of a “band” of two levels;

– n wells induce the formation of a “band” of n levels.

Chapter 2

The Free Electron and State Density Functions

2.1. Overview of the free electron

2.1.1. The model

[2.1]

2.1.2. Parameters to be determined: state density functions in k or energy spaces

With:

[2.2]

equation [2.1] can be written as:

[2.3]

We shall now determine for different depth potential wells, with both symmetric and asymmetric forms, the corresponding solutions for the wave function (Ψ0) and 0 the energy (E0). To each wave function there is a corresponding electronic state (characterized by quantum numbers). It is important in physical electronics to understand the way in which these states determine how energy levels are filled.

2.2. Study of the stationary regime of small scale (enabling the establishment of nodes at extremities) symmetric wells (1D model)

2.2.1. Preliminary remarks

The result is that

[2.4]

2.2.2. Form of stationary wave functions for thin symmetric wells with width (L) equal to several inter-atomic distances (L ≈ a), associated with fixed boundary conditions (FBC)

[2.5]

This limiting condition is equivalent to the physical status of an electron that cannot leave the potential well due to it being infinitely high. The result is that between and there is a zero probability of presence, hence the preceding FBC:

The general stationary solution to equation [2.3] is:

The use of the boundary conditions of equation [2.5] means that:

or

These last two equations result in the two same conditions:

[2.6]

[2.7] , with N being odd.

The normalization condition gives , and the two solutions in equations [2.6] and [2.7] can be brought together in:

[2.8]

For both symmetric and asymmetric solutions, k is of the form

[2.9]

This quantification is restricted to the quantum number N without involving spin. As we already know, spin makes it possible to differentiate between two electrons with the same quantum number N. This is due to a projection of kinetic moment on 1 the z axis which brings into play a new quantum number, namely .

2.2.3. Study of energy

From equation [2.2] we deduce that: . With k given by equation [2.9], 2m we find that the energy is quantified and takes on values given by:

[2.10]

Figure 2.2.Curve E0

The filling of energy levels is carried out from the bottom up. The fundamental level (E1) is filled with electrons in the states and . Each level is thus filled with two electronic states that are differentiated by their spin.

2.2.4. State density function (or “density of states”) in k space

On average, each interval of size can hold 1 orbital (without taking spin into account). In fact, this interval corresponds to 2 orbital states; however, each is shared with the adjacent intervals. For example, in the interval we can place the orbital states and , but is shared with and , just as is shared with and . From this we can see that the resulting average is 1 orbital state per interval .

[2.11]

2.3. Study of the stationary regime for asymmetric wells (1D model) with L ≈ a favoring the establishment of a stationary regime with nodes at extremities

[2.12]

[2.13]

B can be determined using the normalization condition, as in: that which gives , from which

[2.14]

For its part, energy is still deduced from equation [2.2]. With the condition imposed by equation [2.13] on kN, we are brought to the same expression as equation [2.10]:

[2.15]

2.4. Solutions that favor propagation: wide potential wells where L ≈ 1 mm, i.e. several orders greater than inter-atomic distances

2.4.1. Wave function

This problem can be seen as that of a wire, or rather molecular wire, with a given length (L) tied around on itself as shown in Figure 2.3.

Figure 2.3.Molecular wire of length L

[2.16]

[2.17]

That several revolutions are possible means that the solution must be a progressive wave. The amplitude of the free electrons wave function must take the form (see Chapter 1) given by:

[2.18]

[2.19]

Placing these results into equation [2.18] we finally have for the amplitude function:

[2.20]

NOTE.– We can immediately say that for the conditions that favor propagation, we now have , a constant value wherever along (x) an electron might be. The electrons move freely, without any specific localization (i.e. the probability of their presence is constant, whatever the value of x).

2.4.2. Study of energy

By taking the expression for k given in equation [2.19] and placing it into equation [2.2], we obtain:

[2.21]

When taking spin into account, we also find that the degree of degeneracy is four as each energy level can accommodate four electrons, each corresponding to a specific wave function. Thus, at the Nth level these four functions are:

2.4.3. Study of the state density function in k space

[2.22]

To conclude, we can see that with progressive solutions, the number of states that can be placed in a unit interval in reciprocal space is equal to 1/π. One half of that can be placed in stationary solutions, even though the available k space is twice as large. It should be noted that the negative values of N and thus of k must also be taken into account.

2.5. State density function represented in energy space for free electrons in a 1D system

Rigorously speaking, Z(E) should be a discontinuous function as it is defined, a priori, only for discrete values of energy corresponding to the solution of the Schrödinger equations [2.10] and [2.15] or [2.21] as below, respectively for stationary or progressive cases:

[2.21]

A numerical estimation can be carried out to find the typical value for free (conduction) electrons and, in this example, shows that EF ≈ 3 eV (Fermi energy measured with respect to the bottom of the potential wells).

The difference in energy between two adjacent states [N + 1] and N is thus given by:

giving ∆E ≈ 4 × 10 -6 eV.

This holds where EF is small and in effect the conduction electrons show a discrete energy value that can be neglected in an overall representation of electron energies (see below).

To conclude, the energy levels are quantified but the difference in their energies are so small that the function Z(E) as defined above can be considered as being quasi-continuous around EF. Often the term quasi-continuum is used in this situation.

2.5.1. Stationary solution for FBC

Here, as shown in Figure 2.2, only values with k > 0 are physically relevant.E10 pertains to a single value (k1) in k space. Similarly, E20 corresponds to a single value (k2). A consequence of this relationship between energy and k spaces is that for a number of states with energies between E and E + dE there is a corresponding and equal number of states between k and k + dk. This can be written as:

[2.23]

From this it can be deduced that

With E from equation [2.2], i.e. , we also equally have and thus .

From this it can be deduced that, for n(k) given by equation [2.11] (or rather :

[2.24]

Thus, when E increases, Z(E) decreases, as shown in Figure 2.5.

Figure 2.5.State density function for a stationary or progressive 1D system

2.5.2. Progressive solutions for progressive boundary conditions (PBC)

As shown in Figures 2.4 and 2.6, here the interval dE corresponds simultaneously to the intervals dk+ (for k > 0) and dk – (for k < 0).

As in both dk+ and dk– we can place the same number of states, it is possible to state that:

With n (k) given by equation [2.22], , we obtain:

where was calculated in the preceding section.

We again find the same expression for Z(E), as shown in equation [2.24] and thus the same graphical representation as shown in Figure 2.5.

2.5.3. Conclusion: comparing the number of calculated states for FBC and PBC

Stationary waves: FBC

Progressive waves: PBC

Figure 2.7.FBC and PBC states

– k varying from 0 to 4π/L or from 0 to ± 4π/L;

– E varying from 0 .

2.6. From electrons in a 3D system (potential box)

2.6.1. Form of the wave functions

Figure 2.8.A parallelepiped box (direct space)

We assume that the free electrons are closed within a parallelepiped box with sides of length L1, L2, L3 as shown in Figure 2.8. The potential is zero inside the box and infinite outside. The Schrödinger equation is thus given by:

a), so that on making we have:

b), so that on making we have:

c), so by making we have:

2.6.1.1. Case favoring fixed boundary conditions

Here the FBCs are:

– with respect to ,

– with respect to ,

– with respect to .

The use of these boundary limits means that we can solve these differential equations directly from the equivalent 1D system (the boundary limits are identical to those in the 1D system with an origin at an extremity – see section 2.3):

[2.25]

where and nz are positive integers.

Energy is given by:

[2.26]

2.6.1.2. Case favoring progressive boundary conditions

Analogously to the case for limiting conditions, we have:

– with respect to ,

– with respect to ,

– with respect to .

The use of these boundary limits means that we can solve these differential equations directly from the equivalent 1D system (the boundary limits are identical to those in the 1D system where propagation is favored – see section 2.4.1):

[2.27]

in which and where nx, ny and nz are positive or negative integers.

Energy is given by:

[2.28]

2.6.2. Expression for the state density functions in k space

2.6.2.1. Where stationary solutions are favored

Each elemental cell thus has eight nodes, each of which corresponds to eight states that are, in turn, each shared across eight elemental cells. Without taking spin into account, we can assert that state per cell on average. Taking spin states 8 into account we can now place two electronic states into each elemental cell.

The elemental cells have an elemental volume given by:

[2.29]

2.6.2.2. Where progressive solutions are favored

In 3D, we have and the elemental cells are such that each has a node associated with an electronic state represented by a wave function in the form:

The elemental cells thus have an elemental volume given by:

[2.30]

2.6.3. Expression for the state density functions in k space

2.6.3.1. Where stationary solutions are favored

The calculation for the state density function now denoted Z(E) can be carried out using two different routes:

– using the correlation between k and energy spaces;

– via a direct calculation in energy space using quantum numbers.

Here we will use the direct method.

First we can note that with stationary solutions where quantum numbers nx, ny and nz are positive, the quantum number space must be held within the first octet (nx, ny, nz > 0) as shown in Figure 2.8a. If the problem is dealt with in 2D only, then the space (or more exactly the plain) should be within the first quadrant, as indicated in Figure 2.8b.

Figure 2.8.Nodes and states in quantum number space: (a) 3D space with nx, ny, and nzas whole integers; (b) likewise in 2D

In quantum number space, just as in k space, the cells and their nodes are associated with electronic states that are characterized by their electronic wave functions, as in |ψnx ,ny ,nz where each is denoted with respect to its specific quantum number nx ny nz. Once we take spin into account, characterized by the quantum numbers and , these functions in fact give rise to the electronic states and .

Figure 2.9.The relation between the quantum number and energy spaces

Taking spin into account means that we can now place twice the number of states, so that the total number of electronic states is equal to πn²dn. We can thus finally write that:

The result in that: , so that:

[2.31]

The curve thus has a parabolic shape, as shown in Figure 2.10.

Figure 2.10.State density function in 3D

NOTE.– If we use the correlation between k space and energy space, we can write:

With we have on one side , and on the other , from which we once again obtain equation [2.28].

2.6.3.2. Problem: where progressive solutions are favored (see also problems 5 and 6 of this chapter)

In this case, nx, ny, and nz are positive or negative integers and the quantum number spaces – just as that for k – is no longer restricted to the first octet but covers all space.

By using the correlation between k space and energy space, we can thus write:

from which we again find equation [2.31].

2.7. Problems

The reader is advised that if he or she has not yet looked at the basics of statistical thermodynamics, including the use and significance of the Fermi-Dirac function, problems 1, 3 and 4 should be attempted after reading section 4.4.3.2 of this book.

2.7.1. Problem 1: the function Z(E) in 1D

Here we are looking at free electrons in a small one-dimensioned medium with L being equal to several nanometers. The total number of electrons is equal to Nt and the filling of the energy levels at absolute zero is under consideration.

1) What type of boundary conditions should we use?

2) Give the value of N of the last occupied level.

3)

a) If EF(0) ≡ EF represents the energy of the highest fully occupied level, express the value of EF as a function of Nt.

b) From this and as a function of EF, deduce the expression of the total numbers of orbitals [N(EF)] (of electronic states including spin) for electrons with energies lower than EF.

4)

a) Generalize the last expression for any level E that is lower than EF.

b) From this deduce the expression for the function [Z(E)] of the density of energy states.

Answers

1) The medium is of a sufficiently small size so that we can assume that there is a stationary regime and as a consequence the use of fixed boundary conditions, which takes into account the presence of a node at each extremity of the system.

3)

a) We can now write that .