Free Electron Model

12 Key excerpts on "Free Electron Model"

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  eBook - ePub

  Solid State Quantum Information — An Advanced Textbook

  Quantum Aspect of Many-Body Systems

  • Vlatko Vedral, Wonmin Son(Authors)
  • 2018(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  However, the model by Sommerfeld fails in explaining several other features in solid. For example, the Free Electron Model cannot explain the differences between conductors, semiconductors and insulators. The model also cannot describe effectively the magnetic field dependence and magnetoresistance in Hall effect. The existence band structure arising from various shapes of Fermi surface cannot be explained in this model. Therefore, for the purpose of better understanding we need better model than Sommerfeld model.

  The Free Electron Model can be said as semiclassical model because the electron has been treated quantum mechanically. In fact, the electrons in a solid are not free but are usually bounded by the potential energy from the ions in the lattices.

  2.3.Bloch’s Theorem, Free Electrons and Tight-Binding Model

  The problem with the Free Electron Models is to ignore the interaction of electrons with the crystal lattice. The lattice itself is composed of periodically spaced ion cores that gives rise to a periodic potential variation for electrons moving through the crystal. In this section, we will discuss about how to deal with a periodic potential in a crystal which is known as Bloch theorem. From the formalism, it will be shown that the band structure of electron will emerge. The behavior of electrons in the periodic potential is dependent on the interaction strength and the case of weak potential as well as the tight-binding model will be discussed here.

  2.3.1.Bloch theorem: First quantization treatment

  Now we introduce the effect of periodic potential to the free electrons. The underlying translational periodicity of the crystal is defined by the lattice translational vectors

  where ni is integer numbers and āi is primitive lattice vectors. The lattice translational vector describes the shape of ionic lattices in a crystal. Assuming that the potential energy V(r

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  Carbon Nanotube and Graphene Device Physics

  • H.-S. Philip Wong, Deji Akinwande(Authors)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  Since the electrons will be treated as waves, Schrödinger’s equation in its most basic form will be employed to solve for their properties. For this purpose, we assume the reader has at least a basic exposure to quantum mechanics at the level 1 Think of a very, very large number whose precise value is unnecessary to specify at this moment. 2.3 An electron in empty space 21 An electron in space An electron in a finite empty solid An electron in a solid with a simple periodic potential Ab-initio electron models Nearly-Free Electron Model Tight-binding electron model Fig. 2.1 Illustrative flowchart of independent electron models. Arrows indicate increasing complexity. of an undergraduate modern physics course. 2 In general, the quantum mechanics in this section and in the entire book is kept to a minimum, and much of the understanding and intuition developed as we move ahead should still be accessible to readers who are not familiar with quantum mechanics. 2.3 An electron in empty space The simplest possible model of electrons is the model of a free electron in empty space. Understanding this model is the first step towards our goal of gaining deeper insight and understanding of electrons in real crystalline solids. In principle, empty space is essentially the limiting case of an infinite force-free solid. Free electrons are electrons that have no forces or potential acting on them and hence, they are free. Even though we are considering a single free electron in these introductory models, the mathematical formalism developed and knowledge acquired applies directly to realistic solids with a much larger number of electrons, provided the electrons have negligible interaction with one another, which is often the case at room temperature. In this simple pedagogical model of a free electron in space we seek to determine the energy–wavevector relationship or dispersion by solving the time- independent 3 Schrödinger equation.

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  Introductory Solid State Physics with MATLAB Applications

  • Javier E. Hasbun, Trinanjan Datta(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  5   Magnetism II  

  Contents 5.1Introduction 5.2Free One-Dimensional Electron Gas 5.3The Fermi-Dirac Distribution 5.4Free Three-Dimensional Electron Gas 5.5Electron Gas Heat Capacity 5.6Electrical Conductivity (Drude Model) 5.7Electronic Motion in Magnetic Fields and the Classical Hall Effect 5.8The Quantum Hall Effect 5.9Electronic Thermal Conductivity of Metals 5.10Chapter 5 Exercises

    5.1  Introduction

  A free electron gas refers to the electrons in a crystal whose behavior is treated as if they were free from the binding forces that keep them confined to the crystal. These are the same as the valence electrons associated with the atoms in a crystal. They are also the same as the conduction electrons that move about freely within the crystal volume. The “free electron gas” model is used to understand the physical properties of metals. The simplest of which are the alkali metals such as lithium, sodium, potassium, cesium, and rubidium. In the case of lithium, for example, the valence electron is in the 2s state, which becomes a conduction electron in the associated energy band of the crystal. A crystal of lithium is monovalent. It contains N electrons and N positive lithium cores. Each lithium core contains 2 electrons occupying the 1s shell. The extra electron in the 2s state becomes part of the so-called Fermi sea or free electron gas. The electrons in the Fermi sea are described quantum mechanically and obey the Pauli exclusion principle. Below, we start off by describing a free one-dimensional electron gas quantum mechanically.

    5.2  Free One-Dimensional Electron Gas

  Following the introduction, consider an electron of mass m, in a box of length L. The box is described by the potential

  V ( x ) = { 0 , 0 ≤ x ≤ L , ∞ , x < 0 and x > L ,	(5.2.1)

  where the electron is confined to be free only within the boundaries of the one-dimensional box. The Schrodinger equation in one dimension is

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  Electronic Properties of Crystalline Solids

  An Introduction to Fundamentals

  • Richard Bube(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  In some metals such as sodium, magnesium, or a l u m i n u m , the approximation is reasonably good, a n d m o r e accurate theoretical treatments show that the valence electrons in these metals have properties very similar to those of free elec-trons. T h e concepts a n d formalism derived in terms of free-electron theory can also be applied to electrons in conducting states of semiconductors a n d insulators, and so these considerations have a broader overall application than to metals alone. Consideration of free-electron theory is designed to answer three prin-cipal questions: 1. W h a t are the allowed energy levels for free electrons in a crystal? 2. H o w is the density of these levels (i.e., the n u m b e r of levels per unit volume of the crystal) distributed with the energy of the levels? 3. Given a distribution of allowed energy levels, what is the probability that a given level is occupied by an electron at a given temperature? T h e answers to the first two of these questions can be given in terms of two models. T h e first of these is the Sommerfeld model, which is the applica-tion of the three-dimensional particle in a b o x problem, discussed in one dimension in Section 3.2. T h e second is a Hartree self-consistent field calculation using traveling waves a n d periodic boundary conditions, which are m o r e appropriate for describing electrical transport p h e n o m e n a in metals. 4.1 Atomic Energy Levels and the Periodic Table 75 T h e answer to the occupancy of allowed states is given by the Fermi-Dirac statistics, which m a y be derived from a joint consideration of the indistinguishability of electrons a n d the Pauli exclusion principle. A l t h o u g h the free-electron a p p r o x i m a t i o n is inadequate to describe the properties of a metal in detail, a n u m b e r of simple physical p h e n o m e n a can be described to a fair a p p r o x i m a t i o n in terms of the free-electron model.

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  Introductory Matter Physics

  • Francesco Simoni(Author)
  • 2018(Publication Date)
  • WSPC

    (Publisher)

  Chapter 4

  Electrical Properties of Crystals

  4.1. The Free Electron Model

  The electrical properties of matter include a wide range of phenomena that occur when an external electric field is applied in the space volume occupied by the considered material. Actually, electrical properties are mainly concerned with the induced charge movement which leads to current flow and/or to charge polarization. Additionally, the occurrence of electrical current or electrical polarization and their specific features depend on the material’s structure, which also determines the different contributions of electrons and ions to these phenomena. In solid materials, electrical conduction is due to electrons, while in fluids ions can play a major role. In this chapter, we focus on electrical conduction in crystalline solids, but many of the concepts, which we discuss here, can be extended to other materials.

  We know that materials behave in a different way when submitted to an electric field. In some of them, the electrical stimulus gives rise to charge migration on a macroscopic length scale. These materials are called conductors.

  Some basic properties of electrical conductors can be explained following the simple model of the free electron. This starts from considering the valence electrons of each atom participating in the metallic bonding, thus being delocalized in the solid volume where they are able to move on a macroscopic scale. In this volume, they have a constant potential energy (set = 0), and thus they possess only kinetic energy. In accordance with this model, Fig. 4.1 is drawn. It shows that in each lattice site of a metallic conductor is located a positive ion made up of the atomic nucleus (charge eZa , with Za being the atomic number) and the tightly bound electrons (Za − Z), while a negative charge −eNZ is diffused over the crystal volume (Z is the number of valence electrons in each atom, and N the number of atoms in the lattice sites).

  Fig. 4.1. Scheme of the metallic conductor in the Free Electron Model. The black circles represent the proton charge, surrounded by the tightly bound electrons (dashed circles). The gray background represents the cloud made by the conduction electrons.

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  The Electrical Properties of Metals and Alloys

  • J.S. Dugdale(Author)
  • 2016(Publication Date)
  • Dover Publications

    (Publisher)

  3

  Electrons in Solids

  3.1Introduction

  Our ultimate aim is to understand what happens to the conduction electrons in solids under the influence of external fields. But first we must know something about the general properties of electrons at equilibrium in a solid. To do this, let me begin by comparing an atom in a crystalline solid with the same atom when it is free and by itself.

  In an isolated atom the electrons move under the influence of one single dominating field of force, that of the atomic nucleus. But when the atom joins with others to form a solid, other centres of influence become important: in the first place the nuclei of neighbouring atoms and ultimately all the other nuclei in the solid.

  A classical particle is confined to those immediately accessible regions of space where its total energy exceeds its potential energy—it cannot have a negative kinetic energy. A particle, here an electron, subject to the laws of quantum mechanics, is not confined in this way. The electron can tunnel through a classical potential barrier into neighbouring regions of lower potential. This is the situation in a solid. The electron is no longer confined to the neighbourhood of a particular atom; it can, either directly or by tunnelling, have access to all the other atoms in the solid. If the barrier is high and/or broad the probability of tunnelling will be small. This is true of electrons in the deeper X-ray levels of the atoms in solids. On the other hand, electrons in the higher atomic levels can tunnel rather freely throughout the solid and indeed those in the uppermost levels have energies higher than that of the barriers and so move without tunnelling. The important thing is that any electron can travel throughout the crystal.

  So the states in the original free atom which confined the electron to the immediate vicinity of its nucleus now give rise in the crystal to states in which the electron can travel throughout the crystal. If there are N atoms in the crystal, there are now N

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  Band Theory of Metals

  The Elements

  • Simon L. Altmann(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R 2 Free-electron Theory of Metals W E SHALL introduce in §§ 1-3 of this chapter some approximations that will allow us to construct a model of a metal for which some very simple calculations can be performed. Each approximation should properly be justified by an order of magnitude calculation of the error involved, as was done in some instances when the corresponding theories were first presented. We shall not attempt to do this: instead, we shall describe the approximations required as precisely as possible and shall provide an a posteriori justification of the model through the results of the calculations to be performed in the rest of this chapter. It will be seen, nevertheless, that the crude model used here provides an understanding of some important experimental facts. Other phenomena lie outside the scope of the free-electron model and will require the more precise theory of Chapter 3 , but the free-electron model remains a useful first approximation in many cases. It should be clearly understood that the approximations to be introduced in this and the following section are required not just for the free-electron model: they will remain in use throughout this book. Suppose that we have inside a very large container two particles that are so far apart that they do not interact at all. Clearly, we can set up a Schrödinger equation for each particle as if the other one did not exist: 1. The one-particle approximation H(l)9>(l) = 2s l 9

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  Introduction to the Physics and Chemistry of Materials

  • Robert J. Naumann(Author)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  18 Free Electrons in Metals In order to understand the properties of metals, it is necessary to understand the role of electrons in metals. We already know that the metallic bond has to do with electrons that become delocalized from their parent ion cores and are more or less free to move around in the metal. To the fi rst approximation, treating the electrons as an ideal Fermi gas does a fairly good job of explaining most of the properties of metals, but there is more to it as we shall see. 18.1 Drude Theory of Free Electrons in Metals The properties that characterize metals (good electrical conductivity, good thermal conductivity, bright, shiny, highly re fl ective in the visible spectrum) are attributed to the presence of free electrons. Shortly after the discovery of the electron by J.J. Thompson in 1897, P.K.L. Drude formulated a theory in 1900 to explain these properties of metals in terms of an ideal electron gas that permeates the spaces between the positive ion cores that form the metal lattice. These electrons were supposed to be the carriers of heat and electricity that made metals much better conductors than ceramic or polymeric systems. When light was incident on a metal, the electrons in the metal responded to the oscillating E-fi eld of the incoming light wave, thus reradiating a wave of the same frequency which explained the high re fl ectivity exhibited by metals. Even the metal bond could be explained qualitatively by this electron gas, which played essentially the same role as the anions in the rock salt face-centered cubic structure. This simple theory was successful in predicting a number of properties of metals as will be seen in the following sections. 18.1.1 Electrical Conduction Metals, for the most part, obey a simple linear relationship between the applied voltage V and the current I that fl ows through it.

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  Quantum Mechanics For Applied Physics And Engineering

  • Albert T. Jr. Fromhold(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  free-electron motion. This is the approach taken in Chap. 5, §13. Sometimes this approach is referred to as the nearly-free-electron model [Sachs (1963)]. The purely-free-electron model, however, considers the periodic potential to have a very minimal effect on electronic motion; in fact, the choice TT(r) = const (3.1) is made, where the constant is for convenience taken to be zero inside the metal. The remarkable success of such a naive model is due primarily to the fact that it incorporates many of the essential quantum properties of the electrons. 1.5 Wavelike Behavior of Conduction Electrons Because the electronic mass is small, electrons have pronounced wavelike properties, in contrast to particlelike properties normally associated with bulk masses. This follows from the de Broglie relation λ = h/p = h/rnv, which shows that the de Broglie wavelength of an electron is greater than an atom spacing of 4 Ä for speeds less than 1.8 x 10 6 m/sec. This means that a given electron cannot be considered to have a definitely known position in space; instead, it must be considered to have & probability density p(r) = ψ*ψ in space determined by the electronic wave function φ, as discussed in some detail in Chap 1, §6 [also see §1] FREE-ELECTRON GAS IN THREE DIMENSIONS 191 Bloch (1976)]. This means that p(r) gives the relative probability of finding the electron at position r. Let us assume that the wave function φ is appropriately normalized over the region accessible to the electron, so that f p ( r ) * = l . (3.2) We thus can consider the distribution of conduction electrons to be smeared out in space (i.e., much as a continuous sea of negative charge density), instead of conceiving of the conduction electrons as a large group of point particles which zip rapidly from place to place.

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  Electrons in Solids

  An Introductory Survey

  • Richard Bube(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  6 The Free-Electron Model It is often found that the outermost valence electrons in a solid can be treated as if they were essentially free electrons. This is particularly true in that class of materials known as metals. It is indeed the fact that the valence electrons in metals behave like free electrons, which accounts for many of the electrical, thermal, and optical properties of metals. In metallic sodium, e.g., Table 5.2 indicates that the electronic configuration is ls 2 2s 2 2p 6 3s, and the outermost 3s electron can be considered to be essentially free. It is this same 3s electron that sodium gives up in chemical bonding to become a Na + ion. Similarly Table 5.2 shows that when copper gives up its outermost 4s electron, it becomes Cu + ; in this case the existence of Cu 2+ occurs when both the 4s electron and one of the 3d electrons are involved in chemical bonding. These valence electrons in metals are of course not completely free, since they still move in the presence of the positively charged ions that are located on the crystal lattice of the metal. But to an often striking degree it is suffi-cient to consider the valence electrons to be moving in a potential energy of zero, shielded from the positively charged ions by the other electrons. Since these almost-free electrons are confined within the metal, however, it is appropriate as a first approximation to consider them in terms of our model of a particle in a box from the previous chapter. In order to escape from the metal, an electron in the metal needs to acquire sufficient energy to overcome the potential barrier at the surface, known as the work function of the metal. Although this work function is not truly infinite, it is still suffi-68 What Energies are Allowed? 69 ciently large that a particle-in-a-box model can still give reasonable first-order results.

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  Electrons in Solids 2e

  An Introductory Survey

  • Richard Bube(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  6 The Free-Electron Model It is often found that the outermost valence electrons in a solid can be treated as if they were essentially free electrons. This is particularly true in that class of materials known as metals. It is indeed the fact that the valence electrons in metals behave like free electrons, which accounts for many of the electrical, thermal, and optical properties of metals. In metallic sodium, e.g., Table 5.2 indicates that the electronic configuration is ls 2 2s 2 2p 6 3s, and the outermost 3s electron can be considered to be essentially free. It is this same 3s electron that sodium gives up in chemical bonding to become a Na + ion. Similarly Table 5.2 shows that when copper gives up its outermost 4s electron, it becomes Cu + ; in this case the existence of Cu 2+ occurs when both the 4s electron and one of the 3d electrons are involved in chemical bonding. These valence electrons in metals are of course not completely free, since they still move in the presence of the positively charged ions that are located on the crystal lattice of the metal. But to an often striking degree it is sufficient to consider the valence electrons to be moving in a potential energy of zero, shielded from the positively charged ions by the other electrons. Since these almost-free electrons are confined within the metal, however, it is appropriate as a first approximation to consider them in terms of our model of a particle in a box from the previous chapter. In order to escape from the metal, an electron in the metal needs to acquire sufficient energy to overcome the potential barrier at the surface, known as the work function of the metal. Although this work function is not truly infinite, it is still 82 What Energies are Allowed? 83 sufficiently large that a particle-in-a-box model can still give reasonable first-order results.

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  Condensed Matter Physics

  • Michael P. Marder(Author)
  • 2010(Publication Date)
  • Wiley

    (Publisher)

  Neglect of the interactions of electrons with one another is justified by the idea of the Fermi liquid (Section 17.5), which shows that properly chosen linear combinations of electron states behave like noninteracting particles. But these ideas ultimately provide only partial justification for approximation schemes, so there is no choice but to suspend disbelief and begin to calculate. 6.2 Starting Hamiltonian The single-electron model is defined by the Hamiltonian N fc2 v 2 It describes N conduction electrons, each of which interacts with an external po- tential U but does not interact with the other conduction electrons. Equation (6.2) is called the single-electron model because if one finds the eigenfunctions (7 ) for single electrons, obeying ~ H V +U(r)\ ,( = {?), (6.3) y 2m then the eigenfunctions describing many particles are simply obtained from prod- ucts of the one-particle functions. The energy of the many-electron system is just a sum of the energies of the one-electron functions that make it up (Problem 1). That is, although the equation can be used to study large numbers of electrons, their properties can be obtained one electron at a time. Writing down Eq. (6.2) requires severe approximations outlined in Figure 6.1. However for a general potential U it is still impossible to solve in general. To start 158 Chapter 6. The Free Fermi Gas and Single Electron Model making progress, throw away the potential U too. The free Fermi gas, is described by -h 2 N — T ? ( ? · · ? ") = ( ? · · **)· ( 6 · 4 ) 2m f^ It describes N conduction electrons, interacting neither with nuclei nor each other. The eigenvalues and eigenfunctions of this Hamiltonian can be found exactly. No differential equation is completely specified without naming its boundary conditions. A natural choice would be to take to vanish whenever any of its arguments reaches the boundaries of the system, but this choice is not convenient for calculations.

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