Physics
Drude Model
The Drude model is a simple classical theory used to describe the electrical and thermal conductivity of metals. It assumes that electrons in a metal behave like a gas of free particles that collide with ions, leading to electrical resistance and thermal conductivity. The model provides a basic understanding of electron behavior in metals and forms the basis for more advanced quantum mechanical theories.
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8 Key excerpts on "Drude Model"
eBook - PDF- Syeda Ramsha Ali(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
ELECTRONIC CHARACTERISTICS OF SOLIDS 8 CHAPTER CONTENTS 8.1. Drude Model .................................................................................. 178 8.2. Relaxation Time Approximation ...................................................... 180 8.3. Drude Model Failure ....................................................................... 182 8.4. Hall Effect ....................................................................................... 184 8.5. Sommerfeld Model ......................................................................... 186 8.6. Fermi Dirac Function ...................................................................... 189 8.7. Bloch’s Theorem ............................................................................. 192 8.8. Electronic Band Structure ................................................................ 194 8.9. Free Electron Model Introduction .................................................... 194 8.10. Dispersion Relation ...................................................................... 198 Quantum Mechanics for Applied Nanotechnology 178 8.1. Drude Model 8.1.1. Introduction The Drude demonstrate was produced at the turn of the twentieth century by Paul Drude. It came to a couple of years after J.J. Thompson found the electron in 1897. It originates before quantum hypothesis, yet can disclose to us a considerable measure about electrons in metals. As a foundation the model we ought to become more acquainted with the electrons and what number of we are managing. We will keep the valence presumption. This supposition lays on the instinct that the center electrons will be all the more firmly bound to their cores and henceforth won’t be allowed to meander around and add to conduction. Basically, this brings down the quantity of electrons from Z to Z c where Z c is the quantity of conduction electrons.
eBook - ePubMathematics and Physics for Nanotechnology
Technical Tools and Modelling
- Paolo Di Sia(Author)
- 2019(Publication Date)
- Jenny Stanford Publishing(Publisher)
ω) is the time-resolved THz spectroscopy (TRTS), an ultrafast non-contact optical probe; obtained data are normally fitted using Drude–Lorentz, Drude–Smith or effective medium models. In the following, I illustrate an overview of fundamental models that, starting from the Drude Model, analyse and explain the transport phenomenon of the matter at the nanoscale, coming to recent developments concerning variations of Drude–Lorentz-like models.13.2 The Drude ModelThe most meaningful characteristics of metals are their properties of electric and thermal conduction. Over the years, this fact brought to think in terms of models in which the electrons are relatively free and can move under the influence of electric fields. Historically, two important models of the elementary theory of metals are born:(a) The Drude Model, published in 1900 and based on the kinetic theory of an electron gas in a solid; it is assumed that all electrons have the same average kinetic energy Em .(b) A variation of the previous one, integrated with the foundations of quantum mechanics, called Sommerfeld model.In the Drude Model, the valence electrons of atoms are considered free inside the metal; all electrons move as an electronic gas. The ideal gas equations are applied, even if the electron gas is dense. If the electrons try to leave the metal, they are retained by the net positive charge left on the metal; they are then in a potential energy hole of depth W. The uniform distribution of positive charges associated with the metal ions, that make the metal neutral ‘on average’, determine a large value of W. The work L = W − Em has an enough large value to retain the electrons in the solid. The kinetic theory leads to an average energy medium kB T/2 for each degree of freedom (kB
eBook - PDF- Piers Coleman(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
The classical picture of electron conductivity was developed by Paul Drude in 1900, while working at the University of Leipzig [1]. Although his model was introduced before the advent of quantum mechanics, many of his basic concepts carry over to the quan- tum theory of conductivity. Drude introduced the the concept of the electron mean free path l, the mean distance between scattering events. The characteristic time scale between scattering events is called the transport scattering time τ tr . (We use the “tr” subscript to distinguish this quantity from the quasiparticle scattering time τ , because not all scattering events cause the electric current to decay.) In a Fermi gas, the characteristic velocity of electrons is the Fermi velocity and the mean free path and transport scattering time are related by the simple relation l = v F τ tr . (10.10) The ratio of the mean free path to the electron wavelength is determined by the product of the Fermi wavevector and the mean free path. This quantity is the same order of magnitude 334 Electron transport theory as the ratio of the scattering time to the characteristic time scale / F associated with the Fermi energy, so that l λ F = k F l 2π ∼ τ tr / F = τ tr . (10.11) In very pure metals, the mean free path l of Bloch wave electrons can be tens or even hundreds of microns, l ∼ 10 −6 m, so this ratio can become as large as 10 4 or even 10 6 . From this perspective, the rate at which current decays in a good metal is very slow on atomic time scales. There are two important aspects to the Drude Model (see Figure 10.1): • the diffusive nature of density fluctuations • the Lorentzian lineshape of the optical conductivity, σ (ω) = ne 2 m 1 τ −1 tr − iω . (10.12) Drude recognized that, on length scales much larger than the mean free path, multiple scattering events induce diffusion in the electron motion.
eBook - ePubSolid State Quantum Information — An Advanced Textbook
Quantum Aspect of Many-Body Systems
- Vlatko Vedral, Wonmin Son(Authors)
- 2018(Publication Date)
- WSPC (EUROPE)(Publisher)
Chapter 2
Electrical Conductivity in Solid
Electrical conductivity is a measure of a material’s ability to conduct an electric current. When an electrical potential is placed across a conductor, its movable charges flow is giving rise to an electric current. Solids that can carry the electric current have atoms with outer electrons and the electrons are free to move throughout the entire length of the material. The phenomena of electrical conductivity have explained by several different models and they had provided various levels of understanding. They had applied classical, semiclassical theories and theories with first quantized and second quantized versions.We start with simplest classical model of electrical conductivity which is still good for basic understanding. However, the theory is limited so that they had failed to explain the other phenomena in a solid and it had been overcome by more elaborated theories. It is quantum theory which has successfully explained them with consistency.2.1.Classical (Drude) Model
In 1900, Paul Drude proposed a model of electrical conductivity in a metallic material. The model explains the transport properties of electrons in metals with an assumption that the microscopic behavior of electrons in a solid may be treated as classical motion of point particles. The dynamics of electron resembles a pinball machine as like sea of balls bouncing and rebouncing in the lattices that is composed of heavier, relatively immobile positive ions. In the model, several assumption had been made for the matter of simplification and they are as follows:
eBook - PDFPhysics of Electronic Materials
Principles and Applications
- Jørgen Rammer(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
Each cell volume in real space, x c , thus contains electrons occu-pying Gaussian states with different momenta describing electrons moving in all directions as depicted in Figure 4.2 , the picture emulating that of classical physics. 4.8 Drude Transport Theory A conductor, such as a metal in its normal state, exhibits resistance due to its imperfec-tions: foreign atoms substituting for atoms of the crystal, vacancies due to missing atoms, dislocations in the crystal, grain boundaries. These imperfections lead to scattering of con-duction electrons, as does the interaction of electrons with the thermal vibrations of the 86 Standard Conductor Model Figure 4.2 Each spatial volume contains electrons in wave packet states moving in all directions. Figure 4.3 Myriad of impurity scatterings of electrons. ionic lattice and Coulomb interaction between the electrons. To understand the main effect of scattering on the electrical properties of a conductor, it is sufficient to consider the scat-tering by impurities, characterized, as discussed in Section 4.6 , by a short-range impurity potential, Eq. ( 4.78 ). The scattering off the short-range impurity potentials in the presence of an external force leads to a dynamics of the electrons that is like that of classical parti-cles bouncing off hard spheres, except that the scattering is a quantum mechanical process that gives probabilities for scattering into all Gaussian states of the same energy. 7 In a piece of metal there is an astronomically large number of electrons bouncing off the many impurities, as illustrated in Figure 4.3 . In any short time interval, many electrons are scattered, and a statistical theory is called for: it is the average velocity of the multitude 7 Recall the discussion of scattering off a potential barrier in Chapter 2 , where in the considered one-dimensional case only two energy-conserving channels are present, transmission and reflection.
eBook - PDF- Joseph H. Simmons, Kelly S. Potter, Joseph H Simmons(Authors)
- 1999(Publication Date)
- Academic Press(Publisher)
In metals, the charges are the free electrons; therefore, there is no restoring force. The Drude Model uses this approach to give a good fundamental understanding of the behavior of free electrons in a metal under optical illumination, and allows the calculation of the induced polarization. Let us look at a set of iV^ free electrons per unit volume, each with mass m and charge q^, exposed to an oscillatory electric field in the x direction, E^. The differential equation of motion for the free electrons, written from Newton's second law, and its solution are: Letting x(^) = XQC^^^ yields Xo = -^ (2.6) 2,2 Atomistic view: Drude Model 63 Since x{t) is the displacement of the electrons from equilibrium, we assume that a dipole moment is formed with the underlying charge of the ionic cores. This dipole moment, p, is equal to the product of the charge and the displacement: p{t)=ql=q^{t) = -'^^ (2.7) The polarization, P, of the material is simply the sum of dipole moments over all free electrons. In a first approximation, we neglect electron-electron interactions. All the free electrons are subject to the same force, so the polarization becomes Recalling the relation between the electric permittivity, the electric displacement vector, the applied electric field, and the polarization (D = eE = GQE + P ) , we can write en-l^%P-n _ l + _ ^ _ l -_ _ ^ (2.9) The dielectric constant is therefore negative at low frequencies, goes through zero at a point defined as the plasma frequency [G£)(a)p) = 0], and then becomes positive at higher frequencies. The complex refractive index is essentially imaginary below the plasma frequency; thus the electro-magnetic wave cannot propagate in the metal and is reflected. The plasma frequency, cOp, is expressed as follows: «.J=5# (210) This simplifies the expression for the dielectric constant to (o; 2 eo = l -: i (2.11) CO Note that the value of the plasma frequency is dependent only on the number of free electrons and not on the conductivity.
- 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.
eBook - ePub- Jürgen Jahns, Stefan Helfert(Authors)
- 2012(Publication Date)
- Wiley-VCH(Publisher)
In what follows, we will increase the complexity of the structure by first considering metals with a finite height in one direction. Here, we will also discuss the excitement of plasmon waves. Finally, three-dimensional plasmon waveguides will be presented.12.1 Drude Model of Electrons in Metal
To determine the permittivity of the metal, a Drude Model is used. In this Drude Model, it is assumed that an electric field causes a certain force and, with this, a certain acceleration on an electron. On the other hand, the positive charged cores of the atoms lead to a damping force. With this, a differential equation can be derived from which we obtain a relative permittivity of the metal according to(12.1)A more detailed derivation of this expression is given in Section 14.4.1. In the literature, one often finds ε∞ = 1. We should mention that the plus sign in front of the term iγ ω is a direct consequence of the time dependency that we use in the this book according to e−iωt. Changing the sign in the exponent would also change the sign in the Drude expression. For examples used in this section, we take the following values for silver: ε∞ = 3.361 74, ωp = 1.3388 × 1016 s−1 , and γ = 7.075 92 × 1013 s−1 . It must be noted that these parameters are obtained by fitting measured values. Therefore, slightly differing ones can be found in various textbooks. Due to the well known relation between frequency and wavelength (ωλ0 = 2πc0 ), we can alternatively give εm as a function of λ0 . Values in the optical regime (400 nm < λ0 < 1600 nm) are shown in Figure 12.1 .Immediately, a few characteristics are observed:Figure 12.1Permittivity of silver for optical wavelengths.a) The permittivity takes the formwithHence, the real part of εm is negative in the considered wavelength range. This characteristic is important for the excitement of plasmon polariton waves as we will see soon.b) As seen, εm
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