DOCSLIB.ORG

Lorentz Models for Dielectrics and Drude Models for Metals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lorentz Models for Dielectrics and Drude Models for Metals 9/19/2016 ECE 5322 21st Century Electromagnetics Instructor: Dr. Raymond C. Rumpf Office: A‐337 Phone: (915) 747‐6958 E‐Mail: [email protected] Lecture #2 Electromagnetic Properties of Materials – Part I Lorentz and Drude Models Lecture 2 1 Lecture Outline • High level picture of dielectric response • Resonance • Lorentz model for dielectrics • Lorentz model for permeability • Drude model for metals • Generalizations • Other materials models Lecture 2 2 1 9/19/2016 High Level Picture of Dielectric Response Dielectric Slab We wish to understand why a dielectric exhibits an electromagnetic response. Lecture 2 4 2 9/19/2016 Atoms at Rest Without an applied electric field, the electron “clouds” around the nuclei are symmetric and at rest. Lecture 2 5 Applied Wave The electric field of a electromagnetic wave pushes the electrons away from the nuclei producing “clouds” that are offset. Lecture 2 6 3 9/19/2016 Secondary Waves The motion of the charges emits secondary waves that interfere with the applied wave to produce an overall slowing effect on the wave. Lecture 2 7 Resonance 4 9/19/2016 Visualizing Resonance – Low Frequency • Can push object to modulate amplitude • Displacement is in phase with driving force • DC offset Lecture 2 9 Visualizing Resonance – on Resonance • Can push object to large amplitude • Displacement and driving force are 90° out of phase • Peaks of push correspond to nulls of displacement Lecture 2 10 5 9/19/2016 Visualizing Resonance – High Frequency • Vanishing amplitude • Displacement is 180° out of phase Lecture 2 11 A Harmonic Oscillator >> 0 180° Phase Lag Amplitude 90° > 0 0° 0 res Lecture 2 12 6 9/19/2016 Impulse Response of a Harmonic Oscillator Excitation Ball Displacement Damping loss Amplitude Time, t Lecture 2 13 Moving Charges Radiate Waves outward travelling wave Lecture 2 14 7 9/19/2016 Lorentz Model for Dielectrics Lorentz Oscillator Model Lecture 2 16 8 9/19/2016 Maxwell’s Equations with Material Polarization Material polarization is incorporated into the constitutive rel ations. D 0 EP Response of material Response of free space Constitutive relation in terms of relative permittivity and susceptibility. D 000r EE E Comparing the above equations, we see that D EP E E P E 000 0 DE 00rr1 E 1 Lecture 2 17 Equation of Motion 2rr mm mrqE 2 tt2 0 electric force frictional force restoring force damping rate K natural acceleration force (loss/sec) 0 m frequency mm mass of an e electron Lecture 2 18 9 9/19/2016 Fourier Transform 2rr mm mrqE 2 tt2 0 Fourier transform 2 2 mj r m jr mr0 qE Simplify 22 mjmmr 0 qE Lecture 2 19 Displacement 22 mjmmr 0 qE q E r 22 mje 0 r Lecture 2 20 10 9/19/2016 Dipole Moment Definition of Dipole Moment: qr charge distance from center ** Sorry for the confusing notation, but μ here is NOT permeability. q2 E 22 mje 0 r Lecture 2 21 Lorentz Polarizability, Definition of Polarizability: E ** Sorry for the confusing notation, but here is NOT absorption. () is a tensor quantity for anisotropic materials. For simplicity, we will use the scalar form. This is the Lorentz polarizability for a single atom. q2 1 22 mje 0 Lecture 2 22 11 9/19/2016 Polarization per Unit Volume 1 Average dipole moment over all Definition: P atoms in a material. i All billions and trillions of them!!! V V Unpolarized Polarized with some randomness Equivalent uniform polarization Applied Applied E‐Field E‐Field There is some randomness to the polarized atoms so a statistical approach is taken to compute the average. N Number of atoms per unit volume PN Statistical average Lecture 2 23 Susceptibility (1 of 2) Recall the following: PN 0 E E q2 1 22 mje 0 This leads to an expression for the susceptibility: N Nq2 1 22 000mje Lecture 2 24 12 9/19/2016 Susceptibility (2 of 2) Susceptibility of a dielectric material: 2 2 Nq 2 plasma p p frequency 22 0me 0 j q 1.60217646 1019 C 12 0 8.8541878176 10 F m 31 me 9.10938188 10 kg • Note this is the susceptibility of a dielectric which has only one resonance. • The location of atoms is important because they can influence each other. We ignored this. • Real materials have many sources of resonance and all of these must be added together. Lecture 2 25 The Dielectric Function Recall that, D 00r EEPE 000 E 1 E Therefore, The ~ symbol indicates the quantity is complex r 1 vacuum material The dielectric function for a material with a single resonance is then, 2 Nq2 p 2 r 1 22 p 0 j 0me Lecture 2 26 13 9/19/2016 Summary of Derivation 1. We wrote the equation of motion by comparing bound charges to a mass on a spring. 2rr mm mrqE 2 eett2 e0 2. We performed a Fourier transform to solve this equation for r. q E r 22 mje 0 3. We calculated the electric dipole moment of the charge displaced by r. q2 E 22 mje 0 4. We calculated the volume averaged dipole moment to derive the material polarization. 1 PN V i 5. We calculated the material susceptibility. 2 2 2 p 2 Nq p p 22 r 1 00j me 22 0 j 6. We calculated the dielectric function. Lecture 2 27 Real and Imaginary Parts of ε 2 Nq2 p 2 rp122 00j me Split into real and imaginary parts 2 22 p 0 j j 1 rr r 22 22 00jj 22j 2 0 1 p 2 22 22 0 22 220 1 pp22 j 2 2 22 2 2 22 00 22 220 rp1 rp 2 222 22 2 2 22 00 Lecture 2 28 14 9/19/2016 Complex Refractive Index ñ Refractive is like a “density” to an electromagnetic wave. It quantifies the speed of an electromagnetic wave through a material. Waves travel slower through materials with higher refractive index. nnj rr 11 m e For now, we will ignore the magnetic response. n ordinary refractive index nnj r extinction coefficient Converting between dielectric function and refractive index. ñε njrr j 22 r n 2 njrr j r 2n 22 njnjnrr j ε ñ 22 njnj2 rr njrr j Lecture 2 29 Absorption Coefficient (1 of 2) From Maxwell’s equations, a plane wave in a linear, homogeneous, isotropic (LHI) medium as… The wave number is jkz E zEe 0 2 kkn00 k 0 Substituting the complex refractive index into this equation leads to… jk0 n j z kzjknz00 E zEe00 Eee Oscillatory term envelope term The absorption coefficient is defined in terms of the field intensity. z I zIe 0 Lecture 2 30 15 9/19/2016 Absorption Coefficient (2 of 2) The field intensity is related to the field amplitude through 2 I zEz Substituting expressions from the previous slide, the absorption coefficient can be calculated from 2 Iz Ez 2 z kzjknz00 Ie00 Ee e 2 2k0 2 z 2kz0 c0 Ie00 E e ee z 2kz0 2k0 Lecture 2 31 Reflectance (normal incidence) incident reflected air material transmitted The amplitude reflection coefficient r quantifies the amplitude and phase of reflected waves. 1nj r Loss contributes to reflections! 1nj The power reflection coefficient (reflectance) is always positive and between 0 and 1 (for materials without gain). 2 1n 2 Rrr * 2 2 1n Lecture 2 32 16 9/19/2016 Kramers-Kroenig Relations (1 of 3) The susceptibility is essentially the impulse response of a material to an applied electric field. PtEtd 0 e E t 0 t Pt PE 0 Causality requires that (t=0)=P(t=0)=0 From linear system theory, if (t) is a causal, than the real and imaginary parts of its Fourier transform are Hilbert transform pairs. 1 d j 1 d This means that ’ and ’’ are not independent. If we can measure one, we can calculate the other. But how do we measure at negative frequencies? Lecture 2 33 Kramers-Kroenig Relations (2 of 3) From Fourier theory, if (t) is purely real then is an even function is an odd function Applying this symmetry principle to the relations on the previous page leads to 2 d 22 0 2 d 22 0 These equations can be applied to measurements taken over just positive frequencies. Lecture 2 34 17 9/19/2016 Kramers-Kroenig Relations (3 of 3) For dilute media with weak susceptibility (’ and ’’ are small), the complex refractive index can be approximated from the susceptibility as… njj 11 1 22 Comparing the real and imaginary components of ñ and leads to the Kramers‐Kroenig relations for the refractive index and absorption coefficient. c nd 1 0 22 0 For dilute media c n1 0 d 22 0 Lecture 2 35 Summary of Properties Dielectric Function • The dielectric function is most fundamental to Maxwell’s equations. • Imaginary part only exists if there is loss. When there is loss, the real part contributes. • Perhaps more difficult to extract physical meaning from the real and imaginary parts. 22 220 rp1 rp 2222 22 22 22 00 Refractive Index • The refractive index is more closely related to wave propagation. It quantifies both velocity and loss. • Real part is solely related to phase velocity. • Imaginary part is solely related to loss. • In many ways, refractive index is a more physically meaningful parameter. nnj rr 11 m e Lecture 2 36 18 9/19/2016 Typical Lorentz Model for Dielectrics p 4 0 2 0.1 r r n R Lecture 2 37 Example – Salt Water M.
Load more

Recommended publications
  • Chapter 5 Optical Properties of Materials
    Chapter 5 Optical Properties of Materials Part I Introduction Classification of Optical Processes refractive index n() = c / v () Snell’s law absorption ~ resonance luminescence Optical medium ~ spontaneous emission a. Specular elastic and • Reflection b. Total internal Inelastic c. Diffused scattering • Propagation nonlinear-optics Optical medium • Transmission Propagation General Optical Process • Incident light is reflected, absorbed, scattered, and/or transmitted Absorbed: IA Reflected: IR Transmitted: IT Incident: I0 Scattered: IS I 0 IT IA IR IS Conservation of energy Optical Classification of Materials Transparent Translucent Opaque Optical Coefficients If neglecting the scattering process, one has I0 IT I A I R Coefficient of reflection (reflectivity) Coefficient of transmission (transmissivity) Coefficient of absorption (absorbance) Absorption – Beer’s Law dx I 0 I(x) Beer’s law x 0 l a is the absorption coefficient (dimensions are m-1). Types of Absorption • Atomic absorption: gas like materials The atoms can be treated as harmonic oscillators, there is a single resonance peak defined by the reduced mass and spring constant. v v0 Types of Absorption Paschen • Electronic absorption Due to excitation or relaxation of the electrons in the atoms Molecular Materials Organic (carbon containing) solids or liquids consist of molecules which are relatively weakly connected to other molecules. Hence, the absorption spectrum is dominated by absorptions due to the molecules themselves. Molecular Materials Absorption Spectrum of Water
    [Show full text]
  • Einstein and the Early Theory of Superconductivity, 1919–1922
    Einstein and the Early Theory of Superconductivity, 1919–1922 Tilman Sauer Einstein Papers Project California Institute of Technology 20-7 Pasadena, CA 91125, USA [email protected] Abstract Einstein’s early thoughts about superconductivity are discussed as a case study of how theoretical physics reacts to experimental find- ings that are incompatible with established theoretical notions. One such notion that is discussed is the model of electric conductivity implied by Drude’s electron theory of metals, and the derivation of the Wiedemann-Franz law within this framework. After summarizing the experimental knowledge on superconductivity around 1920, the topic is then discussed both on a phenomenological level in terms of implications of Maxwell’s equations for the case of infinite conduc- tivity, and on a microscopic level in terms of suggested models for superconductive charge transport. Analyzing Einstein’s manuscripts and correspondence as well as his own 1922 paper on the subject, it is shown that Einstein had a sustained interest in superconductivity and was well informed about the phenomenon. It is argued that his appointment as special professor in Leiden in 1920 was motivated to a considerable extent by his perception as a leading theoretician of quantum theory and condensed matter physics and the hope that he would contribute to the theoretical direction of the experiments done at Kamerlingh Onnes’ cryogenic laboratory. Einstein tried to live up to these expectations by proposing at least three experiments on the arXiv:physics/0612159v1 [physics.hist-ph] 15 Dec 2006 phenomenon, one of which was carried out twice in Leiden. Com- pared to other theoretical proposals at the time, the prominent role of quantum concepts was characteristic of Einstein’s understanding of the phenomenon.
    [Show full text]
  • Molding of Plasmonic Resonances in Metallic Nanostructures: Dependence of the Non-Linear Electric Permittivity on System Size and Temperature
    Materials 2013, 6, 4879-4910; doi:10.3390/ma6114879 OPEN ACCESS materials ISSN 1996-1944 www.mdpi.com/journal/materials Review Molding of Plasmonic Resonances in Metallic Nanostructures: Dependence of the Non-Linear Electric Permittivity on System Size and Temperature Alessandro Alabastri 1,*, Salvatore Tuccio 1, Andrea Giugni 1, Andrea Toma 1, Carlo Liberale 1, Gobind Das 1, Francesco De Angelis 1, Enzo Di Fabrizio 2,3 and Remo Proietti Zaccaria 1,* 1 Istituto Italiano di Tecnologia, Via Morego 30, Genova 16163, Italy; E-Mails: [email protected] (S.T.); [email protected] (A.G.); [email protected] (A.T.); [email protected] (C.L.); [email protected] (G.D.); [email protected] (F.A.) 2 King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering (PSE) Division, Biological and Environmental Science and Engineering (BESE) Division, Thuwal 23955-6900, Kingdom of Saudi Arabia; E-Mail: [email protected] 3 Bio-Nanotechnology and Engineering for Medicine (BIONEM), Department of Experimental and Clinical Medicine, University of Magna Graecia Viale Europa, Germaneto, Catanzaro 88100, Italy * Authors to whom correspondence should be addressed; E-Mails: [email protected] (A.A.); [email protected] (R.P.Z.); Tel.: +39-010-7178-247; Fax: +39-010-720-321. Received: 10 July 2013; in revised form: 8 October 2013 / Accepted: 10 October 2013 / Published: 25 October 2013 Abstract: In this paper, we review the principal theoretical models through which the dielectric function of metals can be described. Starting from the Drude assumptions for intraband transitions, we show how this model can be improved by including interband absorption and temperature effect in the damping coefficients.
    [Show full text]
  • Lecture Notes in Solid State 3
    Lecture notes in Solid State 3 Eytan Grosfeld Physics Department, Ben-Gurion University of the Negev Classical free electron model for metals: The Drude model Recommended reading: • Chapter 1, Ashcroft & Mermin. The conductivity of metals is described very well by the classical Drude formula, ne2τ σ = (1.1) D m where m is the electronic mass (as decided by the band-structure) and e is the electronic charge. The conductivity is directly proportional to the electronic density n; and, to τ, the mean free time between collisions of the conducting electrons with defects that are generically present in the system: • Static defects that scatter the electron elastically, including: static impu- rities and structural defects. One can dene the elastic mean free time τe. • Dynamical defects which scatter the electron inelastically (as they can carry o energy) including: photons, other electrons, other excitations (such as plasmons). One can accordingly dene the inelastic mean free time τ'. The eciency of the latter processes depends on the temperature T : we expect it to increase as T is increased. At very low temperatures the dominant scattering is elastic, and then τ does not depend strongly on temperature but instead it depends on the amount of disorder (realized as random static impurities). We expect τ to decrease as the temperature or the amount of disorder in the system are increased. The Drude model 1897 - discovery of the electron (J. J. Thomson). 1900 - only three years later, Drude applied the kinetic theory of gases to a metal - considering it to be a gas of electrons. Assumptions: 1.
    [Show full text]
  • Chapter 5 the Drude Theory of Metals
    Chapter 5 The Drude Theory of Metals • Basic assumption of Drude model • DC electrical conductivity of a metal • Hall effect • Thermal conductivity in a metal 1 Basic assumptions of Drude model * A “ gas of conduction electrons of mass m, which move against a background of heavy immobile ions Zρ Electron density n = .0 6022 ×10 24 m A .0 6022 ×10 24 Avogadro’s number ρm Mass density in g/cm 3 A Atomic mass in g/mole Z Number of electron each atom contribute rs Radius of a sphere whose volume is equal to the volume per conduction electron V 1 4 3 3/1 = = πr 3 r = N n 3 s s 4πn r s ~ 2 − 3 in typical metal a0 Bohr radius The density is typically 10 3 times greater than those of a classical gas at normal T and P. 2 * Between collisions the interaction of a given electron, both with others and with the ions, is neglected. * Coliisons in the Drude model are instantaneous events that abruptly alter the velocity of an electron. Drude attributed them to the electrons bouncing off the impenetrable ion cores. 1 * We shall assume that an electron experiences a collision with a probability per unit time τ Probability dt during time interval dt τ τ : relaxation time * Electrons are assumed to achieve thermal equilibrium with their surroundings only through collisons 3 DC Electrical Conductivity of a Metal r n electrons per unit volume all move with velocity v . n(vdt )A electrons will cross an area A perpendicular to the direction of flow.
    [Show full text]
  • Drude-Lorentz Analysis of the Optical Properties of The
    DRUDE-LORENTZ ANALYSIS OF THE OPTICAL PROPERTIES OF THE QUASI-TWO-DIMENSIONAL DICHALCOGENIDES 2H-NbSe2 AND 2H-TaSe2 A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Dinesh Marasinghe August, 2018 DRUDE-LORENTZ ANALYSIS OF THE OPTICAL PROPERTIES OF THE QUASI-TWO-DIMENSIONAL DICHALCOGENIDES 2H-NbSe2 AND 2H-TaSe2 Dinesh Marasinghe Thesis Approved: Accepted: Advisor Dean of the College Dr. Sasa V. Dordevic Dr. Linda Subich Faculty Reader Dean of the Graduate School Dr. Ben Yu-Kuang Hu Dr. Chand Midha Faculty Reader Date Dr. Sergei F. Lyuksyutov Department Chair Dr. Chris Ziegler ii ABSTRACT The reflectivity of a material basically depends on the collisions of free electrons. Existing spectroscopic reflectance data of two dichalcogenide materials, 2H-NbSe2 and 2H-TaSe2, is studied in this work. We analyzed this information with the Drude- Lorentz model for 2D electron gases. Using the program RefFIT, we explored the materials' reflectance and the conductivity and their dependance on temperature across frequencies ranging from infrared to ultraviolet. We examined both the in- plane (ab-plane) and out-of-plane (c-axis) components. In both materials, the ab- plane had greater reflectance and conductivity, with 2H-NbSe2 having the largest of both quantities. The anisotropy-direction dependence-effects are most pronounced in 2H-TaSe2. When frequency increased, collisions of free elections increased, decreasing the reflectance and conductivity. We also determined that the reflectance and the conductivity of the materials decreased with increased temperature for the same reason. Once the plasma edge was reached, interband transition occurred, causing an increase in conduction electrons and a decrease in both reflectance and conductivity.
    [Show full text]
  • Drude Model 1 in 1897, J. J. Thomson Discovered Electrons. in 1905
    Drude Model In 1897, J. J. Thomson discovered electrons. In 1905, Einstein interpreted the photoelectric effect In 1911 - Rutherford proved that atoms are composed of a point-like positively charged, massive nucleus surrounded by a sea of electrons. - Drude constructed his theory of electrical and thermal conduction in metals by (1) considering the electrons to be a gas of negatively charged particles traversing in a medium of uniformly distributed positive ions, and (2) applying the kinetic theory of gas to the electron sea. Below is a schematic diagram of Drude’s model of metals: (In 1922, Bohr was awarded the Nobel Prize for his contribution to the understanding of the structure of atoms. In late 1925, the Schrödinger equation was formulated.) Core electrons Zc e +Ze Zc e +Ze Zc e +Ze Zc e +Ze Positively charged ions Electron sea due to the delocalized valence electrons, (Z – Zc) from each atom. There is one important parameter of the model: (1) Electron number density, n 23 n = N/V = 6.02 x 10 x (Z Zc)m/A, (1.1) Avogadro’s number where N is the total number of electrons in the metal, V is the volume, m is the mass density, and A is the mass number. Note that only the valence electrons ((Z Zc) per atom) contribute to n. (2) Average electron separation, rs. 3 V/N = 1/n = (4/3) rs 1/3 rs = [3/(4n)] (1.2) 1 Drude Model The value of n varies from 0.911022/cm3 for Cs to 24.71022/cm3 for Be among different metals.
    [Show full text]
  • A Metamaterial Path Towards Optical Integrated Nanocircuits
    University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2015 A Metamaterial Path Towards Optical Integrated Nanocircuits Fereshteh Abbasi Mahmoudabadi University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Electrical and Electronics Commons, Electromagnetics and Photonics Commons, and the Optics Commons Recommended Citation Abbasi Mahmoudabadi, Fereshteh, "A Metamaterial Path Towards Optical Integrated Nanocircuits" (2015). Publicly Accessible Penn Dissertations. 1570. https://repository.upenn.edu/edissertations/1570 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1570 For more information, please contact [email protected]. A Metamaterial Path Towards Optical Integrated Nanocircuits Abstract Metamaterials are known to demonstrate exotic electromagnetic and optical properties. The extra control over manipulation of waves and fields afforded by metamaterials can be exploited towards exploring various platforms, e.g., optical integrated circuits. Nanophotonic integrated circuits have been the topic of past and ongoing research in multiple fields including, but not limited o,t electrical engineering, optics and materials science. In the present work, we theoretically study and analyze metamaterial properties that can be potentially utilized in the future design of optical integrated circuits. On this path, we seek inspiration from electronics to tackle multiple issues in developing such layered nanocircuitry. We identify modularity, directionality/isolation and tunability as three useful features of electronics and we theoretically explore mimicking them in nanoscale optics. Using epsilon-near-zero (ENZ) and mu-near- zero (MNZ) properties we propose concepts to transplant some aspects of modular design of electronic passive circuits and filters into nanophotonics. We also exploit ENZ materials to develop “transformer- like” functionality in optical nanocircuits.
    [Show full text]
  • NOTES on the DRUDE MODEL 1. Assumptions and Basics the Drude
    NOTES ON THE DRUDE MODEL KYLE MCELROY Abstract. Here are some notes about the classical theory of metals and the Drude model. The classical theory of electrons in metals is used to describe the basics of metallic behavior. These include their electrical conductivity, heat conductivity, reflection of optical wavelengths, etc. 1. Assumptions and basics The Drude model was developed at the turn of the 20th century by Paul Drude. It came a few years after J.J. Thompson discovered the electron in 1897. It predates quantum theory, but still can tell us a lot about electrons in metals. As background the the model we should get to know the electrons and how many we are dealing with. We're going to keep the valence assumption. This assumption rests on the intuition that the core electrons will be more tightly bound to their nuclei and hence will not be free to wander around and contribute to conduction. Essentially this lowers the number of electrons from Z to Zc where Zc is the number of conduction electrons. So in a sample of metal say Sodium (Na) the density of conduction electrons, n is: (1.1) Z ½ 1e=atom ¢ 1 £ 106g=m3 n = N c m = 6:02 £ 1023atoms=mol = 2 £ 1028e=m3 A A 29g=mol where NA is Avogadro's number, ½m is the density of the metal, A is the atomic number of the element and the numbers are for Na. For the actual model we're going to eliminate all the electron ion interactions and replace them by a single parameter.
    [Show full text]
  • Drude Model for Dielectric Constant of Metals
    Drude Model for dielectric constant of metals. • Conduction Current in Metals • EM Wave Propagation in Metals • Skin Depth • Plasma Frequency Ref : Prof. Robert P. Lucht, Purdue University Drude model z Drude model : Lorenz model (Harmonic oscillator model) without restoration force (that is, free electrons which are not bound to a particular nucleus) Linear Dielectric Response of Matter ConductionConduction CurrentCurrent inin MetalsMetals The equation of motion of a free electron (not bound to a particular nucleus; C = 0), rr r dr2 r m druurruu dv r mCreEmmveE=− −e − ⇒ +γ =− eeedt 2 τ dt dt 1 (τ =≈ : relaxation time 10−14 s) Lorentz model γ (Harmonic oscillator model) If C = 0, it is called Drude model The current density is defined : r ⎡⎤C J=− N evr with units of ⎣⎦⎢⎥sm− 2 Substituting in the equation of motion we obtain : r dJrr⎛⎞ N e2 +=γ JE⎜⎟ dt⎝⎠ me ConductionConduction CurrentCurrent inin MetalsMetals Assume that the applied electric field and the conduction current density are given by : rr rr Local approximation E=− Eexp() itωω J =− J exp () it 00to the current-field relation Substituting into the equation of motion we obtain : r dJ⎡⎤exp − iω t ⎣⎦0 () rrr +−=−−+−γωωωγωJitiJitJitexp() exp () exp () dt 000 ⎛⎞Ne2 r =−⎜⎟Eit0 exp()ω ⎝⎠me Multiplying through byexp()+ iω t : rr⎛⎞Ne2 ()−+iJωγ 00 =⎜⎟ E ⎝⎠me rr⎛⎞Ne2 or equivalently()−+ iωγ J =⎜⎟ E ⎝⎠me ConductionConduction CurrentCurrent inin MetalsMetals For static fields()ω = 0: we obtain rrr⎛⎞Ne22 Ne J==⇒==⎜⎟ Eσσ E static conductivity ⎝⎠mmeeγγ For the general case of an oscillating applied field : rrr⎡⎤σ J==⎢⎥ Eσσωω E = dynamic conductivity ⎣⎦1/− ()iωγ For very low frequencies,1,()ωγ << the dynamicconductivity is purely real and the electrons follow the electric field.
    [Show full text]
  • Condensed Matter Physics I: 2015-2016 Introduction, Drude Model
    Condensed matter physics I: 2015-2016 Introduction, Drude model R. Ganesh 1 Introduction Condensed matter physics (CMP) is the study of systems with many interact- ing particles. The adjective `condensed' should be understood as the opposite of `dilute' { the constituent particles are so close to each other that their interac- tions cannot be ignored. Some typical examples of `condensed' systems include electrons in a metal, spins in an insulating magnet, neutrons in a neutron star, Na atoms in an ultracold atomic gas, He atoms in liquid He, etc. Historically, the field grew out solid state physics. It used to be called 'solid state physics' until 1967 when Phil Anderson and Volker Heine coined the term `condensed matter physics'. CMP serves as a counterpoint to the `reductionism'. In the reductionist ap- proach, complex systems are studied by breaking them down into smaller, funda- mental units. This is exemplified by particle accelerator physics in which atoms are broken down into nucleons, which are further broken down into quarks, etc. In the condensed matter point of view, the whole is greater than the sum of its parts. For example, we may understand how a single He atom behaves according to the laws of quantum mechanics. However, that does not tell us how a beaker with liquid He will behave. Completely unforeseen behaviour (superfluidity in this case) can emerge from the interactions of many constituent particles. The new physics that emerges from collective behaviour at different length scales is the domain of condensed matter physics. There are two broad (not necessarily mutually exclusive) divisions within CMP.
    [Show full text]
  • 1 the Electronic Structure of Solids Uwe Bovensiepen, Silke Biermann, and Luca Perfetti
    j1 1 The Electronic Structure of Solids Uwe Bovensiepen, Silke Biermann, and Luca Perfetti The discussion of dynamics at interfaces is based on the motion of ion cores and electronic excitations that are mostly optically driven. Hence, the electronic structure is of fundamental importance here. In solids such as molecular or ionic crystals, the valence electron distribution is not considerably distorted from the respective isolated atoms, ions, or molecules. Hence, their cohesion is entirely given by the classical potential energy of negligibly deformed electron distributions of bare particles, and van der Waals or Coulomb interactions are responsible for the formation of solid materials. This ceases to be so in metals and covalent crystals because the valence electron distribution plays the decisive role in bonding the constituents to a solid. In turn, the valence electron distribution can be considerably modified from the isolated atom or ion. A general description of solids must, therefore, consider the electronic structure in the first place. Fur- thermore, the dynamical processes discussed in this book are mostly optically excited or electron mediated. This chapter introduces the basic concepts widely used in the description of the electronic structure in solid materials. In Section 1.1, we present the description of the nearly free electron approximation that is motivated by optical excitations of a solid following the Drude model. We introduce the Fermi sphere and the dispersion of electronic bands in momentum space. In Section 1.2, the influence of the periodic potential in a crystal is considered, which leads to the description of the electronic band structure by Blochs theory for delocalized states.
    [Show full text]