DOCSLIB.ORG

E N E M V De Dn ED 2 3 2 2 )( ===Σ − Σ Π Ρ = Ρi + Ρph(T)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
E N E M V De Dn ED 2 3 2 2 )( ===Σ − Σ Π Ρ = Ρi + Ρph(T) Phys 446 Solid State Physics Summary Last Lecture Lecture 8 Free electron model – simplest way to describe electronic properties of (Ch. 4.9 – 4.10, 5.1-5.6) metals: the valence electrons of free atoms become conduction electrons in crystal and move freely throughout the crystal. Last time: Discussed the free electron (Drude) model applied to electronic specific heat and electrical conductivity. Today: Finish with Free electron model. Thermal conductivity. Motion in magnetic field: cyclotron resonance and Hall effect Start new chapter: energy bands in solids Summary Last Lecture (continued) Fermi energy - energy of the highest occupied electronic level at T = 0 K; 3D case: 2 3 1 3 1 3 =2 ⎛ 3π 2 N ⎞ ⎛ 3π 2 N ⎞ = ⎛ 3π 2 N ⎞ ⎜ ⎟ k = ⎜ ⎟ v = ⎜ ⎟ EF = ⎜ ⎟ F ⎜ ⎟ F ⎜ ⎟ 2m ⎝ V ⎠ ⎝ V ⎠ m ⎝ V ⎠ Density of states of 3D free electron gas: Electrical conductivity: 3 2 ne2τ dN V ⎛ 2m ⎞ 1 2 3N σ = − D(E) = = 2 ⎜ 2 ⎟ E = m* dE 2π ⎝ = ⎠ 2E ρ = ρ + ρ (T) Heat capacity of free electron gas i ph at low temperatures kBT << EF : Thermal conductivity: Wiedemann-Franz law 2 π 2 ⎛ k ⎞ K = LσT L = ⎜ B ⎟ 3 ⎝ e ⎠ Motion in a magnetic field: cyclotron resonance Applied magnetic field →the Lorentz force: F = −e[E+(v × B)] Perfect metal, no electric field - the equation of motion is: Let the magnetic field to be along the z-direction. Then dv dv x = −ω v y eB c y = ωcvx where ω = - cyclotron frequency dt dt c m For moderate magnetic absorption fields (~ few kG), ωc ~ few GHz. e.g. for B = 0.1 T, fc = ωc/2π = 2.8 GHz Cyclotron resonance – peak in absorption of electromagnetic waves at ωc Used to measure the effective mass in metals and semiconductors Summary of free electron model Limitations of free electron model Free electron model – simplest way to describe electronic properties of The free electron model gives a good insight into many properties of metals: the valence electrons of free atoms become conduction electrons in metals, such as the heat capacity, thermal conductivity and electrical crystal and move freely throughout the crystal. conductivity. However, it fails to explain a number of important Fermi energy - the energy of the highest occupied electronic level at T = 0 properties and experimental facts, for example: K; 2 3 1 3 1 3 =2 ⎛ 3π 2 N ⎞ ⎛ 3π 2 N ⎞ = ⎛ 3π 2 N ⎞ ⎜ ⎟ k = ⎜ ⎟ v = ⎜ ⎟ • the difference between metals, semiconductors and insulators EF = ⎜ ⎟ F ⎜ ⎟ F ⎜ ⎟ 2m ⎝ V ⎠ ⎝ V ⎠ m ⎝ V ⎠ •It does not explain the occurrence of positive values of the Hall Density of states of 3D free electron gas: Electrical conductivity: coefficient. 3 2 ne2τ dN V ⎛ 2m ⎞ 1 2 3N σ = − •Also the relation between conduction electrons in the metal and the D(E) = = 2 ⎜ 2 ⎟ E = m* dE 2π ⎝ = ⎠ 2E number of valence electrons in free atoms is not always correct. ρ = ρi + ρph(T) Bivalent and trivalent metals are consistently less conductive than the Heat capacity of free electron gas monovalent metals (Cu, Ag, Au) Thermal conductivity: at low temperatures kBT << EF : Wiedemann-Franz law ⇒ need a more accurate theory, which would be able to answer these 2 π 2 ⎛ k ⎞ questions – the band theory K = LσT L = ⎜ B ⎟ 3 ⎝ e ⎠ The problem of electrons in a solid – a many-electron problem Bloch theorem Write the Schrödinger equation the approximation of non-interacting The full Hamiltonian contains not only the one-electron potentials electrons: describing the interactions of the electrons with atomic nuclei, but also pair potentials describing the electron-electron interactions The many-electron problem is impossible to solve exactly ⇒ ψ(r) – wave function for one electron. simplified assumptions needed Independent electrons, which obey a one-electron Schrödinger The simplest approach we have already considered - a free electron equation a periodic potential U(r) = U(r+ T) - Bloch electrons model Bloch theorem: the solution has the form where The next step is an independent electron approximation: assume that all the interactions are described by an effective potential. uk(r)= uk(r+T) - a periodic function with the same period as the lattice One of the most important properties Bloch theorem introduces a wave vector k, which plays the same of this potential - its periodicity: fundamental role in the motion in a periodic potential that the free U(r) = U(r+ T) electron wave vector k plays in the free-electron theory. ħk is known as the crystal momentum or quasi-momentum Another conclusions following from the Bloch theorem: Number of states in a band the wave vector k can always be confined to the first Brillouin zone The number of states in a band within the first Brillouin zone is equal This is because any k' not in the first Brillouin zone can be written as to the number of primitive unit cells N in the crystal. k' = k + G Then, if the Bloch form holds for k', it will also hold for k Consider the one-dimensional case, periodic boundary conditions. Energy bands Allowed values of k form a uniform mesh whose unit spacing is 2π/L Substitute the solutions in the Bloch form into the ⇒ The number of states inside the first zone, whose length is 2π/a, is (2π/a)/(2π/L) = L/a = N, where N is the number of unit cells Schrodinger equation, obtain: A similar argument may be applied in 2- and 3-dimensional cases. with periodic condition: uk(r)= uk(r+T) For any k, find an infinite number of solutions with discrete energies Taking into account two spin orientations, conclude that there are 2N En(k), labeled with the band index n independent states (orbitals) in each energy band. For each n, the set of electronic levels specified by En(k) is called an energy band. The information contained in these functions for different n and k is referred to as the band structure of the solid. Nearly free electron (weak binding) model Now, turn on a weak potential. Consider it as a weak periodic First step: empty-lattice model. When perturbation in Hamiltonian. the potential is zero the solutions of the From perturbation theory have: 2 Schrödinger equation are plane waves: j,k' U i,k E (k) = E 0 (k) + ψ 0 U ψ 0 + 2 2 1 i i i,k i,k ∑ 0 0 0 = k 0 ik⋅r k ', j Ei (k) − E j (k) E (k) = ψ k (r) = e V 2m0 c where index i refers to ith band; 0 refers to empty-lattice model. where the wave function is normalized The first term is the undisturbed free-electron to the volume of unit cell Vc value for the energy. The second term is the mean value of the potential in the state i, k – constant independent of k – can set to zero The third term – the 2nd order correction – vanishes except k' = k + G The third term can be rewritten as There are special k points for which the energy levels become degenerate and the relationship holds for any non-zero value of the potential: UG - Fourier transform of the crystal potential U Finally we obtain for the energy: This conduction implies that k must lie on a Bragg plane bisecting the line joining the origin of k space and the However, the perturbation theory cannot be applied when the reciprocal lattice point G potential cannot be considered as a small perturbation This happens when the magnitude of the potential becomes comparable with the energy separation between the bands, i.e. ⇒ a weak periodic potential has its major effect on those free electron levels whose wave vectors are close to ones at which the Bragg reflection can occur. In this case we have to solve the Schrödinger equation explicitly In order to find the energy levels and the wave functions near these Illustrate this behavior using a one-dimensional lattice points we need to invoke the degenerate perturbation theory. The splitting of the bands at each Bragg plane in the extended-zone The result: scheme results in the splitting of the bands both at the boundaries and at the centre of the first Brillouin zone. This results is particularly simple for point lying on the Bragg plane: Obtain The magnitude of the band gap is equal to twice the Fourier component of the crystal potential. Intermediate Summary Origin of the energy gaps The Bloch theorem: the wave function for an electron in periodic potential can be written in the form: where uk(r) = uk(r+T) - a periodic function with the period of the lattice The energy spectrum of electrons consists of a set of continuous energy bands, separated by regions with no allowed states - gaps Function E(k) satisfies the symmetry properties of a crystal, in particular, the translational invariance: E(k) = E(k +G) This allows considering the first Brillouin zone only. We focused on the energy values Also, inversion symmetry: E(k) = E(-k) got away from the zone edges Nearly free electron model – weak crystal potential. Electron behaves essentially as a free particle, except the wave vectors close and to the boundaries of the zone. near the zone edges. In these regions, energy gaps appear: Eg = 2|UG| Now, let's see how the wave functions are modified by (weak) crystal For simplicity we consider a one-dimensional lattice, for which the potential. From the perturbation theory, have for the first band zone edges are k=½G = π/a (Bragg reflection occurs) (away from the zone edge): The result: 0 U 0 G 1 0 0 1 i()π a x −i()π a x ψ 1(k) =ψ 1 (k) + ψ 2 (k) 0 0 ψ ± = 1/ 2 []ψ 1 (π a) ±ψ 2 (π a) = 1/ 2 ()e ± e E1 (k) − E2 (k) L L Here, again, we leave only the nearest-band (2nd) term, as we did for at the zone edge, the scattering is so strong that the reflected wave the energy levels – because of the large denominator for higher bands.
Load more

Recommended publications
  • Solid State Physics II Level 4 Semester 1 Course Content
    Solid State Physics II Level 4 Semester 1 Course Content L1. Introduction to solid state physics - The free electron theory : Free levels in one dimension. L2. Free electron gas in three dimensions. L3. Electrical conductivity – Motion in magnetic field- Wiedemann-Franz law. L4. Nearly free electron model - origin of the energy band. L5. Bloch functions - Kronig Penney model. L6. Dielectrics I : Polarization in dielectrics L7 .Dielectrics II: Types of polarization - dielectric constant L8. Assessment L9. Experimental determination of dielectric constant L10. Ferroelectrics (1) : Ferroelectric crystals L11. Ferroelectrics (2): Piezoelectricity L12. Piezoelectricity Applications L1 : Solid State Physics Solid state physics is the study of rigid matter, or solids, ,through methods such as quantum mechanics, crystallography, electromagnetism and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic- scale properties. Thus, solid-state physics forms the theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Crystalline solids & Amorphous solids Solid materials are formed from densely-packed atoms, which interact intensely. These interactions produce : the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed , the atoms may be arranged in a regular, geometric pattern (crystalline solids, which include metals and ordinary water ice) , or irregularly (an amorphous solid such as common window glass). Crystalline solids & Amorphous solids The bulk of solid-state physics theory and research is focused on crystals.
    [Show full text]
  • Lecture Notes
    Solid State Physics PHYS 40352 by Mike Godfrey Spring 2012 Last changed on May 22, 2017 ii Contents Preface v 1 Crystal structure 1 1.1 Lattice and basis . .1 1.1.1 Unit cells . .2 1.1.2 Crystal symmetry . .3 1.1.3 Two-dimensional lattices . .4 1.1.4 Three-dimensional lattices . .7 1.1.5 Some cubic crystal structures ................................ 10 1.2 X-ray crystallography . 11 1.2.1 Diffraction by a crystal . 11 1.2.2 The reciprocal lattice . 12 1.2.3 Reciprocal lattice vectors and lattice planes . 13 1.2.4 The Bragg construction . 14 1.2.5 Structure factor . 15 1.2.6 Further geometry of diffraction . 17 2 Electrons in crystals 19 2.1 Summary of free-electron theory, etc. 19 2.2 Electrons in a periodic potential . 19 2.2.1 Bloch’s theorem . 19 2.2.2 Brillouin zones . 21 2.2.3 Schrodinger’s¨ equation in k-space . 22 2.2.4 Weak periodic potential: Nearly-free electrons . 23 2.2.5 Metals and insulators . 25 2.2.6 Band overlap in a nearly-free-electron divalent metal . 26 2.2.7 Tight-binding method . 29 2.3 Semiclassical dynamics of Bloch electrons . 32 2.3.1 Electron velocities . 33 2.3.2 Motion in an applied field . 33 2.3.3 Effective mass of an electron . 34 2.4 Free-electron bands and crystal structure . 35 2.4.1 Construction of the reciprocal lattice for FCC . 35 2.4.2 Group IV elements: Jones theory . 36 2.4.3 Binding energy of metals .
    [Show full text]
  • Energy Bands in Crystals
    Energy Bands in Crystals This chapter will apply quantum mechanics to a one dimensional, periodic lattice of potential wells which serves as an analogy to electrons interacting with the atoms of a crystal. We will show that as the number of wells becomes large, the allowed energy levels for the electron form nearly continuous energy bands separated by band gaps where no electron can be found. We thus have an interesting quantum system which exhibits many dual features of the quantum continuum and discrete spectrum. several tenths nm The energy band structure plays a crucial role in the theory of electron con- ductivity in the solid state and explains why materials can be classified as in- sulators, conductors and semiconductors. The energy band structure present in a semiconductor is a crucial ingredient in understanding how semiconductor devices work. Energy levels of “Molecules” By a “molecule” a quantum system consisting of a few periodic potential wells. We have already considered a two well molecular analogy in our discussions of the ammonia clock. 1 V -c sin(k(x-b)) V -c sin(k(x-b)) cosh βx sinhβ x V=0 V = 0 0 a b d 0 a b d k = 2 m E β= 2m(V-E) h h Recall for reasonably large hump potentials V , the two lowest lying states (a even ground state and odd first excited state) are very close in energy. As V →∞, both the ground and first excited state wave functions become completely isolated within a well with little wave function penetration into the classically forbidden central hump.
    [Show full text]
  • Inorganic Chemistry for Dummies® Published by John Wiley & Sons, Inc
    Inorganic Chemistry Inorganic Chemistry by Michael L. Matson and Alvin W. Orbaek Inorganic Chemistry For Dummies® Published by John Wiley & Sons, Inc. 111 River St. Hoboken, NJ 07030-5774 www.wiley.com Copyright © 2013 by John Wiley & Sons, Inc., Hoboken, New Jersey Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permis- sion of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley. com/go/permissions. Trademarks: Wiley, the Wiley logo, For Dummies, the Dummies Man logo, A Reference for the Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other countries, and may not be used without written permission. All other trade- marks are the property of their respective owners. John Wiley & Sons, Inc., is not associated with any product or vendor mentioned in this book.
    [Show full text]
  • Chapter 6 Free Electron Fermi Gas
    理学院 物理系 沈嵘 Chapter 6 Free Electron Fermi Gas 6.1 Electron Gas Model and its Ground State 6.2 Thermal Properties of Electron Gas 6.3 Free Electrons in Electric Fields 6.4 Hall Effect 6.5 Thermal Conductivity of Metals 6.6 Failures of the free electron gas model 1 6.1 Electron Gas Model and its Ground State 6.1 Electron Gas Model and its Ground State I. Basic Assumptions of Electron Gas Model Metal: valence electrons → conduction electrons (moving freely) ü The simplest metals are the alkali metals—lithium, sodium, 2 potassium, cesium, and rubidium. 6.1 Electron Gas Model and its Ground State density of electrons: Zr n = N m A A where Z is # of conduction electrons per atom, A is relative atomic mass, rm is the density of mass in the metal. The spherical volume of each electron is, 1 3 1 V 4 3 æ 3 ö = = p rs rs = ç ÷ n N 3 è 4p nø Free electron gas model: Suppose, except the confining potential near surfaces of metals, conduction electrons are completely free. The conduction electrons thus behave just like gas atoms in an ideal gas --- free electron gas. 3 6.1 Electron Gas Model and its Ground State Basic Properties： ü Ignore interactions of electron-ion type (free electron approx.) ü And electron-eletron type (independent electron approx). Total energy are of kinetic type, ignore potential energy contribution. ü The classical theory had several conspicuous successes 4 6.1 Electron Gas Model and its Ground State Long Mean Free Path: ü From many types of experiments it is clear that a conduction electron in a metal can move freely in a straight path over many atomic distances.
    [Show full text]
  • Distortion Correction and Momentum Representation of Angle-Resolved
    Distortion Correction and Momentum Representation of Angle-Resolved Photoemission Data Jonathan Adam Rosen 13.Sc., University of California at Santa Cruz, 2006 A THESIS SUBMIYI’ED 1N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Physics) THE UNIVERSITY OF BRETISH COLUMBIA (Vancouver) October 2008 © Jonathan Adam Rosen, 2008 Abstract Angle Resolve Photoemission Spectroscopy (ARPES) experiments provides a map of intensity as function of angles and electron kinetic energy to measure the many-body spectral function, but the raw data returned by standard apparatus is not ready for analysis. An image warping based distortion correction from slit array calibration is shown to provide the relevant information for construction of ARPES intensity as a function of electron momentum. A theory is developed to understand the calculation and uncertainty of the distortion corrected angle space data and the final momentum data. An experimental procedure for determination of the electron analyzer focal point is described and shown to be in good agreement with predictions. The electron analyzer at the Quantum Materials Laboratory at UBC is found to have a focal point at cryostat position 1.09mm within 1.00 mm, and the systematic error in the angle is found to be 0.2 degrees. The angular error is shown to be proportional to a functional form of systematic error in the final ARPES data that is highly momentum dependent. 11 Table .of Contents Abstract ii Table of Contents iii List of Tables v
    [Show full text]
  • Chapter 6 Free Electron Fermi
    Chapter 6 Free Electron Fermi Gas Free electron model: • The valence electrons of the constituent atoms become conduction electrons and move about freely through the volume of the metal. • The simplest metals are the alkali metals– lithium, sodium, potassium, Na, cesium, and rubidium. • The classical theory had several conspicuous successes, notably the derivation of the form of Ohm’s law and the relation between the electrical and thermal conductivity. • The classical theory fails to explain the heat capacity and the magnetic susceptibility of the conduction electrons. M = B • Why the electrons in a metal can move so freely without much deflections? (a) A conduction electron is not deflected by ion cores arranged on a periodic lattice, because matter waves propagate freely in a periodic structure. (b) A conduction electron is scattered only infrequently by other conduction electrons. Pauli exclusion principle. Free Electron Fermi Gas: a gas of free electrons subject to the Pauli Principle ELECTRON GAS MODEL IN METALS Valence electrons form the electron gas eZa -e(Za-Z) -eZ Figure 1.1 (a) Schematic picture of an isolated atom (not to scale). (b) In a metal the nucleus and ion core retain their configuration in the free atom, but the valence electrons leave the atom to form the electron gas. 3.66A 0.98A Na : simple metal In a sea of conduction of electrons Core ~ occupy about 15% in total volume of crystal Classical Theory (Drude Model) Drude Model, 1900AD, after Thompson’s discovery of electrons in 1897 Based on the concept of kinetic theory of neutral dilute ideal gas Apply to the dense electrons in metals by the free electron gas picture Classical Statistical Mechanics: Boltzmann Maxwell Distribution The number of electrons per unit volume with velocity in the range du about u 3/2 2 fB(u) = n (m/ 2pkBT) exp (-mu /2kBT) Success: Failure: (1) The Ohm’s Law , (1) Heat capacity Cv~ 3/2 NKB the electrical conductivity The observed heat capacity is only 0.01, J = E , = n e2 / m, too small.
    [Show full text]
  • Difference Between Angular Momentum and Pseudoangular
    Difference between angular momentum and pseudoangular momentum Simon Streib Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden (Dated: March 16, 2021) In condensed matter systems it is necessary to distinguish between the momentum of the con- stituents of the system and the pseudomomentum of quasiparticles. The same distinction is also valid for angular momentum and pseudoangular momentum. Based on Noether’s theorem, we demonstrate that the recently discussed orbital angular momenta of phonons and magnons are pseudoangular momenta. This conceptual difference is important for a proper understanding of the transfer of angular momentum in condensed matter systems, especially in spintronics applications. In 1915, Einstein, de Haas, and Barnett demonstrated experimentally that magnetism is fundamentally related to angular momentum. When changing the magnetiza- tion of a magnet, Einstein and de Haas observed that the magnet starts to rotate, implying a transfer of an- (a) gular momentum from the magnetization to the global rotation of the lattice [1], while Barnett observed the in- verse effect, magnetization by rotation [2]. A few years later in 1918, Emmy Noether showed that continuous (b) symmetries imply conservation laws [3], such as the con- servation of momentum and angular momentum, which links magnetism to the most fundamental symmetries of nature. Condensed matter systems support closely related con- Figure 1. (a) Invariance under rotations of the whole system servation laws: the conservation of the pseudomomentum implies conservation of angular momentum, while (b) invari- and pseudoangular momentum of quasiparticles, such as ance under rotations of fields with a fixed background implies magnons and phonons.
    [Show full text]
  • Arxiv:2006.09236V4 [Quant-Ph] 27 May 2021
    The Free Electron Gas in Cavity Quantum Electrodynamics Vasil Rokaj,1, ∗ Michael Ruggenthaler,1, y Florian G. Eich,1 and Angel Rubio1, 2, z 1Max Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, 22761 Hamburg, Germany 2Center for Computational Quantum Physics (CCQ), Flatiron Institute, 162 Fifth Avenue, New York NY 10010 (Dated: May 31, 2021) Cavity modification of material properties and phenomena is a novel research field largely mo- tivated by the advances in strong light-matter interactions. Despite this progress, exact solutions for extended systems strongly coupled to the photon field are not available, and both theory and experiments rely mainly on finite-system models. Therefore a paradigmatic example of an exactly solvable extended system in a cavity becomes highly desireable. To fill this gap we revisit Som- merfeld's theory of the free electron gas in cavity quantum electrodynamics (QED). We solve this system analytically in the long-wavelength limit for an arbitrary number of non-interacting elec- trons, and we demonstrate that the electron-photon ground state is a Fermi liquid which contains virtual photons. In contrast to models of finite systems, no ground state exists if the diamagentic A2 term is omitted. Further, by performing linear response we show that the cavity field induces plasmon-polariton excitations and modifies the optical and the DC conductivity of the electron gas. Our exact solution allows us to consider the thermodynamic limit for both electrons and photons by constructing an effective quantum field theory. The continuum of modes leads to a many-body renormalization of the electron mass, which modifies the fermionic quasiparticle excitations of the Fermi liquid and the Wigner-Seitz radius of the interacting electron gas.
    [Show full text]
  • Impact of the Interband Transitions in Gold and Silver on the Dynamics of Propagating and Localized Surface Plasmons
    nanomaterials Article Impact of the Interband Transitions in Gold and Silver on the Dynamics of Propagating and Localized Surface Plasmons Krystyna Kolwas * and Anastasiya Derkachova Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-22-116-33-19 Received: 19 June 2020; Accepted: 14 July 2020; Published: 19 July 2020 Abstract: Understanding and modeling of a surface-plasmon phenomenon on lossy metals interfaces based on simplified models of dielectric function lead to problems when confronted with reality. For a realistic description of lossy metals, such as gold and silver, in the optical range of the electromagnetic spectrum and in the adjacent spectral ranges it is necessary to account not only for ohmic losses but also for the radiative losses resulting from the frequency-dependent interband transitions. We give a detailed analysis of Surface Plasmon Polaritons (SPPs) and Localized Surface Plasmons (LPSs) supported by such realistic metal/dielectric interfaces based on the dispersion relations both for flat and spherical gold and silver interfaces in the extended frequency and nanoparticle size ranges. The study reveals the region of anomalous dispersion for a silver flat interface in the near UV spectral range and high-quality factors for larger nanoparticles. We show that the frequency-dependent interband transition accounted in the dielectric function in a way allowing reproducing well the experimentally measured indexes of refraction does exert the pronounced impact not only on the properties of SPP and LSP for gold interfaces but also, with the weaker but not negligible impact, on the corresponding silver interfaces in the optical ranges and the adjacent spectral ranges.
    [Show full text]
  • Density Functional Theory : What It Does and Doesn’T Do March 20, 2008
    Computational Nanoscience NSE C242 & Phys C203 Spring, 2008 Lecture 18: Density Functional Theory : What it Does and Doesn’t Do March 20, 2008 Elif Ertekin Jeffrey C. Grossman Elif Ertekin & Jeffrey C. Grossman, NSE C242 & Phys C203, Spring 2006, U.C. Berkeley Updates Homework Assignment: Hartree-Fock and DFT on molecules via GAMESS. Will be posted by Friday morning. Due: April 3rd Class Projects: Abstract/Proposal for your class project is due on April 10th. (one or two paragraphs) Bring your laptop for in-class simulation on Tuesday, April 1st. REMEMBER : There is no class next week (so don’t come!) Elif Ertekin & Jeffrey C. Grossman, NSE C242 & Phys C203, Spring 2006, U.C. Berkeley Errors, Lies, and Mistruths Unrestricted vs. Restricted Open Shell Hartree-Fock for O2 1πg 1πg 1πu 1πu 3σg 3σg 3σg 1πu 2σu 2σu 2σu 2σg 2σg 2σg 1σu 1σu 1σg 1σg 1σu 1σg ROHF UHF For open shell systems, UHF is better. The enhanced exchange interaction in the α channel electrons “pushes down” the eigenvalues relative to the β channels. Energy of unrestricted calculation is always lower than or equal to that of restricted. Elif Ertekin & Jeffrey C. Grossman, NSE C242 & Phys C203, Spring 2006, U.C. Berkeley Scaling After Foulkes, Mitas, Needs, and Rajagopal, pg 33. Rev Mod Phys 73 (2001). Method Ecorr Ecoh Scaling Total % error with # time for electrons C10 HF 0 50% N4 14 DFT-LDA N/A 15-25% N3 1 CC 75% 10-15% N7 1500 VMC 85% 2-10% N3+εN4 16 DMC 95% 1-4% N3+εN4 300 Elif Ertekin & Jeffrey C.
    [Show full text]
  • Schrödinger Correspondence Applied to Crystals Arxiv:1812.06577V1
    Schr¨odingerCorrespondence Applied to Crystals Eric J. Heller∗,y,z and Donghwan Kimy yDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 zDepartment of Physics, Harvard University, Cambridge, MA 02138 E-mail: [email protected] arXiv:1812.06577v1 [cond-mat.other] 17 Dec 2018 1 Abstract In 1926, E. Schr¨odingerpublished a paper solving his new time dependent wave equation for a displaced ground state in a harmonic oscillator (now called a coherent state). He showed that the parameters describing the mean position and mean mo- mentum of the wave packet obey the equations of motion of the classical oscillator while retaining its width. This was a qualitatively new kind of correspondence princi- ple, differing from those leading up to quantum mechanics. Schr¨odingersurely knew that this correspondence would extend to an N-dimensional harmonic oscillator. This Schr¨odingerCorrespondence Principle is an extremely intuitive and powerful way to approach many aspects of harmonic solids including anharmonic corrections. 1 Introduction Figure 1: Photocopy from Schr¨odinger's1926 paper In 1926 Schr¨odingermade the connection between the dynamics of a displaced quantum ground state Gaussian wave packet in a harmonic oscillator and classical motion in the same harmonic oscillator1 (see figure 1). The mean position of the Gaussian (its guiding position center) and the mean momentum (its guiding momentum center) follows classical harmonic oscillator equations of motion, while the width of the Gaussian remains stationary if it initially was a displaced (in position or momentum or both) ground state. This classic \coherent state" dynamics is now very well known.2 Specifically, for a harmonic oscillator 2 2 1 2 2 with Hamiltonian H = p =2m + 2 m! q , a Gaussian wave packet that beginning as A0 2 i i (q; 0) = exp i (q − q0) + p0(q − q0) + s0 (1) ~ ~ ~ becomes, under time evolution, At 2 i i (q; t) = exp i (q − qt) + pt(q − qt) + st : (2) ~ ~ ~ where pt = p0 cos(!t) − m!q0 sin(!t) qt = q0 cos(!t) + (p0=m!) sin(!t) (3) (i.e.
    [Show full text]