Solid State Physics NEARLY FREE ELECTRON MODEL
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Lecture Notes: BCS Theory of Superconductivity
Lecture Notes: BCS theory of superconductivity Prof. Rafael M. Fernandes Here we will discuss a new ground state of the interacting electron gas: the superconducting state. In this macroscopic quantum state, the electrons form coherent bound states called Cooper pairs, which dramatically change the macroscopic properties of the system, giving rise to perfect conductivity and perfect diamagnetism. We will mostly focus on conventional superconductors, where the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons. However, in the so- called unconventional superconductors - a topic of intense research in current solid state physics - the pairing can originate even from purely repulsive interactions. 1 Phenomenology Superconductivity was discovered by Kamerlingh-Onnes in 1911, when he was studying the transport properties of Hg (mercury) at low temperatures. He found that below the liquifying temperature of helium, at around 4:2 K, the resistivity of Hg would suddenly drop to zero. Although at the time there was not a well established model for the low-temperature behavior of transport in metals, the result was quite surprising, as the expectations were that the resistivity would either go to zero or diverge at T = 0, but not vanish at a finite temperature. In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a T 2 contribution from electron-electron scattering, and a T 5 contribution from phonon scattering. Thus, the vanishing of the resistivity at low temperatures is a clear indication of a new ground state. Another key property of the superconductor was discovered in 1933 by Meissner. -
Concepts of Condensed Matter Physics Spring 2015 Exercise #1
Concepts of condensed matter physics Spring 2015 Exercise #1 Due date: 21/04/2015 1. Adding long range hopping terms In class we have shown that at low energies (near half-filling) electrons in graphene have a doubly degenerate Dirac spectrum located at two points in the Brillouin zone. An important feature of this dispersion relation is the absence of an energy gap between the upper and lower bands. However, in our analysis we have restricted ourselves to the case of nearest neighbor hopping terms, and it is not clear if the above features survive the addition of more general terms. Write down the Bloch-Hamiltonian when next nearest neighbor and next-next nearest neighbor terms are included (with amplitudes 푡’ and 푡’’ respectively). Draw the spectrum for the case 푡 = 1, 푡′ = 0.4, 푡′′ = 0.2. Show that the Dirac cones survive the addition of higher order terms, and find the corresponding low-energy Hamiltonian. In the next question, you will study under which circumstances the Dirac cones remain stable. 2. The robustness of Dirac fermions in graphene –We know that the lattice structure of graphene has unique symmetries (e.g. 3-fold rotational symmetry of the honeycomb lattice). The question is: What protects the Dirac spectrum? Namely, what inherent symmetry in graphene we need to violate in order to destroy the massless Dirac spectrum of the electrons at low energies (i.e. open a band gap). In this question, consider only nearest neighbor terms. a. Stretching the graphene lattice - one way to reduce the symmetry of graphene is to stretch its lattice in one direction. -
Semiconductors Band Structure
Semiconductors Basic Properties Band Structure • Eg = energy gap • Silicon ~ 1.17 eV • Ge ~ 0.66 eV 1 Intrinsic Semiconductors • Pure Si, Ge are intrinsic semiconductors. • Some electrons elevated to conduction band by thermal energy. Fermi-Dirac Distribution • The probability that a particular energy state ε is filled is just the F-D distribution. • For intrinsic conductors at room temperature the chemical potential, µ, is approximately equal to the Fermi Energy, EF. • The Fermi Energy is in the middle of the band gap. 2 Conduction Electrons • If ε - EF >> kT then • If we measure ε from the top of the valence band and remember that EF lies in the middle of the band gap then Conduction Electrons • A full analysis taking into account the number of states per energy (density of states) gives an estimate for the fraction of electrons in the conduction band of 3 Electrons and Holes • When an electron in the valence band is excited into the conduction band it leaves behind a hole. Holes • The holes act like positive charge carriers in the valence band. Electric Field 4 Holes • In terms of energy level electrons tend to fall into lower energy states which means that the holes tends to rise to the top of the valence band. Photon Excitations • Photons can excite electrons into the conduction band as well as thermal fluctuations 5 Impurity Semiconductors • An impurity is introduced into a semiconductor (doping) to change its electronic properties. • n-type have impurities with one more valence electron than the semiconductor. • p-type have impurities with one fewer valence electron than the semiconductor. -
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. -
Fermi Surface of Copper Electrons That Do Not Interact
StringString theorytheory andand thethe mysteriousmysterious quantumquantum mattermatter ofof condensedcondensed mattermatter physics.physics. Jan Zaanen 1 String theory: what is it really good for? - Hadron (nuclear) physics: quark-gluon plasma in RIHC. - Quantum matter: quantum criticality in heavy fermion systems, high Tc superconductors, … Started in 2001, got on steam in 2007. Son Hartnoll Herzog Kovtun McGreevy Liu Schalm 2 Quantum critical matter Quark gluon plasma Iron High Tc Heavy fermions superconductors superconductors (?) Quantum critical Quantum critical 3 High-Tc Has Changed Landscape of Condensed Matter Physics High-resolution ARPES Magneto-optics Transport-Nernst effect Spin-polarized Neutron STM High Tc Superconductivity Inelastic X-Ray Scattering Angle-resolved MR/Heat Capacity ? Photoemission spectrum Hairy Black holes … 6 Holography and quantum matter But first: crash course in holography “Planckian dissipation”: quantum critical matter at high temperature, perfect fluids and the linear resistivity (Son, Policastro, …, Sachdev). Reissner Nordstrom black hole: “critical Fermi-liquids”, like high Tc’s normal state (Hong Liu, John McGreevy). Dirac hair/electron star: Fermi-liquids emerging from a non Fermi liquid (critical) ultraviolet, like overdoped high Tc (Schalm, Cubrovic, Hartnoll). Scalar hair: holographic superconductivity, a new mechanism for superconductivity at a high temperature (Hartnoll, Herzog,Horowitz) . 7 General relativity “=“ quantum field theory Gravity Quantum fields Maldacena 1997 = 8 Anti de Sitter-conformal -
Periodic Trends Lab CHM120 1The Periodic Table Is One of the Useful
Periodic Trends Lab CHM120 1The Periodic Table is one of the useful tools in chemistry. The table was developed around 1869 by Dimitri Mendeleev in Russia and Lothar Meyer in Germany. Both used the chemical and physical properties of the elements and their tables were very similar. In vertical groups of elements known as families we find elements that have the same number of valence electrons such as the Alkali Metals, the Alkaline Earth Metals, the Noble Gases, and the Halogens. 2Metals conduct electricity extremely well. Many solids, however, conduct electricity somewhat, but nowhere near as well as metals, which is why such materials are called semiconductors. Two examples of semiconductors are silicon and germanium, which lie immediately below carbon in the periodic table. Like carbon, each of these elements has four valence electrons, just the right number to satisfy the octet rule by forming single covalent bonds with four neighbors. Hence, silicon and germanium, as well as the gray form of tin, crystallize with the same infinite network of covalent bonds as diamond. 3The band gap is an intrinsic property of all solids. The following image should serve as good springboard into the discussion of band gaps. This is an atomic view of the bonding inside a solid (in this image, a metal). As we can see, each of the atoms has its own given number of energy levels, or the rings around the nuclei of each of the atoms. These energy levels are positions that electrons can occupy in an atom. In any solid, there are a vast number of atoms, and hence, a vast number of energy levels. -
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. -
Chapter 13 Ideal Fermi
Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k T µ,βµ 1, B � � which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ( ) that we used for the fermionic statistical quantities in the previous chapter. − 13.1 Equation of state Consider a gas ofN non-interacting fermions, e.g., electrons, whose one-particle wave- functionsϕ r(�r) are plane-waves. In this case, a complete set of quantum numbersr is given, for instance, by the three cartesian components of the wave vector �k and thez spin projectionm s of an electron: r (k , k , k , m ). ≡ x y z s Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type ˆ 3 H= �k ck† ck + d r V(r)c r†cr , �k � where thefirst and the second terms are respectively the kinetic and th potential energy. The summation over the statesr (whenever it has to be performed) can then be reduced to the summation over states with different wavevectork(p=¯hk): ... (2s + 1) ..., ⇒ r � �k where the summation over the spin quantum numberm s = s, s+1, . , s has been taken into account by the prefactor (2s + 1). − − 159 160 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We as- sume that the electrons are in a vol- ume defined by a cube with sidesL x, Ly,L z and volumeV=L xLyLz. -
Extent of Fermi-Surface Reconstruction in the High-Temperature Superconductor Hgba2cuo4+Δ
Extent of Fermi-surface reconstruction in the high-temperature superconductor HgBa2CuO4+δ Mun K. Chana,1, Ross D. McDonalda, Brad J. Ramshawd, Jon B. Bettsa, Arkady Shekhterb, Eric D. Bauerc, and Neil Harrisona aPulsed Field Facility, National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA; bNational High Magnetic Field Laboratory, Florida State University, Tallahassee, Forida 32310, USA; cLos Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA; dLaboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 148.3, USA This manuscript was compiled on June 17, 2020 High magnetic fields have revealed a surprisingly small Fermi- change at TH ≈ 20 K. These results attest to the high-quality surface in underdoped cuprates, possibly resulting from Fermi- of the present crystals and confirm prior results indicating surface reconstruction due to an order parameter that breaks trans- Fermi-surface reconstruction in Hg1201 at similar doping lev- lational symmetry of the crystal lattice. A crucial issue concerns els (6,9, 10). the doping extent of this state and its relationship to the principal We have also found a peak in the planar resistivity ρ(T ) pseudogap and superconducting phases. We employ pulsed mag- at high fields, Figs.1C&D. ρ(T ) upturns at a temperature netic field measurements on the cuprate HgBa2CuO4+δ to identify coincident with the steep downturn in Rxy(T ), as shown in signatures of Fermi surface reconstruction from a sign change of the Fig.1D and SI Appendix Fig. S7. The common character- Hall effect and a peak in the temperature-dependent planar resistiv- istic temperatures suggest the features in ρ(T ) are related ity. -
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. -
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. -
Bulk Fermi Surface Coexistence with Dirac Surface State in Bi2se3: a Comparison of Photoemission and Shubnikov–De Haas Measurements
PHYSICAL REVIEW B 81, 205407 ͑2010͒ Bulk Fermi surface coexistence with Dirac surface state in Bi2Se3: A comparison of photoemission and Shubnikov–de Haas measurements James G. Analytis,1,2 Jiun-Haw Chu,1,2 Yulin Chen,1,2 Felipe Corredor,1,2 Ross D. McDonald,3 Z. X. Shen,1,2 and Ian R. Fisher1,2 1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA 2Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University, Stanford, California 94305, USA 3Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA ͑Received 2 February 2010; published 5 May 2010͒ Shubnikov-de Haas ͑SdH͒ oscillations and angle-resolved photoemission spectroscopy ͑ARPES͒ are used to probe the Fermi surface of single crystals of Bi2Se3. We find that SdH and ARPES probes quantitatively agree on measurements of the effective mass and bulk band dispersion. In high carrier density samples, the two probes also agree in the exact position of the Fermi level EF, but for lower carrier density samples discrepan- cies emerge in the position of EF. In particular, SdH reveals a bulk three-dimensional Fermi surface for samples with carrier densities as low as 1017 cm−3. We suggest a simple mechanism to explain these differ- ences and discuss consequences for existing and future transport studies of topological insulators. DOI: 10.1103/PhysRevB.81.205407 PACS number͑s͒: 72.20.My, 03.65.Vf, 71.70.Di, 73.25.ϩi Recently, a new state of matter, known as a topological high level of consistency across all samples measured from insulator, has been predicted to exist in a number of materi- the same batch.