Annett_Chapter 3.pdf

Superconductivity 3.1 Introduction 3.1 Introduction 47 This chapter describes some of the most fundamental experimental facts about 3.2 Conduction in metals 4...

Superconductivity 3.1 Introduction 3.1 Introduction 47 This chapter describes some of the most fundamental experimental facts about 3.2 Conduction in metals 47 superconductors, together with the simplest theoretical model: the London 3.3 Superconducting equation. We shall see how this equation leads directly to the expulsion materials 49 of magnetic fields from superconductors, the Meissner-Ochsenfeld effect, 3.4 Zero-resistivity 51 which is usually considered to be the fundamental property which defines 3.5 The Meissner–Ochsenfeld superconductivity. effect 54 The chapter starts with a brief review of the Drude theory of conduction in 3.6 Perfect diamagnetism 55 normal metals. We shall also show how it is possible to use the Drude theory 3.7 Type I and type II to make the London equation plausible. We shall also explore some of the superconductivity 57 consequences of the London equation, in particular the existence of vortices in 3.8 The London equation 58 superconductors and the differences between type I and II superconductors. 3.9 The London vortex 62 Further reading 64 Exercises 64 3.2 Conduction in metals The idea that metals are good electrical conductors because the electrons move freely between the atoms was first developed by Drude in 1900, only 5 years after the original discovery of the electron. Although Drude's original model did not include quantum mechanics, his formula for the conductivity of metals remains correct even in the modern quantum theory of metals. To briefly recap the key ideas in the theory of metals, we recall that the wave functions of the electrons in crystalline solids obey Bloch's theorem,' See, for example, the text Band theory and ik.r (3.1) electronic properties of solids by J. Singleton Vfnk(r) = unk(r)e , (2001), or other textbooks on Solid State where unk (r) is a function which is periodic, hk is the crystal momentum, and Physics, such as Kittel (1996), or Ashcroft k takes values in the first Brillouin zone of the reciprocal lattice. The energies and Mermin (1976). of these Bloch wave states give the energy bands, Enk, where n counts the different electron bands. Electrons are fermions, and so at temperature T a state with energy E is occupied according to the Fermi-Dirac distribution 1 f(E) = (3.2) efi(6 — P.) +1. The chemical potential, kt, is determined by the requirement that the total density of electrons per unit volume is 2 1 — d3k (3.3) V (27) e fi(ea—kt) +1 48 Superconductivity where the factor of 2 is because of the two spin states of the s = 1/2 electron. Here the integral over k includes all of the first Brillouin zone of the reciprocal lattice and, in principle, the sum over the band index n counts all of the occupied electron bands. In all of the metals that we are interested in here the temperature is such that this Fermi gas is in a highly degenerate state, in which kBT