Properties of the normal state

The normal state is the state of a metallic material that either does not superconduct or is a superconductor at a temperature below its transition temperature Tc. This chapter emphasizes their electrical properties such as the electrical conductivity and its reciprocal the resistivity. Their temperature and frequency dependencies are given. Other properties and phenomena are explained such as the chemical potential, the electron–phonon interaction, thermal conductivity, energy gaps and effective masses, electronic and phonon specific heats, electromagnetic fields, magnetic susceptibilities, and the Hall effect. The Fermi–Dirac distribution function is emphasized. The physical properties of 14 metallic elements are tabulated.

Keywords

Electrical conduction; magnetic properties; relaxation time; Fermi–Dirac statistics; electron collisions; influence of impurities; temperature dependencies

I Introduction

This text is concerned with superconductivity, a phenomenon characterized by certain electrical, magnetic, and other properties, many of which will be introduced in the following chapter. A material becomes superconducting below a characteristic temperature, called the superconducting transition temperature Tc, which varies from very small values (millidegrees or microdegrees) to values above 100 K. The material is called normal above Tc, which merely means that it is not superconducting. Elements and compounds that become superconductors are, generally, conductors—but not good conductors—in their normal state. The good conductors, such as copper, silver, and gold, do not superconduct.

It is helpful to survey some properties of normal conductors before discussing the superconductors, so that we can review some background material and define some of the terms that will be used throughout the text. Many of the normal state properties that will be discussed here are modified in the superconducting state. Much of the material in this introductory chapter will be referred to later in the text.

II Conducting electron transport

The electrical conductivity of a metal may be described most simply in terms of the constituent atoms of the metal. In this representation, the atoms lose their valence electrons, causing a background lattice of positive ions, called cations, to form, and the now delocalized conduction electrons move between these ions. The number density n (electrons/cm3) of conduction electrons in a metallic element of density ρm (g/cm3), atomic mass number A (g/mole), and valence Z is given by

=NAZρmA, (1.1)

where NA is Avogadro’s number. The typical values listed in Table 1.1 are a thousand times greater than those of a gas at room temperature and atmospheric pressure.

Table 1.1

Characteristics of selected metallic elements

Notation: a, lattice constant; ne, conduction electron density; rs=(3/4πne)1/3; ρ, resistivity; τ, Drude relaxation time; Kth, thermal conductivity; L=ρKth/T is the Lorentz number; γ, electronic specific heat parameter; m*, effective mass; RH, Hall constant; ΘD, Debye temperature; ωp, plasma frequency in radians per femtosecond (10−15 s); IP, first ionization potential; WF, work function; EF, Fermi energy; TF, Fermi temperature in kilokelvins; kF, Fermi wavenumber in mega reciprocal centimeters; and υF, Fermi velocity in centimeters per microsecond.

The simplest approximation that we can adopt as a way of explaining conductivity is the Drude model. In this model, it is assumed that the conduction electrons

• do not interact with the cations (“free electron approximation”) except when one of them collides elastically with a cation, which happens, on average, 1/τ times per second, with the result that the velocity υ of the electron abruptly and randomly changes its direction (“relaxation time approximation”);

• maintain thermal equilibrium through collisions, in accordance with Maxwell–Boltzmann statistics (“classical statistics approximation”);

• do not interact with each other (“independent-electron approximation”).

This model predicts many of the general features of electrical conduction phenomena, as we shall see later in the chapter, but it fails to account for many others, such as tunneling, band gaps, and the Bloch T5 law. More satisfactory explanations of electron transport relax or discard one or more of these approximations.

Ordinarily, we abandon the free electron approximation by having the electrons move in a periodic potential arising from the background lattice of positive ions. Figure 1.1 shows an example of a simple potential that is negative near the positive ions and zero between them. An electron moving through the lattice interacts with the surrounding positive ions, which are oscillating about their equilibrium positions, and the charge distortions resulting from this interaction propagate along the lattice, causing distortions in the periodic potential. These distortions can influence the motion of yet another electron some distance away that is also interacting with the oscillating...