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Overview
Author J. S. Dugdale establishes a picture of the current-carrying state of a solid and the associated electron energy states before exploring how departures from crystal perfection scatter electrons. Static imperfections and lattice vibrations receive detailed explanations before the text advances to complex scattering. Emphasis on the behavior of real materials provides readers with a physical understanding of transport properties of transition metals, resistance, and thermoelectric anomalies in dilute magnetic alloys and transport in concentrated alloys and compounds.
Product Details
| ISBN-13: | 9780486814650 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 06/22/2016 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 304 |
| File size: | 26 MB |
| Note: | This product may take a few minutes to download. |
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Read an Excerpt
The Electrical Properties of Metals and Alloys
By J. S. Dugdale
Dover Publications, Inc.
Copyright © 1977 J. S. DugdaleAll rights reserved.
ISBN: 978-0-486-81465-0
CHAPTER 1
Some Bulk Transport Properties
1.1 Introduction
In this book we shall be concerned with understanding some of the electrical properties of comparatively simple solids. We shall consider primarily three groups of crystalline metals or alloys.
(1) simple, non-transition metals, such as potassium, sodium; copper, silver, gold;
(2) transition metals, for example palladium, platinum, nickel;
(3) disordered alloys such as the silver-gold and silver-palladium series.
In order to appreciate more clearly the nature of metals and their properties I shall also contrast these with the corresponding properties of semiconductors but our primary interest is in metals and alloys.
1.2 Electrical resistivity
The first property we will consider is electrical resistivity. Its measurement is, on the whole, straightforward: you measure the resistance R of a specimen of known length L and uniform known cross section A. The resistivity ρ, is then given by
ρ = RA/L (1.1)
In materials with cubic symmetry such as most of those discussed here, ρ and its reciprocal, the conductivity σ, are scalar quantities. (When a magnetic field is applied this is no longer true; ρ and σ become tensor quantities.)
Values for the resistivity of a selection of solids at room temperature are listed in Table 1.1. In the case of the metals and semiconductors these figures refer to highly purified materials.
From the table, you see that Ge and Si the semiconductors have resistivities that are measured in hundreds of ohm centimetres. On the other hand, the resistivities of the metals are typically so small that the ohm centimetre is too big a unit for convenience; the microhm centimetre, a million times smaller, is more appropriate for these materials. For comparison, some insulating materials are included. Their resistivities are as much as 1018 higher than that of the semiconductors and 1025 higher than the metals.
Now what happens to these resistivities at a low temperature, say IK? Some values are listed in Table 1.2 where I have also included values for an alloy, 5 atomic per cent silver in palladium. Whereas the resistivities of the metals and the alloy all go down in value, those of germanium and silicon go up. This is characteristic of metallic behaviour on the one hand and of semiconducting behaviour on the other. This and the difference in magnitudes are two features we shall want to understand.
In its temperature dependence, the alloy shows characteristic behaviour. You will see from the fourth column of Table 1.2, where the ratio of the resistivities at room temperature to those at 1K is listed, that whereas the pure metals change in resistivity by many orders of magnitude the alloy changes by a factor of only two. This is another feature we shall try to understand.
To emphasize and summarize the differences between the resistivities of the various materials, I show in Fig. 1.1(a) on a logarithmic scale how the resistivity of some of them depends on temperature over a wide temperature range. The detailed behaviour of the germanium sample is very sensitive indeed to any impurities it may contain (this property, too, distinguishes the semiconductor from the metal). Nonetheless the main features are clear. The resistivity of the pure metals falls as the temperature falls; the resistivity of the semiconductor rises rapidly as the temperature falls; the resistivity of the alloy changes rather little. Fig. 1.1(b) shows on a linear scale how the resistivity of potassium (the prototype of simple metals) varies with temperature; it is directly proportional to the temperature at high temperatures but varies very much faster (as T7 or more) at the lower temperatures.
We shall now consider briefly two other electrical transport properties which will concern us later: the Hall coefficient and the thermoelectric power.
1.3 The Hall coefficient
To measure the Hall coefficient, you send a known current I through the conductor. At right angles to the direction of the current, a magnetic field H is applied and you measure the e.m.f., ΔV, developed at right angles to both I and H in the specimen. At low fields, ΔV is found to be proportional to both H and I and the constant of proportionality is closely related to the Hall coefficient, RH. In the definition of RH, however, the current density, j, is used rather than the current itself and the transverse electric field, EH, is used instead of ΔV. RH is defined thus:
EH = RHjH (1.2)
This makes RH independent of the size of the specimen as we require. If the breadth of the specimen across which ΔV is measured is b then EH = ΔV/b. If the thickness of the (rectangular) specimen at right angles to to b is d, the cross sectional area is bd so that
j = I/bd
Consequently we have finally
ΔV/b = RHI/bd H (1.3)
or
RH = ΔVd/IH (1.4)
This last relationship shows what quantities must be measured to determine RH. The only dimension required is d, the thickness of the specimen in the direction of H.
The Hall coefficient, as we shall see, can give valuable information about the number of current carriers in the solid. On the other hand it does not have, at least in the simpler metals, a very striking temperature dependence. To a first approximation, at least, RH in the alkali metals is independent of temperature although it does show some variation at the lowest temperatures, as shown in Fig. 1.2. Likewise in the other metallic samples, there is some variation with temperature although nothing very remarkable.
By contrast the semiconductors behave quite differently; their Hall coefficients decrease exponentially with increasing temperature as illustrated in Fig. 1.3.
1.4 Thermoelectric power
The simplest example of a thermoelectric circuit is shown in Fig. 1.4. It consists of two conductors A and B whose junctions are at different temperatures T and T + ΔT.
Under these conditions a potential difference ΔV appears across the terminals 1 and 2; this can be measured by means of, say, a potentiometer or any device that effectively draws no current from the circuit. The thermoelectric power of the circuit is then defined as
SAB = ΔV AB/DELTAT (1.5)
in the limit as ΔT becomes very small. The sign adopted for S is such that the conductor A is positive to B if the current tends to flow from A to B at the cold junction. This quantity SAB is characteristic of the two materials A and B and depends on the temperature.
Moreover we can in fact separate SAB in the following way:
SAB = SA - SB (1.6)
Here SA is characteristic of conductor A alone and SB characteristic of conductor B alone; SA and SB are referred to as the 'absolute' thermoelectric powers of A and B. Indeed as we shall see below S the absolute thermopower (a convenient abbreviation) behaves as if it were an entropy associated with the current carriers in a particular conductor.
The thermoelectric manifestation just described is called, after its discoverer, the Seebeck effect. In addition to this effect, there are two other related effects, also named after their discoverers, Peltier and Thomson. The Peltier coefficient ΠAB defined as the heat reversibly absorbed or given out when unit positive charge passes across the junction from conductor A to conductor B. It too can be split up so that ΠAB = ΠA - ΠB; as before ΠA is characteristic of conductor A alone and ΠB of conductor B alone. The sign convention is such that if heat is given out in this process, A is positive with respect to B; if heat is absorbed A is negative with respect to B. Because the Peltier effect is isothermal and requires measurement at only one junction, it is perhaps the simplest thermoelectric coefficient to think about; for this reason, we shall subsequently make considerable use of the Peltier coefficient in our attempts to understand the nature of thermoelectricity.
Finally the Thomson coefficient, μA or μB, measures the heat absorbed (or given out) reversibly when unit charge passes through unit temperature difference in the conductor concerned. It is sometimes referred to as 'the specific heat of electricity'. μ is defined as positive if heat is absorbed when a positive charge passes through a positive temperature interval.
The thermoelectric circuit, as Thomson pointed out, is analogous to a two-phase circuit, for example, of liquid and vapour as illustrated in Fig. 1.5. In this circuit, unit mass of material goes round instead of unit charge. Then SA and SB of the thermoelectric circuit are analogous to Sv and Si, the specific entropies of vapour and liquid; ΠAB is the analogue of the latent heat of vaporization, L and μA and μB are analogues of the saturated specific heats of vapour and liquid, sv and st. The Thomson thermodynamic relations between the thermoelectric quantities are then the analogues of the Clausius–Clapeyron and Clapeyron equation for ordinary two-phase equilibrium. This leads to Equ. 1.5 and to the relationship:
[MATHEMATICAL EXPRESSION OMITTED] (1.7)
We also get:
[MATHEMATICAL EXPRESSION OMITTED] (1.8)
and
[MATHEMATICAL EXPRESSION OMITTED]
in which, in principle, the integration extends from the absolute zero up to the temperature of interest.
This provides a means of measuring the absolute thermopower of a conductor by suitable calorimetric techniques. However, once the absolute thermopower of one material has been so determined, that of any other material can then be found by e.m.f. measurements on a thermocouple consisting of the reference material and the one under study (see Equ. 1.5).
This has been a rather long discussion of thermoelectric definitions. What is important is to realize that one quantity, e.g. the absolute thermopower, S, characterizes a particular material (at a given temperature, etc.) and if you know S you know all the thermoelectric properties of that material. Because the Peltier coefficient Π is so simply related to S(Π = TS) either Π or S will serve equally well.
We are now in a position to ask: how does the absolute thermopower of a metal, alloy or semiconductor vary with temperature? The answer, for some chosen materials, is given in Fig. 1.6.
Here, then, are some of the main properties that we wish to understand. To do this, we must now look at the structure of these materials at the atomic level and where possible relate the properties of the electrons and ions on the atomic scale to the bulk properties we have just been looking at.
CHAPTER 2Simple Picture of Properties
2.1 Introduction
In this chapter, we take a preliminary look at some electrical properties of a solid from an atomic point of view. The aim is to give here a simple overall picture and to come back to the detailed arguments in later chapters.
First then, we picture our solid (let us take potassium as an example) as a perfect crystal with all the atoms in a periodic arrangement in three dimensions. In potassium the arrangement is such as to form a body-centered cubic array of atoms as shown in Fig. 2.1. The potassium atom has just one valence electron outside the lower lying closed electron shells; in the metal these valence electrons are detached from their parent atoms and form a 'gas' of conduction electrons common to the metal as a whole; the lattice thus consists of an array of positively charged ions, each atom having lost its valence electron.
The valence electrons can move about almost as free particles through the lattice of ions. These electrons are responsible for the electrical properties of the metal, giving rise to its high electrical and thermal conductivity and its characteristic optical properties.
This electron gas in potassium, as in other metals, is very dense since in the atomic volume of the solid (45·5 cm3 for potassium) there exist 6 × 1023 electrons (i.e. one per atom). There are therefore 1·3 × 1022 electrons per cm3 in this metal. Electrons are, of course, subject to the Pauli exclusion principle and so obey Fermi–Dirac statistics. Because the electron gas is so dense, it forms, at all normal temperatures, a highly degenerate Fermi gas; this fact, as we shall see, has very profound consequences for the electrical properties of metals.
2.2 Electrical conductivity
What happens to the electron gas when we apply an electric field, [??], to the metal? If the electrons were free classical particles of charge e we know what would happen. They would begin to accelerate in the direction of the electrostatic force (opposite to that of [??] since the electrons carry a negative charge). In classical terms their acceleration, a, would be given by:
ma = e[??] (2.1)
where m is the mass of the electron. It is the change in velocity that gives rise to the electric current so we focus our attention on this change. If the electron velocity before the field was applied was v0 and if after its application for a time t the velocity was v, the change in velocity is δv = v - v0. So
[MATHEMATICAL EXPRESSION OMITTED] (2.2)
where δv is just the increment of velocity (in the direction of the force) brought about by the application of the field. So in addition to the random velocities of the electrons, a directed drift velocity δv builds up under the influence of [??]. If n is the number of electrons per unit volume, a current of density neδv is established.
Why then do the electrons not continue to accelerate under the influence of the field with a consequent build-up of current?
The answer is that in addition to the field acting on the electrons, there is another mechanism quite independent of [??] that can change the electron velocities. This mechanism is usually called 'scattering' and it can be thought of as arising from collisions of the electrons with each other and with various obstacles that limit their free motion. These 'obstacles' can be impurities or lattice imperfections in the solid or they may be the thermal vibrations of the lattice. Just as in a gas we can think of a mean free path which is determined by the collisions of gas molecules with each other, so here we can think of the electrons having a mean free path determined by these different kinds of scattering processes.
This notion of 'scattering' will be very important to our understanding of the electrical properties of solids. It is the mechanism which, when no disturbances (such as external fields) are present, maintains thermodynamic equilibrium both among the electrons themselves and between them and the lattice. It is also the mechanism which tries to restore equilibrium when some agency such as an external field upsets it. We shall discuss this dynamical, self-balancing aspect of scattering later on. Although it is the mechanism necessary for thermodynamic equilibrium, conventional thermodynamics and statistical mechanics can ignore its details since these disciplines deal only with the equilibrium condition; to them it doesn't matter how equilibrium is maintained. But once you have to deal, as here, with a system not in equilibrium, the scattering process becomes of paramount importance.
I think it can be seen that, if the scattering processes are to be able to prevent the electric field from causing a runaway process, then the rate at which the electrons are scattered back towards their equilibrium energies or speeds must increase in some measure in proportion to the degree of departure from equilibrium. The simplest form such a process could take would be:
[MATHEMATICAL EXPRESSION OMITTED] (2.3)
In this, the rate at which the velocity returns to its equilibrium value, v0, is proportional to (v - v0) the amount that v differs from its equilibrium value. Under these circumstances, any excess velocity imparted by the field would decay exponentially; τ would be the characteristic time involved in this exponential decay, τ is called the electron 'relaxation time'.
(Continues...)
Excerpted from The Electrical Properties of Metals and Alloys by J. S. Dugdale. Copyright © 1977 J. S. Dugdale. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Suitable for advanced undergraduate and graduate students of physics, this classic volume by a prominent authority in the field provides an account of some simple properties of metals and alloys associated with electron transport. Introductory chapters examine the bulk properties of electrical resistivity, the Hall coefficient, and thermoelectric power.
Author J. S. Dugdale establishes a picture of the current-carrying state of a solid and the associated electron energy states before exploring how departures from crystal perfection scatter electrons. Static imperfections and lattice vibrations receive detailed explanations before the text advances to complex scattering. Emphasis on the behavior of real materials provides readers with a physical understanding of transport properties of transition metals, resistance, and thermoelectric anomalies in dilute magnetic alloys and transport in concentrated alloys and compounds.
Dover (2015) republication of the edition published by Edward Arnold (Publishers) Ltd., London, 1977.
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