Electrical conductivity

5.7.1 Electrical conductivity

One of the most important electronic properties of metals is the electrical conductivity, κ, and the reciprocal of the conductivity (known as the resistivity, ρ) is defined by the relation R = ρl/A, where R is the resistance of the specimen, l is the length and A is the cross-sectional area.

A characteristic feature of a metal is its high electrical conductivity, which arises from the ease with which the electrons can migrate through the lattice. The high thermal conduction of metals also has a similar explanation, and the Wiedmann–Franz law shows that the ratio of the electrical and thermal conductivities is nearly the same for all metals at the same temperature.

Since conductivity arises from the motion of conduction electrons through the lattice, resistance must be caused by the scattering of electron waves by any kind of irregularity in the lattice arrangement. Irregularities can arise from any one of several sources, such as temperature, alloying, deformation or nuclear irradiation, since all will disturb, to some extent, the periodicity of the lattice. The effect of temperature is particularly important and, as shown in Figure 5.18, the resistance increases linearly with temperature above about 100 K up to the melting point. On melting, the resistance increases markedly because of the exceptional disorder of the liquid state. However, for some metals such as bismuth, the resistance actually decreases, owing to the fact that the special zone structure which makes bismuth a poor conductor in the solid state is destroyed on melting.

In most metals the resistance approaches zero at absolute zero, but in some (e.g. lead, tin and mercury) the resistance suddenly drops to zero at some finite critical temperature above 0 K. Such metals are called superconductors. The critical temperature is different for each metal but is always close to absolute zero; the highest critical temperature known for an element is 8 K for niobium. Superconductivity is now observed at much higher temperatures in some intermetallic compounds and in some ceramic oxides (see Section 5.7.5).

Physical properties 261

Resistivity

r ∝T

r ∝T 5 0 Temperature

(K)

Figure 5.18 Variation of resistivity with temperature.

An explanation of electrical and magnetic properties requires a more detailed consideration of electronic structure than that briefly outlined in Chapter 1. There the concept of band structure was introduced and the electron can be thought of as moving continuously through the structure with an energy depending on the energy level of the band it occupies. The wave-like properties of the electron were also mentioned. For the electrons the regular array of atoms on the metallic lattice can behave as a three-dimensional diffraction grating, since the atoms are positively charged and interact with moving electrons. At certain wavelengths, governed by the spacing of the atoms on the metallic lattice, the electrons will experience strong diffraction effects, the results of which are that electrons having energies corresponding to such wavelengths will be unable to move freely through the structure. As a consequence, in the bands of electrons, certain energy levels cannot be occupied and therefore there will be energy gaps in the otherwise effectively continuous energy spectrum within a band.

The interaction of moving electrons with the metal ions distributed on a lattice depends on the wavelength of the electrons and the spacing of the ions in the direction of movement of the electrons. Since the ionic spacing will depend on the direction in the lattice, the wavelength of the electrons suffering diffraction by the ions will depend on their direction. The kinetic energy of a moving electron is a function of the wavelength according to the relationship:

E 2 / 2mλ =h 2 . (5.13) Since we are concerned with electron energies, it is more convenient to discuss interaction effects in

terms of the reciprocal of the wavelength. This quantity is called the wave number and is denoted by k. In describing electron–lattice interactions it is usual to make use of a vector diagram in which the direction of the vector is the direction of motion of the moving electron and its magnitude is the wave number of the electron. The vectors representing electrons having energies which, because of diffraction effects, cannot penetrate the lattice, trace out a three-dimensional surface known as a Brillouin zone. Figure 5.19a shows such a zone for a face-centered cubic lattice. It is made up of plane faces which are, in fact, parallel to the most widely spaced planes in the lattice, i.e. in this case the {1 1 1} and {2 0 0} planes. This is a general feature of Brillouin zones in all lattices.

262 Physical Metallurgy and Advanced Materials

Allowed (200) face

Allowed Energy

zone

energy band

1st zone

(b) Figure 5.19 Schematic representation of a Brillouin zone in a metal.

(a)

For a given direction in the lattice, it is possible to consider the form of the electron energies as a function of wave number. The relationship between the two quantities as given from equation (5.13) is

E =h 2 k 2 / 2m, (5.14) which leads to the parabolic relationship shown as a broken line in Figure 5.19b. Because of the

existence of a Brillouin zone at a certain value of k, depending on the lattice direction, there exists

a range of energy values which the electrons cannot assume. This produces a distortion in the form of the E −k curve in the neighborhood of the critical value of k and leads to the existence of a series of energy gaps, which cannot be occupied by electrons. The E–k curve showing this effect is given as a continuous line in Figure 5.19b.

The existence of this distortion in the E–k curve, due to a Brillouin zone, is reflected in the density of states–energy curve for the free electrons. As previously stated, the density of states–energy curve is parabolic in shape, but it departs from this form at energies for which Brillouin zone interactions occur. The result of such interactions is shown in Figure 5.20a, in which the broken line represents the N (E)–E curve for free electrons in the absence of zone effects and the full line is the curve where

a zone exists. The total number of electrons needed to fill the zone of electrons delineated by the full line in Figure 5.20a is 2N , where N is the total number of atoms in the metal. Thus, a Brillouin zone would be filled if the metal atoms each contributed two electrons to the band. If the metal atoms contribute more than two per atom, the excess electrons must be accommodated in the second or higher zones.

In Figure 5.20a the two zones are separated by an energy gap, but in real metals this is not necessarily the case, and two zones can overlap in energy in the N (E)–E curves so that no such energy gaps appear. This overlap arises from the fact that the energy of the forbidden region varies with direction in the

lattice, and often the energy level at the top of the first zone has a higher value in one direction than the lowest energy level at the bottom of the next zone in some other direction. The energy gap in the N (E)–E curves, which represent the summation of electronic levels in all directions, is then closed (Figure 5.20b).

For electrical conduction to occur, it is necessary that the electrons at the top of a band should

be able to increase their energy when an electric field is applied to materials so that a net flow of

Physical properties 263

Density of states

Energy

E Density of states

Figure 5.20 Schematic representation of Brillouin zones. electrons in the direction of the applied potential, which manifests itself as an electric current, can

take place. If an energy gap between two zones of the type shown in Figure 5.20a occurs, and if the lower zone is just filled with electrons, then it is impossible for any electrons to increase their energy by jumping into vacant levels under the influence of an applied electric field, unless the field strength is sufficiently great to supply the electrons at the top of the filled band with enough energy to jump the energy gap. Thus, metallic conduction is due to the fact that in metals the number of electrons per atom is insufficient to fill the band up to the point where an energy gap occurs. In copper, for example, the 4s valency electrons fill only one-half of the outer s-band. In other metals (e.g. Mg) the valency band overlaps a higher energy band and the electrons near the Fermi level are thus free to move into the empty states of a higher band. When the valency band is completely filled and the next higher band, separated by an energy gap, is completely empty, the material is either an insulator or a semiconductor. If the gap is several electron-volts wide, such as in diamond, where it is 7 eV, extremely high electric fields would be necessary to raise electrons to the higher band and the material is an insulator. If the gap is small enough, such as 1–2 eV as in silicon, then thermal energy may be sufficient to excite some electrons into the higher band and also create vacancies in the valency band; such a material is a semiconductor. In general, the lowest energy band which is not completely filled with electrons is called a conduction band and the band containing the valency electrons the valency band. For a conductor the valency band is also the conduction band. The electronic state of a selection of materials of different valencies is presented in Figure 5.20c. Although all metals are relatively good conductors of electricity, they exhibit among themselves a range of values for their resistivities.

There are a number of reasons for this variability. The resistivity of a metal depends on the density of states of the most energetic electrons at the top of the band, and the shape of the N (E)–E curve at this point.

In the transition metals, for example, apart from producing the strong magnetic properties, great strength and high melting point, the d-band is also responsible for the poor electrical conductivity and high electronic specific heat. When an electron is scattered by a lattice irregularity it jumps into a different quantum state, and it will be evident that the more vacant quantum states there are available in the same energy range, the more likely will be the electron to deflect at the irregularity. The high resistivities of the transition metals may therefore be explained by the ease with which electrons can be deflected into vacant d-states. Phonon-assisted s–d scattering gives rise to the non-linear variation of ρ with temperature observed at high temperatures. The high electronic specific heat is also due to the high density of states in the unfilled d-band, since this gives rise to a considerable number of electrons

264 Physical Metallurgy and Advanced Materials at the top of the Fermi distribution which can be excited by thermal activation. In copper, of course,

there are no unfilled levels at the top of the d-band into which electrons can go, and consequently both the electronic specific heat and electrical resistance are low. The conductivity also depends on the degree to which the electrons are scattered by the ions of the metal which are thermally vibrating, and by impurity atoms or other defects present in the metal.

Insulators can also be modified either by the application of high temperatures or by the addition of impurities. Clearly, insulators may become conductors at elevated temperatures if the thermal agitation is sufficient to enable electrons to jump the energy gap into the unfilled zone above.