190 11.Electrical Properties of Materials the material.We can learn from this equation that semiconduc- tors or insulators which have only a small number of free elec- trons (or often none at all)display only very small conductivities. (The small number of electrons results from the strong binding forces between electrons and atoms that are common for insula- tors and semiconductors.)Conversely,metals which contain a large number of free electrons have a large conductivity.Further, the conductivity is large when the average time between two col- lisions,T,is large.Obviously,the number of collisions decreases (i.e.,r increases)with decreasing temperature and decreasing number of imperfections. The above-outlined free electron model,which is relatively sim- ple in its assumptions,describes the electrical behavior of many materials reasonably well.Nevertheless,quantum mechanics provides some important and necessary refinements.One of the refinements teaches us how many of the valence electrons can be considered to be free,that is,how many of them contribute to the conduction process.Equation(11.10)does not provide this distinction.Quantum mechanics of materials is quite involved and requires the solution of the Schrodinger equation,the treat- ment of which must be left to specialized texts.2 Its essential re- sults can be summarized,however,in a few words. Electron Band We know from Section 3.1 that the electrons of isolated atoms Model (for example in a gas)can be considered to orbit at various dis- tances about their nuclei.These orbits constitute different ener- gies.Specifically,the larger the radius of an orbit,the larger the excitation energy of the electron.This fact is often represented in a somewhat different fashion by stating that the electrons are distributed on different energy levels,as schematically shown on the right side of Figure 11.5.Now,these distinct energy levels, which are characteristic for isolated atoms,widen into energy bands when atoms approach each other and eventually form a solid as depicted on the left side of Figure 11.5.Quantum me- chanics postulates that the electrons can only reside within these bands,but not in the areas outside of them.The allowed energy bands may be noticeably separated from each other.In other cases,depending on the material and the energy,they may par- tially or completely overlap.In short,each material has its dis- tinct electron energy band structure.Characteristic band struc- tures for the main classes of materials are schematically depicted in Figure 11.6. 2See,for example,R.E.Hummel,Electronic Properties of Materials,3rd Edition,Springer-Verlag,New York(2001)
190 11 • Electrical Properties of Materials the material. We can learn from this equation that semiconductors or insulators which have only a small number of free electrons (or often none at all) display only very small conductivities. (The small number of electrons results from the strong binding forces between electrons and atoms that are common for insulators and semiconductors.) Conversely, metals which contain a large number of free electrons have a large conductivity. Further, the conductivity is large when the average time between two collisions, , is large. Obviously, the number of collisions decreases (i.e., increases) with decreasing temperature and decreasing number of imperfections. The above-outlined free electron model, which is relatively simple in its assumptions, describes the electrical behavior of many materials reasonably well. Nevertheless, quantum mechanics provides some important and necessary refinements. One of the refinements teaches us how many of the valence electrons can be considered to be free, that is, how many of them contribute to the conduction process. Equation (11.10) does not provide this distinction. Quantum mechanics of materials is quite involved and requires the solution of the Schrödinger equation, the treatment of which must be left to specialized texts.2 Its essential results can be summarized, however, in a few words. We know from Section 3.1 that the electrons of isolated atoms (for example in a gas) can be considered to orbit at various distances about their nuclei. These orbits constitute different energies. Specifically, the larger the radius of an orbit, the larger the excitation energy of the electron. This fact is often represented in a somewhat different fashion by stating that the electrons are distributed on different energy levels, as schematically shown on the right side of Figure 11.5. Now, these distinct energy levels, which are characteristic for isolated atoms, widen into energy bands when atoms approach each other and eventually form a solid as depicted on the left side of Figure 11.5. Quantum mechanics postulates that the electrons can only reside within these bands, but not in the areas outside of them. The allowed energy bands may be noticeably separated from each other. In other cases, depending on the material and the energy, they may partially or completely overlap. In short, each material has its distinct electron energy band structure. Characteristic band structures for the main classes of materials are schematically depicted in Figure 11.6. 2See, for example, R.E. Hummel, Electronic Properties of Materials, 3rd Edition, Springer-Verlag, New York (2001). Electron Band Model
11.1.Conductivity and Resistivity of Metals 191 Energy Electron Band Forbidden Band Energy Levels Electron Band FIGURE 11.5.Schematic representation of energy levels (as for isolated Forbidden Band atoms)and widening of these levels Electron Band into energy bands with decreasing distance between atoms.Energy Solid Gas bands for a specific case are shown Distance between atoms at the left of the diagram. Now,the band structures shown in Figure 11.6 are somewhat simplified.Specifically,band schemes actually possess a fine structure,that is,the individual energy states (i.e.,the possibili- ties for electron occupation)are often denser in the center of a band (Figure 11.7).To account for this,one defines a density of energy states,shortly called the density of states,Z(E). Some of the just-mentioned bands are occupied by electrons while others remain partially or completely empty,similar to a cup that may be only partially filled with water.The degree to which an electron band is filled by electrons is indicated in Fig- ure 11.6 by shading.The highest level of electron filling within a band is called the Fermi energy,Er,which may be compared with the water surface in a cup.(For values of EF,see Appendix II).We notice in Figure 11.6 that some materials,such as insu- lators and semiconductors,have completely filled electron bands. (They differ,however,in their distance to the next higher band.) FIGURE 11.6.Simplified repre- sentation for energy bands for (a)monovalent metals,(b)biva- lent metals,(c)semiconductors, and (d)insulators.For a de- scription of the nomenclature, (a) (b) (c) (d) see Appendix I
11.1 • Conductivity and Resistivity of Metals 191 Now, the band structures shown in Figure 11.6 are somewhat simplified. Specifically, band schemes actually possess a fine structure, that is, the individual energy states (i.e., the possibilities for electron occupation) are often denser in the center of a band (Figure 11.7). To account for this, one defines a density of energy states, shortly called the density of states, Z(E). Some of the just-mentioned bands are occupied by electrons while others remain partially or completely empty, similar to a cup that may be only partially filled with water. The degree to which an electron band is filled by electrons is indicated in Figure 11.6 by shading. The highest level of electron filling within a band is called the Fermi energy, EF, which may be compared with the water surface in a cup. (For values of EF, see Appendix II). We notice in Figure 11.6 that some materials, such as insulators and semiconductors, have completely filled electron bands. (They differ, however, in their distance to the next higher band.) (a) (b) (d) (c) EF EF 3s 3p FIGURE 11.6. Simplified representation for energy bands for (a) monovalent metals, (b) bivalent metals, (c) semiconductors, and (d) insulators. For a description of the nomenclature, see Appendix I. Electron Band Forbidden Band Electron Band Forbidden Band Electron Band Energy Energy Levels Solid Gas Distance between atoms FIGURE 11.5. Schematic representation of energy levels (as for isolated atoms) and widening of these levels into energy bands with decreasing distance between atoms. Energy bands for a specific case are shown at the left of the diagram
192 11.Electrical Properties of Materials FIGURE 11.7.Schematic represen- E tation of the density of electron states Z(E)within an electron en- E ergy band.The density of states is essentially identical to the pop- ulation density N(E)for energies below the Fermi energy,EF (i.e., Valence for that energy level up to which band a band is filled with electrons). Examples of highest electron en- EM一 ergies for a monovalent metal (EM),for a bivalent metal(EB), Z(E) and for an insulator (Ep)are indi- cated. Metals,on the other hand,are characterized by partially filled electron bands.The amount of filling depends on the material, that is,on the electron concentration and the amount of band overlapping. We may now return to the conductivity.In short,according to quantum theory,only those materials that possess partially filled electron bands are capable of conducting an electric current. Electrons can then be lifted slightly above the Fermi energy into an allowed and unfilled energy state.This permits them to be ac- celerated by an electric field,thus producing a current.Second, only those electrons that are close to the Fermi energy partici- pate in the electric conduction.(The classical electron theory taught us instead that all free electrons would contribute to the current.)Third,the number of electrons near the Fermi energy depends on the density of available electron states(Figure 11.7). The conductivity in quantum mechanical terms yields the fol- lowing equation: o=片e2v2N(Er) (11.11) where ve is the velocity of the electrons at the Fermi energy(called the Fermi velocity)and N(EF)is the density of filled electron states(called the population density)at the Fermi energy.The population density is proportional to Z(E);both have the unit J-1m-3 or eV-lm-3.Equation (11.11),in conjunction with Fig- ure 11.7,now provides a more comprehensive picture of electron conduction.Monovalent metals(such as copper,silver,and gold) have partially filled bands,as shown in Figure 11.6(a).Their elec- tron population density near the Fermi energy is high(Figure 11.7),which,according to Eq.(11.11),results in a large con- ductivity.Bivalent metals,on the other hand,are distinguished by an overlapping of the upper bands and by a small electron
192 11 • Electrical Properties of Materials Metals, on the other hand, are characterized by partially filled electron bands. The amount of filling depends on the material, that is, on the electron concentration and the amount of band overlapping. We may now return to the conductivity. In short, according to quantum theory, only those materials that possess partially filled electron bands are capable of conducting an electric current. Electrons can then be lifted slightly above the Fermi energy into an allowed and unfilled energy state. This permits them to be accelerated by an electric field, thus producing a current. Second, only those electrons that are close to the Fermi energy participate in the electric conduction. (The classical electron theory taught us instead that all free electrons would contribute to the current.) Third, the number of electrons near the Fermi energy depends on the density of available electron states (Figure 11.7). The conductivity in quantum mechanical terms yields the following equation: 1 3 e2 v 2 F N(EF) (11.11) where vF is the velocity of the electrons at the Fermi energy (called the Fermi velocity) and N(EF) is the density of filled electron states (called the population density) at the Fermi energy. The population density is proportional to Z(E); both have the unit J1m3 or eV1m3. Equation (11.11), in conjunction with Figure 11.7, now provides a more comprehensive picture of electron conduction. Monovalent metals (such as copper, silver, and gold) have partially filled bands, as shown in Figure 11.6(a). Their electron population density near the Fermi energy is high (Figure 11.7), which, according to Eq. (11.11), results in a large conductivity. Bivalent metals, on the other hand, are distinguished by an overlapping of the upper bands and by a small electron E EI Valence band Z (E) E M EB FIGURE 11.7. Schematic representation of the density of electron states Z(E) within an electron energy band. The density of states is essentially identical to the population density N(E) for energies below the Fermi energy, EF (i.e., for that energy level up to which a band is filled with electrons). Examples of highest electron energies for a monovalent metal (EM), for a bivalent metal (EB), and for an insulator (EI) are indicated.
11.2.Conduction in Alloys 193 concentration near the bottom of the valence band,as shown in Figure 11.6(b).As a consequence,the electron population near the Fermi energy is small(Figure 11.7),which leads to a com- paratively low conductivity.Finally,insulators have completely filled(and completely empty)electron bands,which results in a virtually zero population density,as shown in Figure 11.7.Thus, the conductivity in insulators is virtually zero (if one disregards, for example,ionic conduction;see Section 11.6).These explana- tions are admittedly quite sketchy.The interested reader is re- ferred to the specialized books listed at the end of this chapter. 11.2.Conduction in Alloys The residual resistivity of alloys increases with increasing amount of solute content as seen in Figures 11.3 and 11.8.The slopes of the individual p versus T lines remain,however,essentially con- stant(Figure 11.3).Small additions of solute cause a linear shift of the p versus T curves to higher resistivity values in accordance with the Matthiessen rule;see Eq.(11.8)and Figure 11.8.Vari- ous solute elements might alter the resistivity of the host mate- rial to different degrees.This is depicted in Figure 11.8 for sil- ver,which demonstrates that the residual resistivity increases with increasing atomic number of the solute.For its interpreta- tion,one may reasonably assume that the likelihood for interac- tions between electrons and impurity atoms increases when the solute has a larger atomic size,as is encountered by proceeding from cadmium to antimony. The resistivity of two-phase alloys is,in many instances,the sum of the resistivity of each of the components,taking the vol- ume fractions of each phase into consideration.However,addi- tional factors,such as the crystal structure and the kind of dis- tribution of the phases in each other,also have to be considered. Sb Sn n FIGURE 11.8.Resistivity change Cd of various dilute silver alloys (schematic).Solvent and solute are all from the fifth Ag at.Solute period
11.2 • Conduction in Alloys 193 concentration near the bottom of the valence band, as shown in Figure 11.6(b). As a consequence, the electron population near the Fermi energy is small (Figure 11.7), which leads to a comparatively low conductivity. Finally, insulators have completely filled (and completely empty) electron bands, which results in a virtually zero population density, as shown in Figure 11.7. Thus, the conductivity in insulators is virtually zero (if one disregards, for example, ionic conduction; see Section 11.6). These explanations are admittedly quite sketchy. The interested reader is referred to the specialized books listed at the end of this chapter. The residual resistivity of alloys increases with increasing amount of solute content as seen in Figures 11.3 and 11.8. The slopes of the individual versus T lines remain, however, essentially constant (Figure 11.3). Small additions of solute cause a linear shift of the versus T curves to higher resistivity values in accordance with the Matthiessen rule; see Eq. (11.8) and Figure 11.8. Various solute elements might alter the resistivity of the host material to different degrees. This is depicted in Figure 11.8 for silver, which demonstrates that the residual resistivity increases with increasing atomic number of the solute. For its interpretation, one may reasonably assume that the likelihood for interactions between electrons and impurity atoms increases when the solute has a larger atomic size, as is encountered by proceeding from cadmium to antimony. The resistivity of two-phase alloys is, in many instances, the sum of the resistivity of each of the components, taking the volume fractions of each phase into consideration. However, additional factors, such as the crystal structure and the kind of distribution of the phases in each other, also have to be considered. 11.2 • Conduction in Alloys Ag Sb Sn In Cd at. % Solute FIGURE 11.8. Resistivity change of various dilute silver alloys (schematic). Solvent and solute are all from the fifth period
194 11.Electrical Properties of Materials Some alloys,when in the ordered state,that is,when the solute atoms are periodically arranged in the matrix,have a distinctly smaller resistivity compared to the case when the atoms are ran- domly distributed.Slowly cooled Cu3Au or CuAu are common examples of ordered structures. Copper is frequently used for electrical wires because of its high conductivity(Figure 11.1).However,pure or annealed cop- per has a low strength(Chapter 3).Thus,work hardening(dur- ing wire drawing),or dispersion strengthening (by adding less than 1%Al203),or age hardening (Cu-Be),or solid solution strengthening(by adding small amounts of second constituents such as Zn)may be used for strengthening.The increase in strength occurs,however,at the expense of a reduced conduc- tivity.(The above mechanisms are arranged in decreasing order of conductivity of the copper-containing wire.)The resistance in- crease in copper inflicted by cold working can be restored to al- most its initial value by annealing copper at moderate tempera- tures (about 300C).This process,which was introduced in Chapter 6 by the terms stress relief anneal or recovery,causes the dislocations to rearrange to form a polygonized structure with- out substantially reducing their number.Thus,the strength of stress-relieved copper essentially is maintained while the con- ductivity is almost restored to its pre-work hardened state(about 98%). For other applications,a high resistivity is desired,such as for heating elements in furnaces which are made,for example,of nickel-chromium alloys.These alloys need to have a high melt- ing temperature and also a good resistance to oxidation,partic- ularly at high temperatures. 11.3.Superconductivity The resistivity in superconductors becomes immeasurably small or virtually zero below a critical temperature,T,as shown in Fig- ure 11.9.About 27 elements,numerous alloys,ceramic materials (containing copper oxide),and organic compounds(based,for ex- ample,on selenium or sulfur)have been found to possess this property (see Table 11.1).It is estimated that the conductivity of superconductors below Te is about 1020 1/0 cm (see Figure 11.1). The transition temperatures where superconductivity starts range from 0.01 K (for tungsten)up to about 125 K(for ceramic su- perconductors).Of particular interest are materials whose Te is above 77 K,that is,the boiling point of liquid nitrogen,which is more readily available than other coolants.Among the so-called
194 11 • Electrical Properties of Materials Some alloys, when in the ordered state, that is, when the solute atoms are periodically arranged in the matrix, have a distinctly smaller resistivity compared to the case when the atoms are randomly distributed. Slowly cooled Cu3Au or CuAu are common examples of ordered structures. Copper is frequently used for electrical wires because of its high conductivity (Figure 11.1). However, pure or annealed copper has a low strength (Chapter 3). Thus, work hardening (during wire drawing), or dispersion strengthening (by adding less than 1% Al2O3), or age hardening (Cu–Be), or solid solution strengthening (by adding small amounts of second constituents such as Zn) may be used for strengthening. The increase in strength occurs, however, at the expense of a reduced conductivity. (The above mechanisms are arranged in decreasing order of conductivity of the copper-containing wire.) The resistance increase in copper inflicted by cold working can be restored to almost its initial value by annealing copper at moderate temperatures (about 300°C). This process, which was introduced in Chapter 6 by the terms stress relief anneal or recovery, causes the dislocations to rearrange to form a polygonized structure without substantially reducing their number. Thus, the strength of stress-relieved copper essentially is maintained while the conductivity is almost restored to its pre-work hardened state (about 98%). For other applications, a high resistivity is desired, such as for heating elements in furnaces which are made, for example, of nickel–chromium alloys. These alloys need to have a high melting temperature and also a good resistance to oxidation, particularly at high temperatures. The resistivity in superconductors becomes immeasurably small or virtually zero below a critical temperature, Tc, as shown in Figure 11.9. About 27 elements, numerous alloys, ceramic materials (containing copper oxide), and organic compounds (based, for example, on selenium or sulfur) have been found to possess this property (see Table 11.1). It is estimated that the conductivity of superconductors below Tc is about 1020 1/# cm (see Figure 11.1). The transition temperatures where superconductivity starts range from 0.01 K (for tungsten) up to about 125 K (for ceramic superconductors). Of particular interest are materials whose Tc is above 77 K, that is, the boiling point of liquid nitrogen, which is more readily available than other coolants. Among the so-called 11.3 • Superconductivity