上海交通大学：《材料与文明》课程教学资源（参考资料）Understanding Mater_Chapter 11 - Electrical Properties of Materials.pdf_大学文库

188 11.Electrical Properties of Materials FIGURE 11.4.Schematic representation of an electron path through a conductor (contain- ing vacancies,impu- rity atoms,and a grain boundary)un- der the influence of an electric field.This classical description does not completely describe the resistance in materials. temperatures.At near-zero temperatures,the electrical resistance does not completely vanish,however (except in superconduc- tors).There always remains a residual resistivity,pres (Figure 11.3),which is thought to be caused by "collisions"of electrons (i.e.,by electrostatic interactions)with imperfections in the crys- tal (such as impurities,vacancies,grain boundaries,or disloca- tions),as explained in Chapters 3 and 6.The residual resistivity is essentially temperature-independent. On the other hand,one may describe the electrons to have a wave nature.The matter waves may be thought to be scattered by lattice atoms.Scattering is the dissipation of radiation on small particles in all directions.The atoms absorb the energy of an incoming wave and thus become oscillators.These oscillators in turn re-emit the energy in the form of spherical waves.If two or more atoms are involved,the phase relationship between the individual re-emitted waves has to be taken into consideration. A calculation!shows that for a periodic crystal structure the in- dividual waves in the forward direction are in-phase,and thus interfere constructively.As a result,a wave which propagates through an ideal crystal (having periodically arranged atoms) does not suffer any change in intensity or direction (only its ve- locity is modified).This mechanism is called coherent scattering. If,however,the scattering centers are not periodically arranged (impurity atoms,vacancies,grain boundaries,thermal vibration of atoms,etc.),the scattered waves have no set phase relation- ship and the wave is said to be incoherently scattered.The energy of incoherently scattered waves is smaller in the forward direc- tion.This energy loss qualitatively explains the resistance.In L.Brillouin,Wave Propagation in Periodic Structures,Dover,New York (1953)

188 11 • Electrical Properties of Materials temperatures. At near-zero temperatures, the electrical resistance does not completely vanish, however (except in superconductors). There always remains a residual resistivity, res (Figure 11.3), which is thought to be caused by “collisions” of electrons (i.e., by electrostatic interactions) with imperfections in the crystal (such as impurities, vacancies, grain boundaries, or dislocations), as explained in Chapters 3 and 6. The residual resistivity is essentially temperature-independent. On the other hand, one may describe the electrons to have a wave nature. The matter waves may be thought to be scattered by lattice atoms. Scattering is the dissipation of radiation on small particles in all directions. The atoms absorb the energy of an incoming wave and thus become oscillators. These oscillators in turn re-emit the energy in the form of spherical waves. If two or more atoms are involved, the phase relationship between the individual re-emitted waves has to be taken into consideration. A calculation1 shows that for a periodic crystal structure the individual waves in the forward direction are in-phase, and thus interfere constructively. As a result, a wave which propagates through an ideal crystal (having periodically arranged atoms) does not suffer any change in intensity or direction (only its velocity is modified). This mechanism is called coherent scattering. If, however, the scattering centers are not periodically arranged (impurity atoms, vacancies, grain boundaries, thermal vibration of atoms, etc.), the scattered waves have no set phase relationship and the wave is said to be incoherently scattered. The energy of incoherently scattered waves is smaller in the forward direction. This energy loss qualitatively explains the resistance. In FIGURE 11.4. Schematic representation of an electron path through a conductor (containing vacancies, impurity atoms, and a grain boundary) under the influence of an electric field. This classical description does not completely describe the resistance in materials. 1L. Brillouin, Wave Propagation in Periodic Structures, Dover, New York (1953)

11.1.Conductivity and Resistivity of Metals 189 short,the wave picture provides a deeper understanding of the electrical resistance in metals and alloys. According to a rule proposed by Matthiessen,the total resis- tivity arises from independent mechanisms,as just described, which are additive,i.e.: p=Pth+Pimp Pdef=Pth+Pres. (11.8) The thermally induced part of the resistivity Ph is called the ideal resistivity,whereas the resistivity that has its origin in impuri- ties (pimp)and defects(pder)is summed up in the residual resis- tivity (pres).The number of impurity atoms is generally constant in a given metal or alloy.The number of vacancies,or grain boundaries,however,can be changed by various heat treatments. For example,if a metal is annealed at temperatures close to its melting point and then rapidly quenched into water of room tem- perature,its room temperature resistivity increases noticeably due to quenched-in vacancies,as already explained in Chapter 6. Frequently,this resistance increase diminishes during room tem- perature aging or annealing at slightly elevated temperatures due to the annihilation of these vacancies.Likewise,work hardening, recrystallization,grain growth,and many other metallurgical processes change the resistivity of metals.As a consequence of this,and due to its simple measurement,the resistivity has been one of the most widely studied properties in materials research. Free Electrons The conductivity of metals can be calculated(as P.Drude did at the turn to the 20th century)by simply postulating that the elec- tric force,e.provided by an electric field (Figure 11.2),ac- celerates the electrons (having a charge -e)from the cathode to the anode.The drift of the electrons was thought by Drude to be counteracted by collisions with certain atoms as described above. The Newtonian-type equation(force equals mass times acceler- ation)of this free electron model m变+w=e:g (11.9) leads,after a string of mathematical manipulations,to the con- ductivity: 0、 r.e2.T (11.10) m where v is the drift velocity of the electrons,m is the electron mass,y is a constant which takes the electron/atom collisions into consideration (called damping strength),T=m/y is the average time between two consecutive collisions (called the relaxation time),and Nr is the number of free electrons per cubic meter in

short, the wave picture provides a deeper understanding of the electrical resistance in metals and alloys. According to a rule proposed by Matthiessen, the total resistivity arises from independent mechanisms, as just described, which are additive, i.e.: th imp def th res. (11.8) The thermally induced part of the resistivity th is called the ideal resistivity, whereas the resistivity that has its origin in impurities (imp) and defects (def) is summed up in the residual resistivity (res). The number of impurity atoms is generally constant in a given metal or alloy. The number of vacancies, or grain boundaries, however, can be changed by various heat treatments. For example, if a metal is annealed at temperatures close to its melting point and then rapidly quenched into water of room temperature, its room temperature resistivity increases noticeably due to quenched-in vacancies, as already explained in Chapter 6. Frequently, this resistance increase diminishes during room temperature aging or annealing at slightly elevated temperatures due to the annihilation of these vacancies. Likewise, work hardening, recrystallization, grain growth, and many other metallurgical processes change the resistivity of metals. As a consequence of this, and due to its simple measurement, the resistivity has been one of the most widely studied properties in materials research. The conductivity of metals can be calculated (as P. Drude did at the turn to the 20th century) by simply postulating that the electric force, e , provided by an electric field (Figure 11.2), accelerates the electrons (having a charge e) from the cathode to the anode. The drift of the electrons was thought by Drude to be counteracted by collisions with certain atoms as described above. The Newtonian-type equation (force equals mass times acceleration) of this free electron model m d d v t v e (11.9) leads, after a string of mathematical manipulations, to the conductivity: Nf m e2 , (11.10) where v is the drift velocity of the electrons, m is the electron mass, is a constant which takes the electron/atom collisions into consideration (called damping strength), m/ is the average time between two consecutive collisions (called the relaxation time), and Nf is the number of free electrons per cubic meter in 11.1 • Conductivity and Resistivity of Metals 189 Free Electrons

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

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