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

11 Electrical Properties of Materials One of the principal characteristics of materials is their ability (or lack of ability)to conduct electrical current.Indeed,materi- als are classified by this property,that is,they are divided into conductors,semiconductors,and nonconductors.(The latter are often called insulators or dielectrics.)The conductivity,o,of dif- ferent materials at room temperature spans more than 25 orders of magnitude,as depicted in Figure 11.1.Moreover,if one takes the conductivity of superconductors,measured at low tempera- tures,into consideration,this span extends to 40 orders of mag- nitude (using an estimated conductivity for superconductors of about 1020 1/0 cm).This is the largest known variation in a phys- ical property and is only comparable to the ratio between the di- ameter of the universe (about 1026 m)and the radius of an elec- tron(10-14m). The inverse of the conductivity is called resistivity,p,that is: p= (11.1) The resistance,R of a piece of conducting material is propor- tional to its resistivity and to its length,L,and is inversely pro- portional to its cross-sectional area,A: R=L:P A (11.2) The resistance can be easily measured.For this,a direct current is applied to a slab of the material.The current,I,through the sample (in amperes),as well as the voltage drop,V,on two po- tential probes (in volts)is recorded as depicted in Figure 11.2

11 One of the principal characteristics of materials is their ability (or lack of ability) to conduct electrical current. Indeed, materi￾als are classified by this property, that is, they are divided into conductors, semiconductors, and nonconductors. (The latter are often called insulators or dielectrics.) The conductivity, , of dif￾ferent materials at room temperature spans more than 25 orders of magnitude, as depicted in Figure 11.1. Moreover, if one takes the conductivity of superconductors, measured at low tempera￾tures, into consideration, this span extends to 40 orders of mag￾nitude (using an estimated conductivity for superconductors of about 1020 1/# cm). This is the largest known variation in a phys￾ical property and is only comparable to the ratio between the di￾ameter of the universe (about 1026 m) and the radius of an elec￾tron (1014 m). The inverse of the conductivity is called resistivity, , that is: 
1 . (11.1) The resistance, R of a piece of conducting material is propor￾tional to its resistivity and to its length, L, and is inversely pro￾portional to its cross-sectional area, A: R
L A   . (11.2) The resistance can be easily measured. For this, a direct current is applied to a slab of the material. The current, I, through the sample (in ampères), as well as the voltage drop, V, on two po￾tential probes (in volts) is recorded as depicted in Figure 11.2. Electrical Properties of Materials

186 11.Electrical Properties of Materials SiO, Porcelain Dry wood Fe Doped Si Quartz Rubber Glass Si Ge Mn Ag NaCl Mica GaAs Cu 11 102010-181016101410121010108106101021102.1010 cm Insulators Semiconductors- *-Metals— FIGURE 11.1.Room-temperature conductivity of various materials.(Su- perconductors,having conductivities of many orders of magnitude larger than copper,near 0 K,are not shown.The conductivity of semi- conductors varies substantially with temperature and purity.)It is cus- tomary in engineering to use the centimeter as the unit of length rather than the meter.We follow this practice.The reciprocal of the ohm ( is defined to be 1 siemens(S);see Appendix II.For conducting poly- mers,refer to Figure 11.20. Ohm's Law The resistance (in ohms)can then be calculated by making use of Ohm's law: V=R·I, (11.3) which was empirically found by Georg Simon Ohm (a German physicist)in 1826 relating a large number of experimental ob- servations.Another form of Ohm's law: j=0…8, (11.4) links current density: (11.5) Battery Ampmeter FIGURE 11.2.Schematic representation of an elec- tric circuit to measure the resistance of a conduc- tor. Voltmeter

186 11 • Electrical Properties of Materials The resistance (in ohms) can then be calculated by making use of Ohm’s law: V
R  I, (11.3) which was empirically found by Georg Simon Ohm (a German physicist) in 1826 relating a large number of experimental ob￾servations. Another form of Ohm’s law: j

, (11.4) links current density: j
A I , (11.5) FIGURE 11.1. Room-temperature conductivity of various materials. (Su￾perconductors, having conductivities of many orders of magnitude larger than copper, near 0 K, are not shown. The conductivity of semi￾conductors varies substantially with temperature and purity.) It is cus￾tomary in engineering to use the centimeter as the unit of length rather than the meter. We follow this practice. The reciprocal of the ohm (#) is defined to be 1 siemens (S); see Appendix II. For conducting poly￾mers, refer to Figure 11.20. 10–20 10–18 10–16 10–14 10–12 10–10 10–8 10–6 10–4 10–2 1 102 104 106 Quartz Dry wood NaCl Rubber Porcelain SiO2 Mica Glass GaAs Si Ge Doped Si Mn Fe Ag Cu 1 # cm Insulators Semiconductors Metals Battery Ampmeter I e– – + A L V Voltmeter FIGURE 11.2. Schematic representation of an elec￾tric circuit to measure the resistance of a conduc￾tor. Ohm’s Law

11.1.Conductivity and Resistivity of Metals 187 that is,the current per unit area(A/cm2),with the conductivity o(1/0 cm or siemens per cm)and the electric field strength: 光 (11.6) (V/cm).(We use a script for the electric field strength to dis- tinguish it from the energy.) 11.1.Conductivity and Resistivity of Metals The resistivity of metals essentially increases linearly with increas- ing temperature(Figure 11.3)according to the empirical equation: p2=p[1+a(T2-T1】, (11.7) where a is the linear temperature coefficient of resistivity,and T and T2 are two different temperatures.We attempt to explain this behavior.We postulate that the free electrons (see Chapter 10) are accelerated in a metal under the influence of an electric field maintained,for example,by a battery.The drifting electrons can be considered,in a preliminary,classical description,to occa- sionally collide (that is,electrostatically interact)with certain lat- tice atoms,thus losing some of their energy.This constitutes the just-discussed resistance.In essence,the drifting electrons are then said to migrate in a zig-zag path through the conductor from the cathode to the anode,as sketched in Figure 11.4.Now,at higher temperatures,the lattice atoms increasingly oscillate about their equilibrium positions due to the supply of thermal energy,thus enhancing the probability for collisions by the drift- ing electrons.As a consequence,the resistance rises with higher Cu-3 at Ni Cu-2 at Ni Cu -1 at Ni Cu FIGURE 11.3.Schematic representation of Pres the temperature dependence of the resis- tivity of copper and various copper-nickel 2 alloys.Pres is the residual resistivity

11.1 • Conductivity and Resistivity of Metals 187 that is, the current per unit area (A/cm2), with the conductivity (1/# cm or siemens per cm) and the electric field strength:

V L (11.6) (V/cm). (We use a script
for the electric field strength to dis￾tinguish it from the energy.) The resistivity of metals essentially increases linearly with increas￾ing temperature (Figure 11.3) according to the empirical equation: 2
1[1 (T2  T1)], (11.7) where  is the linear temperature coefficient of resistivity, and T1 and T2 are two different temperatures. We attempt to explain this behavior. We postulate that the free electrons (see Chapter 10) are accelerated in a metal under the influence of an electric field maintained, for example, by a battery. The drifting electrons can be considered, in a preliminary, classical description, to occa￾sionally collide (that is, electrostatically interact) with certain lat￾tice atoms, thus losing some of their energy. This constitutes the just-discussed resistance. In essence, the drifting electrons are then said to migrate in a zig-zag path through the conductor from the cathode to the anode, as sketched in Figure 11.4. Now, at higher temperatures, the lattice atoms increasingly oscillate about their equilibrium positions due to the supply of thermal energy, thus enhancing the probability for collisions by the drift￾ing electrons. As a consequence, the resistance rises with higher 11.1 • Conductivity and Resistivity of Metals res  T Cu – 3 at % Ni Cu – 2 at % Ni Cu – 1 at % Ni Cu FIGURE 11.3. Schematic representation of the temperature dependence of the resis￾tivity of copper and various copper-nickel alloys. res is the residual resistivity.

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 superconduc￾tors). 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 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 calculation1 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 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. 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 resis￾tivity 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 impuri￾ties (imp) and defects (def) is summed up in the residual resis￾tivity (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 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. 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 d d v t v
e 
(11.9) leads, after a string of mathematical manipulations, to the con￾ductivity:
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