Material Science 18

A dielectric material is one that is electrically insulating (nonmetallic) and exhibits or may be made to exhibit an electric dipole structure; that is, there is a separation of positive and negative electrically charged entities on a molecular or atomic level. This concept of an electric dipole was introduced in Section 2.7. As a result of dipole interactions with electric fields, dielectric materials are used in capacitors.

A dipole cannot keep shifting orientation direction when the frequency of the applied electric field exceeds its relaxation frequency and, therefore, will not make a contribution to the dielectric constant. The dependence of r on the field frequency is represented schematically in Figure 18.34 for a dielectric medium that exhibits all three types of polarization; note that the frequency axis is scaled logarithmically. As indicated in Figure 18.34, when a polarization mechanism ceases to function, there is an abrupt drop in the dielectric constant; otherwise, r is virtually frequency independent. Table 18.5 gave values of the dielectric constant at 60 Hz and 1 MHz; these provide an indication of this frequency dependence at the low end of the frequency spectrum

A number of ceramics and polymers are used as insulators and/or in capacitors. Many of the ceramics, including glass, porcelain, steatite, and mica, have dielectric constants within the range of 6 to 10 (Table 18.5). These materials also exhibit a high degree of dimensional stability and mechanical strength. Typical applications include power line and electrical insulation, switch bases, and light receptacles. The titania (TiO2) and titanate ceramics, such as barium titanate (BaTiO3), can be made to have extremely high dielectric constants, which render them especially useful for some capacitor applications

A rapidly evolving information storage technology that uses semiconductor devices is flash memory. As with computer storage, flash memory is programmed and erased electronically, as described in the preceding paragraph. However, unlike computer storage, the flash technology is nonvolatile—that is, no electrical power is needed to retain the stored information.There are no moving parts (as with magnetic hard drives and magnetic tapes, Section 20.11), which makes flash memory especially attractive for general storage and transfer of data between portable devices— such as digital cameras, laptop computers, mobile phones, PDAs, digital audio players, and game consoles. Furthermore, flash technology is packaged in memory cards and USB flash drives. Unlike magnetic memory, flash packages are extremely durable and are capable of withstanding relatively wide temperature extremes as well as immersion in water

A rectifier, or diode, is an electronic device that allows the current to flow in one direction only; for example, a rectifier transforms an alternating current into direct current. Before the advent of the p-n junction semiconductor rectifier, this operation was carried out using the vacuum tube diode. The p-n rectifying junction is constructed from a single piece of semiconductor that is doped so as to be n-type on one side and p-type on the other (Figure 18.21a). If pieces of n- and p-type materials are joined together, a poor rectifier results, because the presence of a surface between the two sections renders the device very inefficient. Also, single crystals of semiconducting materials must be used in all devices because electronic phenomena that are deleterious to operation occur at grain boundarie

Again, polarization is the alignment of permanent or induced atomic or molecular dipole moments with an externally applied electric field. There are three types or sources of polarization: electronic, ionic, and orientation. Dielectric materials ordinarily exhibit at least one of these polarization types depending on the material and also the manner of the external field application.

In intrinsic semiconductors, for every electron excited into the conduction band there is left behind a missing electron in one of the covalent bonds, or in the band scheme, a vacant electron state in the valence band, as shown in Figure 18.6b. 5 Under

An electric current results from the motion of electrically charged particles in response to forces that act on them from an externally applied electric field. Positively charged particles are accelerated in the field direction, negatively charged particles in the direction opposite. Within most solid materials a current arises from the flow of electrons, which is termed electronic conduction. In addition, for ionic materials a net motion of charged ions is possible that produces a current; such is termed ionic conduction. The present discussion deals with electronic conduction; ionic conduction is treated briefly in Section 18.16.

As mentioned previously, most metals are extremely good conductors of electricity; room-temperature conductivities for several of the more common metals are contained in Table 18.1. (Table B.9 in Appendix B lists the electrical resistivities of a large number of metals and alloys.) Again, metals have high conductivities because of the large numbers of free electrons that have been excited into empty states above the Fermi energy. Thus n has a large value in the conductivity expression, Equation 18.8. At this point it is convenient to discuss conduction in metals in terms of the resistivity, the reciprocal of conductivity; the reason for this switch should become apparent in the ensuing discussion. Because crystalline defects serve as scattering centers for conduction electrons in metals, increasing their number raises the resistivity (or lowers the conductivity). The concentration of these imperfections depends on temperature, composition, and the degree of cold work of a metal specimen. In fact, it has been observed experimentally that the total resistivity of a metal is the sum of the contributions from thermal vibrations, impurities, and plastic deformation; that is, the scattering

At high reverse bias voltages, sometimes on the order of several hundred volts, large numbers of charge carriers (electrons and holes) are generated. This gives rise to a very abrupt increase in current, a phenomenon known as breakdown, also shown in Figure 18.22, and discussed in more detail in Section 18.22.

At low temperatures, below about 100 K (Figure 18.17), electron concentration drops dramatically with decreasing temperature and approaches zero at 0 K. Over these temperatures, the thermal energy is insufficient to excite electrons from the P donor level into the conduction band. This is termed the "freeze-out temperature region" inasmuch as charged carriers (i.e., electrons) are "frozen" to the dopant atoms. Finally, at the high end of the temperature scale of Figure 18.17, electron concentration increases above the P content and asymptotically approaches the intrinsic curve as temperature increases. This is termed the intrinsic temperature region because at these high temperatures the semiconductor becomes intrinsic; that is, charge carrier concentrations resulting from electron excitations across the band gap first become equal to and then completely overwhelm the donor carrier contribution with rising temperature.

At this point in the discussion, it is vital that another concept be understood— namely, that only electrons with energies greater than the Fermi energy may be acted on and accelerated in the presence of an electric field. These are the electrons that participate in the conduction process, which are termed free electrons. Another charged electronic entity called a hole is found in semiconductors and insulators. Holes have energies less than Ef and also participate in electronic conduction. As the ensuing discussion reveals, the electrical conductivity is a direct function of the numbers of free electrons and holes. In addition, the distinction between conductors and nonconductors (insulators and semiconductors) lies in the numbers of these free electron and hole charge carriers.

where ni is known as the intrinsic carrier concentration. Furthermore, (18.15) The room-temperature intrinsic conductivities and electron and hole mobilities for several semiconducting materials are also presented in Table 18.3

Because there are two types of charge carrier (free electrons and holes) in an intrinsic semiconductor, the expression for electrical conduction, Equation 18.8, must be modified to include a term to account for the contribution of the hole current. Therefore, we write (18.13) where p is the number of holes per cubic meter and h is the hole mobility. The magnitude of h is always less than e for semiconductors. For intrinsic semiconductors, every electron promoted across the band gap leaves behind a hole in the valence band; thus

Before the application of any potential across the p-n specimen, holes will be the dominant carriers on the p-side, and electrons will predominate in the n-region, as illustrated in Figure 18.21a. An external electric potential may be established across a p-n junction with two different polarities. When a battery is used, the positive terminal may be connected to the p-side and the negative terminal to the n-side; this is referred to as a forward bias. The opposite polarity (minus to p and plus to n) is termed reverse bias. The response of the charge carriers to the application of a forward-biased potential is demonstrated in Figure 18.21b. The holes on the p-side and the electrons on the n-side are attracted to the junction.As electrons and holes encounter one another near the junction, they continuously recombine and annihilate one another, according to

Both cations and anions in ionic materials possess an electric charge and, as a consequence, are capable of migration or diffusion when an electric field is present. Thus an electric current will result from the net movement of these charged ions, which will be present in addition to current due to any electron motion. Of course, anion and cation migrations will be in opposite directions. The total conductivity of an ionic material is thus equal to the sum of both electronic and ionic contributions, as follows: (18.22) Either contribution may predominate depending on the material, its purity, and, of course, temperature. A mobility I may be associated with each of the ionic species as follows:

Concepts relating to electron energy states, their occupancy, and the resulting electron configuration for isolated atoms were discussed in Section 2.3. By way of review, for each individual atom there exist discrete energy levels that may be occupied by electrons, arranged into shells and subshells. Shells are designated by integers (1, 2, 3, etc.),

Electrical and other properties of copper render it the most widely used metallic conductor. Oxygen-free high-conductivity (OFHC) copper, having extremely low oxygen and other impurity contents, is produced for many electrical applications. Aluminum, having a conductivity only about one-half that of copper, is also frequently used as an electrical conductor. Silver has a higher conductivity than either copper or aluminum; however, its use is restricted on the basis of cost.

During the past few years, the advent of microelectronic circuitry, where millions of electronic components and circuits are incorporated into a very small space, has revolutionized the field of electronics. This revolution was precipitated, in part, by aerospace technology, which necessitated computers and electronic devices that were small and had low power requirements. As a result of refinement in processing and fabrication techniques, there has been an astonishing depreciation in the cost of integrated circuitry. Consequently, at the time of this writing, personal computers are affordable to large segments of the population in many countries. Also, the use of integrated circuits has become infused into many other facets of our lives—calculators, communications, watches, industrial production and control, and all phases of the electronics industry.

Each isolated Mg atom has two 3s electrons. However, when a solid is formed, the 3s and 3p bands overlap. In this instance and at 0 K, the Fermi energy is taken as that energy below which, for N atoms, N states are filled, two electrons per state. The final two band structures are similar; one band (the valence band) that is completely filled with electrons is separated from an empty conduction band, and an energy band gap lies between them. For very pure materials, electrons may not have energies within this gap. The difference between the two band structures lies in the magnitude of the energy gap; for materials that are insulators, the band gap is relatively wide (Figure 18.4c), whereas for semiconductors it is narrow (Figure 18.4d). The Fermi energy for these two band structures lies within the band gap—near its center.

Electronic polarization may be induced to one degree or another in all atoms. It results from a displacement of the center of the negatively charged electron cloud

For some applications, such as furnace heating elements, a high electrical resistivity is desirable. The energy loss by electrons that are scattered is dissipated as heat energy. Such materials must have not only a high resistivity, but also a resistance to oxidation at elevated temperatures and, of course, a high melting temperature. Nichrome, a nickel-chromium alloy, is commonly employed in heating elements.

Figure 18.16 plots the logarithm of the intrinsic carrier concentration ni versus temperature for both silicon and germanium. A couple of features of this plot are worth noting. First, the concentrations of electrons and holes increase with temperature because, with rising temperature, more thermal energy is available to excite electrons from the valence to the conduction band (per Figure 18.6b). In addition, at all temperatures, carrier concentration in Ge is greater than for Si. This effect is due to germanium's smaller band gap (0.67 versus 1.11 eV, Table 18.3); thus, for Ge, at any given temperature more electrons will be excited across its band gap.

Figure 18.18 represents the dependence of electron and hole mobilities in silicon as a function of the dopant (both acceptor and donor) content, at room temperature— note that both axes on this plot are scaled logarithmically. At dopant concentrations less than about 1020 m3 , both carrier mobilities are at their maximum levels and independent of the doping concentration. In addition, both mobilities decrease with increasing impurity content. Also worth noting is that the mobility of electrons is always larger than the mobility of holes.

Figure 18.25 illustrates the mechanics of operation in terms of the motion of charge carriers. Because the emitter is p-type and junction 1 is forward biased, large numbers of holes enter the base region. These injected holes are minority carriers in the n-type base, and some will combine with the majority electrons. However, if the base is extremely narrow and the semiconducting materials have been properly prepared, most of these holes will be swept through the base without recombination, then across junction 2 and into the p-type collector. The holes now become a part of the emitter-collector circuit. A small increase in input voltage within the emitter-base circuit produces a large increase in current across junction 2. This large increase in collector current is also reflected by a large increase in voltage across the load resistor, which is also shown in the circuit (Figure 18.24). Thus, a voltage signal that passes through a junction transistor experiences amplification; this effect is also illustrated in Figure 18.24 by the two voltage-time plots. Similar reasoning applies to the operation of an n-p-n transistor, except that electrons instead of holes are injected across the base and into the collector.

For additions of a single impurity that forms a solid solution, the impurity resistivity is related to the impurity concentration ci in terms of the atom fraction (at%/100) as follows:

For some materials, it is on occasion desired to determine the material's majority charge carrier type, concentration, and mobility. Such determinations are not possible from a simple electrical conductivity measurement; a Hall effect experiment must also be conducted. This Hall effect is a result of the phenomenon whereby a magnetic field applied perpendicular to the direction of motion of a charged particle exerts a force on the particle perpendicular to both the magnetic field and the particle motion directions.

For an electron to become free, it must be excited or promoted into one of the empty and available energy states above Ef. For metals having either of the band structures shown in Figures 18.4a and 18.4b, there are vacant energy states adjacent to the highest filled state at Ef. Thus, very little energy is required to promote electrons into the low-lying empty states, as shown in Figure 18.5. Generally, the energy provided by an electric field is sufficient to excite large numbers of electrons into these conducting states. For the metallic bonding model discussed in Section 2.6, it was assumed that all the valence electrons have freedom of motion and form an electron gas, which is uniformly distributed throughout the lattice of ion cores. Although these electrons are not locally bound to any particular atom, nevertheless, they must experience some excitation to become conducting electrons that are truly free. Thus

For insulators and semiconductors, empty states adjacent to the top of the filled valence band are not available.To become free, therefore, electrons must be promoted across the energy band gap and into empty states at the bottom of the conduction band. This is possible only by supplying to an electron the difference in energy between these two states, which is approximately equal to the band gap energy Eg. This excitation process is demonstrated in Figure 18.6.1 For many materials this band gap is several electron volts wide. Most often the excitation energy is from a nonelectrical source such as heat or light, usually the former.

For p-type semiconductors, the Fermi level is positioned within the band gap and near to the acceptor level. Extrinsic semiconductors (both n- and p-type) are produced from materials that are initially of extremely high purity, commonly having total impurity contents on the order of 107 at%. Controlled concentrations of specific donors or acceptors are then intentionally added, using various techniques. Such an alloying process in semiconducting materials is termed doping. In extrinsic semiconductors, large numbers of charge carriers (either electrons or holes, depending on the impurity type) are created at room temperature, by the available thermal energy. As a consequence, relatively high room-temperature electrical conductivities are obtained in extrinsic semiconductors. Most of these materials are designed for use in electronic devices to be operated at ambient conditions.

Four different types of band structures are possible at 0 K.In the first (Figure 18.4a), one outermost band is only partially filled with electrons. The energy corresponding 18.5 Energy Band Structures in Solids • 723 electron energy band Interatomic separation Individual allowed energy states 1s Electron state 2s Electron state 1s Electron energy band (12 states) 2s Electron energy band (12 states) Energy Figure 18.2 Schematic plot of electron energy versus interatomic separation for an aggregate of 12 atoms (N 12). Upon close approach, each of the 1s and 2s atomic states splits to form an electron energy band consisting of 12 states. JWCL187_ch18_719-780.qxd 10/28/09 8:53 PM Page 723 to the highest filled state at 0 K is called the Fermi energy Ef, as indicated. This energy band structure is typified by some metals, in particular those that have a single s valence electron (e.g., copper). Each copper atom has one 4s electron; however, for a solid composed of N atoms, the 4s band is capable of accommodating 2N electrons. Thus only half the available electron positions within this 4s band are filled. For the second band structure, also found in metals (Figure 18.4b), there is an overlap of an empty band and a filled band. Magnesium has this band structure

In addition to their ability to amplify an imposed electrical signal, transistors and diodes may also act as switching devices, a feature used for arithmetic and logical operations, and also for information storage in computers. Computer numbers and functions are expressed in terms of a binary code (i.e., numbers written to the base 2).

In all conductors, semiconductors, and many insulating materials, only electronic conduction exists, and the magnitude of the electrical conductivity is strongly dependent on the number of electrons available to participate in the conduction process. However, not all electrons in every atom will accelerate in the presence of an electric field. The number of electrons available for electrical conduction in a particular material is related to the arrangement of electron states or levels with respect to energy, and then the manner in which these states are occupied by electrons. A thorough exploration of these topics is complicated and involves principles of quantum mechanics that are beyond the scope of this book; the ensuing development omits some concepts and simplifies others

In demonstrating the Hall effect, consider the specimen geometry shown in Figure 18.20, a parallelepiped specimen having one corner situated at the origin of a Cartesian coordinate system. In response to an externally applied electric field, the electrons and/or holes move in the x direction and give rise to a current Ix. When a magnetic field is imposed in the positive z direction (denoted as Bz), the resulting force brought to bear on the charge carriers will cause them to be deflected in the y direction—holes (positively charged carriers) to the right specimen face and electrons (negatively charged carriers) to the left face, as indicated in the figure. Thus, a

In many practical situations the current is alternating (ac); that is, an applied voltage or electric field changes direction with time, as indicated in Figure 18.23a. Now consider a dielectric material that is subject to polarization by an ac electric field. With each direction reversal, the dipoles attempt to reorient with the field, as illustrated in Figure 18.33, in a process requiring some finite time. For each polarization type, some minimum reorientation time exists, which depends on the ease with which the particular dipoles are capable of realignment.A relaxation frequency is taken as the reciprocal of this minimum reorientation time.

Increasing the temperature of either a semiconductor or an insulator results in an increase in the thermal energy that is available for electron excitation.Thus, more electrons are promoted into the conduction band, which gives rise to an enhanced conductivity. The conductivity of insulators and semiconductors may also be viewed from the perspective of atomic bonding models discussed in Section 2.6. For electrically insulating materials, interatomic bonding is ionic or strongly covalent. Thus, the valence electrons are tightly bound to or shared with the individual atoms. In other words, these electrons are highly localized and are not in any sense free to wander throughout the crystal.The bonding in semiconductors is covalent (or predominantly

Inexpensive microelectronic circuits are mass produced by using some very ingenious fabrication techniques. The process begins with the growth of relatively large cylindrical single crystals of high-purity silicon from which thin circular wafers are cut. Many microelectronic or integrated circuits, sometimes called chips, are prepared on a single wafer. A chip is rectangular, typically on the order of 6 mm on a side, and contains millions of circuit elements: diodes, transistors, resistors, and capacitors. Enlarged photographs of and elemental maps of a microprocessor chip are presented in Figure 18.27; these micrographs reveal the intricacy of integrated circuits. At this time, microprocessor chips with densities approaching one billion transistors are being produced, and this number doubles about every 18 months

Intrinsic semiconductors are characterized by the electron band structure shown in Figure 18.4d: at 0 K, a completely filled valence band, separated from an empty conduction band by a relatively narrow forbidden band gap, generally less than 2 eV. The two elemental semiconductors are silicon (Si) and germanium (Ge), having band gap energies of approximately 1.1 and 0.7 eV, respectively. Both are found in Group IVA of the periodic table (Figure 2.6) and are covalently bonded.4 In addition, a host of compound semiconducting materials also display intrinsic behavior. One such group is formed between elements of Groups IIIA and VA, for example, gallium arsenide (GaAs) and indium antimonide (InSb); these are frequently called III-V compounds. The compounds composed of elements of Groups IIB and VIA also display semiconducting behavior; these include cadmium sulfide (CdS) and zinc telluride (ZnTe). As the two elements forming these compounds become more widely separated with respect to their relative positions in the periodic table (i.e., the electronegativities become more dissimilar, Figure 2.7), the atomic bonding becomes more ionic and the magnitude of the band gap energy increases—the materials tend to become more insulative. Table 18.3 gives the band gaps for some compound semiconductors

Ionic polarization occurs only in materials that are ionic. An applied field acts to displace cations in one direction and anions in the opposite direction, which gives rise to a net dipole moment. This phenomenon is illustrated in Figure 18.32b. The magnitude of the dipole moment for each ion pair pi is equal to the product of the relative displacement di and the charge on each ion, or

Let us now make an extrapolation of some of these concepts to solid materials. A solid may be thought of as consisting of a large number, say, N, of atoms initially separated from one another, which are subsequently brought together and bonded to form the ordered atomic arrangement found in the crystalline material. At relatively large separation distances, each atom is independent of all the others and will have the atomic energy levels and electron configuration as if isolated. However, as the atoms come within close proximity of one another, electrons are acted upon, or perturbed, by the electrons and nuclei of adjacent atoms. This influence is such that each distinct atomic state may split into a series of closely spaced electron states in the solid, to form what is termed an electron energy band. The extent of splitting depends on interatomic separation (Figure 18.2) and begins with the outermost electron shells, because they are the first to be perturbed as the atoms coalesce. Within each band, the energy states are discrete, yet the difference between adjacent states is exceedingly small. At the equilibrium spacing, band formation may not occur for the electron subshells nearest the nucleus, as illustrated in Figure 18.3b. Furthermore, gaps may exist between adjacent bands, as also indicated in the figure; normally, energies lying within these band gaps are not available for electron occupancy. The conventional way of representing electron band structures in solids is shown in Figure 18.3a.

Microelectronic circuits consist of many layers that lie within or are stacked on top of the silicon wafer in a precisely detailed pattern. Using photolithographic techniques, for each layer, very small elements are masked in accordance with a microscopic pattern. Circuit elements are constructed by the selective introduction of specific materials [by diffusion (Section 5.6) or ion implantation] into unmasked regions to create localized n-type, p-type, high-resistivity, or conductive areas. This procedure is repeated layer by layer until the total integrated circuit has been fabricated, as illustrated in the MOSFET schematic (Figure 18.26). Elements of integrated circuits are shown in Figure 18.27 and in the (a) chapter-opening photographs for this chapter.

Microelectronic circuits consist of many layers that lie within or are stacked on top of the silicon wafer in a precisely detailed pattern. Using photolithographic techniques, for each layer, very small elements are masked in accordance with a microscopic pattern. Circuit elements are constructed by the selective introduction of specific materials [by diffusion (Section 5.6) or ion implantation] into unmasked regions to create localized n-type, p-type, high-resistivity, or conductive areas. This procedure is repeated layer by layer until the total integrated circuit has been fabricated, as illustrated in the MOSFET schematic (Figure 18.26). Elements ofintegrated circuits are shown in Figure 18.27 and in the (a) chapter-opening photographs for this chapter.

On the other hand, the carrier concentration-temperature behavior for an extrinsic semiconductor is much different. For example, electron concentration versus temperature for silicon that has been doped with 1021 m3 phosphorus atoms is plotted in Figure 18.17. [For comparison, the dashed curve shown is for intrinsic Si (taken from Figure 18.16)].6 Noted on the extrinsic curve are three regions. At

Most polymeric materials are poor conductors of electricity (Table 18.4) because of the unavailability of large numbers of free electrons to participate in the conduction process. The mechanism of electrical conduction in these materials is not well understood, but it is felt that conduction in polymers of high purity is electronic.

Most polymers and ionic ceramics are insulating materials at room temperature and, therefore, have electron energy band structures similar to that represented in Figure 18.4c; a filled valence band is separated from an empty conduction band by a relatively large band gap, usually greater than 2 eV. Thus, at normal temperatures only very few electrons may be excited across the band gap by the available thermal energy, which accounts for the very small values of conductivity; Table 18.4 gives the room-temperature electrical conductivities of several

On occasion, it is necessary to improve the mechanical strength of a metal alloy without impairing significantly its electrical conductivity. Both solid-solution alloying (Section 7.9) and cold working (Section 7.10) improve strength at the expense of conductivity, and thus, a trade-off must be made for these two properties. Most often, strength is enhanced by introducing a second phase that does not have so adverse an effect on conductivity. For example, copper-beryllium alloys are precipitation hardened (Section 11.9); but even so, the conductivity is reduced by about a factor of 5 over high-purity copper

One of the most important electrical characteristics of a solid material is the ease with which it transmits an electric current. Ohm's law relates the current I—or time rate of charge passage—to the applied voltage V as follows: (18.1) where R is the resistance of the material through which the current is passing. The units for V, I, and R are, respectively, volts (J/C), amperes (C/s), and ohms (V/A). The value of R is influenced by specimen configuration, and for many materials is independent of current.The electrical resistivity is independent of specimen geometry but related to R through the expression

One variety of MOSFET8 consists of two small islands of p-type semiconductor that are created within a substrate of n-type silicon, as shown in cross section in Figure 18.26; the islands are joined by a narrow p-type channel. Appropriate metal connections (source and drain) are made to these islands; an insulating layer of silicon dioxide is formed by the surface oxidation of the silicon. A final connector (gate) is then fashioned onto the surface of this insulating layer.

Perhaps the best approach to an explanation of the phenomenon of capacitance is with the aid of field vectors. To begin, for every electric dipole there is a separation between a positive and a negative electric charge, as demonstrated in Figure 18.29. An electric dipole moment p is associated with each dipole as follows: (18.28) where q is the magnitude of each dipole charge and d is the distance of separation between them. In reality, a dipole moment is a vector that is directed from the negative to the positive charge, as indicated in Figure 18.29. In the presence of an electric field e, which is also a vector quantity, a force (or torque) will come to bear on an electric dipole to orient it with the applied field; this phenomenon is illustrated in Figure 18.30. The process of dipole alignment is termed polarization. Again, returning to the capacitor, the surface charge density D, or quantity of charge per unit area of capacitor plate (C/m2 ), is proportional to the electric field. When a vacuum is present, then (18.29) the constant of proportionality being 0. Furthermore, an analogous expression exists for the dielectric case; that is, (18.30) Sometimes, D is also called the dielectric displacement. The increase in capacitance, or dielectric constant, can be explained using a simplified model of polarization within a dielectric material. Consider the capacitor in Figure 18.31a, the vacuum situation, wherein a charge of Q0 is stored on the top plate and Q0 on the bottom one. When a dielectric is introduced and an electric field is applied, the entire solid within the plates becomes polarized (Figure 18.31c).As a result

Plastic deformation also raises the electrical resistivity as a result of increased numbers of electron-scattering dislocations. The effect of deformation on resistivity is also represented in Figure 18.8. Furthermore, its influence is much weaker than that of increasing temperature or the presence of impuritie

Polymeric materials have been synthesized that have electrical conductivities on par with metallic conductors; they are appropriately termed conducting polymers. Conductivities as high as 1.5 107 ( m)1 have been achieved in these materials; on a volume basis, this value corresponds to one-fourth of the conductivity of copper, or twice its conductivity on the basis of weight.

The magnitude of the dielectric constant for most polymers is less than for ceramics, because the latter may exhibit greater dipole moments: r values for polymers generally lie between 2 and 5. These materials are commonly used for insulation of wires, cables, motors, generators, and so on, and, in addition, for some capacitors

Solid materials exhibit an amazing range of electrical conductivities, extending over 27 orders of magnitude; probably no other physical property experiences this breadth of variation. In fact, one way of classifying solid materials is according to the ease with which they conduct an electric current; within this classification scheme there are three groupings: conductors, semiconductors, and insulators. Metals are good conductors, typically having conductivities on the order of 107 . At the other extreme are materials with very low conductivities, ranging between 1010 and 1020 ;these are electrical insulators. Materials with intermediate conductivities,generally from 106 to 104 , are termed semiconductors. Electrical conductivity ranges for the various material types are compared in the bar chart of Figure 1.7

The absorption of electrical energy by a dielectric material that is subjected to an alternating electric field is termed dielectric loss. This loss may be important at electric field frequencies in the vicinity of the relaxation frequency for each of the

The conductivity (or resistivity) of a semiconducting material, in addition to being dependent on electron and/or hole concentrations, is also a function of the charge carriers' mobilities (Equation 18.13)—that is, the ease with which electrons and holes are transported through the crystal. Furthermore, magnitudes of electron and hole mobilities are influenced by the presence of those same crystalline defects that are responsible for the scattering of electrons in metals—thermal vibrations (i.e., temperature) and impurity atoms. We now explore the manner in which dopant impurity content and temperature influence the mobilities of both electrons and hole

The conductivity of the channel is varied by the presence of an electric field imposed on the gate. For example, imposition of a positive field on the gate will drive charge carriers (in this case holes) out of the channel, thereby reducing the electrical conductivity. Thus, a small alteration in the field at the gate will produce a relatively large variation in current between the source and the drain. In some respects, then, the operation of a MOSFET is very similar to that described for the junction transistor. The primary difference is that the gate current is exceedingly small in comparison to the base current of a junction transistor. MOSFETs are, therefore, used where the signal sources to be amplified cannot sustain an appreciable current. Another important difference between MOSFETs and junction transistors is that although majority carriers dominate in the functioning of MOSFETs (i.e., holes for the depletion-mode p-type MOSFET of Figure 18.26), minority carriers do play a role with junction transistors (i.e., injected holes in the n-type base region, Figure 18.25).

The constant of proportionality is called the electron mobility, which is an indication of the frequency of scattering events; its units are square meters per voltsecond (m2 /V s). The conductivity of most materials may be expressed as

The current-voltage characteristics for forward bias are shown on the right-hand half of Figure 18.22. For reverse bias (Figure 18.21c), both holes and electrons, as majority carriers, are rapidly drawn away from the junction; this separation of positive and negative charges (or polarization) leaves the junction region relatively free of mobile charge carriers. Recombination will not occur to any appreciable extent, so that the junction is now highly insulative. Figure 18.22 also illustrates the current-voltage behavior for reverse bias.

The electrical conductivity of the semiconducting materials is not as high as that of the metals; nevertheless, they have some unique electrical characteristics that render them especially useful. The electrical properties of these materials are extremely sensitive to the presence of even minute concentrations of impurities.Intrinsic semiconductors are those in which the electrical behavior is based on the electronic structure inherent in the pure material. When the electrical characteristics are dictated by impurity atoms, the semiconductor is said to be extrinsic.

The energy state of such an electron may be viewed from the perspective of the electron band model scheme. For each of the loosely bound electrons, there exists a single energy level, or energy state, which is located within the forbidden band gap just below the bottom of the conduction band (Figure 18.13a). The electron binding energy corresponds to the energy required to excite the electron from one of these impurity states to a state within the conduction band. Each excitation event (Figure 18.13b) supplies or donates a single electron to the conduction band; an impurity of this type is aptly termed a donor. Because each donor electron is excited from an impurity level, no corresponding hole is created within the valence band. At room temperature, the thermal energy available is sufficient to excite large numbers of electrons from donor states; in addition, some intrinsic valence-conduction band transitions occur, as in Figure 18.6b, but to a negligible degree. Thus, the number

The junction transistor is composed of two p-n junctions arranged back to back in either the n-p-n or the p-n-p configuration; the latter variety is discussed here. Figure 18.24 is a schematic representation of a p-n-p junction transistor along with its attendant circuitry. A very thin n-type base region is sandwiched in between p-type emitter and collector regions. The circuit that includes the emitter-base junction (junction 1) is forward biased, whereas a reverse bias voltage is applied across the base-collector junction (junction 2)

The number of electrons excited thermally (by heat energy) into the conduction band depends on the energy band gap width as well as temperature. At a given temperature, the larger the Eg, the lower the probability that a valence electron will be promoted into an energy state within the conduction band; this results in fewer conduction electrons. In other words, the larger the band gap, the lower the electrical conductivity at a given temperature. Thus, the distinction between semiconductors and insulators lies in the width of the band gap; for semiconductors it is narrow, whereas for insulating materials it is relatively wide

The number of states within each band will equal the total of all states contributed by the N atoms. For example, an s band will consist of N states, and a p band of 3N states. With regard to occupancy, each energy state may accommodate two electrons, which must have oppositely directed spins. Furthermore, bands will contain the electrons that resided in the corresponding levels of the isolated atoms; for example, a 4s energy band in the solid will contain those isolated atoms' 4s electrons. Of course, there will be empty bands and, possibly, bands that are only partially filled. The electrical properties of a solid material are a consequence of its electron band structure—that is, the arrangement of the outermost electron bands and the way in which they are filled with electrons.

The prime objective of this chapter is to explore the electrical properties of materials, that is, their responses to an applied electric field. We begin with the phenomenon of electrical conduction: the parameters by which it is expressed, the mechanism of conduction by electrons, and how the electron energy band structure of a material influences its ability to conduct. These principles are extended to metals, semiconductors, and insulators. Particular attention is given to the characteristics of semiconductors and then to semiconducting devices. Also treated are the dielectric characteristics of insulating materials. The final sections are devoted to the peculiar phenomena of ferroelectricity and piezoelectricity

The rectification process in terms of input voltage and output current is demonstrated in Figure 18.23. Whereas voltage varies sinusoidally with time (Figure 18.23a), maximum current flow for reverse bias voltage IR is extremely small in comparison to that for forward bias IF (Figure 18.23b). Furthermore, correspondence between IF and IR and the imposed maximum voltage ( V0) is noted in Figure 18.22.

The temperature dependences of electron and hole mobilities for silicon are presented in Figures 18.19a and 18.19b, respectively. Curves for several impurity dopant contents are shown for both carrier types; furthermore, both sets of axes are scaled logarithmically. From these plots, note that, for dopant concentrations of 1024 m3 and below, both electron and hole mobilities decrease in magnitude with rising temperature; again, this effect is due to enhanced thermal scattering of the carriers. For both electrons and holes, and dopant levels less than 1020 m3 , the dependence of mobility on temperature is independent of acceptor/donor concentration (i.e., is represented by a single curve). Also, for concentrations greater than 1020 m3 , curves in both plots are shifted to progressively lower mobility values with increasing dopant level. These latter two effects are consistent with the data presented in Figure 18.18

The third type, orientation polarization, is found only in substances that possess permanent dipole moments. Polarization results from a rotation of the permanent moments into the direction of the applied field, as represented in Figure 18.32c. This alignment tendency is counteracted by the thermal vibrations of the atoms, such that polarization decreases with increasing temperature. pi qdi 18.20 Types of Polarization • 763 Electric dipole moment for an ion pair orientation polarization + + + + + + + + + + + + (a) No field Applied field (b) (c) + + + + + + + + + + Figure 18.32 (a) Electronic polarization that results from the distortion of an atomic electron cloud by an electric field. (b) Ionic polarization that results from the relative displacements of electrically charged ions in response to an electric field. (c) Response of permanent electric dipoles (arrows) to an applied electric field, producing orientation polarization. (From O. H. Wyatt and D. Dew-Hughes, Metals, Ceramics and Polymers, Cambridge University Press, 1974.) ionic polarization JWCL187_ch18_719-780.qxd 10/28/09 8:53 PM Page 763 The total polarization P of a substance is equal to the sum of the electronic, ionic, and orientation polarizations (Pe, Pi , and Po, respectively), or (18.34) It is possible for one or more of these contributions to the total polarization to be either absent or negligible in magnitude relative to the others. For example, ionic polarization will not exist in covalently bonded materials in which no ions are present.

The unique electrical properties of semiconductors permit their use in devices to perform specific electronic functions. Diodes and transistors, which have replaced old-fashioned vacuum tubes, are two familiar examples. Advantages of semiconductor devices (sometimes termed solid-state devices) include small size, low power consumption, and no warmup time. Vast numbers of extremely small circuits, each consisting of numerous electronic devices, may be incorporated onto a small silicon chip. The invention of semiconductor devices, which has given rise to miniaturized circuitry, is responsible for the advent and extremely rapid growth of a host of new industries in the past few years

Thus for this bias, large numbers of charge carriers flow across the semiconductor and to the junction, as evidenced by an appreciable current and a low resistivity.

To illustrate how extrinsic semiconduction is accomplished, consider again the elemental semiconductor silicon. An Si atom has four electrons, each of which is covalently bonded with one of four adjacent Si atoms. Now, suppose that an impurity atom with a valence of 5 is added as a substitutional impurity; possibilities would include atoms from the Group VA column of the periodic table (e.g., P, As, and Sb). Only four of five valence electrons of these impurity atoms can participate in the bonding because there are only four possible bonds with neighboring atoms. The extra nonbonding electron is loosely bound to the region around the impurity atom by a weak electrostatic attraction, as illustrated in Figure 18.12a. The binding energy of this electron is relatively small (on the order of 0.01 eV); thus, it is easily removed from the impurity atom, in which case it becomes a free or conducting electron (Figures 18.12b and 18.12c).

Transistors, which are extremely important semiconducting devices in today's microelectronic circuitry, are capable of two primary types of function. First, they can perform the same operation as their vacuum tube precursor, the triode; that is, they can amplify an electrical signal. In addition, they serve as switching devices in computers for the processing and storage of information. The two major types are the junction (or bimodal) transistor and the metal-oxide-semiconductor field-effect transistor (abbreviated as MOSFET)

Virtually all commercial semiconductors are extrinsic; that is, the electrical behavior is determined by impurities, which, when present in even minute concentrations, introduce excess electrons or holes. For example, an impurity concentration of one atom in 1012 is sufficient to render silicon extrinsic at room temperature.

When a voltage is applied across a capacitor, one plate becomes positively charged, the other negatively charged, with the corresponding electric field directed from the positive to the negative. The capacitance C is related to the quantity of charge stored on either plate Q by (18.24) where V is the voltage applied across the capacitor. The units of capacitance are coulombs per volt, or farads (F). Now, consider a parallel-plate capacitor with a vacuum in the region between the plates (Figure 18.28a). The capacitance may be computed from the relationship (18.25) where A represents the area of the plates and l is the distance between them. The parameter 0, called the permittivity of a vacuum, is a universal constant having the value of 8.85 1012 F/m. If a dielectric material is inserted into the region within the plates (Figure 18.28b), then

When an electric field is applied, a force is brought to bear on the free electrons; as a consequence, they all experience an acceleration in a direction opposite to that of the field, by virtue of their negative charge. According to quantum mechanics, there is no interaction between an accelerating electron and atoms in a perfect crystal lattice. Under such circumstances all the free electrons should accelerate as long as the electric field is applied, which would give rise to an electric current that is continuously increasing with time. However, we know that a current reaches a constant value the instant that a field is applied, indicating that there exist what might be termed frictional forces, which counter this acceleration from the external field. These frictional forces result from the scattering of electrons by imperfections in the crystal lattice, including impurity atoms, vacancies, interstitial atoms, dislocations, and even the thermal vibrations of the atoms themselves. Each scattering event causes an electron to lose kinetic energy and to change its direction of motion, as represented schematically in Figure 18.7.There is, however, some net electron motion in the direction opposite to the field, and this flow of charge is the electric current. The scattering phenomenon is manifested as a resistance to the passage of an electric current. Several parameters are used to describe the extent of this scattering; these include the drift velocity and the mobility of an electron. The drift velocity represents the average electron velocity in the direction of the force imposed by the applied field. It is directly proportional to the electric field as follows:

Within this framework, numbers are represented by a series of two states (sometimes designated 0 and 1). Now, transistors and diodes within a digital circuit operate as switches that also have two states—on and off, or conducting and nonconducting; "off" corresponds to one binary number state, and "on" to the other. Thus, a single number may be represented by a collection of circuit elements containing transistors that are appropriately switched.

a One mil 0.001 in. These values of dielectric strength are average ones, the magnitude being dependent on specimen thickness and geometry, as well as the rate of application and duration of the applied electric field. JWCL187_ch18_719-780.qxd 10/28/09 8:53 PM Page 758 where is the permittivity of this dielectric medium, which will be greater in magnitude than 0. The relative permittivity r, often called the dielectric constant, is equal to the ratio (18.27) which is greater than unity and represents the increase in charge storing capacity by insertion of the dielectric medium between the plates. The dielectric constant is one material property that is of prime consideration for capacitor design. The r values of a number of dielectric materials are contained in Table 18.5

a hole in the valence band; a free electron is not created in either the impurity level or the conduction band. An impurity of this type is called an acceptor, because it is capable of accepting an electron from the valence band, leaving behind a hole. It follows that the energy level within the band gap introduced by this type of impurity is called an acceptor state. For this type of extrinsic conduction, holes are present in much higher concentrations than electrons (i.e., p n), and under these circumstances a material is termed p-type because positively charged particles are primarily responsible for electrical conduction. Of course, holes are the majority carriers, and electrons are present in minority concentrations. This gives rise to a predominance of the second term on the right-hand side of Equation 18.13, or

although only a fraction are excited, this still gives rise to a relatively large number of free electrons and, consequently, a high conductivity

and is indicative of the ease with which a material is capable of conducting an electric current. The units for are reciprocal ohm-meters [ , or mho/m]. The following discussions on electrical properties use both resistivity and conductivity. In addition to Equation 18.1, Ohm's law may be expressed as (18.5) in which J is the current density, the current per unit of specimen area I/A, and e is the electric field intensity, or the voltage difference between two points divided by the distance separating them; that is, (18.6) The demonstration of the equivalence of the two Ohm's law expressions (Equations 18.1 and 18.5) is left as a homework exercise.

and subshells by letters (s, p, d, and f). For each of s, p, d, and f subshells, there exist, respectively, one, three, five, and seven states. The electrons in most atoms fill only the states having the lowest energies, two electrons of opposite spin per state, in accordance with the Pauli exclusion principle.The electron configuration of an isolated atom represents the arrangement of the electrons within the allowed states

aterial having a maximum conductivity along the direction of orientation. These conducting polymers have the potential to be used in a host of applications inasmuch as they have low densities, are highly flexible, and are easy to produce. Rechargeable batteries and fuel cells are currently being manufactured that employ polymer electrodes. In many respects these batteries are superior to their metallic counterparts. Other possible applications include wiring in aircraft and aerospace components, antistatic coatings for clothing, electromagnetic screening materials, and electronic devices (e.g., transistors and diodes).

copper, or twice its conductivity on the basis of weight. This phenomenon is observed in a dozen or so polymers, including polyacetylene, polyparaphenylene, polypyrrole, and polyaniline. Each of these polymers contains a system of alternating single and double bonds and/or aromatic units in the polymer chain. For example, the chain structure of polyacetylene is as follows: The valence electrons associated with the alternating single and double chainbonds are delocalized, which means they are shared among the backbone atoms in the polymer chain—similar to the way that electrons in a partially filled band for a metal are shared by the ion cores. In addition, the band structure of a conductive polymer is characteristic of that for an electrical insulator (Figure 18.4c)— at 0 K, a filled valence band separated from an empty conduction band by a forbidden energy band gap. These polymers become conductive when doped with appropriate impurities such as AsF5, SbF5, or iodine. As with semiconductors, conducting polymers may be made either n-type (i.e., free-electron dominant) or p-type (i.e., hole dominant) depending on the dopant. However, unlike semiconductors, the dopant atoms or molecules do not substitute for or replace any of the polymer atoms

covalent) and relatively weak, which means that the valence electrons are not as strongly bound to the atoms. Consequently, these electrons are more easily removed by thermal excitation than they are for insulators.

er atoms. The mechanism by which large numbers of free electrons and holes are generated in these conducting polymers is complex and not well understood. In very simple terms, it appears that the dopant atoms lead to the formation of new energy C C C C H H C H H C C H H Repeat unit C H H # 756 • Chapter 18 / Electrical Properties JWCL187_ch18_719-780.qxd 10/28/09 8:53 PM Page 756 bands that overlap the valence and conduction bands of the intrinsic polymer, giving rise to a partially filled band, and the production at room temperature of a high concentration of free electrons or holes. Orienting the polymer chains, either mechanically (Section 15.7) or magnetically, during synthesis results in a highly anisotropic material having a maximum conductivity along the direction of orientation

in which t , i , and d represent the individual thermal, impurity, and deformation resistivity contributions, respectively. Equation 18.9 is sometimes known as Matthiessen's rule. The influence of each variable on the total resistivity is demonstrated in Figure 18.8, a plot of resistivity versus temperature for copper and several copper-nickel alloys in annealed and deformed states. The additive nature of the individual resistivity contributions is demonstrated at 100 C. Influence of Temperature For the pure metal and all the copper-nickel alloys shown in Figure 18.8, the resistivity rises linearly with temperature above about 200 C. Thus, (18.10) where and a are constants for each particular metal. This dependence of the thermal resistivity component on temperature is due to the increase with temperature in thermal vibrations and other lattice irregularities (e.g., vacancies), which serve as electron-scattering centers.

induced charge from the dielectric ( Q or Q ) may be thought of as nullifying some of the charge that originally existed on the plate for a vacuum (Q0 or Q0).The voltage imposed across the plates is maintained at the vacuum value by increasing the charge at the negative (or bottom) plate by an amount Q , and the top plate by Q . Electrons are caused to flow from the positive to the negative plate by the external voltage source such that the proper voltage is reestablished.And so the charge on each plate is now Q0 Q , having been increased by an amount Q . In the presence of a dielectric, the surface charge density on the plates of a capacitor may also be represented by (18.31) where P is the polarization, or the increase in charge density above that for a vacuum because of the presence of the dielectric; or, from Figure 18.31c, P Q /A, where A is the area of each plate. The units of P are the same as for D (C/m2 ). The polarization P may also be thought of as the total dipole moment per unit volume of the dielectric material, or as a polarization electric field within the dielectric that results from the mutual alignment of the many atomic or molecular dipoles with the externally applied field e. For many dielectric materials, P is proportional to e through the relationship (18.32) in which case r is independent of the magnitude of the electric field. Table 18.6 lists the several dielectric parameters along with their units.

intermediate temperatures (between approximately 150 K and 475 K) the material is n-type (inasmuch as P is a donor impurity), and electron concentration is constant; this is termed the "extrinsic-temperature region".7 Electrons in the conduction band are excited from the phosphorus donor state (per Figure 18.13b), and because the electron concentration is approximately equal to the P content (1021 m3 ), virtually all of the phosphorus atoms have been ionized (i.e., have donated electrons). Also, intrinsic excitations across the band gap are insignificant in relation to these extrinsic donor excitations. The range of temperatures over which this extrinsic region exists will depend on impurity concentration; furthermore, most solid-state devices are designed to operate within this temperature range.

level is shifted upward in the band gap, to within the vicinity of the donor state; its exact position is a function of both temperature and donor concentration. p-Type Extrinsic Semiconduction An opposite effect is produced by the addition to silicon or germanium of trivalent substitutional impurities such as aluminum, boron, and gallium from Group IIIA of the periodic table. One of the covalent bonds around each of these atoms is deficient in an electron; such a deficiency may be viewed as a hole that is weakly bound to the impurity atom. This hole may be liberated from the impurity atom by the transfer of an electron from an adjacent bond as illustrated in Figure 18.14. In essence, the electron and the hole exchange positions. A moving hole is considered to be in an excited state and participates in the conduction process, in a manner analogous to an excited donor electron, as described earlier

nalogous to an excited donor electron, as described earlier. Extrinsic excitations, in which holes are generated, may also be represented using the band model. Each impurity atom of this type introduces an energy level within the band gap, above yet very close to the top of the valence band (Figure 18.15a). A hole is imagined to be created in the valence band by the thermal excitation of an electron from the valence band into this impurity electron state, as demonstrated in Figure 18.15b.With such a transition, only one carrier is produced—

of electrons in the conduction band far exceeds the number of holes in the valence band (or n p), and the first term on the right-hand side of Equation 18.13 overwhelms the second; that is, (18.16) A material of this type is said to be an n-type extrinsic semiconductor. The electrons are majority carriers by virtue of their density or concentration; holes, on the other hand, are the minority charge carriers. For n-type semiconductors, the Fermi

of these materials. (The electrical resistivities of a large number of ceramic and polymeric materials are provided in Table B.9, Appendix B.) Of course, many materials are used on the basis of their ability to insulate, and thus a high electrical resistivity is desirable. With rising temperature, insulating materials experience an increase in electrical conductivity, which may ultimately be greater than that for semiconductors.

of this polarization, there is a net accumulation of negative charge of magnitude Q at the dielectric surface near the positively charged plate and, in a similar manner, a surplus of Q charge at the surface adjacent to the negative plate. For the region of dielectric removed from these surfaces, polarization effects are not important. Thus, if each plate and its adjacent dielectric surface are considered to be a single entity, the

ometimes, electrical conductivity is used to specify the electrical character of a material. It is simply the reciprocal of the resistivity, o

operative dipole types for a specific material. A low dielectric loss is desired at the frequency of utilization. 18.22 DIELECTRIC STRENGTH When very high electric fields are applied across dielectric materials, large numbers of electrons may suddenly be excited to energies within the conduction band. As a result, the current through the dielectric by the motion of these electrons increases dramatically; sometimes localized melting, burning, or vaporization produces irreversible degradation and perhaps even failure of the material. This phenomenon is known as dielectric breakdown.The dielectric strength, sometimes called the breakdown strength, represents the magnitude of an electric field necessary to produce breakdown. Table 18.5 presented dielectric strengths for several materials

relative to the positive nucleus of an atom by the electric field (Figure 18.32a). This polarization type is found in all dielectric materials and, of course, exists only while an electric field is present

the influence of an electric field, the position of this missing electron within the crystalline lattice may be thought of as moving by the motion of other valence electrons that repeatedly fill in the incomplete bond (Figure 18.11). This process is expedited by treating a missing electron from the valence band as a positively charged particle called a hole. A hole is considered to have a charge that is of the same magnitude as that for an electron, but of opposite sign ( 1.6 1019 C). Thus, in the presence of an electric field, excited electrons and holes move in opposite directions. Furthermore, in semiconductors both electrons and holes are scattered by lattice imperfections.

voltage, termed the Hall voltage VH, will be established in the y direction. The magnitude of VH will depend on Ix, Bz, and the specimen thickness d as follows: (18.18) In this expression RH is termed the Hall coefficient, which is a constant for a given material. For metals, wherein conduction is by electrons, RH is negative and equal to (18.19) Thus, n may be determined, inasmuch as RH may be measured using Equation 18.18 and the magnitude of e, the charge on an electron, is known. Furthermore, from Equation 18.8, the electron mobility is just (18.20a) or, using Equation 18.19, (18.20b) Thus, the magnitude of may also be determined if the conductivity has also been measured. For semiconducting materials, the determination of majority carrier type and computation of carrier concentration and mobility are more complicated and will not be discussed here.

where A is a composition-independent constant that is a function of both the impurity and host metals. The influence of nickel impurity additions on the roomtemperature resistivity of copper is demonstrated in Figure 18.9, up to 50 wt% Ni; over this composition range nickel is completely soluble in copper (Figure 9.3a). Again, nickel atoms in copper act as scattering centers, and increasing the concentration of nickel in copper results in an enhancement of resistivity. For a two-phase alloy consisting of and phases, a rule-of-mixtures expression may be used to approximate the resistivity as follows: i V V (18.12) where the Vs and s represent volume fractions and individual resistivities for the respective phases.

where l is the distance between the two points at which the voltage is measured and A is the cross-sectional area perpendicular to the direction of the current. The units for are ohm-meters ( m). From the expression for Ohm's law and Equation 18.2, (18.3) Figure 18.1 is a schematic diagram of an experimental arrangement for measuring electrical resistivity

where n is the number of free or conducting electrons per unit volume (e.g., per cubic meter) and |e| is the absolute magnitude of the electrical charge on an electron (1.6 1019 C). Thus, the electrical conductivity is proportional to both the number of free electrons and the electron mobilit

where nI and DI represent, respectively, the valence and diffusion coefficient of a particular ion; e, k, and T denote the same parameters as explained earlier in the chapter. Thus, the ionic contribution to the total conductivity increases with increasing temperature, as does the electronic component. However, in spite of the two conductivity contributions, most ionic materials remain insulative, even at elevated temperatures.

Material Science 18

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