Electrical Conduction in Metals and Alloys (Electrical Properties of Materials) Part 3

Theory

Attempts to explain superconductivity have been made since its discovery in 1911. One of these theories makes use of the two-fluid model, which postulates superelectrons that experience no scattering, have zero entropy (perfect order), and have long coherence lengths, i.e., an area 1000 nm wide over which the superelectrons are spread. The London theory is semi-phenome-nological and dwells basically on the electrodynamic properties. The BCS theory (which was developed in 1957 by Bardeen, Cooper, and Schrieffer) is capable of explaining the properties of conventional superconductors reasonably well. However, it does not seem to satisfactorily interpret high-temperature (ceramic) superconductors. The BCS theory is quite involved. Phenomenological descriptions of the concepts leading to this theory are probably simplifications of the actual mechanisms which govern superconduction and may thus provide temptations for misleading conclusions. (As is so often the case in quantum mechanics, the mathematics is right—it is only our lack of imagination that holds us back from correctly interpreting the equations.) Nevertheless, a conceptual description of the BCS theory and its results is attempted.

One key to the understanding of the BCS theory is accepting the existence of a pair of electrons (Cooper pair) that has a lower energy than two individual electrons. Imagine an electron in a metal at T = 0 K (no lattice vibrations). This electron perturbs the lattice slightly in its neighborhood. When such an electron drifts through a crystal the perturbation is only momentary, and, after passing, a displaced ion reverts back into its original position. One can consider this ion to be held by springs in its lattice position, so that after the electron has passed by, the ion does not simply return to its original site, but overshoots and eventually oscillates around its rest position. A phonon is created.5 This phonon in turn interacts quickly with a second electron, which takes advantage of the deformation and lowers its energy. Electron 2 finally emits a phonon by itself, which interacts with the first electron and so on. It is this passing back and forth of phonons which couples the two electrons together and brings them into a lower energy state (Fig. 7.15). One can visualize that all electrons on the Fermi surface having opposite momentum and opposite spin (i.e., k " and —k #) form those Cooper pairs (Fig. 7.16), so that these electrons form a cloud of Cooper pairs which drift cooperatively through the crystal. Thus, the superconducting state is an ordered state of the conduction electrons. The scattering on the lattice atoms is eliminated, thus causing a zero resistance, as described similarly in Section 7.5.3 where we observed that ordering of the atoms in a crystal lattice reduces the resistivity.


One further aspect has to be considered. We just mentioned that the electrons of a Cooper pair have a lower energy than two unpaired electrons. Thus, the Fermi energy in the superconducting state may be considered to be lower than that for the nonsuperconducting state. This lower state is separated from the normal state by an energy gap, Eg (Fig. 7.17). The energy gap stabilizes the Cooper pairs against small changes of net momentum, i.e., prevents them from breaking apart. Such an energy gap of about10—4 eV has indeed been observed by impinging IR radiation on a superconductor at temperatures below Tc and observing an onset of absorption of the IR radiation.

Schematic of a Cooper pair.

Figure 7.15. Schematic of a Cooper pair.

Fermi sphere, Fermi surface, and Cooper pair in a metal.

Figure 7.16. Fermi sphere, Fermi surface, and Cooper pair in a metal.

Density of states, Z(E), versus electron energy in the superconducting state.

Figure 7.17. Density of states, Z(E), versus electron energy in the superconducting state.

An alternate method for measuring this gap energy is by utilizing the Josephson effect. The experiment involves two pieces of metal, one in the superconducting state and the other in the normal state. They are separated by a thin insulating film of about 1 nm thickness (Fig. 7.18(a)). A small voltage of proper polarity in the millivolt range applied to this device raises the energy bands in the superconductor. Increasing this voltage eventually leads to a configuration where some filled electron states in the superconductor are opposite to empty states in the normal conductor (Fig. 7.18(b)). Then the Cooper pairs are capable of tunneling across the junction similarly as described in Section 4.3. The gap energy is calculated from the threshold voltage at which the tunneling current starts to flow.

 Josephson junction (a) in the unbiased state (b) with applied voltage across the junction which facilitates tunneling in the indicated direction.

Figure 7.18. Josephson junction (a) in the unbiased state (b) with applied voltage across the junction which facilitates tunneling in the indicated direction.

In closing, we would like to revisit the electron-phonon coupling mechanism, which is believed to be the essential concept for the interpretation of superconduction, at least for metals and alloys. It has been explained above that in the normal state of conduction (above Tc) strong interactions between electrons and phonons would lead to collisions (or scattering of the electron waves), and thus to electrical resistance, whereas at low temperatures the same interactions would cause Cooper pairs to form and thus promote superconduction. This would explain why the noble metals (which have small electron-phonon interactions) are not superconducting. In other words, poor conductors in the normal state of conduction are potential candidates for high-Tc superconductors (and vice versa). Ceramic and organic superconductors fit into this scheme. Still, some scientists believe that phonons are involved in the coupling process only at very low temperatures (e.g., below 40 K). At somewhat higher temperatures, when phonons cause substantial scattering of the electrons, excitons (i.e., electron-hole pairs) may link electrons to form Cooper pairs, as suggested by A. Little for organic superconductors. Still other scientists propose resonating valence bonds as a coupling mechanism for high-Tc superconductors.

Thermoelectric Phenomena

Assume that two different types of materials (e.g., a copper and an iron wire) are connected at their ends to form a loop, as shown in Fig. 7.19. One of the junctions is brought to a higher temperature than the other. Then a potential difference, DV, between these two thermocouples is observed which is essentially proportional to the temperature difference, DT, where

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is called the thermoelectric power, or the Seebeck coefficient (after its inventor, T.J. Seebeck, a German physicist who discovered, in 1821, that a thermoelectric circuit like the one just described deflected a close-by compass needle).

Schematic representation of two thermocouples made of copper and iron which are brought in contact with each other (Seebeck effect).

Figure 7.19. Schematic representation of two thermocouples made of copper and iron which are brought in contact with each other (Seebeck effect).

A thermoelectric power of several microvolts per degree is commonly observed. As an example, the frequently used copper/constantan (Cu-45% Ni) combination yields about 43 mV/K. It has a useful range between _180 and +400°C. For higher temperatures, thermocouples of chromel (90%Ni-10%Cr) and alumel (95%Ni-2%Mn-2%Al) or platinum/ Pt-13%Rh (up to 1700°C) are available. Some semiconductors have See-beck coefficients that reach into the millivolt per degree range, that is, they are one or two orders of magnitude higher than for metals and alloys. Among them are bismuth telluride (Bi2Te3), lead telluride (PbTe), and silicon-30% germanium alloys.

Thermocouples made of metal wires are utilized as rigid, inexpensive, and fast probes for measuring temperatures even at otherwise not easily accessible places. Thermoelectric power generators (utilizing the above-mentioned semiconductors) are used particularly in remote locations of the earth (Siberia, Alaska, etc.). They contain, for example, a ring of thermocouples, arranged over the glass chimney of a kerosene lamp which is concomitantly used for lighting. The temperature difference of 300°C thus achieved yields electric power of a few watts or sometimes more, which can be used for radios or communication purposes. Heat produced by the decay of radioisotopes or by small nuclear reactors yields thermoelectric power for scientific instruments on the moon (e.g., to record moon quakes) and for relaying the information back to earth. In solar thermoelectric generators sunlight is concentrated by concave mirrors on thermocouples. Most of the above-described devices have an efficiency between 5 and 10%.

A reversion of the Seebeck effect is the Peltier effect: A direct electric current that flows through junctions made of different materials causes one junction to be cooled and the other to heat up (depending on the direction of the current); see Fig. 7.20(a). Lead telluride or bismuth telluride in combination with metals are frequently used. One particularly effective device for which temperature differences up to 70° C have been achieved is shown in Fig. 7.20(b). It utilizes n- and p-type semiconductors (see Section 8.3) in conjunction with metals. Cooling occurs on those junctions that are connected to the upper metal plate (1 and 2), whereas heat develops on the lower junctions 3 and 4. The heat on the lower plate is removed by water or air cooling. The above-quoted temperature drop can even be enhanced by cascading several devices, that is, by joining multiple thermoelectric refrigerators for which each stage acts as the heat sink for the next.

Thermoelectric refrigeration devices which make use of the Peltier effect. (a) Principle arrangement. (b) Efficient device utilizing p- and n-type semiconductors (see Section 8.3) in conjunction with metals.

Figure 7.20. Thermoelectric refrigeration devices which make use of the Peltier effect. (a) Principle arrangement. (b) Efficient device utilizing p- and n-type semiconductors (see Section 8.3) in conjunction with metals.

The thermoelectric effects can be explained by applying elements of electron theory as described in the previous sections: When two different types of conducting materials are brought into contact, electrons are transferred from the material with higher Fermi energy (EF) "down" into the material having a lower EF until both Fermi energies are equal. As a consequence, the material that had the smaller EF assumes a negative charge with respect to the other. This results in the above-mentioned contact potential between the materials. The contact potential is temperature-dependent. Specifically, when a material is heated, a substantial number of electrons are excited across the Fermi energy to higher energy levels. These extra electrons drift to the cold junction, which becomes negatively charged compared to the hot junction. The equivalent is true for the Peltier effect: The electrons having a larger energy (that is, those having a higher EF) are caused by the current to transfer their extra energy into the material having a lower Ef, which in turn heats up. Concomitantly, the material having a higher EF is caused to lose energy and thus becomes colder.

Galvanoelectric Phenomena (Batteries)

Primary Cells

The method of obtaining steady electricity, involving two different metals and an electrolyte, goes back to the famous experiment by Luigi Galvani, (an anatomy professor at the Italian City of Bologna), who observed in 1786 that legs from freshly killed frogs twitched, when connected to a copper hook and an iron railing. This experiment was explained in 1790 by Count Alessandro Volta, an Italian physics professor, who postulated that the chemical action of the bodily fluid of frogs and two different metals produced electricity. Based on Galvani’s observation, Volta combined a series of "galvanic cells" to make a battery6 which was named a voltaic pile. He utilized for his battery alternating stacked silver and zinc disks which were separated from each other by paper that was moistened with a salt solution. The more noble metal (e.g. copper and silver in the above-mentioned cases) provides the plus polarity of the galvanic cell (called the cathode), whereas the less noble metal (e.g. iron or zinc) is termed the negative pole or anode.

In 1836 Daniell introduced the copper/zinc galvanic cell in which the electrodes were immersed in sulfate solutions of their respective metals, see Fig. 7.21. To explain the mechanisms involved, the Cu/Zn cell is used as an example. We consider two half-cells which are separated by a semiperme-able membrane which allows the SO4 ions to pass freely. When a load is applied to the cell, an oxidation process occurs at the negative electrode which releases Zn ions into the ZnSO4 solution and provides electrons. As a consequence, the Zn electrode is eventually reduced in size. Concomitantly, the same number of electrons is accepted by the positive electrode (e.g. Cu) which gains in size by picking up Cu-ions from the copper sulfate solution. The pertinent reaction equations are thus as follows:

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The galvanic cell essentially "dies" when all the Zn metal is used up or the electrolyte is exhausted. It should be noted that most galvanic cells contain only one electrolyte.

Schematic representation of a copper-zinc galvanic cell.

Figure 7.21. Schematic representation of a copper-zinc galvanic cell.

The positive electrode of a Leclanche cell (invented ~1860) consists of a mixture of manganese dioxide and carbon powder packed around a carbon rod. A zinc container serves as the negative electrode. The electrolyte is a paste i.e. a watery solution of ammonium chloride and zinc chloride which are thickened by a swelling substance such as flour. This paste permeates a paper separator between the electrodes. The reaction equations are similar, as above with the modification that the electrons combine with the manganese dioxide (MnO2) and the water to form manganese oxide (Mn2O3) and hydroxide ions (OH-):

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(A secondary reaction yields ammonia (NH3) and water when the negative hydroxide ions combine with positive ammonium ions (NH4+) that form when ammonium chloride (NH4Cl) is dissolved in water.) The Leclanche cell, often also called a dry cell or carbon-zinc cell, is the inexpensive workhorse for general purpose applications, such as flashlights, toys, radios, tape recorders, and low power uses. It provides about 1.5 V at open circuit when new. Unfortunately, the electrolyte eventually corrodes the zinc container which could cause leakage of the electrolyte and damage to the device in which the battery is inserted. To prevent corrosion, the zinc has been amalgamated (up to one weight% mercury per cell) which is environmentally questionable when the battery is discarded into a land fill. Today most dry cells (including the alkaline cell, below) utilize corrosion inhibitors like indium, alloyed into zinc, or use ultra-pure Zn instead. Thus, in general these newer batteries can be disposed of after exhaustion without major environmental concerns.

The alkaline cells are in many respects similar to the carbon-zinc batteries with the exception that the negative electrode consists of porous zinc that, because of its larger surface, oxidizes more readily than a solid Zn electrode. Further, the electrolyte consists of highly caustic potassium hydroxide which is a better electron conductor and therefore allows larger currents. Alkaline batteries last about 5 to 8 times longer than Leclanche cells but cost somewhat more. The specific energy is about 20% to 30% higher than that for the Leclanche cell.

The mercury cell consists of a zinc anode, a mercury oxide cathode, and potassium hydroxide as electrolyte. Its main advantage is that the cell voltage remains constant during use. It is therefore primarily utilized for hearing aids and sensitive scientific instruments. Because of environmental concerns with disposal, some countries do not allow sale of mercury-containing batteries or have very stringent recycling requirements.

A somewhat different cell is the zinc-air battery which possesses an up to 5 times higher energy density compared to the devices discussed so far. The reason for this is that oxygen from the atmosphere is the reactant for one of the electrodes (the cathode), whereas in many other systems the oxidant must be contained (packaged) in the cell which adds weight. The overall chemical reaction is accordingly

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The anode consists of a jelly-type mixture of amalgated zinc powder and a highly conductive solution of KOH in water, (which serves as the electrolyte). The cathode is made of catalyzed carbon which reduces oxygen from the air. The air flows into the cell through small holes, drilled into the corrosion resistant nickel can. During shipping and storing, these holes are sealed by an adhesive tape to prevent air penetration. Shortly before service, the tape is removed which activates the cell. Once the tab is peeled off, the cell capacity reduces to 50% of its original value in 3-12 weeks, depending on cell size and temperature. With the seal in place, the cell can be stored for about 3 years. The nominal cell voltage is between 1.4 V and theoretically 1.65 V. Zinc-air batteries are used in hearing aids, medical devices, pagers, and film cameras. Their cost is relatively low.

Finally, the silver-oxide battery, (also called silver-zinc battery) has a 40% longer run time than lithium-ion batteries. It has an open potential of 1.86 V, and a high energy to weight ratio. The cost is, however, large due to the price of silver. The cathode consists of silver oxide and the anode is made of zinc. These electrodes are immersed in an electrolyte of KOH or NaOH. The overall chemical reaction is

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Silver-zinc batteries contain generally about 0.2% mercury to prevent zinc corrosion. However, mercury-free silver-oxide batteries are available since 2004. Silver-oxide batteries are used for button cells (hearing aids) and specialty space applications where price does not play a role.

All taken, primary batteries are designed for one-time use, that is, as a rule, they cannot be recharged and need to be discarded (Exceptions exist). The energy provided by them ranges from about $100 per KW-h (flashlight batteries) to $5,000 per KW-h for batteries used in watches or hearing aids. These figures compare to about $0.14 per KW-h for household currents.

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