Condensed-Matter Physics and Materials Research

BERTRAND I.HALPERIN

The discipline of physics is intimately connected with all aspects of materials science, including the synthesis of new materials, the characterization of their properties, and the development of new materials applications. In materials synthesis, for example, it is clear that complicated nonequilibrium growth processes or annealing processes are important in the formation of many new materials and that a physical understanding of these processes, at least at an empirical level, is essential for improving the production of such materials. It is equally clear that long-term progress in materials synthesis is, in turn, dependent on careful characterization of the materials that are being produced and that developments in experimental physics have greatly enlarged the scope and the accuracy of materials characterization.

Over the past quarter century, new and better materials have played, and will continue to play, an enormous role in expanding the horizons of physics itself. This emphasis reflects in part my need, as a physicist, to thus acknowledge my personal debt to the creators of new materials. However, I also believe that the needs of condensed-matter physicists for improved materials, the interest that physicists have shown in the work of materials developers, and the excitement that has accompanied the periodic discoveries of new physical phenomena in condensed-matter systems, have often been vital motivating forces in the development of new materials.

It is not possible here to discuss systematically even the highlights of recent developments in condensed-matter physics that owe their existence to materials science.1 Rather, this chapter is limited to a few illustrative examples. It necessarily excludes several of the most exciting developments. For example, there has been enormous progress in our ability to calculate from first principles the electronic energies of solids, including the energies



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Advancing Materials Research Condensed-Matter Physics and Materials Research BERTRAND I.HALPERIN The discipline of physics is intimately connected with all aspects of materials science, including the synthesis of new materials, the characterization of their properties, and the development of new materials applications. In materials synthesis, for example, it is clear that complicated nonequilibrium growth processes or annealing processes are important in the formation of many new materials and that a physical understanding of these processes, at least at an empirical level, is essential for improving the production of such materials. It is equally clear that long-term progress in materials synthesis is, in turn, dependent on careful characterization of the materials that are being produced and that developments in experimental physics have greatly enlarged the scope and the accuracy of materials characterization. Over the past quarter century, new and better materials have played, and will continue to play, an enormous role in expanding the horizons of physics itself. This emphasis reflects in part my need, as a physicist, to thus acknowledge my personal debt to the creators of new materials. However, I also believe that the needs of condensed-matter physicists for improved materials, the interest that physicists have shown in the work of materials developers, and the excitement that has accompanied the periodic discoveries of new physical phenomena in condensed-matter systems, have often been vital motivating forces in the development of new materials. It is not possible here to discuss systematically even the highlights of recent developments in condensed-matter physics that owe their existence to materials science.1 Rather, this chapter is limited to a few illustrative examples. It necessarily excludes several of the most exciting developments. For example, there has been enormous progress in our ability to calculate from first principles the electronic energies of solids, including the energies

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Advancing Materials Research of defects and surfaces.2 The importance of this work for the future of materials science should be obvious to readers of this volume. Other areas of research are excluded because they are discussed elsewhere in this volume. (See, for example, the discussion of icosahedral quasicrystals by Cahn and Gratias and the discussion of new techniques in surface science by Plummer et al.) The references cited in this chapter, and the experimental curves displayed in the text, are intended to be illustrative of their subjects. These are not necessarily the most important contributions in each case, and no attempt has been made to construct a balanced list of all the key references in the four subjects discussed. It is hoped that the references given will be sufficient to help the interested reader gain entry to the literature in these areas. This chapter presents four particular examples where materials science and physics have combined to advance our understanding of nature. The fractional quantized Hall effect and the field of heavy-electron systems are two cases where experiments on newly developed materials or structures have yielded results so surprising as to change our understanding of the behavior of interacting electron systems in certain conditions. The two remaining examples, research on novel forms of structural order and on quantum interference effects in electron transport in ultrasmall structures and disordered systems, illustrate the larger symbiotic relationship that exists between physics and materials science. Here we shall find examples where theoretical predictions of unusual properties have led to the experimental investigation of novel materials and structures, and where experimentation has confounded previous expectations. HEAVY-ELECTRON COMPOUNDS One of the most fascinating subjects in condensed-matter physics is heavy-electron compounds.3–9 The conduction electrons in these metallic compounds have effective masses of 100 to 1,000 times the free-electron mass as opposed to values of about 10 for transition metals. It has been said that the carriers in heavy-electron compounds behave more like protons or helium atoms than electrons! The huge effective mass is manifest directly in the large electronic specific heats, at low temperatures, and in the similarly large Pauli paramagnetic susceptibilities. The Fermi degeneracy temperature, which marks the onset of the low-temperature regime for the specific heat and various properties, is on the order of 100 K in many of these compounds as opposed to 10,000 K and higher in ordinary metals. These heavy-electron compounds show many other amazing properties as well. The table on page 133 lists selected heavy-electron compounds, together with their low-temperature specific-heat coefficients, γ=Cel/T. (The corresponding coefficient for free electrons would be about 1 mJ/mol-K2.) To illustrate the variety of properties of the heavy-electron compounds, we note

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Advancing Materials Research Low-Temperature Specific-Heat Coefficients of Selected Heavy-Electron Compounds Compound Low-Temperature Specific-Heat Coefficient (mJ/mol-K2) CeCu2Si2 1,100 UBe13 1,100 UPt3 450 U2Zn17 530 UCd11 840 NpBe13 ≳900 CeAl3 1,600 CeCu6 1,600 CePb3 200 ≲ γ ≲ 1,400 that the first three compounds are superconducting at low temperatures, the next three are magnetically ordered, the next two are neither superconducting nor magnetically ordered, whereas the last compound, CePb3, is antiferromagnetic but becomes superconducting in the presence of a sufficiently large magnetic field!3,5,8,10 The compounds UBe13 and UPt3 differ significantly from ordinary superconductors. They are generally believed to exhibit “triplet superconductivity,” or pairing in a state of odd parity, similar to that which occurs in liquid 3He.5,6,8,9 On the other hand, the superconductivity in CeCu2Si2 is thought to be of the conventional singlet (even-parity) type, as is found in all other known superconducting metals.11 The heavy-electron materials all contain elements such as cerium or uranium in which the outermost f-electron shell is partially filled. The properties of these compounds differ, however, from what was expected on the basis of our previous understanding of f-electron materials. The usual, naive picture is that the f-electrons are localized on individual atoms, so that they have infinite effective mass and only their spin degrees of freedom need be taken into account, and that there should exist, in addition, a band of itinerant electrons, constructed from atomic s and d orbitals, with an effective mass of order 10, as in ordinary transition metals. The primary effect of the weak interaction between the two types of electrons, in this picture, is to give a weak effective interaction between the spins, so that there will be magnetic ordering at low temperatures, generally in some kind of antiferromagnetic state. The spin degrees of freedom will give a large contribution to the specific heat, for temperatures in the vicinity of the magnetic-ordering temperature, but the spin contribution freezes out rapidly below this temperature, leaving only the much smaller contribution of the itinerant s and d electrons. Electrical conductivity by the s and d electrons would persist below the magnetic-ordering temperature and might be expected to increase as ordering reduces the scattering by the spins.

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Advancing Materials Research This description works well, in fact, for most known rare-earth metals and their conducting compounds. It does not apply, however, in the case of the heavy-electron compounds. Somehow, in these materials, the itinerant electrons acquire some of the characteristics of the massive f-electrons, and the localized spins do not exist as separate degrees of freedom, at sufficiently low temperatures. The specific-heat curve for UBe13 shown in Figure 1 is evidence that there exists only one type of electron in heavy-electron compounds. The solid curve is the specific-heat anomaly of an ordinary Bardeen-Cooper-Schrieffer superconductor, rescaled by a factor of 1,000 to match the value of the linear specific-heat coefficient just above the transition temperature Tc. If the electrons responsible for superconductivity were relatively light s and d electrons, while the background specific heat was given by independent spin degrees of freedom, the discontinuity at Tc would be much smaller than the value observed. Further indication of the subtlety of the heavy-electron materials is given by comparing the electrical resistances of the two superconductors UBe13 and UPt3 in the normal state (see Figure 2). Whereas the resistance of UPt3 decreases monotonically from room temperature down to the transition tem- FIGURE 1 Electronic specific heat, divided by temperature, of the heavy-electron superconductor UBe13. From Ott et al.7 Reprinted with permission.

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Advancing Materials Research FIGURE 2 Temperature dependence of the electrical resistivities of UBe13 and UPt3 between their superconducting transitions and 300 K. From Fisk et al.5 Reprinted with permission. perature, about 1 K, the resistance of UBe13 is observed to increase slowly down to a temperature of about 10 K, then rise more sharply to a maximum around 2.5 K before it decreases because of fluctuations associated with the superconducting transition at lower temperatures. The rising resistance in UBe13 is reminiscent of the Kondo effect produced by impurity spins in many dilute alloys, and it has led to the description of the uranium compound as a “Kondo lattice.” Indeed, it seems likely that something related to the Kondo effect is occurring in all of the heavy-electron compounds.8 However, a satisfactory theoretical understanding of these materials has not yet been achieved. In all cases the heavy-electron materials are difficult to work with, and it is difficult to produce samples that are chemically pure and free of defects. Therefore, experimental measurements of these compounds have been strongly dependent on improvements in the production of these materials. THE QUANTIZED HALL EFFECT The discovery of the quantized Hall conductance by von Klitzing, Dorda, and Pepper in 1980 was certainly one of the most surprising developments

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Advancing Materials Research in condensed-matter physics in recent years.12 The discovery two years later of the fractional quantized Hall effect by Tsui, Stormer, and Gossard13 was perhaps equally surprising to workers in the field and has caused, in some ways, an even greater revolution in our theoretical understanding. The original, integral quantized Hall effect, as well as the fractional quantized Hall effect, occurs in two-dimensional electron systems at low temperatures in strong magnetic fields. The original experiments of von Klitzing and co-workers12 were performed on high-quality silicon metal-oxide-semiconductor field effect transistors (MOSFETs). The fractional quantization is seen only in systems with unusually high mobility and was first observed in modulation-doped GaAs-GaAlAs heterojunctions, grown by molecular beam epitaxy techniques. The fractional effect has been subsequently reported in several other types of semiconductor heterojunctions and more recently has been observed in high-mobility silicon MOSFETs as well.14 In any case, the discovery of this effect was made possible only by the remarkable state of the art in materials science in the design and fabrication of semiconductor inversion layers. The experimental discovery of von Klitzing and co-workers was the existence of a series of plateaus in the Hall resistance RH, which satisfy the following simple equation to a high degree of accuracy: (1) where h is Planck’s constant, e is the electron charge, and n is an integer that varies from one plateau to another. Recent experiments have confirmed that the ratios of Hall conductances of various steps in silicon and GaAs inversion layers agree with the ratios of simple integers at a level of 1 part in 30 million,15 and it is believed that Equation (1) is exact, in the limit of zero temperature. For the ranges of magnetic field and carrier concentration where the Hall plateaus are observed, the voltage drop parallel to the current vanishes, in the limit of zero temperature, so that the current flows without any dissipation. We may recall that the Hall resistance is defined as the ratio Vy/Ix, where Ix is the electrical current flowing in the sample, and Vy is the voltage difference in the perpendicular direction, measured between two contacts on opposite edges of the sample. Under the conditions of the quantized Hall effect, the Hall resistance is independent of the precise shape or size of the sample. For the fractional quantized Hall effect, one sees in addition to the integer steps, plateaus in the Hall resistance where the integer n in Equation (1) is replaced by a simple rational fraction, denoted by v. Well-established plateaus have been observed at values equal to 1/3, 2/3, 4/3, 5/3, 2/5, and 3/5. There is indirect evidence of Hall plateaus at various other fractions with odd

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Advancing Materials Research denominators, including 7/3, 8/3, 3/7, 4/7, 4/9, and 5/9. There is no convincing evidence of a plateau at any fraction with an even denominator, although there have been reports of resistance anomalies that could be associated with such plateaus.14 The explanation for the integral quantized Hall effect, though subtle, can be formulated in a single electron picture, where the charge carriers move independently in the potential of random impurities, and electron-electron interactions have only the effect of a weak perturbation.16 The fractional quantized Hall effect, however, is entirely a manifestation of the interaction between the electrons in the layer. The explanation given by Laughlin in 1983, together with subsequent extensions, suggests that the carriers in the fractional quantized Hall effect have essentially entered a new state of highly correlated electron matter.17–21 The ground state of the electrons is an in-compressible liquid, the density of which is locked to the density of magnetic flux in the system. The elementary charged excitations from this state are quasiparticles whose electric charge is a fractional multiple of the electron charge17 and whose collective states resemble those that would be expected for particles obeying “fractional statistics,” intermediate between bosons and fermions.21,22 Although the basic details of the integral and fractional quantized Hall effect appear to be understood, there remain many quantitative questions as well as qualitative observations that are poorly understood.14 NOVEL FORMS OF STRUCTURAL ORDER One of the major areas of interest to condensed-matter physicists in recent years has been the study of novel forms of structural order. A few examples are given here. There has been much experimental and theoretical work in the area of incommensurate crystalline structures. Electronic charge-density wave structures, incommensurate with the atomic lattice but related to the dimensions of the Fermi surface in reciprocal space, have been studied extensively in layered transition-metal chalcogenides and also in quasi one-dimensional materials, including organic conducting salts and NbSe3.23,24 The latter material has been particularly fascinating because the charge-density wave is apparently unpinned from the crystal lattice on application of a relatively weak electric field, resulting in interesting nonlinear effects in the electrical transport. The theory of the transport process and the associated finite-frequency noise remains controversial.24 Incommensurate structures commonly occur when a physisorbed layer solidifies on a crystalline substrate, and similar incommensurate structures are found in intercalated materials.25,26 Incommensurate distortions also occur in certain conventional insulating materials. The most dramatic and surprising

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Advancing Materials Research incommensurate structure, however, is surely the newly discovered icosahedral quasicrystal, observed in rapidly quenched aluminum-manganese alloys and in several other systems27 (see Cahn and Gratias, in this volume). The subject of partially ordered systems is a particularly fascinating area of the study of structural order. It embraces the various liquid-crystal phases, superionic conductors, and so-called plastic crystals, all of which are intermediate between a liquid and a conventional solid.28–30 All these materials have some type of long-range structural order or broken symmetry, in contrast to the disorder of liquids, but the structural order is not enough to specify a unique equilibrium position for each of the constituent atoms. One partially ordered system that illustrates the richness of this field is the hexatic phase of smectic liquid crystals. The existence of this phase was proposed by Birgeneau and Litster31 in 1978 on the basis of theoretical work that Nelson and I did on two-dimensional systems.32 In 1981, Pindak, and co-workers33 found the hexatic phase experimentally through x-ray studies of a material known as n-hexyl-4′-n-pentyloxybiphenyl-4-carboxylate (650BC). The molecules in the hexatic phase are arranged in layers, but in the plane of the layers there is no long-range translational order of the positions of the molecules. Nevertheless, there is long-range order in the orientation of the bonds between neighboring molecules in the layers so that the material possesses sixfold anisotropy in the plane. In other words the hexatic phase has the anisotropy characteristic of a hexagonal crystal but lacks the translational order in the plane of the layers. These properties are manifest in the x-ray diffraction pattern by the appearance of six diffuse spots in the x-y plane, where an ordinary hexagonal crystal would have infinitely sharp Bragg peaks. Many interesting forms of order occur in phases that exist at surfaces or in very thin films, including suspended smectic films, adsorbates on crystal surfaces, and reconstructions of clean crystal surfaces.25,33 There are also forms of “induced order” that may be found at the surface of a bulk liquid or liquid crystal.34 These subjects interest condensed-matter physicists because of the problems posed by the greater importance of fluctuations in some two-dimensional systems than in the analogous three-dimensional systems. And, improvements in experimental techniques and in materials preparation have made many of these systems accessible for the first time. For example, the development of glancing-incidence x-ray diffraction, together with synchrotron x-ray sources, has given us a sensitive and powerful method to study order just inside the surface of a bulk material.34 Many types of nonequilibrium structures can also be properly characterized as novel forms of structural order, in particular, various macroscopic structures, such as dendrites and other complex forms of crystal growth, loose aggregate structures, and structures formed by spinodal decomposition.35–37 On the atomic level, the structure of glasses continues to be of great interest to condensed-matter physicists.38

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Advancing Materials Research However, the various novel forms of structural order pose problems in finding the correct description of different types of order and in understanding the phase transitions between them. Microscopic causes of the different types of order need to be understood to anticipate when particular types of order will occur. And, there are interesting problems concerning the classification of defects in various structures and the description of hydrodynamic modes and of other forms of time-dependent behavior. ELECTRICAL CONDUCTION IN ULTRASMALL STRUCTURES AND QUANTUM INTERFERENCE EFFECTS IN DISORDERED ELECTRON SYSTEMS The interest of ultrasmall structures for materials science and for condensed-matter physics is obvious. At a certain point, a small particle, a narrow wire, or a thin film can no longer be considered just a “small piece” of the constituent material but begins to acquire new properties because of the small size. Similarly, the properties of a composite material containing fine particles, wires, or layers may differ from those of the constituent substances. In the area of electrical conduction, there are particularly interesting finite-size effects, arising from the quantum-mechanical wave nature of the electrons in the material. These finite-size effects are also closely related to subtle quantum interference effects that occur even in bulk disordered electron systems and are related, in turn, to problems of metal-insulator transitions in disordered systems. The problems become even more subtle when there is a superconducting transition in the system under study. A few examples give some indication of the range of these effects. Figure 3 shows the electrical resistance per unit length of a small MOSFET channel plotted as a function of the gate voltage, which is a measure of the carrier concentration in the sample.39 The channel is 200 nm long by 50 nm wide, which is several times the wavelength of the conduction electrons in the inversion layer. There are several runs at each of the indicated temperatures, and although there is a certain amount of variation from run to run, it is clear that the dominant feature of the curves is the existence of an apparently random structure, which is intrinsic to the sample in question and reproducible from run to run. If different samples are examined, however, different structures are observed. This structure may be described as arising from a series of resonances, or more accurately, from the interference of different electron waves, that undergo multiple scattering and multiple reflections along different paths in the sample.40,41 Figure 4 shows the magnetoresistance of an evaporated aluminum film 20 nm thick. The curve labeled “1-D” was obtained from a narrow sample 0.5 µm wide and 500 µm long, and the curve labeled “2-D” was obtained from a sample that was macroscopic in width as well as length, deposited at the

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Advancing Materials Research FIGURE 3 Electrical resistance per unit length of a small MOSFET channel (200 nm long, 50 nm wide). Gate voltage VG controls carrier density sample. From Skocpol et al.39 Reprinted with permission. same time as the 1-D sample.42,43 Both samples are too large, relative to the wavelength of the electrons in the metal, to show a large, random dependence on the magnetic field strength, analogous to the structure in the MOSFET sample in Figure 3. The samples show a systematic dependence on the magnetic field strength, however, which can again be explained as a subtle interference effect between electron waves traveling along the various multiple-scattering paths in the sample. (This interference effect is frequently referred to as weak localization.44) The theoretical interpretation of these data, which is indicated by the solid-line fit to the data points, requires separate consideration of the rates of elastic scattering, inelastic scattering, and spin-orbit scattering, all of which play essential but differing roles in the phenomenon. The contribution of incipient superconducting fluctuations

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Advancing Materials Research FIGURE 4 Magnetoresistance of an evaporated aluminum film 20 nm thick, measured at temperature T=4 K. From Gordon.43 is negligible in these data but must be taken into account to understand the magnetoresistance at lower temperatures.42,43 Figure 5 shows the magnetoresistance of a drawn platinum wire 150 nm in diameter and 2.5 mm long at three temperatures in the range from 1.5 K to 4 K.45 These data again can be explained by the weak localization type of quantum interference effect. Despite the fact that the wire here is narrower than the 1-D sample in Figure 4, the interpretation in this case involves a three-dimensional (bulk) theory, which is a reflection of the difference in materials parameters between the platinum and aluminum samples. It is interesting to note that platinum wires as thin as 8 nm have been made by the same technique as the sample in Figure 5.46 In this procedure, the wire is drawn while it is encased in a silver supporting matrix, which is subsequently etched away. As yet, however, there have been no successful attempts to attach leads to these ultrathin specimens, so the electrical properties have not yet been measured. A quantum interference effect related to the weak localization effects discussed above was first observed in 1981 by Sharvin and Sharvin47 using a thin magnesium film deposited on a fine quartz fiber as shown in Figure 6. The electrical resistance of this sample was found to depend periodically on the magnetic flux through the hole in the magnesium cylinder in a fashion that is reminiscent of the classic Aharanov-Bohm effect. In this case, however, the observed period is equal to h/2e, which is half the normal flux quantum.48 More recently, measurements on small one-dimensional rings

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Advancing Materials Research FIGURE 5 Relative change in resistance of a drawn platinum wire, 150 nm in diameter, in parts per thousand, due to magnetic field H at the three indicated temperatures. From Sacharoff, Westervelt, and Bevk.45 Reprinted with permission. have shown a dependence on the flux through the hole in the ring with a period h/e, in addition to the period h/2e, as is illustrated in Figure 7.49,50 Figure 8 shows the current versus voltage characteristic of a superconducting amorphous tungsten-rhenium line 25 nm wide, 10 nm thick, and 1,000 nm long, which was fabricated using electron-beam lithography and the “contamination resist” technique.51,52 The three steps in the voltage are believed to arise from the entry of individual phase-slip centers into the FIGURE 6 Experimental configuration used by Sharvin and Sharvin47 to study quantum interference effects due to magnetic flux through hole in a wire.

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Advancing Materials Research FIGURE 7 (a) Magnetoresistance of a one-dimensional ring, with inner diameter 784 nm, made from a wire strip 41 nm wide, measured at temperature T=0.01 K. (b) Fourier power spectrum of (a). Peaks are at inverse periods e/h and 2e/h for the flux through the ring. Insert shows electron-micrograph of ring. From Webb et al.49 Reprinted with permission. destruction of superflow in the sample. Thus, one sees here direct evidence for still another kind of small-sample effect. Small-sample effects can appear in superconductivity experiments in cases where the relevant dimensions are almost macroscopic. Figure 9 shows the resistance transition of a two-dimensional array of superconducting-normal-superconducting (SNS) junctions, constructed with islands of lead on a normal copper film.53,54 The separation between the edges of adjacent islands is approximately 6 µm, and the distance between centers is 25 µm. Although the islands become superconducting below a temperature Tcs of approximately 7 K, the array as a whole can carry a supercurrent only below a temperature Tc of approximately 2.4 K. Throughout the intermediate temperature region

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Advancing Materials Research FIGURE 8 Current versus voltage for a superconducting amorphous tungsten-rhenium line 25 nm wide, 10 nm thick, and 1,000 nm long. From Chaudhari et al.51 Reprinted with permission. FIGURE 9 Resistance transition of a two-dimensional array of superconducting Pb islands on a normal-metal Cu film. Data points show voltage drop, for a fixed current of 10 µA, in the temperature range from the transition temperature Tcs of the islands, down to the transition temperature Tc for the cooperative superconductivity of the array. From Abraham.54

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Advancing Materials Research FIGURE 10 Portion of an array of superconducting junctions similar to that used for data in Figure 9. From Abraham.54 shown in Figure 9, the Josephson coupling energy between the neighboring islands of lead is comparable to, or smaller than, the thermal energy unit kT. The solid curve labeled “proximity-effect model” is the fit to a theory in which there is no phase coherence of the superconducting order-parameter between adjacent islands, because the Josephson coupling is very small compared to kT in this region. The curve labeled “vortex model” is a fit to a theory of the phase transition of a superconducting array, based on the theory of Kosterlitz and Thouless, for a two-dimensional superfluid or X-Y spin model.55 In this temperature range, near Tc, the Josephson coupling is comparable to kT, and there is a considerable amount of correlation between the phases of neighboring islands. However, the long-range coherence is still destroyed by the presence of a number of thermally excited vortex defects. It should be noted that when superconducting particles become sufficiently small, much smaller than the islands in the sample in Figure 9, quantum mechanical fluctuations may be large enough to destroy the superconductivity of a coupled array even at zero temperature. This situation has been much

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Advancing Materials Research discussed theoretically, and effects of such quantum fluctuations have been observed recently in experiments.56 A micrograph of a portion of a square array of SNS junctions, similar to the sample used in Figure 9, is shown in Figure 10. Figures 11 and 12 show, respectively, electron micrographs of the MOSFET device used to obtain the curves shown in Figure 3 and of a drawn platinum wire 8 nm in diameter. FIGURE 11 Narrow MOSFET channel of type used to obtain data in Figure 3. S and D denote source and drain contacts; other contacts are voltage probes, which divide channel into samples of various lengths. From Skocpol et al.39 Reprinted with permission.

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Advancing Materials Research FIGURE 12 Drawn platinum wire, 8 nm in diameter. The data in Figure 5 were obtained from a sample made by the same technique but 20 times larger in diameter. From Sacharoff, Westervelt, and Bevk.46 Reprinted with permission. ACKNOWLEDGMENTS In the preparation of this chapter, I have received extensive help from R. Westervelt, Z.Tesanovic, and C.Lobb. I have also benefited greatly from suggestions by H.Ehrenreich, W.Skocpol, R.Laibowitz, T.Geballe, J.D. Weeks, M.Kardar, D.R.Nelson, and P.C.Hohenberg. This work was supported in part by the Harvard Materials Research Laboratory and by National Science Foundation grant DMR 85–14638. NOTES 1.   For an overall view of current problems in condensed-matter physics, see Physics Through the 1990s: Condensed-Matter Physics (National Academy Press, Washington, D.C., 1986). 2.   See, for example, M.Schluter and L.J.Sham, Phys. Today 35, 36 (1982); M.L.Cohen, Phys. Scripta T1, 5 (1982); J.Callaway and N.H.March, in Solid State Physics, edited

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