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6 Defects ant! Diffusion INTRODUCTION The field of defects and diffusion in solid materials is concerned with the structure of possible Haws in otherwise homogeneous materials and with their observable consequences in the properties of materials. Of particular interest are point defects, which are fairly localized on an atomic scale; line defects, such as dislocations; and boundary defects, such as surfaces, stacking faults, and grain boundaries. Although it is one of the older fields in condensed-matter physics and materials sciences, it remains an attractive arena for the observation and description of new physical phenomena and therefore maintains an enduring interest of physicists, in addition to researchers with other scientific orientations. An indication of the vigor that defect concepts still possess is that they mold the physical understanding of many phenomena in appar- ently unrelated fields that are of interest in condensed-matter physics today. Several examples serve to make this point: Solitons made their appearance in defect physics as lattice disloca- tions. Coupled electron-lattice complexes have long been exemplified by self-trapped holes and other polarons. These ideas have recently been combined into the unexpected form of the coupled electron-lattice 127
128 A DECADE OF CONDENSED-MA TTER PHYSICS modes that form the soliton charge carriers that have excited interest in connection with the transport properties of conducting polymers such as polyacetylene. Two-dimensional structures have become the focus of much recent activity for their unusual characteristics with regard to melting and phase transitions. The melting of adsorbed overlayers, or equivalently of impurity planes intercalated in layered compounds, to form a hexatic "floating raft" phase, is described in current theory by the thermally activated dissociation of dislocation dipole pairs. Similarly, the theo- retical building blocks of the roughening transitions of surfaces and interfaces are just the steps and jogs of classical model surfaces. Textures in liquid crystals, which led to dislocation descriptions of defective crystals, echo in the topical description of structure both in solids that support incommensurate charge-density waves, through discommensuration structures, and in the structural characteristics of the magnetic phases of liquid The that exist only below 1 mK. Local tunneling systems have long been the model for inversion of molecules and for the pocket states of centers tunneling among equivalent configurations in crystals. They return in recent advances as the central characteristic causing the linear specific heat and dynamical echo phenomena of apparently all amorphous solids (metals, ceramics, and polymers) and many disordered solids (e.g., the ~ aluminas). Internal fric lion from stress-induced changes of defect structures, first understood for such classic systems as C in Fe and the damping of brass reeds, is now applied to the analysis of both backbone and side-chain effects in polymeric materials and even in natural carbonaceous materials such as coal and amber. While chosen primarily for their critical importance in research fields of current interest in solid-state physics, these examples do indicate the flow of seminal ideas back and forth between areas that are clearly physics and those that are not and the way in which new subfields have emerged from areas of defect physics. Because it links directly to practical materials, the field of defects and diffusion exhibits these interconnections to a high degree, and accordingly presents the great- est problems for concise summary. NEW FIELDS FROM OLD: AN EXAMPLE A wide variety of choice is available to exemplify the growth of new subfields from areas of defect physics. For example, the surfaces of crystals, and the faults, reconstruction, steps, and other configurations adopted by them, now form a separate field discussed in Chapter 7.
DEFECTS AND DIFFUSION 129 The evolution of new vigorous subfields will be exemplified here through brief descriptions of three fertile subfields of particle-beam irradiation of materials. Phase Microstructure and Phase Generation in Radiation Fields In the past few years it has become apparent that when energetic radiation produces atomic displacements in solid materials it not only generates defects, defect aggregates, dislocations, and voids but may also give rise to phases that are not present without prior exposure to radiation. Here we mean phases in the thermodynamic sense of spatially bounded regions with distinct compositions and/or physical properties and with abrupt interphase boundaries. These phases are truly radiation induced, in the sense that they are thermally unstable in the absence of radiation. They thus differ from radiation-enhanced phases, which are merely thermodynamically stable phases that un- dergo more rapid formation when assisted by radiation-induced mix- ~ng. Radiation-induced phases have been widely observed in binary and multicomponent alloys, including semiconductors. They occur also in insulating solids in the form of metal colloids in NaC1 or amorphous islands in crystalline quartz, for example. Several possible causes for their formation can be suggested. One is that the excess point defects produced and maintained by the radiation field may favor a state or solid phase different from the one that is stable in the absence of radiation. Examples of this are the formation of amorphous silicon, of highly supersaturated crystalline solid solutions, and of the disordered state of ordered alloys, all during irradiation. These states are retained after irradiation when the thermal reordering is sufficiently slow. It is not yet known whether systems exist for which a transformation from one crystalline phase to a different phase is made energetically favor- able solely because excess defects are present in either or both of the phases. An alternative mechanism involves the way that radiation- induced segregation typically redistributes the components of a solid on a microstructural scale of 10-3 to 10 ~m. There are two causes for this redistribution: first, persistent defect fluxes are set up during irradiation, and, second, certain components couple preferentially to the defect fluxes. Changes of composition then shift the system locally into a region of the phase diagram that differs from that occupied by the overall homogeneous alloy. A new radiation-modified equilibrium phase may then precipitate locally, or an existing equilibrium phase may dissolve.
130 A DECADE OF CONI>ENSED-MA TTER PHYSICS Surface and Near-Surface Probes Recent years have seen the development of important methods for the microstructural and microchemical analysis of the surface and near-surface constitution of solid materials. Particle-beam methods have so revolutionized analytical science in these areas that the first o 10,000 A (1 ~m) of a material structure can now be analyzed with a chemical sensitivity often approaching 1 part in 107, with a depth resolution of 100 A, or with a lateral resolution of 0.1 Em or better. Different techniques are complementary in depth, spatial, or chemical resolution; experts have therefore learned to use an arsenal of powerful new particle-beam methods to obtain detailed near-surface chemical analysis of materials structures. Ions with energy greater than 100 eV incident on a crystal penetrate the surface and dissipate their energy and momentum. By detecting the x rays caused by the collisions it is possible to detect trace impurities at the level of 1 in 108 in favorable cases. Collisions also eject surface atoms from the material as secondary atoms or charged ions. By detecting the surface chemical species as the material is sputtered away it is then possible to determine the original microchemistry of the material. The detection can be performed by secondary-ion mass spectrometry (SIMS), in which the secondary species are fed into an isotope-imaging mass spectrometer, or by Auger spectroscopy of core levels excited by an auxiliary electron beam. The resolution limit perpendicular to the surface is ~100 A. Lateral resolution is limited at present to about 1 lam in SIMS and 0.05 lam in Auger probes. In chemical sensitivity SIMS can often achieve a level of 1 part in 106 or better. By Auger methods the sensitivity is reduced to 1 part in 103, but the depth resolution may be improved to a few atomic layers since the Auger electrons from deeper layers are scattered and lost. Sputtering then allows a three-dimensional chemical map of the surface region to be acquired. The mixing that takes place through the top 100 A or so while atoms are being sputtered away causes a steady-state nonuniform subsurface concentration profile to develop even in a chemically uniform bulk material. In a second category of technique, the incident and detected fluxes involve the same species. The incident beam, usually H+ or He+, impinges on the surface. Part is reflected by Rutherford backscattering from atoms in the subsurface region and with an energy transfer that depends first on the target mass and second on the depth to which the
DEFECTS AND DIFFUSION 13 1 particle penetrates. The spectrum of reflected particle energies con- tains information about both depth and chemical structure that can be separated to obtain accurate chemical information with a depth reso- lution of about 100 A. Alternatively, channeling methods can be used to probe the position of an atomic species in the lattice structure. When the beam is directed along a crystallographic axis so that the particle range is long, atoms located off crystal lattice sites scatter particles into other channels and into the bulk material. In this way, the location of off-site atoms can be determined with some precision. It is also possible to measure the misfit at the heterojunctions between different crystals by this method. Ion-Beam Microfabrication As described in Appendix C, new types of crystals, compounds, alloys, microstructures, and other materials can bring with them formerly unimagined opportunities for scientific and technological advancement. This is particularly true of microfabrication, in which components are chemically and structurally tailored for particular application on a microscopic scale. Several important methods are based principally on ion-beam methods. Here we mention three areas of major effort: Ion implantation is the process in which foreign dopant or alloying ions are accelerated to energies of typically 1-300 keV and implanted into the near-surface regions of target materials. Implantation depths are typically of the order of 1 ~m, depending on the implantation energy, and the profiles are reasonably well understood. The compo- sitions achieved by implantation are not constrained by usual thermo- dynamic or kinetic limitations. As ions penetrate the solid they create lattice displacements, and the material is damaged by the implanting beam. Further manipulation of the resulting nonequilibrium structure is then often desirable. Ion-beam mixing often involves a surface layer, typically a few hundred angstroms thick, of one material being mixed with a bulk substrate of a second material by means of a penetrating ion beam. The advantage of this procedure is that higher concentrations of new alloy phases may be achieved using only a small fraction of the irradiation fluence normally needed for ion implantation. The effects of sputtering, cascade mixing, radiation-induced segregation, and radiation- enhanced diffusion remain important but are as yet imperfectly under- stood.
132 A DECADE OF CONDENSED-MA TTER PHYSICS Lithography has been used in the semiconductor industry for many years. It proceeds by damaging a chemically resistive material using a photon, electron, or ion beam and then preferentially etching away either the damaged or undamaged region and the substrate beneath it. Complex patterns with resolution of about 0.1 Am may be inscribed into semiconductors for device fabrication by these methods. CALCULATIONS OF DEFECT STRUCTURE A decade ago no reliable procedures were available by which accurate calculations of cohesion could be carried out for most solids. The exceptions to this statement were highly ionic solids and, to some degree, simple metals. It is now possible to make calculations that correctly indicate the small differences of relative energy among alternative bulk crystal structures of metals and covalent compounds. The energy of a surface and its electronic structure can be calculated quite well. A mainstay of defect calculations for two decades has been the modeling of crystal configurations using energies derived from model interactions among atoms. A large crystallite with appropriate bound- ary conditions is used. These methods have played a major role in the development of fundamental ideas about defect structure in simple materials. The practice for metallic systems has been to sum pairwise potentials, and this has some measure of validity although the model clearly lacks a rigorous basis. Nevertheless, qualitatively useful studies even of such complicated defects as impurities bound in mixed- dumbbell interstitials have been forthcoming. For ionic materials, elaborate codes have been developed to add appropriate treatments of core polarizability to the coulombic and core-core interactions, in order to simulate the total energy of a configuration more accurately. By whatever method, the energy as a function of configuration is finally computed, and the result provides the input for calculations of molec- ular dynamics and properties of relaxed point defects or surfaces, for example. If not highly precise, these methods can nevertheless often repro- duce systematic trends in data such as F-center excitation energies and Schottky pair energies with absolute values within 20 percent of those observed. Long experience, the systematic elimination of errors, and fine tuning of the codes have made the procedures reliable for ionic materials such as alkali halide and alkaline earth fluorides, which have large excitation energies and hence stable polarizabilities. Advances over the past decade include reasonable description of defect volumes
DEFEC TS A ND DI FF USI ON 1 3 3 (errors formerly led to large discrepancies) and entropies. A good understanding has been achieved for important model impurity prob- lems involving dopants, both with and without effective charges, in halide and fluoride structures. The case of rare-gas impurity properties in these lattices warrants special mention. Ongoing efforts seek to broaden applications of this general approach to less-ionic materials such as oxides, where added effects of covalency must be simulated. A powerful approach to the properties of metallic and covalent systems, mentioned in Chapter 1, is provided by methods derived from the density functional formalism. Normally this involves iteration of the one-electron Green's function to obtain a self-consistent electron density and hence the energy by integration over position. Various paths to these results employ frozen cores, pseudopotentials, or other strategies to avoid the atomic core. In applications to total energies of extended solids these methods have been highly successful: relative energies, and hence stabilities, of different structures are predicted with a precision that often is better than 0.1 eV. Note that the problem of calculating the total energy of the electron liquid from first principles is circumvented rather than solved in these approaches. Variational and Green's function Monte Carlo approaches to the specific problem of the electron liquid appear to offer feasible future routes. The difficulty in representing excited configurations, particularly those containing inhomogeneities such as charge localization, probably places serious limitations on the applications of the density functional method to areas of excited-state spectroscopy. At the time of writing, applications to the ground state of dilute impurity systems, both in metals and semiconductors, and including transition metal centers, have been completed to provide good insight into the local structure. Typically the approach is to use a small cluster consisting of the atom and its neighbors as a perturbation on the electronic structure of the perfect solid; this is then iterated through to self-consistency and the properties derived. Magnetic systems are treated using an Ansatz for the spin-polarized density functional. The impurity ground state is discussed more successfully than the excitation energy, particularly for deep levels. Vacancies have been treated as relatively simple defect centers in both semiconducting and metallic crystals. The problem of incorporating lattice relaxation into the calculation must be solved before the energies obtained can usefully be compared with experi- ment. An alternative approach, which has developed mostly in connection with insulators, is now finding important applications to metals and covalent materials. This is the unrestricted Hartree-Fock approxima
134 A DECADE OF CONDENSED-MATTER PHYSICS tion. A critical recent breakthrough has been the development of many-body perturbation theory to correct the Fock results for corre- lation by an expansion in pair excitations. Almost 90 percent of the correlation energy is returned by these methods in many problems, so practical calculations can be completed with chemical accuracy, or 0.1 eV. One important advantage of the Hartree-Fock approach is its ability to deal accurately with excited configurations. This is particularly the case for excited configurations that differ in symmetry from the ground state, so that the two remain unconnected by pair excitations in the many-body perturbation theory. FUNDAMENTALS OF ATOMIC MOBILITY Until recently, computer simulation using molecular dynamics has provided the sole method by which jump rates can be calculated quantitatively for a given defect in a model crystal in which atoms interact through a specified potential function. This method can be employed for model crystallites of 102-103 atoms, using periodic boundary conditions that minimize surface erects. Computer runs of 104-105 iterative steps necessarily involve only ~103 vibrational periods in order that the iteration can mimic smooth dynamics. Therefore, the method has been useful only when a number of jump events take place in 103 periods. This has limited investigations to fast-diffusing species and to high-temperature properties. Topics on which attention has focused include diffusion in liquids, in superionic conductors, and on surfaces. Each of these will be mentioned further in what follows. It should be emphasized that existing evaluations of statistical theories describing atomic jump rates have disagreed with the results of correct molecular-dynamics calculations for the identical model sys- tems. This has been a consequence of the insufficient accuracy in the statistical evaluation afforded, for example, by absolute rate theory (ART). For the jump frequencies encountered in real crystals it has been possible to perform dynamical simulations only at the highest temperatures. Consequently there has been an almost complete ab- sence of accurate theoretical information for ordinary materials about the way atomic jump processes are determined by the potential energy of interaction among the atoms. Much discussion has arisen over the past decade about the possible dependence of the energies, entropies, and volumes of migration on temperature and pressure, since they are sensitive to theoretically intractable derivatives of the jump rate with respect to these variables. The meaning of the isotope effect and its
DEFECTS AND DIFFUSION 135 dependence on thermodynamic coordinates have remained equally obscure. One recent advance in this area is to correct ART predictions for those nonrandom return jumps that are inherent in the dynamical system and that therefore cause an intrinsic error in the rate theory formulation. It turns out that in most crystals these amount to only 10 percent of the jumps, even at high temperature, so the correction is small. Apparently absolute rate theory alone is a relatively sound first . . . approximation In many cases. ART is nevertheless valid only for a limited range of jump problems. A second formulation, Brownian rate theory (BRT), is potentially useful under different circumstances from those for ART. In ART it is supposed that the system randomizes completely once a jump occurs; therefore ART requires correction for subsequent dynamical events that occur before randomization. In BRT the inertia of the jumping system is retained but introduces random motion into the remaining system, which causes a viscosity. No clear synthesis of these two approaches has appeared. The physical sense of BRT becomes appar- ent for problems such as adatom diffusion on a smooth surface where the adatom may move many lattice spacings before undergoing a significant collision with the lattice. Under these circumstances the assumption in ART of immediate randomization is incorrect. Diffusion occurs instead by long flights broken by collisions in a way that could conceivably be described by viscosity. No first principles evaluation of the damping has, however, been possible as yet for atomic migration. One striking application of the BRT has been to the breakaway of dislocations in a stress field. Here, thermal activation through pinning points plays a role, as does the inertia of the dislocation at its terminal velocity in the viscous field. An attractive feature of the phenomenon is that the viscous drag from the electronic system can be modified to an observable extent by the superconducting transition, which thus affects mechanical properties. For this example the model holds together in a semiquantitative way. The influence of quantum constraints on atomic jump processes has been the source of a large theoretical literature that has dwarfed the incidence of verifiable experimental reports citing observations of quantum effects. Well-established observations exist for tunneling of even quite heavy atoms among pocket states of asymmetric defect configurations; these results include, for example, off-center Cu sub- stituted in salts and, more recently, Zn-A1 mixed dumbbell interstitials in Al. These systems undergo tunneling transitions among equivalent configurations and have unmistakable signatures of symmetry and
136 A DECADE OF CONDENSED-MATTER PHYSICS tunnel splitting of the energy-level structure. An elegant system that still lacks adequate model treatment is the motion of exceedingly heavy self-interstitials in certain metals. In Pb, for example, the interstitial is observably mobile below 1 K, so it may in fact delocalize rapidly in the perfect crystal at O K. A possible complication in metals, not currently well understood, is coupling to the conduction electron excitation continuum, with excitation density characteristically proportional to excitation energy in the Fermi liquid. In the 1970s, the migration of light impurity atoms through crystals was modeled using polaron concepts to find regimes of multiphonon (thermally activated) hopping at high temperature, power-law few- phonon hopping at intermediate temperature, and (perhaps attainable) propagation at the lowest temperatures. Feynman path integral meth- ods are now thought to present a viable route for more accurate calculations of transition rates, particularly in combination with Monte Carlo treatments of the classical lattice modes. Extensive studies of the most promising systems, principally H interstitials in bee refractory metals, have seemed consistent with this modeling to some workers, although this is not fully accepted. Certainly, large isotope effects are measurable in mechanical relaxation and specific heat, for example. The situation is complicated by H trapping in tunneling levels at other, heavier interstitials and also at dislocations. True migration of a light particle at low temperatures in these materials has proved elusive. Quantum crystals over the best opportunities for examination of atomic mobility modified by quantum-mechanical requirements. Solid HI and Do are complicated by rotational transitions, and Ne, Ar, and CH4 are too heavy; the focus therefore falls almost entirely on the isotopes 3He and 4He. It is possible that exchange delocalizes vacan- cies in these materials and mixes them into the crystal ground state so that they are never absent in equilibrium, even at O K. Explicit measurements have revealed that the vacancy content of hop 4He and bee 3He is below 1 part in 104 as T ~ 0. The defect density is nevertheless sufficient to promote diffusion with a diffusion constant D ~ 10-6 cm' s-', characteristic of the melting temperature of more ordinary solids. It is further observed that the vacancy formation energy is equal to the activation energy for diffusion over a wide range of volumes in bee 3He. Evidently this structure has little or no barrier to migration, although the hop structure at lower molar volume does exhibit activated hopping. With the use of variational and Green's function Monte Carlo methods with model pairwise forces it is cur- rently possible to reproduce cohesive and structural properties of the heliums quite accurately. The possibility therefore exists that theory
DEFECTS AND DIFFUSION 137 may take the leading role in the exploration of these point-defect properties. Quantum crystals may also have unusual extended defects. For example, the liquid-solid interface shows exaggerated lateral mobility in certain circumstances. Also, melting waves are observed at per- turbed liquid-solid interfaces, owing to the high heat conduction permitted by the superfluid. Finally there is an expectation that dislocations may become delocalized. Measurements nevertheless show that the string model of dislocations, pinned at vacancies, describes the mechanical behavior down to temperatures of 1.5 K for solid 4He and to about 0.2 K for solid 3He. For both point and extended defects, therefore, the promise of remarkable defect behavior remains mostly to be verified in future work. COMMENTS ON ACTIVE AREAS What follows are brief descriptions of additional areas that seem particularly noteworthy in the light of past developments or potential future interest. More material concerning surfaces and interface prop- erties will be found in Chapter 7. Point Defects in Simple Solids Steady progress has been made over the past decade in collecting and interpreting values of the formation and migration properties of simple defects in prototypical solids. In some cases the known activa- tion energies, typically ~1 eV, have changed by less than 10 percent from values quoted two decades ago. The intervening years have nevertheless been filled with careful effort. In many cases, activation energies for self-diffusion and for vacancy formation have been mea- sured over large temperature ranges using new techniques. Curvatures of Arrhenius plots are now commonplace; the interpretation in terms of the temperature dependence of mechanisms, hopping parameters, or defect clustering nevertheless remains a difficult and unsettled area. The assistance of definitive theories will probably be required before these subtleties are resolved. To convey the difficulty, one may note that for many bee metals the Arrhenius plot for diffusion has been known for many years to exhibit strong curvature, yet the cause remains imperfectly resolved. One reasonable possibility is the anom- alous behavior of phonons in these metals. Persistent efforts are necessary in this area. The correct character- ization of defect processes in simple metals, salts, and valence solids is
138 A DECADE OF CONDENSED-MATTER PHYSICS essential if progress toward a predictive framework for complex materials of interest for engineering applications is eventually to be forthcoming. In this connection the realm of definitive information is now advancing from noble metals and alkali halides to refractory and other metals, covalent solids, and refractory oxides. This must be recognized as a major achievement of the field. An indicator of future progress is the healthy arsenal of techniques now applied to quantitative characterization of defect structure in metals. In the early 1970s, for example, the existence of the self- interstitial in Cu as a dumbbell configuration was established using ultrasonic attenuation, neutron scattering, and diffuse x-ray scattering. To these techniques have been added positron annihilation, muon spin rotation, perturbed angular correlation spectroscopy, various special NMR techniques, ion channeling, the Mossbauer effect, and other specialized probes. These advance far beyond the resistivity measure- ments and occasional specific heat and Bragg x-ray scattering experi- ments available before 1970, and the prospects for steady future progress are improved accordingly. Surface Diffusion Only at temperatures typically below half the melting temperature does the diffusion of atoms on clean surfaces usually resemble the site-to-site hopping of atoms in the bulk crystal. Most available information for lower temperatures pertains to refractory metals that can be cleaned in ultrahigh vacuum to secure reproducible results. Field ion microscopy provides a microscopic probe of adatom mobility and clustering in the low-mobility regime, and fluctuation spectroscopy of adatom Auger signals is a second, more macroscopic, probe. Reasonable Arrhenius behavior has been observed in a number of experiments covering limited temperature ranges; evidence for quan- tum behavior at low temperatures has recently been reported for H and D adsorbates. Phase transitions of the bulk and surface reconstructions are both expected to change the diffusion characteristics, but as yet these have not been investigated. Anisotropic diffusion is seen to occur, and interesting mechanisms have been deduced, for anisotropic surfaces containing ridges, for example. These have been simulated in computer dynamical calculations. At higher temperatures, simulations indicate fast surface diffusion. Moreover, so many mobile point defects and surface ledges are activated that the surface becomes quite rough and is subject to rapid fluctuations. Interestingly enough, the surface layer itself is observed
DEFECTS AND DIFFUSION 139 in simulations to exhibit liquidlike behavior below the melting point of the solid. These novel dynamics are, of course, confined to a skin on the crystal surface since the bulk cannot melt. None of these high- temperature processes has yet been observed for real crystals. Photochemical Processes It has been known for several decades that optical interband transitions in salts can create point defects. In alkali halides the products are H centers (a negative halogen molecule at an anion site) and F centers (an electron trapped at an anion vacancy). The excitor created in the optical event self-traps, and a nonradiative decay channel leads to the defect production. A number of similar processes warrant mention here. For example' the Jahn-Teller instability of the vacancy in Si transforms electronic recombination energy into the motion required to surmount the barrier to atomic migration; thus, excitation promotes fast migration. Another example is dislocation glide, which occurs at high excitation levels in solid-state quantum-well lasers made, for example, from GaAs and Ga'_~.Al`As. Yet a further example is photodesorption, in which the surface of a crystal absorbs a photon and an ion is subsequently ejected from the surface. It is believed that Auger processes convert what was formerly a negative ion into a positive ion, which then desorbs under the repulsive field of its neighbors. Angle-resolved effects from molecules oriented on the surface may be expected; site symmetries have been elucidated from the paths of desorbing ions. Photochemical processes of this type offer new opportunities for investigation. Interesting progress has been made in recent years using picosecond pulse-probe laser techniques to monitor the decay of excitors into F and H centers. It appears that a V`. center forms after an unresolvable short time and that it evolves into the H center. Nowadays, laser pulses can be created in the 10-fs (1o-~4 S) time domain, which is faster than most lattice vibrations. A special oppor- tunity therefore exists in the future to examine photochemical pro- cesses, including photon-induced point-defect migration, on the time scale of the atomic jump process itself. Molecular Dynamics Computer simulation of dynamical processes in solids has led to vivid insight into complex mechanisms, to the discovery of qualita- tively new processes, and to quantitative mimicking of processes that
140 A DECADE OF CONDENSED-MATTER PHYSICS occur in real solids. As mentioned above, molecular dynamics has served as the only source of numerous insights into the dynamics and stability of defects, for example in surface structure and diffusion. More recent successes that warrant mention here include the accu- rate treatment of ionic motion in fast ion conductors such as AgI, in which the Ag sublattice disorders, and in CaF', where the F sublattice undergoes large fluctuations involving defects and mobility. Model interatomic forces have reproduced observed diffusion rates quantita- tively. Also of major interest are dynamical studies of the early stages of precipitation. In these investigations the embryo is found to nucleate from the disordered state, possessing from the earliest stages the appropriate symmetry as a precursor of the eventual lattice structure. It is not clear what other approaches could possibly provide such direct access to these important phenomena. The future of computer simulation and dynamics contains research problems to match whatever complexity new generations of computers can handle. Direct modeling of the mechanical properties introduced by dislocations may require consideration of ~106 particles in place of today's 103. Bulk mechanical behavior, including grain-boundary structure, may require still more. Future dynamics programs may be coupled to fast quantum chemistry routines to replace pair forces by more realistic solid-state modeling of the crystalline potential energy. Surface treatments, including particle-beam mixing, the resulting nonequilibrium structure in alloys, and the influence of directed heat input by laser processing, all appear readily susceptible to investigation by simulation. It seems highly probable that applications of this type will ensure a significant future role for computer studies of defect properties. Dislocation Motion in Glasses The glide of dislocations in crystals has long been recognized as a determining factor in plastic behavior. It has recently been recognized that many of the same effects occur in glasses also, despite the fact that the geometrical characteristics are somewhat less clear owing to the amorphous structure of the solid. The experimental fact is that slip bands are observed after plastic How in certain metallic glasses. They occur as expected on planes defined by the maximum shear stress and, in general, resemble similar processes in crystals. Computer modeling of dislocations introduced into a Lennard-Jones glass shows that the core structure and the long-range elastic field remain stable after the atoms are allowed to relax. By way of comparison, a vacancy in the
DEFEC TS A ND DI FF USI ON 1 4 1 Lennard-Jones glass disappears into the structure when relaxation is permitted; however, bond models with forces that vary with bond angle can lead to stable vacancies. These initial discoveries establish that glassy materials can, in some cases, support defect structures resembling those in crystals. The area warrants further effort in the future to determine the range of phenom- ena that occur and the way in which such defects move. Defect Imaging at Atomic Resolution A notable development over the last decade has been the refinement of experimental methods that can image crystals and defect structures with spatial resolution at the atomic level. These are not universally applicable probes but instead generally require particular sample characteristics for successful detection of atomic locations and defect geometry. They are nevertheless extremely powerful techniques when handled well. Two such probes that are surface sensitive are the scanning vacuum tunneling microscope and the field ion microscope. These are described in Chapter 7. A third, the high-resolution electron microscope, can image defects on the surface or within the bulk of the crystal. Over the period from 1970 to 1982 the resolution of commercial transmission electron microscopes improved from 4 to about 1.5 A. Since the beam passes through the entire sample, which must therefore be quite thin (~1000 A thick), these methods are naturally adapted to linear and planar defects aligned with the beam. Vivid patterns of atomic distributions in the perfect crystal can be obtained for appro- priate thin films. Dislocation and defect structures at properly aligned grain boundaries can also be imaged. Early high-resolution successes concerned the planar defect structures of stacking faults and polytypes and the defects accommodating nonstoichiometry in oxides. More recent applications involve grain-boundary structures and the geome- try of heterojunctions, for example of semiconducting materials in device configurations (Figure 6.1~. In all cases the apparently clear imaging of atomic positions is at least partly illusory; careful theoretical modeling is required to obtain precise interpretation of the relevant diffraction processes. Most of the necessary theoretical machinery is now widely available. The past decade has also seen the parallel development of scanning transmission electron microscopes, which form an electron probe o focused to about 3 A. Images with atomic resolution are then formed by monitoring such properties as scattered intensity as the beam is
142 A DECADE OF CONDENSED-MATTER PHYSICS FIGURE 6.1 Atomic resolution image of the interface (lines) between Si (left) in the (110) projection and epitaxial NiSi2 (right), which has the fluorite structure. Each dark blob is the image of two projected rows of atoms in a sample about 100 A thick. (Courtesy of J. C. H. Spence, University of Arizona.) rastered in a suitable pattern. Heavy atoms located on carbon films have been imaged individually by these means; other direct uses at atomic resolution include the analysis of small precipitates; chemical analysis at high resolution can be completed by energy-loss methods. SOME DIRECTIONS FOR FUTURE RESEARCH A qualitative understanding of phase microstructure and phase generation in radiation fields is developing, but detailed model descrip- tions and even the basic theoretical framework remain largely to be developed. One phenomenon, radiation-induced homogeneous precip- itation in undersaturated solid solutions, has been described using a simplified quasi-thermodynamic theory. This is a research area in its infancy. It adds a new dimension to the currently active field of phase relations, phase transformations, and the stability of phases. At present there exist no precise methods by which the steady-state nonuniform substrate concentration profiles that develop while atoms are being sputtered away from the solid surfaces can be predicted
DEFECTS AND DIFFUSION 1 43 quantitatively, so accurate analysis awaits improved understanding of the damage, displacement, and diffusion processes in the subsurface region. The structures and energies of point defects are apparently coming within the grasp of ab initio theoretical calculations. Indeed, electronic aspects of calculations for point defects can already be treated compre- hensively, but the rather complicated problem of lattice relaxation and its effect on the electronic system is not generally tractable at present. As with density functional methods, the new capabilities provided by the use of the unrestricted Hartree-Fock approximation, corrected by the use of many-body perturbation theory, open a wide range of problems to future exploration, including chemical pathways, equilib- rium solid-state lattice geometries, and defect properties, although most investigations to date involve surface and adsorbate problems. It is probable that, over the next few years, accurate research on spectro- scopic applications in both pure and defective solids, but particularly when some degree of electronic localization occurs, will make use of this approach. For example, ab initio calculations on the F and FA center excitations of salts are apparently yielding excellent predictions. Recent results of cluster calculation on metals indicate that local electronic excitations of defects may be predictable to within an uncertainty of 0.1 eV. This accuracy is adequate for most practical purposes, so the oppor- tunities for new applications in the future appear particularly inviting. It is fairly clear that the discrepancies between the results of statistical theories and those of molecular-dynamics calculations for atomic jump rates in systems in which diffusion is occurring will be resolved over the next 5 years for model crystals with reasonably simple potential functions. Detailed properties of atomic jumps in model crystals, including isotope effects and thermodynamic deriva- tives of jump rates, will thus become accessible for the first time. Atten- tion will then focus on the ability of the assumed potential functions to simulate that of the real crystal, much as for defect formation. For the future it seems assured that atomic resolution investigations of defect structure in solids will continue to build momentum. Applica- tions of these methods to surface studies in ultrahigh vacuum currently remain in their infancy but hold extraordinary promise. The real-time recording of solid-state reactions at atomic resolution under controlled conditions can be expected to reveal a wealth of detailed mechanisms over the next decade. Therefore seminal contributions may be ex- pected in areas such as precipitation and phase transitions, where the atomic mechanisms and motions at the reaction front are of central interest.
