7
PROCESSING, FABRICATION, AND EVALUATION

INTRODUCTION

Materials processing and fabrication involve not only large-scale changes to materials associated with manufacturing (for example, solidification, sintering, joining, and mechanical modification) but also the creation of artificially structured advanced materials with properties tailored to meet specific criteria. This chapter presents an assortment (that is far from comprehensive) of materials processing, fabrication, and evaluation topics that involve applications of or offer research opportunities in the mathematical sciences. As mentioned in the first chapter, the committee organized this report's topics by various themes. Two topics, processing of semiconductor chips and nonlinear optical (NLO) materials, involve considerable common ground; since the topic of NLO materials does not quite fit with the themes of the previous chapter, it has been included in this one. For additional processing, fabrication, and evaluation background and topics, and a description of general materials research challenges, see Chapter 4 and Appendix B of National Research Council (1989); see also Friedman et al. (1992a) and Szekely (1993).

PROCESSING OF SEMICONDUCTOR CHIPS

The first step in the processing of semiconductor chips is the production of crystals. Typically one inserts a seed crystal into a silicon melt and slowly extracts it from the melt. Under appropriate temperature and dynamic conditions, the crystal grows (Czochralski's growth) as it is pulled out. Other methods of growing crystals are the Bridgman and float-zone methods. The mathematical description of the model is based on the Navier-Stokes equation with a free surface, the boundary of the melt. See Langlois (1985) and Jones (1988); see Brown (1988) for an excellent review article on crystal growth processing; also see the section Morphological Stability in Chapter 4.

The crystals used for producing semiconductor chips are cylindrical with a diameter of 8 to 20 cm. Each crystal is sliced into discs or wafers of a few millimeters thickness. On each wafer, many rectangular chips with side lengths between 0.1 mm and 1 cm will be produced. With present technology, there can be up to several million "devices" on each finished chip.

Laying out the devices on the chip consists of several "photolithographic" steps. These include (Friedman et al., 1992a): (1) oxidization of a thin layer of the silicon wafer; (2) coating uniformly with photoresist, a polymeric fluid sensitive to light; (3) covering the wafer by a patterned mask and exposing to light (the photoresist that is exposed to light "develops"; that is, it polymerizes, or hardens); (4) washing away uncovered areas of the photoresist layer, thereby leaving the silicon oxide layer bare; (5) etching out the bare portions of the silicon dioxide layer, down to the silicon substrate, by hydrofluoric-base acid that attacks neither the hardened photoresist nor the silicon; and (6) stripping off the



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Mathematical Research in Materials Science: Opportunities and Perspectives 7 PROCESSING, FABRICATION, AND EVALUATION INTRODUCTION Materials processing and fabrication involve not only large-scale changes to materials associated with manufacturing (for example, solidification, sintering, joining, and mechanical modification) but also the creation of artificially structured advanced materials with properties tailored to meet specific criteria. This chapter presents an assortment (that is far from comprehensive) of materials processing, fabrication, and evaluation topics that involve applications of or offer research opportunities in the mathematical sciences. As mentioned in the first chapter, the committee organized this report's topics by various themes. Two topics, processing of semiconductor chips and nonlinear optical (NLO) materials, involve considerable common ground; since the topic of NLO materials does not quite fit with the themes of the previous chapter, it has been included in this one. For additional processing, fabrication, and evaluation background and topics, and a description of general materials research challenges, see Chapter 4 and Appendix B of National Research Council (1989); see also Friedman et al. (1992a) and Szekely (1993). PROCESSING OF SEMICONDUCTOR CHIPS The first step in the processing of semiconductor chips is the production of crystals. Typically one inserts a seed crystal into a silicon melt and slowly extracts it from the melt. Under appropriate temperature and dynamic conditions, the crystal grows (Czochralski's growth) as it is pulled out. Other methods of growing crystals are the Bridgman and float-zone methods. The mathematical description of the model is based on the Navier-Stokes equation with a free surface, the boundary of the melt. See Langlois (1985) and Jones (1988); see Brown (1988) for an excellent review article on crystal growth processing; also see the section Morphological Stability in Chapter 4. The crystals used for producing semiconductor chips are cylindrical with a diameter of 8 to 20 cm. Each crystal is sliced into discs or wafers of a few millimeters thickness. On each wafer, many rectangular chips with side lengths between 0.1 mm and 1 cm will be produced. With present technology, there can be up to several million "devices" on each finished chip. Laying out the devices on the chip consists of several "photolithographic" steps. These include (Friedman et al., 1992a): (1) oxidization of a thin layer of the silicon wafer; (2) coating uniformly with photoresist, a polymeric fluid sensitive to light; (3) covering the wafer by a patterned mask and exposing to light (the photoresist that is exposed to light "develops"; that is, it polymerizes, or hardens); (4) washing away uncovered areas of the photoresist layer, thereby leaving the silicon oxide layer bare; (5) etching out the bare portions of the silicon dioxide layer, down to the silicon substrate, by hydrofluoric-base acid that attacks neither the hardened photoresist nor the silicon; and (6) stripping off the

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Mathematical Research in Materials Science: Opportunities and Perspectives hardened photoresist. Devices are fabricated by incorporating dopant impurities either by direct diffusion or ion implantation and annealing. During subsequent processing, further diffusion takes place. Many of the steps described above have been modeled to various degrees of accuracy by partial differential equations that describe the relevant physics. Modeling of diffusion of impurities in a nondeforming single crystal is done today with considerable sophistication that can take into account point defect interactions (Cole et al., 1990; Rorris et al., 1991). The theory of diffusion in a polycrystal, where diffusion occurs mostly along the grain boundaries (Tseng et al., 1992), and in the presence of external stresses, is not, however, adequately developed. Stresses and elastic deformations have long been investigated by applied mathematicians and mechanical engineers. There is a need to combine expertise and develop methods for the cross-coupled problems of diffusion and deformation in complex materials. Modeling oxidation of silicon (Chin et al., 1983; Peng et al., 1991) poses problems of moving boundaries and the generation of plastic flow by stresses. Stresses are inherent in the process because oxidation yields an oxide that would normally occupy more than twice the volume of the silicon that is consumed. This volume expansion leads to flow and a shape change. In turn, the stresses affect the rate of oxidation, the diffusivity of oxygen, and so on. Self-consistent solutions are needed. Under high stresses and high temperatures, the oxide behaves like an incompressible fluid. To date, there is no completely satisfactory mathematical model for calculating the pressure during incompressible fluid flow. There are, however, promising mathematical developments for simpler models (Tayler and King, 1990; King, 1993). Other areas of semiconductor processing have been modeled to different degrees of sophistication, but major problems remain to be solved. Examples include the modeling of etching (Kuiken, 1990), of the exposure of photoresist (Gerber, 1988), and of warping caused by titanium silicide growth (Willemsen et al., 1988; Friedman and Hu, 1992). Overall, modeling of processing steps in microelectronics currently has substantial gaps, especially in the areas of deposition, etching, and lithography. The works referenced above constitute a starting point. Considerable amounts of information are generated by atomic-scale theories, but this knowledge is yet to be folded into a theoretical description at the appropriate mesoscopic length scales of process modeling. There are various physical processes, not well understood, that contribute to variability of the performance of semiconductors. These include dislocations that develop during doping, diffusion of dopants in polycrystals, and elastic strain created during the various steps of depositions. One highly developed theory that seems relevant to semiconductor performance variability is the theory of dislocations (Hirth and Lothe, 1982; Eshelby, 1975). The analysis of one dislocation loop has been studied numerically (Borucki, 1993) and mathematically (Friedman et al., 1992b). However, nucleation of dislocations and the dynamics of the aggregate of dislocations remain open problems that mathematical sciences research could help to solve. There has been a surge of mathematical analysis in the modeling of semiconductor devices, with increasing attention to small devices. A few representative recent papers are Poupaud (1993), Schmeiser and Unterreiter (1993), and Ward et al. (1993).

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Mathematical Research in Materials Science: Opportunities and Perspectives AMORPHOUS SEMICONDUCTORS In polysilicon, the silicon grains are arranged in a random polygonal structure. A typical size of a grain is 1,000 to 10,000 Å, whereas the distance between two neighboring silicon atoms is 2 to 3 Å. In amorphous polysilicon, the atoms are typically in groups of 4 to 6 atoms. In order to pacify the dangling bonds, hydrogen is added (about 20%). Thin-film transistors fabricated from hydrogenated amorphous silicon are now used in flat panel displays such as in fax machines. The mathematical model is an extension of the equations for a silicon semiconductor device; it includes diffusion equations for the trapped holes and electrons, which depend on the density-of-states and the occupation functions (Shaw et al., 1991). There are various experimental power laws suggested by simulations (Shaw and Hack, 1988), but rigorous mathematical justification is needed. CASTING The process of producing an object of a desired shape through the injection of fluid into a mold followed by the solidification of the fluid within the mold is called casting. Casting is a phenomenon that involves two phases, solid and liquid, and the transition between them. There are various theories modeling the phase transition. Perhaps the best known is the Stefan problem formulation that describes the transition from solid to liquid by a sharp boundary. This interface is "free" in the sense that its location in space varies in time and it is one of the unknowns of the problem. There is a considerable amount of mathematical literature dealing with Stefan problems (Kinderlehrer and Stampacchia, 1980; Luckhaus, 1990; Friedman, 1992); see also the discussion of dendrites in Chapter 4 and the references therein, and the section Macroscopic Scale in Chapter 8. Other models consider the boundary between solid and liquid to be a "mushy" region (addressed in the section Mushy Zones in Chapter 4). In order to gain a better understanding of the mushy region where both solid and liquid phases coexist, a molecular theory was developed by J. W. Gibbs; see, for example, Gibbs (1961). In this theory, an order parameter φ is used. The temperature T, together with φ, satisfy a nonlinear parabolic system with various unspecified parameters. As these parameters go to certain limits, one formally obtains a version of the Stefan problem with surface tension appearing in one of the free-boundary conditions (Caginalp, 1989). This new problem is under current mathematical investigation (Chen and Reitich, 1990; Luckhaus, 1990), and it yields more realistic results regarding dendritic growth during the casting process. Nonetheless, this approach does not explain more complicated but important phenomena such as second-order dendritic growth; see also the subsection Dendritic Growth in Chapter 4. Further mathematical models and ideas are needed to tackle this problem. A good review of mathematical modeling for ingot solidification, continuous casting, and rapid solidification is given in Szekely (1990).

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Mathematical Research in Materials Science: Opportunities and Perspectives POLYMER PROCESSING The processing of polymer materials involves mixing, reaction, and flow in the liquid state and shaping in the solid state. Some of these issues are addressed in Chapter 3. Mathematical analysis and computation have played major roles in understanding polymer processing; see references in the Entanglements, Reptation, and Elasticity section of Chapter 3, and the recent reviews in Denn (1988) and Larson (1992). Asymptotic methods are addressed in Pearson (1985), and computational issues are investigated in Pearson and Richardson (1983) and Tucker (1989). The simulation of non-Newtonian fluid flow, heat transfer, and phase-change in a deformation field have received the greatest attention. Outstanding mathematical sciences research issues in these contexts are summarized in Chapter 3. OTHER PROCESSING There are many other materials processing settings where opportunities for mathematical sciences research exist. Optical fiber processing involves several steps that offer challenges: soot deposition (Chen, 1989), sintering (Sherer, 1979), and fiber drawing (Meyers, 1989). In particular, the fiber-drawing step is modeled as a free-boundary problem for a fluid with vorticity. This situation has been recently analyzed for a somewhat idealized flow (Liu, 1993). Another context presenting mathematical research challenges is electrochemical machining. Here, metal is removed from an anodically polarized workpiece via erosion caused by an electrical discharge that is produced in an electrolytic solution. The process, used for smoothing surfaces and for drilling holes in hard materials, is a focus of ongoing research (McGeough and Rasmussen, 1990). In this setting, the mathematical formulation of the shape of the desired product is a free boundary; see Lacey (1990). MIXING Mixing processes are ubiquitous in materials technology. Mixing can involve two or more fluids, often of very different viscosities, and granular or powdered solids. Mixing processes are often accompanied by a breakup of droplets and other fluid microstructures, aggregation of colloidal particles, and diffusion-controlled reactions in complex flows. The range of mixing practices has been surveyed in a number of texts and monographs, for example, in Harnaby et al. (1985). For centuries, developments in the field of mixing were empirical. Recent developments spurred by results in dynamical systems theory are providing a foundation for various aspects of the subject, particularly for slow flows of the type encountered in blending of viscous liquids; see the overviews and references in Ottino (1989, 1990) and in Ottino et al. (1992). There are significant differences between standard studies of dynamical systems and issues of concern in mixing. The studies usually focus on long-time or asymptotic behavior,

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Mathematical Research in Materials Science: Opportunities and Perspectives for example, convergence to attractors in dissipative systems, while the desire for rapid mixing places interest on short-time behavior. It is desirable to have as much chaos as possible, and so perturbations must be large, limiting the usefulness of analytical tools based on small perturbations away from integrability. Finally, global behavior is much more important in mixing than in ordinary studies of dynamical systems. One of the most important problems in mixing is to devise suitable measures for the intuitive concept of ''goodness of mixing.'' A related issue is to connect mixing measures to the fluid mechanics of the mixing process. Such connections can be investigated using tools inspired by the concepts of scaling and multifractals as well as using the statistics of multiplicative processes with weakly correlated steps. The obstacles to a purely computational attack on problems involving chaotic flows are well illustrated by the example of mixing of two highly reactive substances. A direct simulation of the coupled mixing-diffusion-reaction process might be impossible even in flows where the velocity field is relatively simple. The interface of reaction becomes nearly impossible to track when the stretching ratio exceeds 105, and if the reactions are diffusion controlled, the important processes are those within the striations (that is, at the smallest scales) and are lost with coarse graining. MATHEMATICAL MODELING IN QUANTITATIVE NONDESTRUCTIVE EVALUATION Nondestructive evaluation (NDE) is just one example from the class of inversion problems, such as those posed by x-ray and neutron diffraction and electron microscopy, arising from characterization or evaluation techniques of tremendous importance in materials science. Over the last decade, one of the most significant advances in NDE has been the evolution of NDE from a conglomeration of empirical techniques to a well-defined field of interdisciplinary science and engineering. NDE is an inverse problem that is generally ill posed. In the course of this development it has become well recognized that NDE should be based on quantitative models of the measurement processes of the various inspection techniques. The principal purpose of a model is to predict the response of the measurement system to specific anomalies in a given material or structure, for example, cracks, voids, distributed damage, corrosion, and deviation in material properties from specifications. A good model includes the configuration of the probe and the component under inspection as well as a description of the generation, propagation, and reception of the interrogating energy. For ultrasonics, this description requires computation of the transducer radiation pattern, refraction of the beam at surfaces, the beam profile, and the propagation characteristics in the host material, including effects of material anisotropy, attenuation, and diffraction losses. Detailed modeling of field-flaw interactions, which generate the response function of the measurement system, must also be included as well as information on conditions that produce statistical noise and add uncertainty to the results. A well-constructed model should be able to predict specific responses of instruments to anomalies in complex materials and structures as well as to "standard" flaws placed in various calibration blocks.

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Mathematical Research in Materials Science: Opportunities and Perspectives A number of models have been formulated in the past several years for different inspection techniques (Gray et al., 1989). For practical applications, the challenge lies in making approximations that permit the computations to be tractable while retaining sufficient accuracy so that the engineering applications are not compromised. Numerical results based on a reliable model are very helpful in the design and optimization of efficient testing configurations. A good model is also indispensable in the interpretation of experimental data and the recognition of characteristic signal features. Parametric studies can be carried out with relative ease for models, which facilitates an assessment of the probability of detection of anomalies. A model of the measurement system is a virtual requirement for the development of an inverse technique based on quantitative data. Last but not least, a model, the accuracy of which has been tested by comparison with experimental data, provides a practical way of generating a training set for a neural network or a knowledge base for an expert system. An essential component of a measurement system model for quantitative ultrasonics is the modeling of the interaction of ultrasound with a defect. Of special interest are crack-like defects. A mathematical crack is a surface in a solid body that cannot transmit surface tractions (that is, surface tension from, for example, adhesive friction). Under the influence of incident ultrasound, a crack becomes a surface of displacement discontinuity. Considerable progress has been achieved in the mathematical formulation and solution of crack-scattering problems (Achenbach, 1992). The direct problem is well understood, but further progress is required in the inverse problem of determining the size, shape, and orientation of a crack from the scattered field. FUNCTIONALLY GRADIENT MATERIALS Functionally gradient materials (FGMs) are nano-composites, alloys, and intermetallics that are macroscopically homogeneous but have continuously varying (rather than constant) microstructure with special thermomechanical properties. FGMs were conceived as "smart" or "intelligent" improvements over layered materials capable of enduring the steep temperature gradients, protracted exposure to high temperatures, and highly oxidizing environments that Earth-to-orbit winged planes face (Gandhi and Thompson, 1992; Rogers, 1989; Travis, 1993; Nanavati and Fernandez, 1993). Skin temperatures on such aerospace planes reach 2000 K, while the temperature just below the skin may be 1000 K cooler (Niino and Maeda, 1990). Layered materials, such as materials with a ceramic coating to protect a metallic substrate or with directly bonded homogeneous layers, have major disadvantages here: high thermal and residual stresses and poor bond strength (Houck, 1988). However, the recently developed technique of intentionally grading the composition of an interfacial region (Kerrihara et al., 1990) or entire coating (Kawasaki and Watanabe, 1990) seems effective in overcoming such disadvantages. Composition grading also improves the fracture toughness, fatigue, and corrosion crack resistance of thermal barrier coatings (Yamanouchi et al., 1990). Continuous graded composition occurs naturally in interfacial regions of many diffusion-bonded materials (Shiau et al., 1988) and occurs in certain deposition techniques such as plating, sputtering, and plasma spray coating

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Mathematical Research in Materials Science: Opportunities and Perspectives (Houck, 1988). FGMs are macroscopically modeled as inhomogeneous continua, while microscopically they are two-or three-phase composites with continuously varying composition made up of piecewise homogeneous regions. They can be analyzed as inhomogeneous isotropic (linear or nonlinear) elastic solids with known property distributions, including composite media microstructure specifications of desired thermomechanical properties. Inhomogeneous medium problems in potential theory and continuum solid mechanics can be formulated mathematically as a system of partial differential equations with variable coefficients (Olszak, 1958). The mathematical opportunities lie in finding sufficiently general methods for solving specific boundary value problems. Even in the simplest cases (for example, one-dimensional wave and diffusion equations, and two-dimensional equations of potential theory), general methods do not exist. To date, the most comprehensive solution method is based on an inverse method whereby the general (two-dimensional) second-order partial differential equation with variable coefficients is reduced to a system of first-order nonlinear Ricatti ordinary differential equations (Varley and Seymour, 1988). A particularly important mathematical challenge is to develop alternatives to the present approach that, in order to achieve analytically tractable solutions, uses a simple function to represent material nonhomogeneity despite this assumption possibly producing results that are not physically meaningful (Kassir and Chauprasert, 1974). However, see Bakirtas (1980) for a modification that yields some meaningful results. Another mathematical challenge is developing homogenization techniques for the estimation of the effective thermomechanical properties of FGMs from given composition profiles and micromechanical parameters. NONLINEAR OPTICAL MATERIALS Nonlinear optics (NLO) emerged as an independent field in the early 1960s after the invention of the solid-state laser (Shimoda, 1986). NLO encompasses a host of phenomena related to the nonlinear interaction of laser radiation with matter as well as applications and technology based on these phenomena (National Research Council, 1992). Many NLO effects are already understood, but recent advances in both laser technology and materials science offer exciting opportunities for new properties and new devices, which in turn offer new mathematical challenges. Some examples of NLO effects are frequency conversions (such as higher harmonics, generation of sum and difference frequencies, optical parametric oscillations, and stimulated Raman scattering), nonlinear refractive indexes and related self-action effects (such as self-focusing, self-trapping, self-bending, optical bistability, solitons in fibers), nonlinear absorption (in particular, saturable and multiphoton absorption), photorefractivity and related phase-conjugation effects, and diverse multiwave mixings. Most of these effects can be enhanced by resonances between radiation frequencies and quantum transitions in the material. A partial list of current NLO materials applications includes such frequency conversion; control, steering, rectification, and restoration of laser beams; radiation protection and optical limiting; all-optical switching; logic and computer memory operations; general optical signal

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Mathematical Research in Materials Science: Opportunities and Perspectives processing; and soliton-based optical fiber communications. The most promising NLO materials are found in the following areas (Auston et al., 1987): (1) bulk materials (mostly semiconductors, glasses and semiconductor-doped glasses), (2) multiple-quantum wells, (3) photorefractive materials, (4) liquid crystals, (5) inorganic frequency-conversion materials, and (6) organic and polymeric materials. Bulk materials, particularly solid-state crystals, are classic NLO materials for such applications as frequency conversion. In 1965, two mathematicaians discovered the soliton (Zabusky and Kruskal, 1965). Soliton pulses in NLO fibers were first predicted in the early 1970s (Hasegawa and Tappert, 1973) on the basis of soliton solutions of the nonlinear Schrödinger equation that had been discovered a short time earlier (Zakharov and Shabat, 1972). These solutions are some of the finest examples of the contributions of the mathematical sciences to NLO; highly transparent fibers can now propagate soliton pulses for distances exceeding 1000 km. They also exemplify the potential for real-world application of mathematical sciences research in NLO: an undersea trans-Atlantic cable that was recently developed by AT&T Bell Laboratories uses solitons as information carriers. The mathematical sciences can assist in formulating optimal material parameters for fibers used for soliton propagation. The mathematical sciences can provide better theoretical understanding of nonlinear distortions produced by higher-order dispersion and nonlinearity terms and by relaxation (proven to generate the so-called red shift in soliton frequency). A fundamental challenge concerns the "quantum" theory of solitons. Bulk semiconductors are some of the most universal nonlinear optical materials; under proper conditions they can demonstrate almost any possible nonlinear optical effect and can be used for applications from 0.3-to 12-µm wavelengths with a variety of materials systems. The nonlinear phenomenon in bulk semiconductors that has attracted most of the attention in the last decade is the nonlinear refractive index and related effects. For the next decade, the challenges in bulk semiconductors are to develop better physical understanding of the interplay of different nonlinearity mechanisms (Trillo et al., 1986; Fork et al., 1987) and, in applications, to reduce switching energy and switching time. The latter problem involves the ability to cycle these devices at rates of multigigabits per second. Good mathematical models will be required to describe these processes. In the 1980s, many quantum-mechanical calculations of semiconductor nonlinear susceptibilities were done, in particular for generation of higher optical harmonics. Some of the recent calculations agree well with experimental data, but all of the calculations are valid only for static situations—they fail to predict the temporal dynamics of the materials. A major mathematical challenge is to develop a dynamic theory of transient processes in NLO semiconductor systems (and in quantum well devices, which are discussed in the next paragraph), especially for nonlinear refractive indexes. Thus far, the nonstationary models that have been developed have been quite simple. Quantum well structures (QWSs), and multiple QWSs (MQWSs), are semiconductor structures with two-dimensional carrier confinement. They are fundamental to laser sources and have lately received much attention; for example, enhanced QWS exciton resonances led to the development of self-electrooptic effect devices (SEEDs). It has been demonstrated that quantum well structures, and devices based on them, can be used as logic gates and surface emitting lasers. The theory of QWSs and MQWSs involves a great deal

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Mathematical Research in Materials Science: Opportunities and Perspectives of modeling. It combines many-body electron-hole physics with nonlinear propagation effects, and results in nonlinear coupled partial integrodifferential equations, such as the so-called Maxwell-Semiconductor Bloch equations (MSBEs). The general approach to describing optical properties and many-body processes in such highly excited semiconductors is via nonequilibrium Green's functions and projection operator techniques. Both formalisms yield the relevant set of semiconductor Bloch and Boltzmann equations, which describe the kinetic effects of an excited semiconductor system and its optical response function. The equations are nonlinear integrodifferential equations that have to be solved numerically using advanced techniques of parallel computing. The charge-transfer problem for quantum wave functions should be addressed by simultaneous and self-consistent solution of the Poisson and Schrödinger equations. Thus far, only finite-element numerical methods have been used in computational attempts at such a solution. It is desirable to find at least semianalytical methods for solving and/or analyzing the equations for both MSBE and Poisson-Schrödinger problems. Doing so would yield better insight into the physics of the processes and tools for optimizing materials parameters (see the discussion of interatomic potentials in Chapter 2). An important feature of many nonlinear optical systems is that their response is intrinsically dynamical; see, for example, the special issue of Journal of the Optical Society of America, Part B, "Nonlinear Dynamics of Lasers," Vol. 5, May 1988. There are huge time-scale separations between competing physical processes. For example, one has very fast quasi-particle scattering processes that lead to extremely short dephasing rates, often on the order of a few tens of femtoseconds (1 fs = 10-15 s). On the other hand, even after complete dephasing, the system is still far out of its equilibrium state. In many instances, a truncated model for dephasing that eliminates the dynamic response of the material exhibits rather trivial dynamics having little in common with experimentally observed processes. The end result is often highly unpredictable weakly turbulent spatio-temporal pulsations. Nonlinear optics offers numerous manifestations of such space-time complexity and chaos in need of fundamental mathematical analysis. See, for example, Winful and Cooperman (1982), Silberberg and Bar-Joseph (1984), and Firth and Paré (1988); for discussions on optical bistability and chaos, see, for example, National Research Council (1992, 1986) and Gibbs (1985); for information concerning QWSs and chaos, see, for instance, Chemla (1993) and Jensen (1991). There are now quality QWSs and MQWSs based on advanced growth techniques developed in the 1980s, such as molecular beam epitaxy and metallo-organic chemical vapor deposition, and advanced processing techniques such as reactive ion-beam etching and electron-beam lithography. Better theoretical guidance in the simulation of heteroepitaxial growth is sorely needed. Films can grow into fascinating self-similar and fractal structures. Kinetic growth theories based, for example, on random-deposition models and ballistic-deposition models are very attractive from both the physical and the mathematical point of view in that they exhibit dynamic scaling properties governed by simple functions with constant characteristic exponents. However, a great need exists for more sophisticated models able to provide viable tools for predicting properties of the epitaxially grown structures. Another important concern is the mechanical stability of the structures grown, especially when there is a considerable lattice mismatch between neighboring layers; a reliable mathematical theory of this stability would enable researchers to predict mechanical

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Mathematical Research in Materials Science: Opportunities and Perspectives properties of composite structures and thereby design such structures. Within the past few years, the notions of two-dimensional QWSs have been extended to one-dimensional structures (quantum wires) and zero-dimensional structures (quantum dots); see, for instance, Heitmann and Kotthaus (1993). Since one can dilute or concentrate the dots essentially at will, placing quantum dots in glass fibers, for example, allows the creation of local nonlinearities, the magnitudes of which can be manipulated. Theoretically, quantum dots are the ultimate laser material. However, the theory of nonlinearity in these structures is far from satisfactory—analytical solutions have been obtained only for very special cases—and presents another case in which the mathematical topic of chaos arises; for example, see Stone and Bruus (1993). Also, existing theory is valid only for structures with size considerably larger than the lattice spacing; the case where the size is comparable to that spacing offers a worthy mathematical challenge. Photorefractive materials are electro-optic crystals in which absorption of photons triggers a charge migration resulting in a spatial modulation of the material's refractive index through a space-charge-field-induced electro-optic effect. There are some exciting mathematical problems related to the theory of propagation of conjugated waves in photorefractive materials, especially in self-pumped conjugators. Electro-optic crystal chemistry (in particular, altering crystals by cooking them in a specific atmosphere) and crystal growth may greatly benefit from the close attention of mathematical scientists (see Chapter 4). Better understanding of crystal growth in general and electro-optic crystal growth in particular, including adequate mathematical modeling of the processes, may substantially accelerate the technological development of photorefractive materials. Along with quantum-well semiconductor structures, organic and polymeric materials (OPMs) seem to constitute the most promising group of NLO materials (see Chapter 3). One advantage of OPMs is that some have record-high radiation damage thresholds. The main theoretical areas in need of investigation for these materials are quantum calculations of the polarizability of an individual molecule (for example, a molecule oriented by an external field in high temperature can he "frozen" by cooling through the material phase transition) and calculations of macro-susceptibility as determined by microscopic order parameters and by the micro-susceptibility (polarizability). The resulting systems are extremely complicated. The theory involves heavy mathematical machinery (including complicated wave-function calculations and group theory). Further progress in the area will undoubtedly depend on the efforts of applied mathematical scientists. One issue that is common to most of the NLO materials discussed above and that may require a strong mathematical effort is substantially increasing the radiation damage threshold. For many materials, this is an important goal. For materials for a new generation of more powerful lasers, it is very important. A detailed theory of optical breakdown in NLO materials needs to be developed for this; it must include, among other things, a theory of stimulated scatterings (in particular, stimulated Brillouin scattering—SBS) and must address multi-photon absorption (MPA) and plasma formation during the self-focusing process. In particular, it must present a theory that includes SBS and MPA of the collapse of the solution of a three-dimensional nonlinear Schrödinger equation. With the emergence of laser sources of coherent radiation in a fundamentally new x-ray domain—x-ray lasers—a completely new host of NLO materials and effects can be

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Mathematical Research in Materials Science: Opportunities and Perspectives envisioned. Recent results show that third-order NLO phenomena of considerable amplitude (including, for example, nonlinear refractive index) can be expected both in an x-ray laser plasma and, most interestingly, in solid-state materials. The theory of these effects will have to include quantum transitions of (initially) bounded electrons, both between atomic shelves and into a free-electron continuum. This poses new mathematical challenges that are (initially) related to the fast relaxation of hard-driven electrons in the continuum. Lastly, one of the most fascinating recently discovered phenomena in the nonlinear interaction of light with atoms and ions is that high-order harmonic generation (HHG) spectra deviate drastically from perturbation theory predictions. The physics of this phenomenon has two major components: phase-matching conditions and the nonlinear response of individual atoms. It has recently became clear that the major features of HHG, and, in particular, its plateau, result mainly from general properties of nonlinear atomic response. Yet, there exists no simple model or theory that explains even those major features. A number of multiparameter theoretical models of the single-atom response to intensive optical fields in HHG have been suggested; most of them reproduce qualitatively the gross picture of the process. The most successful theoretical approach thus far has been direct numerical simulation using Hartry-Slater approximation of the atomic Schrödinger equations for many-electron atoms. It requires, however, a tremendous amount of calculation, provides little insight into the physics of the process, and hardly allows for general conclusions. A great theoretical challenge is to develop a mathematical and physical model that describes the major features of the phenomenon and identifies the most substantial factors resulting in HHG.