Kinetics in Nanoscale Materials

KING-NING TU, PhD, is Professor in the Department of Materials Science and Engineering at the University of California, Los Angeles. His research focuses on kinetic processes in thin films, metal-silicon interfaces, electromigration, lead-free solder metallurgy, and point contact reactions on silicon nanowires.ANDRIY M. GUSAK, PhD, is Chair and Professor in the Department of Physics at Cherkasy National University. His research explores nanomaterial science and kinetics of nanoscale systems, with an emphasis on the development of microelectronic materials.

• PREFACE ixCHAPTER 1 INTRODUCTION TO KINETICS IN NANOSCALE MATERIALS 11.1 Introduction 11.2 Nanosphere: Surface Energy is Equivalent to Gibbs–Thomson Potential 31.3 Nanosphere: Lower Melting Point 61.4 Nanosphere: Fewer Homogeneous Nucleation and its Effect on Phase Diagram 101.5 Nanosphere: Kirkendall Effect and Instability of Hollow Nanospheres 131.6 Nanosphere: Inverse Kirkendall Effect in Hollow Nano Alloy Spheres 171.7 Nanosphere: Combining Kirkendall Effect and Inverse Kirkendall Effect on Concentric Bilayer Hollow Nanosphere 181.8 Nano Hole: Instability of a Donut-Type Nano Hole in a Membrane 191.9 Nanowire: Point Contact Reactions Between Metal and Silicon Nanowires 211.10 Nanowire: Nanogap in Silicon Nanowires 221.11 Nanowire: Lithiation in Silicon Nanowires 261.12 Nanowire: Point Contact Reactions Between Metallic Nanowires 271.13 Nano Thin Film: Explosive Reaction in Periodic Multilayered Nano Thin Films 281.14 Nano Microstructure in Bulk Samples: Nanotwins 301.15 Nano Microstructure on the Surface of a Bulk Sample: Surface Mechanical Attrition Treatment (SMAT) of Steel 32References 33Problems 35CHAPTER 2 LINEAR AND NONLINEAR DIFFUSION 372.1 Introduction 372.2 Linear Diffusion 382.2.1 Atomic Flux 392.2.2 Fick’s First Law of Diffusion 402.2.3 Chemical Potential 432.2.4 Fick’s Second Law of Diffusion 452.2.5 Flux Divergence 472.2.6 Tracer Diffusion 492.2.7 Diffusivity 512.2.8 Experimental Measurement of the Parameters in Diffusivity 532.3 Nonlinear Diffusion 572.3.1 Nonlinear Effect due to Kinetic Consideration 582.3.2 Nonlinear Effect due to Thermodynamic Consideration 592.3.3 Combining Thermodynamic and Kinetic Nonlinear Effects 62References 63Problems 64CHAPTER 3 KIRKENDALL EFFECT AND INVERSE KIRKENDALL EFFECT 673.1 Introduction 673.2 Kirkendall Effect 693.2.1 Darken’s Analysis of Kirkendall Shift and Marker Motion 723.2.2 Boltzmann and Matano Analysis of Interdiffusion Coefficient 763.2.3 Activity and Intrinsic Diffusivity 803.2.4 Kirkendall (Frenkel) Voiding Without Lattice Shift 843.3 Inverse Kirkendall Effect 843.3.1 Physical Meaning of Inverse Kirkendall Effect 863.3.2 Inverse Kirkendall Effect on the Instability of an Alloy Nanoshell 883.3.3 Inverse Kirkendall Effect on Segregation in a Regular Solution Nanoshell 903.4 Interaction Between Kirkendall Effect and Gibbs–Thomson Effect in the Formation of a Spherical Compound Nanoshell 93References 97Problems 97CHAPTER 4 RIPENING AMONG NANOPRECIPITATES 994.1 Introduction 994.2 Ham’s Model of Growth of a Spherical Precipitate (Cr is Constant) 1014.3 Mean-Field Consideration 1034.4 Gibbs–Thomson Potential 1054.5 Growth and Dissolution of a Spherical Nanoprecipitate in a Mean Field 1064.6 LSW Theory of Kinetics of Particle Ripening 1084.7 Continuity Equation in Size Space 1134.8 Size Distribution Function in Conservative Ripening 1144.9 Further Developments of LSW Theory 115References 115Problems 116CHAPTER 5 SPINODAL DECOMPOSITION 1185.1 Introduction 1185.2 Implication of Diffusion Equation in Homogenization and Decomposition 1215.3 Spinodal Decomposition 1235.3.1 Concentration Gradient in an Inhomogeneous Solid Solution 1235.3.2 Energy of Mixing to Form a Homogeneous Solid Solution 1245.3.3 Energy of Mixing to Form an Inhomogeneous Solid Solution 1265.3.4 Chemical Potential in Inhomogeneous Solution 1295.3.5 Coherent Strain Energy 1315.3.6 Solution of the Diffusion Equation 134References 136Problems 136CHAPTER 6 NUCLEATION EVENTS IN BULK MATERIALS, THIN FILMS, AND NANOWIRES 1386.1 Introduction 1386.2 Thermodynamics and Kinetics of Nucleation 1406.2.1 Thermodynamics of Nucleation 1406.2.2 Kinetics of Nucleation 1436.3 Heterogeneous Nucleation in Grain Boundaries of Bulk Materials 1486.3.1 Morphology of Grain Boundary Precipitates 1506.3.2 Introducing an Epitaxial Interface to Heterogeneous Nucleation 1516.3.3 Replacive Mechanism of a Grain Boundary 1546.4 No Homogeneous Nucleation in Epitaxial Growth of Si Thin Film on Si Wafer 1566.5 Repeating Homogeneous Nucleation of Silicide in Nanowires of Si 1606.5.1 Point Contact Reactions in Nanowires 1616.5.2 Homogeneous Nucleation of Epitaxial Silicide in Nanowires of Si 164References 168Problems 168CHAPTER 7 CONTACT REACTIONS ON Si; PLANE, LINE, AND POINT CONTACT REACTIONS 1707.1 Introduction 1707.2 Bulk Cases 1757.2.1 Kidson’s Analysis of Diffusion-Controlled Planar Growth 1757.2.2 Steady State Approximation in Layered Growth of Multiple Phases 1787.2.3 Marker Analysis 1797.2.4 Interdiffusion Coefficient in Intermetallic Compound 1827.2.5 Wagner Diffusivity 1867.3 Thin Film Cases 1877.3.1 Diffusion-Controlled and Interfacial-Reaction-Controlled Growth 1877.3.2 Kinetics of Interfacial-Reaction-Controlled Growth 1887.3.3 Kinetics of Competitive Growth of Two-Layered Phases 1937.3.4 First Phase in Silicide Formation 1947.4 Nanowire Cases 1967.4.1 Point Contact Reactions 1977.4.2 Line Contact Reactions 2027.4.3 Planar Contact Reactions 208References 208Problems 209CHAPTER 8 GRAIN GROWTH IN MICRO AND NANOSCALE 2118.1 Introduction 2118.2 How to Generate a Polycrystalline Microstructure 2138.3 Computer Simulation of Grain Growth 2168.3.1 Atomistic Simulation Based on Monte Carlo Method 2168.3.2 Phenomenological Simulations 2178.4 Statistical Distribution Functions of Grain Size 2198.5 Deterministic (Dynamic) Approach to Grain Growth 2218.6 Coupling Between Grain Growth of a Central Grain and the Rest of Grains 2258.7 Decoupling the Grain Growth of a Central Grain from the Rest of Grains in the Normalized Size Space 2268.8 Grain Growth in 2D Case in the Normalized Size Space 2298.9 Grain Rotation 2318.9.1 Grain Rotation in Anisotropic Thin Films Under Electromigration 232References 237Problems 238CHAPTER 9 SELF-SUSTAINED REACTIONS IN NANOSCALE MULTILAYERED THIN FILMS 2409.1 Introduction 2409.2 The Selection of a Pair of Metallic Thin Films for SHS 2439.3 A Simple Model of Single-Phase Growth in Self-Sustained Reaction 2459.4 A Simple Estimate of Flame Velocity in Steady State Heat Transfer 2509.5 Comparison in Phase Formation by Annealing and by Explosive Reaction in Al/Ni 2519.6 Self-Explosive Silicidation Reactions 251References 255Problems 256CHAPTER 10 FORMATION AND TRANSFORMATIONS OF NANOTWINS IN COPPER 25810.1 Introduction 25810.2 Formation of Nanotwins in Cu 26010.2.1 First Principle Calculation of Energy of Formation of Nanotwins 26010.2.2 In Situ Measurement of Stress Evolution for Nanotwin Formation During Pulse Electrodeposition of Cu 26410.2.3 Formation of Nanotwin Cu in Through-Silicon Vias 26610.3 Formation and Transformation of Oriented Nanotwins in Cu 26910.3.1 Formation of Oriented Nanotwins in Cu 27010.3.2 Unidirectional Growth of Cu–Sn Intermetallic Compound on Oriented and Nanotwinned Cu 27010.3.3 Transformation of ⟨111⟩ Oriented and Nanotwinned Cu to ⟨100⟩ Oriented Single Crystal of Cu 27410.4 Potential Applications of Nanotwinned Cu 27610.4.1 To Reduce Electromigration in Interconnect Technology 27610.4.2 To Eliminate Kirkendall Voids in Microbump Packaging Technology 277References 278Problems 278APPENDIX A LAPLACE PRESSURE IN NONSPHERICAL NANOPARTICLE 280APPENDIX B INTERDIFFUSION COEFFICIENT Þ D = CBMG′′ 282APPENDIX C NONEQUILIBRIUM VACANCIES AND CROSS-EFFECTS ON INTERDIFFUSION IN A PSEUDO-TERNARY ALLOY 285APPENDIX D INTERACTION BETWEEN KIRKENDALL EFFECT AND GIBBS–THOMSON EFFECT IN THE FORMATION OF A SPHERICAL COMPOUND NANOSHELL 289INDEX 293