FRACTAL CONCEPT S IN SURFACE GROWT H

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1 FRACTAL CONCEPT S IN SURFACE GROWT H Albert-Läszlö Barabäs i H. Eugene Stanley

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3 Preface Notation guide x v xi x PART 1 Introduction 1 1 Interfaces in nature Interface motion in disordered media Deposition processes Biological systems Methods of analysis Discussion Scaling concepts Ballistic deposition Roughening Dynamic scaling Correlations Discussion Fractal concepts Self-similarity Fractal dimension Self-affinity Discussion 3 6 PART 2 Nonequilibrium roughening Random deposition Definition 38

4 4.2 Exact solution Stochastic growth equations Discussion Linear theory Random deposition with surface relaxation Symmetry principles The Edwards-Wilkinson equation Solving the EW equation Discussion 54 6 Kardar-Parisi-Zhang equation Construction of the KPZ equation Excess velocity Scaling arguments Exponents Discussion Renormalization group approach Basic concepts Re-scaling in momentum space Flow equations for the KPZ equation Phase transition in the KPZ equation Exponents for d > Discussion 76 8 Discrete growth models Ballistic deposition Eden model Solid-on-solid models Propagation of interfaces in the Ising model Numerical integration of the KPZ equation Discussion 90 P A R T 3 Interfaces in random media Basic phenomena Depinning transition Interfaces in a disordered medium Scaling arguments Thermal noise Discussion 98

5 10 Quenched noise Universality classes Pinning by directed percolation Isotropic growth models Discussion Experiments Fluid flow in a porous medium Paper wetting Propagation of burning fronts Growth of bacterial colonies Rupture lines in paper sheets Discussion 12 7 PART 4 Molecular beam epitaxy Basic phenomena of MBE Introduction Microscopic processes on crystal surfaces Discussion Linear theory of MBE Surface diffusion Solving the diffusive growth equation Growth with desorption Discussion Nonlinear theory for MBE Surface diffusion : Nonlinear effects Growth with desorption Discussion Discrete models for MBE Irreversible growth models Models with thermal activation Hamiltonian models Discussion MBE experiments Experimental techniques Scaling approach for interface roughening Dynamical properties 170

6 16.4 Discussion Submonolayer deposition Model Scaling theory Rate equations Results from simulations Extensions of the DDA model Experimental results Discussion The roughening transition Equilibrium fluctuations Discrete models and experimental tests Nonequilibrium effects Discussion Nonlocal growth models Diffusion-limited aggregation Sputter deposition Experimental results on sputter deposition Roughening by ion bombardment Discussion Diffusion bias Diffusion bias and instabilities Nonlinear theory Discrete models Experimental support Discussion 23 9 PART 5 Noise Diffusive versus deposition noise Conservative noise Linear theory Scaling regimes Nonlinear theory Discussion Correlated noise Introducing correlated noise 246

7 22.2 Linear theory with correlated noise KPZ equation with spatially-correlated noise KPZ equation with temporally-correlated noise Discussion Rare events Linear theory Nonlinear theory Multi-affinity Discussion 26 1 PART 6 Advanced topics Multi-affine surfaces Hierarchy of scaling exponents A deterministic multi-affine model Brownian motion Local dimensions Variants of the KPZ equation Deterministic KPZ equation Anisotropic KPZ equation Universal amplitudes Discussion Equilibrium fluctuations and directed polymers Discrete model Scaling properties Continuum description Equilibrium theory Discussion 28 4 PART 7 Finale Summary of the continuum growth equations Universality classes Nomenclature ,3 Related problems Discussion Outlook 298

8 A P P E N D I X A Numerical recipes 30 1 A.1 Measuring exponents for self-affine interfaces 30 1 A.2 The coefficient.i of the nonlinear term 30 7 A.3 Intrinsic width 30 9 A.4 Measuring surface diffusion currents 31 0 A.5 Generating noise in simulations 31 1 APP END I X B Dynamic renormalization group 31 5 B.1 Introduction 31 5 B.2 Perturbation expansion 31 6 B.3 Renormalization procedure 32 3 B.4 Calculation of the integrals 32 5 A P PEND I X C Hamiltonian description 33 0 Bibliography 33 2 Index 359