Maximum a posteriori estimation of activation energies that control silicon self-diffusion

a b s t r a c t Self-diffusion in crystalline silicon is controlled by a network of elementary steps whose activation energies are important to know in a variety of applications in microelectronic fabrication. The present work employs maximum a posteriori (MAP) estimation to improve existing values for these activation energies, based on self-diffusion data collected at different values of the loss rates for interstitial atoms to the surface. Parameter sensitivity analysis shows that for high surface loss fluxes, the energy for exchange between an interstitial and the lattice plays the leading role in determining the shape of diffusion profiles. At low surface loss fluxes, the dissociation energy of large-atom clusters plays a more important role. Subsequent MAP analysis provides significantly improved values for these parameters

Nuclear spin warm-up in bulk n-GaAs

We show that the spin-lattice relaxation in n-type insulating GaAs is dramatically accelerated at low magnetic fields. The origin of this effect, that cannot be explained in terms of well-known diffusion-limited hyperfine relaxation, is found in the quadrupole relaxation, induced by fluctuating donor charges. Therefore, quadrupole relaxation, that governs low field nuclear spin relaxation in semiconductor quantum dots, but was so far supposed to be harmless to bulk nuclei spins in the absence of optical pumping can be studied and harnessed in much simpler model environment of n-GaAs bulk crystal.Comment: 5 pages, 4 figure

Modeling of metal nanocluster growth on patterned substrates and surface pattern formation under ion bombardment

This thesis addresses the metal nanocluster growth process on prepatterned substrates, the development of atomistic simulation method with respect to an acceleration of the atomistic transition states, and the continuum model of the ion-beam inducing semiconductor surface pattern formation mechanism. Experimentally, highly ordered Ag nanocluster structures have been grown on pre-patterned amorphous SiO^2 surfaces by oblique angle physical vapor deposition at room temperature. Despite the small undulation of the rippled surface, the stripe-like Ag nanoclusters are very pronounced, reproducible and well-separated. The first topic is the investigation of this growth process with a continuum theoretical approach to the surface gas condensation as well as an atomistic cluster growth model. The atomistic simulation model is a lattice-based kinetic Monte-Carlo (KMC) method using a combination of a simplified inter-atomic potential and experimental transition barriers taken from the literature. An effective transition event classification method is introduced which allows a boost factor of several thousand compared to a traditional KMC approach, thus allowing experimental time scales to be modeled. The simulation predicts a low sticking probability for the arriving atoms, millisecond order lifetimes for single Ag monomers and ≈1 nm square surface migration ranges of Ag monomers. The simulations give excellent reproduction of the experimentally observed nanocluster growth patterns. The second topic specifies the acceleration scheme utilized in the metallic cluster growth model. Concerning the atomistic movements, a classical harmonic transition state theory is considered and applied in discrete lattice cells with hierarchical transition levels. The model results in an effective reduction of KMC simulation steps by utilizing a classification scheme of transition levels for thermally activated atomistic diffusion processes. Thermally activated atomistic movements are considered as local transition events constrained in potential energy wells over certain local time periods. These processes are represented by Markov chains of multi-dimensional Boolean valued functions in three dimensional lattice space. The events inhibited by the barriers under a certain level are regarded as thermal fluctuations of the canonical ensemble and accepted freely. Consequently, the fluctuating system evolution process is implemented as a Markov chain of equivalence class objects. It is shown that the process can be characterized by the acceptance of metastable local transitions. The method is applied to a problem of Au and Ag cluster growth on a rippled surface. The simulation predicts the existence of a morphology dependent transition time limit from a local metastable to stable state for subsequent cluster growth by accretion. The third topic is the formation of ripple structures on ion bombarded semiconductor surfaces treated in the first topic as the prepatterned substrate of the metallic deposition. This intriguing phenomenon has been known since the 1960\'s and various theoretical approaches have been explored. These previous models are discussed and a new non-linear model is formulated, based on the local atomic flow and associated density change in the near surface region. Within this framework ripple structures are shown to form without the necessity to invoke surface diffusion or large sputtering as important mechanisms. The model can also be extended to the case where sputtering is important and it is shown that in this case, certain \\lq magic\' angles can occur at which the ripple patterns are most clearly defined. The results including some analytic solutions of the nonlinear equation of motions are in very good agreement with experimental observation

Modeling of metal nanocluster growth on patterned substrates and surface pattern formation under ion bombardment

This thesis addresses the metal nanocluster growth process on prepatterned substrates, the development of atomistic simulation method with respect to an acceleration of the atomistic transition states, and the continuum model of the ion-beam inducing semiconductor surface pattern formation mechanism. Experimentally, highly ordered Ag nanocluster structures have been grown on pre-patterned amorphous SiO2 surfaces by oblique angle physical vapor deposition at room temperature. Despite the small undulation of the rippled surface, the stripe-like Ag nanoclusters are very pronounced, reproducible and well-separated. The first topic is the investigation of this growth process with a continuum theoretical approach to the surface gas condensation as well as an atomistic cluster growth model. The atomistic simulation model is a lattice-based kinetic Monte-Carlo (KMC) method using a combination of a simplified inter-atomic potential and experimental transition barriers taken from the literature. An effective transition event classification method is introduced which allows a boost factor of several thousand compared to a traditional KMC approach, thus allowing experimental time scales to be modeled. The simulation predicts a low sticking probability for the arriving atoms, millisecond order lifetimes for single Ag monomers and about 1 nm square surface migration ranges of Ag monomers. The simulations give excellent reproduction of the experimentally observed nanocluster growth patterns. The second topic specifies the acceleration scheme utilized in the metallic cluster growth model. Concerning the atomistic movements, a classical harmonic transition state theory is considered and applied in discrete lattice cells with hierarchical transition levels. The model results in an effective reduction of KMC simulation steps by utilizing a classification scheme of transition levels for thermally activated atomistic diffusion processes. Thermally activated atomistic movements are considered as local transition events constrained in potential energy wells over certain local time periods. These processes are represented by Markov chains of multi-dimensional Boolean valued functions in three dimensional lattice space. The events inhibited by the barriers under a certain level are regarded as thermal fluctuations of the canonical ensemble and accepted freely. Consequently, the fluctuating system evolution process is implemented as a Markov chain of equivalence class objects. It is shown that the process can be characterized by the acceptance of metastable local transitions. The method is applied to a problem of Au and Ag cluster growth on a rippled surface. The simulation predicts the existence of a morphology dependent transition time limit from a local metastable to stable state for subsequent cluster growth by accretion. The third topic is the formation of ripple structures on ion bombarded semiconductor surfaces treated in the first topic as the prepatterned substrate of the metallic deposition. This intriguing phenomenon has been known since the 1960s and various theoretical approaches have been explored. These previous models are discussed and a new non-linear model is formulated, based on the local atomic flow and associated density change in the near surface region. Within this framework ripple structures are shown to form without the necessity to invoke surface diffusion or large sputtering as important mechanisms. The model can also be extended to the case where sputtering is important and it is shown that in this case, certain "magic" angles can occur at which the ripple patterns are most clearly defined. The results including some analytic solutions of the nonlinear equation of motions are in very good agreement with experimental observation.:1 Introduction: Atomistic Models 1 1.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Schroedinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 MD algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Lattice Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Thermodynamic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Metropolis Algorithm and limit theorem . . . . . . . . . . . . . . . . . . . . . 15 1.3.3 Kinetic Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.4 Imaginary time reaction diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Cluster Growth on Pre-patterned Surfaces 29 2.1 Nanocluster growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.1 Classical nucleation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.2 Cluster growth on substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.3 Experimental motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Local flux and surface ad-monomer diffusion . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Surface topography and local flux . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Surface gas diffusion under inhomogeneous flux . . . . . . . . . . . . . . . . . 37 2.2.3 Surface migration of ad-monomers . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.4 Simulation vs. experimental gauge . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Nucleation models: Surface gas condensation . . . . . . . . . . . . . . . . . . . . . . 46 2.3.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.3 Evolution of sticking probability . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.4 Evolution of Ag cluster growth . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3.5 Simulation time and system evolution . . . . . . . . . . . . . . . . . . . . . . 57 2.4 Extended cluster growth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.1 Modified setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.2 Simulation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4.3 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3 A Markov chain model of transition states 63 3.1 Acceleration of thin film growth simulation . . . . . . . . . . . . . . . . . . . . . . . 63 3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3 Transition states of Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Local transition events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 The Monte-Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Effective transitions of objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1 Convergence of the local fluctuation . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.2 The importance of individual local transitions . . . . . . . . . . . . . . . . . . 68 3.4.3 The modified algorithm for effective transition states . . . . . . . . . . . . . . 69 3.5 Cluster growth simulation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5.1 The configuration energy and migration barriers . . . . . . . . . . . . . . . . 72 3.5.2 Transition events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5.3 Comparison with Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.5.4 Cluster growth stability evaluation . . . . . . . . . . . . . . . . . . . . . . . . 78 3.6 Stability of modified convergence limit . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.1 Acceleration of convergence to Gibbs field . . . . . . . . . . . . . . . . . . . . 80 3.6.2 Relative convergence speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.3 1D Ag models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6.4 Stability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 Ion beam inducing surface pattern formation 89 4.1 Ion-inducing pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1.1 Bradley-Harper equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1.2 Nonlinear continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.1.3 Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Simulation of surface defects induced by ion beams . . . . . . . . . . . . . . . . . . . 94 4.2.1 MD simulation of single ion impact . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2.2 Monte-Carlo simulations of surface modification . . . . . . . . . . . . . . . . 96 4.2.3 Curvature dependent surface diffusion . . . . . . . . . . . . . . . . . . . . . . 102 4.3 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.2 A travelling wave solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.3 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3.4 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.5 Approximate solutions for other angles . . . . . . . . . . . . . . . . . . . . . . 110 4.4 Contribution of other effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.4.1 Surface diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.4.2 Surface Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5 Summary 119 Appendix 123 A The discrete reaction diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . 123 B The derivation of the solution (2.20) . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 C Contribution of overlapping migration area . . . . . . . . . . . . . . . . . . . . . . . 125 D The RGL potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 E Stability of the traveling wave solution . . . . . . . . . . . . . . . . . . . . . . . . . . 12

A multiscale systems approach to microelectronic processes

Abstract This paper describes applications of molecular simulation to microelectronics processes and the subsequent development of techniques for multiscale simulation and multiscale systems engineering. The progression of the applications of simulation in the semiconductor industry from macroscopic to molecular to multiscale is reviewed. Multiscale systems are presented as an approach that incorporates molecular and multiscale simulation to design processes that control events at the molecular scale while simultaneously optimizing all length scales from the molecular to the macroscopic. It is discussed how design and control problems in microelectronics and nanotechnology, including the targeted design of processes and products at the molecular scale, can be addressed using the multiscale systems tools. This provides a framework for addressing the "grand challenge" of nanotechnology: how to move nanoscale science and technology from art to an engineering discipline