Modeling and control of film porosity in thin film deposition

Systematic methodologies are developed for modeling and control of film porosity in thin film deposition. The deposition process is modeled via kinetic Monte Carlo (kMC) simulation on a triangular lattice. The microscopic events involve atom adsorption and migration and allow for vacancies and overhangs to develop. Appropriate definitions of film site occupancy ratio (SOR), i.e., fraction of film sites occupied by particles over total number of film sites, and its fluctuations are introduced to describe film porosity. Deterministic and stochastic ordinary differential equation (ODE) models are also derived to describe the time evolution of film SOR and its fluctuation. The coefficients of the ODE models are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-square methods and their dependence on substrate temperature is determined. The developed ODE models are used as the basis for the design of model predictive control (MPC) algorithms that include penalty on the film SOR and its variance to regulate the expected value of film SOR at a desired level and reduce run-to-run fluctuations. Simulation results demonstrate the applicability and effectiveness of the proposed film porosity modeling and control methods in the context of the deposition process under consideration.

[1]  Vladimir Mitin,et al.  Monte Carlo simulation of growth and recovery of silicon , 1995 .

[2]  P. Christofides,et al.  Estimation and control of surface roughness in thin film growth using kinetic Monte-Carlo models , 2003 .

[3]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[4]  J. H. Schmid,et al.  Surface morphology of GaAs during molecular beam epitaxy growth: Comparison of experimental data with simulations based on continuum growth equations , 2002 .

[5]  D. Vvedensky,et al.  Stochastic equations of motion for epitaxial growth. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  R. Phaneuf,et al.  Transient evolution of surface roughness on patterned GaAs(001) during homoepitaxial growth. , 2004, Physical review letters.

[7]  P. Maksym.  Fast Monte Carlo simulation of MBE growth , 1988 .

[8]  W. H. Weinberg,et al.  Theoretical foundations of dynamical Monte Carlo simulations , 1991 .

[9]  Peifeng Zhang,et al.  Kinetic Monte Carlo simulation of Cu thin film growth , 2004 .

[10]  Lauritsen,et al.  Noisy Kuramoto-Sivashinsky equation for an erosion model. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  P. Christofides,et al.  Nonlinear Feedback Control of Surface Roughness Using a Stochastic PDE , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[12]  P. Christofides,et al.  Multivariable predictive control of thin film deposition using a stochastic PDE model , 2005, Proceedings of the 2005, American Control Conference, 2005..

[13]  Y. G. Yang,et al.  A Monte Carlo simulation of the physical vapor deposition of nickel , 1997 .

[14]  Kevin Robbie,et al.  State of the art in thin film thickness and deposition rate monitoring sensors , 2005 .

[15]  K. Åström.  Introduction to Stochastic Control Theory , 1970 .

[16]  P. Christofides,et al.  Feedback Control of Surface Roughness Using Stochastic PDEs (R&D Note) , 2005 .

[17]  Vlachos,et al.  Kinetics of faceting of crystals in growth, etching, and equilibrium. , 1993, Physical review. B, Condensed matter.

[18]  Steven W Levine,et al.  A kinetic Monte Carlo study of the growth of Si on Si(100) at varying angles of incident deposition , 1998 .

[19]  P. Christofides,et al.  Model predictive control of nonlinear stochastic partial differential equations with application to a sputtering process , 2008 .

[20]  Payne,et al.  Periodic boundary conditions in ab initio calculations. , 1995, Physical review. B, Condensed matter.

[21]  P. Christofides,et al.  Dynamic output feedback covariance control of stochastic dissipative partial differential equations , 2008 .

[22]  Antonios Armaou,et al.  Control and optimization of multiscale process systems , 2006, Comput. Chem. Eng..

[23]  Panagiotis D. Christofides,et al.  Model parameter estimation and feedback control of surface roughness in a sputtering process , 2008 .

[24]  P. Christofides,et al.  Dynamics and control of thin film surface microstructure in a complex deposition process , 2005 .

[25]  Antonios Armaou,et al.  Time‐steppers and ‘coarse’ control of distributed microscopic processes , 2004 .

[26]  P. Clancy,et al.  Kinetic Monte Carlo simulation of the growth of polycrystalline Cu films , 2001 .

[27]  城塚 正,et al.  Chemical Engineering Scienceについて , 1962 .

[28]  Dionisios G. Vlachos,et al.  Monte Carlo algorithms for complex surface reaction mechanisms: efficiency and accuracy , 2001 .

[29]  Zhang,et al.  Step-density variations and reflection high-energy electron-diffraction intensity oscillations during epitaxial growth on vicinal GaAs(001). , 1992, Physical review. B, Condensed matter.

[30]  Sten Bay Jørgensen,et al.  Parameter estimation in stochastic grey-box models , 2004, Autom..

[31]  Grant M. Kloster,et al.  Porosity effects on low-k dielectric film strength and interfacial adhesion , 2002, Proceedings of the IEEE 2002 International Interconnect Technology Conference (Cat. No.02EX519).

[32]  S. Edwards,et al.  The surface statistics of a granular aggregate , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[33]  Panagiotis D. Christofides,et al.  Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto-Sivashinsky equation , 2005, Comput. Chem. Eng..

[34]  Panagiotis D. Christofides,et al.  Feedback control of surface roughness of GaAs [001] thin films using kinetic Monte-Carlo models , 2004, Proceedings of the 2004 American Control Conference.

[35]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[36]  A. Armaou,et al.  Multiscale optimization using hybrid PDE/kMC process systems with application to thin film growth , 2005 .

[37]  P. Christofides,et al.  Feedback control of growth rate and surface roughness in thin film growth , 2003 .

[38]  J. Villain.  Continuum models of crystal growth from atomic beams with and without desorption , 1991 .

[39]  Antonios Armaou,et al.  Identification of macroscopic variables for low-order modeling of thin-film growth , 2006 .

[40]  Antonios Armaou,et al.  Microscopic/stochastic timesteppers and “coarse” control: A KMC example , 2003 .

[41]  Torsten Bohlin,et al.  Issues in Nonlinear Stochastic Grey-Box Identification , 1994 .

[42]  Stanley,et al.  Stochastic model for surface erosion via ion sputtering: Dynamical evolution from ripple morphology to rough morphology. , 1995, Physical review letters.

[43]  R. Ziff,et al.  Kinetic phase transitions in an irreversible surface-reaction model. , 1986, Physical review letters.

[44]  Steven W Levine,et al.  A simple model for the growth of polycrystalline Si using the kinetic Monte Carlo simulation , 2000 .